]>NUPHB13918S0550-3213(16)30359-510.1016/j.nuclphysb.2016.11.009The Author(s)Quantum Field Theory and Statistical SystemsFig. 1The singular branch lines k ranges (solid lines) and other branch lines k ranges (dashed lines) for which the corresponding exponent ξβσ(k), Eq. (112), is negative and positive, respectively, and the singular boundary lines (dashed–dotted lines) of the weight distribution associated with the ↑ and ↓ one-electron spectral function are plotted in the (k,ω) plane. The curves refer to u=0.1, electronic density ne=0.7, and (a)–(b) spin densities m=0.45 and (c)–(d) m=0.25. The branch line spectra plotted here are defined in Section 4. The c+, c−, and s1 branch-line labels appearing in panels (a) for σ=↑ and (b) for σ=↓ apply to the branch lines with similar topology in panels (c) and (d), respectively. (Online, the c+, c−, and s1 branch lines appear in blue, red, and green, respectively.) The lines represented by sets of diamond symbols contribute to the u→0 one-electron spectrum yet are not branch lines. In the case of σ one-electron UHB addition, only the branch lines that contribute to the u→0 spectral weight are represented.Fig. 1Fig. 2The same singular branch lines k ranges (solid lines) and other branch lines k ranges (dashed lines) as in Fig. 1 for u=1, electronic density ne=0.7, and spin densities (a)–(b) m=0.65 and (c)–(d) m=0.45. (Online, the c+, c−, and s1 branch lines are blue, red, and green, respectively.)Fig. 2Fig. 3The singular branch lines k ranges (solid lines) and other branch lines k ranges (dashed lines) for which the corresponding exponent ξβσ(k), Eq. (112), is negative and positive, respectively, and the singular boundary lines (dashed–dotted lines) of the weight distribution associated with the ↑ and ↓ one-electron spectral function are plotted in the (k,ω) plane. The curves refer to u=1, electronic density ne=0.7, and spin densities (a)–(b) m=0.25 and (c)–(d) m=0.05. (Online, the c+, c−, and s1 branch lines are blue, red, and green, respectively.)Fig. 3Fig. 4The same singular branch lines k ranges (solid lines) and other branch lines k ranges (dashed lines) as in Fig. 3 for u=10, electronic density ne=0.7, and spin densities (a)–(b) m=0.45 and (c)–(d) m=0.25. Note the different ω intervals separated by a horizontal dashed line used for the removal and LHB addition spectral features and the UHB addition branch line, respectively. (Online, the c+, c−, and s1 branch lines are blue, red, and green, respectively.)Fig. 4Fig. 5The singular branch lines k ranges (solid lines) and other branch lines k ranges (dashed lines) for which the corresponding exponent ξβσ(k), Eq. (112), is negative and positive, respectively, and the singular boundary lines (dashed–dotted lines) of the weight distribution associated with the ↑ and ↓ one-electron spectral function are plotted in the (k,ω) plane. The curves refer to u=1, electronic density ne=0.3, and spin densities (a)–(b) m=0.25 and (c)–(d) m=0.05. (Online, the c+, c−, and s1 branch lines are blue, red, and green, respectively.)Fig. 5Fig. 6The exponent ξc+↑(k)=ξc−↑(−k), Eq. (130), that controls the singularities in the vicinity of the c+ branch line whose (k,ω)-plane one-parametric spectrum is defined by Eqs. (126), (128), and (129) is plotted for the σ=↑ one-electron removal and LHB addition spectral function, Eq. (132), as a function of the momentum k/π∈]−1,1[. The curves refer to several u values, electronic density ne=0.7, and spin densities (a) m=0.65, (b) m=0.45, (c) m=0.25, and (d) m=0.05, and for electronic density ne=0.3 and spin densities (e) m=0.25 and (f) m=0.05. The type of exponent line associated with each u value is for all figures the same. Full and dashed vertical lines denote specific momentum values between different subbranches and momenta where the u→0 limiting value of the exponent changes, respectively.Fig. 6Fig. 7The exponent ξc+↓(k)=ξc−↓(−k), Eq. (131), that controls the singularities in the vicinity of the c+ branch line whose (k,ω)-plane shape is defined by Eqs. (126), (128), and (129) is plotted for the σ=↓ one-electron removal and LHB addition spectral function, Eq. (132), as a function of the momentum k/π∈]−1,1[. The curves refer to the same values of u, electronic density ne, and spin density m as in Fig. 6.Fig. 7Fig. 8The exponent ξs1↑(k), Eq. (150), that controls the singularities in the vicinity of the s1 branch line whose (k,ω)-plane shape is defined by Eqs. (145), (147), and (149) is plotted for the σ=↑ one-electron removal and LHB addition spectral function, Eq. (152), as a function of the momentum k/π∈]0,1[. The curves correspond to the same values of u, electronic density ne, and spin density m as in Fig. 6. (For k/π∈]−1,0[ the exponent ξs1↑(k) is given by ξs1↑(k)=ξs1↑(−k) with −k/π∈]0,1[ as plotted here.)Fig. 8Fig. 9The exponent ξs1↓(k), Eq. (151), that controls the singularities in the vicinity of the s1 branch line whose (k,ω)-plane one-parametric spectrum is defined by Eqs. (145) and (148) is plotted for the σ=↓ one-electron removal and LHB addition spectral function, Eq. (152), as a function of the momentum k/π∈]0,1[. The curves refer to the same values of u, electronic density ne, and spin density m as in Fig. 6. (For k/π∈]−1,0[ the exponent ξs1↓(k) is again given by ξs1↓(k)=ξs1↓(−k) with −k/π∈]0,1[ as plotted here.)Fig. 9Fig. 10The exponent ξs1↑(k), Eq. (164), that controls the singularities in the vicinity of the s1 branch line whose (k,ω)-plane one-parametric spectrum is defined by Eq. (161) is plotted for the σ=↑ one-electron UHB addition spectral function, Eq. (165), as a function of the momentum k/π∈]k0/π,1[. Here ]k0/π,1[ with 0<k0<π is a k interval that contains the branch line. The curves correspond to the same values of u, electronic density ne, and spin density m as in Fig. 6. (For k/π∈]−1,−k0/π[ the exponent ξs1↑(k) is given by ξs1↑(k)=ξs1↑(−k) with −k/π∈]k0/π,1[ as plotted here.)Fig. 10Fig. 11The exponent ξc+↓(k)=ξc−↓(−k), Eq. (174), that controls the singularities in the vicinity of the c+ branch line whose (k,ω)-plane one-parametric spectrum is defined by Eq. (171) is plotted for the σ=↓ one-electron UHB addition spectral function, Eq. (175), as a function of the momentum k/π∈]0,1[. The curves refer to the same values of u, electronic density ne, and spin density m as in Fig. 6.Fig. 11One-electron singular spectral features of the 1D Hubbard model at finite magnetic fieldJ.M.P.Carmeloabcd⁎carmelo@fisica.uminho.ptcarmelo@MIT.EDUT.ČadežcbaDepartment of Physics, University of Minho, Campus Gualtar, P-4710-057 Braga, PortugalDepartment of PhysicsUniversity of MinhoCampus GualtarBragaP-4710-057PortugalbCenter of Physics of University of Minho and University of Porto, P-4169-007 Oporto, PortugalCenter of Physics of University of MinhoUniversity of PortoOportoP-4169-007PortugalcBeijing Computational Science Research Center, Beijing 100193, ChinaBeijing Computational Science Research CenterBeijing100193ChinadUniversity of Gothenburg, Department of Physics, SE-41296 Gothenburg, SwedenUniversity of GothenburgDepartment of PhysicsGothenburgSE-41296Sweden⁎Corresponding author.Editor: Hubert SaleurAbstractThe momentum, electronic density, spin density, and interaction dependences of the exponents that control the (k,ω)-plane singular features of the σ=↑,↓ one-electron spectral functions of the 1D Hubbard model at finite magnetic field are studied. The usual half-filling concepts of one-electron lower Hubbard band and upper Hubbard band are defined in terms of the rotated electrons associated with the model Bethe-ansatz solution for all electronic density and spin density values and the whole finite repulsion range. Such rotated electrons are the link of the non-perturbative relation between the electrons and the pseudofermions. Our results further clarify the microscopic processes through which the pseudofermion dynamical theory accounts for the one-electron matrix elements between the ground state and excited energy eigenstates.1IntroductionThe Hubbard model with nearest-neighbor hopping integral t and on-site repulsion U is possibly the most studied lattice system of correlated electrons. It was originally introduced as a toy model to study d-electrons in transition metals [1,2]. The Hubbard model on a one-dimensional (1D) lattice is exactly solvable by the Bethe ansatz (BA). Such a solution was first achieved by the coordinate BA [3,4]. This has followed a similar solution for a related continuous model with repulsive δ-function interaction [5]. For the 1D Hubbard model, the BA solution is also reachable by the inverse-scattering method [6]. In the thermodynamic limit (TL) the imaginary part of its BA complex rapidities simplifies [7].On the one hand, static properties such as the charge and spin stiffnesses of the 1D Hubbard model under periodic boundary conditions can be determined from the use of the response of the energy eigenvalues to an external flux piercing the ring [8,9]. On the other hand, one of the main challenges in the study of the 1D Hubbard model properties is the calculation of dynamical correlation functions. Its BA solution provides the exact spectrum of the energy eigenstates. However, it has been difficult to apply to the derivation of high-energy dynamical correlation functions. (In this paper we use the designation high energy for all energy scales larger than the model low-energy limit associated with the Tomonaga–Luttinger-liquid regime [10–18].) The high-energy dynamical correlation functions of both some integrable models with spectral gap [19–25] and spin lattice systems [26–31] can be studied by the form-factor approach. Form factors of the 1D Hubbard model σ=↑,↓ electron creation and annihilation operators involved in the spectral functions studied in this paper remains though an unsolved problem.The low-energy behavior of the correlation functions of the 1D Hubbard model at finite magnetic field was addressed in Refs. [14–16,32]. In what high-energy behavior of dynamical correlation functions is concerned, the method used in Refs. [33,34] has been a breakthrough to address it for one-electron removal and addition spectral functions at zero magnetic field in the u→∞ limit. In that limit they have been derived over the whole (k,ω) plane. That method relies on the spinless-fermion phase shifts imposed by Heisenberg spins 1/2. Such elementary objects naturally arise from the zero spin density and u→∞ electron wave-function factorization [35–38].A related pseudofermion dynamical theory (PDT) was introduced in Refs. [39,40]. It relies on a representation of the model BA solution in terms of pseudofermions. Those are generated by a unitary transformation from corresponding pseudoparticles [41,42]. It is an extension of the u→∞ method of Refs. [33,34] to the whole u≡U/4t>0 range of the 1D Hubbard model. A key property is that the pseudofermions are inherently constructed to their energy spectrum having no interaction terms. This allows the expression of the one-electron spectral functions in terms of convolutions of pseudofermion spectral functions. The price to pay for the lack of pseudofermion energy spectrum interaction terms is that creation or annihilation of pseudofermions under transitions to excited states imposes phase shifts to the remaining pseudofermions. Within the PDT such phase shifts fully control the one- and two-electron spectral-weight distributions over the (k,ω) plane. That approach has been the first breakthrough for the derivation of analytical expressions of the zero-magnetic-field 1D Hubbard model high-energy dynamical correlation functions for the whole finite u>0 range. Recently a modified form of the PDT was used to study the high-energy spin dynamical correlation functions of the 1D Hubbard model electronic density ne=1 Mott–Hubbard insulator phase [42].After the PDT of the 1D Hubbard model was introduced in Refs. [39,40], a set of novel methods have been developed to also tackle the high-energy physics of 1D correlated quantum problems, beyond the low-energy Tomonaga–Luttinger-liquid limit [43]. In the case of the 1D Hubbard model at zero magnetic field, such methods reach the same results as the PDT. For instance, the momentum, electronic density, and on-site repulsion u=U/4t>0 dependence of the exponents that control the line shape of the one-electron spectral function of the model at zero magnetic field calculated in Refs. [44,45] in the framework of a mobile impurity model using input from the BA solution is exactly the same as that obtained previously by the use of the PDT.However, the applications to the study of the repulsive 1D Hubbard model one-electron spectral functions of both such methods [44,45], those of the PDT [46–49], and the time-dependent density-matrix renormalization group (tDMRG) method [50,51] have been limited to zero magnetic field. The tDMRG studies of Ref. [52] studied the one-electron spectral-weight distributions of the attractive 1D Hubbard model at finite magnetic field. Under the canonical transformation that maps that model into the repulsive 1D Hubbard model, the one-electron spectral-weight distributions plotted in Figs. 1 (c) and 2 of that reference correspond to electronic densities ne=1 and ne=0.9, respectively, and spin density m=1/2. The results refer to a finite system with 40 electrons. While they provide some information on the one-electron spectral-weight distributions, it is not possible to extract from them the momentum dependence of the exponents that in the TL control the line shapes near the σ one-electron spectral functions singularities.The main goal of this paper is to extend the PDT applications to the study of the σ one-electron spectral functions of the repulsive 1D Hubbard model at finite magnetic field h in the TL near their singularities. The corresponding line shapes are controlled by exponents whose momentum, on-site repulsion u=U/4t, electronic density n, and spin density m dependences we study for u>0, n∈[0,1[, and m∈[0,ne]. In addition, the issue of how the σ one-electron creation and annihilation operators matrix elements between the ground state and excited energy eigenstates are accounted for by the PDT introduced in Refs. [39,40] is further clarified in this paper. Beyond the preliminary analysis of these references, the corresponding microscopic processes are shown to involve the rotated electrons as a needed link of the non-perturbative relation between the electrons and PDT pseudofermions.Our studies refer to the TL of the Hubbard model under periodic boundary conditions on a 1D lattice with an even number L→∞ of sites and in a chemical potential μ and magnetic field h,(1)Hˆ=Hˆu+2μSˆηz+2μBhSˆsz,Hˆu=−t∑σ=↑,↓∑j=1L(cj,σ†cj+1,σ+cj+1,σ†cj,σ)+U∑j=1L(cj,↑†cj,σ−1/2)(cj,↓†cj,σ−1/2),Sˆηz=−12(L−Nˆ);Sˆsz=−12(Nˆ↑−Nˆ↓). Here the first and second terms of Hˆu are the kinetic-energy operator and the electron on-site repulsion operator, respectively, the operator cj,σ† (and cj,σ) creates (and annihilates) one spin-projection σ electron at lattice site j=1,...,L, and the electron number operators read Nˆ=∑σ=↑,↓Nˆσ and Nˆσ=∑j=1Lnˆj,σ=∑j=1Lcj,σ†cj,σ. Moreover, μB is the Bohr magneton and Sˆηz and Sˆsz are the diagonal generators of the Hamiltonian Hˆu global η-spin and spin SU(2) symmetry algebras, respectively. We use in general units of lattice constant one, so that the number of lattice sites Na equals the lattice length L. The model properties depend on the ratio U/t. In this paper the corresponding parameter u=U/4t is often used.The lowest weight states (LWSs) and highest weight states (HWSs) of the η-spin (α=η) and spin (α=s) SU(2) symmetry algebras have numbers Sα=−Sαz and Sα=Sαz, respectively. Here Sη is the states η-spin, Ss their spin, and Sηz and Ssz are the corresponding projections, respectively. The latter are the eigenvalues of the spin operators given in Eq. (1).Let {|lr,lηs,u〉} be the complete set of 4L energy eigenstates of the Hamiltonian Hˆ, Eq. (1), associated with the BA solution for u>0. The LWSs of both SU(2) symmetry algebras are here denoted by |lr,lηs0,u〉. The u-independent label lηs is a short notation for the set of quantum numbers,(2)lηs=Sη,Ss,nη,ns;nα=Sα+Sαz=0,1,...,2Sα,α=η,s. Furthermore, the label lr refers to the set of all remaining u-independent quantum numbers needed to uniquely specify an energy eigenstate |lr,lηs,u〉. This refers to occupancy configurations of BA momentum quantum numbers qj=2πLIjβ. Here Ijβ are successive integers, Ijβ=0,±1,±2,... , or half-odd integers, Ijβ=±1/2,±3/2,±5/2,... , according to well-defined boundary conditions. Their allowed occupancies are zero and one. The index β denotes several BA branches of quantum numbers defined below in Section 2.2.We call a Bethe state an energy eigenstate that is a LWS of both SU(2) symmetry algebras. For a Bethe state one then has that nη=ns=0 in Eq. (2), so that lηs0 stands for Sη,Ss,0,0. The non-LWSs |lr,lηs,u〉 can be generated from the corresponding Bethe states |lr,lηs0,u〉 as [53],(3)|lr,lηs,u〉=∏α=η,s(1Cα(Sˆα+)nα)|lr,lηs0,u〉;Cα=(nα!)