Concepts of the complex partition functions and the Fisher zeros provide intrinsic statistical mechanisms for finite temperature and real time dynamical phase transitions. We extend the utility of these complexifications to quantum phase transitions. We exactly identify different Fisher zeros on lines or closed curves and elucidate their correspondence with domain-wall excitations or confined mesons for the one-dimensional transverse field Ising model. The crossover behavior of the Fisher zeros provides a fascinating picture for criticality near the quantum phase transition, where the excitation energy scales are quantitatively determined. We further confirm our results by tensor network calculations and demonstrate a clear signal of deconfined meson excitations from the disruption of the closed zero curves. Our results unambiguously show significant features of Fisher zeros for a quantum phase transition and open up a new route to explore quantum criticality.
Concepts of the complex partition functions and the Fisher zeros provide intrinsic statistical mechanisms for finite temperature and real time dynamical phase transitions. We extend the utility of these complexifications to quantum phase transitions. We exactly identify different Fisher zeros on lines or closed curves and elucidate their correspondence with domain-wall excitations or confined mesons for the one-dimensional transverse field Ising model. The crossover behavior of the Fisher zeros provides a fascinating picture for criticality near the quantum phase transition, where the excitation energy scales are quantitatively determined. We further confirm our results by tensor network calculations and demonstrate a clear signal of deconfined meson excitations from the disruption of the closed zero curves. Our results unambiguously show significant features of Fisher zeros for a quantum phase transition and open up a new route to explore quantum criticality.
By setting the inverse temperature β loose to occupy the complex plane, Fisher showed that the zeros of the complex partition function Z , if approaching the real β 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. Published by the American Physical Society 2024
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.
We demonstrate an alternative approach based on a complex-valued inverse temperature and partition function to probe quantum phases of matter with nontrivial spectra and dynamics. It leverages thermofield dynamics (TFD) to quantitatively characterize quantum and thermal fluctuations, and exploit the correspondence between low-energy excitations and Fisher zeros. Using the quantum XY chain in an external field as a testbed, we show that the oscillatory gap behavior manifests as oscillations in the long-time dynamics of the TFD spectral form factor. We also identify giant bubbles, i.e. large-scale closed lines, of Fisher zeros near the gapless XX limit. They provide a characteristic energy scale that seems to contradict the predictions of the low energy theory of a featureless Luttinger liquid. We identify this energy scale and relate the motion of these giant bubbles with varying external field to the transfer of spectral weight from high to low energies. The deep connection between Fisher zeros, dynamics, and excitations opens up promising avenues for understanding the unconventional gap behaviors in strongly correlated many-body systems.
While the zeros of complex partition functions, such as Lee-Yang zeros and Fisher zeros, have been pivotal in characterizing temperature-driven phase transitions, extending this concept to zero temperature remains an open question. In this work, we propose a solution to this issue by calculating the imaginary-temperature zeros (ITZs), which are defined as the roots of the imaginary-temperature partition function. We illustrate the analytical properties of ITZs in the transverse-field Ising chain, showing that the ITZs' distribution can distinguish between various phases and signify the critical exponents. Universal singular behaviors manifest in such quantities as the edge density of ITZs and the magnetization, with the scaling exponents remarkably differing from those in Lee-Yang theory. We further illuminate the consistency between ITZs and the zeros of the spectral form factor, which offers a practical path for the experimental detection of ITZs.
We consider the scaling behavior of thermodynamic quantities in the one-dimensional transverse field Ising model near its quantum critical point (QCP). Our study has been motivated by the question about the thermodynamical signatures of this paradigmatic quantum critical system and, more generally, by the issue of how quantum criticality accumulates entropy. We find that the crossovers in the phase diagram of temperature and (the nonthermal control parameter) transverse field obey a general scaling ansatz, and so does the critical scaling behavior of the specific heat and magnetic expansion coefficient. Furthermore, the Gr\"uneisen ratio diverges in a power-law way when the QCP is accessed as a function of the transverse field at zero temperature, which follows the prediction of quantum critical scaling. However, at the critical field, upon decreasing the temperature, the Gr\"uneisen ratio approaches a constant instead of showing the expected divergence. We are able to understand this unusual result in terms of a peculiar form of the quantum critical scaling function for the free energy; the contribution to the Gr\"uneisen ratio vanishes at the linear order in a suitable Taylor expansion of the scaling function. In spite of this special form of the scaling function, we show that the entropy is still maximized near the QCP, as expected from the general scaling argument. Our results establish the telltale thermodynamic signature of a transverse-field Ising chain, and will thus facilitate the experimental identification of this model quantum-critical system in real materials.
Quantum Phase Transitions Quantum phase transitions, in contrast to temperature-driven critical phenomena, occur due to a competition between different parameters describing the basic interactions of the system. Their specific feature is the quantum character of the critical fluctuations. This implies, through the uncertainty principle, that energy fluctuations and time are coupled. Then at zero temperature time plays a crucial and fundamental role, the static properties being entangled with the dynamics (Continentino, 1994a; Sondhi et al ., 1997; Sachdev, 1999). In this book we are mainly interested in quantum phase transitions that occur in electronic many-body systems and how scaling concepts can be useful to understand their properties close to these transitions (Continentino, 1994a). Even though a similar approach has been used in the case of interacting bosons (Fisher et al ., 1989) the fermionic problem has its own idiosyncrasies and difficulties. For example, there is no natural order parameter associated with the localisation transition in the electronic case, while bosons at zero temperature are either localised or superfluid so that the superfluid order parameter can be used to distinguish between both phases. If the results of the study of quantum phase transitions were restricted to zero temperature, this would be an interesting but purely academic area of research. What is really exciting about this subject is the effect of quantum critical points (QCP) in the finite temperature phase diagram of actual physical systems, even far away from the QCP (Freitas, 2015). As we will show, there is a special line in this phase diagram, the quantum critical trajectory , where the temperature dependence of the thermodynamic and transport properties is governed by the quantum critical exponents, i.e. those associated with the QCP. This and the observed crossover effects induced by temperature in the non-critical side of the phase diagram of a material with a QCP are sufficient to make the study of quantum phase transitions an inevitable subject. We start this chapter by introducing the scaling theory of quantum critical phenomena, since scaling concepts are used throughout this book.