]>NUPHB13591S0550-3213(16)00010-910.1016/j.nuclphysb.2016.01.009The AuthorsQuantum Field Theory and Statistical SystemsFig. 1(a) The s1 band energy bandwidth Ws1=Ws1p+Ws1h, Eq. (45), and (b) the ratio Ws1p/Ws1 plotted as a function of 1/u for several spin density m values, and (c) Ws1p/Ws1 plotted as a function of m for u=1.Fig. 2The longitudinal spin spectrum ωl(k) for (a) m=0.25 and u=0.5, (b) m=0.75 and u=0.5, (c) m=0.25 and u=10.0, and (d) m=0.75 and u=10.0. The main effect of the on-site repulsion is on the spectrum energy bandwidth. At fixed spin density m its form remains nearly the same for the whole u>0 range.Fig. 3The transverse spin spectrum ωt(k) for (a) m=0.25 and u=0.5, (b) m=0.75 and u=0.5, (c) m=0.25 and u=10.0, and (d) m=0.75 and u=10.0. As in the case of the longitudinal spin spectrum plotted in Fig. 2, the main effect of the on-site repulsion is on the spectrum energy bandwidth.Fig. 4The exponent ξl(k), Eq. (123), that controls the singularities in the vicinity of the lower thresholds of the longitudinal spin spectrum ωl(k) plotted in Fig. 2 as a function of k∈]0,π[ for several values of u and spin densities (a) m=0.25, (b) m=0.50, (c) m=0.75, and (d) m=0.99.Fig. 5The exponent ξt(k), Eq. (123), that controls the singularities in the vicinity of the lower thresholds of the transverse spin spectrum ωt(k) plotted in Fig. 3 as a function of k∈]0,π[ for several values of u and spin densities (a) m=0.25, (b) m=0.50, (c) m=0.75, and (d) m=0.99.Pseudofermion dynamical theory for the spin dynamical correlation functions of the half-filled 1D Hubbard modelJ.M.P.Carmeloabccarmelo@fisica.uminho.ptT.ČadežcabdaDepartment 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 CenterBeijing100193ChinadJožef Stefan Institute, 1000 Ljubljana, SloveniaJožef Stefan InstituteLjubljana1000SloveniaEditor: Hubert SaleurAbstractA modified version of the metallic-phase pseudofermion dynamical theory (PDT) of the 1D Hubbard model is introduced for the spin dynamical correlation functions of the half-filled 1D Hubbard model Mott–Hubbard phase. The Mott–Hubbard insulator phase PDT is applied to the study of the model longitudinal and transverse spin dynamical structure factors at finite magnetic field h, focusing in particular on the singularities at excitation energies in the vicinity of the lower thresholds. The relation of our theoretical results to both condensed-matter and ultra-cold atom systems is discussed.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 features electrons that can hop between nearest-neighboring lattice sites due to the finite hopping integral t. When two electrons are on the same site, they have to pay the energy U due to their mutual repulsion. This introduces additional electronic correlations beyond those due to the Pauli principle. The model properties depend on the ratio u≡U/4t.The calculation of dynamical correlation functions is one of the main challenges in low-dimensional theories. Some systems with spectral gap can be dealt with by the form-factor approach to quantum correlation functions [1–7]. The advantage of this method is that it is in principle not constrained to very low energies. The form-factor approach can also be implemented for spin lattice systems such as the Heisenberg XXX and XXZ chains [8–13].The 1D Hubbard model is solvable by the Bethe ansatz (BA) [14–17]. This technique provides the exact spectrum of the energy eigenstates, yet it has been difficult to apply to the derivation of high-energy dynamical spectral and 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 [18–21].) For instance, form factors of the 1D Hubbard model electronic creation and annihilation operators is an open problem that has not been solved. Even the eventually easier problem of determining form factors of the spin operators in the Hubbard model at finite magnetic field remains as well unsolved. For the model metallic phase, the method used in Refs. [22,23] has been the first breakthrough to address the problem of the high-energy dynamical correlation functions in the u→∞ limit. Specifically, in these references the one-electron spectral functions of the model metallic phase have been derived for 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 u→∞ electron wave-function factorization [24–26].A related pseudofermion dynamical theory (PDT) relying on a representation of the model BA solution in terms of the pseudofermions generated by a unitary transformation from the corresponding pseudoparticles considered in Ref. [27] was introduced in Refs. [28,29]. It is an extension of the u→∞ method of Refs. [22,23] to the whole finite u>0 range of the metallic phase 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 dynamical correlation functions in terms of pseudofermion spectral functions. However, 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.The PDT of Refs. [28,29] has been the first breakthrough for the derivation of analytical expressions of the metallic phase of the 1D Hubbard model high-energy dynamical correlation functions for the whole finite u>0 range. Applications of the 1D Hubbard model metallic-phase PDT to the study of spectral features of actual condensed-matter systems are presented in Refs. [30–33].After the PDT for the metallic phase of the 1D Hubbard model was introduced, 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 [34]. In the case of the 1D Hubbard model such methods reach the same results as the PDT. For instance, the momentum, electronic density ne<1, 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 metallic phase calculated in Refs. [35,36] in the framework of a mobile impurity model using input from the BA solution is exactly the same as that obtained previously by use of the metallic-phase PDT [30–33].However, the latter PDT as reported in Refs. [28,29] does not apply to the study of the spin dynamical correlation functions of the 1D Hubbard model Mott–Hubbard insulator phase, which for the whole u>0 range corresponds to electronic density ne=1. (For that density it is usually called half-filled 1D Hubbard model.) Important examples of such spin dynamical correlation functions are the spin dynamical structure factors,(1)Saa(k,ω)=∑j=1Le−ikj∫−∞∞dte−iωt〈GS|Sˆja(t)Sˆja(0)|GS〉,=∑f|〈f|Sˆka|GS〉|2δ(ω−ωτ(k)). Here a=x,y,z, the spectra read ωτ(k)=Efτ−EGS, Efτ refers to the energies of the excited energy eigenstates that contribute to the τ=l longitudinal and τ=t transverse dynamical structure factors, EGS is the initial ground state energy, and Sˆka are for a=x,y,z the Fourier transforms of the usual local spin operators Sˆja, respectively.The studies of this paper do not address dynamical correlation functions associated with energy gapped excitations of the half-filled 1D Hubbard model such as the one-electron spectral functions and the charge dynamical structure factor [37]. Here we rather consider spin dynamical correlation functions associated with spin gapless excitations of the Mott–Hubbard insulator, such as the spin dynamical structure factors, Eq. (1). Previous studies of these factors focused mainly onto the model at magnetic fields h=0 when Szz(k,ω)=Sxx(k,ω)=Syy(k,ω) [38,39]. The studies of Ref. [38] have proposed an expression for the BA spin-band two-hole form factor and explicitly calculated the corresponding contribution to the spin dynamical structure factor at h=0. Its evolution as a function of u was studied in Ref. [39]. On the other hand, Ref. [40] presents results on one of the few studies about dynamical correlation functions of the 1D Hubbard model Mott–Hubbard insulator phase at finite magnetic field. However it addresses a problem different from that studied here: The low-energy limit of the dynamical density-density response function, which has no threshold singularities.Our present study and results refer to the Mott–Hubbard insulator phase of the 1D Hubbard model at a finite magnetic field h in the thermodynamic limit (TL) for u=U/4t>0 values. For h>0 one has that Szz(k,ω)≠Sxx(k,ω). Here we modify the PDT introduced in Refs. [28,29] for the model metallic phase to study the spin dynamical properties of the half-filled 1D Hubbard model at finite magnetic field h. The Mott–Hubbard insulator phase PDT introduced in this paper is then used to clarify one of the unresolved questions concerning the physics of that model by deriving the exact momentum, repulsive interaction u=U/4t, and spin-density m dependences of the exponents that control the singularities at the spectra lower thresholds in Szz(k,ω) and Sxx(k,ω).The remainder of the paper is organized as follows. In Section 2 the results on the model exact BA solution pseudoparticle and pseudofermion representations needed for our study are presented. The PDT introduced in Refs. [28,29] for the metallic phase of the 1D Hubbard model is suitably modified in Section 3 for the present case of the spin dynamical correlation functions of the half-filled 1D Hubbard model. Such a modified form of the PDT is used in Sec. 4 to derive the longitudinal and transverse dynamical structure factors in the vicinity of their spectra lower thresholds. Finally, Sec. 5 presents the concluding remarks including a brief discussion of the relation of our theoretical results to both condensed-matter [41] and ultra-cold atom [42] systems.2The pseudoparticle and pseudofermion representations of the 1D Hubbard model exact BA solutionIn this section we provide useful information on the pseudoparticle representation of the 1D Hubbard model BA solution needed for its representation in terms of the related pseudofermions, which is also briefly outlined. The Mott–Hubbard insulator PDT introduced in Section 3 relies on the latter representation.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 is given by,(2)Hˆ=tTˆ+UVˆD+2μSˆηz+2μBhSˆsz, where μB is the Bohr magneton,(3)Tˆ=−∑σ=↑,↓∑j=1L(cj,σ†cj+1,σ+cj+1,σ†cj,σ);VˆD=∑j=1Lρˆj,↑ρˆj,↓;ρˆj,σ=cj,σ†cj,σ−1/2, are the kinetic-energy operator in units of t, and the electron on-site repulsion operator in units of U, respectively, and(4)Sˆηz=−12(L−Nˆ);Sˆsz=−12(Nˆ↑−Nˆ↓), are the diagonal generators of the global η-spin and spin SU(2) symmetry algebras, respectively. Here we use in general units of lattice constant one, so that the number of lattice sites Na equals the lattice length L. Moreover, in Eqs. (2) and (3) the operator cj,σ† (and cj,σ) creates (and annihilates) a spin-projection σ electron at lattice site j=1,…,L. The electron number operators read Nˆ=∑σ=↑,↓Nˆσ and Nˆσ=∑j=1Lnˆj,σ=∑j=1Lcj,σ†cj,σ.2.1The model exact BA solution pseudoparticle representation for general densitiesAlthough the studies of this paper refer to the 1D Hubbard model, Eq. (2), at electronic density ne=N/L=1, we start by considering the general case of arbitrary electronic density and spin density m=n↑−n↓ where nσ=Nσ/L. The lowest weight states (LWSs) and highest weight states (HWSs) of the η-spin and spin SU(2) symmetry algebras have numbers Sα=−Sαz and Sα=Sαz, respectively, for α=η,s. Here Sη and Sηz are the states η-spin and η-spin projection and Ss and Ssz their spin and spin projection, respectively. In this paper the LWS formulation of 1D Hubbard model BA solution is used. The model in its full Hilbert space can be described either directly within the BA solution [24,43] or by application onto the LWSs of the η-spin and spin SU(2) symmetry algebras off-diagonal generators [44].The 1D Hubbard model BA equations introduced in Ref. [16] for the TL read in our pseudoparticle momentum distribution functional notation [27],(5)qj=kc(qj)+2L∑n=1∞∑j′=1LsnNsn(qj′)arctan(sinkc(qj)−Λsn(qj′)nu)+2L∑n=1∞∑j′=1LηnNηn(qj′)arctan(sinkc(qj)−Ληn(qj′)nu),j=1,…,L, and(6)qj=δα,η∑ι=±1arcsin(Λαn(qj)−iιu)qj=+2(−1)δα,ηL∑j′=1LNc(qj′)arctan(Λαn(qj)−sinkc(qj′)nu)qj=−1L∑n′=1∞∑j′=1Lαn′Nαn′(qj′)Θnn′(Λαn(qj)−Λαn′(qj′)u),j=1,…,Lαn,α=η,s,n=1,…,∞. The sets of j=1,…,L and j=1,…,Lαn quantum numbers qj in Eqs. (5) and (6), respectively, which are defined below, play the role of microscopic momentum values of different BA excitation branches. In these equations and throughout this paper δα,η is the usual Kronecker symbol and the rapidity function Λαn(qj) is the real part of the following complex rapidity [16],(7)Λαn,l(qj)=Λαn(qj)+i(n+1−2l)u,l=1,…,n, where the rapidity function Λαn(qj) is real, j=1,…,Lαn, α=η,s, and n=1,…,∞.Furthermore, in Eqs. (5) and (6) Θnn′(x) is the function,(8)Θnn′(x)=δn,n′{2arctan(x2n)+∑l=1n−14arctan(x2l)}+(1−δn,n′){2arctan(x|n−n′|)+2arctan(xn+n′)+∑l=1n+n′−|n−n′|2−14arctan(x|n−n′|+2l)}, where n,n′=1,…,∞ and,(9)qj=2πLIjβ,j=1,…,Lβ,β=c,ηn,sn,n=1,…,∞, are the β=c,αn band momentum values. The indices α=η,s and numbers n=1,…,∞ refer to different BA excitation branches that both in Ref. [27] and within the PDT of Refs. [28,29] are associated with β pseudoparticles as defined in the following. The c pseudoparticles and s1 pseudoparticles (often called s pseudoparticles) play a major role in the one-electron and two-electron physics of the metallic phase of the 1D Hubbard model [45,46]. The PDT relies on a representation of the BA solution in terms of β pseudofermions, which are related to the β pseudoparticles by a unitary transformation uniquely defined below in Section 3.For a given energy and momentum eigenstate, the j=1,…,Lβ quantum numbers Ijβ on the right-hand side of Eq. (9) are either integers or half-odd integers according to the following boundary conditions [16],(10)Ijβ=0,±1,±2,…forIβeven,=±1/2,±3/2,±5/2,…forIβodd. Here,(11)Iβ=δβ,cNpsSU(2)+δβ,αn(Lβ−1),α=η,s,n=1,…,∞.The β=c,αn band successive set of momentum values qj, Eq. (9), have only occupancies zero and one and the usual separation, qj+1−qj=2π/L. The number Lβ in Eq. (11) is that of the β-band discrete momentum values given by,(12)Lβ=Nβ+Nβh,β=c,αn,α=η,s,n=1,…,∞. A number Nβ≤Lβ of these momentum values are occupied. The β-band momentum distribution functions Nβ(qj) in Eqs. (5) and (6) thus read Nβ(qj)=1 and Nβ(qj)=0 for occupied and unoccupied discrete momentum values, respectively,Specifically, we call a β pseudoparticle each of the Nββ-band occupied momentum values. The remaining Nβh momentum values are unoccupied, their number reading [16],(13)Nch=L−Nc;Nαnh=2Sα+∑n′=n+1∞2(n′−n)Nαn′,α=η,s,n=1,…,∞. We call β-band holes such unoccupied momentum values.The total number of pseudoparticles Nps, the pseudoparticle number NpsSU(2) appearing in Eq. (11), and the related pseudoparticle number Nαps are given by,(14)Nps=Nc+NpsSU(2);NpsSU(2)=∑α=η,sNαps;Nαps=∑n=1∞Nαnα=η,s. The numbers NpsSU(2) and Nαps obey the following exact sum rules,(15)NpsSU(2)=∑α=η,s∑n=1∞Nαn=12(L−Ns1h−Nη1h);Nαps=∑n=1∞Nαn=12(Lα−Nα1h),α=η,s, where Nα1h is the number of αn-band holes for α=η,s and n=1 and,(16)Lη=Nch=L−Nc;Ls=Nc. Other exact sum rules obeyed by the set of numbers {Nαn} are,(17)MspSU(2)=∑α=η,s∑n=1∞nNαn=12(L−2Ss−2Sη);Mαsp=∑n=1∞nNαn=12(Lα−2Sα),α=s,η, where Mssp=∑n=1∞nNsn is the number of spin-singlet pairs considered below in Sec. 2.2 and Mηsp=∑n=1∞nNηn that of η-spin-singlet pairs, which do not exist for the quantum problem studied in this paper. (The corresponding rotated η-spins 1/2 operators is an issue briefly reported in Appendix A.)The β band discrete momentum values qj, Eq. (9), belong to well-defined domains, qj∈[qβ−,qβ+], where,(18)qc±=±πL(L−1)≈±πforNpsSU(2)odd;qc±=±πL(L−1±1)≈±πforNpsSU(2)even,qαn±=±πL(Lαn−1).The momentum and energy eigenvalues have the following general form for all 4L energy