Miguel Jorge Bernabe Ferreira

arXiv2015

Toric code-like models from the parameter space of $3D$ lattice gauge theories

A state sum construction on closed manifolds á la Kuperberg can be used to construct the partition functions of $3D$ lattice gauge theories based on involutory Hopf algebras, $\mathcal{A}$, of which the group algebras, $\mathbb{C}G$, are a particular case. Transfer matrices can be obtained by carrying out this construction on a manifold with boundary. Various Hamiltonians of physical interest can be obtained from these transfer matrices by playing around with the parameters the transfer matrix is a function of. The $2D$ quantum double Hamiltonians of Kitaev can be obtained from such transfer matrices for specific values of these parameters. A initial study of such models has been carried out in \cite{p1}. In this paper we study other regions of this parameter space to obtain some new and known models. The new model comprise of Hamiltonians which "partially" confine the excitations of the quantum double Hamiltonians which are usually deconfined. The state sum construction allows for parameters depending on the position in obtaining the transfer matrices and thus it is natural to expect disordered Hamiltonians from them. Thus one set of known models consist of the disordered quantum double Hamiltonians. Finally we obtain quantum double Hamiltonians perturbed by magnetic fields which have been considered earlier in the literature to study the stability of topological order to perturbations.

hep-th↗

arXiv2015

Local realizations of anyon exchange symmetries without lattice dislocations

The global $e$-$m$ exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.

cond-mat.str-el↗