Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane

Abstract

Critical behavior of the Ising model is investigated at the center of large scale finite size systems, where the lattice is represented as the tiling of pentagons. The system is on the hyperbolic plane, and the recursive structure of the lattice makes it possible to apply the corner transfer matrix renormalization group method. From the calculated nearest neighbor spin correlation function and the spontaneous magnetization, it is concluded that the phase transition of this model is mean-field like. One parameter deformation of the corner Hamiltonian on the hyperbolic plane is discussed.

Analysis of quantum spin models on hyperbolic lattices and Bethe lattice

“…We have calculated a sequence of the phase transition magnetic fields h (p) t , which is a strictly monotonous and increasing function, which converges exponentially to the asymptotic value h (∞) t . This feature is completely analogous to a fast exponential saturation of the critical temperatures T (p) c we had observed for the classical Ising model on the identical series of hyperbolic lattices in our earlier studies [19,20]. However, we have not found physical interpretation of this phenomenon yet.…”

“…Graphical representation of the lattices with the fixed coordination number equal to four indexed by the lattice parameter p. The hyperbolic lattices (p = 5, 6, 7, and 10) are depicted in the Poincaré disk representation, which maps the infinitesized hyperbolic lattices onto the unitary circle, which leads to the deformation of the uniform and regular polygons toward the circle boundary. [19,20,21,22], we expect fast convergence of the phase transition magnetic field of the quantum TFIM as well as the ground-state energies of the quantum XY and Heisenberg models toward the asymptotic case p → ∞, which represents the Bethe lattice [20]. Numerical results presented in the following sections are in complete agreement with the expectations.…”

Phase transition ofq-state clock models on heptagonal lattices

“…Since it was pointed out that a system may have a novel behavior due to the presence of a nonvanishing boundary [7], there have been ongoing studies to clarify this issue [4,6,[8][9][10][11]. While the boundary effects can be sometimes excluded, for example, by using a periodic boundary condition [12] or by mathematical abstractions [5,[13][14][15], it is often crucial to understand how a boundary affects the physical properties since it may give the most important contribution to an observed behavior as will be explained in this work.…”

Periodic boundary conditions on the pseudosphere

“…Negatively curved spaces have also recently been considered in the context of statistical physics and condensed matter theory. There, the motivations are both practical (describing the behavior of newly designed mesoscopic and nanoscopic objects with exotic shapes, curvatures, and topological properties [10,11,12,13,14,15]) and theoretical (understanding how curvature influences the critical behavior and the phase transitions of classical statistical systems [5,8,16,17,18,19,20,21,22,23,24,25,26,27,28,29]). Still in condensed-matter theory, negatively curved spaces appear in studies of the Quantum Hall effect [10,11,12] and in the framework of "geometrical frustration" [30,31,32].…”