eigenstates,(19)P=∑j=1LqjNc(qj)+∑n=1∞∑j=1LsnqjNsn(qj)+∑n=1∞∑j=1Lηn(π−qj)Nηn(qj)+π2(2Sηz+Lη), and(20)E=∑j=1L(Nc(qj)Ec(qj)+U/4−μη)+∑α=η,s∑n=1∞∑j=1LαnNαn(qj)Eαn(qj)+∑α=η,s2μα(Sα+Sαz), respectively. Here,(21)2μs=2μB|h|;2μη=2|μ|,ne≠1;2μη=2μ0,ne=1, and(22)Ec(qj)=−2tcoskc(qj)−U/2+μη−μs,Eαn(qj)=2nμα+δα,η(4tRe{1−(Ληn(qj)−inu)2}−nU),α=η,s,n=1,…,∞.The energy scale 2μ0 in Eq. (21) is the ne=1 Mott–Hubbard gap [14,15,47]. It is behind the spectra of the one-electron and charge excitations of the half-filled 1D Hubbard model considered in the studies of this paper being gapped. For u>0 it is an even function of m that remains finite for all spin densities, m∈[−1,1]. For instance, in the limits m→0 [14,15] and m→±1 it reads,(23)2μ0=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 [47] are 2μ0≈(8/π)tUe−2π(tU) at m=0 and 2μ0≈U2/8t for m=±1 and the u≫1 behavior is 2μ0≈(U−4t) for the whole m∈[−1,1] range.2.2The Mott–Hubbard insulator phase spin sn pseudoparticle quantum liquidHere we report the information on the spin sector of the half-filled 1D Hubbard model BA solution sn pseudoparticle representation needed for our study. The μ=0;Sη=0;ne=1, and Ss=0;m=0 absolute ground state is both a LWS and HWS of the η-spin and spin SU(2) symmetry algebras. Thus as in the case of other lattices [48], it is both a η-spin and spin singlet. Its c band is full so that it is populated by Nc=Lc pseudoparticles. For u>0 both its one-electron and charge excitations are gapped. Their generation from the absolute ground state involves annihilation of c pseudoparticles that render Nc smaller than L, which for a Nc=L ground state are high-energy processes.In this paper we consider the half-filled 1D Hubbard model in the subspace spanned by the set of energy eigenstates whose c band is full and thus is populated by Nc=Lc pseudoparticles, as in the case of the μ=0;Sη=0;ne=1, and Ss=0;m=0 absolute ground state. We call it Nc=L subspace. Since all states belonging to it have Sη=0 and Sηz=0, for simplicity in the remaining of this paper we call S and Sz their spin Ss and spin projection Ssz, respectively.Moreover, we denote the energy eigenstates that span the Nc=L subspace by |u,lr,S,Sz〉 where, besides the u value, spin S, and spin projection Sz, lr represents the set of Nsn>0 occupied quantum numbers Ijsn, Eq. (10) for β=sn, of all n=1,…,∞ branches with finite occupancy that uniquely specify each state. Concerning our BA representation in terms of energy eigenstates |u,lr,S,−S〉 that are LWSs, which we call Bethe states, the non-LWSs are generated from them as,(24)|u,lr,S,Sz〉=1C(Sˆ+)ns|u,lr,S,−S〉;C=(ns!)∏j=1ns(2S+1−j),ns=1,…,2S, where Sˆ+ is the usual off-diagonal generator of the global spin SU(2) symmetry algebra.A particle subspace (PS) of the Nc=L subspace is spanned by a ne=1 ground state with a value of spin density in the range m∈[−1,1] and the set of excited energy eigenstates generated from it by a finite number of sn pseudoparticle processes that conserve the number Nc=L of c pseudoparticles. For such excited energy eigenstates the corresponding deviation densities δNsn/L and δS/L vanish as L→∞. For a PS there are though no restrictions on the value of the excitation energy and excitation momentum.For simplicity, we consider a PS spanned by a ne=1 ground state that is a spin LWS whose spin density value is thus in the range m∈]0,1] and its excited energy eigenstates. (This is without loss in generality, concerning PSs associated with ne=1 ground states that are spin HWSs with spin densities m∈[−1,0[.) Since the PSs considered in our study are part of the larger Nc=L subspace, they do not contain the gapped excited energy eigenstates associated with the charge and one-electron excitations whose energy spectrum involves the Mott–Hubbard gap 2μ0, Eq. (23).For the spin LWS ground states considered here the β=c,sn band limiting momentum values qβ±, Eq. (18), are given by qβ±≈±qβ where qβ reads [45],(25)qc=π;qs1=kF↑;qsn=(kF↑−kF↓)=πm;qηn=0. Indeed the c momentum band is full, the ηn momentum bands do not exist and thus qηn=0 for n=1,…,∞, the s1 momentum band is partially filled for 0<m<1, and for sn branches such that n>1 the sn momentum band is empty.Hence for the spin LWS ground states under consideration the sn-band pseudoparticle momentum distribution functions read,(26)Ns10(qj)=θ(qFs1−|qj|);Nsn0(qj)=0,n>1, where θ(x) is given by θ(x)=1 for x>0 and θ(x)=0 for x≤0. Within the TL considered here one has that,(27)qFs1±≈±qFs1,qFs1=kF↓=πm↓. The s1-band Fermi momentum values qFs1± including O(1/L) corrections are given in Eqs. (C.9)–(C.11) of Ref. [27]. Such corrections preserve the relation qFs1±=−qFs1∓.Under transitions from a ground state to a PS excited energy eigenstate there may occur shake-up effects. They are generated by transitions to PS excited energy eigenstates under which the boundary conditions of Eq. (10) are changed. This is behind overall β=c-band and β=sn-bands discrete momentum shifts, qj→qj+(2π/L)Φβ0, where Φβ0 is given by,(28)Φc0=0;δNspseven;Φc0=±12;δNspsodd;Φsn0=0;δNsneven;Φsn0=±12;δNsnodd,n=1,…,∞, where δNsps is the deviation in the number Nsps in Eq. (15), which in the present case reads Nsps=∑n=1∞Nsn=(Nc+Ns1h)/2. Note that although in the present case the above transitions preserve the number of c pseudoparticles Nc=L, they may produce c-band overall momentum shifts, qj→qj+(2π/L)Φc0.For the half-filled 1D Hubbard model in the PSs considered here the sn quantum liquid associated with that quantum problem involves the following subset of n=1,…,∞ BA equations,(29)qj=2L∑j′=1Larctan(Λsn(qj)−sinkc(qj′)nu)qj=−1L∑n′=1∞∑j′=1Lsn′Nsn′(qj′)Θnn′(Λsn(qj)−Λsn′(qj′)u),j=1,…,Lsn,n=1,…,∞,kc(qj)=qj−2L∑n=1∞∑j′=1LsnNsn(qj′)arctan(sinkc(qj)−Λsn(qj′)nu),j=1,…,L. Here Lsn is the number Lβ in Eq. (12) for β=sn,(30)Lsn=Nsn+Nsnh;Nsnh=(2S+∑n′=n+1∞2(n′−n)Nsn′),n=1,…,∞, and Λsn(qj) is the real part of the sn complex rapidity [16],(31)Λsn,l(qj)=Λsn(qj)+i(n+1−2l)u,l=1,…,n, where j=1,…,Lsn and n=1,…,∞.Since for the present quantum problem the c-band is full, the momentum rapidity function kc(qj) is fully determined only by the occupancies of the sn pseudoparticles, which are described in Eq. (29) by the corresponding sn-band momentum distribution functions Nsn′(qj′). It turns out that the Nc=Lc pseudoparticles explicit role in the quantum problem physics is through a mere contribution ±π to the excitation momentum when Φc0=±1/2 in Eq. (28).The PS energy functionals are derived from the use in the BA equations, Eq. (29), and general energy spectra, Eq. (20), of distribution functions of form Nsn(qj)=Nsn0(qj)+δNsn(qj), Eq. (32) for β=sn. The combined and consistent solution of such equations and spectra up to second order in the deviations,(32)δNsn(qj)=Nsn(qj)−Nsn0(qj),j=1,…,Lsn,n=1,…,∞, leads to a δE=E−EGS energy spectrum of the following general form,(33)δE=∑n=1∞∑j=1Lβεsn(qj)δNsn(qj)+2μB|h|(S+Sz)+1L∑n=1∞∑n′=1∞∑j=1Lsn∑j′=1Lsn′12fsnsn′(qj,qj′)δNsn(qj)δNsn′(qj′). The sn pseudoparticle energy dispersions εsn(qj) in this equation are given by,(34)εsn(qj)=εsn0(qj)+2μB|h|,j=1,…,Lsn,εsn0(qj)=−2tπ∫−ππdksinkarctan(sink−Λ0sn(qj)nu)+tπ2∫−ππdk∫−B/uB/udrsink2πΦ¯s1,sn(r,Λ0sn(qj)u)1+(sinku−r)2,j=1,…,Lsn. The rapidity functions Λ0sn(qj) appearing here are the solution of Eq. (29) for the sn-band ground-state distribution function distributions, Eq. (26), and the parameter B reads,(35)B≡Λ0s1(kF↓);limm→0B=∞;limm→1B=0.The rapidity dressed phase shift 2πΦ¯s1sn(r,r′) in Eq. (34) is uniquely defined by the integral equation,(36)2πΦ¯s1,sn(r,r′)=δn,12arctan(r−r′2)+(1−δn,1)(2arctan(r−r′n−1)+2arctan(r−r′n+1))+∫−B/uB/udr″G(r,r″)2πΦ¯s1,s1(r″,r′), whose kernel is given by,(37)G(r,r′)=−12π(11+((r−r′)/2)2).The f function in the second-order terms of the energy functional, Eq. (33), reads [45],(38)fsnsn′(qj,qj′)=vsn(qj)2πΦsnsn′(qj,qj′)+vsn′(qj′)2πΦsn′sn(qj′,qj)+12π∑ι=±1vs12πΦs1sn(ιqFs1,qj)2πΦs1sn′(ιqFs1,qj′).Within the TL one may use a continuum q representation for the sn-band discrete momentum values qj such that qj+1−qj=2π/L. Then the deviation values δNsn(qj)=−1 and δNsn(qj)=+1, Eq. (32), become δNsn(q)=−(2π/L)δ(q−qj) and δNsn(q)=+(2π/L)δ(q−qj), respectively. (Here δ(x) denotes the usual Dirac delta-function distribution.) According to Eqs. (9) and (10), under a transition to an excited energy eigenstate the sn band discrete momentum values qj=(2π/L)Ijsn may undergo a collective shift, (2π/L)Φsn0=±π/L. For q at the s1 and ι=±1 Fermi points, ιqFs1=ιkF↓, such an effect is captured within the continuum representation by additional deviations, ±(π/L)δ(q−ιkF↓). For transitions to an excited energy eigenstate for which δLsn≠0, the removal or addition of BA sn band discrete momentum values occurs in the vicinity of the band edges qsn−=−qsn+, Eq. (18). Those are zero-momentum and zero-energy processes.Within the continuum q representation, the β band group velocities appearing in Eq. (38) are given by,(39)vsn(q)=∂εsn(q)∂q,n=1,…,∞;vs1≡vs1(qFs1). Since the ground states are not populated by sn pseudoparticles of n>1 branches, only the s1 pseudoparticles have Fermi points associated with the s1-band Fermi velocity vs1=vs1(qFs1).The momentum dressed phase shift 2πΦsnsn(qj,qj′) in the f function expression, Eq. (38), are of the form,(40)2πΦsnsn′(qj,qj′)=2πΦ¯snsn′(r,r′);r=Λ0sn(qj)/u;r′=Λ0sn′(qj′)/u, where the general rapidity dressed phase shift 2πΦ¯snsn′(r,r′) is for n>1 the solution of the integral equation,(41)2πΦ¯sn,sn′(r,r′)=Θnn′(r−r′)−12π∫−B/uB/udr″2πΦ¯sn,sn′(r″,r′)Θn1[1](r−r″). Here Θnn′(x) is the function given in Eq. (8) and Θnn′[1](x) is its derivative,(42)Θnn′[1](x)=dΘnn′(x)dx=δn,n′{1n[1+(x2n)2]+∑l=1n−12l[1+(x2l)2]}+(1−δn,n′){2|n−n′|[1+(x|n−n′|)2]+2(n+n′)[1+(xn+n′)2]+∑l=1n+n′−|n−n′|2−14(|n−n′|+2l)[1+(x|n−n′|+2l)2]}.The s1 pseudoparticle dispersion εs10(qj), Eq. (34) for sn=s1, defines the spin density curve as follows,(43)h(m)=−εs10(kF↓)2μB|m=1−2kF↓/π∈[0,hc]. Here hc is the critical field for fully polarized ferromagnetism achieved when m→1 and kF↓→0. The corresponding energy scale 2μBhc reads [45],(44)2μBhc=−εs10(0)|m=1=2tπ∫−ππdksinkarctan(sinku)=(4t)2+U2−4t.That as m→1 the spin energy scale 2μB|h|=2μBhc, Eq. (44), and the charge Mott–Hubbard gap 2μ0, Eq. (23), have exactly the same value is because at m=1 the model on-site repulsion has no effects since all electrons have the same spin projection. The equality of these spin and charge energy scales is then associated with a recombination of the charge and spin degrees of freedom, which is only reached at m=1. As reported below in Section 4, this charge-spin recombination leads to a qualitatively different form for the spin dynamical correlation functions for h<hc and at h=hc, respectively.As confirmed below in Section 4, important energy scales that control the u dependence of the (k,ω) plane spectrum on which these functions spectral weight is distributed are the ground-state s1 band energy bandwidth,(45)Ws1=εs(kF↑)−εs(0)=Ws1p+Ws1h, the energy bandwidth Ws1p=εs(kF↓)−εs(0)=Ws1−2μB|h| of the occupied ground-state Fermi sea, and the energy bandwidth Ws1h=εs(kF↑)−εs(kF↓)=2μB|h| of the corresponding unoccupied s1 band. In Fig. 1(a) the s1 band energy bandwidth Ws1, Eq. (45), is plotted as a function of 1/u for several spin density values. It is a decreasing function of the ratio u=U/4t. The occupied Fermi sea energy bandwidths Ws1p has the same type of u dependence behavior, as confirmed from analysis of Fig. 1(b) where the ratio Ws1p/Ws1 is also plotted as a function of 1/u for the same m values as in Fig. 1(a). That ratio is plotted as a function of the spin density m for u=1 in Fig. 1(c).Our present study does not refer to the PS of the ne=1 and m=0 absolute ground state whose physics is qualitatively different from that of the present h>0 spin quantum liquid. For such an absolute ground state the s1 band is full and the holes that emerge in that band under the transitions to the excited energy eigenstates are usually identified with spin-1/2 spinons [38,39,49]. Indeed at h=0 the sn pseudoparticles of n>1 branches created onto the absolute ground state have a sn dispersion with both vanishing momentum and energy bandwidth such that εsn(qj)=0 in Eq. (34). Hence the model h=0 spin SU(2) symmetry allows that the effects of their creation may be incorporated onto phase shifts of a spin-1/2 spinon only representation [49].On the other hand, for h>0 the energy dispersions εsn(qj), Eq. (34), of sn pseudoparticles of n>1 branches have both a finite momentum and energy bandwidth. Hence they become elementary objects that exist in their own right. This renders the h=0 spin-1/2 spinon only representation unsuitable for the h>0 quantum problem considered here. Elsewhere it will be shown that the s1 band holes of the present ne=1 and m>0 ground states and their excited energy eigenstates have scattering properties different from those of spin-1/2 objects, such as the h=0 spin-1/2 spinons.The functional energy spectrum, Eq. (33), rather describes the Mott–Hubbard insulator phase of the half-filled 1D Hubbard model at finite field h as a spin quantum liquid of sn pseudoparticles that have residual interactions associated with the f functions, Eq. (38). The s1 pseudoparticles play an important role in that quantum liquid, as the ground states are not populated by sn pseudoparticles of n>1 branches. Consistently, the specific rapidity dressed phase shift 2πΦ¯s1s1(r,r′) plays a major role in the PDT expressions. Indeed only the s1 pseudoparticles have Fermi points, which are associated with the momentum values ±qFs1=±kF↓. This applies as well to the corresponding s1 pseudofermions generated below from the s1 pseudoparticles by slightly shifting their discrete momentum values. The exponents that appear in the dynamical correlation functions expressions involve the following l=0,1 parameters that are fully controlled by the phase shifts acquired by s1 pseudofermions at the Fermi points under creation or annihilation of other s1 pseudofermions at or very near such points,(46)ξs1,s1l=ξs1,s1l(B/u)=1+Φs1,s1(kF↓,kF↓)+(−1)lΦs1,s1(kF↓,−kF↓),l=0,1. (Due to the Pauli-like occupancies of the s1 band discrete momentum values being zero or one, the two Fermi momentum values in Φs1,s1(kF↓,kF↓) must differ by 2π/L with 2π/L→0 in the present TL.)The l=0,1 parameters in Eq. (46) are such that,(47)ξs1,s10=1ξs1,s11, where for u>0 the parameter ξs1,s11 is an increasing function of m. For general m values it is u dependent whereas in the m→0 and m→1 limits it reaches universal u-independent values. Specifically, for u>0 it reads ξs1,s11=1/2 in the m→0 limit and reaches its maximum value, ξs1,s11=1, in the m→1 limit. In the m→0 limit this follows from the rapidity dressed phase shift 2πΦ¯s1,s1(r,r′) being given by,(48)2πΦ¯s1,s1(r,r′)=ilnΓ(12+i(r−r′)4)Γ(1−i(r−r′)4)Γ(12−i(r−r′)4)Γ(1+i(r−r′)4);r≠±∞=ιπ2;r=ιB=ι∞,ι=±1,r′≠r=ιπ2(3−22);r=r′=ιB=ι∞,ι=±1, where Γ(x) is the usual gamma function. Furthermore, in the m→1 limit its expression becomes,(49)2πΦ¯s1,s1(r,r′)=2arctan(r−r′2).Within the present TL the problem concerning a sn pseudoparticle internal degrees of freedom in terms of electronic spins and that associated with its translational degrees of freedom center of mass motion separate. For u>0 the sn pseudoparticle internal degrees of freedom involve the spins 1/2 of the rotated electrons that occupy singly occupied sites, which we call here rotated spins 1/2. These rotated electrons are generated from the electrons by a