Microscopic description of 2d topological phases, duality and 3d state sums

Doubled topological phases introduced by Kitaev, Levin and Wen supported on two dimensional lattices are Hamiltonian versions of three dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state are shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three dimensional geometrical interpretation are given.Comment: 19 pages, typos corrected, style improved, a final paragraph adde

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

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.Comment: Contributed to Quantum 2008: IV workshop ad memoriam of Carlo Novero 19-23 May 2008 - Turin, Ital

Quantum Transport Enhancement by Time-Reversal Symmetry Breaking

Quantum mechanics still provides new unexpected effects when considering the transport of energy and information. Models of continuous time quantum walks, which implicitly use time-reversal symmetric Hamiltonians, have been intensely used to investigate the effectiveness of transport. Here we show how breaking time-reversal symmetry of the unitary dynamics in this model can enable directional control, enhancement, and suppression of quantum transport. Examples ranging from exciton transport to complex networks are presented. This opens new prospects for more efficient methods to transport energy and information.Comment: 6+5 page

The torus and the Klein Bottle amplitude of permutation orbifolds

The torus and the Klein bottle amplitude coefficients are computed in permutation orbifolds of RCFT-s in terms of the same quantities in the original theory and the twist group. An explicit expression is presented for the number of self conjugate primaries in the orbifold as a polynomial of the total number of primaries and the number of self conjugate ones in the parent theory. The formulae in the $Z_2$ orbifold illustrate the general results.Comment: 8 pages, syntactic corrections to v