The studies of this paper on the spin- 1 / 2 XXZ chain at finite temperatures T > 0 have two complementary goals. The first is to identify the spin carriers of all its S q > 0 energy eigenstates and to show that their spin elementary currents fully control the spin-transport quantities. Here, S q is the q -spin of the continuous S U q ( 2 ) symmetry of the model for anisotropy Δ > 1 . To achieve this goal, our studies rely on a suitable exact valid in throughout the Hilbert space and associated with normal spin operators, rather than fractionalized particles such as spinons. Both the spin stiffness and the zero-field spin-diffusion constant are expressed in terms of thermal expectation values of the square of the elementary currents carried by the spin carriers. Our second goal is to confirm that the zero-field and finite-temperature spin transport is normal diffusive for Δ > 1 . We use two complementary methods that rely on an inequality for the T > 0 spin stiffness and the above thermal expectation values, respectively, to show that the contributions to ballistic spin transport vanish. Complementarily, for T > 0 and Δ > 1 , the spin-diffusion constant is found to be finite and enhanced upon lowering T , reaching its largest yet finite values at low temperatures. Evidence suggests that it diverges in the Δ → 1 limit for T > 0 , consistent with T > 0 anomalous superdiffusive spin transport at Δ = 1 and zero field.
The studies of this paper on the spin- 1 / 2 XXZ chain at finite temperatures T > 0 have two complementary goals. The first is to identify the spin carriers of all its S q > 0 energy eigenstates and to show that their spin elementary currents fully control the spin-transport quantities. Here, S q is the q -spin of the continuous S U q ( 2 ) symmetry of the model for anisotropy Δ > 1 . To achieve this goal, our studies rely on a suitable exact valid in throughout the Hilbert space and associated with normal spin operators, rather than fractionalized particles such as spinons. Both the spin stiffness and the zero-field spin-diffusion constant are expressed in terms of thermal expectation values of the square of the elementary currents carried by the spin carriers. Our second goal is to confirm that the zero-field and finite-temperature spin transport is normal diffusive for Δ > 1 . We use two complementary methods that rely on an inequality for the T > 0 spin stiffness and the above thermal expectation values, respectively, to show that the contributions to ballistic spin transport vanish. Complementarily, for T > 0 and Δ > 1 , the spin-diffusion constant is found to be finite and enhanced upon lowering T , reaching its largest yet finite values at low temperatures. Evidence suggests that it diverges in the Δ → 1 limit for T > 0 , consistent with T > 0 anomalous superdiffusive spin transport at Δ = 1 and zero field.
It is well established that at zero magnetic field h=0, the spin diffusion constant of the spin-21 XXZ chain is for anisotropy $\mathrm{\ensuremath{\Delta}}>1$ finite for both low and high temperatures, implying that the type of spin transport is normal diffusive. Although it is expected that this holds for all finite temperatures T>0, the calculation of the spin diffusion constant for all T is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies $\mathrm{\ensuremath{\Delta}}=1$ and $\mathrm{\ensuremath{\Delta}}>1$. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for T>0. That for $\mathrm{\ensuremath{\Delta}}>1$ such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for T>0. By combining the connection through a Jordan-Wigner transformation of the spin-21 XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ with its Bethe-ansatz solution, where V is the nearest-neighbor Coulomb repulsion and J=2t is twice the hopping integral t, in this paper we also address the issue of the T>0 charge transport of that model at $\ensuremath{\mu}=0$. It is found to be anomalous superdiffusive for V/J=1 and normal diffusive for V/J>1. Our results thus open the door to a key advance in the understanding for all finite temperatures T>0 of the spin transport in the spin-21 XXZ chain at h=0 for anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ and of the charge transport in the 1D lattice spinless fermion model at $\ensuremath{\mu}=0$ for $V/J\ensuremath{\ge}1$.
Abstract The gapped spin- 21 XXZ chain with spin anisotropy Δ>1 is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when Δ>1 , and diverges in the limit Δ→1 . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for Δ>1 , but becomes superdiffusive as Δ→1 . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.
Abstract We present a systematic analytical study of finite-temperature spin diffusion in the gapped spin-21 XXZ chain with easy-axis anisotropy Δ>1 at zero magnetic field. While the finiteness of the spin-diffusion constant Ds(T,Δ) for all T>0 and Δ>1 has recently been established analytically \cite{R01Carmelo2025a,R02Carmelo2025b}, its precise values away from the asymptotic limits remain an outstanding open problem \cite{R03Carmelo2026BJP}. We provide a unified treatment covering three complementary analytical regimes. First, we formulate the two-sided bound Ds∞(Δ)≤Ds(T,Δ)≤DsLT(T,Δ) between the exact high-temperature Bethe-ansatz value and the leading-order low-temperature activated formula; this bound is a conjecture supported by both asymptotic limits and the available numerical data, and it provides qualitative constraints at every temperature. Second, we exploit the high-temperature approximation of Carmelo and Sacramento \cite{R01Carmelo2025a} to obtain quantitative values of Ds throughout the window βJ∈[0,1], tabulated for Δ∈{1.25,1.5,1.75,2.0}. Third, we identify and compute the crossover temperature T∗(Δ) at which the high-temperature approximation meets the low-temperature activated formula; since both expressions are used outside their strict domains of validity at that point, T∗ is a heuristic indicator of the window in which neither closed-form expression is controlled, rather than a sharp regime boundary. We find that T∗(Δ) increases monotonically with anisotropy, ranging from T∗/J≈0.27 at Δ=1.25 to T∗/J≈0.60 at Δ=2.0, and give an explicit expression for it. Finally, we show that the high-temperature series can be resummed exactly into the form Ds∞=(4J/3π)sinhη∑m≥1m/sinh(mη) (Δ=coshη), from which the near-isotropic divergence Ds∞(Δ)≃C0(Δ−1)−1/2 follows with the exact amplitude C0=πJ/(32). These results provide a quantitative analytical picture of spin diffusion over the temperature range accessible to closed-form expressions. The exact high-temperature floor Ds∞(Δ) and the crossover temperature T∗(Δ) serve as reference values against which future tensor-network and generalized-hydrodynamics calculations may be compared.
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.
We present evidence suggesting that spin transport in the gapless phase of the S=1/2 XXZ model is ballistic rather than diffusive. We map the model onto a spinless fermion model whose charge stiffness determines the spin transport of the original model. By means of exact numerical diagonalisation and finite size scaling we study both the stiffness and the level statistics. We show that the stiffness is non-zero at low temperatures so that the transport is ballistic. Our results suggest that the non-zero stiffness is due to the fact that even in the presence of Umklapp scattering a non-zero fraction of states remain degenerate in the thermodynamic limit.
Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.