Algebraically contractible topological tensor network states

S. J. Denny, Jacob Biamonte, D. Jaksch, S. R. Clark

Research output: Contribution to journalArticle

12 Citations (Scopus)

Abstract

We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z2 topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin-S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices.

Original languageEnglish (US)
Article number015309
JournalJournal of Physics A: Mathematical and Theoretical
Volume45
Issue number1
DOIs
StatePublished - Jan 13 2012

Fingerprint

Tensors
Tensor
tensors
Quantum State
contraction
Contraction
Perturbation
Hamiltonians
Lattice Gauge Theory
Hexagonal Lattice
perturbation
Bialgebra
Triangular Lattice
Differentiate
Square Lattice
Ground state
Gages
Ground State
gauge theory
Subsystem

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • Physics and Astronomy(all)

Cite this

Denny, S. J. ; Biamonte, Jacob ; Jaksch, D. ; Clark, S. R. / Algebraically contractible topological tensor network states. In: Journal of Physics A: Mathematical and Theoretical. 2012 ; Vol. 45, No. 1.
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Algebraically contractible topological tensor network states. / Denny, S. J.; Biamonte, Jacob; Jaksch, D.; Clark, S. R.

In: Journal of Physics A: Mathematical and Theoretical, Vol. 45, No. 1, 015309, 13.01.2012.

Research output: Contribution to journalArticle

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