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Journal of Physics A Mathematical and General

New relations between the monomer-dimer and the Yang-Lee models

Yonathan Shapir

Published Aug 1, 1982• 13 citations
Publisher

Abstract

A Hamiltonian approach to the monomer-dimer model is derived by a special truncation of the spin operators in the Yang-Lee model. At its critical point, the negative fugacity dimer problem is proved to belong to the universality class of the Yang-Lee singularity as previously conjectured.
DOI
10.1088/0305-4470/15/8/010

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The density of Lee-Yang zeros, in the thermodynamic limit, for classical nnn-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, $g({h}^{\ensuremath{'}\ensuremath{'}})$, on the imaginary axis of the complex reduced-magnetic-field plane, $h=\frac{H}{{k}_{B}T}={h}^{\ensuremath{'}}+i{h}^{\ensuremath{'}\ensuremath{'}}$, has an algebraic singularity at the edge of the gap in the zero distribution, $g({h}^{\ensuremath{'}\ensuremath{'}})\ensuremath{\sim}{[|{h}^{\ensuremath{'}\ensuremath{'}}|\ensuremath{-}{h}_{0}(T)]}^{\ensuremath{\sigma}}$, then $\ensuremath{\sigma}$ is independent of nnn in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Pad\'e, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates $\ensuremath{\sigma}=\ensuremath{-}0.163\ifmmode\pm\else\textpm\fi{}3$ for d=2d=2d=2 dimensions and $\ensuremath{\sigma}=0.086\ifmmode\pm\else\textpm\fi{}15$ for d=3d=3d=3.
A singularity on the negative fugacity axis of the hard-square lattice gas is investigated in terms of numerical diagonalization of transfer matrices. The location of the singular point zc−z_c^-zc−​ and the critical exponent ννν are accurately determined by the phenomenological renormalization technique as -0.11933888188(1) and 0.416667(1), respectively. It is also found that the central charge ccc and the dominant scaling dimension xσx_σxσ​ are -4.399996(8) and -0.3999996(7), respectively. These results strongly support that this singularity belongs to the same universality class as the Yang-Lee edge singularity (c=−22/5,xσ=−2/5andν=5/12c=-22/5, x_σ=-2/5 and ν=5/12c=−22/5,xσ​=−2/5andν=5/12).
A singularity on the negative-fugacity axis of the hard-core lattice gas is investigated in terms of numerical diagonalization of large-scale transfer matrices. For the hard-square lattice gas, the location of the singular point [Formula: see text] and the critical exponent ν are accurately determined by the phenomenological renormalization technique as -0.11933888188(1) and 0.416667(1), respectively. It is also found that the central charge c and the dominant scaling dimension xσ are -4.399996(8) and -0.3999996(7), respectively. Similar analyses for other hard-core lattice-gas models in two dimensions are also performed, and it is confirmed that the universality between these models does hold. These results strongly indicate that the present singularity belongs to the same universality class as the Yang–Lee edge singularity.
We investigate the critical properties of the Lee-Yang model in less than six spacetime dimensions using truncations of the functional renormalization group flow. We give estimates for the critical exponents, study the dependence on the regularization scheme, and show the convergence of our results for increasing size of the truncations in four and five dimensions. While with our truncations it is numerically challenging to approach the three-dimensional case, we provide a simple approximation which allows us to qualitatively study the Lee-Yang model in two and three dimensions and use it to argue the existence of further nonunitary multicritical theories including one which is relevant for the universality class of the Blume-Capel model.
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.