∏j=1nα(2Sα+1−j),nα=1,...,2Sα,Sˆη+=∑j=1L(−1)jcj,↓†cj,↑†;Sˆs+=∑j=1Lcj,↓†cj,↑. Here Cα where α=η,s are normalization constants. The model in its full Hilbert space can be described either directly within the BA solution [36,54] or by application onto the Bethe states of the η-spin and spin SU(2) symmetry algebras off-diagonal generators, as given in Eq. (3).Relying on the model symmetries, for simplicity and without loss in generality the studies of this paper refer to electronic densities and spin densities in the ranges ne∈[0,1[ and m∈[0,ne], respectively. For such electronic densities and spin densities, the model ground states are LWSs of both the η-spin and spin SU(2) symmetry algebras. Hence we use the LWS formulation of 1D Hubbard model BA solution.The PDT is used in this paper to clarify one of the unresolved questions concerning the physics of the 1D Hubbard model at finite magnetic field, Eq. (1): The dependence of the exponents that control the singularities at the σ one-electron spectral functions on the momentum, repulsive interaction u=U/4t, electron-density ne, and spin-density m. We derive the (k,ω)-plane line shape near the singularities of the following σ one-electron spectral function Bσ,γ(k,ω) such that γ=−1 (and γ=+1) for one-electron removal (and addition),(4)Bσ,−1(k,ω)=∑ν−|〈ν−|ck,σ|GS〉|2δ(ω+(Eν−Nσ−1−EGSNσ))ω≤0,Bσ,+1(k,ω)=∑ν+|〈ν+|ck,σ†|GS〉|2δ(ω−(Eν+Nσ+1−EGSNσ))ω≥0. Here ck,σ and ck,σ† are electron annihilation and creation operators, respectively, of momentum k. |GS〉 denotes the initial Nσ-electron ground state of energy EGSNσ. The ν− and ν+ summations run over the Nσ−1 and Nσ+1-electron excited energy eigenstates, respectively, and Eν−Nσ−1 and Eν+Nσ+1 are the corresponding energies.The remainder of the paper is organized as follows. In Section 2 the σ one-electron lower-Hubbard band (LHB) and upper-Hubbard band (UHB) are defined for u>0 and all densities in terms of quantum numbers associated with the rotated-electron energy eigenstates occupancies. Moreover, the relation of the β pseudoparticle representation to such rotated electrons is an issue also addressed in that section. The electron–rotated-electron unitary operator is uniquely defined in terms of its matrix elements between all model 4L energy and momentum eigenstates. The PDT suitable for the study of the σ one-electron spectral weights is the topic addressed in Section 3. This includes extracting further information beyond that provided in Refs. [39,40] on how the PDT accounts for the matrix elements of the electron operators between the ground state and the excited energy eigenstates. In Section 4 the (k,ω)-plane line shape near the singular spectral features of the σ one-electron spectral functions, Eq. (4), is studied. Finally, the concluding remarks are presented in Section 5.The complexity of the problems studied in this paper requires that some general concepts and theoretical tools used in our analysis are suitably revisited within the specific context of the one-electron problem in a finite magnetic field. Concerning which results are new, the most important such results refer to the expressions of the one-electron spectral functions of the 1D Hubbard model at finite magnetic field near the corresponding (k,ω)-plane high-energy singular features derived in Section 4. The exact relation defined in Section 2 of the c pseudoparticles, rotated spins 1/2, and rotated η-spins 1/2 to the electrons is also new. (The composite sn pseudofermions and composite ηn pseudofermions internal degrees of freedom refer to n=1,...,∞ neutral pairs of such rotated spins 1/2 and rotated η-spins 1/2, respectively.) That relation involves the extension of the unique definition of the electron–rotated-electron unitary operator given in Ref. [42] for a specific subspace of the Mott–Hubbard insulator phase to the model full Hilbert space. The PDT expressions of the leading-order operators that at finite magnetic field contribute to the one-electron spectral functions near high-energy singular features and the precise description of the corresponding microscopic processes reported in Section 3 are new as well. The same applies to the definition in Section 2 of lower- and upper-Hubbard bands for u>0 and electronic densities away from half filling in terms of rotated-electron occupancies.2Lower- and upper-Hubbard bands and the pseudoparticle representation emerging from the rotated electrons associated with the BA solutionImportant concepts for one-electron addition are those of a LHB and a UHB. Such bands are defined in Section 2.1 for the whole u>0 range and all densities in terms of rotated-electron quantum numbers associated with the one-electron addition excited energy eigenstates. In Section 2.2 the corresponding electron–rotated-electron unitary transformation performed by the BA solution is uniquely defined. The separation of the rotated-electron occupancy configurations that generate the exact u>0 energy eigenstates into occupancy configurations of three types of fractionalized particles is an issue also addressed in that section. The latter are the c pseudoparticles without internal degrees of freedom, the rotated spins 1/2, and the rotated η-spins 1/2. The electron–rotated-electron unitary operator definition allows the introduction and expression in Section 2.3 of operators for the c pseudoparticles, rotated spins 1/2, and rotated η-spins 1/2 in terms of the σ rotated-electron creation and annihilation operators. In Section 2.4 the pseudoparticle energy dispersions and other quantities that emerge from the pseudoparticle quantum liquid are introduced. Such quantities are needed for our study. They appear in the expressions of the σ=↑,↓ one-electron spectral functions, Eq. (4), near their (k,ω)-plane singular features derived in Section 4.2.1Definition of σ one-electron lower- and upper-Hubbard bandsThe concept of σ one-electron UHB addition is well established at electronic density ne=1 for u>0 [3,4,55]. Below we define the concepts of a LHB and a UHB for ne≠1 and u>0. At the ne=1 Mott–Hubbard insulator quantum phase there is only σ one-electron UHB addition. For ne≠1 there is both σ one-electron LHB and UHB addition. The Hamiltonian Hˆ, Eq. (1), quantum phases are associated with different ranges of electronic density ne and spin density m and are marked by important energy scales. Those correspond to limiting values of the charge energy scale 2μ=2μ(ne) and magnetic energy scale 2μBh=2μBh(m) involving the chemical potential μ and magnetic field h, respectively.The energy scales 2μ=2μ(ne) and 2μBh=2μBh(m) are odd functions of the hole concentration (1−ne) and spin density m, respectively. They are defined below in Section 2.5 in terms of BA energy dispersions. We consider the ranges ne∈[0,1[ and m∈[0,ne] for which they are positive. The interval ne∈]0,1[ refers for m<ne to a metallic quantum phase. For it 2μ=2μ(ne) is a continuous function of ne. It smoothly decreases from 2μ=(U+4t) for ne→0 to 2μ=2μu for ne→1 where 2μu<(U+4t) is the Mott–Hubbard gap. At ne=1 the chemical potential varies in the range μ∈[−μu,μu]. This is in spite of the electronic density remaining constant, which is a property of the corresponding ne=1 and u>0 Mott–Hubbard insulator quantum phase. The ne=1 Mott–Hubbard gap 2μu is the energy scale associated with the phase transition between the two above mentioned quantum phases. For u>0 it remains finite for all spin densities, m∈[0,1[.For the metallic quantum phase corresponding to the spin density interval m∈[0,ne[ for ne∈[0,1[ the magnetic energy scale 2μBh is a continuous function of m. It smoothly increases from zero at m=0 to 2μBhc for m→ne. Here hc is the critical field for the fully polarized ferromagnetism quantum phase transition. Indeed, for h>hc there is no electron double occupancy, so that the on-site repulsive interaction term in the Hamiltonian, Eq. (1), has no effects and the system is driven into a non-interactive quantum phase. The magnetic energy scale 2μBhc associated with such a quantum phase transition is an even function of the hole concentration (1−ne). Its analytical expression is given below in Section 2.5.The definition of the σ one-electron LHB and UHB addition for the whole u>0 range, electronic densities ne∈[0,1], and spin densities m∈[0,ne] relies on the occupancy configurations of uniquely defined rotated electrons. This involves selecting out of the many choices of u→∞ degenerate 4L energy eigenstates, those obtained from the u>0 Bethe states and corresponding non-LWSs, Eq. (3), as |lr,lηs,∞〉=limu→∞|lr,lηs,u〉.The wave function amplitudes of the u→∞ energy eigenstates |lr,lηs,∞〉 is an interesting issue discussed below in Section 2.2. As further discussed in that section, an important property is that σ electron single occupancy, double occupancy, and unoccupancy are good quantum numbers for such u→∞ energy eigenstates. Hence the numbers of electron ↑ and ↓ singly occupied sites, doubly occupied sites, and unoccupied sites are eigenvalues of corresponding number operators. We call V tower the set of energy eigenstates |lr,lηs,u〉 with exactly the same u-independent quantum numbers lr and lηs and different u values in the range u>0. σ electron single occupancy, electron double occupancy, and electron non-occupancy are not good quantum numbers for the finite-u energy eigenstates |lr,lηs,u〉 belonging to the same V tower. This means that for finite u the numbers of electron ↑ and ↓ singly occupied sites, doubly occupied sites, and unoccupied sites are u-dependent expectation values rather than integer eigenvalues. Consider, for instance, ground states with densities ne∈[0,1] and m∈[0,ne]. In the u→∞ limit they have zero electron double occupancy. Upon decreasing u, there emerges for such ground states belonging to the same V tower a finite electron double occupancy expectation value. For densities ne∈[0,1] and m=0 it reads D0=L(ne/2)2f(ne,u). Here f(ne,u) is a continuous function of ne and u with limiting behaviors limu→0f(ne,u)=1 and f(ne,u)=ln2u2(1−sin(2πne)2πne) for u≫1, respectively [58].For any u>0 value the set of energy eigenstates |lr,lηs,u〉 that belong to the same V tower are generated by exactly the same occupancy configurations of the u-independent quantum numbers lr and lηs given in Eq. (2) and below in Section 2.2, respectively. Hence the Hilbert space is the same for the whole u>0 range. It follows that for any u>0 there is a uniquely defined unitary operator Vˆ=Vˆ(u) such that |lr,lηs,u〉=Vˆ†|lr,lηs,∞〉. This operator Vˆ is the σ electron–rotated-electron unitary operator such that,(5)c˜j,σ†=Vˆ†cj,σ†Vˆ;c˜j,σ=Vˆ†cj,σVˆ;n˜j,σ=c˜j,σ†c˜j,σ,j=1,...,L,σ=↑,↓, are the operators that create and annihilate, respectively, the σ rotated electrons as defined here. Moreover, |lr,lηs,∞〉=Gˆlr,lηs†|0〉 where |0〉 is the electron and rotated-electron vacuum and Gˆlr,lηs† a uniquely defined operator. Hence |lr,lηs,u〉=G˜lr,lηs†|0〉. The generator G˜lr,lηs†=Vˆ†Gˆlr,lηs†Vˆ has the same expression in terms of the σ rotated-electron creation and annihilation operators as Gˆlr,lηs† in terms of σ electron creation and annihilation operators, respectively.Further useful information on the emergence of the rotated electrons associated with the operators, Eq. (5), is provided below in Section 2.2. This includes the unique definition of the electron–rotated-electron unitary operator Vˆ for the whole u>0 range. This is done in terms of its matrix elements between all 4L energy and momentum eigenstates, Eq. (3). The properties of the rotated electrons are found in that section to result from those of the electrons in the u→∞ limit. An important example is that, as reported above, σ electron single occupancy, electron double occupancy, and electron unoccupancy are good quantum numbers for a u→∞ energy eigenstate |lr,lηs,∞〉. This then implies that for all the finite-u energy eigenstates |lr,lηs,u〉 belonging to the same V tower σ rotated-electron single occupancy, rotated-electron double occupancy, and rotated-electron unoccupancy are also good quantum numbers for u>0. This applies to all 4L energy and momentum eigenstates provided that u>0.Ground states with densities ne∈[0,1] and m∈[0,ne] have zero rotated-electron double occupancy for the whole u>0 range. This is a direct consequence of ground states belonging to the same V towers having in u→∞ limit zero electron double occupancy. As confirmed in Section 2.2, the BA quantum numbers are directly related to the numbers of sites singly occupied, doubly occupied, and unoccupied by σ rotated electrons. The σ one-electron LHB addition spectral function Bσ,+1LHB(k,ω) and UHB addition spectral function Bσ,+1UHB(k,ω) are uniquely defined for u>0 as follows,(6)Bσ,+1(k,ω)=Bσ,+1LHB(k,ω)+Bσ,+1UHB(k,ω),Bσ,+1LHB(k,ω)=∑ν0+|〈ν0+|ck,σ†|GS〉|2δ(ω−(Eν0+Nσ+1−EGSNσ))ω≥0,Bσ,+1UHB(k,ω)=∑νD+|〈νD+|ck,σ†|GS〉|2δ(ω−(EνD+Nσ+1−EGSNσ))ω≥0. Here the ν0+ and νD+ summations run over the Nσ+1-electron excited energy eigenstates with zero and D>0, respectively, rotated-electron double occupancy and Eν0+Nσ−1 and EνD+Nσ+1 are the corresponding energy eigenvalues.The σ one-electron spectral functions obey the following sum rules,(7)∑k∫−∞∞dωBσ,−1(k,ω)=Nσ;∑k∫−∞∞dωBσ,+1(k,ω)=L−Nσ,∑k∫−∞∞dωBσ,+1LHB(k,ω)=L−N;∑k∫−∞∞dωBσ,+1UHB(k,ω)=N−Nσ. The first two sum rules are well known and exact for all u values. The Bσ,+1LHB(k,ω) and Bσ,+1UHB(k,ω) sum rules are found to be exact both in the ne→0 and ne→1 limits for u>0. Both in the u≪1 and u≫1 limits they are exact as well for electronic densities ne∈[0,1[ and spin densities m∈[0,ne]. They are likely exact also for intermediate u values yet we could not prove it. If otherwise, they are a very good approximation. Fortunately, clarification of this issue is not needed for our studies. Indeed, it focuses on the line shapes in the vicinity of the singularities of the σ one-electron spectral functions. This does not include the detailed weight distribution over the whole (k,ω) plane. The line shape near the singularities is actually that observed in experiments on actual condensed matter systems and spin 1/2 ultra-cold atom systems. The important point for the present study is rather the definition of σ one-electron LHB and UHB for u>0, ne∈[0,1], and m∈[0,ne], Eq. (6), which follows from the corresponding unique definition of rotated electrons in Section 2.2 in terms of quantities of the exact BA solution.The present definition for u>0 and all densities of the concepts of a LHB and a UHB is directly associated with a global lattice U(1) symmetry of the Hamiltonian Hˆu, Eq. (1), beyond its well-known SO(4)=[SU(2)⊗SU(2)]/Z2 symmetry. The latter contains the η-spin and spin SU(2) symmetries [59–61]. Such a global lattice U(1) symmetry exists for the model on the 1D lattice and on any other bipartite lattice [62]. It is behind its global symmetry being actually larger than SO(4) and given by [SO(4)⊗U(1)]/Z2=[SU(2)⊗SU(2)⊗U(1)]/Z22, which is equivalent to SO(3)⊗SO(3)⊗U(1). (The factor 1/Z22 follows from the total number 4L of independent representations of the group [SU(2)⊗SU(2)⊗U(1)]/Z22 being four times smaller than the dimension 4L+1 of the group SU(2)⊗SU(2)⊗U(1).)That the Hamiltonian Hˆu, Eq. (1), global symmetry is [SO(4)⊗U(1)]/Z2 has direct effects on the 4L energy and momentum eigenstates of the Hamiltonian Hˆ in the presence of a chemical potential and magnetic field also given in Eq. (1). Indeed, these states refer to 4L state representations of the group [SO(4)⊗U(1)]/Z2 in the model full Hilbert space. In the present 1D case, the occurrence of the global lattice U(1) symmetry justifies, for instance, that the spin and charge monodromy matrices of the BA inverse-scattering method have different ABCD and ABCDF forms associated with the spin SU(2) and charge U(2)=SU(2)⊗U(1) symmetries, respectively. (See the definition of such forms in Ref. [6].) Consistently, the latter matrix is larger than the former and involves more fields [6]. If the model global symmetry was SO(4)=[SU(2)⊗SU(2)]/Z2, the charge and spin monodromy matrices would have the same traditional ABCD form, which is that of the spin-1/2 XXX Heisenberg chain [63].What is the relation of the global lattice U(1) symmetry beyond SO(4) to the LHB and UHB as defined here for u>0 and all densities results? The generator of such a symmetry is the operator that counts the number NsR of rotated-electron singly occupied sites. Alternatively, that generator may be chosen to be the operator that counts the number NηR=L−NsR of rotated-electron unoccupied sites plus doubly occupied sites. The relation under consideration is that the UHB exactly originates from transitions to energy eigenstates with a finite number of (i) rotated-electron doubly occupied sites and (ii) rotated-electron unoccupied sites for the electronic density ranges (i) ne∈[0,1] and (ii) ne∈[1,2], respectively.2.2Rotated-electron separation in terms of c pseudoparticles, rotated spins 1/2, and rotated η-spins 1/2The charge-only and spin-only fractionalized particles that emerge in 1D correlated electronic systems are usually identified with holons and spinons, respectively [64]. In 1D integrable correlated electronic models, such holons and spinons are associated with excited-state occupancies of specific quantum numbers of the exact solutions. The use of holon and spinon representations provides a suitable description of these models low-energy physics. Some of such quantum liquids exotic properties survive at higher energies. However, the exponents characterizing the dynamical correlation functions singularities are functions of the momentum. They differ significantly from the predictions of the linear Tomonaga–Luttinger liquid theory [39,43–45]. This applies to the 1D Hubbard model.Furthermore, a careful analysis of the high-energy dynamical correlation functions reveals that their spectral weights are controlled by the scattering of both fractionalized particles without internal degrees of freedom and neutral composite objects. The constituents of the latter are spin-1/2 or η-spin 1/2 fractionalized particles. Both the fractionalized particles without internal degrees of freedom and the composite elementary objects refer to the pseudofermions of the PDT representation used in this paper to study the σ one-electron spectral functions, Eq. (4). Such