Vertex Representation of Hyperbolic Tensor Networks

We propose a vertex representation of the tensor network (TN) for classical spin systems on hyperbolic lattices. The tensors form a network of regular $p$-sided polygons ($p>4$) with the coordination number four. The response to multi-state spin systems on the hyperbolic TN is analyzed for their entire parameter space. We show that entanglement entropy is sensitive to distinguish various hyperbolic geometries whereas other thermodynamic quantities are not. We test the numerical accuracy of vertex TNs in the phase transitions of the first, second, and infinite order at the point of maximal entanglement entropy. The hyperbolic structure of TNs induces non-critical properties in the bulk although boundary conditions significantly affect the total free energy in the thermodynamic limit. Thus developed vertex-type TN can be used for the lowest-energy quantum states on the hyperbolic lattices.