Exact Fisher zeros and thermofield dynamics across a quantum critical point

TL;DR

This paper analytically studies Fisher zeros in the complex temperature plane for the transverse field Ising model, revealing their patterns and connection to quantum critical points, and explores their implications for quantum dynamics and phase transitions.

Contribution

It provides the first comprehensive analytical characterization of Fisher zeros in quantum spin models and links their patterns to quantum criticality and non-unitary dynamics.

Findings

Fisher zeros form continuous lines that change qualitatively at the quantum critical point

Analytical expressions for survival amplitude dynamics at criticality are derived

Patterns of Fisher zeros are observed in other spin models, indicating a universal approach

Abstract

By setting the inverse temperature $\beta$ loose to occupy the complex plane, Fisher showed that the zeros of the complex partition function $Z$, if approaching the real $\beta$ axis, reveal a thermodynamic phase transition. More recently, Fisher zeros were used to mark the dynamical phase transition in quench dynamics. It remains unclear, however, how Fisher zeros can be employed to better understand quantum phase transitions or the non-unitary dynamics of open quantum systems. Here we answer this question by a comprehensive analysis of the analytically continued one-dimensional transverse field Ising model. We exhaust all the Fisher zeros to show that in the thermodynamic limit they congregate into a remarkably simple pattern in the form of continuous open or closed lines. These Fisher lines evolve smoothly as the coupling constant is tuned, and a qualitative change identifies the quantum critical point. By exploiting the connection between $Z$ and the thermofield double states, we obtain analytical expressions for the short- and long-time dynamics of the survival amplitude, including its scaling behavior at the quantum critical point. We point out $Z$ can be realized and probed in monitored quantum circuits. The exact analytical results are corroborated by the numerical tensor renormalization group. We further show that similar patterns of Fisher zeros also emerge in other spin models. Therefore, the approach outlined may serve as a powerful tool for interacting quantum systems.