Voce esta aquiUnderstanding Topological Order Using Hopf Algebras

Understanding Topological Order Using Hopf Algebras

By bertuzzo - Posted on 09 março 2015

Dr. Pramod Padmanabhan (IF-USP)
Terça-feira, 14 Abril, 2015 - 11:00


The problem of classifying phases of matter at low temperatures is an outstanding problem in condensed matter physics. The mean field theory of order parameters due to Landau-Ginzburg, achieves this for a large variety of systems at higher temperatures but fails at lower temperatures where quantum effects come into play giving rise to a new order named topological order. Our current understanding of topological order classifies it into two kinds - systems with short ranged entangled ground states (SPT Phases) and systems with long ranged entangled ground states. The ones with long ranged entangled ground states are said to have intrinsic topological order and are characterized by low energy excitations which have fractional quantum numbers in two dimensions. On the other hand short ranged entangled systems have gapless edge states protected by some global symmetry group. These phases can be described by exactly solvable lattice models which are described by Hamiltonians made up of operators satisfying the quantum double relations or its variants. The quantum double is an example of an Hopf algebra. We will illustrate this through the example of the two dimensional toric code model which is based on the quantum double of the group algebra of the discrete group Z_2 . The toric code model is the simplest example of a long ranged entangled phase in two dimensions and can be thought of as a particular limit of a lattice gauge theory based on the discrete gauge group Z_2 . We will use this fact and construct the transfer matrix of this lattice gauge theory which will then be used to obtain the toric code as a particular limit. The advantage of this approach will be highlighted. We will remark that the SPT phases can be obtained from transfer matrices of gauge and matter fields.