pseudofermions are identical to the pseudoparticles of Ref. [41] except that their momentum values are slightly shifted by a well defined unitary transformation. The direct relation of the corresponding c pseudoparticles without internal degrees of freedom and spin-1/2 or η-spin 1/2 fractionalized particles within the neutral composite pseudoparticles to the rotated electrons whose operators are given in Eq. (5) encodes important physical information. Such a direct relation is actually the needed missing link of the corresponding non-perturbative relation between the electrons and PDT pseudofermions.It is useful for the understanding of the physics behind such relations to revisit some interesting properties of the 1D Hubbard model in the u→∞ limit. As mentioned above, in that limit the number of sites singly occupied by electrons, which we denote by Nc, is a good quantum number. The following related numbers are thus also conserved: The number Ms,±1/2 of sites singly occupied by electrons of spin projection ±1/2, the number Mη,+1/2 of sites unoccupied by electrons, and the number Mη,−1/2 of sites doubly occupied by electrons. These u→∞ electron conserved numbers are such that,(8)Ms=Ms,+1/2+Ms,−1/2=Nc,Mη=Mη,+1/2+Mη,−1/2=L−Nc=Nch,Ms,+1/2−Ms,−1/2=−2Ssz=N↑−N↓,Mη,+1/2−Mη,−1/2=−2Sηz=L−N.In u→∞ limit the model wave function amplitudes provided in Eqs. (2.5)–(2.10) of Ref. [35] are found to be given by Eq. (2.23) of Ref. [36]. The latter are of the general form,(9)(−1)Q[eiπMη,−1/2ψ1(y1d,y2d,...,yMη,−1/2d)]×[(∑P(−1)Pe(i∑j=1NckPjxQjs))ψ2(y1s,y2s,...,yMs,−1/2s)]. Here y1d,y2d,...,yMη,−1/2d are the spatial coordinates of the doubly occupied sites and y1s,y2s,...,yMs,−1/2s those of the down-spin singly occupied sites. Moreover, Q stands for a permutation that arranges the spatial coordinates x1s,x2s,...,xNcs of the singly occupied sites that multiply kP1,kP2,...,kPNc, respectively, into non-decreasing order. There is an additional restriction in the case of two equal spatial coordinates. The Pauli exclusion principle implies that they refer to electrons with different spin projection. The restriction is that the spatial coordinate of the electron with down spin projection must come first in x1s,x2s,...,xNcs. The sum ∑P in Eq. (9) runs over all permutations of the j=1,...,Nc BA real momentum rapidity numbers kj [36].Furthermore, the factor ∑P(−1)Pe(i∑j=1NckPjxQjs) in that equation is a Slater determinant of u→∞ spinless fermions [34,37]. They live on a lattice similar to that of the model. They occupy the Nc sites of spatial coordinates x1s,x2s,...,xNcs. The remaining L−Nc sites correspond to spinless fermion holes. The quantity ψ1(y1d,y2d,...,yMη,−1/2d) is in Eq. (9) the wave function of a u→∞ η-spin 1/2 XXX Heisenberg chain. Within the u→∞ limit, the η-spin SU(2) symmetry is associated with η-spins 1/2 of projection +1/2 and −1/2 that describe the η-spin degrees of freedom of the unoccupied and doubly occupied sites, respectively. The quantity ψ2(y1s,y2s,...,yMs,−1/2s) in that equation is in turn the wave function of a u→∞ spin 1/2 XXX Heisenberg chain. The corresponding Nc spins 1/2 are those of the electrons that singly occupy sites. The charges of these electrons are carried by the Nc spinless fermions. We call these two XXX chains, Heisenberg chains 1 and 2, respectively.On the one hand, the Mη=L−Nc η-spins 1/2 of the u→∞ Heisenberg chain 1 only “see” the L−Nc sites unoccupied and doubly occupied by electrons. Their spatial coordinates are those left over by the Nc spatial coordinates x1s,x2s,...,xNcs of the electron singly occupied sites. On the other hand, the Ms=Nc spins 1/2 of the u→∞ Heisenberg chain 2 only “see” the latter Nc sites. Hence for the u→∞ 1D Hubbard model in fixed-Nc subspaces one can define within the TL a squeezed η-spin effective lattice with Mη=L−Nc sites for the η-spins 1/2 and a corresponding squeezed spin effective lattice with Ms=Nc sites on which the singly-occupied sites spins 1/2 live. Such squeezed spaces are well known from studies of the 1D Hubbard model in that limit [33–38]. The order of the Ms=Nc Heisenberg chain 2 spins 1/2 in the squeezed spin effective lattice is the same as their order in the model lattice [36]. The same applies to the order of the Mη=L−Nc Heisenberg chain 1 η-spins 1/2 in the squeezed η-spin effective lattice.The form of the wave function amplitude, Eq. (9), follows from in the u→∞ limit the degrees of freedom of each 1D Hubbard model lattice site occupancy separating into two degrees of freedom. On the one hand, those of the Nc singly occupied sites separate into lattice/charge degrees of freedom described by the Nc spinless fermions and spin degrees of freedom associated with the Ms=Ms,+1/2+Ms,−1/2=Nc spins 1/2 of the Heisenberg chain 2, respectively. On the other hand, the degrees of freedom of the remaining L−Nc sites separate into lattice/charge degrees of freedom described by the L−Nc spinless fermion holes and η-spin/charge degrees of freedom associated with the Mη=Mη,+1/2+Mη,−1/2=L−Nc η-spins 1/2 of the Heisenberg chain 1, respectively.The electron occupancy configurations that generate the exact energy eigenstates |lr,lηs,∞〉 remain complex even in the corresponding u→∞ limit. It is easiest to express them in terms of spatial lattice occupancy configurations of the Nc spinless fermions, Mη=L−Nc Heisenberg chain 1 η-spins 1/2, and Ms=Nc Heisenberg chain 2 spins 1/2 that naturally emerge from the wave function amplitude, Eq. (9), degrees of freedom separation.The spatial lattice occupancies of the Nc spinless fermions that generate the exact energy eigenstates |lr,lηs,∞〉 can be expressed in terms of occupancy configurations of the BA momentum quantum numbers qj=2πLIjc introduced below. Here Ijc are successive integers Ijc=0,±1,±2,... or half-odd integers Ijc=±1/2,±3/2,±5/2,... according to well-defined boundary conditions. Similarly, the η-spins 1/2 (α=η) and spins 1/2 (α=s) spatial occupancies of their corresponding squeezed effective lattices, respectively, that generate such energy eigenstates can be expressed in terms of occupancy configurations of the BA momentum quantum numbers qj=2πLIjαn also considered below. Here Ijαn are again successive integers Ijαn=0,±1,±2,... or half-odd integers Ijαn=±1/2,±3/2,±5/2,... according to well-defined boundary conditions. Furthermore, n=1,...,∞ is the number of neutral η-spin 1/2 pairs (α=η) and spin 1/2 pairs (α=s) associated with the corresponding αn branches of BA quantum number configurations. Out of the Ms=Nc spins 1/2, an even number Nc−2Ss of them are paired within such spin-singlet configurations. The remaining 2Ss spins 1/2 remain unpaired, contributing to the spin SU(2) multiplet configurations. Similarly, out of the Mη=L−Nc η-spins 1/2, an even number L−Nc−2Sη of them are paired within the above ηn branches η-spin-singlet configurations. The 2Sη η-spins 1/2 left over remain unpaired, contributing to the η-spin SU(2) multiplet configurations. For a LWS, all 2Ss unpaired spins 1/2 have up spin projection and all 2Sη unpaired η-spins 1/2 have up η-spin projection, i.e. correspond to unoccupied sites.An important BA solution property is that for the whole u=U/4t>0 range the exact energy eigenstates |lr,lηs,u〉 remain being generated by occupancy configurations of exactly the same u-independent BA momentum quantum numbers qj=2πLIjc and qj=2πLIjαn where α=η,s and n=1,...,∞. Furthermore, also the spin and η-spin multiplet configurations are exactly the same for the whole u=U/4t>0 interval. For finite U the relation of the occupancy configurations of BA momentum quantum numbers qj=2πLIjc and qj=2πLIjαn to lattice occupancy configurations becomes though much more complex. This is because at finite U electron single occupancy and double occupancy are not good quantum numbers anymore. This is reflected in the much more involved form of the wave function amplitudes. Rather than the simpler form, Eq. (9), for general finite u values they are given by Eqs. (2.5)–(2.10) of Ref. [35].Interestingly, though, there is a uniquely defined unitary transformation under which such u>0 wave function amplitudes become of the simpler form, Eq. (9). That unitary transformation only changes the lattice electron occupancies that generate the exact energy eigenstates. It preserves their individual spins and charges. It actually maps the electrons and their operators into rotated electrons and their operators, as given in Eq. (5). The resulting rotated electrons have the same charge and spin as the corresponding electrons. For them single occupancy and double occupancy are good quantum numbers for the whole u>0 range. As mentioned in Section 2.1, the importance of such rotated electrons is that they are the link between the electrons and the pseudofermions of the PDT representation used in this paper to study the σ one-electron spectral functions, Eq. (4).For the 1D Hubbard model there is an infinite number of transformations that generate rotated electrons from the electrons such that rotated-electron single occupancy is a good quantum number for u>0 [62]. The pseudoparticle representation and corresponding pseudofermion representation refer though to the specific choice of rotated electrons under which the wave function amplitudes provided in Eqs. (2.5)–(2.10) of Ref. [35] become of the simpler form, Eq. (9). Those are thus generated from the electrons by a uniquely defined unitary transformation. Actually, the BA solution performs such a transformation. It is behind the exact energy eigenstates belonging to the same V tower being generated by exactly the same occupancy configurations of u-independent BA momentum quantum numbers for the whole u=U/4t>0 range.The electron–rotated-electron unitary operator Vˆ in Eq. (5) can be defined by its matrix elements between the model 4L energy and momentum eigenstates |lr,lηs,u〉. Fortunately, such matrix elements can be expressed in terms of the well known u>0 BA wave function amplitudes of the Bethe states |lr,lηs0,u〉,(10)flr,lηs0,u(x1σ1,...,xN0σN0)=〈x1σ1,...,xN0σN0|lr,lηs0,u〉. Those are uniquely defined in Eqs. (2.5)–(2.10) of Ref. [35]. In them, |x1σ1,...,xN0σN0〉 denotes a local state in which the N0=L−2Sη electrons with spin projection σ1,...,σN0 are located at sites of spatial coordinates x1,...,xN0, respectively. For a LWS their numbers are N↑0=L/2−Sη+Ss and N↓0=L/2−Sη−Ss. Due to symmetry, the amplitudes of the non-LWSs |lr,lηs,u〉 generated from each Bethe state as given in Eq. (3) differ from it by the trivial phase factor (−1)nη. Here nη=Sη+Sηz is the non-LWS number given in Eq. (2).The amplitudes 〈nη;ns;x1σ1,...,xN0σN0|lr,lηs,u〉 of a non-LWS are given in terms of those of the corresponding Bethe state merely by (−1)nη〈x1σ1,...,xN0σN0|lr,lηs0,u〉 and thus by (−1)nηflr,lηs0,u(x1σ1,...,xN0σN0). Here the local states |x1′σ1′,...,xN0+2nη′σ(N0+2nη)′〉 in which the N0+2nη electrons with spin projection σ1′,...,σ(N0+2nη)′ are located at sites of spatial coordinates x1′,...,xN0+2nη′ have been denoted by |nη;ns;x1σ1,...,xN0σN0〉. Except for the phase factor (−1)nη, this equality follows from the non-LWS amplitudes being insensitive to the nη created electrons pairs and their spatial coordinates. These electrons pairs emerge as a result of the application onto the Bethe state of the η-spin off-diagonal generator Sˆη+ a number of times nη, as given in Eq. (3). Moreover, such amplitudes are insensitive to the spatial coordinates of the ns electrons whose spin has been flipped by the ns spin off-diagonal generators (Sˆs+)ns, Eq. (3). Such insensitivities are behind denoting the local states |x1′σ1′,...,xN0+2nη′σ(N0+2nη)′〉 by |nη;ns;x1σ1,...,xN0σN0〉. They also imply that, as for the Bethe states, for the set of any energy eigenstates corresponding to different finite u values and belonging to the same V tower the general amplitudes flr,lηs,u(x1σ1,...,xN0σN0)=〈nη;ns;x1σ1,...,xN0σN0|lr,lηs,u〉 smoothly and continuously behave as a function of u.One then straightforwardly finds that the electron–rotated-electron unitary operator Vˆ in Eq. (5) is uniquely defined by the set of the following matrix elements between the energy eigenstates,(11)〈lr,lηs,u|Vˆ|lr′,lηs′,u〉=〈lr,lηs,u|lr′,lηs′,∞〉=δlηs,lηs′〈lr,lηs,u|lr′,lηs,∞〉=δlηs,lηs′∑x=1L...∑xN0=1Lflr,lηs0,u⁎(xσ1,...,xN0σN0)flr′,lηs0,∞(xσ1,...,xN0σN0). To arrive to the last expression of 〈lr,lηs,u|Vˆ|lr′,lηs′,u〉 given here, we accounted for the relations Vˆ†|lr′,lηs′,∞〉=|lr′,lηs′,u〉 and thus Vˆ|lr′,lηs′,u〉=|lr′,lηs′,∞〉. Moreover, we introduced in the resulting amplitude 〈lr,lηs,u|lr′,lηs′,∞〉 a decomposition of unity in terms of the complete basis of local states |nη;ns;x1σ1,...,xN0σN0〉.The quantity δl,l′ in Eq. (11) is the usual Kronecker symbol such that δl,l′=1 for l=l′=0,1,2,... and δl,l′=0 for l≠l′. The factor δlηs,lηs′ in that equation then implies that the phase factors (−1)nη always occur in pairs. This gives rise to an overall phase factor (−1)2nη=1. Moreover, flr,lηs0,u(x1σ1,...,xN0σN0) and flr′,lηs0,∞(x1σ1,...,xN0σN0) are in Eq. (11) the amplitude, Eq. (10), for the finite u value under consideration and u→∞, respectively. They are defined by Eqs. (2.5)–(2.10) of Ref. [35] for u>0 and Eq. (2.23) of Ref. [36] for u→∞, respectively. That the latter amplitude is that given in Eq. (9) and thus in Eq. (2.23) of Ref. [36] can be confirmed by expressing limu→∞flr′,lηs0,u(x1σ1,...,xN0σN0), Eqs. (2.5)–(2.10) of Ref. [35], in terms of the u→∞ spatial coordinates x1s,x2s,...,xNcs of the singly occupied sites, y1s,y2s,...,yMs,−1/2s of the down-spin singly occupied sites, and y1d,y2d,...,yMη,−1/2d of the doubly occupied sites.The set of 4L×4L=42L matrix elements of form, Eq. (11), are between all 4L energy and momentum eigenstates that span the model full Hilbert space. This is why they uniquely define the electron–rotated-electron unitary operator Vˆ. That because of symmetries behind the factor δlηs,lηs′ many of the matrix elements vanish simplifies the quantum problem under consideration. Specifically, the electron–rotated-electron unitary operator Vˆ commutes with the three generators of both the global η-spin and spin SU(2) symmetry algebras and the charge density operator. This ensures that the σ rotated electrons have the same charge and spin 1/2 as the σ electrons.From analysis of the relation between (i) the BA quantum numbers and (ii) rotated-electron occupancy configurations, respectively, that generate the finite-u exact energy eigenstates |lr,lηs,u〉=Vˆ†|lr,lηs,∞〉 of any V tower one reaches important physical information. First, the σ rotated-electron spatial occupancy configurations that generate the finite-u energy eigenstates |lr,lηs,u〉=Vˆ†|lr,lηs,∞〉 of any V tower are exactly the same as the σ electron spatial occupancy configurations of the tower u→∞ energy eigenstate |lr,lηs,∞〉. Hence for u>0 the number Ns,±1/2R of spin-projection ±1/2 rotated-electron singly occupied sites, Nη,+1/2R of rotated-electron unoccupied sites, and Nη,−1/2R of rotated-electron doubly occupied sites are conserved. Such numbers obey the sum rules Ns,±1/2R+Nη,−1/2R=N±1/2, NsR+2Nη,−1/2R=N, and NsR+NηR=L. The σ rotated-electron numbers values equal those of the σ electrons. Hence here N±1/2 denotes the number of electrons and rotated electrons of spin projection ±1/2. For finite u values the numbers NsR=Ns,+1/2R+Ns,−1/2R of rotated-electron singly occupied sites and NηR=Nη,+1/2R+Nη,−1/2R of rotated-electron doubly occupied plus unoccupied sites are only conserved for rotated electrons.Second, for u>0 a non-perturbative three degrees of freedom lattice–η-spin–spin separation occurs at all energy scales. Here the lattice–η-spin degrees of freedom separation may be considered as a separation of the charge degrees of freedom. At energy scales lower than 2|μ|, one has that D=Nη,−1/2R=0 (and Nη,+1/2R=0) for ne∈[0,1[ (and ne∈]1,2]). Therefore, the η-spin degrees of freedom are hidden. Hence the three degrees of freedom non-perturbative lattice–η-spin–spin separation is seen as the usual two degrees of freedom charge–spin separation. Within the former general separation the (i) lattice global U(1) symmetry, (ii) η-spin global SU(2) symmetry, and (iii) spin global SU(2) symmetry state representations are in each fixed number NsR of rotated-electron singly occupied sites subspace generated by well-defined occupancy configurations: (i) Those of the Nc=NsR c pseudoparticles without internal degrees of freedom and corresponding Nch=NηR c pseudoparticle holes whose c effective lattice is identical to the original lattice and thus has NsR+NηR=L sites; (ii) Configurations involving Ms,±1/2=Ns,±1/2R spin-1/2 fractionalized particles of spin projection ±1/2 that we call rotated spins 1/2; (iii) Those involving Mη,±1/2=Nη,+1/2R η-spin-1/2 fractionalized particles of η-spin projection ±1/2 that we call rotated η-spins 1/2. (+1/2 and −1/2 η-spin projections refer to η-spin degrees of freedom of rotated-electron unoccupied and doubly occupied sites, respectively.)Third, the properties of the rotated electrons stem for u>0 from those of the electrons in the u→∞ limit. Hence their above numbers equal for u>0 those of the Mη=Mη,+1/2+Mη,−1/2 u→∞ Heisenberg chain 1 η-spins 1/2, Ms=Ms,+1/2+Ms,−1/2 u→∞ Heisenberg chain 2 spins 1/2, and Nc u→∞ spinless fermions given in Eq. (8). As confirmed below in Section 2.3 in terms of operators, the Nc c pseudoparticles, Mη,±1/2 rotated η-spins of η-spin projection ±1/2, and Ms,±1/2 rotated spins of spin projection ±1/2 stem from rotated-electron occupancy configurations degrees of freedom separation. Hence their numbers are fully controlled by those of rotated electrons as follows,(12)Nc=NRs;Nch=NRη;Nc+Nch=NRs+NRη=L,Mα,±1/2=NR,±1/2α;Mα=Mα,+1/2+Mα,−1/2=NRα,α=η,s. The