NUPHB14312S0550-3213(18)30086-510.1016/j.nuclphysb.2018.03.011The Author(s)Quantum Field Theory and Statistical SystemsAbsence of ballistic charge transport in the half-filled 1D Hubbard modelJ.M.P.Carmeloabc⁎carmelo@fisica.uminho.ptcarmelo@MIT.EDUS.NematicbT.ProsendaDepartment 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 CenterBeijing100193ChinadDepartment of Physics, FMF, University of Ljubljana, Jadranska 19, 1000 Ljubljana, SloveniaDepartment of PhysicsFMFUniversity of LjubljanaJadranska 19Ljubljana1000Slovenia⁎Corresponding author.Editor: Hubert SaleurAbstractWhether in the thermodynamic limit of lattice length L→∞, hole concentration mηz=−2Sηz/L=1−ne→0, nonzero temperature T>0, and U/t>0 the charge stiffness of the 1D Hubbard model with first neighbor transfer integral t and on-site repulsion U is finite or vanishes and thus whether there is or there is no ballistic charge transport, respectively, remains an unsolved and controversial issue, as different approaches yield contradictory results. (Here Sηz=−(L−Ne)/2 is the η-spin projection and ne=Ne/L the electronic density.) In this paper we provide an upper bound on the charge stiffness and show that (similarly as at zero temperature), for T>0 and U/t>0 it vanishes for mηz→0 within the canonical ensemble in the thermodynamic limit L→∞. Moreover, we show that at high temperature T→∞ the charge stiffness vanishes as well within the grand-canonical ensemble for L→∞ and chemical potential μ→μu where (μ−μu)≥0 and 2μu is the Mott–Hubbard gap. The lack of charge ballistic transport indicates that charge transport at finite temperatures is dominated by a diffusive contribution. Our scheme uses a suitable exact representation of the electrons in terms of rotated electrons for which the numbers of singly occupied and doubly occupied lattice sites are good quantum numbers for U/t>0. In contrast to often less controllable numerical studies, the use of such a representation reveals the carriers that couple to the charge probes and provides useful physical information on the microscopic processes behind the exotic charge transport properties of the 1D electronic correlated system under study.1IntroductionLikely, the most widely studied correlated electronic model on a lattice in one (spatial) dimension (1D) is the Hubbard model with first neighbor transfer integral t and on-site repulsion U. In spite of being solvable by the Bethe ansatz (BA) [1–10], in the case of electronic density ne=Ne/L=1 its unusual charge transport properties remain poorly understood at finite temperatures T>0 [11–18]. This includes, specifically, some of the behaviors of the real part of charge conductivity at finite temperature T whose general form reads,(1)σ(ω,T)=2πD(T)δ(ω)+σreg(ω,T). Even for the T=0 Mott–Hubbard insulating quantum phase, the related charge dynamic structure factor is a complex problem that is only partially understood [19].The charge stiffness or Drude weight D(T) in Eq. (1) characterizes the response to a static field and σreg(ω,T) describes the absorption of light of frequency ω. For T>0 these quantities can be written as,(2)D(T)=12TL∑νpν∑ν′(ϵν=ϵν′)|〈ν,u|Jˆ|ν′,u〉|2, and(3)σreg(ω,T)=πL1−e−ωTω∑νpν∑ν′(ϵν≠ϵν′)|〈ν,u|Jˆ|ν′,u〉|2δ(ω−ϵν′+ϵν), respectively. In these equations and elsewhere in this paper units of Boltzmann constant kB, Planck constant ħ, and lattice spacing a one are generally used. Moreover, L→∞ denotes the system length in the thermodynamic limit (TL), which within the units of lattice constant one equals the (even) number of lattice sites Na, |ν,u〉 are energy and momentum eigenstates, ν stands for all quantum numbers other than the parameter,(4)u=U4t, needed to uniquely specify each such a state, the sum runs over states with the same energy eigenvalue, ϵν=ϵν′, pν=e−ϵν/T/Z is the usual Boltzmann weight, Z=∑νe−ϵν/T, and Jˆ is the charge current operator. (Its specific expression for the present model is given below in Section 2.)The studies of this paper rely in part on the BA solution of the 1D Hubbard model. It was solved first by the so-called coordinate BA [1,2], which provided the ground state energy and revealed that the model undergoes a Mott metal-insulator transition at electronic density ne=1 whose corresponding critical onsite interaction is U=0. Which are the effects of a finite temperature on such a transition is one of the issues studied in this paper.Following the coordinate BA solution, the ground state properties [20–22] and the excitation spectrum [6,7,23–27] were studied by several authors. The 1D Hubbard model termodynamic Bethe ansatz (TBA) and corresponding ideal strings have been proposed in Ref. [4]. This has allowed the study of the thermodynamic properties of the model [28,29]. The energy spectra of its elementary excitations can be obtained from the TBA equations in the zero temperature limit [9].An important property of the 1D Hubbard model is that its spectrum becomes conformal invariant in the low-energy limit. The corresponding finite-size corrections were obtained in Refs. [30,31]. The relation between the finite-size spectrum and the asymptotic behavior of correlation functions was used to calculate the critical exponents of the general two-point correlation functions [32,33]. The corresponding conformal dimensions have been expressed in terms of dressed phase shifts associated with a preliminary pseudoparticle representation [34–40].The conformal approach is not applicable to the zero-temperature model Mott insulating phase at half filling. In the small-U and scaling limits, dynamical correlation functions at low energies [41–44] can though be computed relying on the methods of integrable quantum field theory [45–47]. The wave functions of the energy eigenstates can be extracted from the coordinate BA solution. An explicit representation for the wave functions was given in Ref. [7].In the u=U/4t→∞ limit the dynamical correlation functions can be computed at zero temperature for all energy scales relying on the simplified form that the BA equations acquire. This was achieved by a combination of analytical and numerical techniques for the whole range of electronic densities [48–59]. In the case of the one-electron spectral function studies of Refs. [54–56], the method relies on the spinless-fermion phase shifts imposed by XXX chain physical spins 1/2. Such fractionalized particles naturally arise from the zero spin density and u→∞ electron wave-function factorization [6,7,48]. A related pseudofermion dynamical theory 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. [60] was introduced in Ref. [61]. It is an extension of the u→∞ method of Refs. [54–56] to the whole u>0 range of the 1D Hubbard model. The use of the mobile quantum impurity model [62,63], which has been developed to also tackle the high-energy physics of both integrable and non-integrable 1D correlated quantum problems, leads in the case of the 1D Hubbard model to the same results as the pseudofermion dynamical theory [64,65]. Further general information on the 1D Hubbard model is given in Ref. [10].Provided that the energy eigenstates |ν,u〉 are as well momentum eigenstates, it is well known [13,66] that for u>0, T>0, and in the TL the charge stiffness expression, Eq. (2), further simplifies to,(5)D(T)=12TL∑νpν|〈ν,u|Jˆ|ν,u〉|2foru>0. Within that limit, this expression is not valid in the T=0 regime though. The T=0 charge stiffness is actually known [35,67], reading D(0)=(2t/π)δU,0 at hole concentration mηz=−2Sηz/L=1−ne=0 (half filling) where the η-spin z component Sηz=−(L−Ne)/2 is the eigenvalue of the diagonal generator of the global η-spin SU(2) symmetry. Hence at T>0 it is finite at U=0 and vanishes for the whole u>0 range.A finite D(T) for T>0 value would imply the occurrence of ballistic charge transport. At T>0 the model can behave as an ideal conductor with ballistic charge transport and thus D(T)>0 or a system without such a ballistic transport, so that D(T)=0. In the latter case there are two scenarios, the system behaving as a normal resistor if D(T)=0 and the diffusive conductivity contribution σ0=limω→0σreg(ω,T) is finite or as an ideal insulator with D(T)=σ0=0 [11,66,68].On the one hand, a D(T) inequality, which is derived from the more general Mazur's inequality [69,70], provides for hole concentrations mηz≠0 a finite lower bound for its value [66,71]. This reveals that D(T)>0 for mηz≠0 and finite temperature [13,14]. On the other hand, at mηz=0 that lower bound vanishes, so that the inequality is inconclusive. Whether in the TL and for u>0 the charge stiffness D(T) vanishes or is finite for T>0 and mηz=0 remains actually an open and controversial issue, as different approaches yield contradictory results [11–18].The results of this paper provide strong evidence that the predictions of Ref. [12] for the 1D Hubbard model charge stiffness for T>0 and mηz=0 are not correct. This is consistent with the numerical results of Ref. [14] and the large-u studies of Ref. [13]. The latter results are reached by two completely different methods: an exact method that does not rely on the BA and a TBA calculation [4], respectively. These studies reveal that the finite charge stiffness expression found in Ref. [12] for mηz=0 and T>0 cannot be correct for large u>0. The results of Refs. [13,14] agree with some preliminary conjectures by Zotos and Prelovšek according to which limu→∞D(T) should be zero for the 1D Hubbard model at mηz=0 and T>0.Recently, a general formalism of hydrodynamics for the 1D Hubbard model and other integrable models was introduced in Refs. [17,18]. By linearizing hydrodynamic equations, the closed-form expressions for the stiffnesses that were conjectured to be valid on the hydrodynamic scale have been accessed. The stiffness is then calculated from the stationary currents generated in an inhomogeneous quench from bipartitioned initial states [17]. Within such an hydrodynamic ansatz for the stiffnesses, the studies of Refs. [17,18] clearly established vanishing at finite temperature of charge or spin Drude weights when the corresponding chemical potentials vanish, irrespective of the interaction strength. In our work we, however, take a different perspective. We start from the standard linear-response expressions for the charge and spin Drude weights and reach conclusions that are consistent with the results of Ref. [18]. Although there is no reasonable doubt that the hydrodynamics ansatz used in Refs. [17,18] is correct, it has, nevertheless, not yet been rigorously justified. Hence we believe that adding our independent and complementary result is a valuable contribution to the solution of this important problem. Actually, both methods rely on the standard assumptions behind the TBA.Our previous results reported in Refs. [72] and [73], which have been obtained by the method used in this paper, provide strong evidence that in the case of the spin stiffness of the spin-1/2 XXX chain the approach of Ref. [12] used in the investigations of Ref. [74] leads to correct results. Specifically, that such a stiffness vanishes in the limit of zero spin density. (The apparent inconsistency that the use of the approach of Ref. [12] leads to misleading results for the 1D Hubbard model and to correct results for that spin chain is an issue discussed below in Section 7.)Our method to compute suitable upper bounds for the charge stiffness relies in part on the properties of the charge current operator matrix elements between energy and momentum eigenstates that follow from the η-spin SU(2) symmetry operator algebra. This is similar to the method used in Refs. [72] and [73] for the spin stiffness of the spin-1/2 XXX chain in what its relation to its spin SU(2) symmetry operator algebra is concerned. The method combines the TBA [4] with stiffness expressions in terms of current operator expectation values. It accounts though for the effects of complex-rapidity string deviations [9] and does not access the charge stiffness through the second derivative of the energy eigenvalues of the TBA relative to a uniform vector potential [12].In the case of energy eigenstates of spin S of the spin-1/2 XXX chain, there are L−2S spins 1/2 that are paired within singlet configurations and 2S spins 1/2 that remain unpaired and contribute to the multiplet configurations. The spin degrees of freedom couple to a vector potential through such 2S unpaired spins 1/2, which are those that contribute to the spin currents.Within the rotated-electron related representation of the 1D Hubbard model used in our studies, there emerge from the rotated-electrons η-spin degrees of freedom basic fractionalized particles of η-spin 1/2 that are associated with η-spin SU(2) symmetry of the model. Again, in the case of energy eigenstates of η-spin Sη there is a number 2Sη of η-spin 1/2 fractionalized particles that couple to a vector potential, which are those that participate in η-spin multiplet configurations and contribute to the charge currents.A trivial result is that at U=0 the global symmetry of the Hubbard model on a bipartite lattice is O(4)=SO(4)⊗Z2. This thus applies to the 1D Hubbard model. Here the factor Z2 refers to the Shiba particle-hole transformation on a single spin under which the Hamiltonian is not invariant for U≠0 and SO(4)=[SU(2)⊗SU(2)]/Z2 contains the two SU(2) symmetries. An exact result of Heilmann and Lieb is that in addition to the spin SU(2) symmetry, also for u>0 the model has a second global SU(2) symmetry [75]. It is generally called η-spin symmetry [76,77]. Yang and Zhang considered the most natural possibility that the SO(4) symmetry inherited from the U=0 Hamiltonian O(4)=SO(4)⊗Z2 symmetry is the model's global symmetry for U>0 [77]. The energy and momentum eigenstates are either lowest weight states (LWSs) or highest weight states with respect to the two SU(2) symmetry algebras [75–78]. The non-LWSs can be generated from the LWSs explicitly accounted for by the BA solution, which confirmed the completeness of the quantum problem [79–81].At half-filling and zero spin density the 1D Hubbard model TBA dressed phase shifts and the corresponding S-matrices have been associated with fractionalized particles called holon, antiholon, and spinon. The holon and antiholon have been inherently constructed to have zero spin and charge +e and −e, respectively. The spinon has been inherently constructed to have no charge and to have spin 1/2 [82,83]. The model SO(4) symmetry group state representations were identified with occupancy configurations of such fractionalized particles.The solution of the model by the quantum inverse scattering method has provided further information on its symmetries. The first steps to obtain such a solution were made in Refs. [84–86]. The model Hamiltoninan was mapped under a Jordan–Wigner transformation into a spin Hamiltonian. It commutes with the transfer matrix of a related covering vertex model [84]. The R-matrix of the spin model was also derived [85,86]. Alternative derivations were carried out by several authors [87–89]. The R-matrix was later shown to satisfy the Yang–Baxter equation [90]. An algebraic BA having as starting point the results of Refs. [84–86] was afterwards constructed in Refs. [91,92]. The expressions for the eigenvalues of the transfer matrix of the two-dimensional statistical covering model were obtained. That problem was also addressed in Ref. [93].The algebraic BA introduced in Refs. [91,92] allowed the quantum transfer matrix approach to the thermodynamics of the 1D Hubbard model [94]. Within it, the thermodynamic quantities and correlation lengths can be calculated numerically for finite temperatures [95,96]. The 1D Hubbard model Hamiltonian was found in the TL to be invariant under the direct sum of two Y(sl(2)) Yangians [97]. The relation of these Yangians to the above R-matrix and the implications of one of these Yangians for the structure of the bare excitations was later clarified [98,99]. More recently, it was demonstrated that the Yangian symmetries of the R-matrix specialize to the Yangian symmetry of the model and that its Hamiltonian has an algebraic interpretation as the so-called secret symmetry [100].It was found in Ref. [101] that for u>0 the 1D Hubbard model global symmetry is actually larger than SO(4) and given by [SO(4)⊗U(1)]/Z2, and thus equivalently to [SU(2)⊗SU(2)⊗U(1)]/Z22. (This applies as well to the model on any bipartite lattice.) Consistently with the model's extended global symmetry, the quantum inverse scattering method spin and charge monodromy matrices were found to have different ABCD and ABCDF forms, respectively. Those are actually associated with the spin SU(2) and charge U(2)=SU(2)⊗U(1) symmetries, respectively [92]. The latter matrix is larger than the former and involves more fields [92]. If the global symmetry was only SO(4)=[SU(2)⊗SU(2)]/Z2, the charge and spin monodromy matrices would have the same traditional ABCD form, which is that of the spin-1/2 XXX chain [102].The exact rotated-electron representation used in our studies is that suitable for the further understanding of this basic similarity between the spin SU(2) symmetry degrees of freedom of the spin-1/2 XXX chain type of configurations that contribute to spin transport