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Europhysics Letters (EPL)

Criticality senses topology

Oleg A. Vasilyev, Anna Maciołek, S. Dietrich

Published Dec 9, 2019• 5 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/128/20002

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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.
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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.
In this paper we study finite-size scaling at criticality in the Gibbs ensemble with Monte Carlo simulations. A recent lattice-gas version of Gibbs-ensemble sampling for the two-dimensional Ising-model magnet with nearest-neighbor ferromagnetic interactions on two subsystems of L\ifmmode\times\else\texttimes\fi{}L lattices is considered. Both square and triangular lattices with a range of lattice sizes are simulated at bulk criticality. We are interested in the question of universality for the fourth-order cumulant of the order-parameter distribution in the Gibbs ensemble. Estimates of -1.954\ifmmode\pm\else\textpm\fi{}0.005 and -1.9543\ifmmode\pm\else\textpm\fi{}0.005 are obtained for the square and triangular lattices, respectively. These values confirm universality within the Gibbs ensemble, but are statistically different from the canonical-ensemble values of \ensuremath{\sim}-1.83. We point out that one must interpret this apparent ensemble dependence with care. Unlike in the canonical ensemble, the order parameter in the Gibbs ensemble is the subsystem magnetization. Its fourth-order cumulant is not simply related to the usual fourth magnetic-field derivative of the total free energy. Standard finite-size scaling analysis of data for the total specific heat, subsystem magnetization, and susceptibility are also considered and found to be in good agreement with finite-size scaling.
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On Critical Temperature and Finite Size Scaling of Continuous Spin 2d2d2d Ising Model
In this paper, we have studied the critical temperature TcT_cTc​ of continuous spin 2d2d2d square-lattice Ising model using Monte-Carlo simulation. We have considered spins sss in a bounded interval, where s∈[−1,+1]s \in [-1,+1]s∈[−1,+1] in square-lattice configuration with periodic boundary condition. We have observed that the critical temperature TcT_cTc​ is approximately 0.9250.9250.925, showing a clear second order phase transition. Considering finite size scaling, we have also obtained the critical exponents associated with susceptibility, specific heat, magnetization and we find that these values are in good agreement with the corresponding values obtained for the standard 2d2d2d Ising universality class.