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Journal of Physics A Mathematical and General

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

C K Harris

Published Jun 1, 1984• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/17/8/031

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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.
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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.
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 calculate perturbatively the tunneling decay rate $\ensuremath{\Gamma}$ of the metastable phase in the quantum d=1d=1d=1 Ising model in a skew magnetic field near the coexistence line $0<{h}_{x}<1, {h}_{z}\ensuremath{\rightarrow}\ensuremath{-}0$ at T=0.T=0.T=0. It is shown that $\ensuremath{\Gamma}$ oscillates in the magnetic field hz{h}_{z}hz​ due to discreteness of the excitation energy spectrum. After mapping of the obtained results onto the extreme anisotropic d=2d=2d=2 Ising model at T<Tc,T<{T}_{c},T<Tc​, we verify in the latter model the droplet theory predictions for the free energy analytically continued to the metastable phase. We also find evidence for the discrete-lattice corrections in this metastable phase free energy.
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.