and the 1D Hubbard model η-spin SU(2) symmetry degrees of freedom type of configurations that contribute to charge transport. The rotated electrons are inherently constructed to their numbers of singly occupied and doubly occupied lattice sites being good quantum numbers for u>0. As further discussed below in Section 2.3, the form of the 1D Hubbard model energy and momentum eigenstates wave function for u→∞ derived in Ref. [7] reveals that in that limit such a model corresponds to a spin-1/2 XXX chain, an η-spin-1/2 XXX chain, and a quantum problem with simple lattice U(1) symmetry, respectively. In terms of the rotated electrons, whose relation to the electrons has been uniquely defined in Ref. [103], the energy and momentum eigenstates wave function has that form for the whole u>0 range.The degrees of freedom of the rotated electrons naturally separate into two fractionalized particles with spin 1/2 and η-spin 1/2, respectively, plus one basic fractionalized particle without internal degrees of freedom [103,104]. (The η-spin projections +1/2 and −1/2 refer to the η-spin degrees of freedom of the rotated-electron unoccupied and doubly-occupied sites, respectively.) The occupancy configurations of these three basic fractionalized particles generate exact state representations of the group associated with the spin SU(2) symmetry, η-spin SU(2) symmetry, and c lattice U(1) symmetry, respectively, in the global [SU(2)⊗SU(2)⊗U(1)]/Z22 symmetry of the model [101].In the case of the spin-1/2 XXX chain, the translational degrees of freedom of the 2S unpaired spins 1/2 that contribute to the spin currents are described by an average number 2S of holes in each TBA n-band with finite occupancy. Here n=1,...,∞ is the number of singlet pairs bound within each of the n-band pseudoparticles considered in Ref. [73] that populate such a band. The n-band pseudoparticles occupancies generate the singlet configurations of the spin SU(2) symmetry group state representations.Also in the case of the 1D Hubbard model charge transport, the translational degrees of freedom of the 2Sη unpaired η-spin 1/2 fractionalized particles that contribute to the charge currents are found in this paper to be described by an average number 2Sη of holes in each TBA ηn-band with finite occupancy. For that model, the corresponding ηn pseudoparticles occupancies generate the η-spin singlet configurations of the η-spin SU(2) symmetry group state representations. The difference relative to the spin-1/2 XXX chain refers to contributions from the holes in the charge band of the above mentioned basic fractionalized particles without internal degrees of freedom whose occupancy configurations generate state representations of the group associated with the c lattice U(1) symmetry. Indeed, an average number 2Sη of such holes holes also contributes to the translational degrees of freedom of the 2Sη unpaired η-spin 1/2 fractionalized particles that couple to the charge probes. This is related to the above mentioned U(2)=SU(2)⊗U(1) symmetry in the model's [SU(2)⊗SU(2)⊗U(1)]/Z22 global symmetry referring to the charge degrees of freedom. (The remaining SU(2) symmetry refers to the spin degrees of freedom.) Indeed, that charge U(2)=SU(2)⊗U(1) symmetry includes the η-spin SU(2) symmetry and the c lattice U(1) symmetry beyond SO(4)=[SU(2)⊗SU(2)]/Z2.The use of the above mentioned holon and spinon representations [15,82,83] provides a suitable description of the model both at low excitation energy relative to a ground state and more generally in subspaces spanned by energy and momentum eigenstates described by a vanishing density of both TBA complex rapidities and η-spin strings of length one [4]. In the case of the 1D Hubbard model, such holons and spinons are different from the three fractionalized particles that naturally emerge from the exact rotated-electron degrees of freedom separation. The latter have operators that have simple expressions in terms of rotated-electron operators and are defined for the 1D Hubbard model in its full Hilbert space [103].The charge stiffness problem under study in this paper involves summations that run over all energy and momentum eigenstates. This is why the holon and/or spinon (and anti-spinon) representations are not suitable to study it. For instance, the phenomenological method in terms of a spinon and anti-spinon particle basis used in Ref. [15] leads to a misleading large spin stiffness for the spin-1/2 XXX chain in the limit of zero spin density. The validity of that result is excluded by the careful investigations of Ref. [71], which indicate that transport at finite temperatures is dominated by a diffusive contribution, the spin stiffness being very small or zero. They are also excluded by the studies of Refs. [72,73] and the TBA results of Ref. [74], which find a vanishing spin stiffness within the zero spin density limit in the TL.We emphasize that the electrons and the rotated electrons are for u>0 related by a mere unitary transformation under which the electronic charge and spin degrees of freedom remain invariant. Hence a rotated electron carries the same charge and has the same spin 1/2 as an electron. Indeed, such a unitary transformation only changes the lattice occupancies and corresponding spatial distributions of the charges and spins 1/2. The relation of the rotated electrons to the rotated spins 1/2, rotated η-spins 1/2, and c pseudoparticles is direct. It is uniquely defined for the full Hilbert space spanned by a complete set of 4L energy and momentum eigenstates [103,104]. The corresponding representation of the 1D Hubbard model in terms of such fractionalized particles is thus faithful in that space.The holons and spinons are related to such fractionalized particles for the 1D Hubbard model in some reduced subspaces mentioned above for which they correspond as well to a faithful representation. However, the representation in terms of holons and spinons is only defined for the model in such subspaces. This is why in our studies we rather use the representation in terms of the fractionalized particles that naturally emerge from the separation of the rotated-electrons degrees of freedom.In the u→∞ limit the rotated electrons become electrons and the c pseudoparticles and rotated spins 1/2 of the representation used in the studies of this paper become the spinless fermions and XXX chain spins 1/2, respectively, of Refs. [48,49,54–56]. As mentioned above, such fractionalized particles naturally emerge from the u→∞ electron wave-function factorization [6,7]. That factorization includes a third factor [7] associated with the η-spin SU(2) symmetry. It corresponds to the u→∞ limit of the rotated η-spins 1/2 of the u>0 representation used in this paper.In summary, there are two main reasons why we use in our study the representation of the rotated-electron related three fractionalized particles. Given their simple and direct relation to the rotated-electrons charge and spin degrees of freedom, it allows a more clear physical description of the microscopic processes that control the charge properties under study. This is consistent with each of the set of 4L energy and momentum eigenstates that span the model Hilbert space being generated from the electron and rotated-electron vacuum by occupancy configurations of the three types of fractionalized particles under consideration that are much simpler than those in terms of electrons. A second reason is that, in contrast to the usual holon and spinon representation, that representation is defined for the model in its full Hilbert space. The holon and spinon representation applies for instance to low-energy problems whereas here we consider all ranges of temperatures.Our study refers to zero spin projection, Ssz=0. It addresses the problem of the charge stiffness of the 1D Hubbard model in the TL within the canonical ensemble at hole concentration mηz=0 and for mηz→0 at temperatures T>0. Within that ensemble for T>0 we find that the charge stiffness vanishes as mηz→0 for fixed total η-spin projection Sηz, including Sηz=0, at least as fast as,(6)D(T)≤cct2L2T(mηz)2, where cc is a L-independent constant that smoothly varies as a function of u for the whole u>0 range. A similar result is also reached for a canonical ensemble near the η-spin fully polarized sector of maximal hole concentration mηz=1,(7)D(T)≤cc′t2L2T(1−mηz)2, where cc′ is found to be independent of u for u>0.That for finite temperatures our results partially resolve the charge stiffness behavior of the 1D Hubbard model as mηz→0 stems in part from the fact that they leave out, marginally, the grand canonical ensemble in which 〈(mηz)2〉=O(1/L). (While for a canonical ensemble one considers that the η-spin density mηz is kept constant, in the case of a grand-canonical ensemble it is the chemical potential μ that is fixed.)However, for the canonical ensemble our study relies on a charge stiffness upper bound whose derivation involves a large overestimation of the elementary charge currents of the energy and momentum eigenstates. Hence accounting for the usual expectation of the equivalence of the canonical and grand canonical ensembles in the TL, one would expect that our results remain valid in the latter grand canonical case for any finite temperature T>0. The canonical-ensemble and grand-canonical ensembles lead indeed in general to the same results in the TL except near a phase transition or a critical point. Since a quantum phase transition from a metallic state to a Mott–Hubbard insulator occurs in the u>0 1D Hubbard model as mηz→0 and μ→μu for (μ−μu)≥0 where 2μu is the Mott–Hubbard gap [1–3], Eq. (A.9) of Appendix A, this issue deserves the careful analysis in these limits carried out in this paper.We have addressed such an issue in the limit of high temperatures T→∞ for which strong evidence is provided that the charge stiffness indeed also vanishes within the grand-canonical ensemble for chemical potential μ such that (μ−μu)≥0 in the μ→μu limit. Specifically, within that ensemble for T→∞ we find that the charge stiffness vanishes as mηz→0, at least as fast as,(8)D(T)≤cgct22T(mηz)2, where cgc is again a L-independent constant that smoothly varies as a function of u. A similar result is also reached for a grand-canonical ensemble near the η-spin fully polarized sector of maximal hole concentration mηz=1,(9)D(T)≤cgc′t22T(1−mηz), where cgc′ is found to be independent of u up to O(u−2) order. That the upper bounds on the right-hand side of Eqs. (6) and (7) have an extra factor L as compared to those on the right-hand side of Eqs. (8) and (9) confirms the large overestimation of the elementary charge currents used in the case of the canonical ensemble. The found lack of ballistic transport in the half-filled 1D Hubbard model indicates that charge transport at finite temperatures is dominated by a diffusive contribution [105].The paper is organized as follows. The 1D Hubbard model, its energy and momentum eigenstates, symmetry, and the rotated-electron representation are the topics addressed in Section 2. In Section 3 useful subspaces for our charge current absolute values upper bounds and charge stiffness upper bounds studies are considered and expressions for the charge current operator expectation values are obtained. Useful upper bounds for absolute values of the charge current are then introduced in Section 4. In Section 5 a related charge stiffness upper bound is constructed within the canonical ensemble. Moreover, a charge stiffness upper bound is introduced in Section 6 within the grand-canonical ensemble for T→∞. Finally, the concluding remarks are presented in Section 7.2The model, energy eigenstates, the rotated-electron representation, and symmetryThe goal of this section is the introduction of the rotated-electron related representation used in our study of the expectation values of the charge current operator and charge stiffness in the 1D Hubbard model. Its relatively large length is justified by the complexity of the problem. However, the use of the representation introduced in this section simplifies the description in later sections of the model charge transport properties. Importantly, it has been inherently constructed to be that suitable to clarify the issue of the microscopic mechanisms behind such exotic properties.2.1The 1D Hubbard model, its energy eigenstates, and the rotated-electron representationWe consider the 1D Hubbard model Hamiltonian under periodic boundary conditions in the TL and in a chemical potential μ,(10)Hˆ=−t∑σ∑j=1L[cj,σ†cj+1,σ+h.c.]+U∑j=1Lρˆj,↑ρˆj,↓+2μSˆηz. It describes Ne electrons in a lattice with Na=L sites. Here cj,σ† creates one electron of spin projection σ=↑,↓ at site j=1,...,L, ρˆj,σ=(nˆj,σ−1/2), nˆj,σ=cj,σ†cj,σ, and Sˆηz=−12∑j=1L(1−nˆj) with nˆj=∑σnˆj,σ is the diagonal generator of the global η-spin SU(2) symmetry.The z-component η-spin current operator Jˆηz and charge current operator Jˆρ are closely related as follows,(11)Jˆηz=(1/2)JˆandJˆρ=(e)Jˆ,whereJˆ=−it∑σ∑j=1L[cj,σ†cj+1,σ−cj+1,σ†cj,σ], and e denotes the electronic charge. Hence, except for a constant pre-factor, the charge current operator Jˆρ equals the η-spin current operator Jˆηz. For simplicity, in several general expressions we use units such that Jˆηz=Jˆρ=Jˆ. We thus call Jˆ, Eq. (11), the charge current operator.Within the exact representation of the u>0 1D Hubbard model in terms of rotated electrons used in our study, the operators that create and annihilate such rotated electrons are related to the corresponding electron operators as follows,(12)c˜j,σ†=Vˆ†cj,σ†Vˆ;c˜j,σ=Vˆ†cj,σVˆ;n˜j,σ=c˜j,σ†c˜j,σwherej=1,...,Landσ=↑,↓. Here Vˆ is the electron – rotated-electron unitary operator uniquely defined in Eq. (11) of Ref. [103] in terms of the 4L×4L matrix elements between a complete set of 4L u>0 energy and momentum eigenstates of the 1D Hubbard model. For all these 4L states the number Ns,±1/2R of spin projection ±1/2 rotated-electron singly occupied sites, Nη,−1/2R of η-spin projection −1/2 rotated-electron doubly occupied sites, and Nη,+1/2R of η-spin projection +1/2 rotated-electron unoccupied sites are good quantum numbers for u>0 [103,104]. Hence the number NsR=Ns,+1/2R+Ns,−1/2R of rotated-electron singly occupied sites and NηR=Nη,+1/2R+Nη,−1/2R of rotated-electron unoccupied plus doubly occupied sites are conserved for u>0 as well.Our choice of energy and momentum eigenstates |ν,u〉 in Eqs. (2)–(5) is different at u=0 and for u>0. For u>0, the energy and momentum eigenstates associated with the exact BA solution are chosen along with those generated from application onto them of the off-diagonal generators of the global η-spin SU(2) and spin SU(2) operator algebras symmetries. As reported in Section 1, for u>0 the 1D Hubbard model global symmetry is [SU(2)⊗SU(2)⊗U(1)]/Z22. Here U(1) refers to the global c lattice U(1) symmetry, which is associated with the lattice degrees of freedom and is independent from the two SU(2) symmetries. Its generator is the operator N˜ηR=∑j=1L(1−∑σ=↑,↓n˜j,σ(1−n˜j,−σ)) that counts the number NηR=0,1,...,L of rotated-electron unoccupied plus doubly occupied sites. (Alternatively, it could be chosen to be the operator N˜sR=∑j=1L∑σ=↑,↓n˜j,σ(1−n˜j,−σ) that counts the number NsR=L−NηR=0,1,...,L of rotated-electron singly occupied sites.) The generator N˜ηR eigenvalues are thus the numbers of rotated-electron unoccupied plus doubly occupied sites. As justified in later sections, the role of such an eigenvalue in several physical quantities that emerge from the interplay of the model's symmetry with its exact BA solution justifies that it is called in this paper Lη, i. e. Lη≡NηR=0,1,...,L.We denote each of the u>0 energy and momentum eigenstates that belong to the subset of such states that span the Ssz=0 subspace considered here by |lr,Lη,Sη,Sηz,u〉. Here lr stands for all quantum numbers other than Lη, Sη, Sηz, and u>0 needed to uniquely specify each such a state. This includes spin Ss, spin projection Ssz, and a well-defined set of u independent TBA quantum numbers. Such states can be written as,(13)|lr,Lη,Sη,Sηz,u〉=[1Cη(Sˆη+)γη]|lr,Lη,Sη,−Sη,u〉, where,(14)γη=Sη+Sηz=0,1,...,2SηandSηz=−(L−Ne)/2. Furthermore, Cη=[γη!]