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

Abstract

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and nonequilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.

Nonequilibrium phase transition in a single-electron micromaser

“…Having characterized each of the two dynamical phases, we now go on to investigate the transition between them. To this end, we make use of Lee-Yang theory [29][30][31][32][33][34][35] by considering the zeros of the factorial moment generating function [56][57][58][59][60][61][62][63][64][65][66][67], which for our purposes plays the role of the partition function in equilibrium statistical mechanics [39][40][41][42][43][44][45][46][47]. However, in contrast to thermal phase transitions, we are not considering transitions between different equilibrium phases such as spin lattices with a vanishing or a finite average magnetization.…”

“…Using these properties, we may invert the relation in Eq. ( 37) to find the real and absolute part of the closest conjugate pair of zeros, s 0 and s * 0 [42][43][44][45], 21).…”

“…To understand such non-equilibrium phase transitions, the Lee-Yang theory of equilibrium phase transition [29][30][31][32][33][34][35] has been adapted to systems out of equilibrium [36][37][38][39][40][41]. In the approach that we follow here, one considers the zeros of the moment generating function of the full counting statistics in the complex plane of the counting field [39][40][41][42][43][44][45][46][47]. The full counting statistics may display signatures of critical behavior [48], and in case of a phase transition, the zeros should reach the origin of the complex plane in the thermodynamic limit.…”

“…The full counting statistics may display signatures of critical behavior [48], and in case of a phase transition, the zeros should reach the origin of the complex plane in the thermodynamic limit. Moreover, the zeros can be determined from measurements of the higherorder cumulants of the full counting statistics [39][40][41][42][43][44][45][46][47]. This idea was realized in the experiment of Ref.…”

Lee-Yang theory of criticality in interacting quantum many-body systems

“…The Lee-Yang formalism has in recent years experienced a surge of interest because of several experiments that have determined the partition function zeros in a variety of physical systems [12][13][14][15] and thereby shown that Lee-Yang zeros are not just a theoretical concept. In fact, they provide an efficient tool to predict and understand phase transitions in interacting many-body systems, also experimentally [16][17][18][19][20]. However, while the focus so far has been on classical systems, a Lee-Yang theory to describe quantum phase transitions is now clearly becoming desirable [21].…”