Weak correlation effects in the Ising model on triangular-tiled hyperbolic lattices

Abstract

The Ising model is studied on a series of hyperbolic two-dimensional lattices which are formed by tessellation of triangles on negatively curved surfaces. In order to treat the hyperbolic lattices, we propose a generalization of the corner transfer matrix renormalization group method using a recursive construction of asymmetric transfer matrices. Studying the phase transition, the mean-field universality is captured by means of a precise analysis of thermodynamic functions. The correlation functions and the density-matrix spectra always decay exponentially even at the transition point, whereas power-law behavior characterizes criticality on the Euclidean flat geometry. We confirm the absence of a finite correlation length in the limit of infinite negative Gaussian curvature.

Analysis of quantum spin models on hyperbolic lattices and Bethe lattice

“…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.…”

“…. , 8}, and only m = 4 for p ∈ {9, 10}, which was sufficient due to exponentially weak correlations caused by the hyperbolic lattice geometry [21]; any further increase of the states kept m has not improved the numerical calculations significantly).…”

Free-energy analysis of spin models on hyperbolic lattice geometries

“…The claim is identical to that of the exact solution of the Ising model on the Bethe lattice, where the analytically derived mean-field exponents on the Bethe lattice have nothing to do with the mean-field approximation of the model at all 14 . Instead, the mean-field-like feature is caused by the hyperbolic lattice geometry, which is accompanied by the absence of the divergent correlation length at the phase transition 13 .…”

“…(i) The iterative expansion process is formulated in terms of the generalized corner transfer matrix notation (for details, see Refs. 10,12,13,19 ), where the corner transfer tensors C j and the transfer tensors T j expand their sizes as the iteration step (indexed by j) increases, i.e., j = 1, 2, 3, . .…”

Critical Phase in Complex Networks: a Numerical Study

“…It already has been proven by mathematicians that a critical phase also exists in equilibrium spin systems on infinite NAGs [19,70,71]. In addition, the Ising model on hyperbolic lattices has been investigated by means of statistical physics, i.e., Monte Carlo simulations [72][73][74] and the transfer-matrix method [75][76][77][78]. A critical phase and the inverted BKT transition has also been found in the Ising model on an inhomogeneous annealed network [79], the decorated flower [64], Hanoi networks [80][81][82], and the Potts model on the HSWN [12].…”