A Density Matrix Algorithm for 3D Classical Models

Nishino, Tomotoshi; Okunishi, Kouichi

doi:10.1143/jpsj.67.3066

Cited by 102 publications

(113 citation statements)

References 30 publications

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“…Here, we remark that the relation between the Hamiltonian and the wavefunction Ψ is quite reminiscent of Eq. (16) for the 2D vertex model. Thus, Ψ in the quantum systems approximately corresponds to ρ in the HOTRG for 2D classical vertex model.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Doubling of entanglement spectrum in tensor renormalization group

2014

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We investigate the entanglement spectrum in HOTRG -tensor renormalization group (RG) method combined with the higher order singular value decomposition-for two-dimensional (2D) classical vertex models. In the off-critical region, it is explained that the entanglement spectrum associated with the RG transformation is described by 'doubling' of the spectrum of a corner transfer matrix. We then demonstrate that the doubling actually occurs for the square-lattice Ising model by HOTRG calculations up to D = 64, where D is the cut-off dimension of tensors. At the critical point, we also find that a non-trivial D scaling behavior appears in the entanglement entropy. We mention about the HOTRG for the 1D quantum system as well.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Doubling of entanglement spectrum in tensor renormalization group

2014

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“…This is particularly true when trying to obtain good accuracies in physical regimes where entanglement is large (e.g., close to a quantum critical point, or in the presence of many nearly-degenerate quantum states), which requires a large D. The computational bottleneck of the method is the calculation of the so-called effective environment, i.e., the effective description of the tensor network surrounding a given site. Technically, effective environments can be computed using different approaches, such as TRG/SRG and HOTRG/HOSRG, 32,36,41,58 iTEBD, 33 corner transfer matrices (CTM), 34,[59][60][61][62][63] or more recently the tensor network renormalization method. 64 Independently of the chosen approach for their calculation, accurate effective environments are required for the so-called full update (FU), which is the accurate scheme proposed to find the iPEPS tensors throughout a (imaginary) time evolution.…”

Section: -9mentioning

confidence: 99%

Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing

Phien¹,

Bengua²,

Tuan³

et al. 2015

Phys. Rev. B

192

206

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The infinite Projected Entangled Pair States (iPEPS) algorithm [J. Jordan et al, PRL 101, 250602 (2008)] has become a useful tool in the calculation of ground state properties of 2d quantum lattice systems in the thermodynamic limit. Despite its many successful implementations, the method has some limitations in its present formulation which hinder its application to some highly-entangled systems. The purpose of this paper is to unravel some of these issues, in turn enhancing the stability and efficiency of iPEPS methods. For this, we first introduce the fast full update scheme, where effective environment and iPEPS tensors are both simultaneously updated (or evolved) throughout time. As we shall show, this implies two crucial advantages: (i) dramatic computational savings, and (ii) improved overall stability. Besides, we extend the application of the local gauge fixing, successfully implemented for finite-size PEPS [M. Lubasch, J. Ignacio Cirac, M.-C. Bañuls, PRB 90, 064425 (2014)], to the iPEPS algorithm. We see that the gauge fixing not only further improves the stability of the method, but also accelerates the convergence of the alternating least squares sweeping in the (either "full" or "fast full") tensor update scheme. The improvement in terms of computational cost and stability of the resulting "improved" iPEPS algorithm is benchmarked by studying the ground state properties of the quantum Heisenberg and transverse-field Ising models on an infinite square lattice.

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“…Tensor network states in D > 1 dimensions, such as PEPS [21][22][23][24][25][26][27][28][29][30] and MERA [33][34][35][36][37][38][39] , are also thought to be capable of accurately describing a large variety of ground states. This is supported both by growing numerical evidence and by the existence of analytical MERA 35,38 and PEPS 57,58 constructions for some topologically ordered ground states in D = 2 dimensions.…”

Section: Introductionmentioning

confidence: 99%

Tensor Network States and Geometry

2011

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Tensor network states are used to approximate ground states of local Hamiltonians on a lattice in D spatial dimensions. Different types of tensor network states can be seen to generate different geometries. Matrix product states (MPS) in D = 1 dimensions, as well as projected entangled pair states (PEPS) in D > 1 dimensions, reproduce the D-dimensional physical geometry of the lattice model; in contrast, the multi-scale entanglement renormalization ansatz (MERA) generates a (D+1)-dimensional holographic geometry. Here we focus on homogeneous tensor networks, where all the tensors in the network are copies of the same tensor, and argue that certain structural properties of the resulting many-body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters. Indeed, the asymptotic decay of correlations in homogeneous MPS and MERA for D = 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries, respectively; whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary lawthat is, again in the relevant geometry. This geometrical interpretation offers a simple and unifying framework to understand the structural properties of, and helps clarify the relation between, different tensor network states. In addition, it has recently motivated the branching MERA, a generalization of the MERA capable of reproducing violations of the entropic boundary law in D > 1 dimensions.

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A Density Matrix Algorithm for 3D Classical Models

Cited by 102 publications

References 30 publications

Doubling of entanglement spectrum in tensor renormalization group

Doubling of entanglement spectrum in tensor renormalization group

Infinite projected entangled pair states algorithm improved: Fast full update and gauge fixing

Tensor Network States and Geometry

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