unitary transformation such that rotated-electron singly occupancy is a good quantum number for u>0 [27,50]. There is an infinite number of such transformations, the specific rotated electrons associated with the BA quantum numbers corresponding to a unitary transformation uniquely defined by the BA.Out of the many choices of u→∞ degenerate energy eigenstates belonging to the Nc=L subspace, that transformation involves those obtained from the u>0 Bethe states and corresponding non-LWSs, Eq. (24), as |∞,lr,S,Sz〉=limu→∞|u,lr,S,Sz〉. We call V tower the set of Nc=L subspace energy eigenstates |u,lr,S,Sz〉 with exactly the same independent-u quantum numbers lr,S,Sz and different u values in the range u>0. The amplitudes,(50)fu,lr,S(x1σ1,…,xLσL)=〈x1σ1,…,xLσL|u,lr,S,−S〉, of Bethe states |u,lr,S,−S〉 belonging to the same V tower smoothly and continuously behave as a function of u. Such amplitudes are uniquely defined in Eqs. (2.5)–(2.10) of Ref. [24] in terms of BA solution quantities. In the amplitudes, Eq. (50), |x1σ1,…,xLσL〉 denotes a local state in which the L electrons with spin projection σ1,…,σL are located at sites of spatial coordinates x,…,xL, respectively. For a LWS their numbers are N↑=L/2+S and N↓=L/2−S.A useful property is that the amplitude 〈x1σ1′,…,xLσL′|u,lr,S,Sz〉 of a non-LWS |u,lr,S,Sz〉 for which a number ns=1,…,2S of spins out of the L spins have been flipped relative to those of the corresponding LWS |u,lr,S,−S〉 obeys the equality 〈x1σ1′,…,xLσL′|u,lr,S,Sz〉=〈x1σ1,…,xLσL|u,lr,S,−S〉. Hence 〈x1σ1′,…,xLσL′|u,lr,S,Sz〉 does not depend on the flipped spins and is given by 〈x1σ1′,…,xLσL′|u,lr,S,Sz〉=fu,lr,S(x1σ1,…,xLσL).For the u→∞ energy eigenstates |∞,lr,S,Sz〉 electron single occupancy is a good quantum number. It is not for the finite-u energy eigenstates |u,lr,S,Sz〉 belonging to the same V tower because upon decreasing u there emerges a finite electron doubly occupancy expectation value, which vanishes for u→∞ [51]. Since for any u>0 value the set of energy eigenstates |u,lr,S,Sz〉 that belong to the same V tower are generated by exactly the same occupancy configurations of the u-independent quantum numbers lr, S, and Sz, the Nc=L subspace is the same for the whole u>0 range. Hence for any u>0 there is a uniquely defined unitary operator Vˆ=Vˆ(u) such that |u,lr,S,Sz〉=Vˆ†|∞,lr,S,Sz〉. This operator Vˆ is the electron – rotated-electron unitary operator such that,(51)c˜j,σ†=Vˆ†cj,σ†Vˆ;c˜j,σ=Vˆ†cj,σVˆ;n˜j,σ=c˜j,σ†c˜j,σ, are the operators that create and annihilate, respectively, the rotated electrons as defined here. Moreover, |∞,lr,S,Sz〉=Gˆlr,S,Sz†|0〉 where |0〉 is the electron vacuum and Gˆlr,S,Sz† a uniquely defined operator. It then follows that |u,lr,S,Sz〉=G˜lr,S,Sz†|0〉 where the generator G˜lr,S,Sz†=Vˆ†Gˆlr,S,Sz†Vˆ has the same expression in terms of the rotated-electron creation and annihilation operators as Gˆlr,S,Sz† in terms of electron creation and annihilation operators, respectively.For the 1D Hubbard model in the Nc=L subspace such an electron – rotated-electron unitary operator Vˆ is uniquely defined by the set of the following matrix elements between the energy eigenstates that span such a subspace,(52)〈u,lr,S,Sz|Vˆ|u,lr′,S′,S′z〉=δS,S′δSz,S′z∑x=1L...∑xL=1Lfu,lr,S⁎(x1σ1,…,xLσL)f∞,lr′,S(x1σ1,…,xLσL). Here fu,lr,S(x1σ1,…,xLσL) and f∞,lr′,S(x1σ1,…,xLσL) are the amplitudes defined by Eqs. (2.5)–(2.10) of Ref. [24] for u>0 and Eq. (2.23) of Ref. [25] for u→∞, respectively.The electron – rotated-electron unitary operator Vˆ commutes with the three generators of the global SU(2) symmetry algebra and the charge density operator. The rotated electrons have the same spins 1/2 and charge as the electrons, the application of Vˆ only changing their spatial distribution. For u>0 and m<1 there is at all energy scales a non-perturbative charge-spin separation such that the L rotated-electron charges are associated with the L c pseudoparticles whose momentum band is full and the L rotated-electron spins 1/2 are the rotated spins 1/2 whose independent occupancy configurations determine the exotic properties of the 1D Mott Hubbard insulator.The rotated spins 1/2 occupancy configurations associated with the sn pseudoparticle internal degrees of freedom are implicitly accounted for by the BA solution. The microscopic details of such spin-singlet configurations are not needed for the studies of this paper. Here we only provide some general information on that interesting issue. The imaginary part of the l=1,…,n rapidities Λsn,l(qj)=Λsn(qj)+i(n+1−2l)u, Eq. (31), with the same real part Λsn(qj) that emerge for n>1 is for a qj value occupied by a sn pseudoparticle associated with a set l=1,…,n of singlet pairs of rotated spins 1/2 and the binding of these pairs within the sn pseudoparticle. Each of such l=1,…,n rapidities refers to one of the l=1,…,n singlet pairs bound within the sn pseudoparticle. For n=1 the rapidity imaginary part vanishes. Indeed, the s1 pseudoparticle internal degrees of freedom refer to a single singlet pair of rotated spins 1/2.For the quantum problem considered here one has that 2Sη=0 and Nηn=0 for n=1,…,∞ in Eq. (17). Hence the first sum rule given in that equation can be written as L=2S+∑n=1∞2nNsn. Each of the original lattice j=1,…,L sites is occupied by one rotated spin 1/2. A number ∑n=1∞2nNsn of such rotated spins 1/2 participate in Mssp=∑n=1∞nNsn spin-singlet pairs, Eq. (17) for α=s. Those are contained in the set of the state sn pseudoparticles whose number is Nsps=∑n=1∞Nsn. The latter number obeys the second sum rule in Eq. (15) for α=s, which for the Nc=L subspace reads Nsps=∑n=1∞Nsn=(L−Ns1h)/2. Similarly, the number Mssp of spin-singlet pairs obeys the second sum rule in Eq. (17) for α=s.The remaining 2S rotated spins out of the system L=2S+∑n=1∞2nNsn rotated spins 1/2 remain unpaired. They are those that participate in the 2S+1 spin multiplet configurations, which are generated by an application of a number ns=1,…,2S of times of the off-diagonal spin operator Sˆ+ onto a LWS, as given in Eq. (24). Application of such an operator leaves the spin-singlet configurations of the ∑n=1∞nNsn spin-singlet pairs contained in sn pseudoparticles unchanged. It merely flips the unpaired rotated spins 1/2. For u>0 the number M±1/2un of unpaired rotated spins of projection ±1/2 are good quantum numbers, which read,(53)M±1/2un=(S∓Sz);Mun=(M−1/2un+M+1/2un)=2S. For the spin LWSs one has that M+1/2un=Mun=2S and M−1/2un=0.The sn pseudoparticle translational degrees of freedom associated with its center of mass motion are important for the PDT introduced below in Section 3, which implicitly accounts for its internal degrees of freedom through the BA quantities that contribute to the dynamical properties. The sn band momentum qj in the argument of the rapidities real part Λsn(qj) is associated with such sn pseudoparticle translational degrees of freedom. That ground states are not populated by sn pseudoparticles containing n>1 singlet pairs plays an important role in the PDT. Within it the dynamical correlation functions spectral weights are described by exotic quantum overlaps expressed in terms of s1 pseudofermion operators generated from s1 pseudoparticle operators by suitable shifts of the momentum values qj. On the other hand, the quantum overlaps stemming from sn pseudoparticles of n>1 branches are trivial to compute. The same applies to the overlaps associated with the unpaired rotated spins flipping processes.For u>0 the s1 pseudoparticles live in the TL on a squeezed s1 effective lattice [23,26,52]. Its number of sites equals that of s1 band discrete momentum values, Eq. (30) for sn=s1,(54)Ls1=Ns1+Ns1h;Ns1h=2S+∑n=2∞2(n−1)Nsn. The Ns1h unoccupied s1 effective lattice sites refer to the 2S=Mun sites occupied in the original lattice by the unpaired rotated spins 1/2 and the sets of 2(n−1) sites of that lattice out of the 2n sites occupied by each sn pseudoparticle of n>1 branches.The line shape near the longitudinal and transverse dynamical structure factors, Eq. (1), spectra lower thresholds is below in Section 4 found to be determined by transitions to excited energy eigenstates that are not populated by sn pseudoparticles of n>1 branches. The unpaired rotated spins 1/2 of these states are used within the s1 pseudoparticle motion as unoccupied sites with which they interchange position. Within the BA solution such processes are accounted for by the s1 band occupancy configurations. Indeed the Mun=2S unpaired rotated spins 1/2 have zero momentum. The energy of a unpaired rotated spin of spin projection ±1/2 relative to the ground state energy level is straightforwardly calculated by combining the LWSs momentum eigenvalues with such states and their non-LWSs transformation laws under the off-diagonal spin SU(2) symmetry algebra generators, respectively, and is given by,(55)ε±1/2=2μs=2μB|h|;εs,∓1/2=0,sgn{m}=∓, for h≠0 and vanishes at h=0. The energy scale 2μB|h| is that required for a spin flip. Spin flips generated by the off-diagonal spin operator Sˆ+ in Eq. (24) are the only processes whose energy is associated with the unpaired rotated spins 1/2. They generate the transitions between the 2S+1 multiplet configurations. An interesting related reference energy scale is that of a S=1;Sz=0 multiplet configuration involving two unpaired rotated spins of opposite spin projection. It is merely additive in the energies, Eq. (55), and reads,(56)ε1/2+ε−1/2=2μs=2μB|h|.In general in this paper we use units of lattice spacing a one, so that the lattice length L equals the number of lattice sites Na. In the TL the s1 effective lattice has j=1,…,Ls1 sites and length L. Hence it is a 1D lattice with spacing,(57)as1=NaLs1a, such that L=Ls1as1.For the present Nc=L subspace the general sum rules in Eqs. (15) and (17) lead to,(58)Nsps=∑n=1∞Nsn=12(L−Ns1h);∑n=1∞2nNsn=L−2S, where Nsps is the total number of sn pseudoparticles of all n=1,…,∞ branches. Since Ns1h=L−2Nsps, the s1 effective lattice number of lattice sites Ls1, Eq. (54), remains unchanged provided that values of Ns1 and Nsps also do.The s1 pseudoparticle translational degrees of freedom center of mass motion may be 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. That each s1 band momentum value qj where j=1,…,Ls1 has only Pauli-like occupancies zero and one is consistent with the local s1 pseudofermion operators obeying a Fermi algebra,(59){fj,s1†,fj′,s1}=δj,j′.Furthermore, one may introduce s1 pseudoparticle operators labeled by the s1 band momentum values,(60)fqj,s1†=1L∑j′=1Ls1eiqjj′fj′,s1†;fqj,s1=1L∑j′=1Ls1e−iqjj′fj′,s1,j=1,…,Ls1.That the pseudoparticle operators provide a faithful representation of the quantum problem and its BA solution and obey a fermionic algebra can be confirmed in terms of their statistical interactions [53]. This is a problem that we address here very briefly. The local operator fj,s1† may be written as fj,s1†=eiϕj,s1gj,s1† where ϕj,s1=∑j′≠jfj′,s1† and 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, Eq. (54). The s1 effective lattice has been constructed inherently to on it that algebra being of hard-core type. Therefore, through a Jordan–Wigner transformation, fj,s1†=eiϕj,s1gj,s1† [54], the operators fj,s1† obey indeed a fermionic algebra, Eq. (59). Besides acting within subspaces spanned by energy eigenstates with fixed Ls1 values, the s1 pseudofermion operators labeled by momentum qj, Eq. (60), also appear in the expressions of the shake-up effects generators that transform such subspaces quantum number values into each other.2.3The pseudofermion representationFrom straightforward yet lengthly manipulations of the BA equations, Eq. (29), one finds that for PS excited energy eigenstates of ne=1 and m>0 ground states the sn-band rapidity functionals Λsn(qj) can be written in terms of the corresponding ground-state rapidity function Λ0sn(qj) as follows,(61)Λsn(qj)=Λ0sn(q¯(qj)),j=1,…,Ls1. Here q¯j=q¯(qj) where j=1,…,Lsn are the following discrete canonical momentum values,(62)q¯j=q¯(qj)=qj+2πLΦsn(qj)=2πL(Ijsn+Φsn(qj)),j=1,…,Lsn. The functional Φsn(qj) in this equation reads,(63)Φsn(qj)=∑n′=1∞∑j′=1Lsn′Φsnsn′(qj,qj′)δNsn′(qj′), where the deviation δNsn′(qj′) and the dressed phase shift 2πΦsnsn′(qj,qj′) are given in Eqs. (32) and (40), respectively. The discrete canonical momentum values q¯j=q¯(qj) have spacing q¯j+1−q¯j=2π/L+h.o. Here h.o. stands for terms of second order in 1/L.We call a sn pseudofermion each of the Nsn occupied sn-band discrete canonical momentum values q¯j [28–30]. We call s1 pseudofermion holes the remaining Nsnh unoccupied sn-band discrete canonical momentum values q¯j of a PS energy eigenstate. There is a pseudofermion representation for each ne=1 and m>0 ground state and its PS.One generates from the s1 pseudoparticle creation and annihilation operators fqj,s1† and fqj,s1, respectively, Eq. (60), the following corresponding s1 pseudofermion operators,(64)f¯q¯j,s1†=fqj+(2π/L)Φ(qj),s1†=(SˆΦ)†fqj,s1†SˆΦ;f¯q¯j,s1=(f¯q¯j,s1†)†. Here SˆΦ denotes the s1 pseudoparticle – s1 pseudofermion unitary operator,(65)SˆΦ=e∑j=1Ls1fqj+(2π/L)Φ(qj),s1†fqj,s1;(SˆΦ)†=e∑j=1Ls1fqj−(2π/L)Φ(qj),s1†fqj,s1. In this equation and in the following we use the notation,(66)Φ(qj)≡Φs1(qj)=∑n′=1∞∑j′=1Lsn′Φ1nsn′(qj,qj′)δNsn′(qj′), for the functional, Eq. (63), of the sn=s1 branch.The sn pseudofermions have the same internal structure as the corresponding sn pseudoparticles. Indeed they differ in their discrete momentum values, which rather refer to the translational degrees of freedom. For the initial ground state one has that q¯j=qj. Hence for that state the sn pseudofermions and sn pseudoparticles are identical objects. Several pseudofermion quantities are expressed in terms of such an initial-state unshifted sn pseudofermion canonical momentum values qj. An example is the function 2πΦsnsn′(qj,qj′), Eq. (40), (and −2πΦsns1(qj,qj′)) in the functional expression, Eq. (63). It is the phase shift acquired by a sn pseudofermion or a sn pseudofermion hole of canonical momentum occupied or unoccupied in the final state, respectively, upon scattering off a sn′ pseudofermion (and s1 pseudofermion hole) created at the initial ground-state momentum qj′ under a transition from the latter state to a PS excited energy eigenstate. (For n>1 the phase shift 2πΦsnsn′(qj,qj′) or −2πΦsns1(qj,qj′) is acquired by a sn pseudofermion of canonical momentum q¯j=qj+(2π/L)Φsn(qj) associated with the initial-state canonical momentum qj, which has also been created under such a transition.) As confirmed below, the functional Φ(qj), Eq. (66), controls the spectral weights of the spin dynamical correlation functions.Within the s1 pseudofermion motion in the s1 effective lattice the Mun=2S unpaired rotated spins 1/2 play the role of unoccupied sites. Such unpaired rotated spins 1/2 are zero-momentum objects that under the transitions from a ground state to its PS excited energy eigenstates do not acquire phase shifts. Furthermore the unpaired rotated spins flips that occur under some of such transitions are scattering-less processes that do not lead to pseudofermion phase shifts. Hence the Mun=2S unpaired rotated spins 1/2 are neither scatterers nor scattering centers.Upon expressing the PS energy functional, Eq. (33), in terms of the discrete canonical momentum values q¯j=q¯(qj), Eq. (62), it reads up to O(1/L) order,(67)δE=∑n=1∞∑j=1Lsnεsn(q¯j)δNsn(q¯j)+2μB|h|(S+Sz),=∑n=1∞∑j=1Lsnεsn(q¯j)δNsn(q¯j)+ε−1/2M−1/2un. Here