following rotated-electron properties valid for u>0 also stem from those in terms of electrons in the u→∞ limit. On the one hand, the degrees of freedom of each rotated-electron occupied site decouple into one c pseudoparticle without internal degrees of freedom that carries the rotated-electron charge and one rotated spin 1/2 that carries its spin. On the other hand, the degrees of freedom of each rotated-electron unoccupied and doubly occupied site decouple into one c pseudoparticle hole and one rotated η-spin 1/2 of projection +1/2 and −1/2, respectively. Hence the rotated-electron on-site separation refers to two degrees of freedom associated with the lattice global U(1) symmetry and one of the two global SU(2) symmetries, respectively. That the rotated-electron occupancy configurations give rise to the independent occupancy configurations of the c pseudoparticles, rotated spins 1/2, and rotated η-spins 1/2 is behind the exotic properties of the corresponding quantum liquid.Fourth, from the further analysis of the relation between the BA quantum numbers and the three degrees of freedom separation of the rotated-electron occupancy configurations one finds that such quantum numbers are directly associated with the occupancy configurations of the three types of fractionalized particles that generate all 4L energy eigenstates, Eq. (3). For the densities ranges ne∈[0,1] and m∈[0,ne] one has that Ns,+1/2R≥Ns,−1/2R and Nη,+1/2R≥Nη,−1/2R. For the corresponding exact Bethe states, there is a number Mssp=Ns,−1/2R of spin-singlet pairs (α=s) and Mηsp=Nη,−1/2R of η-spin-singlet pairs (α=η). All Ns,−1/2R rotated spins of projection −1/2 are paired with an equal number of rotated spins of projection +1/2. Similarly, all Nη,−1/2R rotated η-spins of projection −1/2 are paired with an equal number of rotated η-spins of projection +1/2. Such Mαsp spin-singlet (α=s) and η-spin-singlet (α=η) pairs are found to correspond to the internal structure of a set of composite αn pseudoparticles. Here n=1,...,∞ gives the number of pairs that refer to such an internal structure. One denotes by Nαn the number of such αn pseudoparticles in each energy and momentum eigenstate. The sum rule Mαsp=∑n=1∞nNαn is then obeyed.The remaining Mαun=Nα,+1/2R−Nα,−1/2R=2Sα unpaired rotated spins (α=s) and rotated η-spins (α=η) have for a Bethe state spin and η-spin projection +1/2. For general energy eigenstates, the multiplet configurations of these 2Ss unpaired rotated spins and 2Sη unpaired rotated η-spins generate the spin and η-spin, respectively, towers of non-LWSs. Specifically, the 2Ss unpaired rotated spins and 2Sη unpaired rotated η-spins of the Bethe states are flipped upon the application of the corresponding SU(2) algebras off-diagonal generators, as given in Eq. (3). Application of such generators leaves the spin (α=s) and η-spin (α=η) singlet configurations of the Mαsp=∑n=1∞nNαn pairs in αn pseudoparticles unchanged. Hence for general u>0 energy eigenstates one finds that the number Ms,±1/2un of unpaired rotated spins of projection ±1/2 and Mη,±1/2un of unpaired rotated η-spins of projection ±1/2 are good quantum numbers. They read,(13)Mα,±1/2un=(Sα∓Sαz);Mαun=Mα,−1/2un+Mα,+1/2un=2Sα,α=η,s. For the α=η,s LWSs one has that Mα,+1/2un=Mαun=2Sα and Mα,−1/2un=0 for both α=η,s. The set of Mηsp η-spin-singlet pairs and Mssp spin-singlet pairs of an energy eigenstate contains an equal number of rotated η-spins and rotated spins, respectively, of opposite projection. Hence the total numbers Mη,±1/2 of rotated η-spins of projection ±1/2 and Ms,±1/2 of rotated spins of projection ±1/2 are given by,(14)Mα,±1/2=Mαsp+Mα,±1/2un,α=η,s.Moreover, by combining the above equations one finds that the set of numbers {Nαn} of composite αn pseudoparticles of any u>0 energy eigenstate obey the following exact sum rules concerning the number of Mαsp of spin (α=s) and η-spin (α=η) singlet pairs,(15)Mαsp=∑n=1∞nNαn=12(Lα−2Sα),α=s,η,MspSU(2)≡∑α=η,s∑n=1∞nNαn=12(L−2Ss−2Sη). Here MspSU(2) denotes the total number of both rotated spins and rotated η-spins pairs.The BA solution contains different types of quantum numbers. Their occupancy configurations are within the pseudoparticle representation described by corresponding occupancy configurations of c pseudoparticles without internal degrees of freedom and composite αn pseudoparticles. Complete information on the microscopic details of the latter pseudoparticles internal η-spin (α=η) and spin (α=s) n-pair configurations is encoded within the BA solution. It is not needed for the studies of this paper. Indeed, within the present TL the problem concerning an αn pseudoparticle internal degrees of freedom and that associated with its translational degrees of freedom center of mass motion separate.Here we merely provide some general information on the internal degrees of freedom issue. As further discussed below, in the TL a composite αn pseudoparticle internal degrees of freedom are described by the imaginary part of a set of l=1,...,n BA complex rapidities with the same real part [7],(16)Λαn,l(qj)=Λαn(qj)+i(n+1−2l)u,l=1,...,n. Here α=η,s and n=1,...,∞. The real part Λαn(qj) of these rapidities depends on the αn pseudoparticle momentum qj defined in the following. It associated with the pseudoparticle translational degrees of freedom and may have j=1,...,Lαn different values.The number Lαn of the set j=1,...,Lαn of the αn branch BA quantum numbers {qj} and that Lc of the related set j=1,...,Lc of the c branch BA quantum numbers {qj} are given by,(17)Lαn=Nαn+Nαnh;Nαnh=2Sα+∑n′=n+1∞2(n′−n)Nαn′,α=η,s,n=1,...,∞,Lc=Nc+Nch=NsR+NηR=L, respectively. The real part Λαn(qj) of the complex rapidities, Eq. (16), is the rapidity function that for each u>0 energy eigenstate is the solution of the BA equations introduced in Ref. [7] for the TL. Within the pseudoparticle momentum distribution functional notation [41], these equations have the form given in Eqs. (A.1) and (A.2) of Appendix A. The sets of j=1,...,Lc and j=1,...,Lαn of quantum numbers qj, respectively, in these equations read,(18)qj=2πLIjβ,j=1,...,Lβ,β=c,ηn,sn,n=1,...,∞. Here the j=1,...,Lβ quantum numbers Ijβ are either integers or half-odd integers according to the following boundary conditions [7],(19)Ijβ=0,±1,±2,...forIβeven,=±1/2,±3/2,±5/2,...forIβodd, where the numbers Iβ are given by,(20)Ic=NpsSU(2)≡∑α=η,s∑n=1∞Nαn,Iαn=Lαn−1,α=η,s,n=1,...,∞.The β=c,αn band successive set j=1,...,Lβ of momentum values qj, Eq. (18), have only β pseudoparticle occupancies zero and one and the usual separation, qj+1−qj=2π/L. They play the role of β=c,αn band momentum values. Consistently, within our functional representation the momentum eigenvalues of all u>0 energy and momentum eigenstates are additive in qj. They read,(21)P=∑j=1LqjNc(qj)+∑n=1∞∑j=1LsnqjNsn(qj)+∑n=1∞∑j=1Lηn(π−qj)Nηn(qj)+πMη,−1/2. The β-band momentum distribution functions Nβ(qj) in this equation and BA equations, Eqs. (A.1) and (A.2) of Appendix A, read Nβ(qj)=1 and Nβ(qj)=0 for occupied and unoccupied discrete momentum values, respectively. One finds from the use of Eq. (14) that the momentum contribution πMη,−1/2 can be written as π(Mηsp+Mη,−1/2un). It results from both the paired and unpaired rotated spins 1/2 and rotated η-spins 1/2 of projection ±1/2 having an intrinsic momentum given by,(22)qs,±1/2=qη,+1/2=0;qη,−1/2=π. Furthermore, the ηn pseudoparticle contribution (π−qj) to the momentum eigenvalue, Eq. (21), refers to its translational degrees of freedom. It is associated with the center of mass motion of that composite n-pair object as a whole. That such a contribution to the momentum eigenvalue reads (π−qj) rather than qj is related to each of the Mηsp η-spin singlet pairs having an anti-binding character, as confirmed below in Section 2.5.On the one hand, the c pseudoparticles have no internal structure. On the other hand, the imaginary part i(n+1−2l)u of the set of l=1,...,n complex rapidities, Eq. (16), with the same real part Λαn(qj) refers to the internal degrees of freedom of one composite αn pseudoparticle with n>1 pairs whose center of mass carries αn band momentum qj. Specifically, for α=s the imaginary part of such l=1,...,n rapidities refers to the set l=1,...,n of spin-singlet pairs of rotated spins 1/2. It is associated with a corresponding binding of these pairs within the composite sn pseudoparticle. For α=η it is rather associated with a set l=1,...,n of η-spin-singlet pairs of rotated η-spins 1/2 and the binding of these pairs within the composite ηn pseudoparticle. (The anti-binding character found in Section 2.5 rather refers to the η-spin singlet configuration of a single pair.) Each such l=1,...,n rapidities thus refers to one of the l=1,...,n singlet pairs bound within the composite αn pseudoparticle. For n=1 the rapidity imaginary part vanishes. Indeed, the α1 pseudoparticle internal degrees of freedom correspond to a single singlet pair of rotated spins 1/2 (α=s) or rotated η-spins 1/2 (α=η).By combining the Mαsp sum rule, Eq. (15), with the Nα1h expression, Eq. (17) for n=1, one finds after some straightforward algebra that the following sum rules involving the number Nαps=∑n=1∞Nαn of composite αn pseudoparticles of all n=1,...,∞ branches and the related number NpsSU(2)=∑α=η,sNαps in Eq. (20) are also obeyed,(23)Nsps=∑n=1∞Nsn=12(Nc−Ns1h),Nηps=∑n=1∞Nηn=12(Nch−Nη1h),NpsSU(2)=∑α=η,s∑n=1∞Nαn=12(L−Ns1h−Nη1h). Hence for given fixed Nc and Nch=L−Nc values, that of Nαps is determined by the corresponding value of the number Nα1h of α1-band holes.The c band is populated by Nc=NsR c pseudoparticles. They occupy Nc discrete momentum values out of the c band j=1,...,Lc such momentum values, where Lc=L. Hence the number of c pseudoparticle holes indeed reads Nch=NηR=L−NsR. The number Lαn in Eq. (17) refers in turn to that of αn band j=1,...,Lαn momentum values qj in Eq. (18). For an energy and momentum eigenstate each such a band is populated by a well defined number Nαn of αn pseudoparticles. The corresponding number Nαnh of αn pseudoparticle holes is also a conserved number given in Eq. (17).The set j=1,...,Lβ of β=c,αn bands discrete momentum values qj whose different occupancy configurations generate the energy and momentum eigenstates and determine the corresponding momentum eigenvalues, Eq. (21), belong to well-defined domains, qj∈[qβ−,qβ+]. The corresponding limiting momentum values qβ± read,(24)qc±=±πL(L−1)≈±πforNpsSU(2)odd;qc±=±πL(L−1±1)≈±πforNpsSU(2)even,qαn±=±πL(Lαn−1).The label lr in an energy eigenstate |lr,lηs,u〉, Eq. (3), refers to the specific β bands occupancy configurations of the u-independent momentum values qj that generate the state,(25)lr={Ijβ}such thatNβ(qj)=1whereqj=2πLIjβforj=1,...,Lβ,β=c,ηn,sn,n=1,...,∞. All the energy eigenstates |lr,lηs,u〉 corresponding to different u>0 values and belonging to the same V tower are generated by exactly the same occupancy configurations of u-independent quantum numbers. The latter are associated with the labels lηs, Eq. (2), and lr, Eq. (25).Ground states are neither populated by composite sn pseudoparticles with n>1 spin-singlet pairs nor by composite ηn pseudoparticles with any number n=1,...,∞ of η-spin-singlet pairs. For electronic densities ne∈[0,1] and spin densities m∈[0,ne], ground states are LWSs. Hence they have no unpaired rotated spins of projection −1/2 and no unpaired rotated η-spins of projection −1/2. For them the number Msun=NsR=2Ss of unpaired rotated spins of projection +1/2 and the number Mηun=NηR=2Sη of unpaired rotated η-spins of projection +1/2 equal those Ns1h=NsR=2Ss of s1 pseudoparticle holes and Nch=NηR=2Sη of c pseudoparticle holes, respectively. Within the pseudoparticle representation of the one-electron excitations that contribute to the singularities of the spectral functions, Eq. (4), the unpaired rotated spins play the role of empty sites of the squeezed s1 effective lattice considered below in Section 2.3. Moreover, the unpaired rotated η-spins play the role of empty sites of the c effective lattice. Hence their translational degrees of freedom are accounted for by that representation.The ground-state β band pseudoparticle momentum distribution functions are given by,(26)Nc0(qj)=θ(qj−qFc−)θ(qFc+−qj);Ns10(qj)=θ(qj−qFs1−)θ(qFs1+−qj);Nαn0(qj)=0,αn≠s1, where the distribution θ(x) reads θ(x)=1 for x>0 and θ(x)=0 for x≤0. For the c and s1 bands the momentum distribution functions, Eq. (26), refer to compact and symmetrical occupancy configurations. The corresponding β=c,s1 Fermi points are associated with the Fermi momentum values qFβ± in Eq. (26). Accounting for O(1/L) corrections, they are given in Eqs. (C.4)–(C.11) of Ref. [41]. If within the TL we ignore such corrections, one finds that Nβ0(qj)=θ(qFβ−|qj|) for β=c,s1 where the Fermi momentum values are given by,(27)qFc=2kF=πne;qFs1=kF↓=πne↓.2.3The c pseudoparticle, rotated spin, and rotated η-spin operators in terms of σ rotated-electron operatorsThe c pseudoparticles, rotated spins 1/2, and rotated η-spins 1/2 naturally emerge from the σ rotated-electron onsite occupancy configurations separation. This allows the introduction of local operators for these fractionalized particles in terms of the local rotated-electron creation and annihilation operators, Eq. (5).The simplest case refers to the following l=z,± local operators associated with the rotated spins 1/2 (α=s) and rotated η-spins 1/2 (α=η),(28)S˜j,αl=Vˆ†Sˆj,αlVˆ,l=z,±,α=η,s,S˜j,α±=S˜j,αx±iS˜j,αy,α=η,s. Here Sˆj,αl are the usual unrotated l=z,± local spin (α=s) and η-spin (α=η) operators.The l=z,± local operators S˜j,αl, Eq. (28), have in terms of creation and annihilation rotated-electron operators, Eq. (5), exactly the same expressions as the corresponding unrotated l=z,± local operators Sˆj,αl in terms of creation and annihilation σ electron operators. The spin operators S˜j,sl act onto sites singly occupied by σ rotated electrons. They read S˜j,s−=(S˜j,s+)†=c˜j,↑†c˜j,↓ and S˜j,sz=(n˜j,↓−1/2). The η-spin operators S˜j,ηl act onto sites unoccupied by rotated electrons and sites doubly occupied by rotated electrons. They are given by S˜j,η−=(S˜j,η+)†=(−1)jc˜j,↑c˜j,↓ and S˜j,ηz=(n˜j,↓−1/2).Below the c pseudoparticle creation operator fj,c† and annihilation operator fj,c on the lattice site j=1,...,L are uniquely defined in terms of the local rotated-electron creation and annihilation operators, Eq. (5). (Their c effective lattice is identical to the original lattice.) The c pseudoparticles have inherently emerged from the rotated electrons to the sites singly occupied by the latter being occupied by c pseudoparticles and those unoccupied and doubly occupied by rotated electrons being unoccupied by c pseudoparticles. Hence the c pseudoparticle local density operator n˜j,c≡fj,c†fj,c and the corresponding operator (1−n˜j,c) are the natural projectors onto the subset of NRs=Nc original-lattice sites singly occupied by rotated electrons and onto the subset of NRη=Nch=L−Nc original-lattice sites unoccupied and doubly occupied by rotated electrons, respectively. It then follows that the α=s,η and l=z,± local operators S˜j,αl, Eq. (28), can be written as,(29)S˜j,sl=n˜j,cq˜jl;S˜j,ηl=(1−n˜j,c)q˜jl,l=z,±, respectively. The l=z,± local ηs quasi-spin operators,(30)q˜jl=S˜j,sl+S˜j,ηl,l=±,z, appearing here, such that q˜j±=q˜jx±iq˜jy, have the following expression in terms of rotated-electron creation and annihilation operators,(31)q˜j−=(q˜j+)†=(c˜j,↑†+(−1)jc˜j,↑)c˜j,↓;q˜jz=(n˜j,↓−1/2).The Nc c pseudoparticles live on the NRs=Nc sites singly occupied by the rotated electrons. Hence their occupancy configurations refer to the lattice degrees of freedom associated with the relative positions of the Ms=NRs=Nc sites occupied by rotated spins 1/2 and Mη=NRη=Nch=L−Nc sites occupied by rotated η-spins 1/2. The corresponding three degrees of freedom separation of the rotated-electron occupancy configurations then implies that the rotated-electron operators, Eq. (5), can be written as,(32)c˜j,↑†=(12−S˜j,sz−S˜j,ηz)fj,c†+(−1)j(12+S˜j,sz+S˜j,ηz)fj,c;c˜j,↑=(c˜j,↑†)†,c˜j,↓†=(S˜j,s++S˜j,η+)(fj,c†+(−1)jfj,c),c˜j,↓=(c˜j,↓†)†.The local c pseudoparticle operators fj,c† and fj,c appearing here are then uniquely defined for u>0 in terms of σ rotated-electron creation and annihilation operators, Eq. (5). This is achieved by combining the inversion of the relations, Eq. (32), with the expressions of the l=z,± local operators S˜j,αl, Eq. (28), provided in Eqs. (29)–(31). These operators are associated with the rotated spins 1/2 (α=s) and rotated η-spins 1/2 (α=η). This uniquely gives,(33)fj,c†=(fj,c)†=c˜j,↑†(1−n˜j,↓)+(−1)jc˜j,↑n˜j,↓;n˜j,c=fj,c†fj,c,j=1,...,L, where n˜j,σ is the σ rotated-electron local density operator in Eq. (5).The unitarity of the electron–rotated-electron transformation implies that the rotated-electron operators c˜j,σ† and c˜j,σ, Eqs. (5) and (32), have the same anticommutation relations as the corresponding electron operators cj,σ† and cj,σ, respectively. Straightforward manipulations based on Eqs. (28)–(33) then lead to the following algebra for the local c pseudoparticle creation and annihilation operators,(34){fj,c†,fj′,c}=δj,j′;{fj,c†,fj′,c†}={fj,c,fj′,c}=0. Furthermore, the local c pseudoparticle operators and the l=z,± local rotated quasi-spin operators q˜jl, Eq. (31), commute with each other. The latter l=z,± operators and corresponding rotated η-spin (α=η) and rotated spin (α=s) operators S˜j,αl, Eqs. (28) and (29), obey the usual SU(2) operator algebra.On the one hand, the c pseudoparticle and ηs quasi-spin operator algebras refer to the whole Hilbert space. On the other hand, those of the rotated η-spin and rotated spin operators correspond to well-defined