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arXiv:2006.15125 · Physical review. B./Physical review. B

Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

Aydin Deger, Fredrik Brange, Christian Flindt

Published Nov 12, 2020• 34 citations
arXivHTMLPDF

Abstract

The authors determine the Lee-Yang zeros of an Ising spin lattice in one, two, and three dimensions from the thermodynamic fluctuations that are inherent to critical regions. They can thereby predict the critical behavior of the lattice from remarkably small system sizes and, furthermore, express the large-deviation statistics of the magnetization in terms of the Lee-Yang zeros. The method used by the authors can be applied to a wide range of critical systems both in theory and experiment.

DOI
10.1103/physrevb.102.174418

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The authors determine the Lee-Yang zeros of an Ising spin lattice in one, two, and three dimensions from the thermodynamic fluctuations that are inherent to critical regions. They can thereby predict the critical behavior of the lattice from remarkably small system sizes and, furthermore, express the large-deviation statistics of the magnetization in terms of the Lee-Yang zeros. The method used by the authors can be applied to a wide range of critical systems both in theory and experiment.
The collective behavior of large numbers of interacting particles may give rise to a phase transition. A continuing challenge is to identify the underlying principles of this phenomenon emerging in many important systems in nature and characterize the critical behavior of interacting many-body systems. In this thesis, we present a theoretical and methodological framework for predicting the phase properties of a macroscopic system based on the behavior of just a few of its constituents. To this end, we devise a direct pathway from the detection of partition function zeros by measuring or simulating fluctuating observables in systems of finite size to the characterization of criticality and large deviation statistics in interacting many-body systems. Our approach combines ideas and concepts from the finite-size scaling analysis with the Lee-Yang formalism and theories of high cumulants and large deviations, and it can be applied in a wide range of critical systems from physics, chemistry, and biology, both in theory and experiment. The thesis consists of four publications. In publications I and II, we report a novel method that makes it possible to extract the partition function zeros in interacting many-body systems of finite size solely from the fluctuations of thermodynamic observables without any prior knowledge of the partition function. To illustrate the feasibility of our approach, we use the Fisher zeros and their relation to the energy fluctuations as a tool for probing criticality in Ising models in two and three dimensions. In particular, we suggest an alternative way of extracting the universal critical exponents from measured fluctuations in finite-size systems away from the phase transition. In publications III and IV, we develop a scaling analysis of the partition function zeros to investigate the criticality in higher dimensions where the hyperscaling breaks down. We also show that even if the system does not exhibit a sharp phase transition, the partition function zeros carry important information about the large-deviation statistics of the system and its symmetry properties. To this end, we determine the rare magnetization fluctuations from the asymptotic behavior of the Lee-Yang zeros, i.e., from the Yang-Lee edge singularities. This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
Phase transitions are typically accompanied by nonanalytic behaviors of the free energy, which can be explained by considering the zeros of the partition function in the complex plane of the control parameter and their approach to the critical value on the real axis as the system size is increased.Recent experiments have shown that partition function zeros are not just a theoretical concept.They can also be determined experimentally by measuring fluctuations of thermodynamic observables in systems of finite size.Motivated by this progress, we investigate here the partition function zeros for the Curie-Weiss model of spontaneous magnetization using our recently established cumulant method.Specifically, we extract the leading Fisher and Lee-Yang zeros of the Curie-Weiss model from the fluctuations of the energy and the magnetization in systems of finite size.We develop a finite-size scaling analysis of the partition function zeros, which is valid for mean-field models and which allows us to extract both the critical values of the control parameters and the critical exponents, even for small systems that are away from criticality.We also show that the Lee-Yang zeros carry important information about the rare magnetic fluctuations as they allow us to predict many essential features of the large-deviation statistics of the magnetization.This finding may constitute a profound connection between Lee-Yang theory and large-deviation statistics.
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.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.
Quantum phase transitions are a ubiquitous many-body phenomenon that occurs in a wide range of physical systems, including superconductors, quantum spin liquids, and topological materials. However, investigations of quantum critical systems also represent one of the most challenging problems in physics, since highly correlated many-body systems rarely allow for an analytic and tractable description. Here, we present a Lee-Yang theory of quantum phase transitions including a method to determine quantum critical points which readily can be implemented within the tensor network formalism and even in realistic experimental setups. We apply our method to a quantum Ising chain and the anisotropic quantum Heisenberg model and show how the critical behavior can be predicted by calculating or measuring the high cumulants of properly defined operators. Our approach provides a powerful formalism to analyze quantum phase transitions using tensor networks, and it paves the way for systematic investigations of quantum criticality in two-dimensional systems.