Plaquette renormalization scheme for tensor network states

Wang, Ling; Kao, Ying-Jer; Sandvik, Anders W.

doi:10.1103/physreve.83.056703

Cited by 21 publications

(29 citation statements)

References 25 publications

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“…Eventually, the tensor-network is reduced to only four sites and the double tensor trace for the norm or overlap can be calculated easily. For more detailed description of the TRG method, please see [17,19,20]. We will then adopt the TRG method to numerically evaluate the entanglement measure such as GE for the ground state in the form of TPS on the 2D square spin lattice.…”

Section: Entanglement Measure In 2d Spin Systemsmentioning

confidence: 99%

“…Instead, a method called tensor renormalization group (TRG) is developed in [17,18,19,20] to make the double tensor trace polynomially calculable by merging the sites and truncating the bond dimension of the merging lattices according to the relevance of the components in the Schmidt decomposition. The cutoff of the merging bond dimension is denoted as D cut which controls the accuracy of the computation.…”

Section: Entanglement Measure In 2d Spin Systemsmentioning

confidence: 99%

“…This is mainly due to the lack of the exactly solvable or numerically tractable ground states in 2D case. The situation, however, changes recently because of the new development of the generalization of the 1D density-matrix renormalization group (DMRG) [10] and matrix product states (MPSs) [11,12,13,14] to the 2D tensor product states (TPSs) [15,16,17,18,19,20].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Huang

Lin

2010

Phys. Rev. A

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We compute the multipartite entanglement measures such as the global entanglement of various one-and two-dimensional quantum systems to probe the quantum criticality based on the matrix and tensor product states (MPSs/TPSs). We use infinite time-evolving block decimation (iTEBD) method to find the ground states numerically in the form of MPSs/TPSs, and then evaluate their entanglement measures by the method of tensor renormalization group (TRG). We find these entanglement measures can characterize the quantum phase transitions by their derivative discontinuity right at the critical points in all models considered here. We also comment on the scaling behaviors of the entanglement measures by the ideas of quantum state renormalization group transformations. *

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Section: Entanglement Measure In 2d Spin Systemsmentioning

confidence: 99%

Section: Entanglement Measure In 2d Spin Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Huang

Lin

2010

Phys. Rev. A

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“…This is further compounded by the inherent computational difficulties posed by two-dimensional (2D) quantum systems. Nevertheless, significant progress has been made to develop efficient numerical algorithms to simulate 2D quantum many-body lattice systems in the context of tensor network representations [20][21][22][23][24][25][26][27][28][29][30][31]. The algorithms have been successfully exploited to compute, for example, the ground-state fidelity per lattice site [32][33][34][35][36], which has been established as a universal marker to detect quantum phase transitions in many-body lattice systems.…”

Section: Introductionmentioning

confidence: 99%

Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models

Shi

Wang

et al. 2016

Phys. Rev. A

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Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum manybody lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the q-state quantum Potts model on the square lattice with q ∈ {2, 3, 4, 5} is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 XXX and anisotropic XYX models in an external magnetic field, and the antiferromagnetic spin-1 XXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with non-maximum value at the transition point and the phase transition is continuous. For the models under consideration we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.

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“…Such a decomposition allows to achieve a higher bond dimension after the original truncation. Other approaches work with square plaquettes that include the physical indices [22]. It is also important to notice that the method proposed here is not a variational procedure, though it remains numerically stable.…”

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confidence: 99%

Renormalization group contraction of tensor networks in three dimensions

García-Sáez

Latorre

2013

Phys. Rev. B

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We present a new strategy for contracting tensor networks in arbitrary geometries. This method is designed to follow as strictly as possible the renormalization group philosophy, by first contracting tensors in an exact way and, then, performing a controlled truncation of the resulting tensor. We benchmark this approximation procedure in two dimensions against an exact contraction. We then apply the same idea to a three dimensional system. The underlying rational for emphasizing the exact coarse graining renormalization group step prior to truncation is related to monogamy of entanglement.PACS numbers: 03.67. Mn, 05.10.Cc, 05.50.+q Computing analytically the properties of a quantum model is, in general, not possible. It is then necessary to resort to classical simulations that rely on approximation techniques. From a quantum information perspective, the presence of a small amount of entanglement in the system has been identified as the key ingredient that allows for efficient classical simulations. This has been exploited in one dimension (1D) in a series of algorithms based on a description of the system as a tensor network. Following this approach, condensed matter systems can be simulated using Matrix Product States (MPS, [1,2]) as the testbed for the variational procedure of the density-matrix renormalization group (DMRG,[3]) to study ground states of local 1D Hamiltonians. Beyond the 1D setting, the computation of physical magnitudes using tensor networks is limited by the numerical effort necessary to perform the contraction of the tensor network, i.e. to sum all its indices. The efficiency for performing this task is limited by the area law scaling of the entanglement entropy in the system [4][5][6][7]. To overcome this problem, several strategies aim at finding the best possible approximation to the contraction of tensor networks after identifying the relevant degrees of freedom of the system [8][9][10].Let us briefly recall the key elements of the tensor network representation. Given a quantum state of N particles |ψ = c i1,...,iN |i 1 . . . i N its coefficients can be represented as a contraction of local tensors c i1,...,iN = tr(A 1,i1 . . . A N,iN ), where each local tensor A j carries a physical index i j and ancillary indices (which are not written) that get contracted according to a prescribed geometry. The rank of these ancillary indices, that we shall call χ, controls the amount of entanglement which is captured by the tensor representation. If the tensors are simple matrices on a line, the tensor network is called MPS [1,2]. Other possible geometries are regular squared grids in any dimensions that correspond to PEPS [11], and tree-like structures that go under the name of TTN [12] and MERA [13].In this letter we propose a new strategy to contract tensor networks in general geometries, that we shall illustrate in detail for PEPS in 2D and 3D. The method is based on following as strictly as possible the renormalization group (RG) philosophy. First, an exact contraction of a set of local tenso...

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Plaquette renormalization scheme for tensor network states

Cited by 21 publications

References 25 publications

Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models

Renormalization group contraction of tensor networks in three dimensions

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