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arXiv:1407.7966 · Journal of Statistical Mechanics Theory and Experiment

Critical energy density ofO(n) models ind= 3

Rachele Nerattini, Andrea Trombettoni, Lapo Casetti

Published Dec 4, 2014• 6 citations
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Abstract

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 .

DOI
10.1088/1742-5468/2014/12/p12001

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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 .
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 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.
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin-1/2 Ising model with nearest-neighbour interactions, a spin-1/2 model with nearest-neighbour and third-neighbour interactions, and a spin-1 model with nearest-neighbour interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbour interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), y h =2.4815 (15) and y i =-0.82 (6). The universal ratio Q=(m 2 ) 2 /(m 4 ) is equal to 0.6233 (4) for periodic systems with cubic symmetry. The critical point of the nearest-neighbour spin-1/2 model is K c =0.2216546 (10).
We investigate three Ising models on the simple cubic lattice by means of Monte Carlo methods and finite-size scaling. These models are the spin- 1 2 Ising model with nearest-neighbor interactions, a spin- 1 2 model with nearest-neighbor and third-neighbor interactions, and a spin-1 model with nearest-neighbor interactions. The results are in accurate agreement with the hypothesis of universality. Analysis of the finite-size scaling behavior reveals corrections beyond those caused by the leading irrelevant scaling field. We find that the correction-to-scaling amplitudes are strongly dependent on the introduction of further-neighbor interactions or a third spin state. In a spin-1 Ising model, these corrections appear to be very small. This is very helpful for the determination of the universal constants of the Ising model. The renormalization exponents of the Ising model are determined as y t =1.587 (2), yh =2.4815 (15) and y i = -0.82 (6). The universal ratio Q = #m 2 # 2 /#m 4...
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).
This paper addresses the limitation for routine applications of the proposed Monte Carlo method of direct calculation of absolute free energy by sampling of finite-size dependences and discusses new extensions. The need for multistage sampling is removed and the method is now simpler to apply in practice. To exhibit this, the critical free-energy density for the simple-cubic nearest-neighbor ferromagnetic Ising model is studied for a wide range of lattice sizes, up to 32\ifmmode\times\else\texttimes\fi{}32\ifmmode\times\else\texttimes\fi{}32. The finite-size scaling is demonstrated and the scaling amplitude calculated with higher accuracy than before. These results indicate the possibility that very accurate free-energy density and related quantities such as surface free energy, can be routinely obtained for lattice models from Monte Carlo simulations, even at criticality.