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arXiv:2406.18981 · Physical Review Research

Exact Fisher zeros and thermofield dynamics across a quantum critical point

Yang Liu, Songtai Lv, Yuchen Meng, Zefan Tan, Erhai Zhao, Haiyuan Zou

Published Nov 14, 2024• 10 citations
arXivHTMLPDF

Abstract

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

DOI
10.1103/physrevresearch.6.043139

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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
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.
The far-from-equilibrium dynamics of certain interacting quantum systems still defy precise understanding. One example is the so-called quantum many-body scars (QMBSs), where a set of energy eigenstates evade thermalization to give rise to long-lived oscillations. Despite the success of viewing scars from the perspectives of symmetry, commutant algebra, and quasiparticles, it remains a challenge to elucidate the mechanism underlying all QMBS and to distinguish them from other forms of ergodicity breaking. In this work, we introduce an alternative route to detect and diagnose QMBS based on Fisher zeros, i.e., the patterns of zeros of the analytically continued partition function Z on the complex β (inverse temperature) plane. For systems with scars, a continuous line of Fisher zeros will appear off the imaginary β axis and extend upward, separating the β plane into regions with distinctive thermalization behaviors. This conjecture is motivated from interpreting the complex Z as the return amplitude of the thermofield double state, and it is validated by analyzing two models with QMBS, the P[over ¯]XP[over ¯] model and the Ising chain in external fields. These models also illustrate the key difference between QMBS and strong ergodicity breaking including their distinctive renormalization group flows on the complex β plane. This "statistical mechanics" approach places QMBS within the same framework of thermal and dynamical phase transitions. It has the advantage of spotting scars without exhaustively examining each individual quantum state.
The far-from-equilibrium dynamics of certain interacting quantum systems still defy precise understanding. One example is the so-called quantum many-body scars (QMBSs), where a set of energy eigenstates evade thermalization to give rise to long-lived oscillations. Despite the success of viewing scars from the perspectives of symmetry, commutant algebra and quasiparticles, it remains a challenge to elucidate the mechanism underlying all QMBS and to distinguish them from other forms of ergodicity breaking. In this work, we introduce an alternative route to detect and diagnose QMBS based on Fisher zeros, i.e., the patterns of zeros of the analytically continued partition function ZZZ on the complex βββ (inverse temperature) plane. For systems with scars, a continuous line of Fisher zeros will appear off the imaginary βββ axis and extend upward, separating the βββ-plane into regions with distinctive thermalization behaviors. This conjecture is motivated from interpreting the complex ZZZ as the return amplitude of the thermofield double state, and it is validated by analyzing two models with QMBS, the PˉXPˉ\bar{P}X\bar{P}PˉXPˉ model and the Ising chain in external fields. These models also illustrate the key difference between QMBS and strong ergodicity breaking including their distinctive renormalization group flows on the complex βββ-plane. This ``statistical mechanics" approach places QMBS within the same framework of thermal and dynamical phase transitions. It has the advantage of spotting scars without exhaustively examining each individual quantum state.
A boundary transfer matrix formulation allows to calculate the Loschmidt echo for one-dimensional quantum systems in the thermodynamic limit. We show that nonanalyticities in the Loschmidt echo and zeros for the Loschmidt amplitude in the complex plane (Fisher zeros) are caused by a crossing of eigenvalues in the spectrum of the transfer matrix. Using a density-matrix renormalization group algorithm applied to these transfer matrices, we numerically investigate the Loschmidt echo and the Fisher zeros for quantum quenches in the XXZ model with a uniform and a staggered magnetic field. We give examples---both in the integrable and the nonintegrable cases---where the Loschmidt echo does not show nonanalyticities although the quench leads across an equilibrium phase transition, and examples where nonanalyticities appear for quenches within the same phase. For a quench to the free fermion point, we analytically show that the Fisher zeros sensitively depend on the initial state and can lie exactly on the real axis already for finite system size. Furthermore, we use bosonization to analyze our numerical results for quenches within the Luttinger liquid phase.
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.
Fisher zero lines of the Loschmidt amplitude in non-Hermitian quenches can exhibit anomalous structures in the complex time plane, such as divergence, separation, and the restriction of all branches or a specific branch to the real or imaginary axis. Here, we use the biorthogonal framework to reveal the origin of these structures and determine the corresponding critical momentum conditions by analyzing the values of mk=h^0⋅h^1m_k=\hat{\mathbf{h}}_0\cdot\hat{\mathbf{h}}_1mk​=h^0​⋅h^1​, the dot product between the normalized prequench and postquench complex Bloch vectors. When Fisher zeros continuously cover the real-time axis, the density of Fisher zeros is further introduced to identify the critical times within these critical regions, at which the rate function becomes nonanalytic. We verify our results in a generalized non-Hermitian SSH model by quenching the Peierls phase and the non-Hermitian gain-loss strength, and also use the Jordan form to analyze the dynamics at postquench exceptional points, where the conventional Fisher zero formula is not applicable.