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Springer series in synergetics

Real-Space Renormalization-Group Method for Quantum Systems: Application to Quantum Frustration in Two Dimensions

K. A. Penson, Rémi Jullien, Pierre M. Pfeuty, Katarina Uzelac

Published Jan 1, 1981• 1 citation
Publisher
DOI
10.1007/978-3-642-81703-8_21

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The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.

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We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions.The tensor RG methods have had great success in producing accurate free energy compared with the conventional real-space RG schemes.However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes.We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory.This approach is benchmarked in the context of the Ising models in one dimension and two dimensions.Based on a pure RG argument, the proposed method has potential applications to three-dimensional systems, where the existing bread-and-butter method is inapplicable.

arXiv:2102.08136•Physical Review Research•Apr 16, 2021•15 citations

高精度テンソル繰り込み群法とその応用

Adachi, Daiki

High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏 名 足立 大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.

•Mar 23, 2020•0 citations
The application of real space renormalization group methods to quantum lattice models has become a topic of great interest following the development of the density matrix renormalization group by White.This method has been used to find the ground and low-lying excited state energies and wave functions of quantum spin models in which the form of the ground state is not clear, for instance because the interactions are frustrated.It has also been applied to fermion problems where the tendency for localization due to the strong Coulomb repulsion is opposed by the lowering of the kinetic energy which occurs as a result of electron transfer.The approach is particularly suitable for one-, or quasi-one-dimensional problems.The method involves truncating the Hilbert space in a systematic and optimised manner.Results for the ground state energy are thus variational bounds.The results for low-lying energies and correlation functions for one-dimensional systems have unprecedented accuracy and the method has become the method of choice for solving one-dimensional quantum spin problems.We review the method and results obtained for the spin-1 chain with biquadratic exchange as well as the spin-1/2 model with competing nearest and next nearest neighbour exchange will be described.More recently, the density matrix renormalization group has been applied to reformulate the coupling constant renormalization group approach which is appropriate for the study of critical properties.This approach has been applied to the anisotropic spin-1/2 Heisenberg chain.Finally, we discuss recent work which has borne promising applications in two dimensions -the Ising model and the two-dimensional Hubbard model.
We show a way to perform the canonical renormalization group (RG) prescription in tensor space: write down the tensor RG equation, linearize it around a fixed-point tensor, and diagonalize the resulting linearized RG equation to obtain scaling dimensions.The tensor RG methods have had great success in producing accurate free energy compared with the conventional real-space RG schemes.However, the above-mentioned canonical procedure has not been implemented for general tensor-network-based RG schemes.We extend the success of the tensor methods further to extraction of scaling dimensions through the canonical RG prescription, without explicitly using the conformal field theory.This approach is benchmarked in the context of the Ising models in one dimension and two dimensions.Based on a pure RG argument, the proposed method has potential applications to three-dimensional systems, where the existing bread-and-butter method is inapplicable.
High-accuracy tensor renormalization group algorithms and their applications (高精度テンソル繰り込み群法とその応用) 氏 名 足立 大樹We propose novel real-space renormalization group algorithms for classical and quantum lattice models.Conventionally, the tensor renormalization group (TRG) algorithm proposed by Levin and Nave in 2007 and the higher-order tensor renormalization group (HOTRG) algorithm proposed by Xie et al. in 2012 have been widely used as efficient numerical renormalization methods to study large-scale classical and quantum spin systems.The accuracy of the tensor network algorithms is generally controlled by the bond dimension χ, which is the cutoff of tensor indices.With a fixed bond dimension χ, TRG can renormalize the two-dimensional squarelattice tensor network within O ( χ 5 ) computational time.However, it is difficult to apply the TRG algorithm to models in three or higher dimensions.On the other hand, HOTRG can be applied, at least in principle, to higher-dimensional models, and it has a higher accuracy than TRG when comparing with the same bond dimension, while the computational cost of O ( χ 4d-1 ) for d-dimensional models is higher than TRG even in two dimensions and too high to execute simulations for higher-dimensional models in practice.New algorithms that can perform the real-space renormalization in general dimensions with a reasonable cost are strongly demanded.