∏j=1γη[2Sη+1−j] is a normalization constant and Sˆη+ is the η-spin SU(2) off-diagonal generator,(15)Sˆη+=∑j=1L(−1)jcj,↓†cj,↑†andthusSˆη−=(Sˆη+)†.Except in the u→∞ limit, electron single occupancy, electron double occupancy, and electron non-occupancy are not good quantum numbers for the energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉. For instance, upon decreasing u there emerges for ground states for which mηz≥0 a finite electron double occupancy expectation value, which vanishes for u→∞ [106].We call η-Bethe states the u>0 energy and momentum eigenstates |lr,Lη,Sη,−Sη,u〉 that are LWSs of the η-spin SU(2) algebra, so that Sηz=−Sη and thus γη=0 in their expression, Eq. (13). We call Bethe states the u>0 energy and momentum eigenstates that are both LWSs of the η-spin and spin SU(2) operator algebras for which Sαz=−Sα for α=η,s. However, the η-Bethe states considered in this paper can either be spin LWSs or spin non-LWSs. The designation LWS and non-LWS refers in general in this paper to the η-spin SU(2) operator algebra alone. In the case of the spin SU(2) operator algebra, one always specifies spin LWS and spin non-LWS, respectively.For η-Bethe states |lr,Lη,Sη,−Sη,u〉 and general energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉 the electron numbers are given by,(16)Ne0=L−2SηandNe=Ne0+2γη, respectively, where γη=Sη+Sηz=0,1,...,2Sη, Eq. (14).In the case of η-Bethe states, the u>0 charge current operator expectation values 〈lr,Lη,Sη,−Sη,u|Jˆ|lr,Lη,Sη,−Sη,u〉, which are such states charge currents, can be expressed in terms of the BA solution momentum rapidity and rapidity functionals, Eqs. (A.10) and (A.11) of Appendix A. For each u>0 η-Bethe state, such functions are uniquely defined by the TBA equations, Eqs. (A.1) and (A.2) of that Appendix. Furthermore, we rely on exact symmetry relations to express the charge currents of general energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉, Eq. (13), in terms of that of the corresponding η-Bethe state |lr,Lη,Sη,−Sη,u〉 on the right-hand side of that equation.A V tower is within the rotated-electron representation the set of energy eigenstates |lr,Lη,Sη,Sηz,u〉 with exactly the same u-independent quantum numbers lr, Lη, Sη, and Sηz and different u values in the range u>0 [103]. The set of energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉 that belong to the same V tower are for any u>0 value generated by exactly the same occupancy configurations of the u-independent quantum numbers as the corresponding u=∞ energy and momentum eigenstate |lr,Lη,Sη,Sηz,∞〉=limu→∞|lr,Lη,Sη,Sηz,u〉. Out of the many choices of u=∞ energy and momentum eigenstates, the states |lr,Lη,Sη,Sηz,∞〉 are those obtained from the finite-u energy and momentum eigenstates, Eq. (13), whose LWSs are the η-Bethe states, as limu→∞|lr,Lη,Sη,Sηz,u〉.The Hilbert space remains the same for the whole u>0 range. For any fixed u>0, there is thus a uniquely defined unitary operator Vˆ=Vˆ(u) such that |lr,Lη,Sη,Sηz,u〉=Vˆ†|lr,Lη,Sη,Sηz,∞〉. This operator Vˆ is the electron – rotated-electron unitary operator appearing in Eq. (12). It is uniquely defined in Eq. (11) of Ref. [103]. The σ=↑,↓ electron single occupancy, electron double occupancy, and electron non-occupancy are good quantum numbers for a u→∞ energy and momentum eigenstate |lr,Lη,Sη,Sηz,∞〉. This is why for all the finite-u energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉 belonging to the same V tower the rotated-electron numbers Ns,±1/2R, Nη,±1/2R, NsR=L−Lη, and NηR=Lη are conserved as well.2.2Effects of the symmetry on the charge degrees of freedomOne of the few rigorous results for the Hubbard model on any bipartite lattice refers to its global symmetry. As was mentioned in Section 1, it is well known that on such a lattice the Hamiltonian has two global SU(2) symmetries [75–78]. Consistently, in the early nineties of the past century it was found that for u≠0 the Hubbard model on a bipartite lattice has at least a SO(4)=[SU(2)⊗SU(2)]/Z2 symmetry, which contains the η-spin and spin SU(2) symmetries [76,77]. More recently it was found in Ref. [101] that for u≠0 and on any bipartite lattice its global symmetry is actually larger and given by [SO(4)⊗U(1)]/Z2=[SU(2)⊗SU(2)⊗U(1)]/Z22. (The 1/Z22 factor in the u>0 model global symmetry refers to the number 4L of its independent representations being four times smaller than the dimension 4L+1 of the group SU(2)⊗SU(2)⊗U(1).)The origin of the u>0 global [SU(2)⊗SU(2)⊗U(1)]/Z22 symmetry is a local gauge SU(2)⊗SU(2)⊗U(1) symmetry of the U>0 Hamiltonian t=0 term first identified in Ref. [107]. This u−1=0 local gauge symmetry becomes for finite u=U/4t a group of permissible unitary transformations. (The corresponding local U(1) canonical transformation is not the ordinary gauge U(1) subgroup of electromagnetism. It is rather a “nonlinear” transformation [107].) The related global c lattice U(1) symmetry beyond SO(4) found in Ref. [101], which is associated with the lattice degrees of freedom and does not exist at U=0, emerges at any arbitrarily small u value.Importantly, the rotated-electrons charge and spin 1/2 are the same as those of the corresponding electrons and thus remain invariant under the electron – rotated-electron unitary transformation. That transformation only changes the lattice occupancies and corresponding spatial distributions of the charges and spins 1/2. Furthermore, in the u→∞ limit the electron – rotated-electron unitary operator Vˆ becomes the unit operator. This is why in such a limit the rotated electrons become electrons.That in the u−1→0 limit the rotated electrons become electrons and for u>0 they have the same charge and spin 1/2 as the electrons reveals that for finite u they are quasiparticles whose “noninteracting” limit is u−1=0. In terms of the onsite repulsion, this is thus a type of turned upside-down “Fermi liquid”. Its exotic properties follow in part from at such u−1=0 “noninteracting” point the degrees of freedom of the electron occupancy configurations that generate from the electron vacuum the u−1=0 energy and momentum eigenstate |lr,Lη,Sη,Sηz,∞〉 separating into three types of configurations. Those refer to state representations of the two SU(2) symmetries and U(1) symmetry in the u−1=0 model local gauge SU(2)⊗SU(2)⊗U(1) symmetry. As reported below in Section 2.3, this three degrees of freedom separation persists at finite u in terms of the rotated-electron occupancy configurations that generate from the electron vacuum the energy and momentum eigenstate |lr,Lη,Sη,Sηz,u〉=Vˆ†|lr,Lη,Sη,Sηz,∞〉. At finite u values this is related though to the three symmetries in the u>0 model global [SU(2)⊗SU(2)⊗U(1)]/Z22 symmetry that stems from its u−1=0 local gauge SU(2)⊗SU(2)⊗U(1) symmetry.Furthermore and as reported above, for u>0 and at U=0 the global symmetry is different and given by [SO(4)⊗U(1)]/Z2 and SO(4)⊗Z2, respectively. The factor Z2 in the U=0 global symmetry corresponds to a discretely generated symmetry associated with a well-known transformation that exchanges spin and η-spin. It is an exact symmetry of the U=0 and t≠0 Hamiltonian. However, it changes the sign of U when U≠0. That the global symmetry is different at U=0 and for u=U/4t>0 plays an important role in the quantum transition that occurs for mηz=0 at U=Uc=0. It separates two qualitatively different types of transport of charge. It may as well play an important role in the charge transport properties for T>0.Another important symmetry property that has effects on the transport of charge is that the U(2)=SU(2)⊗U(1) and SU(2) symmetries in the [SU(2)⊗SU(2)⊗U(1)]/Z22 global symmetry refer to the charge and spin degrees of freedom, respectively. We recall that the charge U(2)=SU(2)⊗U(1) symmetry includes the η-spin SU(2) symmetry and the c lattice U(1) symmetry beyond SO(4). The state representations of the groups associated with these two symmetries are found in this paper to contribute to the charge current of the u>0 energy and momentum eigenstates.That the charge and spin global symmetries are U(2)=SU(2)⊗U(1) and SU(2), respectively, has in the present 1D case direct effects on the model's exact BA solution. For instance and as reported in Section 1, it is behind the charge and spin monodromy matrices of the BA inverse-scattering method [8,86] having different ABCD and ABCDF forms [8].2.3The rotated-electron degrees of freedom separationThe rotated-electron degrees of freedom naturally separate for u>0 into occupancy configurations of three basic fractionalized particles that generate exact state representations of the groups associated with the independent spin and η-spin SU(2) symmetries and the c lattice U(1) symmetry, respectively, in the model global [SU(2)⊗SU(2)⊗U(1)]/Z22 symmetry. This refers namely to a number Ls=NsR of rotated spins 1/2, Lη=NηR of rotated η-spins 1/2, and Nc=L−Lη=NsR of c pseudoparticles without internal degrees of freedom, respectively.The corresponding numbers of rotated spins of spin projection ±1/2 and rotated η-spins of η-spin projection ±1/2 are denoted by Ls,±1/2 and Lη,±1/2, respectively. They are determined by corresponding numbers of rotated-electrons as they read Ls,±1/2=Ns,±1/2R and Lη,±1/2=Nη,±1/2R. There are in addition Nch=Lη=NηR c pseudoparticle holes. Lη=NηR and Ls=NsR=L−Lη are as well the number of sites of the η-spin and spin effective lattices, respectively, introduced in the following. The state representations of the groups associated with the η-spin and spin SU(2) symmetries that are generated by rotated η-spins 1/2 and rotated spins 1/2 occupancy configurations, respectively, of such effective lattices are similar to those of an η-spin-1/2 and a spin-1/2 XXX chain on a lattice with Lη and Ls sites, respectively. This justifies the notations Lη and Ls for NηR and NsR, respectively.The concept of a squeezed effective lattice is well known in 1D correlated systems [48,56,108]. In the present case, the rotated η-spins 1/2 only “see” the set of Lη=NηR sites unoccupied and doubly occupied by rotated electrons. The rotated η-spins 1/2 thus live in an η-spin squeezed effective lattice with Lη sites that corresponds to an η-spin-1/2 XXX chain. The rotated spins 1/2 only “see” the set of Ls=L−Lη sites singly occupied by rotated electrons. They live in a spin squeezed effective lattice with Ls=L−Lη=NsR sites that corresponds to a spin-1/2 XXX chain. The c pseudoparticles live on an effective lattice identical to the original model lattice. In the case of the electron representation of the 1D Hubbard model, these lattices are known in the u→∞ limit in which the rotated electrons become electrons [48,56,108].The spatial coordinates of the Nc=L−Lη=Ls sites occupied by c pseudoparticles and those of the corresponding Nch=Lη unoccupied sites (c pseudoparticle holes) fully define the relative positions in the model's original lattice of the spin squeezed effective lattice sites and η-spin squeezed effective lattice sites, respectively. The role of the c lattice U(1) symmetry is actually to preserve the independence of the spin and η-spin SU(2) symmetries and corresponding squeezed effective lattices occupancy configurations, which do not “see” each other. This is fulfilled by the state representations of the c lattice U(1) symmetry group by storing full information on the relative positions in the model's original lattice of the spin and η-spin squeezed effective lattices sites, respectively.The following relations between the numbers of the three types of fractionalized particles hold,(17)Ls=Ls,+1/2+Ls,−1/2=Nc,Lη=Lη,+1/2+Lη,−1/2=L−Nc=Nch,Ls,+1/2−Ls,−1/2=−2Ssz=Ne↑−Ne↓,Lη,+1/2−Lη,−1/2=−2Sηz=L−Ne, where Neσ is the number of σ=↑,↓ electrons, which equals that of σ=↑,↓ rotated electrons. The u>0 good quantum numbers Lη=NηR and Ls=NsR naturally emerge within the BA solution as Lη=L−Nc and Ls=Nc, respectively [103]. Here Nc is our notation for the number called N−M′ in Ref. [4], which is the number of real charge rapidities kj of a Bethe state.On the one hand, the electron – rotated-electron unitary transformation changes the lattice occupancies and corresponding spatial distributions of the rotated-electrons charges and spins 1/2. On the other hand, the rotated-electrons charge and spin 1/2 are the same as those of the corresponding electrons and thus remain invariant under that transformation. This ensures that the rotated spins 1/2, which are the spins of the rotated electrons that singly occupy sites, are physical spins 1/2. The same applies to the c pseudoparticles that carry the charges of these rotated electrons and to the rotated η-spins 1/2 of η-spin projection +1/2 and −1/2 that describe the η-spin degrees of freedom of the rotated-electron unoccupied and doubly occupied sites, respectively. Consistently, the operators associated with the rotated spins 1/2, c pseudoparticles, and rotated η-spins 1/2 have explicit expressions in terms of the rotated-electron creation and annihilation operators, Eq. (12). Moreover, the corresponding electron – rotated-electron unitary operator is uniquely defined in Eq. (11) of Ref. [103]. Specifically, the local SU(2) operators associated the rotated η-spins 1/2 and rotated spins 1/2 are expressed in terms of rotated-electron creation and annihilation operators in Eqs. (29)–(31) of that reference. The c pseudoparticle creation and annihilation operators are expressed in terms of those of the rotated electrons in Eq. (33) and in Eq. (38) for β=c of Ref. [103].For u>0 energy and momentum eigenstates of η-spin Sη and spin Ss, a number Lη−2Sη of rotated η-spins 1/2 out of Lη such η-spins and a number Ls−2Ss of rotated spins 1/2 out of Ls such spins are part of Πη=(Lη−2Sη)/2 η-spin singlet pairs and Πs=(Ls−2Ss)/2 of spin-singlet pairs, respectively. Subsets of n=1,...,∞ such pairs refer to the internal structure of neutral composite ηn and sn pseudoparticles, respectively [103]. The occupancy configurations of the fractionalized particles and related composite particles that generate the exact energy and momentum eigenstates from the electron vacuum are found to be labelled by the quantum numbers emerging from the model TBA solution [103,104]. This is a generalization of the representation in terms of spins 1/2 and n-band pseudoparticles used for the spin-1/2 XXX chain in Refs. [72,73] to address the related problem of that model spin stiffness [74].The rotated-electron creation and annihilation operators, Eq. (12), have been inherently constructed from those of the electrons to the form of the 1D Hubbard model energy and momentum eigenstates wave function in terms of rotated electrons being for u>0 similar to that of the wave function in terms of electrons for u→∞. The latter is given in Eq. (2.23) of Ref. [7]. It is a product of an η-spin 1/2 XXX chain wave function φ1, a spin 1/2 XXX chain wave function φ2, and a Slater determinant of fermions without internal degrees of freedom. Hence this confirms that in the u→∞ limit the 1D Hubbard model corresponds to an η-spin-1/2 XXX chain, a spin-1/2 XXX chain, and a quantum problem with simple U(1) symmetry, respectively. The same applies to the whole u>0 range within the rotated-electron representation.Note though that for finite u values this applies only to the u-independent lr, Lη, Sη, and Sηz quantum number values that label the exact energy and momentum eigenstates |lr,Lη,Sη,Sηz,u〉, which includes the momentum operator eigenvalues, as well as to the occupancy configurations that generate such states, which in terms of rotated electrons remain the same for the whole u>0 range, and to the corresponding rotated-electron wave functions. The energy eigenvalues, Eqs. (A.5) and (A.7) of Appendix A, and for instance the charge current operator expectation values, Eqs. (A.10) and (A.11) of that Appendix, of the energy and momentum eigenstates are though dependent on u. They have a different form from the corresponding u→∞ energy eigenvalues and charge current operator expectation values. In the case of the latter this stems from the exotic overlap that occurs within matrix elements between energy and momentum eigenstates of the charge current operator expressed in terms of electron creation and annihilation operators, Eq. (11), with the rotated-electron occupancy configurations that generate such states [103].2.4Relation to the Bethe-ansatz solution quantum numbersThe studies of Ref. [103] have considered the relation between the TBA quantum numbers and the three degrees of freedom separation of the rotated-electron occupancy configurations. This confirms that such quantum numbers are directly associated with the occupancy configurations of the above considered three types of fractionalized particles that generate all Bethe states. Upon application onto those of the off-diagonal generators of the model's two SU(2) symmetries, one then generates all 4L u>0 energy and momentum eigenstates, as given in Eq. (13).The exact Bethe states are populated by Ls=L−Lη rotated spins 1/2 and Lη rotated η-spins 1/2. As mentioned above, out of those, a number Ls−2Ss of rotated spins 1/2 are part of a number Πs=(Ls−2Ss)/2 of spin-singlet pairs (α=s) and a number Lη−2Sη of rotated η-spins 1/2 are part of a number Πη=(Lη−2Sη)/2 of η-spin singlet pairs (α=η). Such Πα spin-singlet (α=s) and η-spin singlet (α=η) pairs are bound within a set of α n-pairs configurations each of which refers to the internal degrees of freedom of one neutral composite αn pseudoparticle. Here n=1,...,∞ gives the number of pairs bound within each such pseudoparticles.Consistently with TBA corresponding results, the following exact sum rules then hold for all u>0 energy and momentum eigenstates, Eq. (13),(18)Πα=∑n=1∞nNαn=12(Lα−2Sα)whereα=s,η,Π≡∑α=η,sΠα=∑α=η,s∑n=1∞nNαn=12(L−2Ss−2Sη). Here Nαn is the number of αn pseudoparticles and Π denotes the total number of both rotated-spin and rotated-η-spin pairs.For a Bethe state, the remaining Mα=2Sα unpaired rotated spins (α=s) and rotated η-spins (α=η) have spin and η-spin projection +1/2, respectively. For general u>0 energy and momentum eigenstates, the multiplet configurations of these Ms=2Ss unpaired rotated spins and Mη=2Sη unpaired rotated η-spins generate the spin and η-spin, respectively, SU(2) symmetry towers of non-LWSs. The SU(2) symmetry