This article is cited in 4 papers

Chain of interacting $SU(2)_4$ anyons and quantum $SU(2)_k\times\overline{SU(2)_k}$ doubles

V. A. Verbus^a, L. Martina^bc, A. P. Protogenov<sup>d

a</sup> Institute for Physics of Microstructures, RAS, Nizhny Novgorod, Russia
^b INFN, Sezione di Lecce, Lecce, Italy
^c Dipartimento di Fisica, Università del Salento, Lecce, Italy
^d Institute of Applied Physics, RAS, Nizhny Novgorod, Russia

Abstract: We consider a chain of $SU(2)_4$ anyons with transitions to a topologically ordered phase state. For half-integer and integer indices of the type of strongly correlated excitations, we find an effective low-energy Hamiltonian that is an analogue of the standard Heisenberg Hamiltonian for quantum magnets. We describe the properties of the Hilbert spaces of the system eigenstates. For the Drinfeld quantum $SU(2)_k \times\overline{SU(2)_k}$ doubles, we use numerical computations to show that the largest eigenvalues of the adjacency matrix for graphs that are extended Dynkin diagrams coincide with the total quantum dimensions for the levels $k=2,3,4,5$. We also formulate a hypothesis about the reason for the universal behavior of the system in the long-wave limit.

Keywords: modular tensor category, quantum double, anyon.

Received: 23.06.2011

DOI: 10.4213/tmf6658

 English version: Theoretical and Mathematical Physics, 2011, 167:3, 843–855

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