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Physical Review A

Square-tiling model for the glass transition: Transfer-matrix approach for the competing energy

S. Rao, Somendra M. Bhattacharjee

Published Jan 1, 1992• 3 citations
Publisher

Abstract

We introduce the correlation length for the square-tiling model for the glass transition through the transfer-matrix approach developed earlier [S. M. Bhattacharjee and E. Helfand, Phys. Rev. A 36, 3332 (1987)] for this model. We show that this transfer-matrix approach, in the limit when the competing internal energy reduces the model to the hard-square gas problem with nearest- and next-nearest-neighbor exclusion, is equivalent to the conventional row-to-row transfer matrix. Based on this, we propose that the transitions for other values of the coupling constant for the competing term are also nonuniversal, as is known for the hard-square gas problem.

DOI
10.1103/physreva.45.670

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We introduce the correlation length for the square-tiling model for the glass transition through the transfer-matrix approach developed earlier [S. M. Bhattacharjee and E. Helfand, Phys. Rev. A 36, 3332 (1987)] for this model. We show that this transfer-matrix approach, in the limit when the competing internal energy reduces the model to the hard-square gas problem with nearest- and next-nearest-neighbor exclusion, is equivalent to the conventional row-to-row transfer matrix. Based on this, we propose that the transitions for other values of the coupling constant for the competing term are also nonuniversal, as is known for the hard-square gas problem.
We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is known to undergo a phase transition to one out of four partially ordered phases. We find that this transition displays strong finite-size corrections to scaling and that the correlation functions deviate from isotropy to rather large distances. In contrast with an earlier study, we find that the critical temperature exponent of the transition is not Ising-like.
We present a transfer-matrix scaling study of a hard-square lattice-gas model with an exclusion shell extending up to second-nearest neighbors, and with a uniaxial finite repulsion active at third-nearest-neighbor separations. The phase diagram, as a function of temperature and chemical potential, shows three distinct regions bounded by two merging critical lines. At low temperature the system undergoes a transition to an ordered phase whose c(4\ifmmode\times\else\texttimes\fi{}2) structure is characterized---within this calculation---by the asymptotic degeneracy of the four largest eigenvalues of the transfer matrix. At high temperature this phase is reached from the disordered state via an intermediate, degenerate phase which, on average, exhibits long-range order in one dimension only. These two transition regimes are found to be separated by a multicritical point which is likely to arise from the competition between interaction anisotropy and thermal-disordering effects. Evidence of anisotropic critical behavior is found in the correlation lengths exponent ratios, ${\ensuremath{\nu}}_{?}$/${\ensuremath{\nu}}_{\ensuremath{\perp}}$, which appear to vary continuously along both transition lines. However, the accuracy of the predictions we make for the thermodynamics of the model is still limited by sensitive finite-size effects which are presumably emphasized by the relatively extended range of the interactions.
An analytical expression for the boundary free energy of the Ising square lattice with nearest- and next-nearest-neighbor interactions is derived. The Ising square lattice with anisotropic nearest-neighbor ( J x and J y ) and isotropic next-nearest-neighbor ( J d ) interactions has an order-disorder phase transition at a temperature T = T c given by the condition e −2 J x / k b T c + e − 2 J y / k b T c + e −2( J x + J y )/ k b T c (2 − e −4 J d / k b T c ) = e 4 J d / k b T c . The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase is in excellent agreement with series expansion, finite scaling of transfer matrix and Monte Carlo results. For a vanishing next-nearest-neighbor interaction Onsager's famous result, i.e. sinh (2 J x / k b T c )sinh (2 J y / k b T c ) = 1, is recaptured.
Baxter has solved a restricted class of square lattice gas models with nearest-neighbor exclusion (thus "hard squares") and next-nearest-neighbor interactions. Arguments are presented which demonstrate that the subspace spanned by his exact solution contains the line of tricritical points and the associated surface of first-order transitions of hard squares with attractive next-nearest-neighbor interactions. The tricritical exponents so identified confirm those obtained by Nienhuis for a dilute Ising model.
Phase transitions in centered-rectangular- and square-lattice-gas models have been studied by transfer-matrix methods, with the use of finite-size scaling ideas. The transition lines and critical exponents have been obtained for the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ structures for the centered-rectangular model with several pairwise interactions and a three-body interaction. This model is of interest in that the $p(2\ifmmode\times\else\texttimes\fi{}1)$ and $p(2\ifmmode\times\else\texttimes\fi{}2)$ transitions belong to the universality class of the XY\mathrm{XY}XY model with cubic anisotropy and hence the critical exponents are nonuniversal. As well, it is a simple model of O on W(110). The transition lines have also been obtained for a square-lattice-gas model with competing interactions, for the (2\ifmmode\times\else\texttimes\fi{}1) and (2\ifmmode\times\else\texttimes\fi{}2) ordered structures. The importance of finite-size effects is demonstrated for both models.
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.