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A hierarchy of topological tensor network states

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10.1063/1.4773316

### Abstract

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf *C**-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [Ann. Phys.303, 2 (Year: 2003)10.1016/S0003-4916(02)00018-0]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing an entanglement renormalization flow. Furthermore, we argue that the hierarchy states are related to each other by the condensation of topological charges.

© 2013 American Institute of Physics

Received 28 October 2011
Accepted 19 November 2012
Published online 23 January 2013

Acknowledgments: We gratefully acknowledge discussions with David Pérez-García, J. Ignacio Cirac, Norbert Schuch, and Sergey Bravyi. For valuable comments on an earlier version of this manuscript we would like to thank Liang Kong and Joost Slingerland. M.A. thanks Jürgen Fuchs for an illuminating introduction to Hopf algebras. Part of this work was carried out while M.C. was affiliated with the Faculty of Physics at the University of Munich in Germany. M.C. is supported by the Swiss National Science Foundation (Grant No. PP00P2-128455), the German Science Foundation (Grant Nos. CH 843/1-1 and CH 843/2-1), and the National Centre of Competence in Research “Quantum Science and Technology.”

Article outline:

I. INTRODUCTION

A. Main results

B. Outline

II. PRELIMINARIES

A. Quantum double models from groups

B. Tensornetwork representations

C. Hopf algebras

D. The Hopf algebra*H* _{8}

E. Quantum doubles as bicrossed products

III. CONSTRUCTING QUANTUM DOUBLE MODELS FROM HOPF ALGEBRAS

A. Graph representations of quantum doubles

B. Hilbert space

C. Hamiltonian

D. Comparison with quantum double models from groups

IV. A HIERARCHY OF TOPOLOGICAL TENSORNETWORK STATES

A. Tensor traces

B. Quantum states

C. Hierarchy

D. Projected entangled-pair states

V. CALCULATING THE TOPOLOGICAL ENTANGLEMENT ENTROPY

A. Isometries

B. Transforming hierarchy states

C. Entanglement entropy

D. Boundary configurations

VI. DISCUSSION

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2013-01-23

2014-04-17

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