Entanglement renormalization in free bosonic systems: real-space versus momentum-space renormalization group transforms

Evenbly, Glen; Vidal, G.

doi:10.48550/arxiv.0801.2449

Cited by 9 publications

(16 citation statements)

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“…2 as a binary 1D MERA, since it becomes a binary tree when we remove the disentanglers. Most of the previous work for 1D lattices [1,5,6,7,16,17,18] has been done using the binary 1D MERA. However, there are many other possible choices.…”

Section: A Quantum Circuitmentioning

confidence: 99%

“…Given the Hamiltonian H for an infinite lattice at a quantum critical point, where we expect the system to be invariant under rescaling, we can use a scale invariant MERA to represent its ground state [1,5,6,7,10,11].…”

Section: Scale Invariant Systemsmentioning

confidence: 99%

“…The MERA, based in turn on a class of quantum circuits, is particularly successful at describing ground states at quantum criticality [1,5,6,7,8,9,10,11] or with topological order [12,13]. From the computational viewpoint, the key property of the MERA is that it can be manipulated efficiently, due to the causal structure of the underlying quantum circuit [5].…”

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confidence: 99%

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Algorithms for entanglement renormalization

2009

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We describe an iterative method to optimize the multi-scale entanglement renormalization ansatz (MERA) for the low-energy subspace of local Hamiltonians on a D-dimensional lattice. For translation invariant systems the cost of this optimization is logarithmic in the linear system size. Specialized algorithms for the treatment of infinite systems are also described. Benchmark simulation results are presented for a variety of 1D systems, namely Ising, Potts, XX and Heisenberg models. The potential to compute expected values of local observables, energy gaps and correlators is investigated.

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Section: A Quantum Circuitmentioning

confidence: 99%

Section: Scale Invariant Systemsmentioning

confidence: 99%

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Algorithms for entanglement renormalization

2009

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“…Two dimensional systems have been explored in Refs. [10,11,12,13,14,15,16], including scalable simulations in interacting systems [13,14] and analytical results for systems with topological order [15,16]. Scale invariant systems, including non-critical and critical fixed points of the RG flow, have been studied in Refs.…”

Section: Introductionmentioning

confidence: 99%

“…Scale invariant systems, including non-critical and critical fixed points of the RG flow, have been studied in Refs. [5,6,9,10,11,15,16,17,18,19,20,21]. Extensions to two-dimensional systems with fermionic and anyonic degrees of freedom have been proposed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Finite Temperature Dissipation and Transport Near Quan- tum Critical Points

2010

Understanding Quantum Phase Transitions

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We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block. DOI: 10.1103/PhysRevLett.99.220405 PACS numbers: 05.30.ÿd, 02.70.ÿc, 03.67.Mn, 05.50.+q Renormalization group (RG), one of the conceptual pillars of statistical mechanics and quantum field theory, revolves around the idea of coarse-graining and rescaling transformations of an extended system [1]. These so-called RG transformations are not only a key theoretical element in the modern formulation of critical phenomena and phase transitions, but also the basis of important computational methods for many-body problems.In the case of quantum systems defined on a lattice, Wilson's real-space RG methods [2], based on the truncation of the Hilbert space, replaced Kadanoff's spinblocking ideas [3] with a concise mathematical formulation and an explicit prescription to implement RG transformations. But it was not until the advent of White's density matrix renormalization group (DMRG) algorithm [4] that RG methods became the undisputed numerical approach for systems on a 1D lattice. More recently, such techniques have gained renewed momentum under the influence of quantum information science. By paying due attention to entanglement, algorithms to simulate timeevolution in 1D systems [5] and to address 2D systems [6] have been put forward.The practical value of DMRG and related methods is unquestionable. And yet, they are based on a RG transformation that notably fails to satisfy a most natural expectation, namely, to have scale invariant systems as fixed points. Instead, for such systems, the size of an effective site (as measured by the dimension of its Hilbert space) increases with each iteration of the RG transformation. This fact does not only conflict with the very spirit of RG theory, but it also has important computational implications: after a sufficiently large number of iterations, the effective sites are unaffordably large, and the numerical method is no longer viable.In this Letter, we propose a real-space RG transformation for quantum systems on a D-dimensional lattice that, by renormalizing the amount of entanglement in the system, aims to eliminate the growth of the site's Hilbert space dimension along successive rescaling transformations. In particular, wh...

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Zero modes and entanglement entropy

Yazdi

2017

J. High Energ. Phys.

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Ultraviolet divergences are widely discussed in studies of entanglement entropy. Also present, but much less understood, are infrared divergences due to zero modes in the field theory. In this note, we discuss the importance of carefully handling zero modes in entanglement entropy. We give an explicit example for a chain of harmonic oscillators in 1D, where a mass regulator is necessary to avoid an infrared divergence due to a zero mode. We also comment on a surprising contribution of the zero mode to the UV-scaling of the entanglement entropy.

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Entanglement renormalization in free bosonic systems: real-space versus momentum-space renormalization group transforms

Cited by 9 publications

References 0 publications

Algorithms for entanglement renormalization

Algorithms for entanglement renormalization

Finite Temperature Dissipation and Transport Near Quan- tum Critical Points

Zero modes and entanglement entropy

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