Spin hydrodynamics in the $S=\frac{1}{2}$ anisotropic Heisenberg chain

We study the finite-temperature dynamical spin susceptibility of the one-dimensional (generalized) anisotropic Heisenberg model within the hydrodynamic regime of small wave vectors and frequencies. Numerical results are analyzed using the memory-function formalism with the central quantity being the spin-current decay rate $\gamma (q,\omega )$. It is shown that in a generic nonintegrable model the decay rate is finite in the hydrodynamic limit, consistent with normal spin-diffusion modes. On the other hand, in the gapless integrable model within the XY regime of anisotropy $\Delta <1$ the behavior is anomalous with vanishing $\gamma (q,\omega =0)\propto \left|q\right|$, in agreement with dissipationless uniform transport. Furthermore, in the integrable system the finite-temperature $q=0$ dynamical conductivity $\sigma (q=0,\omega )$ reveals besides the dissipationless component a regular part with vanishing ${\sigma }_{\text{reg}}(q=0,\omega \to 0)\to 0$.

Figure 1

$\gamma (q,\omega =0)$ vs $q$ (in real units) for $\Delta =0.5$, $s=0$, as calculated for a integrable system of $L=22,24,26$, and 28 sites.

Figure 2

High-$T$ (a) spin-relaxation function ${\Phi }^{\prime \prime }(q,\omega )$, (b) projected conductivity $\tilde{\sigma }(q,\omega )$, and (c) spin-current decay rate $\gamma (q,\omega )$ for a nonintegrable AHM at $s=0$, $T=10$, $\Delta =0.5$, ${\Delta }_2=0.6$, as calculated for a system of $L=28$ sites. With dots we present the comparison with a $q=0$ perturbation theory.

Figure 3

Decay rate $\gamma (q,\omega =0)$ vs momentum $q$ for the case as in Fig. 2 with ${\Delta }_2=0.6$, but for different $\Delta =0.5,0.8,1.0$, evaluated for $L=28$. Horizontal lines represent $q=0$ results from the perturbation theory.

Figure 4

High-$T$ (a) ${\Phi }^{\prime \prime }(q,\omega )$, (b) $\tilde{\sigma }(q,\omega )$, and (c) $\gamma (q,\omega )$ for the integrable case at $s=0$, $\Delta =0.5$, evaluated for $L=28$ and $T=10$. Points represent the analytical $\Delta =0$ results for the same parameters and $q$.

Figure 5

Low-$T$ (a) ${\Phi }^{\prime \prime }(q,\omega )$ and (b) $\gamma (q,\omega )$ for the integrable case as calculated for $L=26$, $s=0$, $\Delta =0.5$, and $T=0.7$. Points represent the analytical $\Delta =0$ results for the same parameters and $q$.

Figure 6

$\gamma (q,\omega =0)$ vs $q$ for $s=0$ and different $\Delta \le 1$: (a) low $T=0.7$ and $L=26$ and (b) $T\gg 1$ and $L=28$. Gray dashed lines are guides to the eye.

Figure 7

$\gamma (q,\omega =0)$ vs $q$ for $s=1/4$ and different $\Delta \le 1$: (a) low $T=0.7$ and $L=32$ and (b) $T\gg 1$ and $L=36$. Gray dashed lines are guides to the eye.

Figure 8

$\gamma (q,\omega =0)$ vs $T$ for $\Delta =0.5$, $s=0$, as calculated for a system of $L=26$ sites. Inset: For $\Delta =1$ the comparison between $\gamma (q_1,\omega =0)$ (dashed, red curve) and the bosonization result in Eq. (21) (solid, orange curve).

Figure 9

Sketch of two possible scenarios for (a) ${\sigma }^{\prime }\left(\omega \right)$ and (b) the normalized integrated spectrum $I\left(\omega \right)$.

Figure 10

Low-$\omega$ dependence of (a) ${\sigma }_{\text{reg}}^{\prime }\left(\omega \right)$ and (b) normalized $I\left(\omega \right)$.