Tensor Network States and Geometry View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2011-11

AUTHORS

G. Evenbly, G. Vidal

ABSTRACT

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 law—that 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. More... »

PAGES

891-918

References to SciGraph publications

  • 1992-03. Finitely correlated states on quantum spin chains in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 2006-08-21. Aspects of holographic entanglement entropy in JOURNAL OF HIGH ENERGY PHYSICS
  • 1997. Conformal Field Theory in NONE
  • 2004-08. Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture in JOURNAL OF STATISTICAL PHYSICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s10955-011-0237-4

    DOI

    http://dx.doi.org/10.1007/s10955-011-0237-4

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1022635674


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    324 https://www.grid.ac/institutes/grid.420198.6 schema:alternateName Perimeter Institute
    325 schema:name Perimeter Institute for Theoretical Physics, N2L 2Y5, Waterloo, Ontario, Canada
    326 School of Mathematics and Physics, the University of Queensland, 4072, Brisbane, Australia
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