Topological phases and nonequilibrium dynamics

Abstract

Topological behavior manifests itself through non-local effects at low energies in condensed matter physics. They lead to new phases of matter which are beyond the symmetry breaking scheme of Landau's classification. The systems exhibiting such phases are called topologically ordered systems. These phases are described by topological field theories in the continuum and in the discrete setting by their Hamiltonian realizations. Such Hamiltonians can be found by constructing state sum models such as the Turaev-Viro models which were originally introduced to study 3-manifold invariants. Over the last two years we (Pramod Padmanabhan (IFUSP) along with his supervisor Prof. Paulo Teotonio-Sobrinho (IFUSP) developed such state sum models using the Kuperberg invariant. By generalizing Kuperberg's construction we systematically constructed transfer matrices of lattice gauge theories based on involutory Hopf algebras of which group algebras are a special case. The operators obtained can be seen as tensor network representations. This procedure helped us embed Kitaev's toric code, a special phase of $\mathbb{Z}_2$ lattice gauge theory, in an enlarged parameter space. We then went beyond lattice gauge theories which helped us derive the Hamiltonian realizations of the Dikgraaf-Witten topological gauge theories. We then went further by including matter fields to find transfer matrices describing lattice theories with gauge and matter fields. Another useful aspect of these methods is that they gave us a recipe to construct exactly soluble lattice models. Thus we can obtain effective theories describing various topological phases using these methods. In this project we want to further develop this program of generalizing Kuperberg's constructions to look for models describing new topological phases. Such extensions can embed the two dimensional string-net models-systems realizing bulk anyons coming from a general unitary fusion category-in a bigger parameter space. In addition we will use these techniques to develop models showing symmetry enriched topological phases (SET phases) which are long ranged entangled phases acted upon an external symmetry (such symmetries include spatial symmetries, time reversal invariance, parity among others), and short ranged phases protected by symmetries (SPT phases) such as crystalline symmetries which give rise to topological crystalline insulators (TCI's). Prof. Siddharth Parameswaran (UCI) is an expert in the above fields. He already has several important works covering these topics. His network consists of several important people who are among the pioneers in these areas. Along with his group we plan to introduce disorder into these Hamiltonians to study the thermalization properties of these many body states. In particular we will be interested in those states which retain the topological order at high energies. These are called as many-body localized states (MBL) and they are examples of topological order being preserved by localization at high temperatures. Prof. Parameswaran has been recently active in this area where he studied not just systems with topological order but also other strongly correlated systems. Studies in this direction are important for practical reasons. The manifestation of these properties at higher temperatures is essential for using these topological properties to build a quantum computer. (AU)