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We investigate the phase transition of the four-dimensional Ising model withtwo types of tensor network scheme, one is the higher-order tensorrenormalization group and the other is the anisotropic tensor renormalizationgroup. The results for the internal energy and magnetization obtained by theformer algorithm with the impure tensor method, enlarging the lattice volume upto $1024^4$, are consistent with the weak first-order phase transition. For thelater algorithm, our implementation successfully reduces the execution timethanks to the parallel computation and the results provided by ATRG seemscomparable to those with HOTRG.

Phase transition of four-dimensional Ising model with tensor network scheme