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arXiv:cond-mat/0503733 · Physical Review E

Density of states of classical spin systems with continuous degrees of freedom

Andreas Richter, Michel Pleimling, Alfred Hüller

Published May 25, 2005• 7 citations
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Abstract

In the last years different studies have revealed the usefulness of a microcanonical analysis of finite systems when dealing with phase transitions. In this approach the quantities of interest are exclusively expressed as derivatives of the entropy S=ln Omega where Omega is the density of states. Obviously, the density of states has to be known with very high accuracy for this kind of analysis. Important progress has been achieved recently in the computation of the density of states of classical systems, as new types of algorithms have been developed. Here we extend one of these methods, originally formulated for systems with discrete degrees of freedom, to systems with continuous degrees of freedom. As an application we compute the density of states of the three-dimensional XY model and demonstrate that critical quantities can directly be determined from the density of states of finite systems in cases where the degrees of freedom take continuous values.

DOI
10.1103/physreve.71.056705

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In the last years different studies have revealed the usefulness of a microcanonical analysis of finite systems when dealing with phase transitions. In this approach the quantities of interest are exclusively expressed as derivatives of the entropy S=ln Omega where Omega is the density of states. Obviously, the density of states has to be known with very high accuracy for this kind of analysis. Important progress has been achieved recently in the computation of the density of states of classical systems, as new types of algorithms have been developed. Here we extend one of these methods, originally formulated for systems with discrete degrees of freedom, to systems with continuous degrees of freedom. As an application we compute the density of states of the three-dimensional XY model and demonstrate that critical quantities can directly be determined from the density of states of finite systems in cases where the degrees of freedom take continuous values.
The basic quantity for the description of the statistical properties of physical systems is the density of states or equivalently the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are for example related to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions already in finite systems. In this paper we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities which announce a continuous phase transition in the infinite system are characterised by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analyzing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
A highly efficient Monte Carlo method for the calculation of the density of states of classical spin systems is presented. As an application, we investigate the density of states ΩN(E, M) of two- and three-dimensional Ising models with N spins as a function of energy E and magnetization M. For a fixed energy lower than a critical value Ec,N the density of states exhibits two sharp maxima at M = ± Msp(E) which define the microcanonical spontaneous magnetization. An analysis of the form Msp(E) ∝ (Ec, ∞ - E)βε yields very good results for the critical exponent βε, thus demonstrating that critical exponents can be determined by analyzing directly the density of states of finite systems.
The basic quantity for the description of the statistical properties of physical systems is the density of states or, equivalently, the microcanonical entropy. Macroscopic quantities of a system in equilibrium can be computed directly from the entropy. Response functions such as the susceptibility are related, for example, to the curvature of the entropy surface. Interestingly, physical quantities in the microcanonical ensemble show characteristic properties of phase transitions in finite systems. In this article we investigate these characteristics for finite Ising systems. The singularities in microcanonical quantities, which signal a continuous phase transition in infinite systems, are characterized by classical critical exponents. Estimates of the non-classical exponents which emerge only in the thermodynamic limit can nevertheless be obtained by analysing effective exponents or by applying a microcanonical finite-size scaling theory. This is explicitly demonstrated for two- and three-dimensional Ising systems.
A relation between a class of stationary points of the energy landscape of continuous spin models on a lattice and the configurations of an Ising model defined on the same lattice suggests an approximate expression for the microcanonical density of states. Based on this approximation we conjecture that if a O(n) model with ferromagnetic interactions on a lattice has a phase transition, its critical energy density is equal to that of the n=1 case, i.e., an Ising system with the same interactions. The conjecture holds true in the case of long-range interactions. For nearest-neighbor interactions, numerical results are consistent with the conjecture for n=2 and n=3 in three dimensions. For n=2 in two dimensions (XY model) the conjecture yields a prediction for the critical energy of the Berežinskij-Kosterlitz-Thouless transition, which would be equal to that of the two-dimensional Ising model. We discuss available numerical data in this respect.
A relation between O ( n ) models and Ising models has been recently conjectured (Casetti et al 2011 Phys. Rev. Lett. 106 057208 ). Such a relation, inspired by an energy landscape analysis, implies that the microcanonical density of states of an O ( n ) spin model on a lattice can be effectively approximated in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. Were this relation exact, it would imply that the critical energy densities of all the O ( n ) models (i.e. the average values per spin of the O ( n ) Hamiltonians at their respective critical temperatures) should be equal to that of the corresponding Ising model. It is therefore worth investigating how different the critical energies are and how this difference depends on n . We compare the critical energy densities of O ( n ) models in three dimensions in some specific cases: the O (1) or Ising model, the O (2) or XY model, the O (3) or Heisenberg model, the O (4) model and the O (∞) or spherical model, all defined on regular cubic lattices and with ferromagnetic nearest-neighbor interactions. The values of the critical energy density in the n = 2, n = 3 and n = 4 cases are derived through a finite-size scaling analysis of data produced by means of Monte Carlo simulations on lattices with up to 128 3 sites. For n = 2 and n = 3 the accuracy of previously known results has been improved. We finally derive an interpolation formula showing that the difference between the critical energy densities of O ( n ) models and that of the Ising model is smaller than 1% if n < 8 and never exceeds 3% for any n .
Abstract The continuous ferromagnetic-paramagnetic phase transition in the two-dimensional Ising model has already been excessively studied by conventional canonical statistical analysis in the past. We use the recently developed generalized microcanonical inflection-point analysis method to investigate the least-sensitive inflection points of the microcanonical entropy and its derivatives to identify transition signals. Surprisingly, this method reveals that there are potentially two additional transitions for the Ising system besides the critical transition.