subspaces spanned by energy eigenstates whose value of the number NsR=Nc of rotated-electron singly occupied sites and thus of c pseudoparticles is fixed. This ensures that the value of the corresponding rotated η-spin number Mη=NηR=L−Nc and rotated spin number Ms=NsR=Nc is fixed as well.The degrees of freedom separation, Eq. (32), is such that the c pseudoparticle operators, Eq. (33), rotated-spin 1/2 and rotated-η-spin 1/2 operators, Eq. (29), and the related ηs quasi-spin operators, Eqs. (30) and (31), emerge from the rotated-electron operators by an exact local transformation that does not introduce constraints.That, as given in Eq. (26), ground states are only populated by c and s1 pseudoparticles plays an important role in the PDT. On the one hand and as mentioned above, for u>0 the c pseudoparticles live on a c effective lattice identical to the original lattice. It thus has j=1,...,L sites and length L. On the other hand, the s1 pseudoparticles live on a squeezed s1 effective lattice [34,37,38]. Its j=1,...,Ls1 sites number Ls1 equals that of s1 band discrete momentum values, Eq. (17) for αn=s1. The squeezed s1 effective lattice has length L. Hence its spacing is in general larger than a. In the TL considered in this paper it is given by,(35)as1=NaLs1a. This ensures that indeed L=Ls1as1. (Except in Eq. (35), in this paper we use units of lattice spacing a one.)The s1 pseudoparticle translational degrees of freedom center of mass motion are described by operators fj,s1† (and fj,s1) that create (and annihilate) one s1 pseudoparticle at the s1 effective lattice site xj=as1j where j=1,...,Ls1. Such local s1 pseudoparticle creation and annihilation operators obey a fermionic algebra. This is consistent with the β=c,s1 band momentum value qj having only occupancies zero and one.The s1 pseudoparticle operator representation is valid for the 1D Hubbard model in subspaces spanned by energy eigenstates with fixed Ls1 value, Eq. (17) for αn=s1. That in such subspaces the local s1 pseudoparticle operators obey a fermionic algebra, can be confirmed in terms of their statistical interactions [65]. This is a problem that we address here very briefly. The local s1 pseudoparticle creation and annihilation operators may be written as,(36)fj,s1†=eiϕj,s1gj,s1†;fj,s1=(fj,s1†)†,j=1,...,Ls1. Here ϕj,s1=∑j′≠jfj′,s1† and the operator gj,s1† obeys a hard-core bosonic algebra. This algebra is justified by the corresponding statistical interaction vanishing for the model in subspaces spanned by energy eigenstates with fixed Ls1 value. The s1 effective lattice has been constructed inherently to that algebra being of hard-core type for the operators gj,s1† and gj,s1. Therefore, through a Jordan–Wigner transformation, fj,s1†=eiϕj,s1gj,s1† [66], the operators fj,s1† and fj,s1=(fj,s1†)† in Eq. (36) obey indeed a fermionic algebra,(37){fj,s1†,fj′,s1}=δj,j′;{fj,s1†,fj′,s1†}={fj,s1,fj′,s1}=0.Each of the Ns1 occupied s1 effective lattice sites corresponds to a spin-singlet pair. It thus involves two model lattice sites occupied by rotated spins 1/2 of opposite spin projection. For the densities ne∈[0,1[ and m∈[0,ne], the line shape in the vicinity of the singular features of the σ one-electron spectral functions, Eq. (4), studied in Sections 3 and 4 is controlled by ground state transitions to excited energy eigenstates for which Nsn=0 for n>1. For both the ground states and such excited states, the number Ns1h of unoccupied s1 effective lattice sites, Eq. (17) for αn=s1, reads Ns1h=2Ss. For such states the s1 effective lattice unoccupied sites refer to the Msun=Ms,+1/2un=2Ss sites occupied in the original lattice by the unpaired rotated spins 1/2. Such unpaired rotated spins 1/2 are used within the s1 pseudoparticle motion as unoccupied sites with which they interchange position.The β=c,s1 pseudoparticle operators labeled by the β=c,s1 band momentum values defined in Eqs. (18) and (19) play a key role in these studies. They read,(38)fqj,β†=1L∑j′=1Lβeiqjxj′fj′,β†;fqj,β=(fqj,β†)†,j=1,...,Lβ,β=c,s1. The j′=1,...,Lβ local operators fj′,β† appearing in this expression are those given in Eqs. (33) and (36) for β=c and β=s1, respectively.The s1 pseudofermion operators labeled by momentum qj, Eq. (38) for β=s1, act within subspaces spanned by energy eigenstates with fixed Ls1 values. In addition, they also appear in the expressions of the shakeup effects generators that transform such subspaces quantum number values into each other.The expressions of the σ one-electron LHB and UHB addition spectral functions, Eq. (6), near their singularities studied in Sections 3 and 4 are controlled by transitions to excited energy eigenstates with Nη1=0 and Nη1=1, respectively. Such states are not populated by composite αn pseudoparticles with n>1 pairs. Moreover, they have no unpaired rotated spins of projection −1/2 and no unpaired rotated η-spins of projection −1/2.As for the s1 pseudoparticles, one introduces anti-commuting operators fqj,η1† and fqj,η1 for the η1 pseudoparticles. Such η1 pseudofermion operators appear in the one-electron expressions derived below in Section 3.2. As justified in Section 3, their explicit use is not though required in what the computation of the corresponding UHB one-electron matrix elements involving creation of one η1 pseudofermion with momentum values ±(π−2kF) is concerned. Only the c and s1 pseudofermion operators generated from the β=c,s1 pseudoparticle operators, Eq. (38), are needed for the computation of the one-electron matrix elements considered in our study.2.4Needed quantities associated with the β pseudoparticle quantum liquidThe quantities associated with the β pseudoparticle quantum liquid briefly revisited in this section are needed for the σ one-electron spectral functions expressions studied in Sections 3 and 4.A particle subspace (PS) is spanned by one ground state and the set of excited energy eigenstates generated from it by a finite number of β pseudoparticle processes. For the densities ne∈[0,1[ and m∈[0,ne] considered in this paper, ground states are LWSs of both the spin and η-spin SU(2) symmetry algebras. The deviation densities δNβ/L, δSs/L, and δSη/L of their PS excited energy eigenstates vanish in the TL as L→∞. For a PS there are though no restrictions on the value of the excitation energy and excitation momentum.The β pseudoparticle quantum liquid shortly reported here refers to general PSs whose finite occupancies may correspond to more β=c,αn bands than those of the PSs directly involved in our study. It is often convenient within the TL to replace the β=c,αn band discrete momentum values qj, Eq. (18), such that qj+1−qj=2π/L, by a corresponding continuous momentum variable, q. It belongs to a domain q∈[qβ−,qβ+] whose limiting momentum values qβ± are given in Eq. (24). For the β=αn bands the relation qαn−=−qαn+ is exact, as given in that equation. Ignoring 1/L corrections as L→∞, one finds qβ±≈±qβ where for all β=c,αn bands qβ has simple expressions for the ground states and their PS excited energy eigenstates. For the present densities ranges they read [41],(39)qc=π;qs1=kF↑;qsn=(kF↑−kF↓)=πm,n>1;qηn=(π−2kF)=π(1−ne).The β=c,αn momentum band distribution functions of the PS excited energy eigenstates are of the general form Nβ(qj)=Nβ0(qj)+δNβ(qj). The ground-state β band pseudoparticle momentum distribution functions Nβ0(qj) appearing here are given in Eq. (26). Several physical quantities are then expressed as functionals of the corresponding β=c,αn momentum band distribution function deviations,(40)δNβ(qj)=Nβ(qj)−Nβ0(qj),j=1,...,Lβ,β=c,αn,n=1,...,∞. For transitions to an excited energy eigenstate for which the number Ls1 of BA s1 band momentum values changes, their removal or addition occurs in the vicinity of the s1 band edges qs1−=−qs1+, Eq. (24) for αn=s1. Those are zero-momentum and zero-energy processes.Under transitions from a ground state to its PS excited energy eigenstates, there may occur a β band momentum qj shakeup effect. It is an overall β-band discrete momentum shift, qj→qj+2πΦβ0/L, where Φβ0 reads,(41)Φc0=0;δNpsSU(2)even;Φc0=±12;δNpsSU(2)odd;Φαn0=0;δNc+δNαneven;Φαn0=±12;δNc+δNαnodd,α=η,s,n=1,...,∞. Here δNpsSU(2) is the deviation in the number NpsSU(2), Eq. (23). The shakeup effect results from a collective shift, (2π/L)Φβ0=±π/L, that all β band discrete momentum values qj=(2π/L)Ijβ may undergo due to a change in the boundary conditions that determine the qj values, Eqs. (18) and (19).Within the continuum q representation, the deviation values δNβ(qj)=−1 and δNβ(qj)=+1, Eq. (40), become δNβ(q)=−(2π/L)δ(q−qj) and δNβ(q)=+(2π/L)δ(q−qj), respectively. Here and throughout this paper, δ(x) denotes the usual Dirac delta-function distribution. Within such a representation the ground state occupancy q∈[−qFβ,qFβ] becomes a continuum compact distribution. Hence a β band shakeup effect is felt mostly by q values at the four β=c,s1 and ι=±1 Fermi points, ιqFβ. This effect is captured within that representation by additional deviations, ±(π/L)δ(q−ιqFβ). Their signs ± are those of (2π/L)Φβ0=±π/L.The PS energy functionals are derived from the use in the TBA equations, Eqs. (A.1)–(A.2) of Appendix A, and general energy spectra, Eq. (A.4) of that Appendix, of distribution functions of general form Nβ(qj)=Nβ0(qj)+δNβ(qj) for the excited energy eigenstates. The combined and consistent solution of such equations and spectra up to second order in the deviations, Eq. (40), leads to [57],(42)δE=∑β∑j=1Lβεβ(qj)δNβ(qj)+1L∑β∑β′∑j=1Lβ∑j′=1Lβ′12fββ′(qj,qj′)δNβ(qj)δNβ′(qj′)+∑α=η,sεα,−1/2Mα,−1/2un. For the present densities ranges, the unpaired rotated η-spin (α=η) and unpaired rotated spin (α=s) energies in this expression read,(43)εα,−1/2=2μα;εα,+1/2=0,α=η,s, where the energy scales 2μα are given by,(44)2μη=2|μ|;2μs=2μB|h|. The latter expression applies to general electronic and spin densities. It reads 2μη=2μ and 2μs=2μBh for the densities ranges ne∈[0,1[ and m∈[0,ne] which Eq. (43) refers to. For the ne=1 Mott–Hubbard insulator phase, the unpaired rotated η-spin energy rather is given by εη,∓1/2=(μu±μ) for μ∈[−μu,μu]. The ne=1 Mott–Hubbard gap 2μu appearing in that range is behind the spectra of the one-electron and charge excitations of the half-filled 1D Hubbard model being gapped [3,4,56].The β=c,αn band energy dispersions εβ(qj) in Eq. (42) are given by,(45)εβ(qj)=Eβ(qj)+tπ∫−QQdk2πΦ¯c,β(sinku,Λ0β(qj)u)sink,j=1,...,Lβ. Here Eβ(qj) stands for the β=c,ηn,sn energy spectra, Eq. (A.5) of Appendix A, with the rapidity functions in their expressions given by the ground-state rapidity functions k0c(qj) and Λ0β(qj). These functions are the solution of Eqs. (A.1) and (A.2) of that Appendix for the β-band ground-state distribution function distributions, Eq. (26). The parameter Q also appearing in Eq. (45) and related parameters B, rc0, and r0s read,(46)Q≡kc0(2kF);B≡Λ0s1(kF↓);rc0=sinQu;rs0=Bu.Furthermore, the rapidity dressed phase shift 2πΦ¯c,β(r,r′) in Eq. (45) is a particular case of the more general rapidity dressed phase shifts 2πΦ¯β,β′(r,r′) uniquely defined by the set of integral equations given in Eqs. (A.8)–(A.22) of Appendix A. The general expression of the f functions in the second-order terms of the energy functional, Eq. (42), is provided in Eq. (A.24) of that Appendix. It involves the related momentum dressed phase shifts,(47)2πΦβ,β′(qj,qj′)=2πΦ¯β,β′(r,r′);r=Λ0β(qj)/u;r′=Λ0β′(qj′)/u. Such a f function expression also involves the β band group velocities vβ(qj). Within the TL continuum q representation, they read,(48)vβ(q)=∂εβ(q)∂q,β=c,ηn,sn,n=1,...,∞;vβ≡vβ(qFβ),β=c,s1. The β band energy dispersions appearing here are given in Eq. (45).The following overall dressed phase shift functional,(49)2πΦβ(qj)=∑β′∑j′=1Naβ′2πΦβ,β′(qj,qj′)δNβ′(qj′),j=1,...,Lβ,β=c,s1, plays a key role in the PDT dynamical correlation functions expressions. It involves the momentum dressed phase shifts, Eq. (47). The summation ∑β′ in Eq. (49) refers to β′=c,s1 for σ one-electron removal and LHB addition and to β′=c,s1,η1 for σ one-electron UHB addition. The deviation δNβ′(qj′) is defined in Eq. (40).The functional energy spectrum, Eq. (42), describes the 1D Hubbard model as a quantum liquid of c, ηn, and sn pseudoparticles. They have residual interactions associated with the f functions, Eqs. (A.24). On the one hand, the general energy spectrum, Eq. (A.4) of Appendix A, gives the energy eigenvalues. On the other hand, that given in Eq. (42) provides the excitation energies. Those are given by the excited-state energy eigenvalues minus the ground state energy. The second term of the energy dispersion, Eq. (45), and the f-function terms in Eq. (42) are absent from Eq. (A.4) of Appendix A. Indeed, they stem from the latter energies difference. This justifies why that energy dispersion term and the f-function expressions involve dressed phase shifts, Eq. (47): Those emerge under the transitions from the ground state to energy eigenstates of excitation energy, Eq. (42).The spectra of the σ one-electron spectral functions near their singular features are expressed in Sections 3 and 4 in terms of the c and s1 band energy dispersions, Eq. (45) for β=c,s1. The spectrum of a particular type of such features involves as well the β pseudoparticle group velocities, Eq. (48). The exponents that control the line shape in the vicinity of singular features called branch lines also considered in these sections are expressed in terms of momentum dressed phase shifts, Eq. (47). Hence in Appendix B useful limiting behaviors of these quantum-liquid quantities are provided.2.5Binding and anti-binding character of the rotated spins 1/2 (α=s) and rotated η-spins 1/2 (α=η) pairing, respectively, and important energy scalesOn the one hand, for general electronic densities ne≠1 and all spin densities m the energy of two unpaired rotated η-spins (α=η) and two unpaired rotated spins (α=s) of opposite projection reads,(50)2μα=εα,−1/2+εα,+1/2,α=η,s, where the energy scale 2μα is given in Eq. (44). For ne=1 and m∈[−1,1] this expression remains valid for α=s. For α=η it is replaced by 2μu=εη,−1/2+εη,+1/2 and thus rather involves the ne=1 Mott–Hubbard gap 2μu.On the other hand, the αn pseudoparticle energy dispersion, Eq. (45) for β=αn, may be written as,(51)εαn(qj)=n2μα+εαn0(qj),α=η,s,n=1,...,∞. The term n2μα in this energy dispersion is merely additive in the bare energy 2μα, Eq. (50). Indeed, that bare η-spin-triplet (α=η) and spin-triplet (α=s) pair energy also applies to an η-spin-singlet (α=η) and spin-singlet (α=s) pair, respectively. This requires that εαn(qj)=n2μα and thus εαn0(qj)=0 in Eq. (51). This occurs when each single pair configuration has no binding or anti-binding character.The internal degrees of freedom of an α1 pseudoparticle correspond to a single pair. The energy εα10(qj) in its energy dispersion εα1(qj)=2μα+εα10(qj), Eq. (51) for n=1, refers to a binding or anti-binding character if εα10(qj)<0 or εα10(qj)>0, respectively. One finds that εs10(qj)<0 for |qj|<qs1 and εη10(qj)>0 for |qj|<qη1. This reveals that the spin-singlet s1-pair configuration and the η-spin-singlet η1-pair configuration have a binding and an anti-binding character, respectively. Interestingly, though, one finds that εα10(±qα1)=0 at the α=s,η limiting momenta qj=±qα1. Hence at these two band edge momenta the two spins 1/2 (α=s) or two η-spins 1/2 (α=η) remain in a singlet configuration, yet become unbound. The energy εα1(±qα1)=2μα then reduces to the intrinsic pair energy, Eq. (50).In the case of a composite αn pseudoparticle with n>1 pairs bound within it, one finds as well that εsn0(qj)<0 for |qj|<qsn and εηn0(qj)>0 for |qj|<qηn. However, the maximum absolute value of the αn-pair configuration binding (α=s) and anti-binding (α=η) energy per pair, |εαn0(0)|/n, is found to strongly decrease upon increasing n. This reveals that the binding of the n>1 pairs within the composite αn pseudoparticle tends to suppress the single-pair binding (α=s) and anti-binding (α=η) energy of each such pairs.Moreover, one finds that εαn0(±qαn)=0 and thus εαn(±qαn)=n2μα, as for the α1 pseudoparticles. Hence at the αn band limiting values qj=±qαn given in Eq. (39) the energy, Eq. (51), becomes additive in the intrinsic energy 2μα, Eq. (50). As discussed below in Section 4.3, this is due to a symmetry that is behind the σ one-electron UHB addition singular spectral features being for ne∈[0,1[ and under the transformations k→π−k and ω→2μ−ω similar to those of the corresponding σ¯ one-electron removal singular spectral features. (Here σ¯ denotes the spin projection opposite to σ.)The magnetic-field energy scale 2μBh=2μBh(m) and the energy scale 2μ=2μ(ne) associated with the chemical potential μ are given by [57],(52)2μBh(m)=−εs10(qFs1)∈[0,2μBhc]forqFs1=kF↓=π2(ne−m)wherem∈[0,ne]at fixedne,2μ(ne)=−2εc0(qFc)−εs10(qFs1)∈[2μu,(U+4t)]forqFc=2kF=πneandqFs1=π2(ne−m)wherene∈[0,1[at fixedm<ne. Here εs10(qj)=εs1(qj)−2μs is the s1 band binding energy in Eq. (51) and,(53)εc0(qj)=εc(qj)−12(2μη−2μs). In some limits the energy scales 2μBh=2μBh(m) and 2μ=2μ(ne), Eq. (52), are associated with the quantum phase transitions considered in Section 2.1.On the one hand, the ne=1 Mott–Hubbard gap 2μu associated with the phase transition between the metallic and Mott–Hubbard insulator quantum phases reads 2μu=limne→12μ(ne). For u>0 it remains finite for all spin densities, m∈[0,1[. In the limits m→0 [3,4,56] and m→1 it is found to read,(54)2μu=U−4t+8t∫0∞dωJ1(ω)ω(1+e2ωu)=16t2U∫1∞dωω2−1sinh(2πtωU),m→0,=(4t)2+U2−4t,m→1, respectively. Its u≪1 limiting behaviors are 2μu≈(8/π)tUe−2π(tU) for m→0 [56] and 2μu≈U2/8t for m→1. For u≫1 it behaves as 2μu≈(U−4t) for the whole spin density range m∈[0,1].On the