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arXiv:1911.12954 · International Symposium on Lattice Field Theory (LATTICE)

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

Shinichiro Akiyama, Y. Kuramashi, Takumi Yamashita, Yusuke Yoshimura

Published Jan 4, 2020• 9 citations
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Abstract

We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to $1024^4$, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.

DOI
10.22323/1.363.0138

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We investigate the phase transition of the four-dimensional Ising model with two types of tensor network scheme, one is the higher-order tensor renormalization group and the other is the anisotropic tensor renormalization group. The results for the internal energy and magnetization obtained by the former algorithm with the impure tensor method, enlarging the lattice volume up to 102441024^410244, are consistent with the weak first-order phase transition. For the later algorithm, our implementation successfully reduces the execution time thanks to the parallel computation and the results provided by ATRG seems comparable to those with HOTRG.
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 ${\ensuremath{\phi}}_{d=4}^{4}$ theory. We investigate the phase transition of this model with the higher-order tensor renormalization group enlarging the lattice size up to 10244102{4}^{4}10244 with parallel computation. The results for the internal energy and the magnetization are consistent with the weak first-order phase transition.
We propose a different tensor renormalization group algorithm, anisotropic tensor renormalization group (ATRG), for lattice models in arbitrary dimensions. The proposed method shares the same versatility with the higher-order tensor renormalization group (HOTRG) algorithm, i.e., it preserves the lattice topology after the renormalization. In comparison with HOTRG, both the computation cost and the memory footprint of our method are drastically reduced, especially in higher dimensions, by renormalizing tensors in an anisotropic way after the singular value decomposition. We demonstrate the ability of ATRG for the square lattice and the simple cubic lattice Ising models. Although the accuracy of the present method degrades when compared with HOTRG of the same bond dimension, the accuracy with fixed computation time is improved greatly due to the drastic reduction of the computation cost.
We investigate the phase transition of the four-dimensional single-component ${\ensuremath{\phi}}^{4}$ theory on the lattice using the tensor renormalization group method. We have examined the hopping parameter dependence of the bond energy and the vacuum condensation of the scalar field $⟨\ensuremath{\phi}⟩$ at a finite quartic coupling $\ensuremath{\lambda}$ on large volumes up to V=10244V=102{4}^{4}V=10244 in order to detect the spontaneous breaking of the Z2{\mathbb{Z}}_{2}Z2​ symmetry. Our results show that the system undergoes the weak first-order phase transition at a certain critical value of the hopping parameter. We also make a comparative study of the three-dimensional ${\ensuremath{\phi}}^{4}$ theory and find that the properties of the phase transition are consistent with the universality class of the three-dimensional Ising model.
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The development of tensor renormalization group (TRG) algorithm in higher dimensions is an important and urgent task, as the TRG is expected to provide a way to overcome the sign problem in lattice quantum chromodynamics (QCD) calculations at finite density. One possible approach that enables faster computations in four-dimensional lattice theories is the anisotropic tensor renormalization group (ATRG). However, the computational cost remains substantial and requires significant computational resources. In this paper, we propose a novel algorithm, called the triad-ATRG, which is based on the ATRG and other improved TRG variants with triad network representation. This method achieves lower scaling with respect to the bond dimension, while minimizing the loss of accuracy in the free energy and other physical quantities. We also present parallel implementations of both the ATRG and triad-ATRG on multiple GPUs, which significantly improve performance compared to CPU-based calculations for the four-dimensional system.
We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension D=70D=70D=70, the absolute errors of the critical temperature Tc{T}_{c}Tc​ and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.