Abstract
We apply the higher-order tensor renormalization group (HOTRG) to the four-dimensional ferromagnetic Ising model, which has been attracting interests in the context of the triviality of the scalar $\phi^4_{d=4}$ theory. We investigate the phase transition of this model with HOTRG enlarging the lattice size up to $1024^4$ with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.
Highlights
It is well known that the critical behavior of the Ising model on the higher-dimensional hypercubic lattice is well explained with the mean-field theory
Since the Ising model is specified by the infinite coupling limit of the single-component scalar φ44 theory, the model in four dimensions has been attracting the interest of particle physicists for a long time in the context of the triviality of the scalar φ44 theory, which is related to the scalar sector of the standard model describing the generation of gauge boson and fermion mass through the Higgs mechanism [4,5,6,7,8,9,10]
We first evaluate the free energy with the plain higherorder TRG (HOTRG) method
Summary
It is well known that the critical behavior of the Ising model on the higher-dimensional hypercubic lattice is well explained with the mean-field theory. At the upper critical dimension, multiplicative logarithmic corrections are added to the leading scaling behavior of the mean-field theory Some of these corrections were derived by the perturbative calculation with the renormalization group method [3]. From the viewpoint of numerical calculation, it could be possible that there remain some unrevealing aspects in the phase transition of this model, and it should be worth trying different approaches other than the Monte Carlo method For this purpose, we employ the tensor network scheme to investigate the four-dimensional classical Ising model. We employ the tensor network scheme to investigate the four-dimensional classical Ising model This scheme has various types of numerical algorithms [24], which can be divided into two streams: a Hamiltonian approach and a Lagrangian one.
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