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arXiv:2205.02773 · Journal of the Physical Society of Japan

Tensor Network Calculation of the Logarithmic Correction Exponent in the XY Model

Seongpyo Hong, DH Kim

Published Jul 27, 2022• 9 citations
arXivHTMLPDF

Abstract

We study the logarithmic correction to the scaling of the first Lee-Yang (LY) zero in the classical $XY$ model on square lattices by using tensor renormalization group methods. In comparing the higher-order tensor renormalization group (HOTRG) and the loop-optimized tensor network renormalization (LoopTNR), we find that the entanglement filtering in LoopTNR is crucial to gaining high accuracy for the characterization of the logarithmic correction, while HOTRG still proposes approximate bounds for the zero location associated with two different bond-merging algorithms of the higher-order singular value decomposition and the oblique projectors. Using the LoopTNR data computed up to the system size of $L=1024$ in the $L \times L$ lattices, we estimate the logarithmic correction exponent $r = -0.0643(9)$ from the extrapolation of the finite-size effective exponent, which is comparable to the renormalization group prediction of $r = -1/16$.

DOI
10.7566/jpsj.91.084003

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We study the logarithmic correction to the scaling of the first Lee-Yang (LY) zero in the classical XYXYXY model on square lattices by using tensor renormalization group methods. In comparing the higher-order tensor renormalization group (HOTRG) and the loop-optimized tensor network renormalization (LoopTNR), we find that the entanglement filtering in LoopTNR is crucial to gaining high accuracy for the characterization of the logarithmic correction, while HOTRG still proposes approximate bounds for the zero location associated with two different bond-merging algorithms of the higher-order singular value decomposition and the oblique projectors. Using the LoopTNR data computed up to the system size of L=1024L=1024L=1024 in the L×LL \times LL×L lattices, we estimate the logarithmic correction exponent r=−0.0643(9)r = -0.0643(9)r=−0.0643(9) from the extrapolation of the finite-size effective exponent, which is comparable to the renormalization group prediction of r=−1/16r = -1/16r=−1/16.
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 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.
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We explicitly compute the critical exponents associated with logarithmic corrections (the so-called hatted exponents) starting from the renormalization group equations and the mean field behavior for a wide class of models at the upper critical behavior (for short and long range φ n -theories) and below it.This allows us to check the scaling relations among these critical exponents obtained by analysing the complex singularities (Lee-Yang and Fisher zeroes) of these models.Moreover, we have obtained an explicit method to compute the ϙ exponent [defined by ξ ∼ L(log L) ϙ] and, finally, we have found a new derivation of the scaling law associated with it.
Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.
Using the tensor renormalization group method based on the higher-order singular value decomposition, we have studied the thermodynamic properties of the continuous XY model on the square lattice. The temperature dependence of the free energy, the internal energy, and the specific heat agree with the Monte Carlo calculations. From the field dependence of the magnetic susceptibility, we find the Kosterlitz-Thouless transition temperature to be 0.8921(19), consistent with the Monte Carlo as well as the high temperature series expansion results. At the transition temperature, the critical exponent δ is estimated as 14.5, close to the analytic value by Kosterlitz.