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arXiv:1406.5566 · Journal of Physics A Mathematical and Theoretical

Integrability versus non-integrability: hard hexagons and hard squares compared

Michael Assis, Jesper Lykke Jacobsen, Iwan Jensen, J.-M. Maillard, Barry M. McCoy

Published Oct 20, 2014• 5 citations
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Abstract

In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free–free boundary conditions up to sizes 40 × 40 and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point of hard squares, where all eigenvalues have unit modulus, and we give several conjectures for the value at of the partition functions.

DOI
10.1088/1751-8113/47/44/445001

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In this paper we compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. We consider partition functions for toroidal, cylindrical, and free–free boundary conditions up to sizes 40 × 40 and transfer matrices up to 30 sites. For all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest (and equal) moduli and exhibits much greater structure than the corresponding density of hard hexagons. We also study the special point of hard squares, where all eigenvalues have unit modulus, and we give several conjectures for the value at of the partition functions.
The Ising model at magnetic field H=0 is one of the most important exactly solved models in statistical mechanics, solved on the square lattice in 1944 by Onsager, and later by others on the triangular lattice; its magnetic susceptibility at H=0 continues to be an unsolved aspect of the model, however. The susceptibility can either be viewed as a sum over all correlation functions of the integrable model at H=0, or else as a second derivative of the free energy of the non-integrable Ising model in a field. Therefore, its analytic properties, though derived from the non-integrable model, can be studied through series expansion of the integrable correlation functions. We begin this process by analyzing the first four terms in the form factor expansion of the diagonal correlation functions, and after summing over the diagonal form factor expansion, the first four terms of the diagonal susceptibility expansion. We have been able to reduce the form factor and susceptibility expansion terms, given as multi-dimensional integrals, to closed-form functions in all cases. Under the limit of H going to infinity with the interaction energy E going to minus infinity, the isotropic Ising model becomes a hard particle lattice gas model, with exclusion of nearest neighbor lattice sites. In this limit, the triangular lattice Ising model becomes the exactly solved hard hexagon model, while the square lattice becomes the non-integrable hard squares model. We study in detail these models for finite lattice sizes in order to understand the differences between integrable and non-integrable models. We consider the partition functions zeros and their density for different boundary conditions, and find notable differences in the density which is attributed to an extra factorization in the transfer matrices of hard hexagons which is absent in hard squares. We also study the special point at fugacity z=-1 of hard squares where all eigenvalues of the transfer matrix are equimodular and where the grand partition function's value depends on boundary conditions. | 300 pages
Earlier this year Chan extended the low-density series for the hard-squares partition function κ(z)κ(z)κ(z) to 92 terms. Here we analyse this extended series focusing on the behaviour at the dominant singularity zdz_dzd​ which lies on on the negative fugacity axis. We find that the series has a confluent singularity of order 2 at zdz_dzd​ with exponents θ=0.83333(2)θ=0.83333(2)θ=0.83333(2) and θ′=1.6676(3)θ'= 1.6676(3)θ′=1.6676(3). We thus confirm that the exponent θθθ has the exact value 56\frac5665​ as observed by Dhar.
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).
Abstract Yang–Baxter integrable dense A 1 ( 1 ) and dilute A 2 ( 2 ) loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions ( h , v ) is applied in the horizontal and vertical directions with h , v = 0 and 1 for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by α and β respectively where β is simply related to the crossing parameter λ . At roots of unity, when λ / π ∈ ℚ , these models are the dense L M ( p , p ′ ) and dilute D L M ( p , p ′ ) logarithmic minimal models with p , p ′ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with d defects and deduce the conformal partition functions Z dense ( h , v ) ( α ) and Z dilute ( h , v ) ( α )
The authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next-nearest neighbour (diagonal) interactions on the special surface corresponding to regimes III and IV of the generalised hard hexagon model. The interfacial tension, correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row-to-row and corner transfer matrices. The associated critical exponents are found to be mu =v= 5 / 4 , beta = 3 / 32 in regime III and mu '=v'= 5 / 2 , beta '= 1 / 4 in regime IV. In particular, their results confirm the recent proposal by Huse that regime III is the first-order coexistence surface (separating the disordered fluid phase from the square ordered solid phases) and that the regime III/regime IV boundary is a line of tricritical points.
Functional equations are derived for the eigenvalues of the row-to-row transfer matrix of the generalised hard hexagon model. These equations are exact for a lattice with N columns and are solved in the limit N to infinity . The partition function per site is rederived without the previous analyticity assumptions. The authors are also able to calculate the interfacial tension and the correlation length; the associated critical exponents are mu = nu = nu '= 5 / 6 in agreement with the scaling relations.