### Abstract

We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z_{2} 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 language | English (US) |
---|---|

Article number | 015309 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - Jan 13 2012 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*45*(1), [015309]. https://doi.org/10.1088/1751-8113/45/1/015309

}

*Journal of Physics A: Mathematical and Theoretical*, vol. 45, no. 1, 015309. https://doi.org/10.1088/1751-8113/45/1/015309

**Algebraically contractible topological tensor network states.** / Denny, S. J.; Biamonte, J. D.; Jaksch, D.; Clark, S. R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Algebraically contractible topological tensor network states

AU - Denny, S. J.

AU - Biamonte, J. D.

AU - Jaksch, D.

AU - Clark, S. R.

PY - 2012/1/13

Y1 - 2012/1/13

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=83455163823&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=83455163823&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/45/1/015309

DO - 10.1088/1751-8113/45/1/015309

M3 - Article

AN - SCOPUS:83455163823

VL - 45

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 1

M1 - 015309

ER -