Mean-field universality class induced by weak hyperbolic curvatures

Abstract

Order-disorder phase transition of the ferromagnetic Ising model is investigated on a series of twodimensional lattices that have negative Gaussian curvatures. Exceptional lattice sites of coordination number seven are distributed on the triangular lattice, where the typical distance between the nearest exceptional sites is proportional to an integer parameter n. Thus, the corresponding curvature is asymptotically proportional to −n −2 . Spontaneous magnetization and specific heat are calculated by means of the corner transfer matrix renormalization group method. For all the finite n cases, we observe the mean-field-like phase transition. It is confirmed that the entanglement entropy at the transition temperature is linear in (c/6) ln n, where c = 1/2 is the central charge of the Ising model. The fact agrees with the presence of the typical length scale n being proportional to the curvature radius.

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.…”

Free-energy analysis of spin models on hyperbolic lattice geometries

“…3 including the case of the Euclidean (4, 4) lattice, which serves as as a benchmark since this case is exactly solvable. We have shown 12,13,19 that the Ising model on certain types of the hyperbolic lattices belongs to the mean-field universality class. Now, we expand our analysis for arbitrary (p ≥ 4, q ≥ 4) lattices.…”

“…(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, . .…”

“…A direct numerical calculation of the free energy per site in Eq. (19) frequently results in an extremely fast divergence of the partition function F (p,q) on hyperbolic lattices whenever k 10. Therefore, the numerical operations with the tensors C k and T k require to consider an appropriate norm (normalization) in each iteration step k. Two types of the norm are at hand.…”

What are the limits of universality?

“…We remark that there is an extensive literature investigating critical behaviour on hyperbolic lattices including e.g. [52,3,36,51,42,9,24]. These lattices are very different from the non-Euclidean lattices we consider in this paper.…”