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arXiv:1011.4164 · Physical Review Letters

Microcanonical Relation between Continuous and Discrete Spin Models

Lapo Casetti, Cesare Nardini, Rachele Nerattini

Published Feb 3, 2011• 14 citations
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Abstract

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.

DOI
10.1103/physrevlett.106.057208

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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 ) lattice spin models and Ising models defined on the same lattice was recently put forward (Casetti et al 2011 Phys. Rev. Lett. 106 057208). Such a relation, inspired by an energy landscape analysis, implies that the density of states of an O ( n ) spin model on a lattice can be effectively approximated, at least close to the phase transition, in terms of the density of states of an Ising model defined on the same lattice and with the same interactions. In this paper we show that such a relation exactly holds, albeit in a slightly modified form, in the special cases of the mean-field XY model and the one-dimensional XY model. We also discuss the possible consequences of this result for the general case.
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 .
The one-dimensional Ising model in a random field and with power-law decaying ferromagnetic bonds is studied at zero temperature. Comparing the scaling of the energy contributions of the ferromagnetic domain wall flip and of the random field \`a la Imry-Ma, a threshold value for the power $\ensuremath{\rho}$ of the long-range interaction can be determined, beyond which no critical behavior occurs. The critical threshold value is ${\ensuremath{\rho}}_{c}=3/2$, at a difference with the zero field model in which ${\ensuremath{\rho}}_{c}=2$. This prediction is analyzed by numerical computation of the ground states below, at, and above this threshold value. The analogy between the critical behavior of long-range one-dimensional systems and in short-range DDD-dimensional systems is investigated. At the critical threshold value of $\ensuremath{\rho}$, corresponding to the lower critical dimension, numerical evidence is found for a zero temperature transition at a finite critical field. Possible finite size crossover effects are discussed in this case. Some implications for the critical behavior of spin glasses in a field are conjectured.
The Ising model was introduced in 1920 to describe a uniaxial system of magnetic moments, localized on a lattice, interacting via nearest-neighbour exchange interaction. It is the generic model for a continuous phase transition and arguably the most studied model in theoretical physics. Since it was solved for a two-dimensional lattice by Onsager in 1944, thereby representing one of the very few exactly solvable models in dimensions higher than one, it has served as a testing ground for new developments in analytic treatment and numerical algorithms. Only series expansions and numerical approaches, such as Monte Carlo simulations, are available in three dimensions. This review focuses on Monte Carlo simulation. We build upon a data set of unprecedented size. A great number of quantities of the model are estimated near the critical coupling. We present both a conventional analysis and an analysis in terms of a Puiseux series for the critical exponents. The former gives distinct values of the high- and low-temperature exponents; by means of the latter we can get these exponents to be equal at the cost of having true asymptotic behaviour being found only extremely close to the critical point. The consequences of this for simulations of lattice systems are discussed at length.
We report the results of a microcanonical simulation of the two- and three-dimensional Ising models at criticality. We also present a microcanonical algorithm that allows a simultaneous simulation of a lattice spin system at different energies, in our case the number of different energies is 32. The critical behavior of the systems was studied via a recently proposed microcanonical renormalization-group technique that yields independent estimates for the critical energy and the critical temperature, as well as for three critical exponents providing a direct test of hyperscaling. Our results in two dimensions are in good agreement with exact results. In the three dimensions our quoted values are consistent with Monte Carlo estimates recently reported in the literature. We obtain uc{\mathit{u}}_{\mathit{c}}uc​=-0.991(1), Tc{\mathit{T}}_{\mathit{c}}Tc​=4.5112(3), \ensuremath{\beta}/\ensuremath{\nu}=0.541(1), \ensuremath{\nu}=0.630(3), and \ensuremath{\alpha}=0.109(1).
The properties of the pure-site clusters of spin models, i.e., the clusters which are obtained by joining nearest-neighbor spins of the same sign, are here investigated. In the Ising model in two dimensions it is known that such clusters undergo a percolation transition exactly at the critical point. We show that this result is valid for a wide class of bidimensional systems undergoing a continuous magnetization transition. We provide numerical evidence for discrete as well as for continuous spin models, including SU(N)\mathrm{SU}(N)SU(N) lattice gauge theories. The critical percolation exponents do not coincide with the ones of the thermal transition, but they are the same for models belonging to the same universality class.