algebras off-diagonal generators that flip such unpaired rotated spins and unpaired rotated η-spins, which for η-spin are given in Eq. (15), do not affect though the spin (α=s) and η-spin (α=η) singlet configurations of the Πα=∑n=1∞nNαn pairs contained in neutral composite αn pseudoparticles. Those remain unchanged.For general u>0 energy and momentum eigenstates, the number Ms,±1/2 of unpaired rotated spins of projection ±1/2 and Mη,±1/2 of unpaired rotated η-spins of projection ±1/2 are good quantum numbers given by,(19)Mα,±1/2=Sα∓SαzandMα=Mα,−1/2+Mα,+1/2=2Sαwhereα=η,s. For the Bethe states, one has that Mα,+1/2=Mα=2Sα and Mα,−1/2=0 for both α=η,s. The rotated η-spins (α=η) and rotated spins (α=η) numbers Lα and Lα,±1/2 in Eq. (17) can be written as,(20)Lα=2Πα+Mα=2Πα+2Sα,Lα,±1/2=Πα+Mα,±1/2=Πα+Sα∓Sαzwhereα=η,s, respectively.Another important symmetry property is that the spatial lattice occupancies of the Mα=2Sα unpaired rotated spins (α=s) and unpaired rotated η-spins (α=η) remain invariant under the electron – rotated-electron unitary transformation. This means that their lattice occupancy configurations are for the whole u>0 range exactly the same as those of the corresponding electrons occupancy configurations. That invariance plays an important role in the transport of charge and spin. Indeed and as reported below for the present case of charge transport, the electronic degrees of freedom couple to charge and spin probes through only such Mη=2Sη unpaired physical η-spins and Ms=2Ss unpaired physical spins, respectively.Note though that the paired rotated spins 1/2 and paired rotated η-spins 1/2 are also physical spins 1/2 and physical η-spins 1/2 in what their spin and η-spin degrees of freedom, respectively, are concerned. Only their lattice spatial occupancies are changed under the electron – rotated-electron unitary transformation. Nevertheless, to stress that the lattice spatial occupancies of the unpaired spins 1/2 and unpaired η-spins 1/2 remain invariant under that transformation we omit the term rotated from their designation. We use more often in the following the designation physical for them.The TBA solution contains different types of quantum numbers. Their occupancy configurations are within the pseudoparticle representation described by corresponding occupancy configurations of c pseudoparticles with no internal degrees of freedom and composite αn pseudoparticles plus a number Mη=2Sη of unpaired physical η-spins and Ms=2Ss of unpaired physical spins. The c branch momentum rapidity functional kc(qj) and set of αn branches rapidity functionals Λαn(qj) where α=η,s and n=1,...,∞ are solutions of the coupled TBA integral equations introduced in Ref. [4], which are given in functional form in Eqs. (A.1) and (A.2) of Appendix A. For n>1, the rapidity functionals Λαn(qj) are the real part of corresponding l=1,...,n complex rapidities given below.The c branch TBA quantum numbers {qj} in the argument of the momentum rapidity functional kc(qj) and corresponding c rapidity functional Λc(qj)≡sin(kc(qj)) and αn branch BA quantum numbers {qj} in the argument of the rapidity functionals Λαn(qj) are given by,(21)qj=2πLIjβforj=1,...,Lβwhereβ=c,ηn,snandn=1,...,∞. Here {Ijβ} are the quantum numbers {qj} in units of 2π/L that are successive integers or half-odd integers according to the following boundary conditions,(22)Ijβ=0,±1,±2,...forIβeven,=±1/2,±3/2,±5/2,...forIβodd. The β=c,ηn,sn numbers Iβ in this equation read,(23)Ic=Nps≡∑α=η,s∑n=1∞Nαn,Iαn=Lαn−1whereα=η,sandn=1,...,∞. Moreover, Lβ=Nβ+Nβh is the number of β=c,αn-band discrete momentum values qj of which for a given state Nβ are occupied and Nβh are unoccupied. They read,(24)Lc=Nc+Nch=L,Nch=L−Nc=Lη=2Sη+∑n=1∞2nNηn,Lαn=Nαn+Nαnhwhereα=η,sandn=1,...,∞,Nαnh=2Sα+∑n′=n+1∞2(n′−n)Nαn′=Lα−∑n′=1∞(n+n′−|n−n′|)Nαn′.The momentum eigenvalues can be written as,(25)P=∑j=1LqjNc(qj)+∑n=1∞∑j=1LsnqjNsn(qj)+∑n=1∞∑j=1Lηn(π−qj)Nηn(qj)+πLη,−1/2, where Nβ(qj) are for β=c,ηn,sn pseudoparticle branches the β-band momentum distribution functions. They are such that Nβ(qj)=1 and Nβ(qj)=0 for occupied and unoccupied qj values, respectively. For the c and αn branches, such values have intervals qj∈[qc−,qc+] and qj∈[−qαn,qαn] where ignoring 1/L corrections within the TL the c-band limiting momentum values are such that qc±=±qc. Here the limiting momentum values qc and qαn are given by,(26)qc=πandqαn=π(Lαn−1)/L, respectively.That the momentum eigenvalues, Eq. (25), are additive in the quantum numbers qj in Eq. (21) is consistent with they playing the role of β=c,αn band momentum values. The momentum contribution πLη,−1/2=π(Πη+Mη,−1/2) in Eq. (25) follows from both the paired and unpaired rotated η-spins of projection −1/2 having an intrinsic momentum given by,(27)qη,−1/2=π. For a η-Bethe state one has that πLη,−1/2=πΠη.2.5Internal degrees of freedom of the composite αn pseudoparticles and u→0 n>1 pairs unbindingAs for the spin-neutral composite n-band pseudoparticles of the spin-1/2 XXX chain [73], the problem concerning an αn pseudoparticle internal degrees of freedom and that associated with its translational degrees of freedom center of mass motion separate within the TL.On the one hand, the αn-band momentum qj, Eq. (21) for β=αn, is associated with the latter. On the other hand, for n>1 the internal degrees of freedom are related to the imaginary part of the αn rapidities,(28)Λαn,l(qj)=Λαn(qj)+i(n+1−2l)uwherel=1,...,n, j=1,...,Lαn, α=η,s, and n=1,...,∞.Each set of l=1,...,n complex rapidities Λαn,l(qj) with the same real part Λαn(qj) is associated with the l=1,...,n η-spin-singlet pairs (α=η) or spin-singlet pairs (α=s) bound for n>1 within a neutral composite αn pseudoparticle. Each of such l=1,...,n rapidities actually describes one of the Πα=∑n=1∞nNαn, Eq. (18), spin-singlet pairs (α=s) or η-spin-singlet pairs (α=η) of the Bethe state under consideration. The real part Λαn(qj) is the rapidity functional that as reported above is for each Bethe state the solution of the coupled Eqs. (A.1) and (A.2) of Appendix A.For n=1, the rapidity Λα1,1(qj), Eq. (28) for l=n=1, refers to a single pair and is real. We call unbound spin-singlet pairs (α=s) and unbound η-spin singlet pairs (α=η) of a Bethe state the corresponding Nα1 pairs, each referring to a single n=1 pair configuration. Otherwise, the n>1 rapidities Λαn,l(qj) imaginary part i(n+1−2l)u of a u>0 Bethe state is finite. The corresponding set of l=1,...,n complex rapidities with the same real part then describes the binding of the n>1 pairs within an αn-pair configuration. Such a configuration describes the internal structure of a neutral composite αn pseudoparticle. We call bound spin-singlet pairs (α=s) and bound η-spin singlet pairs (α=η) the Πα−Nα1 pairs that are bound within n>1 αn-pair configurations. All this is again exactly as for the spin n-pairs configurations of the spin 1/2-XXX chain [73].In contrast to that chain, for n>1 the imaginary part i(n+1−2l)u of each set of the l=1,...,n rapidities with the same real part depends on the interaction u=U/4t and thus vanishes as u→0. As discussed in more detail in Appendices B and C, such an unbinding in that limit of the l=1,...,n pairs within each u>0 αn-pair configuration marks the qualitatively different physics of the U=0 and u>0 quantum problems, respectively. It is associated with the rearrangement of the η-spin and spin degrees of freedom in terms of the noninteracting electrons occupancy configurations that generate the U=0 common eigenstates of the Hamiltonian, momentum operator, and current operator. Indeed, as the imaginary part i(n+1−2l)u of each set of l=1,...,n rapidities, the commutator of the charge current operator and the 1D Hubbard model Hamiltonian,(29)[Jˆ,Hˆ]=iu4t2∑σ∑j=1Na[cj,σ†(cj+1,cj,σ−cj−1,σ)+(cj+1,σ†−cj−1,σ†)cj,σ]nˆj,−σ, also vanishes as u→0.The form i(n+1−2l)u of that imaginary part and of that commutator, Eq. (29), confirms that the u>0 physics survives for any arbitrarily small value of u. Indeed, the l=1,...,n pairs unbinding and commutator [Jˆ,Hˆ] vanishing occur only in the u→0 limit. The rearrangement of the η-spin and spin degrees of freedom in terms of the noninteracting electrons occupancy configurations that occurs within the unbinding of the l=1,...,n pairs within each u>0 ηn-pair configuration has most severe consequences on the transport of charge at hole concentration mηz=0. The effects of the u→0 transition on the charge dynamic structure factor at mηz=0 is a problem addressed in Ref. [19]. The mechanisms behind the corresponding qualitatively different types of transport associated with the occurrence at hole concentration mηz=0 of charge ballistic transport at U=0 and its absence found in this paper for u>0 is an interesting issue discussed in Appendix B for mηz→0 and mηz=0 and in Appendix C for mηz∈[0,1].Within the usual TBA notation, the set of l=1,...,n complex rapidities Λαn,l(qj) with the same real part is called an αn string. Specifically, a charge ηn string and a spin sn string. It thus refers to an αn-pair configuration involving l=1,...,n pairs. Hence the number Nα=∑n=1∞Nαn of composite αn pseudoparticles of all n=1,...,∞ branches of a u>0 energy and momentum eigenstate equals that of corresponding TBA αn-strings of all lengths n=1,...,∞. Such a number obeys an exact sum rule given by [103],(30)Nα=∑n=1∞Nαn=12(Lα−Nα1h)whereα=η,s,Nps=∑α=η,s∑n=1∞Nαn=12(L−Ns1h−Nη1h). Here Nps is the number of both α=η and α=s composite αn pseudoparticles of all n=1,...,∞ branches also appearing in Eq. (23) and Nα1h is that of α1-band holes, Eq. (24) for α=η,s and n=1. Hence Nps is as well the number of both ηn-strings and sn-strings of all lengths n=1,...,∞.The TBA solution performs the electron – rotated-electron unitary transformation. Consistently, it accounts for the 1D Hubbard model related symmetries and conserved rotated-electron numbers through the set of α=η,s TBA αn strings of length n numbers {Nαn} and c branch number Nc. This follows from the generators that produce all 4L energy and momentum eigenstates, Eq. (13), from the electron and thus rotated-electron vacuum being naturally expressed in terms of the three fractionalized particles operators that emerge from the rotated-electron three degrees of freedom separation. The latter is associated with the two independent SU(2) symmetries and the independent U(1) symmetry in the model's global symmetry.Specifically, the conserved rotated-electron numbers of unoccupied sites Nη,+1/2R and of doubly occupied sites Nη,−1/2R can be expressed in terms of energy and momentum eigenstates η-spin Sη, η-spin projection Sηz, and set of TBA charge ηn strings of length n numbers {Nηn} as Nη,±1/2R=∑n=1∞nNηn+Sη∓Sηz. Similarly, the conserved spin projection ±1/2 rotated-electron number of singly occupied sites Ns,±1/2R can be expressed in terms of the energy and momentum eigenstates spin Ss, spin projection Ssz, and set of TBA spin sn strings of length n numbers {Nan} as Ns,±1/2R=∑n=1∞nNsn+Ss∓Ssz. Furthermore, the eigenvalue Lη=L−Ls of the generator of the c lattice U(1) symmetry group such that Lη=Nη,+1/2R+Nη,−1/2R and Lη=L−Ns,+1/2R−Ns,−1/2R appears in Eq. (24) within the TBA solution through the numbers Nc=L−Lη and Nch=Lη.As for the spin-1/2 XXX chain [72,73], for a large finite system some of the 1D Hubbard model αn strings of length n>1 deviate from their TBA ideal form, Eq. (28). As discussed in Appendix D, the effects of such string deviations [9] are in the TL though not important for the problem studied in this paper.On the one hand, for u>0 the imaginary part of the n>1 rapidities with the same real part, Eq. (28), describes the binding of the l=1,...,n pairs within the corresponding αn-pair configuration. On the other hand, in Ref. [103] it is shown that the configuration of the two rotated spins within each such unbound spin-singlet pair and that of the two rotated η-spins within each unbound η-spin singlet pair has for u>0 a binding and anti-binding character, respectively.3Charge current operator expectation values and useful subspacesOur study of the charge stiffness refers to the hole concentration interval mηz∈[0,1], yet the limit of particular interest for the clarification of the main issue under consideration is that of mηz→0. This applies to that stiffness. In most cases the charge properties of physical systems are studied at zero spin density, msz=−Ssz/L=0. This is why for simplicity in the remaining of this paper we consider the 1D Hubbard model in the Ssz=0 subspace. For such quantum problem only η-spin SU(2) symmetry state representations for which Sη=0,1,2,… is an integer are allowed, so that the results presented in this and following sections refer to integer η-spin values. However, concerning the charge quantities studied in the following similar results are obtained within the TL for η-spin half-integer values and |Ssz|=1/2.For the 1D Hubbard model in the Ssz=0 subspace one has that Ms,+1/2=Ms,−1/2 in Eq. (20), so that Ls,+1/2=Ls,−1/2=Πs+Ss where Πs=∑n=1∞nNsn=(L−Lη−2Ss)/2. The dimension of Ssz=0 subspaces spanned by states populated by fixed numbers Ls=L−Lη of rotated spins and Lη=Nch=L−Nc of rotated η-spins is given in Eq. (E.5) of Appendix E.3.1Three exact properties of the charge current operator expectation valuesThe following commutators play a major role in our evaluation of the charge current operator off-diagonal matrix elements and expectation values that contribute to the real part of the charge conductivity, Eq. (1),(31)[Jˆ,Sˆηz]=0;[Jˆ,(S→ˆη)2]=Jˆ+Sˆη−−Sˆη+Jˆ−,[Jˆ,Sˆη±]=[Sˆηz,Jˆ±]=±Jˆ±;[Jˆ±,Sˆη∓]=±2Jˆ. Here as usual, (S→ˆη)2=(Sˆηz)2+12(Sˆη+Sˆη−+Sˆη−Sˆη+), and the current operators Jˆ± read,(32)Jˆ+=i2t∑j=1L(−1)j(cj,↓†cj+1,↑†+cj+1,↓†cj,↑†)andJˆ−=(Jˆ+)†. They are related to the transverse η-spin current operators as Jˆ±,η=(1/2)Jˆ±. The commutators given in Eq. (31) have exactly the same form as those associated with the spin current operator and corresponding spin SU(2) symmetry algebra operators considered in related studies of the spin-1/2 XXX chain spin stiffness [72,73].For simplicity, we denote the η-Bethe states charge currents by 〈JˆLWS(lr,Lη,Sη,u)〉≡〈lr,Lη,Sη,−Sη,u|Jˆ|lr,Lη,Sη,−Sη,u〉 and the charge currents of general u>0 energy and momentum eigenstates by 〈Jˆ(lr,Lη,Sη,Sηz,u)〉≡〈lr,Lη,Sη,Sηz,u|Jˆ|lr,Lη,Sη,Sηz,u〉. By combining the systematic use of the commutators given in Eq. (31) with the transformation laws,(33)Sˆη−|lr,Lη,Sη,−Sη,u〉=0andSˆη+|lr,Lη,0,0,u〉=Sˆη−|lr,Lη,0,0,u〉=0, we reach the following general useful result for the current operator matrix elements between Sηz=0 energy and momentum eigenstates,(34)〈lr,Lη,Sη,0,u|Jˆ|lr,Lη,Sη+δSη,0,u〉=0forδSη≠±1. This selection rule is useful for the discussion in Appendix B of the Sηz=0 and T>0 transition that is found to occur at U=Uc=0, similarly to the T=0 quantum Mott–Hubbard insulator - metal transition. It separates two qualitatively different types of finite-temperature charge transport. For Sηz=0 energy and momentum states whose generation from metallic LWSs involves small γη=Sη+Sηz values, the calculations to reach the result, Eq. (34), are straightforward. They become lengthly as the γη value increases, yet remain straightforward.In the following we report three exact properties that play a major role in our study. The first property refers to the identification of the carriers that within the exact rotated-electron representation couple to charge probes. The 1D Hubbard model in a uniform vector potential Φ/L whose Hamiltonian is given in Eq. (4) of Ref. [109] remains solvable by the BA. The TBA equations for the model in a uniform vector potential are given in Eq. (9) of that reference. The only difference relative to the Φ=0 case is the c band and ηn band momentum values qj being shifted to qj+Φ/L and qj−2nΦ/L, respectively, whereas the sn band momentum values remain unchanged.Concerning the coupling of the charge degrees of freedom to the vector potential, one finds that the η-Bethe states momentum eigenvalues, P(Φ), have the general form,(35)P(Φ/L)=P(0)−(Lη−∑n2nNηn)ΦL=P(0)−2SηΦL=P(0)−MηΦL. Here the Φ=0 momentum eigenvalue P(0) is given in Eq. (25) with Lη,−1/2=Πη for the present η-Bethe states. The sum rule ∑n=1∞2nNηn=Lη−2Sη involving the number Lη−2Sη of paired rotated η-spins 1/2 has been used in Eq. (35). (Such a sum rule follows from that of the corresponding η-spin singlet pairs, Eq. (18) for α=η.)On the one hand, the charge currents of the Φ→0 η-Bethe states can be derived from the Φ/L dependence of the energy eigenvalues E(Φ/L) as 〈Jˆ〉=−dE(Φ/L)/d(Φ/L)|Φ=0, as given in Eqs. (A.10) and (A.11) of Appendix A. On the other hand, dP(Φ/L)/d(Φ/L)|Φ=0 gives the number of charge carriers that couple to the vector potential. The natural candidates are the numbers Lη=Nch of rotated η-spins 1/2. Within the TBA, their translational degrees of freedom are described by c band and ηn bands particle-hole processes. The form of the exact momentum eigenvalues, Eq. (35), reveals that only the Mη=2Sη unpaired physical η-spins 1/2 contributing to the η-spin multiplet configurations couple to the vector potential Φ/L. Since the Lη−2Sη rotated η-spins 1/2 left over are those within the Πη=(Lη−2Sη)/2 neutral η-spin singlet pairs, this exact result is physically appealing. Consistently with