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Application of a constrained-transfer-matrix method to metastability in the d = 2 Ising ferromagnet

Günther, C.C.A., Rikvold, P.A., M. A. Novotny

Published Jan 1, 1994• 10 citations
OpenAlex

Abstract

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.

Related papers

Numerical transfer-matrix study of metastability in thed=2 Ising model

C. C. A. Günther, Per Arne Rikvold, M. A. Novotny

We apply a generalized transfer-matrix method to the d=2 Ising ferromagnet at H\ensuremath{\ne}0 and T${\mathit{T}}_{\mathit{c}}$ 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.

arXiv:cond-mat/9308019•Physical Review Letters•Dec 13, 1993•35 citations

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

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

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.

arXiv:cond-mat/9402007•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Apr 1, 1994•11 citations

Metastability in the (1+1)D Ising model: a primitive droplet model calculation

C K Harris

Using a primitive droplet model in conjunction with recent exact results for the surface free energies of Ising droplets on a square lattice with anisotropic couplings, the author obtains an expression for the principal non-universal amplitude appearing in the imaginary part of the small field metastable state energy of the Hamiltonian field theory version of the Ising model in (1+1)D. The calculation is in very good numerical agreement with the results of a recent finite lattice study.

Journal of Physics A Mathematical and General•Jun 1, 1984•2 citations

Calculation of the Critical Temperature for the Anisotropic Two-Layer Ising Model Using the Transfer Matrix Method

Mehrdad Ghaemi, Mohammad Ghannadi, Behrouz Mirza

A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r -wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K x and K y (where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K z, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.

The Journal of Physical Chemistry B•Dec 18, 2002•12 citations

Method to study relaxation of metastable phases: Macroscopic mean-field dynamics

Jooyoung Lee, M. A. Novotny, Per Arne Rikvold

We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter m through the restricted free energy F(m) and are designed to give the correct equilibrium distribution for m. The connection between macroscopic dynamics and the underlying microscopic dynamic is considered in the context of a projection-operator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime 〈\ensuremath{\tau}〉 on the inverse of the applied field H, and the observation of distinct field regions in which \ensuremath{\Lambda}\ensuremath{\equiv}d ln〈\ensuremath{\tau}〉/d\ensuremath{\Vert}H${\mathrm{\ensuremath{\Vert}}}^{1\mathrm{\ensuremath{-}}\mathit{d}}$ depends differently on \ensuremath{\Vert}H\ensuremath{\Vert}. In addition, at very low temperatures we observe oscillatory behavior of \ensuremath{\Lambda} with respect to \ensuremath{\Vert}H\ensuremath{\Vert}, which is due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.

arXiv:cond-mat/9503073•Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics•Jul 1, 1995•43 citations

Ising ferromagnets and antiferromagnets in an imaginary magnetic field

Roman Krčmár, Andrej Gendiar, Ladislav Šamaj

We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.

arXiv:2112.09536•Physical review. E•May 6, 2022•2 citations
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 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.
A new finite-size scaling approach based on the transfer matrix method is developed to calculate the critical temperature of an anisotropic two-layer Ising ferromagnet on strips of r -wide sites of square lattices. The reduced internal energy per site has been accurately calculated for the ferromagnetic case, with the nearest-neighbor couplings K x and K y (where K x and K y are the nearest-neighbor interactions within each layer in the x and y directions, respectively) and with interlayer coupling K z, using different size-limited lattices. The calculated energies for different lattice sizes intersect at various points when plotted versus the reduced temperature. It is found that the location of the intersection point versus the lattice size can be fitted to a power series in terms of the lattice sizes. The power series is used to obtain the critical temperature of the unlimited two-layer lattice. The results obtained are in good agreement with the accurate values reported by others.
We propose two different macroscopic dynamics to describe the decay of metastable phases in many-particle systems with local interactions. These dynamics depend on the macroscopic order parameter m through the restricted free energy F(m) and are designed to give the correct equilibrium distribution for m. The connection between macroscopic dynamics and the underlying microscopic dynamic is considered in the context of a projection-operator formalism. Application to the square-lattice nearest-neighbor Ising ferromagnet gives good agreement with droplet theory and Monte Carlo simulations of the underlying microscopic dynamic. This includes quantitative agreement for the exponential dependence of the lifetime 〈\ensuremath{\tau}〉 on the inverse of the applied field H, and the observation of distinct field regions in which \ensuremath{\Lambda}\ensuremath{\equiv}d ln〈\ensuremath{\tau}〉/d\ensuremath{\Vert}H${\mathrm{\ensuremath{\Vert}}}^{1\mathrm{\ensuremath{-}}\mathit{d}}$ depends differently on \ensuremath{\Vert}H\ensuremath{\Vert}. In addition, at very low temperatures we observe oscillatory behavior of \ensuremath{\Lambda} with respect to \ensuremath{\Vert}H\ensuremath{\Vert}, which is due to the discreteness of the lattice and in agreement with rigorous results. Similarities and differences between this work and earlier works on finite Ising models in the fixed-magnetization ensemble are discussed.
We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.