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テンソルくりこみ群を用いた相関関数の評価

智洋 中本, Norihiro Nakamoto

Published Sep 26, 2016• 0 citations
OpenAlex

Abstract

Computation of correlation functionsby tensor renormalization group method) 金沢大学大学院自然科学研究科数物科学専攻 中本 智洋 1 概要 We demonstrate how to compute the correlation function by using the tensor renormalization group for 2D Ising model.From the correlation function around the critical temperature, we extract the correlation length and the spontaneous magnetization and then determine some critical exponents.Attractive feature of our method to extract the critical exponents is that numerical derivative, that causes a loss of significant digits, and additional simulation parameters, that demand more computational cost and complicated analysis, are not required.On the other hand, our approach requires an additional treatment of impurity tensors, but the additional cost is still tolerable.

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Computation of correlation functionsby tensor renormalization group method) 金沢大学大学院自然科学研究科数物科学専攻 中本 智洋 1 概要 We demonstrate how to compute the correlation function by using the tensor renormalization group for 2D Ising model.From the correlation function around the critical temperature, we extract the correlation length and the spontaneous magnetization and then determine some critical exponents.Attractive feature of our method to extract the critical exponents is that numerical derivative, that causes a loss of significant digits, and additional simulation parameters, that demand more computational cost and complicated analysis, are not required.On the other hand, our approach requires an additional treatment of impurity tensors, but the additional cost is still tolerable.
We demonstrate how to compute the correlation function by using the tensor renormalization group for 2D Ising model. From the correlation function around the critical temperature, we extract the correlation length and the spontaneous magnetization and then determine some critical exponents. Attractive feature of our method to extract the critical exponents is that numerical derivative, that causes a loss of significant digits, and additional simulation parameters, that demand more computational cost and complicated analysis, are not required. On the other hand, our approach requires an additional treatment of impurity tensors, but the additional cost is still tolerable.
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.
The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ, and δ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.
The locality of field theories strongly constrains the possible behaviors of symmetry-twisted partition functions, and thus they serve as order parameters to detect low-energy realizations of global symmetries, such as spontaneous symmetry breaking (SSB). We demonstrate that the tensor renormalization group (TRG) offers an efficient framework to compute the symmetry-twisted partition functions, which enables us to detect the symmetry-breaking transition and also to study associated critical phenomena. As concrete examples of SSB, we investigate the two-dimensional (2D) classical Ising model and the three-dimensional (3D) classical O ( 2 ) nonlinear sigma model, and we identify their critical points solely from the twisted partition function. By employing the finite-size scaling argument, we find the critical temperature T c = 2.2017 ( 2 ) with the critical exponent ν = 0.663 ( 33 ) for the 3D O ( 2 ) model. In addition, we also study the Berezinskii–Kosterlitz–Thouless (BKT) criticality of the 2D classical O ( 2 ) model by extracting the helicity modulus from the twisted partition functions, and we obtain the BKT transition temperature, T BKT = 0.8928 ( 2 ) .
The tensor renormalization-group method, developed by Levin and Nave, brings systematic improvability to the position-space renormalization-group method and yields essentially exact results for phase diagrams and entire thermodynamic functions. The method, previously used on systems with no quenched randomness, is extended in this study to systems with quenched randomness. Local magnetizations and correlation functions as a function of spin separation are calculated as tensor products subject to renormalization-group transformation. Phase diagrams are extracted from the long-distance behavior of the correlation functions. The approach is illustrated with the quenched bond-diluted Ising model on the triangular lattice. An accurate phase diagram is obtained in temperature and bond-dilution probability for the entire temperature range down to the percolation threshold at zero temperature.