the sn pseudofermion energy dispersions εsn(q¯j) have exactly the same form as those given in Eq. (34) with the momentum qj replaced by the corresponding canonical momentum, q¯j=q¯(qj). Moreover, ε−1/2M−1/2un where ε−1/2=2μB|h| corresponds for m>0 to the energy associated with spin flipping a number M−1/2un of unpaired rotated spins 1/2.If in Eq. (67) one expands the sn band canonical momentum q¯j=qj+(2π/L)Φsn(qj) around qj and considers all energy contributions up to O(1/L) order, one arrives after some algebra to the energy functional, Eq. (33), which includes terms of second order in the deviations δNsn(qj). Their absence from the corresponding energy spectrum, Eq. (67), results from the functional Φsn(qj), Eq. (63), being incorporated in the sn band canonical momentum, Eq. (62).It follows that in contrast to the equivalent energy functional, Eq. (33), that in Eq. (67) has no energy interaction terms of second-order in the deviations δNsn(q¯j). This property simplifies the expression of the spin dynamical correlation functions in terms of s1 pseudofermion spectral functions. Specifically, their spectral weights can be expressed as Slater determinants of s1 pseudofermion operators. In the case of the PDT suitable for the metallic phase of the 1D Hubbard model [28,29], that property also allows the dynamical correlation functions being expressed as a convolution of c and s1 pseudofermion spectral functions. Such convolutions are absent though from the modified PDT introduced in the following.That within the s1 pseudofermion representation the functional Φ(qj)≡Φs1(qj), Eq. (66), is incorporated in the canonical momentum, Eq. (62), has also consequences on the form of the above mentioned Slater determinants of the s1 pseudofermion operators, Eq. (64), which can be written as,(68)f¯q¯j,s1†=1L∑j′=1Ls1eiq¯jj′f¯j′,s1†;f¯q¯j,s1=1L∑j′=1Ls1e−iq¯jj′f¯j′,s1,j=1,…,Ls1. As in the case of the corresponding s1 pseudoparticle operators, Eq. (60), the operator f¯j′,s1† (and f¯j′,s1) creates (and annihilates) one s1 pseudofermion at the s1 effective lattice site xj′=as1j′ where j′=1,…,Ls1. Indeed, the s1 pseudofermions also live in the squeezed s1 effective lattice. And as in Eq. (59) for the local s1 pseudoparticle operators, the corresponding local s1 pseudofermion operators obey the Fermi algebra,(69){f¯j,s1†,f¯j′,s1}=δj,j′.Consider two s1 pseudofermions of canonical momentum q¯j and q¯j′, respectively. Here q¯j and q¯j′=qj′ refer to a PS excited-energy-eigenstate and the corresponding initial ground-state s1 band, respectively. From the use of Eqs. (62) for sn=s1, (68), and (69) one then finds the anticommutators,(70){f¯q¯j,s1†,f¯q¯j′,s1}={f¯q¯j,s1†,f¯qj′,s1}=1Le−i(q¯j−qj′)/2ei2πΦT(qj)/2sin(2πΦT(qj)/2)sin((q¯j−qj′)/2),ΦT(qj)=Φs10+Φ(qj), and {f¯q¯j,s1†,f¯q¯j′,s1†}={f¯q¯j,s1,f¯q¯j′,s1}=0. Here ΦT(qj) is the overall phase shift acquired by a s1 pseudofermion of momentum qj under the transition from the ground state to the PS excited-energy-eigenstate, Φs10, Eq. (28), is the corresponding non-scattering part of that phase shift, and Φ(qj), Eq. (66), is its scattering part.For 2πΦT(qj)→0 the anticommutator relation, Eq. (70), would be the usual one, {fq¯j,s1†,fqj′,s1}=δq¯j,q¯j′. That such an anticommutator relation has not that simple form is the price to pay to render the s1 pseudofermions non-interacting objects associated with an energy spectrum of form, Eq. (67). Indeed this is achieved by incorporating the functional Φ(qj), Eq. (66), in the s1 band canonical momentum, Eq. (62) for sn=s1. The unusual form, Eq. (70), of that anticommutator relation is behind the phase-shift functional Φ(qj) controlling the spectral weight distributions of spin dynamical correlation functions, as confirmed below.The unitarity of the s1 pseudoparticle – s1 pseudofermion transformation preserves the s1 pseudoparticle operator algebra provided that the canonical momentum values q¯j and q¯j′ belong to the s1 band of the same energy eigenstate. The exotic form of the anticommutator, Eq. (70), follows from q¯j and q¯j′=qj′ corresponding in it rather to the excited-energy-eigenstate s1 band and the ground-state s1 band, respectively.3The Mott–Hubbard insulator pseudofermion dynamical theoryThe Mott–Hubbard insulator PDT considered here profits from the sn pseudofermions having no energy interactions, as given in Eq. (67), and accounts for such elementary objects scattering events. Its aim is the evaluation of finite-ω spin dynamical correlation functions of general form,(71)B(k,ω)=∑f|〈f|Oˆ(k)|GS〉|2δ(ω−(Ef−EGS)),ω>0. Here Oˆ(k) is a corresponding spin operator whose application onto the ground state conserves the BA number Nc=L of c pseudoparticles, |GS〉 is that ground state, and |f〉 denotes its PS excited energy eigenstates contained in the excitation Oˆ(k)|GS〉.In the present case of the spin dynamical correlation functions of the half-filled 1D Hubbard model, the elementary processes that generate such excited energy eigenstates from ground states with spin densities 0<m<1 can be classified into three (A)–(B) classes:(A) High-energy elementary s1 pseudofermion (and sn≠s1 pseudofermion) and (if any) unpaired rotated spins flipping processes. The pseudofermion processes involve creation or annihilation (and creation) of one or a finite number of s1 pseudofermions (and sn≠s1 pseudofermions) with canonical momentum values q¯j≠±q¯Fs1 (and canonical momentum values q¯j∈[qsn−,qsn+]).(B) Zero-energy and finite-momentum processes that change the number of s1 pseudofermions at the ι=+1 right and ι=−1 left s1 Fermi points.(C) Low-energy and small-momentum elementary s1 pseudofermion – s1 pseudofermion-hole processes in the vicinity of their right (ι=+1) and left (ι=+1) Fermi points, relative to the excited-state s1 pseudofermion momentum occupancy configurations generated by the above elementary processes (A) and (B).3.1Pseudofermion representation of the matrix elements of the spin operators between the ground state and the excited energy eigenstatesWithin the PDT, the matrix elements 〈f|Oˆ(k)|GS〉 in Eq. (71) of the spin operators between the ground state and the excited energy eigenstates are expressed in the pseudofermion representation. The S>0 ground state in these matrix elements has in such a representation the simple form,(72)|GS〉=∏q¯=−kF↓kF↓∏q¯′=−ππf¯q¯,s1†f¯q¯′,c†|0〉=∏j=1N↓∏j′=1Lf¯q¯j,s1†f¯q¯j′,c†|0〉f¯q¯,β†=fq,β†,β=c,s1, where |0〉 stands for the electron and rotated-electron vacuum, the ground-state generator has been written in terms of s1 and c pseudofermion creation operators, and the corresponding s1 and c band momentum values q¯=q=q¯j=qj and q¯′=q′=q¯j′=qj′, respectively, are those of the corresponding occupied ground-state Fermi seas. The s1 and c band discrete momentum values of any energy eigenstate are uniquely defined in Eqs. (10) and (11). (We recall that the pseudofermion representation has been inherently constructed to q¯=q for a PS initial ground state.)To express the matrix elements 〈f|Oˆ(k)|GS〉 appearing in Eq. (71) in the pseudofermion representation, one must introduce the local spin operators associated with the rotated spins 1/2 whose singlet pairs refer to the internal degrees of freedom of the n-pair sn pseudofermions. On the other hand and as discussed below, within the pseudofermion representation the Mun=2S, Eq. (53), unpaired rotated spins are used by the s1 pseudofermions as unoccupied sites of their s1 effective lattice. Hence all the system L rotated spins 1/2 are accounted for by the pseudofermion representation.The first step to express the matrix elements 〈f|Oˆ(k)|GS〉 in the pseudofermion representation is to express the corresponding one- and two-electron operator Oˆ(k) in terms of creation and annihilation rotated-electron operators, Eq. (51). Indeed, the rotated spins 1/2 are the spins of the rotated electrons generated from the electrons by the specific unitary operator Vˆ defined in Eq. (52). Some of the procedures followed to achieve that expression differ from those used within the metallic-phase PDT of Refs. [28,29]. Indeed, the latter refers to operators Oˆ(k) that change the c pseudoparticle occupancies, which except for possible overall ±π/L momentum shifts of all Nc=Lc pseudofermions remain unchanged for the quantum problem considered here.The use of the Baker–Campbell–Hausdorff formula leads to the following expression of a general operator Oˆ in terms of the creation and annihilation rotated-electron operators, Eq. (51),(73)Oˆ=∑i=0∞Oˆi=O˜+[O˜,S˜]+12[[O˜,S˜],S˜]+…,Oˆi=[O˜,S˜]i=[[O˜,S˜]i−1,S˜],i=1,…,∞;[O˜,S˜]0=O˜=Vˆ†OˆVˆ, where the operators O˜ and S˜ have the same expression in terms of creation and annihilation rotated-electron operators as Oˆ and Sˆ, respectively, in terms of creation and annihilation electron operators.There are two qualitatively different situations that follow from symmetry. The first refers to operators Oˆ that commute with the electron – rotated-electron unitary operator Vˆ, Eq. (52). It follows from Eq. (73) that Oˆ=O˜=Vˆ†OˆVˆ, so that such operators have exactly the same expression in terms of creation/annihilation electron and rotated-electron operators, respectively. Trivial examples are Vˆ=eSˆ and Sˆ, which commute with theirself, and thus according to the general formulas, Eq. (73), are such that Vˆ=eSˆ=V˜=eS˜ and Sˆ=S˜, respectively. Other examples of such operators are the six generators of the model spin and η-spin SU(2) symmetry algebras and the momentum operator whose eigenvalues are given in Eq. (19).On the other hand, for most operators it holds that [Oˆ,Sˆ]≠0. Their expressions have in terms of creation and annihilation rotated-electron operators an infinite number of terms, Oˆ=∑i=0∞Oˆi, as given in Eq. (73). This is the case of the model Hamiltonian and most one- and two-electron operators Oˆ(k) in dynamical correlation functions Lehmann representations, Eq. (71). Except in the u→∞ limit, the same applies to the three l=z,± local spin operators Sˆjl, which when expressed in terms of rotated-electron operators have thus an infinite number of terms,(74)Sˆjl=∑i=0∞Sˆj,il=S˜jl+[S˜jl,S˜]+12[[S˜jl,S˜],S˜]+…,l=z,±. Interestingly,(75)S˜jl=Vˆ†SˆjlVˆ,l=z,±;S˜j±=S˜jx±iS˜jy, are here the l=z,± local operators associated with the rotated spins 1/2.The creation and annihilation rotated-electron operators, Eq. (51), can be uniquely expressed in terms of such three l=z,± rotated-spin-1/2 operators, three corresponding rotated-η-spin-1/2 operators, and c pseudoparticle operators, and vice versa. Here we are interested in the 1D Hubbard model in the subspace for which Nc=L. For that quantum problem, the number of rotated-electron doubly occupied sites vanishes for u>0, so that the η-spin degrees of freedom are frozen and one may omit the rotated-η-spin-1/2 operators from all operational expressions. The corresponding more general operational expressions valid for the 1D Hubbard model in its full Hilbert space of the rotated-spin 1/2, rotated-η-spin 1/2, and c pseudoparticle operators in terms of rotated-electron operators and of the latter operators in terms of the former are given in Appendix A. Those provided in the following are particular cases of the general expressions in that Appendix.The three l=z,± local rotated spin operators S˜jl, which in Appendix A are denoted by S˜j,sl, and corresponding three l=z,± generators Sˆl=S˜l of the global spin SU(2) symmetry algebra, which commute with the electron – rotated-electron unitary operator Vˆ, Eq. (52), can be written in terms of the rotated-electron operators, Eq. (51), as follows,(76)S˜j−=(S˜j+)†=c˜j,↑†c˜j,↓;S˜jz=(n˜j,↓−1/2);S˜l=∑j=1LS˜jl=∑j=1Lc˜j,↑†c˜j,↓,l=z,±, where the operator n˜j,↓ is given in Eq. (51) for σ=↓.Moreover, the c pseudoparticle operators labeled by the discrete momentum values qj′=(2π/L)Ij′c, Eqs. (9) and (10) for β=c, are given by,(77)fqj′,c†=(fqj′,c)†=1L∑j=1Le+iqj′jfj,c†,j′=1,…,L, where for the present case of the 1D Hubbard model in the Nc=L subspace the general expression of the operator fj,c†=(fj,c)† in terms of creation and annihilation rotated-electron operators, Eq. (A.1) of Appendix A, simplifies to,(78)fj,c†=(fj,c)†=c˜j,↑†(1−n˜j,↓);nj,c=fj,c†fj,c,j=1,…,L. The operators fj,c† and fj,c create and annihilate, respectively, one c pseudoparticle at the j=1,…,L site of the c effective lattice, which is identical to the model original lattice.Furthermore, the expression in terms of rotated-electron operators given in Eq. (A.2) of Appendix A for c pseudofermion operators, Eq. (64) for β=c, becomes for the model in the Nc=L subspace,(79)f¯q¯j,c†=1L∑j′=1Le+iq¯jj′c˜j′,↑†(1−n˜j′,↓);f¯q¯j,c=(f¯q¯j,c†)†,j=1,…,L. In the u→∞ limit, the ground-state momentum rapidity function k0c(qj) simplifies to k0c(qj)=qj. Hence, according to Eq. (61), for its PS excited energy eigenstates such a function reads, kc(qj)=q¯j. The u→∞ spinless fermions of Refs. [22,23] have been constructed inherently to carry the momentum rapidity kj=kc(qj)=q¯j. This reveals that such spinless fermions are the c pseudofermions as defined here in the u→∞ limit. Indeed, f¯q¯j,c†=Vˆ†bkj†Vˆ and f¯q¯j,c=Vˆ†bkjVˆ where 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. [23]. Such a relation between u→∞ spinless fermions and u>0c pseudofermions holds provided that Vˆ is the electron – rotated-electron unitary operator defined in terms of its matrix elements in Eq. (52).Inversion of the relations, Eqs. (76) and (78), leads to the following simplified form of the general expressions given in Eq. (A.6) of Appendix A,(80)c˜j,↑†=fj,c†(12−S˜jz);c˜j,↑=(c˜j,↑†)†,c˜j,↓†=fj,c†S˜j+;c˜j,↓=(c˜j,↓†)†, where (S˜j+)†=S˜j−. The rotated-electron degrees of freedom separation, Eq. (80), is such that the rotated-spin 1/2 operators, Eq. (76), and the c pseudoparticle operators, Eq. (78), emerge from the rotated-electron operators by an exact local transformation that does not introduce constraints.The unitarity of the electron – rotated-electron transformation implies that the rotated-electron operators c˜j,σ† and c˜j,σ, Eq. (51), have the same anticommutation relations as the corresponding electron operators cj,σ† and cj,σ, respectively. From the combination of that result with the use of Eq. (76), one confirms that the SU(2) algebra obeyed by the local rotated-spin operators s˜jl is the usual one,(81)[S˜j,s+,S˜j′,s−]=δj,j′2S˜j,sz;[S˜j±,S˜j′,sz]=∓δj,j′S˜j,s±;[S˜jl,S˜j′l]=0,l=z,±. Furthermore, straightforward manipulations based on Eqs. (76) and (78) lead to the operator algebra {fj,c†,fj′,c}=δj,j′ and {fj,c†,fj′,c†}={fj,c,fj′,c}=0 for the c pseudoparticle operators, Eq. (78), and,(82)[fj,c†,S˜j′l]=[fj,c,S˜j′l]=0,l=z,±, for the c pseudoparticle operators and the local spin operators, Eq. (76).The 1D Hubbard model is a non-perturbative quantum problem in terms of electron processes. This is behind the computation of its dynamical correlation functions, Eq. (71), which involve a given one- or two-electron operator Oˆ(k), being a very complex many-electron problem. On the other hand, a property that plays a central role in the PDT follows from expressing the operator Oˆ(k) in the terms of the rotated-electron operators, Eq. (51), generated by application of the unitary operator Vˆ defined in Eq. (52) as Oˆ(k)=∑i=0∞Oˆi(k), Eq. (73). It is that the latter expression renders the computation of the dynamical correlation functions, Eq. (71), a perturbative problem in terms of the processes involving the