Boundary quantum critical phenomena with entanglement renormalization

Evenbly, Glen; Pfeifer, Robert N. C.; Picó, V.; Iblisdir, Sofyan; Tagliacozzo, Luca; McCulloch, Ian P.; Vidal, Guifré

doi:10.1103/physrevb.82.161107

Cited by 40 publications

(55 citation statements)

References 30 publications

(71 reference statements)

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“…II B, was initially introduced and tested in Ref. 11. Here we shall both reproduce and expand upon the results in that paper.…”

Section: Single Boundary (Semi-infinite Chain)mentioning

confidence: 66%

“…This form of modular MERA for boundary problems, boundary MERA, was first proposed and tested in Ref. 11. There, however, no theoretical justification of its remarkable success was provided.…”

Section: B Boundariesmentioning

confidence: 99%

“…Entanglement renormalization operates in real space (it does not rely on Fourier space analysis) and it is a non-perturbative approach (that is, it can handle interactions of any strength). As a result, it has a wide range of applicability, from quantum criticality [4][5][6][7][8][9][10][11][12][13][14][15] to emergent topological order [16][17][18][19][20][21][22] , from frustrated antiferromagnets [23][24][25] to interacting fermions [26][27][28] and even to interacting anyons 29,30 . Entanglement renormalization produces an efficient (approximate) representation of the ground state of the system in terms of a variational tensor network, the multi-scale entanglement renormalization ansatz (MERA) 31 , from which one can extract expectation values of arbitrary local observables.…”

Section: Entanglement Renormalizationmentioning

confidence: 99%

“…In particular, the MERA offers an extremely compact description of ground states of homogeneous systems at fixed points of the RG flow, that is, in systems with both translation invariance and scale invariance. These encompass both stable (gapped) RG fixed points, which include topologically ordered systems [16][17][18][19][20][21][22] , and unstable (gapless) RG fixed points, corresponding to quantum critical systems [4][5][6][7][8][9][10][11][12][13][14][15] . Indeed, translation invariance leads to a position-independent coarse-graining transformation U , made of copies of a single pair of tensors {u, w}, whereas scale invariance implies that the same U can be used at all scales.…”

Section: B Mera and Quantum Criticalitymentioning

confidence: 99%

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Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces

Evenbly

Vidal

2014

J Stat Phys

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We propose algorithms, based on the multi-scale entanglement renormalization ansatz, to obtain the ground state of quantum critical systems in the presence of boundaries, impurities, or interfaces. By exploiting the theory of minimal updates [Ref. 1: G. Evenbly and G. Vidal, arXiv:1307.0831], the ground state is completely characterized in terms of a number of variational parameters that is independent of the system size, even though the presence of a boundary, an impurity, or an interface explicitly breaks the translation invariance of the host system. Similarly, computational costs do not scale with the system size, allowing the thermodynamic limit to be studied directly and thus avoiding finite size effects e.g. when extracting the universal properties of the critical system.

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“…II B, was initially introduced and tested in Ref. 11. Here we shall both reproduce and expand upon the results in that paper.…”

Section: Single Boundary (Semi-infinite Chain)mentioning

confidence: 66%

Section: B Boundariesmentioning

confidence: 99%

Section: Entanglement Renormalizationmentioning

confidence: 99%

Section: B Mera and Quantum Criticalitymentioning

confidence: 99%

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Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces

Evenbly

Vidal

2014

J Stat Phys

Self Cite

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“…This property can be ultimately traced back to the existence of distinct energy scales in the Hamiltonian. According to the principle of minimal updates, an impurity initially localized in space thus remains localized under coarse-graining, which leads to a very efficient MERA description of systems with boundaries, impurities, or interfaces 40,41 . The relevant degrees of freedom for an impurity are found to be exactly those living at the boundary of the causal cone.…”

