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arXiv:2509.08741 · Physical review. B./Physical review. B

Finite-temperature transport in the gapped spin- 1 2 XXZ chain and one-dimensional lattice spinless fermion model

J. M. P. Carmelo, Pedro D Sacramento

Published Aug 26, 2025• 3 citations
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Abstract

It is well established that at zero magnetic field $h=0$, the spin diffusion constant of the spin-$\frac{1}{2}$ 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-$\frac{1}{2}$ 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-$\frac{1}{2}$ 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 $V/J\ensuremath{\ge}1$.

DOI
10.1103/x2zc-xdkb

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It is well established that at zero magnetic field h=0h=0h=0, the spin diffusion constant of the spin-12\frac{1}{2}21​ XXZ chain is for anisotropy \ensuremathΔ>1\mathrm{\ensuremath{\Delta}}>1\ensuremathΔ>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>0T>0T>0, the calculation of the spin diffusion constant for all TTT is typically intractable. Here we consider a class of energy eigenstates that exist both for anisotropies \ensuremathΔ=1\mathrm{\ensuremath{\Delta}}=1\ensuremathΔ=1 and \ensuremathΔ>1\mathrm{\ensuremath{\Delta}}>1\ensuremathΔ>1. We show that at the isotropic point their contributions are behind the diffusion constant being infinite, spin transport being anomalous superdiffusive for T>0T>0T>0. That for \ensuremathΔ>1\mathrm{\ensuremath{\Delta}}>1\ensuremathΔ>1 such states do not contribute to the diffusion constant is shown to imply it is finite, spin transport being normal diffusive for T>0T>0T>0. By combining the connection through a Jordan-Wigner transformation of the spin-12\frac{1}{2}21​ XXZ chain to the one-dimensional (1D) lattice spinless fermion model at zero chemical potential \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0, for V/J\ensuremath≥1V/J\ensuremath{\ge}1V/J\ensuremath≥1 with its Bethe-ansatz solution, where VVV is the nearest-neighbor Coulomb repulsion and J=2tJ=2tJ=2t is twice the hopping integral ttt, in this paper we also address the issue of the T>0T>0T>0 charge transport of that model at \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0. It is found to be anomalous superdiffusive for V/J=1V/J=1V/J=1 and normal diffusive for V/J>1V/J>1V/J>1. Our results thus open the door to a key advance in the understanding for all finite temperatures T>0T>0T>0 of the spin transport in the spin-12\frac{1}{2}21​ XXZ chain at h=0h=0h=0 for anisotropy \ensuremathΔ\ensuremath≥1\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1\ensuremathΔ\ensuremath≥1 and of the charge transport in the 1D lattice spinless fermion model at \ensuremathμ=0\ensuremath{\mu}=0\ensuremathμ=0 for V/J\ensuremath≥1V/J\ensuremath{\ge}1V/J\ensuremath≥1.
Abstract The gapped spin- 12{1\over 2}21​ XXZ chain with spin anisotropy Δ>1\Delta > 1Δ>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\Delta > 1Δ>1 , and diverges in the limit Δ→1\Delta \rightarrow 1Δ→1 . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for Δ>1\Delta > 1Δ>1 , but becomes superdiffusive as Δ→1\Delta \rightarrow 1Δ→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 Generalized hydrodynamics (GHD) has successfully addressed several open questions regarding finite-temperature ( T > 0) transport in one-dimensional (1D) integrable models, including the mechanisms behind anomalous superdiffusive transport. However, disagreements persist concerning finite-temperature charge transport in the 1D Hubbard model at zero chemical potential ( µ = 0). These disagreements do not arise from the GHD framework itself but rather from differing treatments of the model’s global symmetry at the h = μ = 0 point, where h is the magnetic field. Specifically, the controversy concerns whether finite-temperature charge transport at µ = 0 is anomalous superdiffusive, as predicted using GHD when only the SU (2) × SU (2) symmetries are considered, or instead normal diffusive, as found when employing a representation that fully accounts for the model’s global [ SU (2) × SU (2) × U (1)]/ Z 2 2 symmetry at h = μ = 0 . This work aims to contribute to resolving this issue and advancing the understanding of transport in such systems. We begin by briefly reviewing GHD and other methods predictions regarding anomalous superdiffusive spin transport at finite temperature in gapless 1D integrable models such as the spin- 1 / 2 XXX chain at h = 0 and the 1D Hubbard model at h = μ = 0 . We then review T > 0 transport results in gapped 1D integrable models, focusing on charge transport in the 1D Hubbard model at h = μ = 0 and spin transport in the spin- 1 / 2 XXZ chain at h = 0 with anisotropy Δ > 1 . We identify and discuss the origin of the aforementioned discrepancies, showing how their resolution leads to a deeper understanding of anomalous superdiffusion.