rotated spins 1/2 that emerge from the rotated electrons and the sn pseudofermions within which the corresponding rotated-spin singlet pairs are contained. Indeed, the inconvenience that in terms of rotated-electron operators the one- or two-electron operator Oˆ(k)=∑i=0∞Oˆi(k), Eq. (73), has an infinite number of terms is the price that must be paid to render the computation of such dynamical functions a perturbative problem.The next step of our program consists in rewriting the rotated-electron expression Oˆ(k)=∑i=0∞Oˆi(k) within a related uniquely defined pseudofermion representation as,(83)Oˆ(k)=∑i′=0∞Gˆi′(k)OˆGS⊙. The new index i′=0,1,…,∞ refers here to sn pseudofermions processes and OˆGS⊙ is a generator that transforms the initial ground state |GS〉 into a state with the same electron and rotated-electron numbers N↑ and N↓ and compact symmetrical s1 band momentum occupancies as the intermediate final ground state, which we call |GSf〉. The only difference between the states OˆGS⊙|GS〉 and |GSf〉 is the s1 band discrete momentum values of the former state being those of the initial ground state, q¯′=q′. (For one- or two-electron operators that conserve the numbers N↑ and N↓, one has that Oˆ(k)=∑i′=0∞Gˆi′(k) in Eq. (83).)Each term of index i′=0,1,…,∞ in Eq. (83) may have contributions from several terms of different index i=0,1,…,∞ in Oˆ(k)=∑i=0∞Oˆi(k), Eq. (73). Fortunately, one can compute the operational form in terms of pseudofermion operators of the leading i′=0,1,…,∞ orders of Oˆ(k)=∑i′=0∞Gˆi′(k)OˆGS⊙ from the transformation laws of the ground state |GS〉, Eq. (72), upon acting onto it the related operators Oˆi(k) in the expression Oˆ(k)=∑i=0∞Oˆi(k).Note that both the expressions Oˆ(k)=∑i=0∞Oˆi(k) and Oˆ(k)=∑i′=0∞Gˆi′(k)OˆGS⊙ are not small-parameter expansions. Consistently, the perturbative character of the sn pseudofermions processes refers to the spectral weight contributing to the dynamical correlation 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. (72), of operators in ∑i′=0∞Gˆi′(k)OˆGS⊙ with an increasingly large value of the index i′=0,1,…,∞.The perturbative character of the 1D Hubbard model upon expressing the one- or two-electron operators Oˆ(k) in dynamical correlation functions, Eq. (71), in terms of rotated-spins 1/2 operators and corresponding sn pseudofermion operators follows from the exact energy eigenstates being generated by occupancy configurations of these elementary objects. The non-perturbative character of the problem in terms of electrons results from their relation to the rotated-spins 1/2 and corresponding sn pseudofermions having as well a non-perturbative nature, 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 any one- or two-electron operator Oˆ(k) by Gˆ(k). Such a i′=0 leading-order operator term in the one- or two-electron operator expression,(84)Oˆ(k)=(Gˆ(k)+∑i′=1∞Gˆi′(k))OˆGS⊙, plays a key role in our study.As a particularly simple yet very convenient example of the expression of the matrix elements 〈f|Oˆ(k)|GS〉 in Eq. (71) in the pseudofermion representation, we consider in the following the spin dynamical correlation functions studied in this paper, Eq. (1). As often below in Sec. 4, we chose as corresponding operators Oˆ(k) the three l=z,± spin operators Sˆkl associated with the local spin operators Sˆjl. The corresponding spin dynamical correlation functions are Szz(k,ω), S+−(k,ω), and S−+(k,ω), which are such that the spin dynamical structure factors Sxx(k,ω)=Syy(k,ω) in Eq. (1) read Sxx(k,ω)=Syy(k,ω)=14(S+−(k,ω)+S−+(k,ω)). For the model in the Nc=L subspace, the c pseudofermion operators in the rotated-electron expressions, Eq. (80), do not play any active role.Importantly, all the singular spectral features in the spin dynamical structure factors Szz(k,ω), S+−(k,ω), and S−+(k,ω) studied below in Sec. 4 are produced by application onto the ground state of the corresponding leading-order operators Gˆ(k)OˆGS⊙. Since application of these operators onto that state does not generate n>1 composite sn pseudofermions, in the subspaces spanned by the excited energy eigenstates generated by these operators the number M+1/2un=2S, Eq. (53), of up-spin unpaired rotated spins equals that of Ns1h=2S unoccupied sites of the squeezed s1 effective lattice and thus of s1-band Ns1h=2Ss1 pseudofermion-holes. Indeed, upon moving in the s1 effective lattice, the s1 pseudofermions use the Ns1h=2S up-spin unpaired rotated spins as unoccupied sites. This is why the unpaired rotated spins are implicitly accounted for in the s1 pseudofermion representation through the s1 pseudofermion-holes.Upon acting onto subspaces spanned by energy eigenstates populated by sn pseudofermions with n>1 singlet pairs, the s1 pseudofermions use as well the M+1/2un=2S original-lattice sites occupied by unpaired rotated spins as s1 effective lattice unoccupied sites. Similarly to the s1 pseudoparticles, in that case the s1 pseudofermions use also as s1 effective lattice unoccupied sites the 2(n−1) sites out of the 2n original-lattice sites occupied by the 2n rotated spins 1/2 in the n singlet pairs bound within each n>1sn pseudofermion. This justifies the form of the general Ns1h expression, Eq. (54).Concerning the relation between the rotated-spin 1/2 representation Sˆkl=∑i=0∞Sˆk,il and the pseudofermion representation, we recall that the internal degrees of freedom of each s1 pseudofermion that populates a S>0 ground-state Fermi sea, Eq. (72), refer to a singlet pair of two rotated spins 1/2. Moreover, all ground-state ↓ rotated spins are paired with ↑ rotated spins, which gives Ns1=N↓s1 pseudofermion singlet pairs. The M+1/2un=N↑−N↓ ground-state ↑ rotated spins left over remain unpaired.Hence a ↑–↓ spin-flip process onto a S>0 ground state transforms two ↑ unpaired rotated spins into one spin-singlet pair, which leads to deviations δM+1/2un=−2 and δMssp=1. Such a process thus “annihilates” two ↑ unpaired rotated spins. As discussed below, to leading order the deviation δMssp=1 refers to creation of one s1 pseudofermion, i.e. δNs1=1.Conversely, a ↓–↑ spin-flip process onto a S>0 ground state leads to deviations δMssp=−1 and δM+1/2un=2, since one s1 pseudofermion rotated-spin singlet pair is broken under it. This thus gives rise to the annihilation of one s1 pseudofermion and creation of two ↑ unpaired rotated spins. A deviation δM+1/2un=2 in the number of ↑ unpaired rotated spins leads to a deviation δNs1h=δM+1/2un=2 in the number of the s1 band holes. In the usual condensed-matter bands, annihilation of one particle gives rise to creation of one hole. In contrast, here annihilation of one s1 pseudofermion upon a ↓–↑ spin-flip process leads to creation of two s1 band holes. Indeed, the annihilation of the s1 pseudofermion results from a pair breaking process of the two rotated spins 1/2 within it, which trasform into two ↑ unpaired rotated spins that play the role of unoccupied sites of the squeezed s1 effective lattice, so that δNs1h=δM+1/2un=2.These transformation processes are behind the squeezed s1 effective lattice and corresponding s1 momentum band being exotic, since their number of sites and discrete momentum values, respectively, which both are given by Ls1=N1+N1h, has different values for different subspaces. Hence within the s1 pseudofermion operator algebra, one distinguishes the s1-band holes created and annihilated under processes within which one s1 pseudofermion is annihilated and created, respectively, from the s1-band holes created and annihilated upon changing the number Ls1=N1+N1h of squeezed s1 effective lattice sites, which equals that of s1-band discrete momentum values. (For S>0 states such exotic Ls1 variations only lead to N1h variations.) Specifically, the former processes are described by application of the operators f¯q¯,s1 and f¯q¯,s1†, respectively, onto the initial state. On the other hand, the latter N1h variations that do not conserve Ls1=N1+N1h result from vanishing energy and vanishing momentum processes within which discrete momentum values are added to and removed from one of the s1 band limiting momentum values qs1±, Eq. (18) 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, since the process must leave invariant the s1 band symmetrical relation qs1+=−qs1− for the final state.In the case of the leading-order processes generated by application of the operators Sˆk+ and Sˆk− onto a S>0 ground state considered below, one discrete momentum is removed from and added to, respectively, the s1 band limiting momentum values. Such vanishing energy and vanishing momentum processes are not generated by the s1 pseudofermion operators in the expressions given below for the three l=z,± spin operators Sˆkl. Nonetheless, they are implicitly accounted for by the pseudofermion representation through the s1 band discrete momentum values of the final states, which are uniquely defined.In the following we use the transformation laws of the ground state, Eq. (72), upon acting onto it with the i=0,1,…,∞ operators on the right-hand side of the equations, Sˆkl=∑i=0∞Sˆk,il, for the three l=z,± spin operators to derive the expression of the corresponding leading-order operators Gˆ(k)OˆGS⊙, Eq. (84), in terms of s1 pseudofermion operators. Our goal is to approximate the expression of these rotated-spin operators in terms of pseudofermion operators by the corresponding leading-order term, Sˆkl≈Gˆ(k)OˆGS⊙.First, in the case of the operator Sˆkz=∑i=0∞Sˆk,iz such ground-state transformation laws leave invariant both the number M+1/2un=2S, Eq. (53), of ↑ unpaired rotated spins and Mssp=∑n=1∞nNsn, Eq. (17) for α=s, of spin-singlet pairs. Hence the leading-order operator Gˆ(k)OˆGS⊙=Gˆ(k) in the expression Oˆ(k)=∑i′=0∞Gˆi′(k) generates one s1 pseudofermion – s1 pseudofermion-hole processes. The i′>0 operator terms generate additional s1 pseudofermion – s1 pseudofermion-hole processes and transform s1 pseudofermions into n>1sn pseudofermions. Within such transformation processes, the creation of each new n>1sn pseudofermion involves the annihilation of a number n of s1 pseudofermions, so that M+1/2un=2S and Mssp=∑n=1∞nNsn are conserved. Both the additional s1 pseudofermion – s1 pseudofermion-hole processes and the transformation of s1 pseudofermions into n>1sn pseudofermions preserve such numbers values and give rise to very little spectral weight. For the present operator one has that OˆGS⊙=1, so that to leading order,(85)Sˆkz≈Gˆ(k)=∑q=−kF↓kF↓θ(kF↑−|k+q|)θ(|k+q|−kF↓)f¯q¯(k+q),s1†f¯q¯(q),s1, where q¯(q)=q+2πLΦs1(q) here and in the following operator expressions.Second, the transformation laws of the ground state, Eq. (72), upon acting onto it with the i=0,1,…,∞ operators on the right-hand side of the equation, Sˆk+=∑i=0∞Sˆk,i+, lead to deviations δM+1/2un=−2 and δMssp=1 in the number values of ↑ unpaired rotated spins and spin-singlet pairs, respectively. In this case the leading-order operator Gˆ(k)OˆGS⊙ generates deviations δNs1=1 and δNs1h=−2 and corresponding one s1 pseudofermion – s1 pseudofermion-hole processes. All i′>0 operator terms also generate deviations δM+1/2un=−2 and δMssp=1 along with additional s1 pseudofermion – s1 pseudofermion-hole processes and the transformation of s1 pseudofermions into n>1sn pseudofermions. The latter processes originate again very little spectral weight. One then finds the following leading-order expression,(86)Sˆk+≈Gˆ(k)OˆGS⊙;OˆGS⊙=f¯±kF↓,s1†,Gˆ(k)=∑q=−kF↓kF↓θ(kF↑−|π−k−q|)θ(|π−k−q|−kF↓)f¯q¯(k−π+q),s1†f¯q¯(q),s1.Third, in the case of the i=0,1,…,∞ operators on the right-hand side of Sˆk−=∑i=0∞Sˆk,i−, the corresponding ground-state transformation laws give rise to deviations δM+1/2un=2 and δMssp=−1 in the number values of ↑ unpaired rotated spins and spin-singlet pairs, respectively. One then finds that the leading-order operator Gˆ(k)OˆGS⊙ generates two s1 pseudofermion-holes processes such that δNs1h=2 and δNs1=−1. All i′>0 operator terms generate also deviations δM+1/2un=2 and δMssp=−1 together with additional s1 pseudofermion – s1 pseudofermion-hole processes and the transformation of s1 pseudofermions into n>1sn pseudofermions. As for the previous two spin operators, the latter processes produce very little spectral weight. The leading-order expression found for this operator reads,(87)Sˆk−≈Gˆ(k)OˆGS⊙;OˆGS⊙=f¯∓kF↓,s1,Gˆ(k)=∑q=−kF↓kF↓θ(kF↑−|π−k−q|)θ(|π−k−q|−kF↓)×f¯q¯F↓−,s1†f¯q¯F↓+,s1†f¯q¯(π−k−q),s1f¯q¯(q),s1.In the above expressions, the s1 pseudofermion momentum values ±kF↓ appearing in the operators OˆGS⊙ belong to the initial ground state s1 band whereas the s1 pseudofermion momentum values q¯(q)=q+2πLΦs1(q) in the operators Gˆ(k) expressions belong to the excited energy eigenstates s1 band. As further discussed in Sec. 4, in the case of the Sˆk+ and Sˆk− expressions, Eqs. (86) and (87), respectively, there occurs under the transitions from the ground state to the excited energy eigenstates generated by these operators an overall s1-band discrete momentum shift, qj→qj+(2π/L)Φβ0, for which Φs10=∓1/2 in Eq. (28). This leads to a shift of the whole s1 band occupied Fermi sea that gives rise to an overall momentum ∓kF↓ given by ∓(2π/L)Φs10N↓=πN↓/L. Such an overall momentum ∓kF↓ exactly cancels that of the operators OˆGS⊙=f¯±kF↓,s1† and OˆGS⊙=f¯∓kF↓,s1 in such Sˆk+ and Sˆk− expressions, respectively.Concerning the general dynamical correlation functions, Eq. (71), which in the pseudofermion representation read,(88)B(k,ω)=∑i′=0∞∑f|〈f|Gˆi′(k)OˆGS⊙|GS〉|2δ(ω−(Ef−EGS)),ω>0, following the above properties one approximates them by their leading-order term,(89)B(k,ω)≈B⊙(k,ω)=∑f|〈f|Gˆ(k)OˆGS⊙|GS〉|2δ(ω−(Ef−EGS)),ω>0, where for the three spin dynamical correlation functions studied in Sec. 4 the corresponding operators Gˆ(k)OˆGS⊙ are given in Eqs. (85)–(87). Since the properties studied in the following are valid for operators that preserve the number value Nc=L other than those in Sec. 4, we consider a general situation within which the form of the operators Oˆ(k) and Gˆ(k)OˆGS⊙ is not specified.For such general operators and as in the case of those considered above, both the generator onto the electron vacuum of the initial ground state in Eq. (72) and the operator OˆGS⊙ in OˆGS⊙|GS〉 are written in terms of s1 pseudofermion creation operators, Eqs. (64) and (68), whose discrete canonical momentum values equal the corresponding momentum values qj, Eqs. (9) and (10), of that initial ground state. (The only role of the c pseudofermion sea of that ground state also present in Eq. (72) is providing an overall momentum contribution π upon processes that change the number value Nsps=∑n=1∞Nsn in Eq. (15) by an odd integer number.) On the other hand, both the operator Gˆ(k) and the generators onto the electron vacuum of the excited energy eigenstates |f〉 are written in terms of s1 pseudofermion operators and sn pseudofermion operators of n>1 branches whose discrete canonical momentum values q¯j, Eq. (62), are those of these excited energy eigenstates. These two types of s1 band discrete canonical momentum values that correspond to the initial ground state and excited energy eigenstates subspaces, respectively, account for the Anderson orthogonality catastrophe [55] occurring in the s1 band under the transitions to the excited energy eigenstates |f〉.Such an Anderson orthogonality catastrophe is behind the exotic character of the quantum overlaps that control the spin dynamical correlation functions of the