results reported in the following, one finds that in the case of general energy and momentum eigenstates, P(Φ/L) rather reads(36)P(Φ/L)=P(0)−(Mη,+1/2−Mη,−1/2)ΦL, with P(0) given now by the general expression provided in Eq. (25). This reveals as expected that the coupling to the vector potential of unpaired physical η-spins 1/2 with opposite η-spin projections ±1/2 has opposite sign.The total flux −2SηΦ=−MηΦ in Eq. (35), has been found within the u→∞ limit in Ref. [13] directly from the solution of the TBA equations of 1D Hubbard model in a uniform vector potential, Eq. (9) of Ref. [109]. Since the lattice occupancy spatial distributions of the Mη=2Sη unpaired physical η-spins 1/2 that couple to the vector potential remain invariant under the electron – rotated-electron unitary transformation, these results hold as well for the whole u>0 range, as found here from the use of the momentum eigenvalues, Eqs. (35) and (36).A second exact property is related to only the Mη=2Sη unpaired physical η-spins 1/2 coupling to the charge vector potential also holding for non-LWSs, as given in Eq. (36). For a η-Bethe state carrying an η-spin current 〈JˆLWS(lr,Lη,Sη,u)〉 all Mη=2Sη unpaired physical η-spins have projection +1/2. The following exact relation that refers to the charge current of general energy and momentum eigenstates, Eq. (13), holds,(37)〈Jˆ(lr,Lη,Sη,Sηz,u)〉=∑σ=±1/2jη,σMη,σ, where the elementary currents jη,±1/2 are given by,(38)jη,±1/2=±〈JˆLWS(lr,Lη,Sη,u)〉2Sη=±〈JˆLWS(lr,Lη,Sη,u)〉Mη.The exact expression, Eqs. (37) and (38), is derived by combining the systematic use of the commutators given in Eq. (31) with the energy and momentum eigenstates transformation laws under the η-spin SU(2) symmetry operator algebra, Eq. (33). After a suitable handling of such an operator algebra and transformation laws involving commutator manipulations, one finds,(39)〈Jˆ(lr,Lη,Sη,Sηz,u)〉=−SηzSη〈JˆLWS(lr,Lη,Sη,u)〉, where as in Eq. (14), Sηz=−Sη+γη and γη=1,...,2Sη. The relation, Eq. (39), can then be exactly rewritten as given in Eq. (37). For non-LWSs whose generation from η-Bethe states in Eq. (13) involves small γη=Sη+Sηz values the calculations to reach the relation, Eq. (39), is straightforward, and remains so as the γη value increases yet becomes lengthly.The exact relation, Eqs. (37), (38), and (39), confirms that also for non-LWSs the Mη=Mη,+1/2+Mη,−1/2 unpaired physical η-spins 1/2 control the η-spin current values. For each elementary η-spin flip process generated by application of the off-diagonal η-spin generator Sˆη+, Eq. (15), (and Sˆη−=(Sˆη+)†) onto an energy and momentum eigenstate with finite numbers Mη,+1/2 and Mη,−1/2, the η-spin current exactly changes by a current quantum 2jη,−1/2 (and 2jη,+1/2). Hence each unpaired physical η-spin with η-spin projection ±1/2 carries an elementary current jη,±1/2, Eq. (38). For a η-Bethe state one has that Mη,+1/2=2Sη and Mη,−1/2=0, so that 〈JˆLWS(lr,Lη,Sη,u)〉=jη,+1/2×Mη=jη,+1/2×2Sη.That in the present case of charge only the Mη=mηL unpaired physical η-spins 1/2 couple to the vector potential implies that all η-spin currents exactly vanish as mη→0. This exact property by itself can be used to confirm that within the canonical ensemble at fixed value of Sηz, in the TL, and for nonzero temperatures the charge stiffness vanishes as mηz→0. In addition, in the case of high temperature T→∞ that result is extended in this paper to the grand-canonical ensemble.The third exact property concerns the processes that contribute to the charge currents 〈JˆLWS(lr,Lη,Sη,u)〉 on the right-hand side of Eq. (39) of general η-Bethe states described by groups of charge c band real momentum rapidities, charge η1 real rapidities, and n>1 charge ηn complex rapidities. The third property reported in the following is a direct consequence on the β=c,ηn band occupancy configurations that within the TBA describe such η-spins 1/2 translational degrees of freedom of only the Mη=2Sη unpaired physical η-spins 1/2 coupling to charge probes.It is shown in Appendix A that within the TBA the η-Bethe states charge currents can be written in the TL in terms of c-band holes and ηn-band holes occupancies as follows,(40)〈JˆLWS(lr,Lη,Sη,u)〉=∑j=1LNch(qj)Jch(qj)+∑n=1∞∑j=1LηnNηnh(qj)Jηnh(qj), where the hole current spectra Jch(qj) and Jηnh(qj) read,(41)Jch(qj)=−Jc(qj)=2tsinkc(qj)2πρc(kc(qj))forqj∈[−π,π]andJηnh(qj)=−Jηn(qj)=4nt∑ι=±1Ληn(qj)−iιnu2πσηn(Ληn(qj))1−(Ληn(qj)−iιnu)2forqj∈[−qηn,qηn], respectively. Here Nch(qj)=1−Nc(qj), Nηnh(qj)=1−Nηn(qj), and the related c- and ηn-bands current spectra Jc(qj) and Jηn(qj), respectively, are given in Eq. (A.11) of Appendix A. Moreover, qηn=π(Lηn−1)/L, Eq. (26), and the rapidity momentum functional kc(qj) and rapidity functionals Ληn(qj) are obtainable for each η-Bethe state from solution of the TBA equations, Eqs. (A.1) and (A.2) of Appendix A. Such equations also involve spin rapidity functionals Λsn(qj) associated with distributions 2πσsn(Λj), besides the distributions 2πρc(kj) and 2πσηn(Λj) explicitly appearing in Eq. (41). The general distributions 2πρc(kj) and 2πσαn(Λj) are defined in Eq. (A.4) of Appendix A. (The functionals qc(k) and qαn(Λ) in that equation stand for the inverse functions of the rapidity momentum functional kc(q) and rapidity functionals Λαn(q), respectively.)That the lattice occupancy spatial distributions of the Mη=2Sη unpaired physical η-spins 1/2 that couple to the charge probes remain invariant under the electron – rotated-electron unitary transformation implies that such η-spins with η-spin projection +1/2 and −1/2 refer for the whole u>0 range to the η-spin degrees of freedom of original lattice sites unoccupied by bare electrons and onsite spin-singlet pairs of bare electrons, respectively. Their translational degrees of freedom are within the TBA solution described by an average number 2Sη of c band holes out of that band Nch=2Sη+∑n=1∞2nNηn holes, Eq. (24), and by an average number 2Sη of holes out of the Nηnh=2Sη+∑n′=n+1∞2(n′−n)Nηn′ holes, Eq. (24) for α=η, of each of the n=1,...,∞ ηn bands for which Nηn>0 in the energy and momentum eigenstates under consideration.Hence in terms of the exact solution quantum numbers, the local processes that generate the charge currents of the energy and momentum eigenstates refer to the relative occupancy configurations of the Nch=Lη holes and corresponding Nc=L−Lη c pseudoparticles and Nηnh holes and corresponding Nηn ηn pseudoparticles in each ηn band for which Nηn>0. Consistently, the charge currents 〈JˆLWS(lr,Lη,Sη,u)〉 on the right-hand side of the charge current expression, Eq. (39), of general energy and momentum eigenstates can alternatively be expressed in terms of c-band and ηn-band holes, as given in Eq. (40), or of c and ηn pseudoparticles, Eq. (A.10) of Appendix A.The third exact property refers to a total and a partial virtual elementary current cancelling occurring in the β=c,ηn bands of Sη=0 and Sη>0, respectively, η-Bethe states for which Nβh>2Sη and Nβ>0. (Unoccupied β-bands for which Nβ=0 do not contribute to the charge current.) Such a cancelling is encoded within the interplay of the current expressions, Eq. (41) and Eq. (A.10) of Appendix A, with the TBA equations, Eqs. (A.1) and (A.2) of that Appendix. It also affects the charge current expression, Eq. (39), of general energy and momentum eigenstates, which in the case of non-LWSs involves the charge currents 〈JˆLWS(lr,Lη,Sη,u)〉 of the η-Bethe states from which such states are generated in Eq. (13).On the one hand, Sη=0 η-Bethe states whose charge current is zero lack unpaired physical η-spins 1/2 to couple to charge probes. Their numbers of band holes Nch and Nηnh are given by Nch=Lη=∑n=1∞2nNηn and Nηnh=∑n′=n+1∞2(n′−n)Nηn′, respectively, as reported in Eq. (24) for Sη=0. Consistently with the lack of unpaired physical η-spins 1/2, the virtual elementary currents carried by a number ∑n=1∞nNηn of c band holes and ∑n′=n+1∞(n′−n)Nηn′ of ηn-bands holes exactly cancel those carried by an equal number ∑n=1∞nNηn of remaining c band holes and ∑n′=n+1∞(n′−n)Nηn′ of remaining ηn-band holes, respectively. Such two sets of β=c,ηn bands holes describe the translational degrees of freedom of two corresponding sets of paired rotated η-spins 1/2 of opposite η-spin projection. Indeed, this exact total elementary currents cancelling involves the opposite η-spin projections within each η-spin singlet pair. As in the case of the unpaired physical η-spins 1/2 in the η-spin multiplet configurations that contribute to the charge currents, Eq. (37), paired rotated η-spins with opposite η-spin projection carry virtual elementary currents of opposite sign.On the other hand, within the β=c,ηn bands of Sη>0 η-Bethe states for which Nβh>2Sη and Nβ>0 there is a corresponding partial virtual elementary current cancellation. For such c and ηn bands the number of holes, Eq. (24), are given by Nch=Lη=2Sη+∑n=1∞2nNηn and Nηnh=2Sη+∑n′=n+1∞2(n′−n)Nηn′, respectively. There is in average in these bands a number 2Sη of β-band holes that describe the translational degrees of freedom of the Mη=2Sη unpaired physical η-spins 1/2. Hence their elementary currents contribute to the η-Bethe states charge currents, Eq. (40). The virtual elementary currents carried by average numbers ∑n=1∞nNηn and ∑n=1∞nNηn of two sets of c band holes and ∑n′=n+1∞(n′−n)Nηn′ and ∑n′=n+1∞(n′−n)Nηn′ of two sets of ηn-bands holes that describe the translational degrees of freedom of two sets of paired rotated η-spins 1/2 of opposite η-spin projection remain though cancelling each other.As mentioned above, the η-Bethe states virtual elementary charge currents cancellation is encoded in the interplay of the current expressions, Eqs. (40) and (41), with the TBA equations, Eqs. (A.1) and (A.2) of Appendix A. Only within the present exact rotated-electron related representation is that virtual elementary currents cancellation described in terms of explicit physical processes. The main role of such virtual elementary current cancelling processes is to control the dependence on the density mη of unpaired physical η-spins 1/2 of the charge currents of the η-Bethe states. The mη dependence of such charge currents is smooth and continuous.The virtual elementary charge currents partial cancelling does not occur within β=c,ηn bands occupancies for which Nβh=2Sη=Mη. For such β=c,ηn bands of a Sη>0 η-Bethe state all their Nβh=2Sη=Mη holes fully contribute to charge currents. Indeed, all such β-band holes describe the translational degrees of freedom of the corresponding η-Bethe state Mη=2Sη unpaired physical η-spins.3.2Simplified stiffness expression and subspaces of the fixed-Sηz and Ssz=0 subspacesThe use of the exact relation, Eq. (39), in the charge stiffness expression, Eq. (5), leads to the following simplified stiffness expression in terms of only η-Bethe states current operator expectation values that is exact in the TL and valid for T>0 and u>0,(42)D(T)=(2Sηz)22LT∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑lrplr,Lη,Sη,Sηz|〈JˆLWS(lr,Lη,Sη,u)〉|2(2Sη)2. In the present case of the Ssz=0 subspace, the available η-spin projection absolute values are integers, |Sηz|=0,1,2,...,L/2. Hence the summations on the right-hand side of Eq. (42) run over even integers Lη=2|Sηz|,2|Sηz|+2,2|Sηz|+4,...,L and integers Sη=|Sηz|,|Sηz|+1,|Sηz|+2,...,Lη/2, respectively.The charge stiffness upper bounds constructed in this paper rely on the use in the general expression, Eq. (42), of corresponding upper bounds for the absolute values of η-Bethe states charge currents 〈JˆLWS(lr,Lη,Sη,u)〉. Such charge currents result from microscopic processes that are actually easiest to be described in terms of original lattice occupancy configurations. Each c pseudoparticle and ηn pseudoparticle is associated with the charge degrees of freedom of one and 2n sites of that lattice, respectively. We call charge pseudoparticles to both the c and ηn pseudoparticles, their number reading,(43)Nρ=Nc+Nη=L−2Sη−∑n=1∞(2n−1)Nηn.The charge only flows along the original lattice provided that the unpaired physical η-spins 1/2 that couple to charge probes interchange site positions in it with the charge c and ηn pseudoparticles. This occurs upon the latter moving along the original lattice. Hence one can consider that such charge pseudoparticles, whose current spectra Jc(qj)=−Jch(qj) and Jηn(qj)=−Jηnh(qj) are given in Eq. (A.11) of Appendix A, play the role of charge carriers. This is consistent with, for a η-Bethe state, the charge pseudoparticles carrying all L−2Sη electronic charges, with each c pseudoparticle and ηn pseudoparticle carrying one and 2n such elementary charges, respectively.Within the canonical ensemble, the general charge stiffness expression, Eq. (42), refers to one of the fixed-Sηz and Ssz=0 subspaces contained in the larger Ssz=0 subspace. The hole concentrations of such subspaces belong to the interval,(44)mηz=−2SηzL=Mη,+1/2−Mη,−1/2L∈[0,1].It is useful for the study of the charge currents and the introduction of suitable upper bounds for their absolute values to consider the subspaces contained in each fixed-Sηz and Ssz=0 subspace. The definition of such subspaces requires a careful account for the summations on the right-hand side of Eq. (42). The summations ∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2 run in that equation over different η-spin SU(2) multiplet towers that refer to energy and momentum eigenstates with the same Sηz value and different Sη=|Sηz|,|Sηz|+1,|Sηz|+2,...,Lη/2 values. Their currents, 〈Jˆ(lr,Lη,Sη,Sηz,u)〉=(−Sηz/Sη)〈JˆLWS(lr,Lη,Sη,u)〉, Eq. (39), are for Sη>|Sηz| expressed in terms of η-Bethe states currents, 〈JˆLWS(lr,Lη,Sη,u)〉, whose η-spin projection S′ηz=−Sη such that −S′ηz>−Sηz is different from their η-spin projection Sηz.The η-spin flip processes that upon successive applications of the η-spin SU(2) symmetry off-diagonal generator Sˆη+ onto each Sη>0 η-Bethe state generate the non-LWSs in Eq. (13) only change the η-spin projections of the η-Bethe state Mη=2Sη unpaired physical η-spins 1/2. Such processes do not change the c pseudoparticle occupancies, η-spin-singlet configurations, and Ssz=0 spin-singlet and spin-multiplet configurations, which remain those of the η-Bethe state. On the one hand, the summations ∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2 on the right-hand side of Eq. (42) run over energy and momentum eigenstates with the same Sηz value. On the other hand and in spite of that, this symmetry invariance allows that the summation ∑lr1=dsubspaceLWS(Lη,Sη) where dsubspaceLWS(Lη,Sη) is the dimension, Eq. (E.7) of Appendix E, can run over c pseudoparticle occupancy configurations, η-spin-singlet configurations, and spin-singlet and spin-multiplet configurations of η-Bethe states with the same Lη and Sη values that have a η-spin projection S′ηz=−Sη different from the η-spin projection Sηz of such energy and momentum eigenstates. Indeed the latter configurations are exactly the same as those of the corresponding non-LWSs with fixed −Sηz<−S′ηz that contribute to the charge stiffness, Eq. (42).An exact property reported above of major importance for our study is that only the charge degrees of freedom of the Mη=2Sη unpaired physical η-spins whose lattice spatial occupancy distributions remain invariant under the electron – rotated-electron unitary transformation couple to a uniform vector potential. It is thus convenient to divide each fixed-Sηz and Ssz=0 subspace into a set of fixed-Sη and Ssz=0 subspaces that we call SzS subspaces, such that Sη≥−Sηz. Each η-spin value Sη=|Sηz|,|Sηz|+1,|Sηz|+2,...,Lη/2 in the summation ∑Sη=|Sηz|Lη/2 on the right-hand side of Eq. (42) corresponds to one such a SzS subspace. Its dimension corresponds to the summation ∑Lη=2|Sη|L∑lr where ∑Lη=2|Sη|L is for Sη>|Sηz| only a part of the overall summation ∑Lη=2|Sηz|L in Eq. (42) and ∑lr is a summation that runs over c pseudoparticle occupancy configurations, η-spin-singlet configurations, and Ssz=0 spin-singlet and spin-multiplet configurations. Those are associated with spin values Ss=0,1,...,(L−Lη)/2 of η-Bethe states with the same Lη and Sη values. We emphasize that although such η-Bethe states have an η-spin projection S′ηz=−Sη different from the η-spin projection Sηz of the corresponding non-LWSs, due to the above reported symmetry invariance their configurations associated with the summation ∑Lη=2|Sη|L∑lr are identical to those of the latter states.Accounting for that symmetry invariance, the SzS subspaces are defined here as being spanned by η-Bethe states with fixed density mη=Mη/L of unpaired physical η-spins 1/2 that belongs to the interval,(45)mη=MηL=2SηL∈[mηz,1], where the maximum density is reached for states for which lη=Lη/L=1. Each density mη in that interval corresponds to one SzS subspace.Each SzS subspace can be further divided into smaller subspaces spanned by η-Bethe states with fixed total number Lη of rotated η-spins. Such subspaces have Lη values in the interval Lη∈[2Sη,L]. Their dimension corresponds to the above mentioned summation ∑lr that runs over c pseudoparticle occupancy configurations, η-spin-singlet configurations, and spin-singlet and spin-multiplet configurations of η-Bethe states with the same Lη and Sη values. We call them SzSL subspaces. The