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. We subsequently generalize the CTMRG to a hyperbolic lattice with dodecahedral cells, which is an infinite-dimensional lattice. We analyze the spontaneous magnetization, von Neumann entropy, and correlation length to find a continuous noncritical phase transition on the dodecahedral lattice. We estimate the phase-transition temperature and find the magnetic critical exponents $\beta =0.4999$ and $\delta =3.007$, which confirm the mean-field universality class, in accord with predictions from Monte Carlo and high-temperature series expansions. The algorithm can be applied to arbitrary multistate spin models.

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• Continuous phase transition
• Critical phenomena
• Classical spin models
• Density matrix methods
• Numerical approximation & analysis
• Tensor network methods
• Tensor network renormalization