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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-size scaling of the density of zeros of the partition function in first- and second-order phase transitions

Richard J. Creswick, Seung‐Yeon Kim

Published Sep 1, 1997• 35 citations
Publisher

Abstract

The finite-size scaling form for the density of zeros of the partition function in first- and second-order phase transitions is derived. Using the finite-size scaling of the density of zeros, the order of a phase transition can be easily determined and the order parameter calculated from finite-size data. We illustrate the scaling theory using exact values for the zeros of the partition function of the two-dimensional Ising model in the complex magnetic-field plane.

DOI
10.1103/physreve.56.2418

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The finite-size scaling form for the density of zeros of the partition function in first- and second-order phase transitions is derived. Using the finite-size scaling of the density of zeros, the order of a phase transition can be easily determined and the order parameter calculated from finite-size data. We illustrate the scaling theory using exact values for the zeros of the partition function of the two-dimensional Ising model in the complex magnetic-field plane.
The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.
The critical properties of an infinitely long Ising strip with finite width L joined periodically or antiperiodically are investigated by analyzing the distribution of partition function zeros. For periodic boundary condition, the the leading finite-size scaling of partition function zeros and its corrections are given. For antiperiodic boundary condition, the critical point of 2D Ising transition is one of the loci of the zeros, and the associated non-analyticity is identified as a first-order phase transition. The exact amount of the latent heat released by the transition is 4/L.
Critical finite-size scaling functions for the order-parameter distribution of the two- and three-dimensional Ising model are investigated. Within a recently introduced classification theory of phase transitions the universal part of critical finite-size scaling functions has been derived by employing a scaling limit which differs from the traditional finite-size scaling limit. In this paper the analytical predictions are compared with Monte Carlo simulation results. We find good agreement between the analytical expression and the simulation results. The agreement is consistent with the possibility that the functional form of the critical finite-size scaling function for the order-parameter distribution is determined uniquely by only a few universal parameters, most notably the equation of state exponent.
We formulate a finite-size scaling theory for the partition function. This leads to a scaling description of the complex-temperature-plane zeros. The asymptotic form of the scaling function for large complex arguments is conjectured and used to calculate explicitly the location of an unbounded number of zeros. Scaling predictions are checked against the exactly solvable generalized infinite-range models which exhibit multicritical mean-field behavior. New mathematical results are reported for the asymptotic behavior of integrals representing finite-size scaling functions of infinite-range models.
We use the transfer-matrix method to calculate density-of-states (DOS) functions for fully finite Ising-model systems. In this way, we obtain expressions for the corresponding partition functions for systems of sizes up to 12\ifmmode\times\else\texttimes\fi{}12 in zero field and up to 8\ifmmode\times\else\texttimes\fi{}8 in nonzero external magnetic field. We also establish a way of representing the DOS functions, in terms of the suitably scaled variables, that reveals a remarkably rapid convergence of these functions towards their thermodynamic limiting values as the system size increases. This rapid convergence implies that results obtained for systems of small sizes contain a surprisingly large amount of information about the behavior of infinite systems. In particular, we demonstrate that the finite-size-scaling-theory predictions are satisfied for small systems (treated exactly via this method) comparably well as for much larger systems (treated approximately via the Monte Carlo method). In addition, applying the method to the square Ising model in a nonzero field, we discover that the DOS functions for small systems display a regularity that one would expect to appear only for very large systems.
Using thermodynamic fluctuation theory, we study the finite-size rounding of anomalies occurring at first-order phase transitions of the corresponding infinite system. Explicit expressions for thermodynamic functions are derived both for "symmetric transitions" (such as the jump of the spontaneous magnetization in the Ising model from +Msp+{M}_{\mathrm{sp}}+Msp​ to $\ensuremath{-}{M}_{\mathrm{sp}}$ as the field changes from 0+{0}^{+}0+ to ${0}^{\ensuremath{-}}$) as well as for asymmetric cases, but restricting attention to (hyper)cubic system shapes. As an explicit example for the usefulness of these considerations in Monte Carlo simulations where it may be a problem to (i) locate a phase transition and (ii) distinguish first-order from second-order transitions, we present numerical results for the two-dimensional nearest-neighbor Ising ferromagnet in a field, both below the critical temperature Tc{T}_{c}Tc​ and at Tc{T}_{c}Tc​. The numerical results are found to be in very good agreement with the phenomenological theory and it is shown that one may extract the magnitudes of jumps occurring at first-order phase transitions in a well-defined and accurate way.