### Abstract

Original language | English |
---|---|

Article number | 015309 |

Number of pages | 22 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - 6 Dec 2011 |

### Fingerprint

### Keywords

- quant-ph
- cond-mat.other
- math-ph
- math.MP

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*45*(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

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

}

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 - 2011/12/6

Y1 - 2011/12/6

N2 - 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 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 $Z_2$ 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 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.

KW - quant-ph

KW - cond-mat.other

KW - math-ph

KW - math.MP

UR - http://dx.doi.org/10.1088/1751-8113/45/1/015309

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

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

M3 - Article

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 -