Braiding and entanglement in spin networks: a combinatorial approach to topological phases

TL;DR

This paper introduces a combinatorial approach to topological phases using spin networks, braiding, and entanglement, connecting quantum topology with condensed matter physics and quantum information storage.

Contribution

It proposes a novel state sum model based on Turaev-Viro invariants for SU(2)_q triangulations, linking boundary lattice models and topological quantum computation.

Findings

Hamiltonian of Levin-Wen model matches Turaev-Viro amplitude

Supports boundary 2D lattice models with braiding relations

Provides a combinatorial framework for topological phases

Abstract

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q= root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a 'double' quantum Chern-Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom. We propose here a novel combinatorial approach based on `state sum' models of the Turaev-Viro type associated with SU(2)_q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin-Wen condensed string net model in a surface Sigma coincides with the corresponding Turaev-Viro amplitude on Sigma x [0,1] presented in the last section.