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arXiv:hep-lat/9602025 · Physical review. B, Condensed matter

Universality of the Ising model on spherelike lattices

Christian Hoelbling, Christian B. Lang

Published Aug 1, 1996• 33 citations
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Abstract

We study the two-dimensional (2D) Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a 3D cube, a slab, and a sphere, respectively. Systems of volumes ranging up to O(${10}^{5}$) sites are simulated and finite-size scaling is analyzed. The partition function zeros and the values of various cumulants at their respective peak positions are determined and they agree with the scaling behavior expected from universality with the Onsager solution on the torus (\ensuremath{\nu}=1). For the pseudocritical values of the coupling we find significant anomalies indicating a shift exponent \ensuremath{\ne}1 for spherelike lattice topology. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.3434

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We study the two-dimensional (2D) Ising model on three different types of lattices that are topologically equivalent to spheres. The geometrical shapes are reminiscent of the surface of a 3D cube, a slab, and a sphere, respectively. Systems of volumes ranging up to O(105{10}^{5}105) sites are simulated and finite-size scaling is analyzed. The partition function zeros and the values of various cumulants at their respective peak positions are determined and they agree with the scaling behavior expected from universality with the Onsager solution on the torus (\ensuremath{\nu}=1). For the pseudocritical values of the coupling we find significant anomalies indicating a shift exponent \ensuremath{\ne}1 for spherelike lattice topology. \textcopyright{} 1996 The American Physical Society.
We numerically investigate the Ising model near quasi-criticality on finite two-dimensional (2D) manifolds formed by triangular and square lattices which exhibit the topology of the surface of a sphere, the projective plane, the torus, and the Klein bottle. We find that the finite-size scaling function of the magnetic susceptibility is the same for the surface of the torus and of the Klein bottle (Euler's characteristic 0 for both of them) but differs from the one for the projective plane (Euler's characteristic 1) and for the surface of the sphere (Euler's characteristic 2). This indicates that the universal properties of the continuous order-disorder phase transition in the Ising model for 2D manifolds depend on their topology.
We perform high-statistics Monte Carlo simulations of three three-dimensional Ising spin glass models: the ± J Ising model for two values of the disorder parameter p , p = 1/2 and 0.7, and the bond-diluted ± J model for bond-occupation probability p b = 0.45. A finite-size scaling analysis of the quartic cumulants at the critical point shows conclusively that these models belong to the same universality class and allows us to estimate the scaling-correction exponent ω related to the leading irrelevant operator, ω = 1.0(1). We also determine the critical exponents ν and η. Taking into account the scaling corrections, we obtain ν = 2.53(8) and η = −0.384(9).
We present detailed calculations for the partition function and the free energy of the finite two-dimensional square lattice Ising model with periodic and antiperiodic boundary conditions, variable aspect ratio, and anisotropic couplings, as well as for the corresponding universal free energy finite-size scaling functions. Therefore, we review the dimer mapping, as well as the interplay between its topology and the different types of boundary conditions. As a central result, we show how both the finite system as well as the scaling form decay into contributions for the bulk, a characteristic finite-size part, and – if present – the surface tension, which emerges due to at least one antiperiodic boundary in the system. For the scaling limit we extend the proper finite-size scaling theory to the anisotropic case and show how this anisotropy can be absorbed into suitable scaling variables.
PACS. 02.70.Lq(Monte Carlo and statistical methods) PACS. 05.50.+q(Lattice theory and statistics; Ising problems) PACS. 68.35.Rh(Phase transitions and critical phenomena) Abstract. We perform Monte Carlo simulations of the hard-sphere lattice gas on the body-centred cubic lattice with nearest neighbour exclusion. We get the critical exponents, β/ν = 0.311(8) and γ/ν = 2.38(2), where β, γ, and ν are the critical exponents of the staggered density, the staggered compressibility, and the correlation length, respectively. The values of the hard-sphere lattice gas on the simple cubic lattice agree with them but those of the threedimensional Ising model do not. This supports that the hard-sphere lattice gas does not fall into the Ising universality class in three dimensions. In this letter we study the hard-sphere lattice gas whose atoms interact with infinite repulsion of nearest neighbour pairs. The grand partition function is ΞV (z) = ∑ z N ZV (N), (1) N where z is an activity and ZV (N) is the number of configurations in which there are N atoms in the lattice of V sites. There are many studies on:
A Monte Carlo method is applied to simulate the static critical behavior of a cubic-lattice 3D Ising model for systems with quenched disorder. Numerical results are presented for the spin concentrations of p = 1.0, 0.95, 0.9, 0.8, 0.6 on L × L × L lattices with L = 20–60 under periodic boundary conditions. The critical temperature is determined by the Binder cumulant method. A finite-size scaling technique is used to calculate the static critical exponents α, β, γ, and ν (for specific heat, susceptibility, magnetization, and correlation length, respectively) in the range of p under study. Universality classes of critical behavior are discussed for three-dimensional diluted systems.
We study the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter DDD on the simple cubic lattice. To this end we perform Monte Carlo simulations using a hybrid of the local Metropolis, the single cluster and the wall cluster algorithm. Using finite size scaling we determine the value ${D}^{\ensuremath{\ast}}=0.656(20)$ of the parameter DDD, where leading corrections to scaling vanish. We find $\ensuremath{\omega}=0.832(6)$ for the exponent of leading corrections to scaling. In order to compute accurate estimates of critical exponents, we construct improved observables that have a small amplitude of the leading correction for any model. Analyzing data obtained for D=0.641D=0.641D=0.641 and 0.655 on lattices of a linear size up to L=360L=360L=360 we obtain $\ensuremath{\nu}=0.63002(10)$ and $\ensuremath{\eta}=0.03627(10)$. We compare our results with those obtained from previous Monte Carlo simulations and high-temperature series expansions of lattice models, by using field-theoretic methods and experiments.