Dynamically enriched topological orders in driven two-dimensional systems

Abstract

Time-periodic driving of a quantum system can enable new dynamical topological phases of matter that could not exist in thermal equilibrium. We investigate two related classes of dynamical topological phenomena in 2D systems: Floquet symmetry protected topological phases (FSPTs), and Floquet enriched topological orders (FETs). By constructing solvable lattice models for a complete set of 2D bosonic FSPT phases, we show that bosonic FSPTs can be understood as topological pumps which deposit loops of 1D SPT chains onto the boundary during each driving cycle, which protects a non-trivial edge state by dynamically tuning the edge to a self-dual point poised between the 1D SPT and trivial phases of the edge. By coupling these FSPT models to dynamical gauge fields, we construct solvable models of FET orders in which anyon excitations are dynamically transmuted into topologically distinct anyon types during each driving period. These bosonic FSPT and gauged FSPT models are classified by group cohomology methods. In addition, we also construct examples of "beyond cohomology" FET orders, which can be viewed as topological pumps of 1D topological chains formed of emergent anyonic quasi-particles. CONTENTS

Classification of Matrix-Product Unitaries with Symmetries

“…In fact, the cohomology class and the SPIs (including the index) are all characters of the net symmetry-charge current q ϱ − q x . This picture unifies all the related previous works as special situations, such as G ¼ feg [37,69] and Trx g ¼ Trϱ g ¼ δ ge dim ϱ [53,54]. Remarkably, this picture gives an intuition into Eq.…”

“…Examples of MPUs with nontrivial cohomology classes are already found in Refs. [53,54] as the edge dynamics of 2D intrinsic Floquet SPT phases [55]. Initialized as a symmetric state, a nontrivial 1D edge evolves from one SPT phase into another after each Floquet period, reminiscent of the discrete time crystals that toggle between different symmetry-broken phases [56][57][58][59].…”

Topology and Broken Symmetry in Floquet Systems

“…A d-dimensional SPT phase is characterized by the property that its ground state can be obtained from a trivial ground state by the action of a symmetry-preserving local unitary, which nevertheless cannot be locally generated at any finite depth. Remarkably, one finds that in all dimensions d, Floquet evolutions can be constructed which behave as the identity in the bulk, and whose edge unitary is a (d − 1)-dimensional pump which on successive application pumps a trivial product edge state through the set of topological SPT ground states [83,110]. This provides (at least a partial) classification of Floquet SPT phases in d-dimensions, which is the same as the classification of static SPT phases in d − 1 dimensions.…”

Minimalist approach to the classification of symmetry protected topological phases

“…[107] for detail. Example G. 28. The category Set of sets has as objects the class of all sets, and as morphisms the class of all functions between sets.…”