Tensor-Network study of Ising model on infinite hyperbolic dodecahedral lattice

TL;DR

The paper develops a tensor-network CTMRG framework to study the classical Ising model on an infinite hyperbolic lattice built from regular dodecahedra. It extends CTMRG from 2D to 3D and adapts it to the hyperbolic $(5,3,4)$ lattice to extract a phase-transition temperature and mean-field exponents. The main findings are a continuous non-critical transition with $T_{ m pt} \approx 4.66$ and mean-field exponents $\beta = 1/2$, $\delta = 3$, consistent with MC and HT expansions. The method generalizes to arbitrary multi-state spin models on infinite hyperbolic spaces and enables future exploration of Potts and quantum tensor-network models.

Abstract

We propose a tensor-network-based algorithm to study the classical Ising model on an infinitely large hyperbolic lattice with a regular 3D tesselation of identical dodecahedra. We reformulate the corner transfer matrix renormalization group (CTMRG) algorithm from 2D to 3D to reproduce the known results on the cubic lattice. Consequently, we generalize the CTMRG to the hyperbolic dodecahedral lattice, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous non-critical phase transition on the dodecahedral lattice. The phase transition temperature is estimated to be $T_{\rm pt} \approx 4.66$. We find the magnetic critical exponents $β= 0.4999$ and $δ=3.007$ that confirm the mean-field universality class in accord with predictions of Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multi-state spin models.

Figures (15)

• Figure 1: The regular dodecahedron (on the left) serves as a basic cell for constructing the hyperbolic lattice through the uniform 3D tessellation of an infinite number of identical dodecahedra. Around each dodecahedral edge and vertex, there are four and eight dodecahedra, respectively, without leaving free space. Such a generalized 3D tessellation of the infinite lattice is embedded in the infinite-dimensional space. The local visualization from the inside of the hyperbolic dodecahedral lattice is shown on the right and is denoted as a $(5,3,4)$ order-$4$ dodecahedral (honeycomb) lattice. Notice that the standard cubic lattice, denoted as $(4,3,4)$, satisfies the identical rules, the basic cells are identical cubes, and thus the cubic lattice is embedded in three dimensions.
• Figure 2: Visualization of the four tensors required to construct the 3D cubic lattice in vertex representation: (a) rank-$6$vertex tensor ${\cal V}$, (b) rank-$5$face tensor ${\cal F}$, (c) rank-$4$edge tensor ${\cal E}$, (d) rank-$3$corner tensor ${\cal C}$. Index contraction of the physical spin $\sigma$, denoted by a black filled circle, follows from Eqs. \ref{['Tensors']}. An example of the $3\times3\times3$ cubic lattice (e) and explicit tensor structure of the middle and bottom layers (f). In the following text, we omit the black circles that denote the spins.
• Figure 3: Visualization of extended tensors in the cubic lattice: (a) ${\tilde{\cal F}_{j+1}}$, (b) ${\tilde{\cal E}_{j+1}}$, and (c) ${\tilde{\cal C}_{j+1}}$. The spins are located in the vertices and are omitted. The connected lines correspond to tensor contractions according to Eq. \ref{['extcube']} in the simplified notation (without indices). The not-connected lines with open ends are the tensor indices. For more details, see Appendix \ref{['ApA']}.
• Figure 4: Graphical visualization of the two types of reduced density matrices for the cubic lattice: (a) linear$\rho^{~}_{{\rm L}_{j+1}}$ and (b) planar$\rho^{~}_{{\rm P}_{j+1}}$ both of the are depicted as the two parallel thicker lines and squares in gray color, respectively.
• Figure 5: Renormalization scheme of the extended tensors ${\tilde{\cal F}_{j+1}} \to {{\cal F}_{j+1}}$ (a), ${\tilde{\cal E}_{j+1}} \to {{\cal E}_{j+1}}$ (b), and ${\tilde{\cal C}_{j+1}} \to {{\cal C}_{j+1}}$ (c) as in Eqs. \ref{['renormcubic']} after applying the extension scheme from Eqs. \ref{['extcube']}. This renormalization scheme maps them back onto the tensors with their original ranks using the isometries. They also reduce the bond dimensions to the selected values $m_{\rm L}$ and $m_{\rm P}$. This is graphically depicted in gray color either by the doubled thick lines for $U^{~}_{\rm L}$ or by the doubled thick squares for $U^{~}_{\rm P}$. For details, see App. \ref{['ApA']}.