Anomalous nonequilibrium current fluctuations in the Heisenberg model

We study fluctuation properties of a one-dimensional anisotropic Heisenberg model out of equilibrium, focusing in particular on the gapped regime. Within the open-system setting we study large-deviation properties of the spin current. Numerically evaluating the first four current cumulants in a nonequilibrium stationary state at high energies, we find that the first two cumulants scale with the system size in a diffusive way, while the third and the fourth cumulants do not. This means that the model is not an ordinary diffusive spin conductor.

Figure 1

Illustration of nonequilibrium current distribution: (a) the number of transferred particles $N_t$ for three realizations of a stochastic process. (b) The current $J=N_t/t$ for the same three realizations. Gray shading in (a) and (b) is theoretical long-time standard deviation obtained from the LD theory. In (b) ${\tau }_c\approx 2.4$ marks the convergence time towards a long-time LD behavior; the horizontal dashed line is at theoretical $J_1\approx 0.150$. (c) Histograms of current distribution at three times and the theoretical prediction (full red curves). System length is $L=6$, driving $\mu =0.5$, and $\Delta =1.5$.

Figure 2

Second (a) and fourth (b) current cumulants in equilibrium for different anisotropies $\Delta$. Different points are different system sizes $L$. With $\Delta$ three different phases occur, differing in the asymptotic dependence of cumulants on system size $L$. (c) Convergence of $J_{2,4}$ to zero for $\Delta =1$; the full black line indicates $J_2\sim 0.75/\sqrt{L}$.

Figure 3

Scaling of a nonequilibrium current variance $J_2$ with system size $L$. Full circles are obtained by exact diagonalization; empty circles (with small error bars) are from tDMRG calculation. The full line is $\approx 1.05/L,\mu =0.1$, and $\Delta =1.5$.

Figure 4

Scaling of a nonequilibrium current cumulant $J_3$ (a), and $J_4$ (b), with system size $L$. Full circles are obtained by exact diagonalization; empty circles (with error bars) are from tDMRG calculation. Driving is $\mu =0.1$, and $\Delta =1.5$.

Figure 5

A minimalistic Markov model of diffusion. Full vertical arrows denote transition rates due to a coupling with a bath (Lindblad operators $L_3$ and $L_4)$, while dashed arrows describe in an effective way the effect of diffusive $\sim 1/L$ current. Parameter $\mu$ is the driving strength.