Publisher's Note: Algorithms for entanglement renormalization [Phys. Rev. B<b>79</b>, 144108 (2009)]

Evenbly, Glen; Vidal, G.

doi:10.1103/physrevb.79.149903

Cited by 79 publications

(209 citation statements)

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“…Examples of tensor network states for one-dimensional systems include the matrix product state [1][2][3] (MPS), which results naturally from both Wilson's numerical renormalization group 4 and White's density-matrix renormalization group [5][6][7][8] (DMRG) and is also used as a basis for simulation of time evolution, e.g., with the time evolving block decimation (TEBD) [9][10][11] algorithm and variations thereof, often collectively referred to as time-dependent DMRG; 9-14 the tree tensor network 15 (TTN), which follows from coarsegraining schemes where the spins are blocked hierarchically; and the multiscale entanglement renormalization Ansatz [16][17][18][19][20][21] (MERA), which results from a renormalization-group procedure known as entanglement renormalization. 16,21 For twodimensional (2D) lattices there are generalizations of these three tensor network states, namely projected entangled pair states [22][23][24][25][26][27][28][29][30][31] (PEPS), 2D TTN, 32,33 and 2D MERA, [34][35][36][37][38][39][40] respectively.…”

Section: Introductionmentioning

confidence: 99%

Tensor network states and algorithms in the presence of a global U(1) symmetry

2011

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Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [Phys. Rev. A 82, 050301 (2010)] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, associated, e.g., with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1)-symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multiscale entanglement renormalization Ansatz.

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Section: Introductionmentioning

confidence: 99%

Tensor network states and algorithms in the presence of a global U(1) symmetry

2011

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“…Optimization algorithms to approximate ground states, and to evaluate local expectation values and correlators, are present in the literature. 11,15 The numerical cost of finding an expectation value or performing a single optimization iteration using the binary MERA scales as O(χ 9 ln L) for a translation-invariant system. For more complex MERA structures, such as those representing two-dimensional (2D) lattices, the power of χ for the cost increases dramatically.…”

Section: A Mera Expectation Values and Causal Conesmentioning

confidence: 99%

“…8(b). Here, we have used MERA wave functions previously optimized using standard techniques 11 without Monte Carlo sampling, that is, sampling is only employed to extract the expectation values.…”

Section: A Expectation Valuesmentioning

confidence: 99%

“…(15) Multiple approaches are possible for updating the MERA to minimize the energy and find a good approximation to the ground state. One approach often used iteratively optimizes each tensor in the MERA according to the following algorithm 11 :…”

Section: Energy Minimization With Samplingmentioning

confidence: 99%

“…This is accomplished by a steepest descent method within the set of unitary tensors, which is much more robust to statistical noise than the singular value decomposition methods employed in MERA algorithms without sampling. 11 We demonstrate the performance of our approach by computing an approximation to the ground state of a finite Ising chain with transverse magnetic field. For the binary MERA under consideration, sampling lowers the costs of elementary contractions from O(χ 9 ) to O(χ 5 ).…”

Section: Introductionmentioning

confidence: 99%

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Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Ferris

Vidal

2012

Phys. Rev. B

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Monte Carlo sampling techniques have been proposed as a strategy to reduce the computational cost of contractions in tensor network approaches to solving many-body systems. Here, we put forward a variational Monte Carlo approach for the multiscale entanglement renormalization ansatz (MERA), which is a unitary tensor network. Two major adjustments are required compared to previous proposals with nonunitary tensor networks. First, instead of sampling over configurations of the original lattice, made of L sites, we sample over configurations of an effective lattice, which is made of just ln(L) sites. Second, the optimization of unitary tensors must account for their unitary character while being robust to statistical noise, which we accomplish with a modified steepest descent method within the set of unitary tensors. We demonstrate the performance of the variational Monte Carlo MERA approach in the relatively simple context of a finite quantum spin chain at criticality, and discuss future, more challenging applications, including two-dimensional systems.

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Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz

Evenbly

Vidal

2013

Springer Series in Solid-State Sciences

132

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The goal of this manuscript is to provide an introduction to the multi-scale entanglement renormalization ansatz (MERA) and its application to the study of quantum critical systems. Only systems in one spatial dimension are considered. The MERA, in its scale-invariant form, is seen to offer direct numerical access to the scale-invariant operators of a critical theory. As a result, given a critical Hamiltonian on the lattice, the scale-invariant MERA can be used to characterize the underlying conformal field theory. The performance of the MERA is benchmarked for several critical quantum spin chains, namely Ising, Potts, XX and (modified) Heisenberg models, and an insightful comparison with results obtained using a matrix product state is made. The extraction of accurate conformal data, such as scaling dimensions and operator product expansion coefficients of both local and non-local primary fields, is also illustrated.

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Publisher's Note: Algorithms for entanglement renormalization [Phys. Rev. B79, 144108 (2009)]

Cited by 79 publications

References 0 publications

Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network states and algorithms in the presence of a global U(1) symmetry

Variational Monte Carlo with the multiscale entanglement renormalization ansatz

Quantum Criticality with the Multi-scale Entanglement Renormalization Ansatz

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