Large-deviation statistics of a diffusive quantum spin chain and the additivity principle

Using the large-deviation formalism, we study the statistics of current fluctuations in a diffusive nonequilibrium quantum spin chain. The boundary-driven $XX$ chain with dephasing consists of a coherent bulk hopping and a local dissipative dephasing. We analytically calculate the exact expression for the second current moment in a system of any length and then numerically demonstrate that in the thermodynamic limit, higher-order cumulants and the large-deviation function can be calculated using the additivity principle or macroscopic hydrodynamic theory. This shows that the additivity principle can also hold in systems that are not purely stochastic, and can in particular be valid in quantum systems. We also show that in large systems, the current fluctuations are the same as in the classical symmetric simple exclusion process.

Figure 1

Convergence of numerically computed $J_3$ and $J_4$ in the $XX$ dephasing chain toward the additivity prediction (25) shown with full curves. In (a) we use $\mu +\overline{\mu }=1$, in (b) $\mu =0.2$, and in both cases $\gamma =\Gamma =1$.

Figure 2

For equilibrium driving $\mu =0$ and zero average magnetization $\overline{\mu }=0$, fluctuations are Gaussian (for any other driving parameters they are not). (a) The ratio $J_4/J_2$ decays to 0 in the thermodynamic limit. The inset show the same data on a log-log scale. (b) Cumulant generating function $\Lambda \left(s\right)$ converges to theoretical Gaussian (17) (full black curve). We use $\Gamma =\gamma =1$ and $\overline{\mu }=\mu =0$.

Figure 3

Large deviation function for different driving $\mu$, and $\overline{\mu }$. Points are obtained from numerically calculated $\Lambda \left(s\right)$ for $L=10$, while full curves are the asymptotic $L\to \infty$ theory obtained from Eq. (26). We use $\Gamma =\gamma =1$.

Figure 4

Finite-size effect at maximal driving $\mu =1$ and $\overline{\mu }=0$. Dashed colored curves are numerical results; the full black curve is the exact result in the TDL, Eq. (26), growing as $\sim s^2$ for large $s$. Inset: finite-$L$ numerical data (overlapping dashed colored curves) grow with $s$ exponentially (full black line). We use $\gamma =\Gamma =1$.

Figure 5

Current fluctuations $J_2$ for the $XX$ dephasing model. The full curve in both subfigures represents the theory given by Eq. (A1). (a) Points are numerical $J_2$ obtained by exact diagonalization; data are for $\mu =1$, $\gamma =\Gamma =1$, and $\overline{\mu }=0$, for which the theory (full curve) simplifies to $J_2=(3+L+6L^2+2L^3)/\left[3L{(1+L)}^3\right]$. (b) Dependence on $\Gamma$ for $L=40$, $\gamma =1$, $\mu =0.1$, and $\overline{\mu }=0$. Points represent tDMRG data. Observe that the amplitude of variation with $\Gamma$ is a subleading $\sim 1/L^2$ correction.

Figure 6

Schematic diagram of the tilted Liouvillian $\tilde{\mathcal{L}}\left(s\right)$ for a boundary-driven $XX$ chain with dephasing, expressed as a non-Hermitian ladder. Double lines along rungs (blue) are ${\sigma }^{\mathrm{z}}{\tau }^{\mathrm{z}}$ coupling due to dephasing, and two springs at the two boundaries (red) are due to boundary driving that involves tilting by $e^s$ at the right end; see Eq. (B3) for details.