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Journal of Physics A Mathematical and Theoretical

Infinite temperature spin dc conductivity of the spin-1/2 XXZ chain

Shinya Ae

Published Sep 23, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1751-8121/ad7e64

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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.
Using the Bethe ansatz method and the TBA 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 an anisotropy 0≤Δ<10 \le Δ<10≤Δ<1. In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies, which allows the exact evaluation of the infinite temperature spin dc conductivity L\mathcal{L}L. We find that L\mathcal{L}L is discontinuous at all rational numbers of the anisotropy parameter p0=π/cos⁡−1Δp_0=π/\cos^{-1}Δp0​=π/cos−1Δ in the region p0≥2p_0 \ge 2p0​≥2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ=1Δ=1Δ=1 point is exceptional. Close to this point, L\mathcal{L}L slowly increases in proportion to the first power of the magnetization. On the other hand L\mathcal{L}L is proportional to the second power of the magnetization when p0p_0p0​ approaches irrational numbers.
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 . . . . . . . . . .
Motivated by recent investigations of transport properties of strongly correlated 1D models and thermal conductivity measurements of quasi 1D magnetic systems, we present results for the integrable spin-½ XXZ chain. The thermal conductivity κ(ω) of this model has ℜ κ(ω) = δ(ω), i.e. it is infinite for zero frequency ω. The weight of the delta peak is calculated exactly by a lattice path integral formulation. Numerical results for wide ranges of temperature and anisotropy are presented. The low- and high-temperature limits are studied analytically.
Based on the ballistic macroscopic fluctuation theory, the integration of the spin correlation function (spin conductivity) is analyzed for the spin-1/2 XXZ chain in the critical regime. In the time when the magnetization of an infinite spin chain fluctuates from an initial state with a wavelength as long as the infinite length NNN, the equal-time two-point spin correlation function is scaled up to O(1/N)O(1/N)O(1/N). In the state where the ballistic spin transport decays at high temperature TTT, the diffusive transport remains on a large scale. We show that the spin conductivity is proportional to 1/T1/T1/T in the limit T→∞T\to\inftyT→∞ and its high temperature proportionality constant diverges in the case where one-quasiparticle magnetization is infinitely large. This analysis informs that the superdiffusive spin transport is driven by the 1/N1/N1/N-scaled long-range spin correlation and sheds a light on the dynamic scaling in spin transport at the isotropic point.
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 $\mathrm{\ensuremath{\Delta}}>1$ 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>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 $\mathrm{\ensuremath{\Delta}}=1$ 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>0T>0T>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>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 $\ensuremath{\mu}=0$, for $V/J\ensuremath{\ge}1$ 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 $\ensuremath{\mu}=0$. 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 $\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$ 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- 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.