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Physical Review A

Finite-size effect for the critical point of an anisotropic dimer model of domain walls

Somendra M. Bhattacharjee, John F. Nagle

Published May 1, 1985• 46 citations
Publisher

Abstract

The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=${N}^{2}$/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by ${\mathrm{MN}}^{2}$t/(M+${N}^{2}$), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and ${\ensuremath{\nu}}_{M}$ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and ${\ensuremath{\nu}}_{m}$ by ${\ensuremath{\nu}}_{N}$ which takes the value ${\ensuremath{\nu}}_{N}$=(1/2). According to finite-size-scaling theory ${\ensuremath{\nu}}_{M}$ and ${\ensuremath{\nu}}_{N}$ may be interpreted to be the critical exponents ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$, respectively, of the divergent length scales in the two principal directions. Our exact values of ${\ensuremath{\nu}}_{m}$ and ${\ensuremath{\nu}}_{N}$ are in agreement with the values of ${\ensuremath{\nu}}_{\mathrm{y}}$ and ${\ensuremath{\nu}}_{\mathrm{x}}$ predicted for general anisotropic domain-wall models.

DOI
10.1103/physreva.31.3199

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The finite-size effect is studied in the Kasteleyn model of dimers on the brick lattice. This model is isomorphic to an anisotropic domain-wall model. Asymptotic analysis of the exact Pfaffian solution for the specific heat establishes that finite-size-scaling theory is valid near the critical point of this model. The finite-size-scaling function is a function of a scaled temperature variable \ensuremath{\tau} and a shape factor scrr=N2{N}^{2}N2/M, where 2N is the number of lattice points in the direction perpendicular to the preferred axis for the domain walls and 2M is the number of lattice points parallel to the preferred axis. The scaled temperature variable \ensuremath{\tau} is given by MN2{\mathrm{MN}}^{2}MN2t/(M+N2{N}^{2}N2), where t is the reduced temperature. As a function of \ensuremath{\tau} the scaling function scrP(\ensuremath{\tau},scrr) is a sequence of \ensuremath{\delta} functions in the limit scrr=0 and a smooth single-peaked function in the limit scrr=\ensuremath{\infty}. In the latter case the specific heat per lattice site can be written as ,\ensuremath{\infty}), where \ensuremath{\alpha} is the bulk specific-heat exponent with the known value of (1/2) and \ensuremathνM{\ensuremath{\nu}}_{M}\ensuremathνM​ is found to have the value of 1. In the case scrr=0, the specific heat per lattice site can be written in an equivalent form by replacing M by N and \ensuremathνm{\ensuremath{\nu}}_{m}\ensuremathνm​ by \ensuremathνN{\ensuremath{\nu}}_{N}\ensuremathνN​ which takes the value \ensuremathνN{\ensuremath{\nu}}_{N}\ensuremathνN​=(1/2). According to finite-size-scaling theory \ensuremathνM{\ensuremath{\nu}}_{M}\ensuremathνM​ and \ensuremathνN{\ensuremath{\nu}}_{N}\ensuremathνN​ may be interpreted to be the critical exponents \ensuremathνy{\ensuremath{\nu}}_{\mathrm{y}}\ensuremathνy​ and \ensuremathνx{\ensuremath{\nu}}_{\mathrm{x}}\ensuremathνx​, respectively, of the divergent length scales in the two principal directions. Our exact values of \ensuremathνm{\ensuremath{\nu}}_{m}\ensuremathνm​ and \ensuremathνN{\ensuremath{\nu}}_{N}\ensuremathνN​ are in agreement with the values of \ensuremathνy{\ensuremath{\nu}}_{\mathrm{y}}\ensuremathνy​ and \ensuremathνx{\ensuremath{\nu}}_{\mathrm{x}}\ensuremathνx​ predicted for general anisotropic domain-wall models.
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We analyze the partition function of the dimer model on an $\mathcal{M}\ifmmode\times\else\texttimes\fi{}\mathcal{N}$ triangular lattice wrapped on a torus obtained by Fendley, Moessner, and Sondhi [Phys. Rev. B 66, 214513 (2002)]. From a finite-size analysis we have found that the dimer model on such a lattice can be described by a conformal field theory having a central charge $c=\ensuremath{-}2$. The shift exponent for the specific heat is found to depend on the parity of the number of lattice sites N\mathcal{N}N along a given lattice axis: e.g., for odd N\mathcal{N}N we obtain the shift exponent $\ensuremath{\lambda}=1$, while for even N\mathcal{N}N it is infinite ($\ensuremath{\lambda}=\ensuremath{\infty}$). In the former case, therefore, the finite-size specific-heat pseudocritical point is size dependent, while in the latter case it coincides with the critical point of the thermodynamic limit.
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Lattice models are useful for understanding behaviors of interacting complex many-body systems. The lattice dimer model has been proposed to study the adsorption of diatomic molecules on a substrate. Here we analyze the partition function of the dimer model on a 2M×2N checkerboard lattice wrapped on a torus and derive the exact asymptotic expansion of the logarithm of the partition function. We find that the internal energy at the critical point is equal to zero. We also derive the exact finite-size corrections for the free energy, the internal energy, and the specific heat. Using the exact partition function and finite-size corrections for the dimer model on a finite checkerboard lattice, we obtain finite-size scaling functions for the free energy, the internal energy, and the specific heat of the dimer model. We investigate the properties of the specific heat near the critical point and find that the specific-heat pseudocritical point coincides with the critical point of the thermodynamic limit, which means that the specific-heat shift exponent λ is equal to ∞. We have also considered the limit N→∞ for which we obtain the expansion of the free energy for the dimer model on the infinitely long cylinder. From a finite-size analysis we have found that two conformal field theories with the central charges c=1 for the height function description and c=-2 for the construction using a mapping of spanning trees can be used to describe the dimer model on the checkerboard lattice.
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