# From entanglement renormalisation to the disentanglement of quantum double models

## Abstract

We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models in which the Hamiltonian is gradually simplified along a parallel simplification of the connectivity of the lattice. We consider the case of Kitaev's quantum double models, both Abelian and non-Abelian, and we obtain a rederivation of the known map of the toric code to two Ising chains; we pay particular attention to the non-Abelian models and discuss their space of states on the torus. Ultimately, the construction is universal for such models and its essential feature, the lattice simplification, may point towards a renormalisation of the metric in continuum theories. - Highlights: > The toric code is explicitly mapped to two Ising chains and their diagonalisation. > The procedure uses tensor network ideas, notably entanglement renormalisation. > The construction applies to all of Kitaev's non-Abelian quantum double models. > The algebraic structure of non-Abelian models is thoroughly discussed. > The construction is universal and may work on the metric in the continuum limit.

- Authors:

- Publication Date:

- OSTI Identifier:
- 21583332

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Physics (New York)

- Additional Journal Information:
- Journal Volume: 326; Journal Issue: 9; Other Information: DOI: 10.1016/j.aop.2011.07.007; PII: S0003-4916(11)00116-3; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Journal ID: ISSN 0003-4916

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAINS; HAMILTONIANS; ISING MODEL; METRICS; QUANTUM ENTANGLEMENT; RENORMALIZATION; SPECTRA; TENSORS; TOPOLOGY; CRYSTAL MODELS; MATHEMATICAL MODELS; MATHEMATICAL OPERATORS; MATHEMATICS; QUANTUM OPERATORS

### Citation Formats

```
Aguado, Miguel, E-mail: miguel.aguado@mpq.mpg.de.
```*From entanglement renormalisation to the disentanglement of quantum double models*. United States: N. p., 2011.
Web. doi:10.1016/j.aop.2011.07.007.

```
Aguado, Miguel, E-mail: miguel.aguado@mpq.mpg.de.
```*From entanglement renormalisation to the disentanglement of quantum double models*. United States. doi:10.1016/j.aop.2011.07.007.

```
Aguado, Miguel, E-mail: miguel.aguado@mpq.mpg.de. Thu .
"From entanglement renormalisation to the disentanglement of quantum double models". United States. doi:10.1016/j.aop.2011.07.007.
```

```
@article{osti_21583332,
```

title = {From entanglement renormalisation to the disentanglement of quantum double models},

author = {Aguado, Miguel, E-mail: miguel.aguado@mpq.mpg.de},

abstractNote = {We describe how the entanglement renormalisation approach to topological lattice systems leads to a general procedure for treating the whole spectrum of these models in which the Hamiltonian is gradually simplified along a parallel simplification of the connectivity of the lattice. We consider the case of Kitaev's quantum double models, both Abelian and non-Abelian, and we obtain a rederivation of the known map of the toric code to two Ising chains; we pay particular attention to the non-Abelian models and discuss their space of states on the torus. Ultimately, the construction is universal for such models and its essential feature, the lattice simplification, may point towards a renormalisation of the metric in continuum theories. - Highlights: > The toric code is explicitly mapped to two Ising chains and their diagonalisation. > The procedure uses tensor network ideas, notably entanglement renormalisation. > The construction applies to all of Kitaev's non-Abelian quantum double models. > The algebraic structure of non-Abelian models is thoroughly discussed. > The construction is universal and may work on the metric in the continuum limit.},

doi = {10.1016/j.aop.2011.07.007},

journal = {Annals of Physics (New York)},

issn = {0003-4916},

number = 9,

volume = 326,

place = {United States},

year = {2011},

month = {9}

}