Mott–Hubbard insulator. Concerning that orthogonality catastrophe and the corresponding shake-up effects, there is a major difference relative to the metallic-phase PDT of Refs. [28–30]. Indeed for it the Anderson orthogonality catastrophe occurs both in the c and s1 bands under the transitions to the excited energy eigenstates |f〉. On the other hand, for the half-filled 1D Hubbard model at finite field h in the PSs considered here only the s1 band has Fermi points.Besides the ground states not being populated by sn pseudofermions of n>1 branches, for S>0 the leading-order operator Gˆ(k) does not give rise to the transformation of s1 pseudofermions onto such n>1sn pseudofermions. Moreover, for S>0 the spectral weight generated by such transformation processes, which results from application of i′>0 higher order operators Gˆi′(k)OˆGS⊙ onto the ground state, is very small. For simplicity, in the following we ignore such small-weight contributions, which as discussed below can also be accounted for by the PDT.There is always an exact excited energy eigenstate |fG〉 of |GSf〉 such that,(90)|fG〉=Gˆ(k)|GSf〉. The excitation Gˆ(k)OˆGS⊙|GS〉 has then finite overlap with such a specific energy eigenstate, which gives,(91)〈fG|Gˆ(k)OˆGS⊙|GS〉=〈GSfex|OˆGS⊙|GS〉=〈0|f¯q¯Ns1⊙,s1…f¯q¯2,s1f¯q¯1,s1f¯q′1,s1†f¯q′2,s1†…f¯q′Ns1⊙,s1†|0〉=〈0|f¯q′Ns1⊙,s1…f¯q′2,s1f¯q′1,s1f¯q¯1,s1†f¯q¯2,s1†…f¯q¯Ns1⊙,s1†|0〉⁎, where |GSfex〉 is a state with the same s1 pseudofermion occupancy as |GSf〉 but whose s1 band discrete momentum values are those of its excited energy eigenstate |fG〉=Gˆ(k)|GSf〉 and Ns1⊙ is the number of s1 pseudofermions of the states OˆGS⊙|GS〉 and |GSf〉The discrete canonical momentum values q′1, q′2, …,q′Ns1⊙ in Eq. (91) equal the corresponding initial ground state discrete momentum values whereas q¯1, q¯2,…,q¯Ns1⊙ are the discrete canonical momentum values of the excited energy eigenstate |fG〉, Eq. (90). In contrast to a Fermi liquid, such two sets of discrete momenta have different values, so that their relative canonical momentum shifts give rise to the Anderson orthogonality catastrophe. It consists in excited energy eigenstates of general form,(92)|fGC〉=GˆC(m+1,m−1)Gˆ(k)|GSf〉, which result from application onto the state |fG〉, Eq. (90), of the generator GˆC(m+1,m−1) of the low-energy and small-momentum processes (C), also having overlap with the excitation Gˆ(k)OˆGS⊙|GS〉. Hence,(93)〈fG|GˆC(m+1,m−1)†Gˆ(k)OˆGS⊙|GS〉=〈GSfex|GˆC(m+1,m−1)†OˆGS⊙|GS〉=〈0|f¯q¯Ns1⊙,s1…f¯q¯2,s1f¯q¯1,s1GˆC(m+1,m−1)†f¯q′1,s1†f¯q′2,s1†…f¯q′Ns1⊙,s1†|0〉=〈0|f¯q′Ns1⊙,s1…f¯q′2,s1f¯q′1,s1GˆC(m+1,m−1)f¯q¯1,s1†f¯q¯2,s1†…f¯q¯Ns1⊙,s1†|0〉⁎. The number of elementary s1 pseudofermion – s1 pseudofermion-hole processes (C) of momentum ±2π/L in the vicinity of the s1;ι=±1 Fermi points of |GSf〉 is denoted here and in the following by mι=1,2,3,… . Such processes conserve the number Ns1⊙ of s1 pseudofermions, so that the matrix elements, Eq. (93), have the same form as that in Eq. (91) but with the excited-state occupied discrete canonical momentum values q¯1, q¯2, …,q¯Ns1⊙ in the vicinity of the s1 band Fermi points being slightly different from those in that equation.In the case of the general dynamical correlation function expression in the pseudofermion representation, Eq. (88), there are also exact excited energy eigenstates |fG(i′)〉 of |GSf〉 such that,(94)|fG(i′)〉=Gˆi′(k)|GSf〉,i′=0,1,…,∞. These exact excited energy eigenstates may be populated by sn pseudofermions of n>1 branches. Their small contribution to the general dynamical correlation functions is simpler to compute than that from the s1 pseudofermions. The reason is that the initial ground state is not populated by sn pseudofermions of n>1 branches. Since as above for the i′=0 operator Gˆ(k), the sn pseudofermion operators in the expression of any i′≥0 operator Gˆi′(k) that appears both in the dynamical correlation function expression, Eq. (88), and in Eq. (94) have discrete canonical momentum values that belong to the excited energy eigenstate sn band, one finds that,(95)〈fG|Gˆi′(k)OˆGS⊙|GS〉=〈GSf|Gˆi′†(k)Gˆi′(k)OˆGS⊙|GS〉=〈GSfex(i′)|OˆGS⊙|GS〉, where |GSfex(i′)〉 is a state with the same s1 pseudofermion occupancy as |GSf〉 but whose s1 band discrete momentum values are those of its excited energy eigenstate |fG(i′)〉=Gˆi′(k)|GSf〉.Hence the quantum overlaps resulting from the excited energy eigenstates sn pseudofermion occupancies associated with Gˆi′†(k)Gˆi′(k) in Eq. (95) are Fermi-liquid like due to the lack of such occupancies in the ground states |GSf〉 and |GS〉. Indeed, the matrix elements 〈GSfex(i′)|OˆGS⊙|GS〉 that result from such overlaps only involve s1 pseudofermion operators and have the same general form as that in Eq. (91). However, |〈GSfex(i′)|OˆGS⊙|GS〉| strongly decreases upon increasing the index i′=0,1,…,∞, most of the spectral weight being associated with the i′=0 matrix element 〈GSfex(0)|OˆGS⊙|GS〉=〈GSfex|OˆGS⊙|GS〉, Eq. (91). This is why in the following we approximate the general dynamical correlation function expression, Eq. (88), by that given in Eq. (89).3.2The dynamical correlation functions and corresponding state summationsThe energy and momentum spectra,(96)δE⊙=E⊙−EGS;δP⊙=P⊙−PGS, of the excited energy eigenstates |fG〉, Eq. (90), generated by the processes (A) and (B), which have finite quantum overlap with the excitation Gˆ(k)OˆGS⊙|GS〉, are important pieces of the present PDT dynamical correlation function expressions. Within the theory, the function B⊙(k,ω), Eq. (89), can be written as follows,(97)B⊙(k,ω)=∑fΘ(Ω−δωf)Θ(δωf)Θ(|vf|−vs1)BQ(δωf,vf), where the Θ distribution Θ(x) is different from θ(x) at x=0, being given here and in the following by Θ(x)=1 for x≥0 and Θ(x)=0 for x<0. A difference relative to the metallic-phase PDT is that the s1 band Fermi velocity vs1 in Eq. (97) is in that dynamical theory replaced by a velocity vβ¯=min{vc,vs1}. Here vc is the c band Fermi velocity, which is absent for the ne=1 ground states and their PS excited energy eigenstates considered here.The summation ∑f on the right-hand side of Eq. (97) runs over PS excited energy eigenstates |fGC〉, Eq. (92), which are generated by processes (A), (B), and (C) at fixed values of k and ω. Such states have excitation energy and momentum, Eq. (96), in the ranges δEf⊙∈[ω−Ω,ω] and δPf⊙∈[k−Ω/vf,k] where,(98)δωf=(ω−δEf⊙)=(ω−Ef⊙+EGS);δkf=k−δPf⊙,δEf=δEf⊙+δωf=ω;Pf=δPf⊙+δkf=k, and the velocity vf in Eqs. (97) and (98) is defined as,(99)vf=δωf/δkf. The energy deviation δEf=ω and momentum deviation δPf=k denote in Eq. (98) the excitation energy and momentum, respectively, of the excited energy eigenstates. Moreover, in Eq. (97) Ω is the energy range of the elementary processes (C). That energy scale is self-consistently determined as that for which the velocity vf, Eq. (98), remains nearly unchanged.The function BQ(δωf,vf) in Eq. (97) is the pseudofermion spectral function BQ(k′,ω′) given below. Another important difference relative to the metallic-phase PDT of Refs. [28–30] is that for it the function BQ(δωf,vf) is replaced in Eq. (97) by a convolution of c and s1 pseudofermion spectral functions. In the present case such a function is only controlled by the Anderson orthogonality catastrophe associated with the s1 band shake-up effects in the s1 pseudofermion spectral function BQ(k′,ω′). Those involve all s1 Fermi-sea pseudofermions and result from the s1 band discrete canonical momentum value shifts, (2π/L)ΦT(qj), under the transitions to the excited energy eigenstates. They are behind a large number of small-momentum and low-energy s1 pseudofermion – s1 pseudofermion-hole processes (C) in the linear part of the s1 pseudofermion energy dispersions leading to finite spectral-weight contributions. (In a Fermi liquid there are no such discrete momentum shifts, so that only a single quasiparticle–quasihole process contributes.)Creation of sn≠s1 pseudofermions are processes (A). They do not contribute to the Anderson orthogonality catastrophes, since the corresponding quantum overlaps are non-interacting like. Creation of sn≠s1 pseudofermions is accounted for both by their contribution to the spectra δE⊙ and δP⊙, Eq. (96), and the phase shifts acquired by the s1 pseudofermions upon scattering off the created sn≠s1 pseudofermions.The present dynamical theory s1 pseudofermion spectral functions on the right-hand side of Eq. (97) have a general form similar to that of the metallic-phase PDT of Refs. [28–30]. Their expression involves sums that run over the processes (C) numbers mι=1,2,3,… and reads,(100)BQ(k′,ω′)=L2π∑m+1;m−1A(0,0)a(m+1,m−1)×δ(ω′−2πLvs1∑ι=±1(mι+Δι))δ(k′−2πL∑ι=±1ι(mι+Δι)), where the lowest peak weight A(0,0) is associated with a transition from the ne=1 and m>0 ground state to a PS excited energy eigenstate generated by processes (A) and (B), the relative weights a=a(m+1,m−1) are generated by additional processes (C), and Δι refers to the functional 2Δι defined below.The weights A(0,0)a(m+1,m−1) in Eq. (100) are reached after the quantum overlaps stemming from creation of sn pseudofermions of n>1 branches and/or unpaired rotated spin flip processes are trivially computed. They are associated with matrix elements of general form, Eq. (91), and thus only involve s1 pseudofermion operators and read,(101)|〈0|f¯q′Ns1⊙,s1…f¯q′2,s1f¯q′1,s1f¯q¯1,s1†f¯q¯2,s1†…f¯q¯Ns1⊙,s1†|0〉|2, where Ns1⊙ is the PS excited energy eigenstate number of s1 pseudofermions and |0〉 denotes the electron vacuum. The matrix element square, Eq. (101), can be expressed in terms of a corresponding Slater determinant of s1 pseudofermion operators that involves the s1 pseudofermion anticommutators, Eq. (70), as follows,(102)In the case of the lowest peak weight A(0,0) associated with a transition to an excited energy eigenstate generated by processes (A) and (B) the use of the s1 pseudofermion anticommutators, Eq. (70), in Eq. (102) leads after some suitable algebra to,(103)A(0,0)=(1L)2Ns1⊙∏j=1Ls1sin2(π2(1−(1−2ΦT(qj))Ns1⊙(qj)))∏j=1Ls1−1(sin(πjL))2(Ls1−j)×∏i=1Ls1∏j=1Ls1θ(j−i)×sin2(π2(1−(1−(2(j−i)+2ΦT(qj)−2ΦT(qi))L)Ns1⊙(qj)Ns1⊙(qi)))×∏i=1Ls1∏j=1Ls11sin2(π2(1−(1−2(j−i)+2ΦT(qj)L)Ns1⊙(qi)Ns1⊙(qj))). The numbers of s1 band discrete momentum values, Ls1, s1 pseudofermions, Ns1⊙=∑j=1Ls1Ns1⊙(qj), and the corresponding s1 band momentum distribution function, Ns1⊙(qj), are in this expression those of the excited energy eigenstate generated by the processes (A) and (B) and ΦT(qj) is the phase-shift functional in Eq. (70).The general expression of the relative weights a=a(m+1,m−1) in Eq. (100), which are associated with the tower of excited energy eigenstates generated by the processes (C) and corresponding matrix elements, Eq. (93), reads [28],(104)a(m+1,m−1)=(∏ι=±1aι(mι))(1+O(lnL/L)), where,(105)aι(mι)=∏j=1mι(2Δι+j−1)j=Γ(mι+2Δι)Γ(mι+1)Γ(2Δι),ι=±1. For mι=1, Eq. (105) leads to,(106)aι(1)=2Δι=(δq¯Fs1ι(2π/L))2,ι=±1.That for the metallic-phase PDT of Refs. [28–30] there are four β=c,s1;ι=±1 pseudofermion Fermi points whereas here there are only two ι=±1s1 pseudofermion Fermi points implies that for the present Mott–Hubbard insulator PDT there are only two functionals 2Δι, Eq. (106). They play a major role in the half-filled 1D Hubbard model spin dynamical properties, being fully controlled by the excited-state canonical momentum ι=±1 Fermi-point deviations δq¯Fs1ι such that δq¯Fs1ι/(2π/L)=ιδNs1,ιF+Φ(ιqFs1). Here δNs1,ιF=δNs1,ι0,F+ιΦs10 and thus δq¯Fs1ι/(2π/L)=ιδNs1,ι0,F+ΦT(ιqFs1) where the bare deviation δNs1,ι0,F accounts for the number of s1 pseudofermions created or annihilated at the right (ι=+1) and left (ι=+1) s1 band Fermi points.The two functionals, Eq. (106), can be written as,(107)2Δι=(δq¯Fs1ι(2π/L))2=(ιδNs1,ι0,F+ΦT(ιqFs1))2=(ι2ξs1,s11δNs1F+ξs1,s11δJs1F+∑n=1∞∑j=1LsnΦs1sn(ιqFs1,qj)δNsnNF(qj))2. In this expression ξs1,s11 is the two-pseudofermion phase-shift parameter, Eqs. (46) and (47), δNs1F=δNs1,+1F+δNs1,−1F, and 2Js1F=δNs1,+1F−δNs1,−1F. The deviations δNsnNF(qj′) refer to sn band momentum values qj, which for the s1 branch are away from the s1 Fermi points. (The s1 pseudofermion creation or annihilation at and in the vicinity of such points is rather accounted for by the deviations δNs1F and δJs1F in the second expression of Eq. (107).) The form of the ι=±1 functionals, Eq. (107), confirms that the s1 pseudofermion phase shifts acquired upon scattering off the sn pseudofermions of n>1 branches created under transitions to the PS excited energy eigenstates contribute to the spin dynamical properties. In addition, creation of sn pseudofermions with n>1 singlet pairs is accounted for in the dynamical correlation functions energy and momentum. On the other hand, the unpaired rotated spins flipping processes do not lead to phase shifts and thus do not contribute to the ι=±1 functionals, Eq. (107).The occurrence of four Fermi points in the metallic-phase PDT implies that the expression of its four functionals that play the role of those in Eq. (107) involves, instead of a single two-Fermi-points phase-shift parameter ξs1,s11, the four entries ξcc1, ξcs11, ξs1c1, ξs1s11 of the 2×2 dressed-charge matrix Z1 [20,45]. For the c and s1 pseudofermion representation used within the PDT of Refs. [28–30] such entries are combinations of two-pseudofermion phase shifts whose two momentum values are at the Fermi points, as here in Eq. (46).On the one hand, the lack of s1 pseudofermion interaction terms in the energy spectrum, Eq. (67), is associated with the weights A(0,0)a(m+1,m−1) in Eq. (100) being merely of the form, Eqs. (101) and (102). On the other hand, the lack of such interaction terms is reached by incorporating in the s1 band canonical momentum the functional Φ(qj), Eq. (66). This leads to the unusual form of the anticommutator relation, Eq. (70), which is behind the phase-shift functional ΦT(qj)=Φs10+Φ(qj) and related functional 2Δι=(ιδNs1,ι0,F+ΦT(ιqFs1))2 controlling the weights A(0,0)a(m+1,m−1), as confirmed by the form of the expressions given in Eqs. (103) and (104)–(106).According to Eq. (106), the functional, Eq. (107), is the relative weight of the s1,ι pseudofermion spectral function mι=1 peaks. They correspond to relative weights, Eq. (104),(108)a(1,0)=2Δ+1;a(0,1)=2Δ−1.The δ-functions in the pseudofermion spectral function expression, Eq. (100), impose that ((L/4πvs1)(ω′+ιvs1k′)−Δι)=mι. In the present TL, the k′ and ω′ values for which the quantity ((L/4πvs1)(ω′+ιvs1k′)−Δι) equals the integer numbers mι of elementary processes (C) near both the ι=±1 Fermi points refer to a dense distribution of (k′,ω′) points. Moreover, in the TL the factor (L/4πvs1) in ((L/4πvs1)(ω′+ιvs1k′)−Δι)=mι ensures that for any arbitrarily small k′ and ω′ values for which 0<(ω′+ιvs1k′)/(4πvs1)≪1 the corresponding values of the ι=±1 integer numbers mι=((L/4πvs1)(ω′+ιvs1k′)−Δι) of elementary processes (C) are such that mι≫1. Hence within the TL one can use a large-mι expansion for the relative weight expression in Eq. (105). To derive it one uses the asymptotic expansion Γ(x)≈e−xxx2πx(1+1/(12x)+…) of the Γ(x) function valid for x≫1 in the ratio Γ(mι+2Δι)/Γ(mι+1) appearing in Eq. (105). This gives to leading order [23], Γ(mι+2Δι)/Γ(mι+1)≈(mι+Δι)−1+2Δι. Its further use in the ι=±1 relative weight expression, Eq. (105), then leads to the following asymptotic behavior for that weight, which in the TL is used in the derivation of the corresponding exact spin dynamical correlation function expressions near their spectra