Section: From Euclidean Path Integral To Uniform Mpsmentioning

confidence: 99%

Matrix product state renormalization

et al. 2016

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The truncation or compression of the spectrum of Schmidt values is inherent to the matrix product state (MPS) approximation of one-dimensional quantum ground states. We provide a renormalization group picture by interpreting this compression as an application of Wilson's numerical renormalization group along the imaginary time direction appearing in the path integral representation of the state. The location of the physical index is considered as an impurity in the transfer matrix and static MPS correlation functions are reinterpreted as dynamical impurity correlations. Coarse-graining the transfer matrix is performed using a hybrid variational ansatz based on matrix product operators, combining ideas of MPS and the multi-scale entanglement renormalization ansatz. Through numerical comparison with conventional MPS algorithms, we explicitly verify the impurity interpretation of MPS compression, as put forward by V. Zauner et al. [New J. Phys. 17, 053002 (2015)] for the transverse-field Ising model. Additionally, we motivate the conceptual usefulness of endowing MPS with an internal layered structure by studying restricted variational subspaces to describe elementary excitations on top of the ground state, which serves to elucidate a transparent renormalization group structure ingrained in MPS descriptions of ground states.

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Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

Cotler

Mozaffar

Mollabashi

et al. 2019

Fortschritte der Physik

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We develop techniques to systematically construct local unitaries which map scale-invariant, product state wavefunctionals to the ground states of weakly interacting, continuum quantum field theories. More broadly, we devise a "quantum circuit perturbation theory" to construct local unitaries which map between any pair of wavefunctionals which are each Gaussian with arbitrary perturbative corrections. Further, we generalize cMERA to interacting continuum field theories, which requires reworking the existing formalism which is tailored to non-interacting examples. Our methods enable the systematic perturbative calculation of cMERA circuits for weakly interacting theories, and as a demonstration we compute the 1-loop cMERA circuit for scalar ϕ 4 theory and analyze its properties. In this case, we show that Wilsonian renormalization of the spatial momentum modes is equivalent to a local position space cMERA circuit. This example provides new insights into the connection between position space and momentum space renormalization group methods in quantum field theory. The form of cMERA circuits derived from perturbation theory suggests useful ansatzes for numerical variational calculations. RG flows. [9] These "entanglement renormalization" tensor networks are comprised of sequences of spatially local quantum gates.1. Why unitaries with quadratic generators are special; 2. How to systematically treat unitaries with higher-order generators perturbatively in the higher order generators (i.e., "quantum circuit perturbation theory"); and 3. Special circumstances required for us to treat unitaries with higher-order generators non-perturbatively in the higher order generators.By developing quantum circuit perturbation theory, we can systematically construct unitaries which map scale-invariant product states to the ground states of weakly interacting quantum field theory, to any fixed order in perturbation theory. We can also construct unitaries which map between any pair of states that are each Gaussian with arbitrary perturbative corrections. In very special cases, unitaries between non-Gaussian states (or between a Gaussian state and a non-Gaussian state) can be constructed non-perturbatively, although we will not discuss such cases here. Next, we generalize cMERA to interacting fields, which involves generalizing our methods above to unitaries created by path-ordered exponentials of Hermitian generators. We discover new features of cMERA which are absent for the free field examples which have previously been studied. Our techniques enable systematic perturbative calculations of cMERA circuits for interacting field theories, and we calculate the 1-loop cMERA circuit for ϕ 4 theory as an example. Furthermore, the disentangler is constructed to disentangle spatial momentum modes in a manner exactly corresponding to perturbative Wilsonian RG (on spatial momentum modes), and yet in position space disentangles spatially local subsystems. This relationship between disentangling degrees of freedom in momentum space and position s...

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Boundary quantum critical phenomena with entanglement renormalization

Cited by 40 publications

References 30 publications

Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces

Algorithms for Entanglement Renormalization: Boundaries, Impurities and Interfaces

Matrix product state renormalization

Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

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