Fisher information approach to nonequilibrium phase transitions in a quantum XXZ spin chain with boundary noise

We investigated quantum critical behaviors in the nonequilibrium steady state of a XXZ spin chain with boundary Markovian noise using Fisher information. The latter represents the distance between two infinitesimally close states, and its superextensive size scaling witnesses a critical behavior due to a phase transition since all the interaction terms are extensive. Perturbatively, in the noise strength, we found superextensive Fisher information at anisotropy $\left|\mathrm{\Delta }\right|⩽1$ and irrational $\frac{arccos\mathrm{\Delta }}{\pi }$ irrespective of the order of two noncommuting limits, i.e., the thermodynamic limit and the limit of sending $\frac{arccos\mathrm{\Delta }}{\pi }$ to an irrational number via a sequence of rational approximants. From this result we argue the existence of a nonequilibrium quantum phase transition with a critical phase $\left|\mathrm{\Delta }\right|⩽1$. From the nonsuperextensivity of the Fisher information of reduced states, we infer that this nonequilibrium quantum phase transition does not have local order parameters but has nonlocal ones, at least at $\left|\mathrm{\Delta }\right|=1$. In the nonperturbative regime for the noise strength, we numerically computed the reduced Fisher information which lower bounds the full-state Fisher information and is superextensive only at $\left|\mathrm{\Delta }\right|=1$. From the latter result, we derived local order parameters at $\left|\mathrm{\Delta }\right|=1$ in the nonperturbative case. The existence of critical behavior witnessed by the Fisher information in the phase $\left|\mathrm{\Delta }\right|<1$ is still an open problem. The Fisher information also represents the best sensitivity for any estimation of the control parameter, in our case the anisotropy $\mathrm{\Delta }$, and its superextensivity implies enhanced estimation precision which is also highly robust in the presence of a critical phase.

Figure 1

Semilog plot of the coefficient $\xi$ for $\frac{\eta }{\pi }=\frac{f_{m+1}}{f_m}$, with ${\left\{f_m\right\}}_m$ being the Fibonacci sequence. The inset shows the log-log plot of $\xi$ as a function of $f_m$, which is perfectly fitted by $(0.0107\pm 0.0004)f_m^{3.993\pm 0.007}$.

Figure 2

Log-log plots of the contribution $\frac{J^2}{{\lambda }^2{\mu }^2}{\tilde{F}}_{\mathrm{\Delta }}$ to the Fisher information as a function of $n$ for irrational $\frac{\eta }{\pi }$: $\eta =\pi \phi$, with $\phi =\frac{1+\sqrt{5}}{2}$ being the golden ratio (solid black line), $\eta =\pi \frac{\sqrt{3}}{2}$ (dotted line), $\eta ={\pi }^2$ (dashed line), and $\eta =\pi e$ (dot-dashed line). For comparison we also plot the slopes of power laws ${10}^{-2}{n}^2$ and $n^3$ (solid gray lines).

Figure 3

Top: The Fisher information $F_{\mathrm{\Delta }=1}^{\left\{j\right\}}$ of the $j\mathrm{th}$ spin reduced state as a function of $n$ and $j$. The superextensivity is manifest from the comparison with the plane ${10}^{8}n$. Bottom: Log-log plots of $F_{\mathrm{\Delta }=1}^{\left\{j\right\}}$ as a function of $n$, set to powers of 2, for $\lambda =1$ and $j=1$ (circles), $j=⌊\frac{n}{4}⌋$ (squares), and $j=⌊\frac{n}{2}⌋$ (diamonds). The solid lines are the corresponding fits (see Table 2), excluding the first three points of each line, which clearly deviate from the large-$n$ behavior.

Figure 4

Plot of the magnetization profile $\langle {\sigma }_j^z\rangle$ of the $j\mathrm{th}$ spin as a function of $j$ and $\mathrm{\Delta }$ for $n=1000$ and $\lambda =1$.