Yang-Lee Edge Singularity of the Ising Model on a Honeycomb Lattice in an External Magnetic Field
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Yang-Lee Edge Singularity of the Ising Model on a Honeycomb Lattice in an External Magnetic Field Seung-Yeon Kim∗ School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 27469, Korea (Received 12 June 2020; accepted 30 June 2020) The Ising model in an external magnetic field is one of the most outstanding and intriguing unsolved problems. One of the important properties of the Ising model in an external magnetic field is the Yang-Lee edge singularity. The zeros in the complex magnetic-field plane of the grand partition function of the Ising model on a honeycomb lattice in an external magnetic field are exactly calculated by using the exact integer values for the density of states on L × 2L honeycomb lattices (up to L = 14). The critical line of the Yang-Lee edge singularity of the Ising model on a honeycomb lattice in an external magnetic field is accurately investigated from its Yang-Lee zeros. Keywords: External magnetic field, Partition function zeros, Yang-Lee edge singularity DOI: 10.3938/jkps.77.271
I. INTRODUCTION Phase transitions and critical phenomena are the most ubiquitous phenomena in nature. Understanding phase transitions and critical phenomena is very important in modern science. However, understanding phase transitions and critical phenomena is difficult because various physical quantities are singular and divergent in response to small changes of external parameters (such as temperature T , pressure p, chemical potential μ, and magnetic field H) and traditional analytical methods are not applicable [1]. The simplest magnetic spin system showing phase transitions and critical phenomena is the Ising model in two dimensions, which has played an essential role in developing the modern theory of phase transitions and critical phenomena [1]. However, the Ising model in two dimensions in an external magnetic field is one of the most outstanding and intriguing unsolved problems. The theory of partition function zeros is one of the most efficient and powerful methods to understand phase transitions and critical phenomena, directly investigating the singularities and divergences of various physical quantities [2]. Yang and Lee [3] introduced the zeros of the grand partition function Ξ(T, z) in the complex magnetic-field or fugacity (z = e−2H/kB T for spin systems or z = eμ/kB T for fluid systems) plane. Since then these zeros have been simply called Yang-Lee zeros, and Yang-Lee zeros have yielded a mathematically rigorous mechanism for the occurrence of phase transitions and critical phenomena. Also, Lee and Yang [4] proved the famous circle theorem, stating that Yang-Lee zeros of the ∗ E-mail:
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pISSN:0374-4884/eISSN:1976-8524
Ising model in an external magnetic field lie on the unit circle z0 = eiθ in the complex magnetic-field (z = e−2βH ) plane (with inverse temperature β = 1/kB T ). The LeeYang circle theorem provided monumental insight into the unsolved problem of the Ising model in an external magnetic field. Even with the Lee-Yang circle theorem, the general distribu
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