Whether in the thermodynamic limit, vanishing magnetic field h→0, and nonzero temperature the spin stiffness of the spin-1/2 XXX Heisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for h→0 in the thermodynamic limit of chain length L→∞, at high temperatures T→∞. Our approach uses a representation in terms of the L physical spins 1/2. For all configurations that generate the exact spin-S energy and momentum eigenstates such a configuration involves a number 2S of unpaired spins 1/2 in multiplet configurations and L−2S spins 1/2 that are paired within Msp=L/2−S spin–singlet pairs. The Bethe-ansatz strings of length n=1 and n>1 describe a single unbound spin–singlet pair and a configuration within which n pairs are bound, respectively. In the case of n>1 pairs this holds both for ideal and deformed strings associated with n complex rapidities with the same real part. The use of such a spin 1/2 representation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-1/2 XXX Heisenberg chain in the thermodynamic limit, for high temperatures T→∞, vanishing magnetic field h→0 and within the grand-canonical ensemble.
Whether in the thermodynamic limit, vanishing magnetic field h→0, and nonzero temperature the spin stiffness of the spin-1/2 XXX Heisenberg chain is finite or vanishes within the grand-canonical ensemble remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an upper bound on the stiffness and show that within that ensemble it vanishes for h→0 in the thermodynamic limit of chain length L→∞, at high temperatures T→∞. Our approach uses a representation in terms of the L physical spins 1/2. For all configurations that generate the exact spin-S energy and momentum eigenstates such a configuration involves a number 2S of unpaired spins 1/2 in multiplet configurations and L−2S spins 1/2 that are paired within Msp=L/2−S spin–singlet pairs. The Bethe-ansatz strings of length n=1 and n>1 describe a single unbound spin–singlet pair and a configuration within which n pairs are bound, respectively. In the case of n>1 pairs this holds both for ideal and deformed strings associated with n complex rapidities with the same real part. The use of such a spin 1/2 representation provides useful physical information on the problem under investigation in contrast to often less controllable numerical studies. Our results provide strong evidence for the absence of ballistic transport in the spin-1/2 XXX Heisenberg chain in the thermodynamic limit, for high temperatures T→∞, vanishing magnetic field h→0 and within the grand-canonical ensemble.
Whether at the zero spin density m=0 and finite temperatures T>0 the spin stiffness of the spin-1/2XXX chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we explicitly compute the stiffness at m=0 and find strong evidence that it vanishes. In particular, we derive an upper bound on the stiffness within a canonical ensemble at any fixed value of spin density m that is proportional to m2L in the thermodynamic limit of chain length $L\ensuremath{\rightarrow}\ensuremath{\infty}$, for any finite, nonzero temperature, which implies the absence of ballistic transport for T>0 for m=0. Although our method relies in part on the thermodynamic Bethe ansatz (TBA), it does not evaluate the stiffness through the second derivative of the TBA energy eigenvalues relative to a uniform vector potential. Moreover, we provide strong evidence that in the thermodynamic limit the upper bounds on the spin current and stiffness used in our derivation remain valid under string deviations. Our results also provide strong evidence that in the thermodynamic limit the TBA method used by X. Zotos [Phys. Rev. Lett. 82, 1764 (1999)] leads to the exact stiffness values at finite temperature T>0 for models whose stiffness is finite at T=0, similar to the spin stiffness of the spin-1/2 Heisenberg chain but unlike the charge stiffness of the half-filled 1D Hubbard model.
Whether at zero spin density m=0 and finite temperatures T>0 the spin stiffness of the spin-1/2XXX chain is finite or vanishes remains an unsolved and controversial issue, as different approaches yield contradictory results. Here we provide an exact upper bound on the stiffness within a canonical ensemble at any fixed value of spin density m and show that it is proportional to m2L in the thermodynamic limit of chain length L→∞, for any finite, nonzero temperature. Moreover, we explicitly compute the stiffness at m=0 and confirm that it vanishes. This allows us to exactly exclude the possibility of ballistic transport within the canonical ensemble for T>0.
Whether 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.
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.