thresholds,(109)aι(mι)≈1Γ(2Δι)(mι+Δι)−1+2Δι,2Δι≠0,ι=±1.A relation also useful for such a derivation involves the lowest peak weight A(0,0), Eq. (103), in the s1 pseudofermion spectral function BQ(k′,ω′), Eq. (100), which can be written as,(110)A(0,0)=F(0,0)(1LS0)−1+2Δ+1+2Δ−1. Here F(0,0) and S0 are in the TL independent of L and 2Δ+1 and 2Δ−1 are the two functionals, Eq. (107).In the general case in which these two functionals are finite the s1 pseudofermion spectral function BQ(k′,ω′), Eq. (100), can be written as [28,30],(111)BQ(k′,ω′)=L4πvs1A(0,0)∏ι=±1aι(L4πvs1(ω′+ιvs1k′)−Δι)≈F(0,0)4πS0vs1∏ι=±1Θ(ω′+ιvs1k′)Γ(2Δι)(ω′+ιvs1k′4πS0vs1)−1+2Δι. To reach the second expression given here, which in the TL is exact, Eqs. (109) and (110) were used.On the other hand, when 2Δι>0 and 2Δ−ι=0 the s1 pseudofermion spectral function has a different form given by [30],(112)BQ(k′,ω′)=A(0,0)vs1aι(L2πvs1ω′−Δι)δ(k′−ιω′vs1)≈F(0,0)vs1Γ(2Δι)Θ(ιω′)(ω′2πS0vs1)−1+2Διδ(k′−ιω′vs1). Again the second expression provided in this equation is obtained from the use of Eqs. (109) and (110).Finally, when 2Δι=2Δ−ι=0 such a function reads,(113)BQ(k′,ω′)=2πLA(0,0)δ(k′)δ(ω′)≈2πF(0,0)S0δ(k′)δ(ω′).The numerical computation of the momentum and state summations in Eqs. (88) and (89) needed to access the corresponding finite-u spectral-weight distributions over the whole (k,ω) plane is an extremely difficult technical task. Fortunately, the use of Eqs. (111)–(113) for the spectral function BQ(δωf,vf) in Eq. (89), enables partially performing the summations in the latter equation for the (k,ω)-plane vicinity of important singular spectral features.The more important of such features is a branch line. A particle (and hole) branch line is generated by elementary processes (A) where one pseudofermion is created (and annihilated) at an initial-ground-state s1 band momentum value qj outside the Fermi points ±kF↓ plus elementary processes (B). In the cases of a particle and hole branch line, the set of such transitions scans the whole corresponding range |qj|∈[kF↓,kF↑] and qj∈[−kF↓,kF↓], respectively. On the other hand, all remaining s1 pseudofermions are created or annihilated at the s1 Fermi points ±qFs1=±kF↓, Eq. (27). Such a feature may also involve unpaired rotated spins flipping processes. Moreover, the sn≠s1 pseudofermions (if any) are created at their sn band limiting values, ±qsn, Eq. (25). This gives a (k,ω)-plane branch line defined by the following equations,(114)ω(k)=ω0+c0εs1(qj);k=k0+c0qj,c0=±1, where c0=+1 and c0=−1 refers to a particle and hole branch line, respectively, εs1(qj) is the s1 band energy dispersion, Eq. (34), and,(115)ω0=2μB|h|(M−1/2un+Mssp−Ns1);k0=π2(1−(−1)δMssp)+2kF↓2Js1F. The only contribution from the full c band corresponds to the momentum (π/2)(1−(−1)δMssp), which reads 0 or π when the deviation δMssp is an even or odd integer, respectively.For simplicity we consider the more general case for which the two ι=±1 parameters 2Δι, Eq. (107), are finite, so that the corresponding pseudofermion spectral function has the form given in Eq. (111). We then consider a (k,ω)-plane point located just above the branch line whose momentum k expression is of the form given in Eq. (114) and the energy ω is such that (ω−ω(k)) is small and positive. The spectral-weight distribution expression in the vicinity of that point is controlled by the elementary processes (C), which generate from the initial excited energy eigenstates corresponding to the branch line a set of tower states whose momentum and energy relative to the ground state are precisely k and ω.By performing the summations in the general expression, Eq. (89), over all PS excited energy eigenstates generated from the ground state by the elementary processes (A) and (B), which correspond to branch-line points in the vicinity of the (k,ω)-plane point, and then accounting for the elementary processes (C) that combined with the former elementary processes (A) and (B) determine the line shape at that point, one finds from the use of manipulations similar to those reported in Appendix B of Ref. [28] for the metallic-phase PDT the following spectral-weight distribution expression for the line shape in the vicinity of the branch line,(116)B(k,ω)∝(ω−ω(k))ξ(k);(ω−ω(k))≥0,ξ(k)=−1+∑ι=±12Δι(qj)|qj=c0(k−k0). That in the vicinity of a branch line the above state summations can be partially performed follows in part from the lowest peak weights A(0,0), Eq. (103), for the corresponding set of states generated by elementary processes (A) and (B) having nearly the same magnitude.In Eq. (116) the 2Δι(qj) momentum qj dependence stems from a phase-shift contribution, c02πΦβ′β(ιqFs1,qj), within the scattering phase shift 2πΦ(ιkF↓), Eq. (66). The momentum qj is that of the s1 pseudofermion created or annihilated under the transition. Its qj value spans the whole above corresponding particle or hole branch-line range |qj|∈[kF↓,kF↑] or qj∈[−kF↓,kF↓], respectively. In the case of the spin dynamical correlation functions studied in this paper the general expression, Eq. (116), is exact for branch lines that coincide with the lower thresholds of such functions spectra.The corresponding high-energy dynamical correlation functions line shapes are beyond the reach of the techniques associated with the low-energy Tomonaga–Luttinger liquid [18–20,29]. In the limit of low-energy the PDT considered here describes the well-known behaviors predicted by such techniques. This refers specifically to the vicinity of (k,ω)-plane points (k0,0) of which (k0,ω0), Eq. (115), is a generalization for ω0>0. Alike for the metallic-phase PDT [29], near them the spectral-function behavior is,(117)B(k,ω)∝(ω−ω0)ζ,(ω−ω0)≥0,ζ=−2+∑ι=±12Δι,(ω−ω0)≠±vs1(k−k0),B(k,ω)∝(ω−ω0∓vs1(k−k0))ζ±,(ω−ω0∓vs1(k−k0))≥0,ζ±=−1+2Δs1±,(ω−ω0)≈±vs1(k−k0). The expressions given here apply to the finite-weight region above the (k,ω) plane point.4The longitudinal and transverse dynamical structure factors in the vicinity of their spectra lower thresholdsIn this section we use the Mott–Hubbard insulator PDT to study the line shape behavior of the spin dynamical structure factors Szz(k,ω) and Sxx(k,ω), Eq. (1), in the vicinity of their spectra lower thresholds at finite fields h>0. As discussed in Section 1, previous studies of these factors focused mainly onto magnetic fields h=0 when Szz(k,ω)=Sxx(k,ω)=Syy(k,ω) [38,39]. For large u values the spin degrees of freedom of the half-filled 1D Hubbard model can be mapped onto a spin-1/2 XXX chain [56]. Previous studies on that model spin dynamical structure factors Szz(k,ω) and Sxx(k,ω) refer to finite systems and rely on numerical diagonalizations [57], evaluation of matrix elements between BA states [56,58], and the form-factor method [8–13].The singularities that dominate the line shape for small excitation energy values (ω−ωτ(k)) near the lower thresholds ωτ(k) of the longitudinal (τ=l) and transverse (τ=t) dynamical structure factors Szz(k,ω) and Sxx(k,ω) spectra, respectively, are for h>0 controlled by well-defined types of excited energy eigenstates. In the case of the related spin-1/2 XXX chain, such states have already been identified within the pioneering study of Ref. [56] and later investigations, such as those reported in Refs. [10,58]. Such states have a one to one correspondence with the excited energy eigenstates |fG〉=Gˆ(k)|GSf〉, Eq. (90), of the half-filled 1D Hubbard model in the PSs considered in this paper for which Nc=L, whose generators Gˆ(k) are given in Eqs. (85)–(87). As confirmed below, the effects of u are mainly onto the spin dynamical correlation functions spectra energy bandwidths, the form of these spectra remaining for finite fields the same for the whole u>0 range. Also the line-shape singularities occurring in the vicinity of their lower thresholds are found to have the same qualitative behavior for the whole u>0 range.Our study, based on the Mott–Hubbard insulator PDT approach introduced in Section 3, confirms that in the vicinity of such lower thresholds the singularities are determined by transitions from the ne=1 and m>0 ground state to a class (ii) of excited energy eigenstates populated only by s1 pseudofermions and thus described by real BA rapidities. They are the excited energy eigenstates |fG〉=Gˆ(k)|GSf〉, Eq. (90), whose generators, Eqs. (85)–(87), give rise to specific leading-order elementary processes (A) and (B). (Here we have used the classification of Ref. [56] for the related spin-1/2 XXX chain, according to which class (ii) excitations are |Sz|=S excited states.)Consistently with the general form of the dynamical correlation functions in the pseudofermion representation, Eq. (88), higher-order class (ii) excitations described by real BA rapidities and generated by additional higher-order s1 pseudofermion elementary processes (A) also contribute to the dynamical structure factors. However, they lead to contributions in (k,ω)-plane regions other than the vicinity of the lower thresholds ωτ(k) of the longitudinal and transverse dynamical structure factors spectra and thus do not change the momentum dependent exponents obtained in this section. Moreover, class (ii) excitations described by complex BA rapidities and thus associated with excited energy eigenstates populated by sn pseudofermions of n>1 branches are gapped for h>0, their energy gap being for the spin density values considered in the studies of this section larger than the maximum lower threshold energy. Except for very small magnetic fields h these excitations have nearly vanishing spectral weight. For instance, at spin density m=0.5 and large u one estimates from the results of Ref. [12] for a directly related model that their contributions correspond to a relative intensity not larger than 10−6. This holds as well for smaller u values. All higher-order class (ii) excitations, including those described only by real BA rapidities and both by real and complex BA rapidities, respectively, refer within the pseudofermion representation to the i′>0 energy eigenstates |fG(i′)〉=Gˆi′(k)|GSf〉, Eq. (94),For simplicity, our study focuses mainly on the u>0 and m>0.25 region for which the contribution of class (ii) excited energy eigenstates populated by sn pseudofermions of n>1 branches is negligible and their energy gap is larger than the maximum lower threshold energy. (This applies as well for m>0.15.) Hence in this section we limit our analysis to h>0 subspaces spanned by energy eigenstates described by real spin rapidities Λs1(qj). For h>0 the transitions from the ground state to such excited energy eigenstates fully control the singularities of the longitudinal and transverse dynamical structure factors.In the case of the transverse dynamical structure factor, Sxx(k,ω)=14(S+−(k,ω)+S−+(k,ω)), we must consider the transitions to excited energy eigenstates that determine the line shape in the vicinity of the lower thresholds of both the dynamical structure factors S+−(k,ω) and S−+(k,ω), respectively. Indeed, the corresponding transverse dynamical structure factor spectrum ωt(k), is here expressed as the superposition of the spectra ω+−(k) and ω−+(k).The spectra ω∓±(k) that contain most of the dynamical structure factors S−+(k,ω) and S+−(k,ω) spectral weight refer to excited energy eigenstates |fG〉=Gˆ(k)|GSf〉, Eq. (90), whose generators Gˆ(k) are given in Eqs. (86) and (87), respectively. Such states are generated from the ne=1 and m>0 ground state by high-energy and finite-momentum elementary processes (A) and zero-energy and finite-momentum processes (B) that involve a δNs1,ι0,F=±1 deviation at a ι=±1 Fermi point and an overall s1 band momentum shift δqj=∓(2π/L)Φs10=∓ιπ/L where Φs10=ι/2 is the shift parameter Φsn0 given in Eq. (28) for n=1. As discussed in Sec. 3.1, the specific elementary processes (A) generated by the operators Gˆ(k) in Eqs. (86) and (87) that are associated with the spectra ω−+(k) and ω+−(k) are one s1 pseudofermion – s1 pseudofermion-hole elementary processes and two s1 pseudofermion-hole elementary processes, respectively.For spin densities m∈]0,1] the spectra generated by such processes (A) and (B) read,(118)ω−+(k)=εs1(q2)−εs1(q1);k=π+q2−q1∈]0,π[;ω+−(k)=−εs1(q1)−εs1(q2);k=π−q1−q2∈]0,π[, where εs1(q) is the energy dispersion, Eq. (34), q1∈[−kF↓,kF↓], and q2∈[−kF↑,−kF↓] for the −+ spectrum and q2∈[−kF↓,kF↓] for the +− spectrum.On the other hand, for the longitudinal dynamical structure factor Szz(k,ω), the exact line shape in the vicinity of its spectrum lower thresholds is within the Mott–Hubbard insulator PDT determined by transitions to excited energy eigenstates |fG〉=Gˆ(k)|GSf〉, Eq. (90), whose generator Gˆ(k) is given in Eq. (85). These states are generated from the ne=1 and m>0 ground state by high-energy one s1 pseudofermion – s1 pseudofermion-hole elementary processes (A) that conserve the number of down spins. The corresponding energy spectrum, ωl(k)=ωl(−k), which contains most of the longitudinal dynamical structure factor spectral weight, is for spin densities m∈]0,1] of the form,(119)ωl(k)=−εs1(q1)+εs1(q2);k=q2−q1∈]0,π[. Here q1∈[−kF↓,kF↓] and q2∈[kF↓,kF↑]. The PDT high-energy and finite-momentum elementary processes (A) associated with the dominant contributions to the longitudinal dynamical structure factor line shape are the one s1 pseudofermion – s1 pseudofermion-hole elementary processes associated with the spectrum, Eq. (119).For u>0 and both spin densities m→0 and m>m⁎≈0.15, the lower threshold of ωl(k) (and ωt(k)) coincides with a hole branch line for k∈[0,2kF↓] (and k∈[π−2kF↓,π]) and with a particle branch line for k∈[2kF↓,π] (and k∈[0,π−2kF↓]). On the other hand, for 0<m<m⁎≈0.15, the lower threshold of the spectrum ωl(k) (and ωt(k)) does not coincide with the hole branch line for a small momentum width near k=0 (and k=π). As mentioned above and for simplicity, we consider mostly spin densities m→0 and m>0.25 for which ωτ(k) coincides with branch lines and the Mott–Hubbard insulator PDT gives the exact momentum and spin density dependence of the exponents that control the line shape in its vicinity.The use of that dynamical theory reveals that the lower threshold singularities of Sxx(k,ω) are those of S−+(k,ω) near the particle branch line and of S+−(k,ω) near the hole branch line. Accounting for the s1 band energy dispersion vanishing at the Fermi points, εs1(±kF↓)=0, the longitudinal Szz(k,ω) and transverse Sxx(k,ω) hole branch lines spectra read,(120)ωhτ(k)=−εs1(q),τ=l,t,k=kF↓−q∈]0,2kF↓[,τ=l,k=π−kF↓−q∈]π−2kF↓,π[,τ=t, where q∈[−kF↓,kF↓]. The corresponding particle branch lines spectra are given by,(121)ωpτ(k)=εs1(q),τ=l,t,k=kF↓+q∈]2kF↓,π[,τ=l,k=π−kF↓+q∈]0,π−2kF↓[,τ=t, with q∈[kF↓,kF↑] and q∈[−kF↑,−kF↓] for the l and t particle branch lines, respectively.From the use of Eq. (116) one finds that the spectral-weight distribution expression for the line shape in the vicinity of the longitudinal and transverse dynamical structure factors spectrum lower threshold is of the general form,(122)Saa(k,ω)=Cτ(ω−ωτ(k))ξτ(k),k∈]0,π[, where the momentum dependent exponents are given by,(123)ξτ(k)=−1+∑ι=±1(ιδNs1F2ξs1,s11+ξs1,s11δJs1F+c0Φs1,s1(kF↓ι,q))2. Here α=z for τ=l, α=x for τ=t, and Cτ is a k and ω independent constant. Moreover, q=kF↓−k and q=−kF↓+k for the Szz(k,ω) hole and particle branch lines, respectively, whereas q=π−kF↓−k for the hole branch line and q=−π+kF↓+k for the particle branch line of Sxx(k,ω).The behavior, Eq. (122), is valid for