Fisher information approach to non-equilibrium phase transitions in quantum XXZ spin chain with boundary noise

Abstract

We investigated quantum critical behaviours in the non-equilibrium steady state of a $XXZ$ spin chain with boundary Markovian noise using the Fisher information. The latter represents the distance between two infinitesimally close states, and its superextensive size scaling witnesses a critical behaviour due to a phase transition, since all the interaction terms are extensive. Perturbatively in the noise strength, we found superextensive Fisher information at anisotropy $|\Delta |⩽1$ and irrational $\frac{\arccos \Delta }{\pi }$ irrespective of the order of two non-commuting limits, i.e. the thermodynamic limit and the limit of sending $\frac{\arccos \Delta }{\pi }$ to an irrational number via a sequence of rational approximants. From this result we argue the existence of a non-equilibrium quantum phase transition with a critical phase $|\Delta |⩽1$. From the non-superextensivity of the Fisher information of reduced states, we infer that this non-equilibrium quantum phase transition does not have local order parameters but has non-local ones, at least at $|\Delta |=1$. In the non-perturbative 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 $|\Delta |=1$. Form the latter result, we derived local order parameters at $|\Delta |=1$ in the non-perturbative case. The existence of critical behaviour witnessed by the Fisher information in the phase $|\Delta |<1$ 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 $\Delta$, and its superextensivity implies enhanced estimation precision which is also highly robust in the presence of a critical phase.

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