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arXiv:cond-mat/9402007 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite-range-scaling analysis of metastability in an Ising model with long-range interactions

Bryan M. Gorman, Per Arne Rikvold, M. A. Novotny

Published Apr 1, 1994• 11 citations
arXivHTMLPDF

Abstract

We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation f\ifmmode \tilde{}\else \~{}\fi{} of the free energy across the first-order transition, assuming that the system escapes the metastable state by the nucleation of noninteracting droplets. We find that corrections to the field dependence are substantial, and, by solving the Euler-Lagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfer-matrix method, we associate with the subdominant eigenvectors of the transfer matrix a complex-valued ``constrained'' free-energy density ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, ${\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}$ exhibits finite-range scaling behavior in agreement with f\ifmmode \tilde{}\else \~{}\fi{} over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.

DOI
10.1103/physreve.49.2711

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We apply both a scalar field theory and a recently developed transfer-matrix method to study the stationary properties of metastability in a two-state model with weak, long-range interactions: the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising model. Using the field theory, we find the analytic continuation f\ifmmode \tilde{}\else ~{}\fi{} of the free energy across the first-order transition, assuming that the system escapes the metastable state by the nucleation of noninteracting droplets. We find that corrections to the field dependence are substantial, and, by solving the Euler-Lagrange equation for the model numerically, we have verified the form of the free-energy cost of nucleation, including the first correction. In the transfer-matrix method, we associate with the subdominant eigenvectors of the transfer matrix a complex-valued ``constrained'' free-energy density f\ensuremathα{\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}f\ensuremathα​ computed directly from the matrix. For the eigenvector with an associated magnetization most strongly opposed to the applied magnetic field, f\ensuremathα{\mathit{f}}_{\mathrm{\ensuremath{\alpha}}}f\ensuremathα​ exhibits finite-range scaling behavior in agreement with f\ifmmode \tilde{}\else ~{}\fi{} over a wide range of temperatures and fields, extending nearly to the classical spinodal. Some implications of these results for numerical studies of metastability are discussed.
Applying a numerical transfer-matrix formalism, we obtain complex-valued constrained free energies for the two-dimensional square-lattice nearest-neighbor Ising ferromagnet below its critical temperature and in an external magnetic field. In particular, we study the imaginary part of the constrained free-energy branch that corresponds to the metastable phase. Although droplets are not introduced explicitly, the metastable free energy is obtained in excellent agreement with field-theoretical droplet-model predictions. The finite-size scaling properties are different in the weak-field and intermediate-field regimes, and we identify the corresponding different critical-droplet shapes. For intermediate fields, we show that the surface free energy of the critical droplet is given by a Wulff construction with the equilibrium surface tension. We also find a prefactor exponent in complete agreement with the field-theoretical droplet model. Our results extend the region of validity for known results of this field-theoretical droplet model, and they indicate that this transfer-matrix approach provides a nonperturbative numerical continuation of the equilibrium free energy into the metastable phase.
We apply a generalized transfer-matrix method to the d=2 Ising ferromagnet at H\ensuremath{\ne}0 and TTc{\mathit{T}}_{\mathit{c}}Tc​ to obtain complex constrained free energies. Although the method does not explicitly introduce droplets, the imaginary parts of those constrained free energies that correspond to the metastable state are accurately predicted by a field-theoretical droplet model. The free energy of a critical droplet is given by an equilibrium Wulff construction, and we find evidence for Goldstone modes on its surface. These results indicate that our approach provides a nonperturbative numerical continuation of the free energy around the essential singularity at H=0.
We develop a critical finite-force-range scaling theory for D-dimensional scalar ${\mathrm{\ensuremath{\varphi}}}^{\mathit{n}}$ field theories that is based on a scaling ansatz equivalent to a Ginzburg criterion. To investigate its relationship to other scaling theories we derive equivalent results from renormalization groups and from finite-size crossover scaling for systems with weak long-range forces. By comparing our finite-range scaling relations with finite-size scaling relations for hypercylindrical systems above the upper critical dimension Dc{\mathit{D}}_{\mathit{c}}Dc​, we arrive at a criterion of critical equivalence that provides an asymptotic mapping between the two kinds of systems. We apply our scaling relations to a ${\mathrm{\ensuremath{\varphi}}}^{4}$ Ginzburg-Landau Hamiltonian, to the one-dimensional Kac model with exponentially decaying interactions, and to the N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty} quasi-one-dimensional Ising (Q1DI) model, in which each spin interacts with O(N) others. Near the Gaussian mean-field critical point the Ginzburg-Landau Hamiltonians for all three models become identical, but for the Q1DI model this requires a length rescaling. For the Kac model the resulting scaling relations are those of a D=1 quartic field theory, and for the Q1DI model they are those of a cylindrical Ising system above Dc{\mathit{D}}_{\mathit{c}}Dc​. Results of specialized numerical scaling techniques applied to transfer-matrix calculations for the Q1DI model with N\ensuremath{\le}1024 strongly support our theoretically obtained scaling relations.
Develops a method of analytic continuation at first-order phase transitions and applies it to the d=2 Ising model in an external field. The method employs a function built of transfer matrix eigenvalues, which provides rapidly convergent approximations to the stable free energy f and its derivatives for all H>or=0. Recent series analysis results on the existence of an essential singularity at H=0 are confirmed. There is also an indication of a spinodal line, and H sp (T)<O is estimated in the range O<T<or=0.6T c .
The fermionic version of the two-dimensional Ising model in a uniform magnetic field is studied. Its approximate but rather accurate solution was obtained recently in a variational procedure based on the BCS ansatz for the ground state. Here, this solution is continued analytically into the complex H-plane in the vicinity of the origin. In the ferromagnetic phase an essential singularity in the free-energy is observed at the point H=0 of the type predicted by the droplet condensation theory. Simple phenomenological estimates for the radius of the critical droplet and for the rate of decay of metastable states are confirmed by microscopic calculations.