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.
Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that in the one-dimensional Hubbard model spin and charge transport are for all temperatures T > 0 anomalous superdiffusive at zero magnetic field, h = 0, and zero chemical potential, μ = 0, respectively. However, this contradicts recent exact results that at very low temperature charge transport rather is normal diffusive. In this Letter we identify the mechanisms that control the different types of temperature dependence of the h = 0 spin and μ = 0 charge transport and find that the latter is normal diffusive for all finite temperatures T > 0, in contrast to the hydrodynamic theory and KPZ scaling predictions.
We identify a class of one-dimensional spin and fermionic lattice models that display diverging spin and charge diffusion constants, including several paradigmatic models of exactly solvable, strongly correlated many-body dynamics such as the isotropic Heisenberg spin chains, the Fermi-Hubbard model, and the t-J model at the integrable point. Using the hydrodynamic transport theory, we derive an analytic lower bound on the spin and charge diffusion constants by calculating the curvature of the corresponding Drude weights at half-filling, and demonstrate that for certain lattice models with isotropic interactions some of the Noether charges exhibit superdiffusive transport at finite temperature and half-filling.
We study magnetization transport in anisotropic spin-12\frac{1}{2}21​ chains governed by the integrable XXZ model with and without integrability-breaking perturbations at high temperatures $(T\ensuremath{\rightarrow}\ensuremath{\infty})$ using a hybrid approach that combines exact sum rules with judiciously chosen Ans\"atze. In the integrable XXZ model we find (i) superdiffusion at the isotropic (Heisenberg) point, with frequency dependent conductivity ${\ensuremath{\sigma}}^{\ensuremath{'}}(\ensuremath{\omega}\ensuremath{\rightarrow}0)\ensuremath{\sim}{|\ensuremath{\omega}|}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=\ensuremath{-}3/7$ in close numerical agreement with recent ttt-DMRG computations; (ii) a continuously drifting exponent from $\ensuremath{\alpha}=\ensuremath{-}{1}^{+}$ in the XY (gapless) limit of the model to $\ensuremath{\alpha}>0$ within the Ising (gapped) regime; and (iii) a diffusion constant saturating in the XY coupling deep in the Ising limit. We consider two kinds of next-nearest-neighbor integrability breaking perturbations---a simple spin-flip term (J2)({J}_{2})(J2​) and a three-spin assisted variant (t2)({t}_{2})(t2​), natural in the fermion particle representation of the spin chain. In the first case we discover a remarkable sensitivity of ${\ensuremath{\sigma}}^{\ensuremath{'}}(\ensuremath{\omega})$ to the sign of J2{J}_{2}J2​, with enhanced low frequency spectral weight and a pronounced upward shift in the magnitude of $\ensuremath{\alpha}$ for J2>0{J}_{2}>0J2​>0. Perhaps even more surprising, we find subdiffusion $(\ensuremath{\alpha}>0)$ over a range of J2<0{J}_{2}<0J2​<0. By contrast, the effects of the ``fermionic'' three-spin perturbation are sign symmetric; this perturbation produces a clearly observable hydrodynamic relaxation. At large strength of the integrability breaking term ${J}_{2}\ensuremath{\rightarrow}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$ the problem is effectively noninteracting (fermions hopping on odd and even sublattices) and we find $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{-}1$ behavior reminiscent of the XY limit of the integrable XXZ chain. Exact diagonalization studies largely corroborate these findings over accessible frequencies.