Scaling of diffusion constants in the spin-$\frac{1}{2}$ XX ladder

We study the dynamics of spin currents in the spin-$\frac{1}{2}$ XX ladder at finite temperature. Within linear response theory, we numerically calculate autocorrelation functions for quantum systems larger than what is accessible with exact diagonalization using the concept of dynamical quantum typicality. While the spin Drude weight vanishes exponentially quickly with increasing system size, we show that this model realizes standard diffusive dynamics. Moreover, we unveil the existence of three qualitatively different dependencies of the spin-diffusion coefficient on the rung-coupling strength, resulting from a crossover from exponential to Gaussian dissipation as the rung coupling increases, in agreement with analytical predictions. We further discuss the implications of our results for experiments with cold atomic gases.

Figure 1

Spin-current autocorrelation function $C\left(t\right)$ for $r=J_{\perp }/J_{\parallel }=1$ and $\beta =0$: (a) Relaxation curve for large systems $N\le 36$ and times $t{J}_{\parallel }\le 50$. (b) Saturation at a very small Drude weight is only visible in a semilog plot of (a) for very long times $t{J}_{\parallel }\gg 50$. (c) Since the Drude weight $\overline{C}\lesssim O(1\%)$, $D\left(t\right)\to \text{const}$ for large times.

Figure 2

High-temperature spin Drude weight $\overline{C}(t_1,t_2)$, extracted at very long times $[t_1{J}_{\parallel },t_2{J}_{\parallel }]=[300,400]$, for different coupling ratios $r=J_{\perp }/J_{\parallel }=0.25$, $0.5$, $0.75$, and 1 (symbols). The finite-size scaling follows an exponential $A\left(r\right)e^{-\gamma N}$ [solid line, $\gamma \ne \gamma \left(r\right)$ and $A\left(r\right)=1/r$], suggesting $\overline{C}\to 0$ for $N\to \infty$.

Figure 3

Spin-diffusion constant $D$ vs $r=J_{\perp }/J_{\parallel }$ for $\beta \to 0$. There are apparently three scaling regimes: (i) $r\ll 1$: $D\propto 1/r^2$, (ii) $1\lesssim r\lesssim 2$: $D\propto 1/r$, and (iii) $r\gg 1$: $D=\text{const}$. The $1/r$ curve results from Eq. (7); see text.

Figure 4

Spin-current autocorrelation function $C\left(t\right)$ for $\beta \to 0$: qualitatively different regimes, depending on $r=J_{\perp }/J_{\parallel }$. (a) Weak-$r$ regime: $C\left(t\right)$ decays exponentially, resulting in $D\propto 1/r^2$, as expected from perturbation theory. (b) Strong-$r$ regime: The decay curve agrees with the Gaussian prediction of Eq. (7), in line with the generic behavior $D\propto 1/r$ suggested in Refs. [39, 40]. For large $r$, revivals occur, resulting in $D=\text{const}$.

Figure 5

Spin-current autocorrelation function $C\left(t\right)$ in the XX ladder for rung couplings (a) $r=0.5$, (b) $r=1.5$, and (c) $r=4$ and different temperatures $\beta J_{\parallel }\le 1$. In all cases, $N=28$. Apparently, $C\left(t\right)$ is qualitatively the same for temperatures down to $\beta J_{\parallel }\sim 0.5$.

Figure 6

Spin-current autocorrelation function $C\left(t\right)$ in the XX ladder for rung couplings (a) $r=0.25$, (b) $r=1.5$, and (c) $r=4$ in the high-temperature limit $\beta \to 0$ and $\langle S^z\rangle =0$, $S^z=0$, $S^z=1$. In all cases, $N=34$. Apparently, the cases of half filling $S^z=0$ and almost half filling $S^z=1$ are already practically identical for finite $N$. No significant difference from $\langle S^z\rangle =0$ is visible for $r=0.25$, while differences for larger $r$ are finite-size effects, as illustrated in Fig. 7.

Figure 7

Spin-current autocorrelation function $C\left(t\right)$ in the XX ladder for (a) $\langle S^z\rangle =0$ and (b) $S^z=0$ for a strong rung coupling $r=J_{\perp }/J_{\parallel }=1.5$ in the high-temperature limit $\beta \to 0$. (c) The resulting time-dependent diffusion coefficient, as extracted from $C\left(t\right)$ depicted in (a) and (b). Although finite-size results for $\langle S^z\rangle =0$ and $S^z=0$ differ from each other, they seem to converge to the same value in the thermodynamic limit. Apparently, the convergence of the $\langle S^z\rangle =0$ data is much faster in time, and there are no finite-size effects up to times $t{J}_{\parallel }\sim 10$ when comparing $N=28$ and $N=34$. Hence, our choice of $t_2{J}_{\parallel }=8.3$ for extracting $D$ from $D\left(t\right)$ is reasonable.