other hand, the magnetic energy scale 2μBhc associated with the quantum phase transition to fully polarized ferromagnetism is given by 2μBhc=2μBh(m)|m=ne=2μBh(ne). It is found to have the following closed-form expression in terms of u=U/4t and the electronic density ne valid for the whole range ne∈[0,1[ [57],(55)2μBhc=2t[1+u2(1−2πarccot(1+u2utan(πne)))−2une−2πcos(πne)arctan(sin(πne)u)]. In the ne→0 and ne→1 limits this gives,(56)2μBhc=0,ne→0,=(4t)2+U2−U,ne→1, respectively. It behaves as 2μBhc=4tsin2(πne/2) for u→0 whereas for u≫1 its behavior is 2μBhc=(2tne/u)[1−sin(2πne)/(2πne)].At fixed electronic density ne<1, the magnetic-field energy scale 2μBh=2μBh(m), Eq. (52), is an increasing continuous function of the spin density m∈[0,ne]. It smoothly increases from 2μBh(0)=0 for m→0 to 2μBh(ne)=2μBhc for m→ne. Here 2μBhc is the magnetic energy scale, Eq. (55). At fixed spin density m<ne, the chemical potential energy scale 2μ=2μ(ne), Eq. (52), is a decreasing continuous function of the electronic density ne∈[0,1]. It smoothly decreases from 2μ(0)=U+4t for ne→0 to 2μ(1−)=2μu for ne→1. Here 2μu is the Mott–Hubbard gap, Eq. (54).3The pseudofermion dynamical theory microscopic processes that account for the σ one-electron spectral weightsHere the PDT quantities and concepts needed for our study on the one-electron spectral functions, Eq. (4), are introduced. This involves new needed information beyond that provided in Refs. [39,40] on the specific microscopic processes that control the σ one-electron spectral weights at finite magnetic field. This includes how the PDT accounts through such processes for the matrix elements of the σ electron creation or annihilation operators between the initial ground state and the excited energy eigenstates.Specifically, in Section 3.1 we briefly revisit the pseudofermion representation. The σ one-electron operators used in our study are expressed in terms of pseudofermion operators in Section 3.2. Section 3.3 addresses issues related to the matrix elements of these operators. In addition, the σ one-electron spectral functions are expressed in terms of β=c,s1 pseudofermion spectral functions. The effects of the small higher-order pseudofermion contributions to the σ one-electron spectral weight are discussed in Section 3.4. In Section 3.5 the involved state summations problem is addressed. Analytical expressions of the σ one-electron spectral functions are obtained by partially performing such state summations for (k,ω)-plane regions near these functions singular features. The relation of the PDT to conformal-field theory and finite-size scaling is the issue discussed in Section 3.6. In Section 3.7 the validity of the expressions for the line shape near σ one-electron singular spectral features is discussed. Finally, the general effects of symmetry on the one-electron spectral functions is the issue addressed in Section 3.8.3.1Pseudofermion representation to be used for the σ electron operators matrix elementsHere we consider the 1D Hubbard model at a finite magnetic field in a PS as defined in Section 2.4. For that quantum problem the c and s1 rapidity functions of the excited energy eigenstates can be expressed in terms of those of the corresponding initial ground state. This is given in Eq. (A.7) of Appendix A. The set of j=1,...,Lβ values q¯j=q¯(qj) in such excited energy eigenstates rapidity expressions Λc(qj)=Λ0c(q¯(qj)) and Λs1(qj)=Λ0s1(q¯(qj)) are the β=c,s1 band discrete canonical momentum values. They read,(57)q¯j=q¯(qj)=qj+2πΦβ(qj)L=2πL(Ijβ+Φβ(qj)),j=1,...,Lβ,β=c,s1. Here Φβ(qj) stands for the dressed phase-shift functional, Eq. (49), in units of 2π. The discrete canonical momentum values, Eq. (57), have spacing q¯j+1−q¯j=2π/L+h.o. where h.o. stands for contributions of second order in 1/L.We call a β=c,s1 pseudofermion each of the Nβ occupied β-band discrete canonical momentum values q¯j [39,40]. We call a β pseudofermion hole the remaining Nβh unoccupied β-band discrete canonical momentum values q¯j of a PS energy eigenstate. There is a pseudofermion representation for each initial ground state and its PS. This holds for all electronic and spin densities.The β=c,s1 pseudofermion creation and annihilation operators are generated from the corresponding β=c,s1 pseudoparticle creation and annihilation operators, Eq. (38), as follows,(58)f¯q¯j,β†=fqj+2πΦβ(qj)/L,β†=(SˆβΦ)†fqj,β†SˆβΦ;f¯q¯j,β=(f¯q¯j,β†)†,SˆβΦ=e∑j=1Lβfqj+2πΦβ(qj)/L,β†fqj,β;(SˆβΦ)†=e∑j=1Lβfqj−2πΦβ(qj)/L,β†fqj,β. In these expressions SˆβΦ is the β pseudoparticle–β pseudofermion unitary operator.The c and s1 pseudofermions live on exactly the same c and s1 effective lattices, respectively, as the corresponding pseudoparticles. The canonical-momentum β=c,s1 pseudofermion operators, Eq. (58), are related to local β=c,s1 pseudofermion operators f¯j′,β† and f¯j′,β. Those create and annihilate, respectively, one β=c,s1 pseudofermion at the β=c,s1 effective lattice site xj′=aβj′. Here j′=1,...,Lβ. The relation reads,(59)f¯q¯j,β†=1L∑j′=1Ls1eiq¯jxj′f¯j′,β†;f¯q¯j,β=1L∑j′=1Ls1e−iq¯jxj′f¯j′,β,j=1,...,Lβ,β=c,s1.By combining Eq. (33) with Eq. (59) for β=c, the c pseudofermion operator, Eq. (58), can be formally expressed in terms of rotated-electron operators as,(60)f¯q¯j,c†=1L∑j′=1Le+iq¯jj′(c˜j′,↑†(1−n˜j′,↓)+(−1)j′c˜j′,↑n˜j′,↓);f¯q¯j,c=(f¯q¯j,c†)†.On the one hand, the c pseudofermions have no internal structure. On the other hand, the s1 pseudofermions have exactly the same internal structure as the corresponding s1 pseudoparticles. They only differ in their discrete momentum values. Those rather refer to the translational degrees of freedom associated with their center of mass motion.In the present pseudofermion operator representation, a PS ground state has the simple form,(61)|GS〉=∏q¯=−kF↓kF↓∏q¯′=−ππf¯q¯,s1†f¯q¯′,c†|0〉=∏j=1N↓∏j′=1Lf¯q¯j,s1†f¯q¯j′,c†|0〉. That representation has been inherently constructed to q¯=q for a PS ground state. Indeed, for it the dressed phase-shift functional 2πΦβ(qj), Eq. (49), vanishes. Hence here the s1 and c band momentum values q¯=q=q¯j=qj and q¯′=q′=q¯j′=qj′, respectively, are those of the corresponding s1 and c pseudoparticle occupied ground-state Fermi seas. Moreover, |0〉 stands in Eq. (61) for the electron and rotated-electron vacuum. The ground-state generator onto that vacuum has been written in terms of s1 and c pseudofermion creation operators, Eqs. (58) and (59).The c pseudofermions as defined here refer to an extension to finite u of the usual u→∞ spinless fermions [33,34,37] considered in the discussions of Section 2.2. Indeed, in the u→∞ limit the momentum rapidity function of the ground state k0c(qj) simplifies to k0c(qj)=qj. The use of the exact relation, Eq. (A.7) of Appendix A, then leads to kc(qj)=q¯j for the PS excited energy eigenstates generated from the initial ground state under consideration. The u→∞ spinless fermions of Refs. [33,34] have been constructed inherently to carry such a momentum rapidity, kj=kc(qj)=q¯j. This confirms that the spinless fermions are the c pseudofermions as defined here in the u→∞ limit. The relations f¯q¯j,c†=Vˆ†bkj†Vˆ and f¯q¯j,c=Vˆ†bkjVˆ then hold. Here Vˆ is the electron–rotated-electron unitary operator defined in terms of its matrix elements in Eq. (11). Moreover, bkj† and bkj stand for the u→∞ spinless fermions creation and annihilation operators that appear in the anti-commutators given in the first equation of Section IV of Ref. [34].There is a one-to-one correspondence defined by Eq. (57) between a canonical momentum value q¯j and the corresponding bare momentum value qj. It enables the expression of several q¯j-dependent pseudofermion quantities in terms of the corresponding bare momentum qj. This applies to the dressed phase shift 2πΦβ,β′(qj,qj′), Eq. (47). Actually such a phase shift has a precise physical meaning within the pseudofermion representation: 2πΦβ,β′(qj,qj′) (and −2πΦβ,β′(qj,qj′)) is the phase shift acquired by a β pseudofermion or β pseudofermion hole of canonical momentum q¯j=q¯(qj) upon scattering off a β′ pseudofermion (and β′ pseudofermion hole) of canonical momentum value q¯j′=q¯(qj′) created under a transition from the ground state to a PS excited energy eigenstate.It then follows that the important functional 2πΦβ(qj), Eq. (49), in the β=c,s1 canonical momentum expression q¯j=qj+2πLΦβ(qj), Eq. (57), is the phase shift acquired by a β pseudofermion or β pseudofermion hole of canonical momentum value q¯j=q¯(qj) upon scattering off the set of β′ pseudofermions and β′ pseudofermion holes created under such a transition. Hence the β pseudofermion phase shift 2πΦβ(qj) has a specific value for each ground-state–excited-state transition.The expression of the σ one-electron UHB addition spectral function near its singular features has a contribution from the creation of a single η1 pseudoparticle at one of the η1 band limiting momentum values qj=±qη1=±(π−2kF), Eq. (39). η1 band canonical momentum values q¯j=qj+2πΦη1(qj)/L can be introduced, as in Eq. (57) for the β=c,s1 bands. Interestingly, one finds that 2πΦη1(qj)=0 at the η1 band limiting momentum values qj=±(π−2kF), so that q¯j=qj. This reveals that an η1 pseudofermion with canonical momentum values q¯j=±(π−2kF) does not acquire phase shifts under transitions from the ground state to the PS excited energy eigenstates. This is as for the unpaired rotated spins 1/2 and unpaired rotated η-spins 1/2. This behavior follows from a symmetry associated with the invariance under the η1 pseudoparticle–η1 pseudofermion unitary transformation of an η1 pseudoparticle with momentum values qj=±(π−2kF). An η1 pseudoparticle and an η1 pseudofermion of momenta ±(π−2kF) are indeed the same quantum object. Such a symmetry is behind the vanishing at qj=±qη1=±(π−2kF) of the η1 pseudoparticle anti-binding energy εη10(qj) on the right-hand side of Eq. (51) for αn=η1. The same applies to the corresponding η1 pseudofermion anti-binding energy εη10(q¯j), which also vanishes at q¯j=±(π−2kF).One can introduce a creation operator fqj,η1† for the η1 pseudoparticles. At qj=ι(π−2kF) where ι=±1 it is identical to the corresponding η1 pseudofermion creation operator,(62)f¯q¯j,η1†=fqj,η1†atq¯j=qj=ι(π−2kF),ι=±1. In the present case, f¯q¯j,η1† creates one η1 pseudofermion at the canonical momentum values q¯j=±(π−2kF).Such an η1 pseudofermion does not acquire phase shifts of its own. However, the β=c,s1 pseudofermions of canonical momentum q¯j acquire a phase shift 2πΦβ,η1(qj,±(π−2kF)), Eq. (47) for β′=η1 and qj′=±(π−2kF). This occurs under that η1 pseudofermion creation within a transition from the ground state to a PS excited energy eigenstate. After some manipulations relying on the use of Eqs. (A.9) and (A.15) of Appendix A for ηn=η1, one finds that it can be written as,(63)2πΦβ,η1(qj,±(π−2kF))=±12(δβ,c2π+2πΦβ,c(qj,2kF)−2πΦβ,c(qj,−2kF)),β=c,s1,ι=±1. Except for the factor 1/2, creation of one η1 pseudofermion at the canonical momentum values ±(π−2kF) is thus felt by a β=c,s1 pseudofermion as the creation and annihilation of two c pseudofermions at opposite c band Fermi points, respectively.The momentum dependent exponents that control the σ one-electron spectral weight in the (k,ω)-plane vicinity of a type of singular features called branch lines play an important role in the PDT. As reported below in Section 3.3, the expression of these exponents involves pseudofermion phase shifts 2πΦc,β(±2kF,qj) and 2πΦs1,β(±kF↓,qj) where β=c,s1. Such phase shifts are acquired by c and s1 pseudofermions, respectively, at the corresponding Fermi points. They result though from high-energy processes within which one c or s1 pseudofermion is created or annihilated at a canonical momentum q¯j=q¯(qj) associated with a momentum qj outside the c or s1 Fermi points, respectively. Furthermore, such exponents expression also involves the following related γ=0,1 Fermi-points phase-shift parameters,(64)ξββ′γ=δβ,β′+∑ι=±1(ι)γΦβ,β′(qFβ,ιqFβ′),β,β′=c,s1,γ=0,1. (For the particular case of β=β′ and ι=1 in Eq. (64), the present notation assumes that the two β=c,s1 Fermi momenta in the argument of the β pseudofermion phase shift, 2πΦβ,β(qFβ,qFβ), differ by 2π/L; For identical momentum values one has that 2πΦβ,β(qj,qj)=0.)In the particular case of β=c,s1 pseudofermions with momentum qj=ιqFβ at the ι=±1 Fermi points, the phase shift 2πΦβ,η1(qj,ι′(π−2kF)), Eq. (63), can be expressed in terms of the γ=1 parameters, Eq. (64), as follows,(65)Φβ,η1(ιqFβ,ι′(π−2kF))=ι′ξβc12,β=c,s1,ι,ι′=±1.The pseudofermion phase-shift related anti-symmetrical ξββ′1 and symmetrical ξββ′0 parameters, Eq. (64), emerge naturally from the pseudofermion representation. Their limiting behaviors are given in Appendix B. They are actually the entries of the low-energy conformal-field theory dressed-charge matrix and of the transposition of its inverse matrix [14,15,40,57],(66)Z1=[ξcc1ξcs11ξs1c1ξs1s11];Z0=((Z1)−1)T=[ξcc0ξcs10ξs1c0ξs1s10], respectively. (Here the dressed-charge matrix definition of Ref. [14] has been used, which is the transposition of that of Ref. [15].)As mentioned previously, for densities ne∈[0,1[ and m∈[0,ne] the PS excited energy eigenstates that contribute to the σ one-electron spectral functions expressions near their singularities are populated only by c and s1 pseudofermions. In case of the UHB one-electron addition, they are populated as well by a single η1 pseudofermion of canonical momentum ±(π−2kF). For such PSs, the pseudoparticle representation general energy functional, Eq. (42), simplifies to,(67)δE=∑β=c,s1∑j=1Lβεβ(qj)δNβ(qj)+1L∑β=c,s1∑β′=c,s1,η1∑j=1Lβ∑j′=1Lβ′12fββ′(qj,qj′)δNβ(qj)δNβ′(qj′)+2μNη1. Expression of this functional in the pseudofermion representation involves the β=c,s1 bands discrete canonical momentum values q¯j=q¯(qj), Eq. (57). One finds after some algebra that in such a representation it reads up to O(1/L) order,(68)δE=∑β=c,s1∑j=1Lβεβ(q¯j)δNβ(q¯j)+2μNη1. Here δNβ(q¯j)=δNβ(qj) and the β=c,s1 pseudofermion energy dispersions εβ(q¯j) have exactly the same form as those given in Eq. (45) with the momentum qj replaced by the corresponding canonical momentum, q¯j=q¯(qj).The pseudofermion energy functional, Eq. (68), can be expressed in terms of the momentum qj upon expanding the β=c,s1 band canonical momentum q¯j=qj+2πΦβ(qj)/L around qj and considering all energy contributions up to O(1/L) order. Upon performing such an expansion, one arrives after some algebra to the energy functional, Eq. (67). It includes terms of second order in the deviations δNβ(qj). Their absence from the corresponding energy spectrum, Eq. (68), follows from the β=c,s1 pseudofermion phase shifts 2πΦβ(qj), Eq. (49), being incorporated in the β=c,s1 band canonical momentum, Eq. (57).That in contrast to the equivalent energy functional, Eq. (67), that in Eq. (68) has no energy interaction terms of second-order in the deviations δNβ(q¯j) has a deep physical meaning. It is that the β=c,s1 pseudofermions have no such interactions up to O(1/L) order. Within the present TL, only finite-size corrections up to that order are relevant for the spectral functions expressions. The property that the excitation energy spectrum, Eq. (68), has no pseudofermion energy interactions is found below to simplify the expression of the σ one-electron spectral functions. They can be expressed in terms of a sum of convolutions of c and s1 pseudofermion spectral functions. Moreover, the spectral weights of the latter spectral functions can be expressed as Slater determinants of pseudofermion operators.3.2Expression of the σ one-electron problem in terms of pseudofermion operatorsWithin the PDT of Refs. [39,40], the β=c,s1 pseudofermion phase shifts control the dynamical correlation functions spectral-weight distributions. Here we provide additional specific information relative to that given in these references about how that dynamical theory accounts for the matrix elements 〈ν−|ck,σ|GS〉 and 〈ν+|ck,σ†|GS〉 in the one-electron spectral functions, Eq. (4), for the model at finite magnetic field. For such spectral functions the elementary processes that generate the excited energy eigenstates from ground states with densities in the ranges ne∈[0,1[ and m∈[0,ne] can be classified into three classes (A)–(C):(A) High-energy and finite-momentum elementary β=c,s1 pseudofermion processes. Specifically, creation or annihilation of one or a finite number of β=c,s1 pseudofermions with canonical momentum values q¯j≠±q¯Fβ outside the corresponding Fermi points;(B) Finite-momentum processes of excitation energy zero that change the number of β=c,s1 pseudofermions at the ι=+1 right and ι=−1 left β=c,s1 Fermi points and finite-momentum processes of high energy 2μ that involve creation of one η1 pseudofermion at one of the limiting momenta qη1±=±(π−2kF);(C) Low-energy and small-momentum elementary pseudofermion particle–hole processes in the vicinity of the β=c,s1 bands right (ι=+1) and left (ι=+1) Fermi points, relative to the excited-state β=c,s1 pseudofermion momentum occupancy configurations generated by the above elementary processes (A) and (B).The processes (B) of high energy 2μ contribute to the line shape near the σ one-electron UHB spectral function singular features. Their high excitation energy 2μ is the minimal energy for creation of one rotated-electron doubly occupied site. It stems from the first term of the spectrum Eη1(qj), Eq. (A.5) of Appendix A for αn=η1. Such a spectrum is part of the η1 energy dispersion εη1(qj), Eq. (45) for β=η1. Such processes refer to transitions from ground states with densities ne<1. For ne=1 initial ground states the σ one-electron UHB involves instead transitions to excited energy eigenstates populated by one unpaired rotated