Recently there has been a renewed interest in the spectra and role in dynamical properties of excited states of the spin-1/2 Heisenberg antiferromagnetic chain in longitudinal magnetic fields associated with Bethe strings. The latter are bound states of elementary magnetic excitations described by Bethe-ansatz complex non-real rapidities. Previous studies on this problem referred to finite-size systems. Here we consider the thermodynamic limit and study it for the isotropic spin-1/2 Heisenberg XXX chain in a longitudinal magnetic field. We confirm that also in that limit the most significant spectral weight contribution from Bethe strings leads to (k,ω)-plane gapped continua in the spectra of the spin dynamical structure factors S+−(k,ω) and Sxx(k,ω)=Syy(k,ω). The contribution of Bethe strings to Szz(k,ω) is found to be small at low spin densities m and to become negligible upon increasing that density above m≈0.317. For S−+(k,ω), that contribution is found to be negligible at finite magnetic field. We derive analytical expressions for the line shapes of S+−(k,ω), Sxx(k,ω)=Syy(k,ω), and Szz(k,ω) valid in the (k,ω)-plane vicinity of singularities located at and just above the gapped lower thresholds of the Bethe-string states's spectra. As a side result and in order to provide an overall physical picture that includes the relative (k,ω)-plane location of all spectra with a significant amount of spectral weight, we revisit the general problem of the line-shape of the transverse and longitudinal spin dynamical structure factors at finite magnetic field and excitation energies in the (k,ω)-plane vicinity of other singularities. This includes those located at and just above the lower thresholds of the spectra that stem from excited states described by only real Bethe-ansatz rapidities.
Spin chains are correlated quantum models of great interest in quantum systems and materials exhibiting quasi-one-dimensional magnetic properties. Here, we review results on quantum problems associated with spin chains that are beyond the usual spinon paradigm. Alternatively, we use a representation valid in the thermodynamic limit, N→∞, in terms of the N spin-1/2 physical spins of the spin-1/2XXZ chain in its whole Hilbert space. It was originally introduced for the isotropic point in Carmelo et al. [Phys. Rev. B 92, 165133 (2015)], co-authored by David, and more recently extended to spin anisotropies Δ>1 in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)]. The physical-spins representation accounts for the spin-1/2XXZ chain's continuous SUq(2) symmetry parameterized by q=Δ+Δ2-1∈]1,∞] and associated with q-spin Sq. Specifically, in this review we consider two quantum problems that are beyond the spinon representation: (a) Spin Bethe strings of length n that have no spinon representation, contribute to the dynamical properties of the spin-1/2XXZ chain with anisotropy Δ>1 and for n=1,2,3 were experimentally identified and realized in the zigzag materials SrCo2V2O8 and BaCo2V2O8; (b) The spin stiffness associated with ballistic spin transport at arbitrary finite temperature, which involves a huge number of energy eigenstates, many of which are generated in the thermodynamic limit from ground states by an infinite number of elementary processes. As found in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)], the use of the continuous SUq(2) symmetry reveals that for anisotropy Δ>1 the Bethe strings of length n=1,2,3,… describe a number n of physical-spins Sq=0 singlet pairs that for n>1 are bound within a Sq=0 singlet configuration. Their contribution to the spin dynamical structure factor of both the spin-1/2XXZ chain in a longitudinal magnetic field and the spin chains in SrCo2V2O8 is one of the issues addressed in this paper. In addition, the SUq(2) symmetry imposes that only 2Sq out of the N physical spins are the spin carriers. We also review recent results of J. M. P. Carmelo and P. D. Sacramento ["Diffusive spin transport of the spin-1/2 XXZ chain in the Ising regime at zero magnetic field and finite temperature," (submitted) (2024)] concerning the vanishing of the contributions to finite-temperature ballistic spin transport at zero magnetic field. Within the physical-spins representation, this merely follows from the absolute value of the elementary spin currents carried by the M=2Sq spin carriers of all finite-Sq states that contribute to the spin stiffness being finite. Finally, we discuss deviations of the zigzag materials BaCo2V2O8 and SrCo2V2O8 from the one-dimensional physics described the spin-1/2XXZ chain due to selective interchain couplings.