small positive values (ω−ωτ(k)) in the vicinity of the lower thresholds ωτ(k)>0. In that equation 2Δτι(q) are the ι=±1 functionals, Eq. (107), whose specific expression 2Δτι(q)=(ιδNs1F/(2ξ1)+ξ1δJs1F+c0Φ(ιkF↓,q))2 appearing in Eq. (123) is that suitable for the line shape near the branch lines of the τ=l,t dynamical structure factors. Here ι=±1 refers to the two s1 band Fermi points, the momentum q is that of the s1 pseudofermion created (c0=1) or annihilated (c0=−1) under the transitions to the excited energy eigenstates, and the q value was above expressed in terms of the corresponding k value, which is given in Eqs. (120) and (121). Moreover, the s1 band Fermi points number deviations read δNs1F=−c0 and δJs1F=12 for the c0=1 particle and c0=−1 hole l branch lines and δNs1F=0 and δJs1F=12 for the t branch lines.It follows that for the longitudinal and transverse dynamical structure factors the ι=±1 functionals, Eq. (107), in the exponent expression, Eq. (123), suitable for the line shape near their spectrum lower threshold are given by,(124)2Δlι(q)=((ξs1,s11)2−ιc02ξs1,s11+c0Φs1,s1(ιkF↓,q))2;2Δtι(q)=(ξs1,s112+c0Φs1,s1(ιkF↓,q))2, respectively. Here q∈[−kF↓,kF↓] for c0=−1 and τ=l,t, q∈[kF↓,kF↑] for c0=1 and τ=l, and q∈[−kF↑,−kF↓] for c0=1 and τ=t.Higher-order processes (A) and (B) beyond those considered here associated with excited energy eigenstates described both only by real rapidities and real and complex rapidities, respectively, that contribute to the dynamical correlation functions overall spectral weight lead to contributions in (k,ω)-plane regions other than those in the vicinity of the (k,ω)-plane lower thresholds. Since for the spin densities m>m⁎≈0.15 the dynamical structure factors branch lines, Eqs. (120) and (121), have no (k,ω)-plane spectral weight below them, for that spin density range their corresponding line shape expressions, Eq. (122), and momentum dependent exponents, Eq. (123), are exact.The longitudinal spin spectrum ωl(k), Eq. (119), and the transverse spin spectrum ωt(k) that results from combination of the spectra ω−+(k) and ω+−(k), Eq. (118), are plotted in Figs. 2 and 3, respectively, for several values of the spin density m and on-site repulsion u. The main effect of the on-site repulsion is on these spectra energy bandwidths. Indeed and as illustrated in these figures, at fixed spin density m their form remains nearly the same for the whole u>0 range. Such on-site repulsion effects are controlled by the u dependence of the s1 energy dispersion bandwidths plotted in Fig. 1(a) and (b) as a function of 1/u for several spin density values.The corresponding exponents ξl(k) and ξt(k), Eq. (123), that control the singularities in the vicinity of the lower threshold of the longitudinal spin spectrum ωl(k) of Fig. 2 and transverse spin spectrum ωt(k) of Fig. 3 are plotted in Figs. 4 and 5, respectively, as a function of the momentum k∈]0,π[ for several values of u and spin density m. The exponent ξl(k), Eq. (123) for τ=l, is negative for k>0 at any u and m values, whereas the exponent ξt(k) given in that equation for τ=t is negative for an u and m-dependent range k∈[kt,π] where the momentum kt is for u>0 an increasing function of m. Furthermore, analysis of Fig. 4 reveals that the negative exponent ξl(k) is an increasing and decreasing function of u for the momentum ranges k∈[0,kl] and k∈[kl,π], respectively. Here kl is a spin density dependent momentum at which the exponent ξl(k) has similar value for the whole u>0 range.For the ranges of the momentum k for which the exponent ξτ(k) is negative, there are lower threshold singularity cusps in Saa(k,ω), Eq. (122). Hence analysis of Figs. 4 and 5 provides valuable information on the k ranges for which there are singularities in the lower thresholds of the dynamical structure factors Szz(k,ω) and Sxx(k,ω)=Syy(k,ω).In the m→0 limit, the spectra ωl(k) and ω+−(k), Eq. (118), reduce to their lower thresholds. At finite m, the thresholds of these two spectra correspond to different (k,ω)-plane lines. On the other hand, as m→0 they become the same (k,ω)-plane line. For finite m values the lower threshold of the spectrum ω+−(k), Eq. (118), coincides with that of ω−+(k) for k∈[π−2kF↓,π], whereas for k∈[0,π−2kF↓] it does not exist. In the m→0 limit the lower threshold of the spectrum ω+−(k) extends to the whole k∈[0,π] range and coincides with those of ωl(k) and ω−+(k). However, in contrast to the latter spectra, ω+−(k) does not reduce in that limit to its lower threshold. The spectrum of the class (ii) two s1 pseudofermion-hole excited energy eigenstates populated by s1 pseudofermions and a single s2 pseudofermion and thus described by real and complex rapidities is gapped for m>0, but in the m→0 limit becomes gapless and degenerate with that of ω−+(k).At h=0 the spectrum ω+−(k) is also that of the Sz=0 and S=1 two s1 pseudofermion-hole excitations of class (i), which due to a selection rule [56] do not contribute to the spin dynamical structure factors at h>0. Hence upon smoothly turning off h there is for the whole u>0 range a large weight transfer from |Sz|=S class (ii) excitations for h→0 to degenerate Sz=0 and S=1 class (i) two s1 pseudofermion-hole excited energy eigenstates at h=0. While the spectra change smoothly upon turning off h, from the point of view of the type of excitations behind the corresponding spin spectral weight distribution the h→0 limit is singular.For u>0 and the m→0 limit one finds from the use of Eq. (122),(125)Szz(k,ω)=Sxx(k,ω)=C(ω−ω(k))−1/2, for k∈]0,π[ where the lower thresholds ωl(k)=ωt(k)=ω(k) coincide with that of the u>0 and m=0 two – s1 pseudofermion-hole spectrum. Consistently, ξτ(k)=−1/2 is also the value of the known exponent that controls the line shape in the vicinity of the lower threshold of the latter spectrum [38,39].In the opposite limit, m→1, the lower thresholds ωτ(k) coincide with the particle branch line for all k values. Furthermore, Szz(k,ω)→0 as h→hc in the TL, the two-component Sxx(k,ω) and Szz(k,ω) dynamical structure factor being dominated by Sxx(k,ω). Here hc is the critical field associated with the spin energy scale 2μBhc, Eq. (44), at which fully polarized ferromagnetism is achieved. Due to the charge-spin recombination occurring at h=hc when the spin energy scale 2μB|h|=2μBhc, Eq. (44), and the charge Mott–Hubbard gap 2μ0, Eq. (23), reach exactly the same value, (4t)2+U2−4t, the PDT expression given in Eq. (122) for the spin dynamical structure factor is not valid, being replaced by a δ-function like distribution,(126)Sxx(k,ω)=π2δ(ω−εs1(π−k)),k∈]0,π[, for αα=xx and by Szz(k,ω)=0 for αα=zz where the energy dispersion εs1(q) reads,(127)εs1(q)=−2tπ∫−ππdksinkarctan(sink−Λs10(q)u)+(4t)2+U2−U,q=−1π∫−ππdkarctan(sink−Λs10(q)u), and the second expression given here defines the ground-state rapidity function Λs10(q) in terms of its inverse function.Finally, the line shapes, Eq. (122), and corresponding exponents ξτ(k), Eq. (123), do not apply at and near the ω=0 lower threshold soft modes such as (k0τ,0) where k0l=2kF↓ and k0t=π−2kF↓ in the (k,ω)-plane. In this low-energy case the spin part of the half-filled 1D Hubbard model spectrum can be described by a Gaussian field theory with central charge c=1 [20]. The present more general Mott–Hubbard insulator PDT reaches in the low-energy limit the same results as that Gaussian field theory. (Also the metallic-phase PDT describes the known low-energy expressions [29].) Indeed, near the points (k0τ,0) the PDT expressions given in Eq. (117) apply, the two ι=±1 functionals, Eq. (107), becoming the ι=±1 operator dimensions of that theory,(128)2Δlι=(ξs1,s11)2;2Δtι=(ι2ξs1,s11−ξs1,s11)2.For the case of the longitudinal and transverse spin dynamical structure factors considered in this section the general expressions provided in Eq. (117) lead for the finite-weight region above the (k0τ,0) plane points for which the low excitation energy ω is not in the vicinity of the low-energy thresholds ω≈±vs1(k−k0τ), to the following line shape,(129)Saa(k,ω)=C0τ|ω|ζ0τ,ω≠±vs1(k−k0τ), where C0τ is a constant and the exponent reads ζ0l=−2+∑ι=±12Δlι, which gives,(130)ζ0l=−2(1−(ξs1,s11)2);ζ0t=−2(1−14(ξs1,s11)2−(ξs1,s11)2).On the other hand, according to the general expressions given in Eq. (117) for low excitation energy ω near the low-energy thresholds, ω≈±vs1(k−k0τ), the spin dynamical correlation functions line shape is rather of the form,(131)Saa(k,ω)=C−1τ(ω+vs1(k−k0τ))ζ−1τ,k<k0τ,τ=l,t, for ω≈−vs1(k−k0τ) and,(132)Saa(k,ω)=C+1τ(ω−vs1(k−k0τ))ζ+1τ,k>k0τ,τ=l,t, for ω≈+vs1(k−k0τ) where C±1τ are constants and the exponents are given by ζ±1τ=−1+2Δl±1 and thus read,(133)ζ±1l=−1+(ξs1,s11)2;ζ±1t=−1∓1+14(ξs1,s11)2+(ξs1,s11)2.5Concluding remarksThe PDT reported in Refs. [28–30] for the metallic-phase of the 1D Hubbard model does not apply to the spin dynamical correlation functions of the half-filled 1D Hubbard model. In this paper we have introduced a modified PDT that applies to the latter problem. This has allowed to study the line shape of singularities in the vicinity of the lower thresholds of the model longitudinal and transverse dynamical spin structure factors, Eq. (1).Specifically, the exact momentum dependence of the exponents, Eq. (123), that control such line shapes in the TL was derived. The corresponding exact line-shape of the structure form factors Szz(k,ω) and Sxx(k,ω) is reported in Eq. (122). Importantly, for the k ranges for which the τ=l,t exponents ξτ(k) given in Eq. (123) (which are plotted in Figs. 4 and 5) are negative, there are lower threshold singularity cusps in the corresponding τ=l,t spin dynamical structure factors. Our results on these form factors of the Mott–Hubbard insulator phase of the 1D Hubbard model at finite magnetic fields h>0 refer to the TL. To our knowledge, no previous investigations accessed the corresponding exact spin density and momentum dependence of the exponents that control the singularities of the longitudinal and transverse spin structure form factors in the vicinity of their lower thresholds.Mott–Hubbard insulators are in 1D a paradigm for the importance of strong correlations and are known to exhibit a wide variety of unusual physical phenomena. Experimental realizations within condensed matter include inelastic neutron scattering in chain cuprates and a number of organic compounds [41,59]. In the limit of very strong on-site repulsion U the spin degrees of freedom of such condensed-matter systems are commonly modeled by the spin-1/2 XXX chain [41]. However, for general Mott–Hubbard insulating materials there is no reason for the on-site repulsion to be much stronger than the electron hopping amplitude t. This situation is realized in the Bechgaard salts [59]. A question that arises then is how electron itinerancy affects the spin dynamics. As discussed in Section 4, the analysis of the u dependences of the dynamical spin structure factor spectra and corresponding exponents that control the line shape near such spectra lower thresholds plotted in Figs. 2–5 provides important information on that issue.An interesting experimental possibility is the potential observation of the spin dynamical structure factors peaks we predict in inelastic neutron scattering experiments on actual spin-chain compounds. The structure form factors Szz(k,ω) and Sxx(k,ω) may be investigated separately in h>0 experiments on such compounds by using a carefully oriented crystal. If the crystal is misoriented, or if a micro crystalline sample is used, the Szz(k,ω) and Sxx(k,ω) spectral features should appear superimposed. Such superimposition changes the excitations lower thresholds and leads to the broadening of the singularities, Eq. (122). However, this does not occur at h=0, since Szz(k,ω)=Sxx(k,ω). These two different situations are clearly seen in the magnetic scattering intensity measured at zero- and finite-field inelastic neutron scattering experiments of Ref. [41] on a spin-chain compound, respectively, (See Figs. 2 (a)–(c) of that reference.) We suggest that more demanding h>0 experiments with a carefully oriented crystal be carried out on spin-chain compounds. This should yield separately Szz(k,ω) and Sxx(k,ω) whose magnetic scattering intensities are expected to display the cusp singularities found theoretically in this paper.On the other hand, the recent progress in implementing the present repulsive Hubbard model with ultra-cold atoms on optical lattices has led to the observation of the Mott–Hubbard insulating state studied in this paper [60]. Another interesting program would be the observation of the spin spectral weight distributions over the (k,ω) plane associated with the dynamical correlation functions studied in this paper in systems of ultra-cold atoms on optical lattices.AcknowledgementsWe thank D.K. Campbell, A. Moreno, and P.D. Sacramento for illuminating discussions and the support by the Beijing CSRC and the FEDER through the COMPETE Program and the Portuguese FCT in the framework of the Strategic Projects PEST-C/FIS/UI0607/2013, PEst-OE/FIS/UI0091/2014, and UID/CTM/04540/2013.Appendix AThree fractionalized particles emerging from the rotated-electron separation for the 1D Hubbard model in its full Hilbert spaceHere general operational expressions valid for the 1D Hubbard model in its full Hilbert space of the rotated-spin 1/2, rotated-η-spin 1/2, and c pseudoparticle operators in terms of rotated-electron operators and of the latter operators in terms of the former are provided.The rotated-electron operators, Eq. (51), can also be defined for the 1D Hubbard model in its full Hilbert space. The corresponding extension of the electron–rotated-electron unitary operator Vˆ definition in Eq. (52) accounts for the occupancy configurations associated with the η-spin degrees of freedom [61]. The c pseudoparticle operators that emerge from such generalized rotated-electron operators are then given by,(A.1)fj,c†=(fj,c)†=c˜j,↑†(1−n˜j,↓)+(−1)jc˜j,↑n˜j,↓;nj,c=fj,c†fj,c,j=1,…,L, where n˜j,σ is given in Eq. (51).The creation operator fj,c† and annihilation operator fj,c create and annihilate, respectively, one c pseudoparticle at the j=1,…,L site of the c effective lattice, which is identical to the model original lattice. The corresponding momentum-dependent c pseudoparticle operators fqj,c†=(fqj,c)† are then given by Eq. (77) but with the operator fj′,c† being that in Eq. (A.1). Furthermore, on combining Eqs. (A.1) and (77), the c pseudofermion operators, Eq. (64) for β=c, can be formally expressed in terms of rotated-electron operators as,(A.2)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†)†.The three electron-rotated local operators S˜j,ηl and three electron-rotated local operators S˜j,sl and corresponding six generators S˜αl of the global η-spin and spin SU(2) symmetry algebras may be written as,(A.3)S˜j,ηl=(1−nj,c)q˜jl;S˜j,sl=nj,cq˜jl;S˜αl=∑j=1Ls˜j,αl,α=η,s,l=z,±, respectively. Here nj,c is the c pseudoparticle local density operator, Eq. (A.1). The ηs quasi-spin operators,(A.4)q˜jl=S˜j,sl+S˜j,ηl,l=±,z, such that q˜j±=q˜jx±iq˜jy and q˜jz, where x,y,z denote the Cartesian coordinates, have the following expressions in terms of rotated-electron creation and annihilation operators,(A.5)q˜j−=(q˜j+)†=(c˜j,↑†+(−1)jc˜j,↑)c˜j,↓;q˜jz=(n˜j,↓−1/2).Inversion of the relations, Eqs. (A.1) and (A.5), along with the use of Eq. 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