Tensor network states and geometry

Evenbly, G. and Vidal, G. (2011) Tensor network states and geometry. Journal of Statistical Physics, 145 4: 891-918. doi:10.1007/s10955-011-0237-4

Author Evenbly, G.
Vidal, G.
Title Tensor network states and geometry
Journal name Journal of Statistical Physics   Check publisher's open access policy
ISSN 0022-4715
Publication date 2011-11
Sub-type Article (original research)
DOI 10.1007/s10955-011-0237-4
Volume 145
Issue 4
Start page 891
End page 918
Total pages 28
Place of publication New York, NY, United States
Publisher Springer New York
Collection year 2012
Language eng
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.
Keyword Quantum many-body
Tensor network states
Matrix product state
Multiscale entanglement renormalization ansatz
Q-Index Code C1
Q-Index Status Confirmed Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
Official 2012 Collection
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Citation counts: TR Web of Science Citation Count  Cited 50 times in Thomson Reuters Web of Science Article | Citations
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