η-spin 1/2 of η-spin projection −1/2. This also amounts for creation of one rotated-electron doubly occupied site.The first two steps to express in the pseudofermion representation the matrix elements 〈ν−|ck,σ|GS〉 and 〈ν+|ck,σ†|GS〉 in the spectral functions, Eq. (4), are: (i) To express the σ electron creation or annihilation operator in terms of σ rotated electron creation and annihilation operators, Eq. (5); (ii) To express the latter operators in terms of rotated spin 1/2 operators, rotated η-spin 1/2 operators, and c pseudofermion operators. This is accomplished by accounting for the relation between the c pseudoparticle and c pseudofermion operators, Eq. (58) for β=c. In addition one uses the σ rotated electron creation and annihilation operators expressions in terms of rotated spin 1/2 operators, rotated η-spin 1/2 operators, and c pseudoparticle operators, Eqs. (32) and (71).The momentum k dependent σ electron operators in the spectral functions Lehmann representation, Eq. (4), are related to the corresponding local operators as,(69)ck,σ=1L∑j=1Leikxjcj,σ;ck,σ†=(ck,σ)†,σ=↑,↓. To write the operators ck,σ and ck,σ† in terms of σ rotated electron creation and annihilation operators, Eq. (5), we use the Baker–Campbell–Hausdorff formula. This allows rewriting the relation, Eq. (5), as follows,(70)ck,σ=∑i=0∞ck,σ,i=c˜k,σ+11![c˜k,σ,S˜]+12![[c˜k,σ,S˜],S˜]+...;ck,σ†=(ck,σ)†,σ=↑,↓,ck,σ,i=[c˜k,σ,S˜]i=1i![[c˜k,σ,S˜]i−1,S˜],i=1,...,∞;[c˜k,σ,S˜]0=c˜k,σ=Vˆ†ck,σVˆ,Vˆ=eSˆ=eS˜. Here the operator S˜=Sˆ commutes with Vˆ and thus has the same expression in terms of creation and annihilation rotated-electron operators and electron operators, respectively. Moreover, the momentum operators c˜k,σ†=Vˆ†ck,σ†Vˆ and c˜k,σ=Vˆ†ck,σVˆ can be written in terms of the local operators c˜j,σ† and c˜j,σ, respectively, in Eqs. (5) and (32) as,(71)c˜k,σ†=1L∑j=1Leikxjc˜j,σ†;c˜k,σ=(c˜k,σ†)†,σ=↑,↓.The next step of our program consists in rewriting the rotated-electron expression ck,σ=∑i=0∞ck,σ,i within its uniquely defined β pseudofermion representation as,(72)ck,σ=∑i′=0∞gˆi′(k)cˆ⊙. The new index i′=0,1,...,∞ refers here to β pseudofermions processes. Furthermore, cˆ⊙ is a generator that transforms the initial ground state |GS〉 into a state with the same electron and rotated-electron numbers as the ground state of the final PS, which we call |GSf〉. It has the same compact symmetrical c and s1 bands momentum occupancies as that ground state. The only difference between the states cˆ⊙|GS〉 and |GSf〉 lays in their c and s1 band discrete momentum values: They are those of the initial ground state, q¯′=q′, and of the excited-energy eigenstate ∑i′=0∞gˆi′(k)|GSf〉, q¯≠q, respectively.Each term of index i′=0,1,...,∞ in Eq. (72) may have contributions from several terms of different index i=0,1,...,∞ in ck,σ=∑i=0∞ck,σ,i, Eq. (70). Fortunately, one can compute the operational form in terms of β pseudofermion operators of the leading i′=0,1,...,∞ orders of ck,σ=∑i′=0∞gˆi′(k)cˆ⊙ from the transformation laws of the ground state |GS〉, Eq. (61). This is achieved upon action of the related operators ck,σ,i in the expression ck,σ=∑i=0∞ck,σ,i onto that state.The 1D Hubbard model is a non-perturbative quantum problem in terms of σ electron processes. This is behind the computation of the one-electron spectral functions, Eq. (4), being a very complex many-electron problem. A property that plays key role in our study follows from expressing the electron operator ck,σ in the terms of pseudofermion operators as ck,σ=∑i′=0∞gˆi′(k)cˆ⊙, Eq. (72). Indeed, this renders the computation of the σ one-electron spectral functions, Eq. (4), a perturbative problem.Note that both the expressions ck,σ=∑i=0∞ck,σ,i and ck,σ=∑i′=0∞gˆi′(k)cˆ⊙ are not small-parameter expansions. Consistently, the perturbative character of the β pseudofermions processes refers to the spectral weight contributing to the spectral functions being dramatically suppressed upon increasing the number of corresponding elementary processes of classes (A) and (B). Those are generated by application onto the ground state, Eq. (61), of operators in ∑i′=0∞gˆi′(k)cˆ⊙ with an increasingly large value of the index i′=0,1,...,∞.The perturbative character of the 1D Hubbard model upon expressing the σ electron creation or annihilation operators in the spectral functions, Eq. (4), in terms of pseudofermion operators follows from the exact energy eigenstates being generated by occupancy configurations of such pseudofermions. The non-perturbative character of the problem in terms of electrons results from their relation to the pseudofermions having as well a non-perturbative nature. It is qualitatively different from that of the electrons to the quasiparticles of a Fermi liquid.For simplicity, in the following we denote the i′=0 operator gˆ0(k) associated with the σ one-electron operator ck,σ (or ck,σ†) by gˆ(k). Such an i′=0 leading-order operator term in the one- or two-electron operator expression,(73)ck,σ=(gˆ(k)+∑i′=1∞gˆi′(k))cˆ⊙, plays a key role in our study.In the present case, the leading-order operators gˆ(k)cˆ⊙ are selected inherently to all the singular spectral features in the σ one-electron spectral functions, Eq. (4), being produced by their application onto the ground state. Here we list and define such leading-order pseudofermion processes (A) and (B). After being dressed by low-energy and small-momentum elementary β=c,s1 pseudofermion particle–hole processes (C) in the vicinity of their right (ι=+1) and left (ι=+1) Fermi points, they control the line shape near the singular features of the σ one-electron spectral functions, Eq. (4). Importantly, for the whole u>0 range the creation or annihilation of one σ electron gives rise to the creation or annihilation, respectively, of exactly one σ rotated electron. The leading-order pseudofermion processes (A) and (B) under consideration are the following:(1) Removal of one ↑ electron is a process that involves the recombination of one c pseudofermion and one unpaired rotated spin 1/2 of projection ↑. This leads to the emergence in the system of one ↑ rotated electron. It is annihilated under the ↑ electron removal. This process thus involves one c pseudofermion annihilation. It leads to a deviation δNc=−1. The annihilation of the unpaired rotated spin 1/2 of projection ↑ leaves the number Ns1 s1 pseudofermions unchanged. It leads to a deviation δNs1h=−1 in the number of s1 band holes.(2) LHB addition of one ↑ electron is a process that involves creation of one ↑ rotated electron. It separates into one c pseudofermion and one unpaired rotated spin 1/2 of projection ↑. Hence δNc=1. The creation of the unpaired rotated spin 1/2 leaves the number Ns1 s1 pseudofermions unchanged. It gives rise to a deviation δNs1h=1 in the number of s1 band holes.(3) UHB addition of one ↑ electron is a process that involves combination of one c pseudofermion with one rotated spin 1/2 of projection ↓. This gives rise to the emergence in the system of one ↓ rotated electron. The rotated spin 1/2 originates from one s1 pseudofermion spin-singlet pair breaking. Such processes thus involve annihilation of one c pseudofermion and one s1 pseudofermion and creation of one unpaired rotated spin 1/2 of projection ↑. As described below, one η1 pseudofermion is also created. Hence δNc=−1, δNs1=−1, and δNη1=1. The above emerging ↓ rotated electron pairs with the ↑ rotated electron that also emerges in the system under the ↑ electron creation. This gives rise to a rotated-electron doubly occupied site. The rotated η-spin 1/2 of projection −1/2 that describes the η-spin degrees of freedom of such a doubly occupied site combines with one ground-state unpaired rotated η-spin 1/2 of projection +1/2. This originates the η1 pseudofermion and its η-spin-singlet pair. The creation of one unpaired rotated spin 1/2 of projection ↑ leads to a deviation δNs1h=1 in the number of s1 band holes.(4) Removal of one ↓ electron is a process that involves the recombination of one c pseudofermion and one rotated spin 1/2 of ↓ projection. This gives rise to the emergence in the system of one ↓ rotated electron. The rotated spin 1/2 originates from one s1 pseudofermion spin-singlet pair breaking. Such processes thus involve annihilation of one c pseudofermion and one s1 pseudofermion and creation of one unpaired rotated spin 1/2 of projection ↑. This leads to the deviations δNc=−1 and δNs1=−1. The annihilation of the emerging ↓ rotated electron gives rise to the ↓ electron removal. The creation of the rotated spin 1/2 of projection ↑ leads to a deviation δNs1h=1 in the number of s1 band holes.(5) LHB addition of one ↓ electron is a process that involves the creation of one ↓ rotated electron. It separates into one c pseudofermion and one rotated spin 1/2 of ↓ projection. The latter combines with one unpaired rotated spin 1/2 of projection ↑. This gives rise to the formation of one s1 pseudofermion spin-singlet pair and annihilation of the unpaired rotated spin 1/2 of projection ↑. The corresponding deviations thus read δNc=1 and δNs1=1. The annihilation of the unpaired rotated spin 1/2 of projection ↑ leads to a deviation δNs1h=−1 in the number of s1 band holes.(6) UHB addition of one ↓ electron is a process that involves the recombination of one c pseudofermion with one unpaired rotated spin 1/2 of projection ↑. This gives rise to the emergence in the system of one ↑ rotated electron. That process thus involves the annihilation of one c pseudofermion and one unpaired rotated spin 1/2 of projection ↑. As described in the following, it involves as well the creation of one η1 pseudofermion. The corresponding deviations are thus given by δNc=−1 and δNη1=1. The above emerging ↑ rotated electron pairs with the ↓ rotated electron that also emerges in the system as a result of the ↓ electron creation. This gives rise to a doubly occupied site. The rotated η-spin 1/2 of projection −1/2 that describes the η-spin degrees of freedom of such a doubly occupied site combines with one ground-state unpaired rotated η-spin 1/2 of projection +1/2. This originates the creation of one η1 pseudofermion η-spin singlet pair. The annihilation of one unpaired rotated spin 1/2 leaves the number Ns1 s1 pseudofermions unchanged. It leads to a deviation δNs1h=−1 in the number of s1 band holes.The above elementary processes involving s1 pseudofermion pair breaking and s1 pseudofermion pair formation are behind the squeezed s1 effective lattice and corresponding s1 momentum band being exotic. Their number of sites and discrete momentum values, respectively, both given by Ls1=Ns1+Ns1h, have different values for different subspaces. Hence within the s1 pseudofermion operator algebra one distinguishes two types of variations in the number of s1-band holes: The s1-band holes created and annihilated by processes that conserve the number Ls1=Ns1+Ns1h, under which one s1 pseudofermion is annihilated and created, respectively; The s1-band holes created and annihilated by processes that do not conserve the number Ls1=Ns1+Ns1h. (For Ss>0 states such exotic Ls1 variations only lead to Ns1h variations.)The former processes are described by application of the operators f¯q¯,s1 and f¯q¯,s1†, respectively, onto the initial state. The processes leading to the latter Ns1h variations that do not conserve Ls1=Ns1+Ns1h result from vanishing energy and vanishing momentum processes. Under such processes discrete momentum values are added to and removed from one of the s1 band limiting momentum values qs1±, Eq. (24) for αn=s1. Whether such an addition or removal occurs at the left limiting momentum qs1− or at right limiting momentum qs1+ is uniquely defined. Only one of these two choices leaves invariant the s1 band symmetrical relation qs1+=−qs1− for the final state. Such a relation must hold for all energy eigenstates.In the present cases of (i) ↑ one-electron removal processes (1) and ↓ one-electron UHB addition processes (6) and (ii) ↑ one-electron LHB addition processes (2) a single discrete momentum value is (i) removed from and (ii) added to, respectively, the s1 band limiting momentum values. Such vanishing energy and vanishing momentum processes are implicitly accounted for by the pseudofermion representation. This occurs through the s1 band discrete momentum values of the final states, which are uniquely defined.In the following we derive the expression of the leading-order operators gˆ(k)cˆ⊙, Eq. (73), in terms of c and s1 pseudofermion operators for the processes (1), (2), (4), and (5). In the case of the σ one-electron UHB addition processes (3) and (6), they are expressed in terms of c, s1, and η1 pseudofermion operators. This is achieved by the use of the transformation laws of the ground state, Eq. (61), upon acting onto it with the i=0,1,...,∞ operators on the right-hand side of the equation, ck,σ=∑i=0∞ck,σ,i (and ck,σ†=∑i=0∞ck,σ,i†), whose first terms are given in Eq. (70).Within the PDT, the σ electron creation and annihilation operators are approximated by the corresponding pseudofermion representation leading-order terms, gˆ(k)cˆ⊙. This is justified by the perturbative nature of that representation. It ensures that the use of such leading-order terms provides the correct expressions of the corresponding σ one-electron spectral functions near their singular features.In the case of the ↑ one-electron removal processes (1), one finds the following leading-order operator expression,(74)ck,↑≈gˆι(k)cˆ⊙,cˆ⊙=f¯±2kF,c;Φc0=0;Φs10=ι/2,ι=±1,gˆι(k)=f¯q¯(±2kF),c†f¯q¯(ιkF↓),s1∑q=−2kF2kFΘ(kF↓−|k+q|)f¯q¯(q),cf¯q¯(k+q),s1†. The shift parameters Φβ0 are here those in Eq. (41) for β=c,s1 and q¯(q)=q+2πΦβ(q)/L. The capital-Θ distribution Θ(x) is given in this expression and in the following by Θ(x)=1 for x≥0 and Θ(x)=0 for x<0. A momentum ∓kF↓ results from the s1 pseudofermion annihilation at q¯(±kF↓). It exactly cancels the momentum ±kF↓ stemming from the overall s1 band momentum shift qj→qj±π/L. The latter results from Φs10=±1/2 in Eq. (74) for the transitions under consideration.Within a k extended zone scheme, the ω<0 spectrum generated by application of the ↑ one-electron removal leading-order generator, Eq. (74), onto the ground state reads −ω=−εc(q)+εs1(k+q). It has two branches whose spectra are of the form,(75)−ω(k)=−εc(q)+εs1(q′);k=−q+q′,k∈[−kF↑,(2kF+kF↑)];q∈[−2kF,2kF];q′∈[kF↓,kF↑],branchA,k∈[−(2kF+kF↑),kF↑];q∈[−2kF,2kF];q′∈[−kF↑,−kF↓],branchB.In the case of the ↑ one-electron LHB addition processes (2), the leading-order operator is given by,(76)ck,↑†≈gˆι(k)cˆ⊙,cˆ⊙=f¯±2kF,c†;Φc0=0;Φs10=ι/2,ι=±1,gˆι(k)=f¯q¯(±2kF),cf¯q¯(−ιkF↓),s1†(∑q=−π−2kF+∑q=2kFπ)Θ(kF↓−|k−q|)f¯q¯(q),c†f¯q¯(−k+q),s1. A momentum ∓kF↓ results from the s1 pseudofermion creation at q¯(∓kF↓). It exactly cancels again the momentum ±kF↓ stemming from an overall s1 band momentum shift qj→qj±π/L.The ω>0 spectrum generated by application of the ↑ one-electron LHB addition leading-order generator, Eq. (76), onto the ground state reads ω=εc(q)−εs1(k−q). Within a k extended zone scheme, it has again two branches. Their spectra are of the form,(77)ω(k)=εc(q)−εs1(q′);k=q−q′,k∈[kF↑,(π+kF↓)];q∈[2kF,π];q′∈[−kF↓,kF↓],branchA,k∈[−(π+kF↓),−kF↑];q∈[−π,−2kF];q′∈[−kF↓,kF↓],branchB.In the case of the ↑ one-electron UHB addition processes (3), the leading-order operator reads,(78)ck,↑†≈gˆι(k)cˆ⊙,cˆ⊙=f¯ι2kF,cf¯±kF↓,s1f¯−ι(π−2kF),η1†;Φc0=Φs10=0,ι=±1,gˆι(k)=f¯q¯(ι2kF),c†f¯q¯(±kF↓),s1†∑q=−2kF2kFΘ(kF↓−|k−ι(π−2kF)+q|)×f¯q¯(q),cf¯q¯(−k+ι(π−2kF)−q),s1. In this case one has Nη1(qj)=1 where qj=−ι(π−2kF) and Mη,−1/2=1 for the excited energy eigenstates in the general momentum expression, Eq. (21). The momentum πMη,−1/2=π then combines with (π−qj)Nη1(qj)=π−qj. This gives 2π−qj=−qj=ι(π−2kF).Within a k extended zone scheme, the ω>0 spectrum generated by application of the ↑ one-electron UHB addition leading-order generator, Eq. (78), onto the ground state reads ω=2μ−εc(q)−εs1(k−ι(π−2kF)+q). It has two branches associated with the two values of the index ι=±1. Their spectra are of the form,(79)ω(k)=2μ−εc(q)−εs1(q′);k=ι(π−2kF)−q−q′;q∈[−2kF,2kF];q′∈[−kF↓,kF↓],k=(π−2kF)−q−q′∈[(π−4kF−kF↓),(π+kF↑)],branchA,k=−(π−2kF)−q−q′∈[−(π+kF↑),−(π−4kF−kF↓)],branchB.In the case of the ↓ one-electron removal processes (4), the leading-order operator is given by,(80)ck,↓≈gˆι(k)cˆ⊙,cˆ⊙=f¯ι2kF,cf¯−ιkF↓,s1;Φc0=ι/2;Φs10=0,ι=±1,gˆι(k)=f¯q¯(ι2kF),c†f¯q¯(−ιkF↓),s1†∑q=−2kF2kFΘ(kF↓−|k−ι2kF+q|)×f¯q¯(q),cf¯q¯(−k+ι2kF−q),s1. The operator f¯ι2kF,c in cˆ⊙ leads to a momentum −ι2kF. On the one hand, it exactly cancels the momentum ι2kF stemming from the overall c band momentum shift associated with Φc0=ι/2. On the other hand, the operator f¯q¯(ι2kF),c† in gˆι(k) leads to a momentum contribution that restores such a momentum ι2kF.The ω<0 spectrum generated by application of the ↓ one-electron removal leading-order generator, Eq. (80), onto the ground state reads −ω=−εc(q)−εs1(k−ι2kF+q). It has two branches associated with the two values of the index ι=±1 whose spectra are of the form,(81)ω(k)=−εc(q)−εs1(q′);k=ι2kF−q−q′;q∈[−2kF,2kF];q′∈[−kF↓,kF↓],k=2kF−q−q′∈[−kF↓,(4kF+kF↑)],branchA,k=−2kF−q−q′∈[−(4kF+kF↑),kF↓],branchB.In the case of the ↓ one-electron LHB addition processes (5), the leading-order operator reads,(82)ck,↓†≈gˆι(k)cˆ⊙,cˆ⊙=f¯−ι2kF,c†f¯ιkF↓,s1†;Φc0=ι/2;Φs10=0,ι=±1,gˆι(k)=f¯q¯(−ι2kF),cf¯q¯(ιkF↓),s1×(∑q=−π−2kF+∑q=2kFπ)δ−ι,sgn{k−ι2kF−q}Θ(kF↑−|k−ι2kF−q|)×Θ(|k−ι2kF−q|−kF↓)f¯q¯(q),c†f¯q¯(k−ι2kF−q),s1†. Here and throughout this paper the sign distribution reads sgn{x}=1 for x>0, sgn{x}=−1 for x<0, and sgn{x}=0 for x=0. The operator f¯−ι2kF,c† in the expression of the operator cˆ⊙ leads to a momentum −ι2kF. It exactly cancels the momentum ι2kF stemming