Abstract The gapped spin- $${1\over 2}$$ XXZ chain with spin anisotropy $$\Delta > 1$$ 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 $$\Delta > 1$$ , and diverges in the limit $$\Delta \rightarrow 1$$ . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for $$\Delta > 1$$ , but becomes superdiffusive as $$\Delta \rightarrow 1$$ . 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 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.
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 $V/J\ensuremath{\ge}1$.
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.
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.
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.
Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we derive functional relations (T -and Y -systems) satisfied by the T -functions and certain combinations of them (Y -functions).Through analytical arguments, the Y -system is reduced to a set of non-linear integral equations, equivalent to the TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain 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 reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, similar to the spin-1/2 case.In our approach, the solutions to the TBA equations (i.e., the Y -functions) can be explicitly written in terms of the T -functions, thus allowing for a systematic calculation of the hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .
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.