Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Rams, Marek M.; Zauner, Valentin; Bal, Matthias; Haegeman, Jutho; Verstraete, Frank

doi:10.1103/physrevb.92.235150

Cited by 23 publications

(29 citation statements)

References 40 publications

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“…〉can be represented by a cMPS with free hamiltonian K given by (25) and interaction given by (26). We should emphasize that it is the final state for which we are providing an efficient description, and not the dynamics.…”

Section: A Toy Example: Cmps Representation Of the Final States Of Phmentioning

confidence: 99%

Continuum tensor network field states, path integral representations and spatial symmetries

et al. 2015

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A natural way to generalize tensor network variational classes to quantum field systems is via a continuous tensor contraction. This approach is first illustrated for the class of quantum field states known as continuous matrix-product states (cMPS). As a simple example of the path-integral representation we show that the state of a dynamically evolving quantum field admits a natural representation as a cMPS. A completeness argument is also provided that shows that all states in Fock space admit a cMPS representation when the number of variational parameters tends to infinity. Beyond this, we obtain a well-behaved field limit of projected entangled-pair states (PEPS) in two dimensions that provide an abstract class of quantum field states with natural symmetries. We demonstrate how symmetries of the physical field state are encoded within the dynamics of an auxiliary field system of one dimension less. In particular, the imposition of Euclidean symmetries on the physical system requires that the auxiliary system involved in the class' definition must be Lorentzinvariant. The physical field states automatically inherit entropy area laws from the PEPS class, and are fully described by the dissipative dynamics of a lower dimensional virtual field system. Our results lie at the intersection many-body physics, quantum field theory and quantum information theory, and facilitate future exchanges of ideas and insights between these disciplines.

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Section: A Toy Example: Cmps Representation Of the Final States Of Phmentioning

confidence: 99%

Continuum tensor network field states, path integral representations and spatial symmetries

et al. 2015

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“…27, beyond the analytical results obtained for the XY model in Ref. 29. Having a layered decomposition of a MPS ground state at our disposal, we benchmark our method on the Ising model, propose an ansatz for the structure of MPS fixed point reduced density matrices, and study the effect of restricting a variational MPS ansatz for elementary excitations to a subspace of variational parameters.…”

Section: Application Of Wilsons Numerical Renormalization Group (Nrg)mentioning

confidence: 99%

“…IV A, different implementations of the implicit infrared cutoff for gapless states, yield different variational MPS approximations, which has recently been discussed in Ref. 29.…”

Section: A Mps Ansatz From Wilson Mpomentioning

confidence: 99%

“…To that end, we construct the isometry U reduced density operator of the MPS on a half-infinite chain (our initial A i in Fig. 1) and finding dominant modes in its diagonal basis 29 . To that end, we look at the singular values of U † svd U s imp , which directly show how well the dominant modes in the entanglement spectrum are preserved during the compression.…”

Section: Compressing the Transfer Matrixmentioning

confidence: 99%

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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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“…In this paper, we motivate and introduce the use for entanglement scaling of a different (inverse) length scale δ defined in terms of the gaps in the full spectrum of arXiv:1907.08603v1 [cond-mat.stat-mech] 19 Jul 2019 (inverse) correlation lengths, as obtained from the (negative) logarithm of the eigenvalues of the transfer matrix [21]. A careful study of the nature of the MPS approximation [21][22][23] indicates that these gaps are a direct consequence of the finite bond dimension and go to zero for D → ∞, regardless whether the system is gapped or not. In particular, the correlation length ξ (D) can itself be scaled in terms of δ, as was first illustrated by Rams et al for gapped systems [24].…”

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confidence: 99%

Scaling Hypothesis for Matrix Product States

et al. 2019

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We revisit the question of describing critical spin systems and field theories using matrix product states, and formulate a scaling hypothesis in terms of operators, eigenvalues of the transfer matrix, and lattice spacing in the case of field theories. Critical exponents and central charge are determined by optimizing the exponents such as to obtain a data collapse. We benchmark this method by studying critical Ising and Potts models, where we also obtain a scaling ansatz for the correlation length and entanglement entropy. The formulation of those scaling functions turns out to be crucial for studying critical quantum field theories on the lattice. For the case of λφ 4 with mass µ 2 and lattice spacing a, we demonstrate a double data collapse for the correlation length δξ(µ, λ, D) = ξ (α − αc)(δ/a) −1/ν with D the bond dimension, δ the gap between eigenvalues of the transfer matrix, and αc = µ 2 R /λ the parameter which fixes the critical quantum field theory.

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Truncating an exact matrix product state for the XY model: Transfer matrix and its renormalization

Cited by 23 publications

References 40 publications

Continuum tensor network field states, path integral representations and spatial symmetries

Continuum tensor network field states, path integral representations and spatial symmetries

Matrix product state renormalization

Scaling Hypothesis for Matrix Product States

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