from the c band overall momentum shift. The operator f¯q¯(−ι2kF),c in gˆι(k) leads to a momentum contribution that restores such a momentum ι2kF.Within a k extended zone scheme, the ω>0 spectrum generated by application of the ↓ one-electron LHB addition leading-order generator, Eq. (82), onto the ground state reads ω=εc(q)+εs1(k−ι2kF−q). It has four branches. Their spectra are of the form,(83)ω(k)=εc(q)+εs1(q′);k=ι2kF+q+q′;sgn{q′}=−ιforq′≠0,k=2kF+q+q′∈[(4kF+kF↑),(π+2kF+kF↑)],branchA,q∈[2kF,π];q′∈[kF↓,kF↑],k=2kF+q+q′∈[−(π−2kF−kF↓),kF↑],branchB,q∈[−π,−2kF];q′∈[kF↓,kF↑],k=−2kF+q+q′∈[−(π+2kF+kF↑),−(4kF+kF↑)],branchA′,q∈[−π,−2kF];q′∈[−kF↑,−kF↓],k=−2kF+q+q′∈[−kF↑,(π−2kF−kF↓)],branchB′,q∈[2kF,π];q′∈[−kF↑,−kF↓].In the case of the UHB addition of one ↓ electron processes (6), the leading-order operator is given by,(84)cˆk,↓†≈gˆ(k)cˆ⊙,cˆ⊙=f¯ι2kF,cf¯−ι(π−2kF),η1†;Φc0=ι/2;Φs10=±1/2,ι=±1,gˆ(k)=f¯q¯(ι2kF),c†f¯q¯(±kF↓),s1∑q=−2kF2kFΘ(kF↓−|k−ιπ+q|)f¯q¯(q),cf¯q¯(k−ιπ+q),s1†. The operator f¯ι2kF,c in cˆ⊙ leads to a momentum −ι2kF. It exactly cancels the momentum ι2kF stemming from the c band overall momentum shift. The operator f¯q¯(ι2kF),c† in gˆι(k) leads to a momentum contribution that restores such a momentum ι2kF. The latter momentum is finally canceled by the momentum −ι2kF from the second term of the momentum ι(π−2kF). It stems from the operator f¯−ι(π−2kF),η1†. As in the case of the ↑ one-electron UHB addition processes (3), Eq. (78), one has Nη1(qj)=1 where qj=−ι(π−2kF) and Mη,−1/2=1 for the excited energy eigenstates in the general momentum expression, Eq. (21). The momentum πMη,−1/2=π then combines with (π−qj)Nη1(qj)=π−qj. This gives 2π−qj=−qj=ι(π−2kF). Moreover, the momentum ∓kF↓ resulting from the s1 pseudofermion annihilation at q¯(±kF↓) exactly cancels the momentum ±kF↓ stemming from the s1 band overall momentum shift.The ω>0 spectrum generated by application of the ↓ one-electron UHB addition leading-order generator, Eq. (84), onto the ground state reads ω=2μ−εc(q)+εs1(k−ιπ+q). Within a k extended zone scheme, it has two branches whose spectra are of the form,(85)ω(k)=2μ−εc(q)+εs1(q′);k=ιπ−q+q′=π−q+q′,k∈[(π−kF↑),(π+2kF+kF↑)];q∈[−2kF,2kF];q′∈[kF↓,kF↑],branchA,k∈[(π−2kF−kF↑),(π+kF↑)];q∈[−2kF,2kF];q′∈[−kF↑,−kF↓],branchB.On the one hand, the c and/or s1 pseudofermion momentum values ±2kF and ±kF↓, respectively, in the operators cˆ⊙ appearing in the above expressions belong to the initial ground state β=c,s1 band. On the other hand, the β pseudofermion momentum values q¯(q)=q+2πΦβ(q)/L in the operators gˆ(k) expressions belong to the excited energy eigenstates β=c,s1 bands.3.3The σ one-electron operators matrix elements between the ground state and the excited energy eigenstates and corresponding spectral functions in terms of β=c,s1 pseudofermion spectral functionsThe σ one-electron spectral functions, Eq. (4), can be written in the pseudofermion representation as follows,(86)Bγ(k,ω)=∑i′=0∞∑ν|〈ν|gˆi′(k)cˆ⊙|GS〉|2δ(ω−γ(Eν−EGS)),γω>0. For simplicity, we have here omitted from Bσ,γ(k,ω) the label σ. The excited-state indices ν− and ν+ have been denoted generally by ν.Following the properties reported in the previous section regarding the perturbative nature of the pseudofermion representation, one approximates the general spectral function, Eq. (86), by its pseudofermion leading-order term. It involves the operators given in Eqs. (74), (76), (78), (80), (82), and (84). This gives,(87)Bγ(k,ω)≈Bγ⊙(k,ω)=∑ν|〈ν|gˆ(k)cˆ⊙|GS〉|2δ(ω−γ(Eν−EGS)),γω>0.Both the generator onto the electron vacuum of the initial ground state in Eq. (61) and the operator cˆ⊙ in cˆ⊙|GS〉 are written in terms of c and s1 pseudofermion creation and/or annihilation operators, Eqs. (58) and (59). Their discrete canonical momentum values equal the corresponding momentum values qj, Eqs. (18) and (19), of that initial ground state. In the case of the σ one-electron UHB addition operators in Eqs. (78) and (84), the expression of the operator cˆ⊙ includes as well an η1 pseudofermion creation operator of canonical momentum ±(π−2kF).Both the operator gˆ(k) and the generators onto the electron vacuum of the excited energy eigenstates |ν〉 are written in terms of c and s1 pseudofermion operators. Their discrete canonical momentum values q¯j, Eq. (57), are those of these excited energy eigenstates. A useful property is that there is always an exact excited energy eigenstate |fG〉 of the final Nσ±1 ground state |GSf〉such that,(88)|fG〉=gˆ(k)|GSf〉.In the case of the c and s1 bands, the two types of discrete canonical momentum values that correspond to the initial ground state and excited energy eigenstates, respectively, account for the Anderson orthogonality catastrophes [34,67]. They occur in these bands under the transitions to the excited energy eigenstates |ν〉. On the one hand, such c and s1 bands Anderson orthogonality catastrophes are behind the exotic character of the quantum overlaps that control the one-electron spectral functions. On the other hand, the UHB one-electron operators matrix elements overlaps involving creation of one η1 pseudofermion do not involve Anderson orthogonality catastrophes. Such overlaps are straightforwardly computed. This follows in part from the initial ground state not being populated by η1 pseudofermions. A second reason is the symmetry behind the invariance of an η1 pseudoparticle with band limiting momentum values ±(π−2kF) under the pseudoparticle–pseudofermion unitary transformation. Hence the corresponding η1 pseudofermion canonical momentum has exactly the same values, ±(π−2kF). Indeed, such an η1 pseudofermion does not acquire phase shifts under the transitions to the excited states.The excitation gˆ(k)cˆ⊙|GS〉 in the matrix elements of the spectral function expression, Eq. (87), has finite overlap with the corresponding specific energy eigenstate, Eq. (88). This gives,(89)〈fG|gˆ(k)cˆ⊙|GS〉=〈GSfex|cˆ⊙|GS〉=〈0|∏β=c,s1f¯q¯Nβ⊙,β...f¯q¯2,βf¯q¯1,βf¯q′1,β†f¯q′2,β†...f¯q′Nβ⊙,β†|0〉=〈0|∏β=c,s1f¯q′Nβ⊙,β...f¯q′2,βf¯q′1,βf¯q¯1,β†f¯q¯2,β†...f¯q¯Nβ⊙,β†|0〉⁎. Here 1,...,Nβ⊙ labels the occupied β=c,s1 band discrete canonical momentum values and |GSfex〉 is a state with the same c and s1 pseudofermion occupancy as |GSf〉. Its β=c,s1 band discrete momentum values are though those of the excited energy eigenstate |fG〉=gˆ(k)|GSf〉.On the one hand, the β=c,s1 bands discrete canonical momentum values q′1, q′2, ..., q′Nβ⊙ in Eq. (89) equal the corresponding initial ground state discrete momentum values. On the other hand, q¯1, q¯2, ..., q¯Nβ⊙ are the discrete canonical momentum values of the excited energy eigenstate |fG〉, Eq. (88). Since these two sets of discrete momenta have different values, an Anderson orthogonality catastrophe occurs. It is such that the excited energy eigenstates of general form,(90)|fGC〉=∏β=c,s1gˆC(mβ,+1,mβ,−1)gˆ(k)|GSf〉=∏β=c,s1gˆC(mβ,+1,mβ,−1)|fG〉,β=c,s1,ι=±1, also have overlap with the excitation gˆ(k)cˆ⊙|GS〉. These states are originated by the application onto the state |fG〉, Eq. (88), of the β=c,s1 generators gˆC(mβ,+1,mβ,−1) of the low-energy and small-momentum processes (C).One then finds,(91)〈fG|∏β=c,s1gˆC†(mβ,+1,mβ,−1)gˆ(k)cˆ⊙|GS〉=〈GSfex|∏β=c,s1gˆC†(mβ,+1,mβ,−1)cˆ⊙|GS〉=〈0|∏β=c,s1f¯q¯Nβ⊙,β...f¯q¯2,βf¯q¯1,βgˆC†(mβ,+1,mβ,−1)f¯q′1,β†f¯q′2,β†...f¯q′Nβ⊙,β†|0〉=〈0|∏β=c,s1f¯q′Nβ⊙,β...f¯q′2,βf¯q′1,βgˆC†(mβ,+1,mβ,−1)f¯q¯1,β†f¯q¯2,β†...f¯q¯Nβ⊙,s1†|0〉⁎. The number of elementary β=c,s1 pseudofermion–pseudofermion-hole processes (C) of momentum ±2π/L in the vicinity of the β;ι=±1 Fermi points of |GSf〉 is denoted here and in the following by mβ,ι=1,2,3,.... Such processes conserve the number Nβ⊙ of β=c,s1 pseudofermions. Hence the matrix elements, Eq. (91), have the same form as that in Eq. (89). Their excited-state occupied discrete canonical momentum values q¯1, q¯2, ..., q¯Nβ⊙ in the vicinity of the β=c,s1 bands Fermi points are though slightly different from those in that equation.The function B⊙(k,ω), Eq. (87), is below expressed in terms of a sum of terms. Each of them is a convolution of c and s1 pseudofermion spectral functions. The expression of such pseudofermion spectral functions involves sums that run over the processes (C) numbers mβ,ι=1,2,3,.... It reads,(92)BQβ(k′,ω′)=L2π∑mβ,+1;mβ,−1Aβ(0,0)aβ(mβ,+1,mβ,−1)δ(ω′−2πLvβ∑ι=±1(mβ,ι+Δβι))×δ(k′−2πL∑ι=±1ι(mβ,ι+Δβι)),β=c,s1. The β=c,s1 lowest peak weights Aβ(0,0) appearing here are associated with a transition from the ground state to a PS excited energy eigenstate generated by processes (A) and (B). The corresponding β=c,s1 relative weights aβ=aβ(mβ,+1,mβ,−1) are generated by additional processes (C). Their β=c,s1 generators gˆC(mβ,+1,mβ,−1) are those in Eq. (90). The quantity Δβι in Eq. (92) refers to a functional given by 2Δβι=(ιδNβ,ιF+Φβ(ιqFβ))2. That functional involves the β=c,s1 pseudofermion number deviation δNβ,ιF at the ι=±1 Fermi points and corresponding phase shift 2πΦβ(ιqFβ), Eq. (49), in units of 2π. These phase shifts are acquired by the β=c,s1 pseudofermions with momenta ιqFβ=±qFβ under the above transition. This functional plays a key role in the PDT. It is found below to emerge naturally in the β=c,s1 pseudofermion spectral weights.In the case of σ one-electron UHB addition, the β=c,s1 weights Aβ(0,0)aβ(mβ,+1,mβ,−1) in Eq. (92) are reached after the quantum overlap stemming from creation of the η1 pseudofermion is trivially computed. For all the σ one-electron removal, LHB addition, and UHB addition processes that contribute to the spectral functions in the vicinity of their singular features the β=c,s1 weights Aβ=Aβ(0,0)aβ(mβ,+1,mβ,−1) have the general form,(93)Aβ=|〈0|f¯q′Nβ⊙,β...f¯q′2,βf¯q′1,βf¯q¯1,β†f¯q¯2,β†...f¯q¯Nβ⊙,β†|0〉|2,β=c,s1. Here Nβ⊙ stands for the number of β=c,s1 pseudofermions of the excited energy eigenstate generated by the processes (A) and (B). Such matrix element square can be expressed in terms of a Slater determinant of β=c,s1 pseudofermion operators, Eqs. (58) and (59), as follows,(94) The matrix elements 〈0|f¯q′Nβ⊙,β...f¯q′2,βf¯q′1,βf¯q¯1,β†f¯q¯2,β†...f¯q¯Nβ⊙,β†|0〉 in Eq. (93) of the β=c,s1 pseudofermion operators are associated with the two factors of the product ∏β=c,s1 in the matrix elements, Eq. (89).The function Bγ⊙(k,ω), Eq. (87), can be written as follows,(95)Bγ⊙(k,ω)=∑νΘ(Ω−δων)Θ(δων)Θ(|vν|−vβ¯)B˘ν⊙(δων,vν). The summation ∑ν runs here over excited energy eigenstates of the form given in Eq. (90), |fGC〉=∏β=c,s1gˆC(mβ,+1,mβ,−1)|fG〉, generated by processes (A), (B), and (C) with exactly the same values of the excitation momentum k and excitation energy ω.The excitation energy and momentum of the corresponding excited energy eigenstates |fG〉, Eq. (88), generated by processes (A) and (B) are given by,(96)δEν⊙=Eν⊙−EGS≥0;δPν⊙=Pν⊙−PGS, respectively. Such states have finite quantum overlap with the excitation gˆ(k)cˆ⊙|GS〉.The spectra of the excited states |fGC〉, Eq. (90), generated from those by processes (C), whose excitation momentum k and excitation energy ω are fixed under the summation ∑ν in Eq. (95), read,(97)δEν=δEν⊙+δων=γω≥0;δPν=δPν⊙+δkν=k,δων=γω−δEν⊙∈[0,Ω];δkν=k−δPν⊙∈[0,Ω/vν],δEν⊙=γω−δων∈[ω−Ω,Ω];δPν⊙=k−δkν∈[k−Ω/vν,k]. Here δων and δkν are their excitation energy and momentum, respectively, relative to those of the corresponding states |fG〉, Eq. (88). Their intervals given here are controlled by the processes (C) energy range Ω. Its value is self-consistently determined as that for which the corresponding velocity vν,(98)vν=δων/δkν;vβ¯=min{vc,vs1};vβ=max{vc,vs1}, remains nearly unchanged. The related β=c,s1 velocities vβ¯ and vβ in Eq. (95) are also defined here in terms of the β=c,s1 Fermi velocities vc and vs1, Eq. (48). For each fixed values of k and ω, the summation ∑ν in Eq. (95) runs over excited energy eigenstates |fG〉, Eq. (88), generated by processes (A) and (B). Their excitation energy and momentum vary under such a summation in corresponding intervals δEν⊙∈[ω−Ω,Ω] and δPν⊙∈[k−Ω/vν,k], respectively, as given in Eq. (97).The lack of c and s1 pseudofermion interaction terms in the PS finite-u energy spectrum, Eq. (68), enables the function B˘ν⊙(δων,vν) in Eq. (95) being expressed as the following convolution of c and s1 pseudofermion spectral functions, Eq. (92),(99)B˘ν⊙(δων,vν)=sgn(vν)2π∫0δωνdω′∫−sgn(vν)δων/vβ+sgn(vν)δων/vβdk′BQβ¯(δων/vν−k′,δων−ω′)BQβ(k′,ω′). Here β¯=c,s1 and β=s1,c, respectively, are chosen according to the criterion, Eq. (98), concerning the relative magnitudes of the two c and s1 Fermi velocities, Eq. (48).The spectral-function matrix-element overlap associated with the creation of one η1 pseudofermion of canonical momentum ±(π−2kF) is non-interacting like. Its creation leads to contributions 2μ and ∓(π−2kF) to the excitation energy and excitation momentum spectra δE⊙ and δP⊙, Eq. (96), respectively.The Slater determinant of β=c,s1 pseudofermion operators, Eq. (94), involves the pseudofermion anti-commutators. The apparent simplicity of such a Slater determinant masks the complexity of the main technical problem of the PDT. It lays in performing the state summations in the spectral functions Lehmann representation, Eq. (4). As discussed in the following, that problem stems from the involved form of such anti-commutators and thus of the corresponding Slater determinants of β=c,s1 pseudofermion operators.The unitarity of the pseudoparticle–pseudofermion transformation implies that the local β=c,s1 pseudofermion operators f¯j′,β† and f¯j′,β in Eq. (59) obey the following fermionic algebra,(100){f¯j,β†,f¯j′,β}=δj,j′,β=c,s1. It is similar to that in Eqs. (34) and (37) for the corresponding local β=c,s1 pseudoparticle operators.Consider two β=c,s1 pseudofermions of canonical momentum q¯j and q¯j′, respectively. Here q¯j and q¯j′=qj′ refer to the β=c,s1 bands of a PS excited energy eigenstate and the corresponding ground state, respectively. The β=c,s1 pseudofermion phase-shift functional 2πΦβ(qj), Eq. (49), is incorporated in the canonical momentum, Eq. (57). One then straightforwardly finds from the use of Eqs. (59) and (100) that the anti-commutator of f¯j′,β† and f¯j′,β reads,(101){f¯q¯j,β†,f¯q¯j′,β}=1Lβe−i(q¯j−q¯j′)/2ei2πΦβT(qj)/2sin(2πΦβT(qj)/2)sin([q¯j−q¯j′]/2);ΦβT(qj)=Φβ0+Φβ(qj),β=c,s1, whereas {f¯q¯j,β†,f¯q¯j′,β†}={f¯q¯j,β,f¯q¯j′,β}=0. Here 2πΦβT(qj) is the overall phase shift acquired by a β=c,s1 pseudofermion of momentum qj under the transition from the ground state to the PS excited energy eigenstate. The quantities 2πΦβ0, Eq. (41), and 2πΦβ(qj), Eq. (49), are the corresponding non-scattering and scattering part, respectively, of that phase shift.For 2πΦβT(qj)→0 the anti-commutator relation, Eq. (101), would be the usual one, {fq¯j,β†,fqj′,β}=δq¯j,q¯j′. That such an anti-commutator relation has not that simple form is the price to pay to render the β=c,s1 pseudofermions without interaction terms in their energy spectrum, Eq. (68). This is achieved by incorporating the β pseudofermion scattering phase shift 2πΦβ(qj), Eq. (49), in the β=c,s1 band canonical momentum, Eq. (57). As confirmed below, the unusual form, Eq. (101), of that anti-commutator relation is behind such a scattering phase shift controlling the spectral weight distributions of the σ one-electron spectral functions, Eq. (4).The unitarity of the pseudoparticle–pseudofermion transformation would preserve the pseudoparticle operator algebra provided that the sets of β=c,s1 band j=1,...,Lβ and j′=1,...,Lβ canonical momentum values {q¯j} and {q¯j′}, respectively, were the same. The exotic form of the anti-commutator, Eq. (101), follows from q¯j and q¯j′ corresponding to different sets of slightly shifted canonical momentum values. This is due to the shakeup effects introduced by the state-dependent β=c,s1 pseudofermion phase-shift functional 2πΦβT(qj), Eq. (101).The derivation of the β=c,s1 spectral weights Aβ=Aβ(0,0)aβ(mβ,+1,mβ,−1) in the β=c,s1 pseudofermion spectral functions, Eq. (92), proceeds much as for the corresponding u→∞ spinless fermion spectral function in Ref. [34]. Following the procedures of such a reference, after some algebra that involves the use of the pseudofermion operators anti-commutators, Eq. (101), in the pseudofermion operators Slater determinant representation of these weights, Eq. (94), one arrives to their expressions given in Eqs. (A.25)–(A.27) of Appendix A.Also the corresponding computation of the one-electron spectral-weight (k,ω)-plane distributions follows steps similar to those used in Ref. [34]. The PDT is indeed an extension to finite u of the method used in that reference for u→∞ [39]. Note though that the mapping to a Heisenberg chain used in that reference to deal with the spin part of the problem is valid only at m=0 and u≫1. In our case for which u is finite and m∈[0,ne], the alternative use of the s1 pseudofermion representation renders the treatment of the corresponding rotated spins 1/2 formally similar to that of the related c pseudofermion representation.For mβ,ι=1, the four β=c,s1 and ι=±1 relative weights given in Eq. (A.27) of Appendix A play a key role in the one-electron spectral-weight distributions. They are actually the four functionals 2Δβι that appear in the c and s1 pseudofermion spectral function expression, Eq. (92). They read,(102)2Δβι≡aβ,ι(1)=(δq¯Fβι(2π/L))2=(ιδNβ,ιF+Φβ(ιqFβ))2,β=c,s1,ι=±1. They are merely the square of the β=c,s1 and ι=±1 Fermi canonical momentum deviations δq¯Fβι=(ι2πδNβ,ιF+2πΦβ(ιqFβ))/L in units of 2π/L. Here δNβ,ιF=δNβ,ι0,F+ιΦβ0 so that δq¯Fβι=(ι2πδNβ,ι0,F+2πΦβT(ιqFβ))/L. Such functionals are thus controlled by the β=c,s1 and ι=±1 Fermi-points pseudofermion scattering phase shifts 2πΦβ(ιqFβ), Eq. (49). The bare deviation δNβ,ι0,F accounts for the number of β=c,s1 pseudofermions created or annihilated at the right (ι=+1) and left (ι=+1)