Matrix product state renormalization

Bal, Matthias; Rams, Marek M.; Zauner, Valentin; Haegeman, Jutho; Verstraete, Frank

doi:10.1103/physrevb.94.205122

Cited by 18 publications

(23 citation statements)

References 56 publications

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“…While the current discussion was limited to exactly solvable system, it motivates further studies of the intrinsic structure of the MPS matrices. Such an analysis is indeed done in [31], where a general tensor network algorithm based on the impurity picture, discussed in Sec. III.C, is designed.…”

Section: Discussionmentioning

confidence: 99%

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Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

et al. 2015

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We discuss how to analytically obtain an -essentially infinite -Matrix Product State (MPS) representation of the ground state of the XY model. On the one hand this allows to illustrate how the Ornstein-Zernike form of the correlation function emerges in the exact case using standard MPS language. On the other hand we study the consequences of truncating the bond dimension of the exact MPS, which is also part of many tensor network algorithms, and analyze how the truncated MPS transfer matrix is representing the dominant part of the exact quantum transfer matrix. In the gapped phase we observe that the correlation length obtained from a truncated MPS approaches the exact value following a power law in effective bond dimension. In the gapless phase we find a good match between a state obtained numerically from standard MPS techniques with finite bond dimension, and a state obtained by effective finite imaginary time evolution in our framework. This provides a direct hint for a geometric interpretation of Finite Entanglement Scaling at the critical point in this case. Finally, by analyzing the spectra of transfer matrices, we support the interpretation put forward by [V. Zauner at. al., New J. Phys. 17, 053002 (2015)] that the MPS transfer matrix emerges from the quantum transfer matrix though the application of Wilson's Numerical Renormalisation Group along the imaginary-time direction.

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Section: Discussionmentioning

confidence: 99%

“…This picture is further validated in Ref. 31, where a general tensor-network ansatz based on this idea is constructed.…”

Section: Truncation As Effective Description Of Impuritymentioning

confidence: 99%

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

et al. 2015

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“…The third class of tensor networks involves the ones with an additional dimension which plays the role of scale in a renormalization group approach; those are called MERA (Multiscale Entanglement Renormalization Ansatz) [33,34]. All three of those classes can be shown to arise naturally from a compression of the path integral representation of the quantum state |ψ limτ→∞ exp(−τ H) |Ω [35,36].…”

Section: Introductionmentioning

confidence: 99%

Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods

Haegeman

Verstraete

2017

Annu. Rev. Condens. Matter Phys.

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Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix product state methods for finding extremal eigenvectors of matrix product operators.

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“…Evaluating these quantities is reduced to the contraction of a multidimensional tensor network. In the two dimensional case, many algorithms [8,32,[37][38][39][40][41]43,[45][46][47][48][49][50][53][54][55][56][57] have been developed to implement the approximate tensor contractions. Among these, the tensor renormalization group approach introduced by Levin and Nave [38] and its generalizations [8,22,39,[43][44][45][46][47]55,56,61] have unique features: the tensor contraction is based on a fully isotropic coarse-graining procedure.…”

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confidence: 99%

Loop Optimization for Tensor Network Renormalization

2017

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We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the tensors on each loop. In this way, we remove short-range entanglement at each iteration step and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model. DOI: 10.1103/PhysRevLett.118.110504 Introduction.-In recent years, the tensor network (TN) approach [1,2] has become a powerful theoretical and computational [8, tool for studying condensed matter systems. Many physical quantities, including the partition function of a classical system, the Euclidean path integral of a quantum system, and the expectation value of physical observables, can be expressed in terms of tensor networks. Evaluating these quantities is reduced to the contraction of a multidimensional tensor network. In the two dimensional case, many algorithms [8,32,[37][38][39][40][41]43,[45][46][47][48][49][50][53][54][55][56][57] have been developed to implement the approximate tensor contractions. Among these, the tensor renormalization group approach introduced by Levin and Nave [38] and its generalizations [8,22,39,[43][44][45][46][47]55,56,61] have unique features: the tensor contraction is based on a fully isotropic coarse-graining procedure. Moreover, when applying the method to a system on a finite torus, the computational cost is lower than those based on matrix product states (MPS) [32,37,41,[48][49][50]53,54].However, the Levin-Nave tensor network renormalization (TRG, also referred as LN-TNR here) [38] is based on the singular value decomposition (SVD) of local tensors, which only minimizes the truncation errors of tree tensor networks. Several improvements [45][46][47] have taken into account the effect of the environments, but they are still essentially based on tree tensor networks. These approaches cannot completely remove short-range entanglements during the coarse graining process. For example, in the 2D TN calculation of a partition function (or a path integral) TNR based on simple SVD cannot simplify the corner-double-line (CDL) tensor [38], despite the CDL tensor describing a product state that should be simplified to a one-dimensional tensor. In Ref.[8], this issue was seriously discussed. The authors pointed out that to further remove short-range entanglement, it is crucial to optimize the tensor configurations that contain a loop. However, due to the computational cost, only a crude iterative method is

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Matrix product state renormalization

Cited by 18 publications

References 56 publications

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Diagonalizing Transfer Matrices and Matrix Product Operators: A Medley of Exact and Computational Methods

Loop Optimization for Tensor Network Renormalization

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