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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 nonunitary 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. 

Abstract Quantum phase transitions are a fascinating area of condensed matter physics. The extension through complexification not only broadens the scope of this field but also offers a new framework for understanding criticality and its statistical implications. This mini review provides a concise overview of recent developments in complexification, primarily covering finite temperature and equilibrium quantum phase transitions, as well as their connection with dynamical quantum phase transitions and non-Hermitian physics, with a particular focus on the significance of Fisher zeros. Starting from the newly discovered self-similarity phenomenon associated with complex partition functions, we further discuss research on self-similar systems briefly. Finally, we offer a perspective on these aspects. 

Abstract Neural network quantum states provide a novel representation of the many-body states of interacting quantum systems and open up a promising route to solve frustrated quantum spin models that evade other numerical approaches. Yet its capacity to describe complex magnetic orders with large unit cells has not been demonstrated, and its performance in a rugged energy landscape has been questioned. Here we apply restricted Boltzmann machines (RBMs) and stochastic gradient descent to seek the ground states of a compass spin model on the honeycomb lattice, which unifies the Kitaev model, Ising model and the quantum 120° model with a single tuning parameter. We report calculation results on the variational energy, order parameters and correlation functions. The phase diagram obtained is in good agreement with the predictions of tensor network ansatz, demonstrating the capacity of RBMs in learning the ground states of frustrated quantum spin Hamiltonians. The limitations of the calculation are discussed. A few strategies are outlined to address some of the challenges in machine learning frustrated quantum magnets.