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arXiv:cond-mat/0510278 · International Journal of Modern Physics B

STATISTICAL MECHANICS OF EQUILIBRIUM AND NONEQUILIBRIUM PHASE TRANSITIONS: THE YANG–LEE FORMALISM

IOANA BENA, MICHEL DROZ, Adam Lipowski

Published Nov 20, 2005• 171 citations
arXivHTMLPDF

Abstract

Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang–Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions.

DOI
10.1142/s0217979205032759

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Showing that the location of the zeros of the partition function can be used to study phase transitions, Yang and Lee initiated an ambitious and very fruitful approach. We give an overview of the results obtained using this approach. After an elementary introduction to the Yang–Lee formalism, we summarize results concerning equilibrium phase transitions. We also describe recent attempts and breakthroughs in extending this theory to nonequilibrium phase transitions.
Determining the phase diagram of interacting quantum many-body systems is an important task for a wide range of problems such as the understanding and design of quantum materials. For classical equilibrium systems, the Lee-Yang formalism provides a rigorous foundation of phase transitions, and these ideas have also been extended to the quantum realm. Here, we develop a Lee-Yang theory of quantum phase transitions that can include thermal fluctuations caused by a finite temperature, and it thereby provides a link between the classical Lee-Yang formalism and recent theories of phase transitions at zero temperature. Our methodology exploits analytic properties of the moment generating function of the order parameter in systems of finite size, and it can be implemented in combination with tensor-network calculations. Specifically, the onset of a symmetry-broken phase is signaled by the zeros of the moment generating function approaching the origin in the complex plane of a counting field that couples to the order parameter. Moreover, the zeros can be obtained by measuring or calculating the high cumulants of the order parameter. We determine the phase diagram of the two-dimensional quantum Ising model and thereby demonstrate the potential of our method to predict the critical behavior of two-dimensional quantum systems at finite temperatures.
Lee-Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.
Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros-complex singularities of the free energy in systems of finite size-have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize nonequilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such nonequilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.
We study the Yang-Lee theory in quantum phase transitions from the perspective of quantum entanglement in one-dimensional many-body systems. We primarily focus on the distribution of Yang-Lee zeros and its associated Yang-Lee edge singularity of two prototypical models: the Su-Schrieffer-Heeger model and the XXZ spin chain. By taking the zero-temperature limit, we show how the Yang-Lee zeros approach the quantum phase transition points on the complex plane of parameters. To characterize the edge singularity induced by Yang-Lee zeros in a quantum phase transition, we introduce the entanglement entropy of the ground state to show that the edges of Yang-Lee zeros lead to the ground-state entanglement transition. We further show that our results are also applicable to the general noninteracting parity-time-symmetric Hamiltonians.