Phase transition of four-dimensional Ising model with higher-order tensor renormalization group

We apply the higher-order tensor renormalization group to the four-dimensional ferromagnetic Ising model, which has been attracting interest in the context of the triviality of the scalar ${\varphi }_{d=4}^4$ theory. We investigate the phase transition of this model with the higher-order tensor renormalization group 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.

Figure 1

Convergence behavior of $\ln Z_N$ as a function of bond dimension $D_{\mathrm{cut}}$ at $T=6.64250$ in the vicinity of the transition temperature. $L$ is a linear extent of the lattice defined as $N=L^4$.

Figure 2

$X^{(n)}$ at the $n$th coarse-graining step with $D_{\mathrm{cut}}=13$. The red line corresponds to the disordered phase, and the blue one corresponds to the ordered one.

Figure 3

Transition temperature as a function of bond dimension. Error bars are within symbols.

Figure 4

Internal energy at the $n$th coarse-graining step with $D_{\mathrm{cut}}=13$. The red line corresponds to the disordered phase, and blue one corresponds to the ordered phase. The inset graph magnifies the $n$ dependence beyond $n=20$.

Figure 5

The internal energy as a function of temperature for various lattice sizes with $D_{\mathrm{cut}}=13$. $T_{\mathrm{c}}(D_{\mathrm{cut}}=13)$ estimated by $X^{(n)}$ of Eq. (15) is within the gray band.

Figure 6

Magnetization at the $n$th coarse-graining step with $D_{\mathrm{cut}}=13$ and $h=1.0\times {10}^{-9}$. The red line corresponds to the disordered phase, and the blue one corresponds to the ordered phase.

Figure 7

Spontaneous magnetization in the thermodynamic limit with $D_{\mathrm{cut}}=13$. Error bars, provided by extrapolation, are within symbols. $T_{\mathrm{c}}(D_{\mathrm{cut}}=13)$ estimated by $X^{(n)}$ of Eq. (15) is within the gray band.