Exact realization of integer and fractional quantum Hall phases in models in

Abstract

In this work we present a set of microscopic U (1)× U (1) models which realize insulating phases with a quantized Hall conductivity σ xy . The models are defined in terms of physical degrees of freedom, and can be realized by local Hamiltonians. For one set of these models, we find that σ xy is quantized to be an even integer. The origin of this effect is a condensation of objects made up of bosons of one species bound to a single vortex of the other species. For other models, the Hall conductivity can be quantized as a rational number times two. For these systems, the condensed objects contain bosons of one species bound to multiple vortices of the other species. These systems have excitations carrying fractional charges and non-trivial mutual statistics. We present sign-free reformulations of these models which can be studied in Monte Carlo, and we use such reformulations to numerically detect a gapless boundary between the quantum Hall and trivial insulator states. We also present the broader phase diagrams of the models.(1)The J 1 variables are integer-valued conserved currents residing on a cubic lattice (the 'direct' lattice) labeled with the index r. They satisfy ∇ · J 1 = 0, which means that they form closed 2

Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order

“…In a recent breakthrough, analogous states were shown to exist in higher dimensions [6][7][8][9][10], which could potentially be realized as ground states of frustrated magnetic insulators or ultracold bosonic atoms [19][20][21][22]. The simplest example is a (2 + 1)-dimensional [(2 + 1)D] bosonic phase with a gapped bulk but c − = 8n (n an integer) edge modes that all propagate in the same direction [11,23].…”

Stroboscopic symmetry-protected topological phases

“…On the other hand, when the amplitude of the drive is chosen to scale with the frequency ω, the stroboscopic Hamiltonian can acquire a nontrivial form that is different from H 0 [27]. In this work, we show that the stroboscopic Hamiltonian (4) describes microscopic models of SPT states with Z 2 × Z 2 [7,20,21] and Z 2 [7,17] symmetries, respectively, for one-and twodimensional driven systems. We refer to the phases generated in this way as stroboscopic SPT (SSPT) phases.…”

“…Following the classification of weakly-interacting fermionic SPT states [8][9][10], there has been a vast amount of recent effort to classify strongly-interacting SPT phases [7,[11][12][13][14][15] as well as to construct models supporting them [7,[16][17][18][19][20][21][22][23][24][25]. In light of this effort, it is highly desirable to identify controlled mechanisms capable of bringing SPT states into realization.…”

Rokhsar-Kivelson models of bosonic symmetry-protected topological states

“…6,7,9,15,16,[19][20][21][22][23][24][25][26] The purpose of this paper is to provide a framework for constructing microscopic models capable of describing bosonic SPT states. As we shall see, some of the exactly solvable models previously studied 20,21,26 will be identified as special cases of a large class of models to be constructed here. We shall also be able to construct parent Hamiltonians for two dimensional Z 2 × Z 2 paramagnets, whose effective edge theory was shown in Ref.…”