Spin Drude weight for the integrable XXZ chain with arbitrary spin

Abstract

Using generalized hydrodynamics (GHD), we exactly evaluate the finite-temperature spin Drude weight at zero magnetic field for the integrable XXZ chain with arbitrary spin and easy-plane anisotropy. First, we construct the fusion hierarchy of the quantum transfer matrices ($T$-functions) and 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 thermodynamic Bethe ansatz (TBA) equations. Then, employing GHD, we calculate the spin Drude weight at arbitrary finite temperatures. As a result, a characteristic fractal-like structure of the Drude weight is observed at arbitrary spin, 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 high-temperature limit of the Drude weight.

Figures (4)

• Figure 1: The QTM $T_{\sigma}(u,v)$ and the double-row transfer matrix $T^{\textrm{DR}}_{\sigma}(u,v)$ defined on the $N\times L$ square lattice, where $N\in 2\mathbb{N}_+$ is the Trotter number.
• Figure 2: Temperature-dependence of the spin Drude weight (divided by $\sigma^2$) $D_\textrm{s}/\sigma^2$ for $J=1$, $S = \sigma / 2$ ($\sigma = 1, 2, 3, 4$) and various anisotropies $\Delta=\cos(\pi/p_0)$. $D_\textrm{s}/\sigma^2$ smoothly varies with respect to $T$ and approaches zero at the isotropic limit $p_0 \to \infty$ ($\Delta\to 1$). For $\sigma=4$, the admissible region of $p_0$ is separated into two intervals: $p_0\in(2,3)$ and $p_0\in(4,\infty)$ as in Table \ref{['interval']}. The behavior of $D_\textrm{s}/\sigma^2$ in each region distinctly differs, reflecting the fact that the structure of the energy spectrum characterized by the strings is essentially different in each interval.
• Figure 3: Popcorn structure of the Drude weight (divided by temperature $\beta=1/T$ and $\sigma^2$) $D_\textrm{s}/(\sigma^2\beta)$ for $J=1$, $S = \sigma / 2$ ($\sigma = 1, 2, 3, 4$). $D_\textrm{s}/(\sigma^2\beta)$ exhibits discontinuities throughout the anisotropy $\Delta=\cos(\pi/p_0)$. The physically admissible region for $\Delta$ depends on $S$. Refer to \ref{['restriction1']} and Table \ref{['interval']} for details.
• Figure 4: High-temperature limit of the spin Drude weight (divided by $\beta = 1/T$) with $J=\pm 1$ and $S = \sigma / 2$ ($1 \le \sigma \le 8$) for the anisotropy dependence. The graph displays popcorn structures, indicating everywhere discontinuity with respect to $\Delta$. For the physically admissible region of $\Delta$, refer to Eq. \ref{['restriction1']} and Table \ref{['interval']}.

Theorems & Definitions (2)

• Lemma B.1
• Lemma C.1