SzSL subspaces contained in a given SzS subspace are thus spanned by η-Bethe states with fixed densities lη=Lη/L that vary in the interval,(46)lη=LηL∈[mη,1]. Each pair of densities mη,lη in the ranges, Eqs (45) and (46), respectively, refers to one SzSL subspace.The related dependent densities associated with the numbers Nc=L−Lη of c pseudoparticles, Πη=(Lη−2Sη)/2 of η-spin singlet pairs, and Ls=L−Lη of rotated spins 1/2, are also fixed for a SzSL subspace. For different SzSL subspaces, such densities hence vary in the ranges,(47)nc=NcL=1−lη∈[0,(1−mη)],πη=ΠηL=12(lη−mη)∈[0,(1−mη)/2],ls=LsL=1−lη∈[0,(1−mη)].Each SzSL subspace can be further divided into smaller subspaces we call SzSLN subspaces. They are spanned by η-Bethe states whose total number of ηn pseudoparticles Nη=∑n=1∞Nηn∈[0,Πη] is fixed. (That Nη=0 implies that Πη=0.) The SzSLN subspaces contained in a given SzSL subspace can have densities nη=Nη/L and nρ=Nρ/L in the intervals,(48)nη=NηL∈[0,πη]=[0,(lη−mη)/2],nρ=nc+nη∈[(1−lη),(2−lη−mη)/2].Each SzSLN subspace can be further divided into smaller subspaces we call SzSLNS subspaces. They are spanned by η-Bethe states with a fixed total number Ms=2Ss=L−Lη−2Πs of unpaired physical spins 1/2. The related dependent density πs=Πs/L=(1−lη−ms)/2 of spin-singlet pairs is also fixed for a SzSLNS subspace. The SzSLNS subspaces contained in a SzSLN subspace can have densities ms=Ms/L=2S/L and πs in the ranges,(49)ms=MsL=1−lη−2πs∈[0,(1−lη)],πs=ΠsL=(1−lη−ms)/2∈[0,(1−lη)/2].Each SzSLNS subspace can be further divided into smaller subspaces that we call SzSLNSN subspaces. They are spanned by η-Bethe states with a fixed overall number of sn pseudoparticles Ns=∑n=1∞Nsn∈[0,Πs]. The SzSLNSN subspaces contained in a SzSLNS subspace can have densities ns in the interval,(50)ns=NsL∈[0,(1−lη−ms)/2].Each SzSLNSN subspace can be further divided into smaller subspaces that are spanned by η-Bethe states with fixed numbers of ηn and sn pseudoparticles for all n=1,...,∞ branches. (For the subspaces of more interest for our study these numbers are finite only for a finite number of n=1,...,∞ branches.)Within the above notations used in this paper to designate the subspaces contained in each fixed-Sηz and Ssz=0 subspace of more interest for its studies, Sz, S, and L refer to the charge conserved numbers Sηz, Sη, and Lη, respectively, whereas S and N refer to the spin conserved numbers Ss and Ns, respectively. The designations SzS, SzSL, SzSLN, SzSLNS, and SzSLNSN only include the corresponding subset of these numbers that are fixed for the subspaces under consideration.4Useful current absolute values upper boundsThe upper bound procedures of our study are initiated in this section. Specifically, the charge currents 〈JˆLWS(lr,Lη,Sη,u)〉, Eq. (40), of η-Bethe states in the stiffness expression, Eq. (42), with largest absolute values are identified. First, the type of c and ηn bands occupancy configurations that maximize such absolute values is considered. The second issue addressed in the following is that of the largest charge current absolute value of the reference SzSLNSN subspaces contained in each SzS subspace spanned by η-Bethe states as defined in Section 3.2. Finally, useful further information about the charge currents of the selected reference SzSLNSN subspace is provided.4.1Compact c and αn bands occupancy configurationsIt is straightforward to confirm from manipulations of the TBA equations, Eqs. (A.1) and (A.2) of Appendix A, general η-Bethe-states charge current expression, Eq. (40), and corresponding c and ηn-band holes current functional spectra, Eq. (41), that for SzSL subspaces the class of η-Bethe states that reach the largest current absolute values |〈JˆLWS(lr,Lη,Sη,u)〉| have asymmetrical compact hole β=c,ηn band distributions for Nβh<Nβ and asymmetrical compact pseudoparticle β=c,ηn band distributions for Nβh>Nβ. Such a general charge current expression, Eq. (40), does not directly depend on the type of sn band distributions. In general it rather depends on the corresponding densities ms and ns through the dependence on it of the c and ηn-band holes current functional spectra, Eq. (41). Hence for simplicity we consider in general symmetrical compact sn bands distributions.The general form of the general compact β=c,ηn,sn bands distributions of the class of η-Bethe states with largest current absolute values is thus,(51)ForNβ≤Nβhwhereβ=c,ηn:Nβ,A(qj)=0andNβ,Ah(qj)=1forqj∈[qβ−,qFβ,+−]andqj∈[qFβ,++,qβ+]Nβ,A(qj)=1andNβ,Ah(qj)=0forqj∈[qFβ,+−,qFβ,++]ForNβh≤Nβwhereβ=c,ηn:Nβ,A(qj)=1andNβ,Ah(qj)=0forqj∈[qβ−,qFβ,−−]andqj∈[qFβ,−+,qβ+]Nβ,A(qj)=0andNβ,Ah(qj)=1forqj∈[qFβ,−−,qFβ,−+]Forsnbands:Nsn,S(qj)=1forqj∈[qFsn−,qFsn+]otherwiseNsn,S(qj)=0. The two limiting occupancy momenta of each band are related to each other and are such that,(52)qFβ,+−∈[qβ−,qβ+−2πnβ],qFβ,++=qFβ,+−+2πnβforNβ≤NβhqFβ,−−∈[qβ−,qβ+−2πnβh],qFβ,−+=qFβ,+−+2πnβhforNβh≤Nβwhereβ=c,ηnqFsn±=±π(Nsn−1)/L≈±πnsn.The use of both the compact momentum distributions of general form, Eq. (51), and of the distributions 2πρc(k)=∂qc(k)/∂k and 2πσαn(Λ)=∂qαn(Λ)/∂Λ, Eq. (A.4) of Appendix A, in the general current expression, Eqs. (40) and (41), straightforwardly leads to the following simplified form of the charge currents of the η-Bethe states associated with such compact momentum distributions,(53)〈JˆLWS(lr,Lη,Sη,u)〉=Ltπ∑ι=±(ι)(τcoskc(qFc,τι)+∑n=1∞τn2n∑ι′=±11−(Ληn(qFηn,τι)−iι′nu)2). The indices,(54)τ=+forNc≤Nch=−forNch≤Nc,τn=+forNηn≤Nηnh=−forNηnh≤Nηn, refer here to the β=c,ηn bands particle-like and hole-like asymmetric compact distributions of such η-Bethe states. The simplified current expression, Eq. (53), involves the β=c,ηn rapidity functional at merely the two occupancy limiting momenta qFβ,τ±, Eq. (52).Another type of compact distributions considered in our study refers to η-Bethe states for which they are symmetrical for all β=c,ηn,sn branches,(55)Nβ,S(qj)=1forqj∈[qFβ−,qFβ+]otherwiseNβ,S(qj)=0whereqFβ±=±πnβ, where that qFβ±=±πnβ holds in the TL upon ignoring π/L corrections.Useful quantities are the β=c,ηn bands holes elementary currents jβh(qj) and β=c,ηn pseudoparticle elementary currents jβ(qj)=−jβh(qj) of a η-Bethe state generated from a reference η-Bethe state with compact distributions of form Eq. (51) or (55) by small β=c,ηn band distribution deviations. They are defined as the deviations in the charge current 〈JˆLWS(lr,Lη,Sη,u)〉, Eq. (40), upon addition onto a reference η-Bethe state with compact distributions of form, Eq. (51), of one β-band hole of momentum qj and one β pseudoparticle of momentum qj, respectively. Relying on techniques similar to those used in Ref. [35] for the excited η-Bethe states of a ground state, one finds that such β=c,ηn elementary currents read,(56)jch(qj)=−jc(qj)=vc(qj)+12π∑ι=±(ι)(τfcc(qj,qFc,τι)+∑n=1∞τn2nfcηn(qj,qFηn,τnι)),jηnh(qj)=−jηn(qj)=−2nvηn(qj)−12π∑ι=±(ι)(τfηnc(qj,qFc,τι)+∑n′=1∞τn′2n′fηnηn′(qj,qFηn′,τn′ι)). The expressions of the β=c,ηn fββ′ functions and group velocities vβ(qj) appearing here are defined in Eqs. (F.1) and (F.2) of Appendix F, respectively. In that Appendix all quantities involved in such expressions are also defined.For a given η-Bethe state, the β=c,ηn bands holes current spectra Jβh(qj)=−Jβ(qj), Eq. (41), and the β=c,ηn bands holes elementary currents jβh(qj)=−jβ(qj), Eq. (56), are related yet in general different quantities. Indeed, they are generated from the different energy spectra Eβ(qj), Eq. (A.7) of Appendix A, and εβ(qj)=Eβ(qj)+εβc(qj), Eq. (F.3) of Appendix F, respectively. Therefore, jβh(qj) can be written as jβh(qj)=Jβh(qj)+δJβh(qj) where δJβh(qj) is a well-defined quantity that vanishes in some finite-u subspaces and more generally for u→∞.While the β=c,ηn band current spectra Jβh(qj)=−Jβ(qj), Eq. (41), refer to the charge current of a η-Bethe state, Eq. (40), the β=c,ηn band elementary currents jβh(qj)=−jβ(qj), Eq. (56), are associated with the difference of the charge currents of two η-Bethe states whose β=c,ηn bands occupancies differ in the TL only in those of a finite number of β=c,ηn pseudoparticles. In terms of β=c,ηn pseudoparticle elementary currents such current deviations read,(57)δ〈JˆLWS(lr,Lη,Sη,u)〉=∑j=1LδNc(qj)jc(qj)+∑n=1∞∑j=1LηnδNηn(qj)jηn(qj). In the present case this refers within the TL to the charge current deviation of a given η-Bethe state relative to that of the η-Bethe state with compact β=c,ηn bands distributions from which it is generated by a finite number of β=c,ηn pseudoparticle processes.It follows from the exact properties considered in Section 3.1 that the charge currents vanish both in the Mη=2Sη→0 and Nρ=(Nc+Nη)→0 limits, respectively. That Nρ=(Nc+Nη)→0 and thus Nc+∑n=1∞Nηn→0 implies that Nc+∑n=1∞2nNηn→0 and thus that Nc+2Πη=(L−2Sη)→0. The η-Bethe states corresponding to these limits have compact distributions. We consider two types of states. Namely, η-Bethe states that are generated from mη→0 states by creation of a finite number of unpaired physical η-spins 1/2. Moreover, η-Bethe states that are generated from lη→1 states and thus mη→1 states by creation of a finite number of charge pseudoparticles. One finds within the TL from the use of Eq. (57) that the charge currents of both such two types of states can be written as,(58)〈JˆLWS(lr,Lη,Sη,u)〉=∑j=1LNc(qj)jc(qj)+∑n=1∞∑j=1LηnNηn(qj)jηn(qj). Hence within the TL this charge current expression is valid for both η-Bethe states for which (i) mη≪1 and (ii) mη→1 provided that lη→1, respectively. Its validity implies that the qj sums in Eq. (A.10) of Appendix A and Eq. (58), respectively, lead to exactly the same charge current. It does not imply though that the β=c,ηn pseudoparticle current spectra Jβ(qj), Eq. (A.11) of Appendix A, and β=c,ηn pseudoparticle elementary currents jβ(qj)=−jβh(qj), Eq. (56), in these sums, respectively, are equal.The ground state associated with each canonical ensemble is not populated by ηn pseudoparticles and sn pseudoparticles with n>1 spin-singlet pairs. The corresponding distributions refer to a particular case of those given in Eq. (55). Within the TL, it has c and s1 bands compact and symmetrical distributions,(59)NcGS(qj)=1forqj∈[qFc−,qFc+]otherwiseNβA(qj)=0Ns1GS(qj)=1forqj∈[qFs1−,qFs1+]otherwiseNβA(qj)=0. where, except for π/L corrections, qFc±=±2kF and qFs1±=±kF for mηz≥0 and msz=0.4.2The reference subspaces largest charge current absolute valueThe T>0 charge stiffness expression, Eq. (42), depends on the charge currents of η-Bethe states belonging to SzS subspaces. For each fixed density mη in the range mη∈[|mηz|,1] there is a large number of SzSLNS subspaces as defined in Section 3.2. They are spanned by a set of η-Bethe states with fixed values of lη, nη, ns, and ms in the intervals lη∈[mη,1], nη∈[0,(lη−mη)/2], ns∈[0,(1−lη)/2], and ms∈[0,(1−lη−ns)], respectively.The use of the simplified current expression, Eq. (53), of the η-Bethe states with compact distributions, Eq. (51), plays a key role in the present analysis. From it one finds that each SzSLNS subspace largest charge current absolute value |〈JˆLWSmax(lr,Lη,Sη,u)〉| has the general form,(60)|〈JˆLWSmax(lr,Lη,Sη,u)〉|lη,nη,ms,ns=Clη,nη,ms,nstLmη(1−mη), where the coefficient Clη,nη,ms,ns depends on u and on the densities lη, nη, ms, and ns.A SzSL subspace contains a set of SzSLN subspaces, one for each density lη in the interval lη∈[mη,1]. Within the procedures used in this paper to derive suitable upper bounds, it is convenient to consider three limiting reference SzSLN subspaces, which we call reference SzSLN subspace 1, 2, and 3, respectively. On the one hand, each SzSL subspace only contains one reference SzSLN subspace 1 for which lη→mη and thus nη→0. On the other hand, it contains a set of SzSLN subspaces for which lη→1, each corresponding to a fixed nη density in the interval nη∈[0,(1−mη)/2]. Out of those, it only contains one reference SzSLN subspace 2 and one reference SzSLN subspace 3 for which nη→(1−mη)/2 and nη→0, respectively.Since lη→1 implies that ls→0 and thus that ms→0 and ns→0, the reference SzSLN subspaces 2 and 3 only contain one SzSLNSN subspace each, which we call reference SzSLNSN subspaces 2 and 3, respectively. Indeed, they are at the same time SzSLN subspaces, SzSLNS subspaces, and SzSLNSN subspaces. In contrast, a reference SzSLN subspace 1 contains a set of SzSLNS subspaces, one for each density ms in the interval ms∈[0,(1−mη)]. Furthermore, inside each of the latter subspaces there is in general a set of SzSLNSN subspaces for each density ns in the range ns∈[0,(1−mη−ms)/2]. The reference SzSLNSN subspaces 1A and 1B considered here have a fixed density ms in the interval ms∈[0,(1−mη)] and a maximum and a minimum density ns→(1−mη−ms)/2 and ns→0, respectively.The β=c,ηn bands for which Nβ>0 of the η-Bethe states that span the four reference SzSLNSN subspaces 1A, 1B, 2, and 3 under consideration have occupancies such that Nβh=2Sη=mηL. Hence the contributions to the charge current of such β=c,ηn bands are free of virtual elementary charge currents cancelling, which much simplifies the calculation of that current. The main effect of the virtual elementary current cancelling processes occurring for u>0 in the remaining set of SzSLNSN subspaces of a SzS subspace corresponding to intermediate values of the densities lη∈[mη,1], nη∈[0,(lη−mη)/2], ms∈[0,(1−lη)], and ns∈[0,(1−lη−ms)/2] that label these subspaces is that the corresponding set of largest charge current absolute values, Eq. (60), are in the TL continuous functions of such densities. They smoothly vary between the largest charge current absolute values of the reference SzSLNSN subspaces 1A, 1B, 2, and 3 considered here.For such limiting reference subspaces the use of the TBA equations, Eqs. (A.1) and (A.2) of Appendix A, allows the derivation of the general simplified current expression, Eq. (53), for η-Bethe states with β=c,ηn,sn bands compact distributions of the general form, Eq. (51). Often such current expressions have though only simple analytical form in the u→0 and u≫1 limits. Within the set of limiting reference SzSLNSN subspaces 1A, 1B, 2, and 3, the problem is most complex for the reference SzSLNSN subspace 1A, its analysis being addressed in more detail below in Section 4.3. In addition to the direct use of the equivalent charge current 〈JˆLWS(lr,Lη,Sη,u)〉 expressions, Eq. (53) and Eq. (A.10) of Appendix A, to derive the coefficient Clη,nη,ms,ns in Eq. (60), one can use its limiting expressions, Eq. (58), which are valid in the mη≪1 limit and in the (1−mη)≪1 limit provided that lη→1.For a reference SzSLNSN subspace 1A of a SzS subspace as defined above one has that nc→(1−mη), nηn→0 for n=1,...,∞, ns1→(1−mη−ms)/2, and nsn→0 for n>1. Hence the coefficient Clη,nη,ms,ns, Eq. (60), only depends on u and on the subspace fixed densities mη and ms and is thus called here Cmη,ms. As further discussed in Section 4.3, one finds that in this subspace that coefficient has the limiting behaviors,(61)Cmη,ms=4formη→0andms→0=2formη∈[0,1]andms→1−mη=2formη→1andms→0, for u→0 and,(62)Cmη,ms=2(1+12(ln2u)2+O(u−4))formη→0andms→0=2(1+12(1−msu)2+O(u−4))formη→0andms→1=2formη→1andms→0, up to u−3 order, which is a good approximation for approximately u>3/2. (Corresponding expansions of the coefficient Cmη,ms up to u−3 order and valid for the whole mη∈[0,1] interval are given in Section 4.3.) For u>0 and ms∈[0,(1−mη)], the coefficient Cmη,ms smoothly increases upon increasing mη from mη=0 until reaching a maximum value at an u-dependent intermediate density mη. Upon further increasing mη, it is a continuous decreasing function of mη. The largest Cmη,ms value refers for any density mη∈[0,1] to the reference SzSLNSN subspace 1 for which ms→0.The reference SzSLNSN subspace 1B of a SzS subspace is such that nc→(1−mη), nηn→0 for n=1,...,∞, Nsn=1 for n=(L−2Sη−2Ss)/2, and nsn′→0 for n′=1,...,∞ (including for n′=n within the TL). In the case of this subspace, the coefficient Clη,nη,ms,ns, Eq. (60), is for u>0, mη∈[0,1], and ms∈[0,(1−mη)] found to be independent of u and ms, so that it is here denoted by Cmη. It is found to read,(63)Cmη=2πsin(πmη)mη(1−mη)formη∈[0,1]andms∈[0,(1−mη)]. It increases and decreases upon increasing mη within the ranges mη∈[0,1/2] and mη∈[1/2,1], respectively, reaching a maximum value 8/π at mη=1/2. Its limiting behaviors are,(64)Cmη=2formη→0andms∈[0,1]=8/πformη=1/2andms∈[0,1/2]=2formη→1andms→0.For the reference SzSLNSN subspace 2 of a SzS subspace one has that nc→0, nη1→(1−mη)/2, nηn→0 for n>1, and nsn→0 for n=1,...,∞. The corresponding coefficient Clη,nη,ms,ns, Eq. (60), only depends on u and mη, so that we call it Cmη. Its limiting behaviors are found to be given by,(65)Cmη=2foru→0,mη→0,andms→0=2foru→0,mη→1,andms→0=π2uforu≫1,mη→0,andms→0=1uforu≫1,mη→1,andms→0. For u>0 the coefficient Cmη is a continuous function of mη with a maximum value at a u-dependent intermediate density mη.The reference SzSLNSN subspace 3 of a SzS subspace is such that nc→0, Nηn=1 for n=(L−2Sη)/2, nηn′→0 for n′=1,...,∞ (including for n′=n within the TL), and nsn→0 for n=1,...,∞. The coefficient Clη,nη,ms,ns, Eq. (60), again only depends on u and mη. It is here denoted by Cmη. Its values in the u→0 and u≫1 limits are for the whole mη∈[0,1] range given by,(66)Cmη=2sin(π2mη)mηforu→0,mη∈[0,1],andms→0=O(1/L)=0intheTLforu≫1,mη∈[0,1[,andms→0, respectively. In