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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Yang-Lee circle theorem for an ideal pseudospin-1/2 Bose gas in an external magnetic field

Xian Zhi Wang

Published Mar 21, 2001• 8 citations
Publisher

Abstract

The Yang-Lee circle theorem is extended to an ideal pseudospin-1/2 Bose gas in an external magnetic field. It is found that the zeros of the canonical partition function are located on the unit circle in the complex activity plane if the temperature is above the critical temperature of ideal Bose-Einstein condensation. No zeros exist if the temperature is below the critical temperature.

DOI
10.1103/physreve.63.046103

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The Yang-Lee circle theorem is extended to an ideal pseudospin-1/2 Bose gas in an external magnetic field. It is found that the zeros of the canonical partition function are located on the unit circle in the complex activity plane if the temperature is above the critical temperature of ideal Bose-Einstein condensation. No zeros exist if the temperature is below the critical temperature.
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle theorem. Distinct features are observed at different temperatures---below, above, and at the critical temperature. Application of the tensor network approach is also made to the qqq-state Potts models in both two and three dimensions and a previous debate on whether, in the thermodynamic limit, the Yang-Lee zeros lie on a unit circle for q>2q>2q>2 is resolved: they clearly do not lie on a unit circle except at the zero temperature. For the Potts models (q=3,4,5,6)(q=3,4,5,6)(q=3,4,5,6) investigated in two dimensions, as the temperature is lowered the radius of the zeros at a fixed angle from the real axis shrinks exponentially towards unity with the inverse temperature.
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 Lee-Yang circle theorem revolutionized our understanding of phase transitions in ferromagnetic systems by showing that the complex zeros of partition functions lie on the unit circle, with criticality arising as these zeros approach the real axis in the thermodynamic limit. However, in frustrated systems such as antiferromagnets and spin glasses, the zeros deviate from this structure, making it challenging to extend the Lee-Yang theory to disordered systems. In this work, we establish a new circle theorem for two-dimensional Ising spin glasses, proving that the square of the partition function exhibits zeros densely packed along the unit circle. Numerical simulations on the square lattice confirm our theoretical predictions, demonstrating the validity of the circle law for quenched disorder. Furthermore, our results uncover a finite-temperature crossover in ±J\pm J±J spin glasses, characterized by the emergence of a spectral gap in the angular distribution of zeros. This result extends the Lee-Yang framework to disordered systems, offering new insights into spin-glass criticality.
It is found that the singularity of the thermodynamic potential of an ideal Bose gas is connected with the physical root z0=1{z}_{0}=1z0​=1 of the inverse of the grand partition function. The critical nature is determined solely by the behavior of the root distribution of the inverse of the grand partition function near z0=1{z}_{0}=1z0​=1. This is quite similar to the situation of phase transition described by Yang-Lee theory.