the u→0 limit it is thus a decreasing function of mη with limiting values,(67)Cmη=πforu→0,mη→0,andms→0=2foru→0,mη→1,andms→0.As mentioned above, the largest charge current absolute value, Eq. (60), and thus the corresponding coefficient Clη,nη,ms,ns of all remaining reference SzSLNSN subspaces contained in a SzS subspace is a continuous and smooth function of the densities lη∈[mη,1], nη∈[0,(lη−mη)/2], ms∈[0,(1−lη)], and ns∈[0,(1−lη−ms)/2]. Such a coefficient Clη,nη,ms,ns varies between the limiting values of those of the limiting reference SzSLNSN subspaces 1A, 1B, 2, and 3. From analysis and comparison of the whole corresponding set of largest charge current absolute value |〈JˆLWSmax(lr,Lη,Sη,u)〉|, Eq. (60), one finds that the larger coefficients Clη,nη,ms,ns are, on the one hand concerning the density lη∈[mη,1], reached for some of the reference SzSLNSN subspaces contained in the reference SzSLN 1 for which lη→mη. Concerning the density ns∈[0,(1−mη−ms)/2] of the subset of reference SzSLNSN subspaces contained in the reference SzSLN 1 for which lη→mη and thus nη→0 and ms∈[0,(1−mη)], one finds in turn that the larger coefficients Clη,nη,ms,ns in Eq. (60) are reached for the SzSLNSN subspaces 1A for which ns→(1−mη−ms)/2 where ms∈[0,(1−mη)]. This largest charge current absolute value of each SzS subspace is thus that used in the upper bound procedures considered in the following and within the canonical ensemble in Section 5. Actually, the absolute largest coefficient Clη,nη,ms,ns is found to be that of the SzSLNSN subspace 1A of all SzS subspaces for which ms→0.In Appendix C the effect of varying u on the physical microscopic processes behind the largest charge current absolute value of a SzS subspace reported in this section is addressed. (That such an effect is discussed in an Appendix follows from the remaining studies of this paper accounting for it but not needing its detailed analysis, which requires a relative long account that would affect the information flow on the main issues addressed in the following. The information provided in Appendix C is though physically important, its presentation in that Appendix contributing to the further understanding of the microscopic mechanisms behind the 1D Hubbard model charge transport properties.)As discussed in Appendix B for mηz=0 and in Appendix C for mηz∈[0,1], the physics is very different (i) at u=0 and in the u→0 limit and (ii) for finite u. In the latter Appendix it is shown that due to the u→0 unbinding of the η-spin singlet pairs that at finite u are bound within the composite ηn pseudoparticles, the charge carriers that interchange position with the Mη unpaired physical η-spins 1/2 that couple to charge probes are different in the u→0 limit and for finite u. Their numbers read Nc+2Πη=L−2Sη and Nρ=Nc+Nη, respectively.Another issue discussed in Appendix C refers to the similarities and differences relative to the case of the spin stiffness and currents of the spin-1/2 XXX chain, which are studied in Refs. [72,73] by the upper-bound method used in this paper. There is an advantage of that upper-bound method relative to more common numerical approximations used to address the stiffness problem and the charge currents contributing to it. It is that such a method directly refers to a representation in terms of which the microscopic mechanisms under consideration in the discussions of Appendix C refer to elementary processes of the fractionalized particles whose configurations generate the exact energy and momentum eigenstates. Indeed, in terms of processes of the physical particles, the electrons, the present quantum problem is non-perturbative, so that the microscopic mechanisms that control the charge currents corresponds to a much more complex many-particle problem. This is one of the reasons why such mechanisms remain hidden under the use of standard numerical techniques, which usually rely on the electron representation of the problem.Other techniques that rely on the direct use of the BA quantum numbers without accounting for their relation to the integrable models physical particles [12,15] also pause technical problems. This occurs for instance within the use of phenomelogical spinon and antispinon representations of the BA quantum numbers whose relation to the physical particles remains undefined [15]. Moreover, in the case of some integrable models such as the present 1D Hubbard model, divergences emerge at the densities at which the Mazur's inequality is inconclusive in the stiffness expressions obtained from the second derivative of the energy eigenvalues relative to the uniform vector potential obtained from the BA. These divergences can though be avoided. This is accomplished if one rather expresses the stiffness in terms of charge current operator expectation values, as within the method used in this paper.4.3The SzS subspaces largest charge current absolute valueIn this section we provide further useful information on the largest charge current absolute value |〈JˆLWSmax(lr,Sη,u)〉| of a reference SzSLNSN subspace 1A, which for any fixed density ms is the largest charge current absolute value of the corresponding SzS subspace. (Here Lη was removed from |〈JˆLWS(lr,Lη,Sη,u)〉| because Lη=2Sη for a reference SzSLN subspace 1 where the SzSLNSN subspaces 1A are contained.)For the 1D Hubbard model in a reference SzSLN subspace 1 the general η-Bethe-state charge current expression, Eq. (40), simplifies to,(68)〈JˆLWS(lr,Sη,u)〉=∑j=1LNch(qj)Jch(qj)whereJch(qj)=2tsinkc(qj)2πρc(kc(qj))forqj∈[−π,π]. Moreover, for η-Bethe states with c and sn bands compact distributions of general form, Eq. (51), belonging to the reference SzSLN subspace 1, the charge current expression, Eq. (53), further simplifies for densities |mηz|∈[0,1] and mη∈[|mηz|,1],(69)〈JˆLWS(lr,Sη,u)〉=τLtπ∑ι=±(ι)coskc(qFc,τι). Thus this applies to any SzSLNSN subspace contained in a reference SzSLN subspace 1 whose densities ms∈[0,(1−mη)] and ns∈[0,(1−mη−ms)]/2 have fixed values.In the following we consider the reference SzSLNSN subspaces 1A of interest for our upper-bound procedures for which ns→(1−mη−ms)/2 where the density ms has a fixed value in the interval ms∈[0,(1−mη)]. The derivation of the c-band current spectrum Jch(qj) in Eq. (68) involves the solution of the TBA equations, Eqs. (A.1) and (A.2), to derive the momentum rapidity function kc(qj) and related distribution 2πρc(kc(qj)) of the η-Bethe states that span the reference SzSLN subspaces 1A of a SzS subspace. This is simplest to be accomplished in terms of u−1 expansions of such a function and distribution, which for densities mη∈[0,1] and ms∈[0,(1−mη)] leads to the following universal expansion for Jch(qj) up to u−2 order,(70)Jch(qj)=2tsinqj−2tnηsusin2qj+6t(nηsu)2(1−32sin2qj)sinqj. Here,(71)nηs=(1−mη−ms)gswheregs=ln2forms→0andgs=1forms→1−mη. gs=gs(ms)∈[ln2,1] is in these equations a continuous increasing function of the spin density ms.For j>2 orders u−j the calculations become a more complex technical problem. Analysis of the interplay of the TBA equations with those that define the charge operator expectation values reveals that the c-band current spectrum Jch(qj) expansion terms of such j>2 orders are state dependent. As an example, two current spectra expansions up to u−3 order are derived in Appendix G for η-Bethe states that span two reference SzSLNSN subspaces 1A with limiting spin densities ms=0 and ms→1−mη, respectively. For the former spin density, the obtained expansion refers to η-Bethe states with compact distributions of general form, Eq. (51), belonging to the reference SzSLNSN subspaces 1A under consideration and η-Bethe states generated from those by a finite number of c-band particle-hole processes. It reads,(72)Jch(qj)=2tsinqj−2t(1−mη)ln2usin2qj+6t((1−mη)ln2)2u2(1−32sin2qj)sinqj−4t((1−mη)ln2)3u3(1−83sin2qj)sin2qj+3tζ(3)16u3((1−mη)(1+43sin2qj)+3τ2π∑ι=±(ι)cos(qFc,τι)(sinqj−13sin(qFc,τι)))sin2qj. That its terms up to u−2 order and of u−3 order do not depend and depend on the limiting momenta qFc,τι associated with the states with compact and asymmetrical c-band distributions considered here is consistent with their state independence and state dependence, respectively.The expansion up to u−3 order of Jch(qj) obtained in Appendix G for the reference SzSLNSN subspaces 1A with spin ms→1−mη is of second order in (1−mη−ms)≪1 and is given by,(73)Jch(qj)=2tsinqj−2t(1−mη−ms)usin2qj+6t(1−mη−ms)2u2(1−32sin2qj)sinqj+4t3(1−mη−ms)u3sin2qjsin2qj+O((1−mη−ms)3). The reference SzSLNSN subspaces 1A associated with the (1−mη−ms)≪1 limit is the only one for which the terms up to second order in (1−mη−ms) of Jch(qj) are state independent for any u>0 value. Note that the terms of u−3 order in Eqs. (72) and (73), respectively, have a completely different form for the two reference SzSLNSN subspaces 1A for which ms=0 and ms→1−mη, respectively.The use in the general expansion, Eq. (70), of Jch(qj) up to u−2 order valid for the densities intervals mη∈[0,1] and ms∈[0,(1−mη)] of the reference SzSLN subspaces 1A of the nηs=(1−mη)ln2 and nηs=(1−mη−ms) values of the function gs given in Eq. (71) specific to its ms=0 and ms→1−mη reference SzSLNSN subspaces 1A recovers the terms up to u−2 order in Eqs. (72) and Eq. (73), respectively. The calculations reported in Appendix G to derive the expansions given in these equations are more complex for the ms=0 reference SzSLNSN subspaces 1A than for that for which ms→1−mη. For the former ms=0 subspace, the distribution 2πρc(k) in the Jch(qj) expression in Eq. (68) and the function qc(k) that is the inverse of the momentum rapidity functional kc(q) also appearing in that expression are in Appendix G expanded in powers of u−j for all j=1,...,∞ orders.From the use of Eq. (70) in the general expression for 〈JˆLWS(lr,Sη,u)〉 in Eq. (68) with compact distributions of general form, Eq. (51), one finds the following expansion up to u−2 order of the charge current expression, Eq. (69), valid for the reference SzSLN subspace 1A,(74)〈JˆLWS(lr,Sη,u)〉=τLtπ∑ι=±(ι)cos(qFc,τι)+τLtπnηsu∑ι=±(ι)sin2(qFc,τι)−τ3Lt2π(nηsu)2∑ι=±(ι)sin2(qFc,τι)cos(qFc,τι). Here qFc,τι with τ=± and ι=± are the c band limiting occupancy momenta in Eq. (52) for β=c. Terms of u−3 order of the charge current expression, Eq. (69), are derived in Appendix G for the ms=0 and ms→1−mη reference SzSLNSN subspaces 1A and the whole mη∈[0,1] range, with the results,(75)〈JˆLWS(3)(lr,Sη,u)〉=τ2Ltπ((1−mη)ln2u)3∑ι=±(ι)(1−43sin2(qFc,τι))sin2(qFc,τι)−τ3ζ(3)Lt32πu3∑ι=±(ι){(1−mη)(1+23sin2(qFc,τι))−τ2π∑ι′=±(ι′)cos(qFc,τι′)(sin(qFc,τι′)−2sin(qFc,τι))}sin2(qFc,τι)forms=0, and(76)〈JˆLWS(3)(lr,Sη,u)〉=−τ(1−mη−ms)Lt3πu3∑ι=±(ι)sin4(qFc,τι)+O((1−mη−ms)3)for(1−mη−ms)≪1, respectively. (The expansion term of u−3 order, Eq. (76), only includes contributions up to second order in (1−mη−ms)≪1.)The following expansion of the limiting occupancy momenta qFc,τι maximizes the u−3 order expansion of the charge current absolute value, Eq. (69), for all densities ranges |mηz|∈[0,1], mη∈[|mηz|,1], and ms∈[0,(1−mη)] of the reference SzSLN subspaces 1A,(77)qFc,−ι=π2+ιπmη+2nηsu+O(u−3)formη∈[0,12−δηsu]=π−(1−ι)πmηformη∈[12−δηsu,12],qFc,+ι=π−(1−ι)π(1−mη)formη∈[12,12+δηsu]=π2+ιπ(1−mη)+2nηsu+O(u−3)formη∈[12+δηsu,1], where ι=± and,(78)δηsu=(1−ms)gsπu+O(u−3). The corresponding largest charge current absolute value of general form, Eq. (60), of such a reference SzSLN subspace can be written as,(79)|〈JˆLWSmax(lr,Sη,u)〉|=Cmη,mstLmη(1−mη), where Cmη,ms stands for the coefficient whose limiting values are provided in Eqs. (61) and (62). It reaches as a function of mη and for u>0 and ms∈[0,(1−mη)] a maximum value at an u-dependent intermediate density mη, being a continuous increasing and decreasing function of mη below and above that density, respectively. At fixed mη, its largest value is reached for the ms=0 reference SzSLNSN subspace 1A.The coefficient Cmη,ms limiting values valid for u→0, which from Eq. (61) read,(80)Cmη,ms=4formη→0andms→0=2formη∈[0,1]andms→1−mη=2formη→1andms→0, follow from those in Eq. (G.21) of Appendix G for the largest charge current absolute value derived in that Appendix for mη≪1 and (1−mη)≪1.Moreover, from the use of the expansion up to u−2 order of |〈JˆLWSmax(lr,Sη,u)〉|, Eq. (G.22) of Appendix G, one finds that up to that order the coefficient Cmη,ms in Eq. (79) is for densities mη∈[0,1] and ms∈[0,(1−mη)] of reference SzSLN subspaces 1A given by,(81)Cmη,ms=2πsin(πmη)mη(1−mη)(1−72(nηsu)2(1−87cos(πmη)−37sin2(πmη)))formη∈[0,12−δηsu]andms∈[0,(1−mη)]=2πsin2(πmη)mη(1−mη)(1+2nηsu(1+32nηsucos(2πmη))cos2(πmη))formη∈[12−δηsu,12+δηsu]andms∈[0,(1−mη)]=2πsin(πmη)mη(1−mη)(1−72(nηsu)2(1+87cos(πmη)−37sin2(πmη)))formη∈[12+δηsu,1]andms∈[0,(1−mη)]. Both for mη∈[0,1/2−δηsu] and mη∈[1/2+δηsu,1] the terms of orders u−1, u−3, and remaining odd orders u−j where j=5,7,… of this coefficient expansion exactly vanish.Finally, from the use of Eqs. (G.23) and (G.24) of Appendix G one finds that the terms of u−3 order of the coefficient Cmη,ms in Eq. (79) are for the ms=0 and ms→1−mη reference SzSLNSN subspaces 1A and the whole density interval mη∈[0,1] given by,(82)Cmη,ms(3)=0+O(u−4)formη∈[0,12−δηsu]andms=0=sin2(2πmη)πmη(1−mη){2((1−mη)ln2u)3(1−43sin2(2πmη))+3ζ(3)16u3((1−mη)(1+23sin2(2πmη))+1π(1−12cos(2πmη))sin(2πmη))}+O(u−4)formη∈[12−δηsu,12+δηsu]andms=0=0+O(u−4)formη∈[12+δηsu,1]andms=0, and(83)Cmη,ms(3)=0+O(u−4)formη∈[0,12−δηsu]andms→1−mη=sin4(2πmη)3πu−3(1−mη−ms)mη(1−mη)+O((1−mη−ms)3)formη∈[12−δηsu,12+δηsu]andms→1−mη=0+O(u−4)formη∈[12+δηsu,1]andms→1−mη, respectively. (The terms in Eq. (83) only include the contributions up to second order in (1−mη−ms)≪1.)5Charge stiffness upper bounds within the canonical ensembleThe function,(84)FUB(mη,u)≡|〈JˆLWSmax(lr,Sη,u)〉|2Sη=t(1−mη)Cmη,ms, where Cmη,ms is the coefficient in Eqs. (79)–(83), is a continuous and decreasing function of mη for u>0, mη∈[0,1], and ms∈[0,(1−mη)]. It has limiting behaviors,(85)FUB(mη,u)=4t(1−mη)formη→0,ms→0,andu→0=2t(1−mη)formη∈[0,1],ms→1−mη,andu→0=2t(1−mη)formη→1,ms→0,andu→0=2tsin(πmη)πmηformη∈[0,1],ms∈[0,(1−mη)],andu→∞. Its derivative with respect to mη is such that,(86)∂FUB(mη,u)∂mη=0formη→0,ms∈[0,1],andu>0,∂FUB(mη,u)∂mη<0formη∈]0,1],ms∈[0,(1−mη)],andu>0. It has the limiting behaviors,(87)∂FUB(mη,u)∂mη=0formη→0,ms∈[0,1],andu→0,=−4tformη→1,ms→0,andu→0,=−2tmη(sin(πmη)πmη−cos(πmη))formη∈[0,1],ms∈[0,(1−mη)],andu→∞.A first stiffness upper bound,(88)D⁎(T)=(2Sηz)22LT∑Sη=|Sηz|L/2∑lrplr,Sη,Sηz(FUB(mη,u))2=t2(mηz)2L2T∑Sη=|Sηz|L/2∑lrplr,Sη,SηzCmη,ms2(1−mη)2, is obtained within the canonical ensemble by replacing the moduli of the expectation values 〈JˆLWS(lr,Lη,Sη,u)〉 of η-Bethe states with the same Sη value in the stiffness expression, Eq. (42), by the upper bound of the largest absolute value of the charge current, Eq. (79).For each fixed-Sηz and Ssz=0 canonical ensemble, the largest value of FUB(mη,u) in the Sη summation of Eq. (88) is that referring to the minimum Sη and Ss values, Sη=|Sηz|=mηzL/2 and Ss=|Ssz|=0, respectively, such that mη=mηz and ms=msz=0. This follows from the function FUB(mη,u) smoothly decreasing upon increasing mη. The same applies upon increasing ms at finite u. A second stiffness upper bound is then reached by replacing in Eq. (88) the function FUB(mη,u) by its largest value,(89)FUB(mηz,u)=tCmηz,0(1−mηz)formηz∈[0,1]. Here Cmηz,0 is obtained by replacing mη and ms by mηz and msz=0, respectively, in the expression of the coefficient Cmη,ms. The state summations in Eq. (88) can then be performed exactly for all finite temperatures T>0. Indeed, the probability distribution plr,Sη,Sηz in each fixed-Sηz canonical ensemble is normalized as,(90)∑Sη=|Sηz|L/2∑lrplr,Sη,Sηz=1. Such state summations account for the subspace dimensions and thus as well for the full Ssz=0 subspace dimension. For T>0 the resulting (larger) upper bound D⁎⁎(T)≥D⁎(T)≥D(T), then becomes,(91)D⁎⁎(T)=t2Cmηz,02L2T(1−mηz)2(mηz)2formηz∈[0,1]andmsz=0.For mηz≪1 and msz=0 its values continuously vary from,(92)D⁎⁎(T)=16t2L2T(mηz)2, for u→0 to,(93)D⁎⁎(T)=4t2L2T(mηz)2, for u≫1 whereas for (1−mηz)≪1 and msz=0 it is given by,(94)D⁎⁎(T)=4t2L2T(1−mηz)2, for all u>0 values.In the u→∞ limit, it has the following simple expression for msz=0 and the whole mηz∈[0,1] interval,(95)D⁎⁎(T)=(2t/π)2sin2(πmηz)L2T=(2t/π)2sin2(π(1−mηz))L2Tformηz∈[0,1]andmsz=0.The charge stiffness of the 1D Hubbard model was studied in Ref. [13] for large u, where it was shown to exactly vanish in the mηz→0 limit. As found in that reference, for mηz finite the charge stiffness of the 1D Hubbard and that for spinless fermions alone are different even in the u→∞ limit, as illustrated in Fig. 1 of that reference for a finite system. Such a different behavior persists in the TL and is due to the spinless-fermion phase shifts imposed by the spins 1/2 [56].The coefficient cc in the upper bound, Eq. (6), then smoothly varies from cc=16 for u→0 to cc=4 for u≫1 whereas the coefficient cc′ in the upper bound, Eq. (7), reads cc′=4 for the whole u>0 range. This completes our finding of a vanishing charge stiffness in the TL, L→∞, within the canonical ensemble for any fixed range or even distribution of Sηz, or any distribution of mηz shrinking sufficiently fast that 〈(mηz)2〉L→0.6Stiffness upper bounds within the grand-canonical ensemble for T→∞The average value of the square of the charge current |〈Jˆ(lr,Lη,Sη,Sηz,u)〉|2, Eq. (39), in a fixed-Sηz and Ssz=0 subspace that contains the set of SzS subspaces with η-spin values Sη=|Sηz|,|Sηz|+1,|Sηz|+2,…, reads,(96)〈|〈Jˆ(lr,Lη,Sη,Sηz,u)〉|2〉Sηz=(2Sηz)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑lr|〈JˆLWS(lr,Lη,Sη,u)〉|2(2Sη)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2dsubspaceLWS(Lη,Sη)=(2Sηz)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑lr|〈JˆLWS(lr,Lη,Sη,u)〉|2(2Sη)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑Ss=0(L−Lη)/2(LLη)×((LηLη/2−Sη)−(LηLη/2−Sη−1))×((L−Lη(L−Lη)/2−Ss)−(L−Lη(L−Lη)/2−Ss−1))=(2Sηz)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑lr|〈JˆLWS(lr,Lη,Sη,u)〉|2(2Sη)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑Ss=0(L−Lη)/2(LLη)×(∑{Nηn}∏n=1∞(LηnNηn))×(∑{Nsn′}∏n′=1∞(Lsn′Nsn′)). The subspaces dimensions appearing here are defined in Appendix E and useful information on the corresponding states summations was given in Section 3.2.We denote by |〈JˆLWS〉|A(lη,mη,nρ) a SzSLN subspace current absolute value average. It is given by,(97)|〈JˆLWS〉|A(lη,mη,nρ)=∑lnρ⋆|〈JˆLWS(lnρ⋆,Lη,Sη,u)〉|∑Ss=0(L−Lη)/2(LLη)×(∑{(Nηn)lη,mη,nρ}∏n=1∞(LηnNηn))×((L−Lη(L−Lη)/2−Ss)−(L−Lη(L−Lη)/2−Ss−1)). Here lnρ⋆ denotes lr⋆ for the set of η-Bethe states with fixed values for the densities lη, mη, and nρ that span a SzSLN subspace and the sum ∑lnρ⋆ runs over all c-band, ηn-bands of n=1,...,∞ branches, and sn-band occupancy configurations of spin Ss=0,1,...,(L−Lη)/2 that generate such η-Bethe states. They have the same numbers Mη=2Sη of unpaired η-spins 1/2 and Nρ=Nc+Nη, Eq. (43), of charge pseudoparticles where Nc=L−Lη. The SzSLN subspace dimension ∑lnρ⋆1 is given in the denominator on the right-hand side of Eq. (97). Hence the summation ∑{(Nηn)lη,mη,nρ} runs over all sets of ηn pseudoparticle numbers {Nηn} that obey both the sum rules ∑n=1∞nNηn=(Lη−2Sη)=Πη, Eq. (18) for α=η, and Nη=∑n=1∞Nηn=(Lη−Nη1h)/2, Eq. (30) for α=η.For finite u a reference SzSLN subspace largest charge current absolute value is proportional to MηNρ. That in Eq. (60) it is written as proportional to mη(1−mη) and thus to 2Sη(L−2Sη) follows from the expression given in that equation applying both to u→0 and to finite u. Indeed and as justified in Appendix C, the carriers of charge are different for u→0 and finite u, respectively. As a result, such a largest charge current absolute value can be written as proportional to 2Sη(L−2Sη) and MηNρ for u→0 and finite u, respectively. (If one requires it to apply both to the u→0 limit and to finite u, then it should be written as given in Eq. (60).)That each SzSLN subspace of a SzS subspace is spanned by η-Bethe states with exactly the same number Nρ=Nc+Nη of charge pseudoparticles simplifies the form of the current absolute values average, Eq. (97). Its expression can for u>0 be written in the general form,(98)|〈JˆLWS〉|A(lη,mη,nρ)=Jρ4tmη2Nρ=1LJρ4tMη2Nρ. The coefficient Jρ obeys the inequality Jρ≤1, being of the order of unity. While for u>0 the largest charge current absolute value of a reference SzSLN subspace is proportional to MηNρ, such a subspace average current absolute value, Eq. (97), is proportional to MηNρ. That |〈JˆLWS〉|A(lη,mη,nρ)∝MηNρ stems from the energy and momentum eigenstates that span the SzSLN subspace being generated by all possible occupancy configurations of the Nρ=Nc+Nη charge pseudoparticles.For all SzSLN subspaces contained in a SzS subspace the density mη=mηz in Eq. (98) has a fixed value. This combined with Jρ≤1 being of the order of the unity reveals that the SzSLN subspace in a SzS subspace whose current absolute value average, Eq. (97), is largest is that for which Nρ reaches its maximum value. One finds from Nρ=L−(Lη+Nη1h)/2 where Nη1h=2Sη+∑n=2∞2(n−1)Nηn, Eq. (24) for αn=η1, that the latter maximum value refers to the SzS subspace minimum Nη1h value, which for the corresponding fixed η-spin Sη reads Nη1h=2Sη. This gives Nρ=L−(Lη+2Sη)/2. For general SzSLN subspaces of a SzS subspace for which Nη1h=2Sη and thus Nρ=L−Πη one has that Nηn=0 for n>1, so that Nρ=L−(Lη+2Sη)/2=Nc+Nη1 where Nc=L−2Sη−2Nη1 and thus Nρ=L−2Sη−Nη1. Further maximizing Nρ at the SzS subspace fixed Sη value corresponds to minimizing Lη, which gives Lη=2Sη and thus Nη1=0. This corresponds to reference SzSLN subspace 1 of the SzS subspace under consideration for which lη→mη and thus nη→0, so that it is indeed spanned by a subset of η-Bethe states for which Nρ=Nc. For such states Nρ reaches its maximum value, Nρ=L−2Sη. A reference SzSLN subspace 1 current absolute value average, Eq. (97), can be written as,(99)|〈JˆLWS〉|A(mη)=Jρ14tmη2Nρ=1LJρ14tMη2Nρ. Since lη=mη and nρ=1−lη=1−mη, the index A(lη,mη,nρ) in the general SzSLN subspace current absolute value average, Eq. (98), was for the particular case of the SzSLN subspace 1 denoted by A(mη) in Eq. (99).The spin degrees of freedom do not couple directly to charge probes and the charge currents do not depend on the spin-singlet sn pseudoparticle occupancy configurations associated with the sn-bands momentum distribution functions Nsn(qj). However, for finite u the charge current spectra of the η-Bethe states that span a reference SzSLN subspace 1 depend on the spin density ms and overall spin sn pseudoparticle density ns. As a consequence, the corresponding coefficient Jρ in Eq. (98) also depends on the densities ms and ns.One finds that finite-u η-Bethe states contained in a reference SzSLN subspace 1 with exactly the same c pseudoparticle occupancy configurations have for any fixed density ms in the interval ms∈[0,(1−mη)] the largest charge current absolute values for the SzSLNSN subspaces 1A for which the density ns∈[0,(1−mη−ms)/2] in Eq. (48) has its largest value, ns=nsmax=(1−mη−ms)/2. Only in the u→∞ limit in which all spin configurations are degenerate have these states the same charge currents absolute values. Limiting examples are (i) the SzSLNSN subspace 1A and (ii) the SzSLNSN subspace 1B. Both such SzSLNSN subspaces of a reference SzSLN subspace 1 have fixed densities lη→mη, nη→0, mη∈[0,1], and ms∈[0,(1−mη)]. Their density ns is given by (i) its maximum value ns→(1−mη−ms)/2 and (ii) minimum value ns→0, respectively.We thus consider here a subspace contained in a reference SzSLN subspace 1 that we call SzSLNN1 subspace. It is spanned by η-Bethe states with spin values Ss=0,1,...,L−Lη whose overall number of sn pseudoparticles reads Ns=Ns1=(L−Lη−2Ss)/2 for each such a spin value. Hence the SzSLNN1 subspace corresponds to the set of reference SzSLNSN subspaces 1A, each with a fixed density ms∈[0,(1−mη)]. Its current absolute value average thus reads,(100)|〈JˆLWS〉|A1N(mη)=∑lnρ⁎|〈JˆLWS(lnρ⁎,Lη,Sη,u)〉|∑Ss=0(L−Lη)/2(LLη)×(∑{(Nηn)lη,mη,nρ}∏n=1∞(LηnNηn))×((L−2Sη+2Ss)/22Ss), where ((L−2Sη+2Ss)/22Ss) is the number of independent s1-band occupancy configurations for each of the spin values Ss=0,1,...,L−Lη. The SzSLNN1 dimension ∑lnρ⁎1 is given by the denominator on the right-hand side of Eq. (100). The current absolute value average, Eq. (100), can be written as,(101)|〈JˆLWS〉|A(mη,nsmax)=Jρ1N4tmη2Nρ=1LJρ1N4tMη2Nρ. The difference relative to the current absolute value average, Eq. (98), of the reference SzSLN subspace 1 where the SzSLNN1 subspace is contained is that Jρ1N≥Jρ1.Each SzS subspace only contains one reference SzSLN subspace 1. Since a reference SzSLN subspace 1 only contains one SzSLNN1 subspace, a SzS subspace also only contains one SzSLNN1 subspace. Let 〈Jˆ(lr⋄,Sη,Sηz,u)〉 denote the currents of the energy and momentum eigenstates that span a reduced subspace of the fixed-Sηz and Ssz=0 subspace obtained by replacing each of its SzS subspaces by the corresponding SzSLNN1 subspace. Here lr⋄ stands for all quantum numbers other than Sη, Sηz, and u>0 needed to uniquely define each such an energy and momentum eigenstate. An important quantity for our upper-bound procedures is the average value of the current square |〈Jˆ(lr⋄,Sη,Sηz,u)〉|2 in the reduced subspace under consideration, which reads,(102)〈|〈Jˆ(lr⋄,Sη,Sηz,u)〉|2〉Sηz1N=(2Sηz)2∑Sη=|Sηz|L/2∑lr⋄|〈JˆLWS(lr⋄,Sη,u)〉|2(2Sη)2∑Sη=|Sηz|L/2∑Ss=0(L−2Sη)/2(L2Sη)×((L−2Sη+2Ss)/22Ss). Here ∑lr⋄1=∑Ss=0(L−2Sη)/2(L2Sη)×((L−2Sη+2Ss)/22Ss) is the dimension of that reduced subspace.Jρ1=max{Jρ} is in Eq. (99) for the reference SzSLN subspace 1 the largest coefficient Jρ in Eq. (98) of all SzSLN subspaces contained in a SzS subspace with density mη=mηz. Moreover, the inequality Jρ1N≥Jρ1 involving the coefficients of the current absolute value averages in Eqs. (99) and (101) is valid for all fixed densities mη∈[0,1] of the corresponding reference SzSLN subspace 1 and SzSLNN1 subspace belonging to the same SzS subspace. A consequence of such properties is that the following inequality involving the average values of the square of the charge current in Eqs. (96) and (102) holds,(103)〈|〈Jˆ(lr⋄,Sη,Sηz,u)〉|2〉Sηz1N≥〈|〈Jˆ(lr⋆,Lη,Sη,Sηz,u)〉|2〉Sηz.For high temperature T→∞, the T>0 expression of the charge stiffness, Eq. (42), simplifies to,(104)D(T)=(2Sηz)22LT∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2∑lr|〈JˆLWS(lr,Lη,Sη,u)〉|2(2Sη)2∑Lη=2|Sηz|L∑Sη=|Sηz|Lη/2dsubspaceLWS(Lη,Sη)=〈|〈Jˆ(lr,Lη,Sη,Sηz,u)〉|2〉Sηz2LT. A high temperature T→∞ charge stiffness upper bound,(105)D⋄(T)=(2Sηz)22LT∑Sη=|Sηz|L/2∑lr⋄|〈JˆLWS(lr⋄,Sη,u)〉|2(2Sη)2∑Sη=|Sηz|L/2∑Ss=0(L−2Sη)/2(L2Sη)×((L−2Sη+2Ss)/22Ss)=〈|〈Jˆ(lr⋄,Lη,Sη,Sηz,u)〉|2〉Sηz1N2LT, such that D(T)≤D⋄(T) then follows from the inequality, Eq. (103).As mentioned above, a SzSLNN1 subspace can be divided into a set of reference SzSLNSN subspaces 1A, each with a fixed density ms∈[0,(1−mη)]. In Appendix H it is shown that a corresponding charge stiffness upper bound only involving the Ss=0 contributions from the ms=0 reference SzSLNSN subspace 1A is larger than that given in Eq. (105). This gives our ultimate charge stiffness upper bound within the grand canonical ensemble for the TL and high temperature T→∞,(106)D⋄⋄(T)=(2Sηz)22LT∑Sη=|Sηz|L/2∑lr⋄|〈JˆLWS(lr⋄,Sη,u)〉|2(2Sη)2∑Sη=|Sηz|L/2(L2Sη), where Jch(qj) is the general current spectrum in Eq. (68) for Ss=0. Up to u−2 order it is given in Eq. (70) for Ss=0. It thus reads,(107)Jch(qj)=2tsinqj−2t(1−mη)ln2usin2qj+6t((1−mη)ln2u)2(1−32sin2qj)sinqj. The general current spectrum in Eq. (68) has up to u−2 order the same universal form for all η-Bethe states that span the reference SzSLNSN subspaces 1A, Eq. (70) for ms∈[0,(1−mη)] and Eq. (107) at ms=0.In Appendix H the current spectrum, Eq. (107), is used in the charge stiffness upper bound, Eq. (106), to derive the following exact expansion up to u−2 order of that upper bound valid in the TL for mηz∈[0,1/2],(108)D(T)≤D⋄⋄(T)=cgct22T(mηz)2wherecgc=2π2(1+(ln22u)2).On the one hand, this expression applies to the mηz≪1 limit. The charge stiffness Mazur's lower bound has been derived for T→∞ in Ref. [66]. In the Ssz=0 case considered in the upper-bound studies of this paper, one finds that the charge stiffness Mazur's lower bound DMz(T) can be written as given in Eq. (H.17) of Appendix H. From the combined use of that equation and Eq. (108) one finds that in the mηz≪1 limit of more interest for our study and up to O(u−2) order the charge stiffness is of the form D(t)=cut22T(mηz)2 where the coefficient cu obeys the double inequality,(109)2(1−(1/22u)2)≤cu≤2π2(1+(ln22u)2)formηz≪1. Here 1/2≈0.707 and ln2≈0.693 have near values.On the other hand, the use of the current spectrum, Eq. (107), in the upper bound, Eq. (106), trivially leads in the ne=(1−mηz)≪1 limit to,(110)D(T)=D⋄⋄(T)=cgc′t22T(1−mηz)wherecgc′=2. In the ne=(1−mηz)≪1 limit the upper bound, Eq. (106), equals up to O(u−2) order the charge stiffness, so that the expression, Eq. (110), gives the exact asymptotic behavior in that limit of the charge stiffness for T→∞ in the TL.On the one hand, the coefficient cgc in the upper bound, Eq. (8), smoothly slightly increases from cgc=2π2≈19.74 for u→∞ upon decreasing u at least down to u≈3/2 within the u>3/2 range for which its O(u−2) order expansion remains a good approximation. At u≈3/2 it reads cgc=2π2(1+(ln2/3)2)≈20.79. On the other hand, the coefficient cgc′ in the upper bound, Eq. (9), reads cgc′=2 up to O(u−2).This completes our finding of a vanishing charge stiffness in the TL, L→∞, within the grand-canonical ensemble for T→∞ and any fixed range or even distribution of Sηz, or any distribution of mηz shrinking sufficiently fast that 〈(mηz)2〉→0.7Concluding remarksAt U=0 the charge stiffness D(T) of the 1D Hubbard model is a simple problem in terms of the non-interacting electron representation. It is found to be finite at mηz=0, both at zero and finite temperature. D(T)>0 reaches a maximum value at T=0, maxD(T)=D(0)=2t/π, behaving for low and high temperature T as [D(0)−D(T)]∝T2>0 and D(T)∝1/T, respectively. (The qualitative difference of the U=0 and u>0 physics and the related T>0 transition that occurs at U=Uc=0 is an issue discussed in Appendix B for mηz→0 and mηz=0 and in Appendix C for mηz∈[0,1].)In this paper strong evidence is provided that the charge stiffness of the 1D Hubbard model vanishes at mηz=0 for T>0 and the whole u>0 range in the TL within the canonical ensemble. For finite temperatures this leaves out, marginally, the grand canonical ensemble in which 〈(mηz)2〉=O(1/L). However, the following properties lead us to expect that our prediction remains valid at finite temperatures in the grand-canonical ensemble case, in accord with the usual expectation of the equivalence of ensembles in the TL.First, we have specifically confirmed the validity of this expectation in the limit of very high temperature T→∞. The corresponding high-temperature charge stiffness upper bound, Eq. (108), confirms that for T→∞ the charge stiffness of the 1D Hubbard model vanishes in the TL in the chemical potential μ→μu limit where (μ−μu)≥0 and 2μu is the Mott–Hubbard gap, Eq. (A.9) of Appendix A. That upper bound was computed up to u−2 order, which applies for approximately u>3/2, yet it is expected that similar results apply for u>0.Second, at zero temperature and mηz=0 the charge Drude weight is given in the TL by D(0)=2t/π at U=0 and vanishes for u>0 [35,67]. That it vanishes at T=0 for u>0 reveals that a finite charge stiffness D(T) for T>0 at mηz=0 would result from thermal fluctuations alone. That such fluctuations are largest at high temperature thus provides strong evidence that our T→∞ results within the grand-canonical ensemble apply as well to all temperatures T≥0.Third, the large overestimate of the charge elementary currents we used in deriving the charge stiffness upper bound, Eq. (91), is consistent with such an expectation. Our canonical-ensemble charge stiffness upper bounds in Eqs. (91)–(94) are also valid for T→∞. Their comparison with those provided in Eqs. (108) and (110) within the grand-canonical ensemble confirms an average charge stiffness upper bound overestimation factor coe2=O(L). For instance, coe2 changes for u≫1 from coe2≈(2/π2)L for mηz≪1 to coe2=nρL=Nρ for mηz→1. (For the SzSLN subspace 1 that dominates the contributions to the charge stiffness, one has that 1−mηz=nc=nρ.) In terms of the charge current absolute values upper bounds derived for the canonical ensemble relative those constructed for the grand canonical ensemble, this means for general SzSLN subspaces an average overestimation factor coe≈L that for u≫1 varies from coe≈(2/π2)L for mηz≪1 to coe≈nρL=Nρ for mηz→1. This huge overestimate of the charge elementary BA currents used in the computation of the charge Drude weight upper bound, Eq. (91), provides additional strong evidence that, as for high temperature T→∞, the charge stiffness vanishes for finite temperatures within the grand-canonical ensemble in the TL for chemical potential μ→μu where (μ−μu)≥0.The use of the general formalism of hydrodynamics introduced in Refs. [17,18] provides further strong evidence that the charge or spin stiffnesses vanish at finite temperatures within the grand canonical ensemble when the corresponding chemical potentials vanish. (Within our notation, in the case of the charge degrees of freedom the chemical potential of such references refers to (μ−μu).) The analysis of Refs. [17,18] accounts for in the 1D Hubbard model the entire space of macro-states being in a one-to-one correspondence with particle-hole invariant commuting (fused) transfer matrices, pertaining to a discrete family of unitary irreducible representations of the underlying quantum symmetry. According to the authors of these references, this readily implies vanishing finite-temperature charge or spin Drude weights when the corresponding chemical potentials vanish, irrespective of the interaction strength.The problem studied in this paper refers though to a controversial issue, as different approaches yield contradictory results [11–18,110–112]. This includes different methods based on the same TBA. Indeed we believe that the problem is not the TBA but rather how to use the TBA to access the stiffness of each specific solvable model. As mentioned in Section 1, our u>0 and mηz=0 predictions for D(T) agree with the conjectures of Ref. [11] and the exact u-large results of Ref. [13]. The latter disagree with the prediction of Ref. [12] that D(T) should be finite in the TL for u>0, T>0, and mηz=0. The exact large-u results of Ref. [13], which find that D(T)