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arXiv:1603.05109 · Physical review. B./Physical review. B

Bridging coupled wires and lattice Hamiltonian for two-component bosonic quantum Hall states

Yohei Fuji, Yin-Chen He, Subhro Bhattacharjee, Frank Pollmann

Published May 20, 2016• 20 citations
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Abstract

Identifying topological phases in strongly interacting many-body systems is a very challenging task, especially beyond fine-tuned exactly solvable models. The construction of topological phases from coupled quantum wires provides a controllable and systematic theoretical handle to those interactions. The authors show an explicit connection between the coupled wire construction and a lattice Hamiltonian of interacting bosons, which host a $U$(1) symmetry-protected topological phase (also referred to as the bosonic integer quantum Hall state).

DOI
10.1103/physrevb.93.195143

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•Dec 10, 2012•0 citations
Identifying topological phases in strongly interacting many-body systems is a very challenging task, especially beyond fine-tuned exactly solvable models. The construction of topological phases from coupled quantum wires provides a controllable and systematic theoretical handle to those interactions. The authors show an explicit connection between the coupled wire construction and a lattice Hamiltonian of interacting bosons, which host a UUU(1) symmetry-protected topological phase (also referred to as the bosonic integer quantum Hall state).
Interactions are crucial in driving topological phenomena in bosonic systems. Here, the authors construct exactly solvable models of topological phases of interacting bosons that preserve electric charge. They focus on the E8{E}_{8}E8​ bosonic integer quantum Hall state and show how its intrinsic E8{E}_{8}E8​ symmetry allows for the emergence of fractional topological phases of bosons. They detail the charge and statistics of the quasiparticle excitations of the resulting fractional quantum Hall states.
We study interacting two-component hard-core bosons on a square lattice for which, in the presence of staggered magnetic flux, its ground state is a bosonic integer quantum Hall (BIQH) state. By using a coupled-wire bosonization approach, we show analytically that this model exhibits a BIQH state at total charge half filling associated with a symmetry-protected topological phase under U(1)\text{U}(1)U(1) charge conservation. These theoretical expectations are verified, using the infinite density matrix renormalization group method, by providing numerical evidence for (i) a quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=\ifmmode\pm\else\textpm\fi{}2$, and (ii) two counterpropagating gapless edge modes. Our model is a bosonic cousin of the fermionic Haldane model and serves as an additional case of analogy between bosonic and fermionic quantum Hall states.
We study topological phases of interacting systems in two spatial dimensions in the absence of topological order (i.e., with a unique ground state on closed manifolds and no fractional excitations). These are the closest interacting analogs of integer quantum Hall states, topological insulators, and superconductors. We adapt the well-known Chern-Simons KKK-matrix description of quantum Hall states to classify such ``integer'' topological phases. Our main result is a general formalism that incorporates symmetries into the KKK-matrix description. Remarkably, this simple analysis yields the same list of topological phases as a recent group cohomology classification, and in addition provides field theories and explicit edge theories for all these phases. The bosonic topological phases, which only appear in the presence of interactions and which remain well defined in the presence of disorder, include (i) bosonic insulators with a Hall conductance quantized to even integers, (ii) a bosonic analog of quantum spin Hall insulators, and (iii) a bosonic analog of a chiral topological superconductor, whose KKK matrix is the Cartan matrix of Lie group E8{E}_{8}E8​. We also discuss interacting fermion systems where symmetries are realized in a projective fashion, where we find the present formalism can handle a wider range of symmetries than a recent group super-cohomology classification. Lastly, we construct microscopic models of these phases from coupled one-dimensional systems.
In two dimensions strongly interacting bosons in a magnetic field can realize a bosonic integer quantum Hall state, the simplest two-dimensional example of a symmetry-protected topological phase. We propose a realistic implementation of this phase using an optical flux lattice. Through exact diagonalization calculations, we show that the system exhibits a clear bulk gap and the topological signature of the bosonic integer quantum Hall state. In particular, the calculation of the many-body Chern number leads to a quantized Hall conductance in agreement with the analytical predictions. We also study the stability of the phase with respect to some of the experimentally relevant parameters.
Coupled-wire constructions have been widely applied to quantum Hall systems and symmetry-protected topological (SPT) phases. In this Letter, we use the coupled one-dimensional nonchiral Luttinger liquids with domain-wall structured mass terms as quantum wires to construct crystalline higher-order topological superconductors (HOTSCs) in two-dimensional (2D) interacting fermionic systems by two representative examples: a D4{D}_{4}D4​-symmetric class-D HOTSC and a C4{C}_{4}C4​-symmetric class-BDI HOTSC, with Majorana corner modes on the edge. Furthermore, based on the coupled-wire constructions, the quantum phase transitions between different phases of 2D HOTSCs by tuning the interwire coupling are investigated in a straightforward way.
We propose a general approach to construct symmetry protected topological (SPT) states i.e the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state of a spin-1 model, a SO(3) SPT state of a boson system with spin-1 bosons and spinless bosons, and a SU(2) SPT state of a spin-1/2 boson system. By applying the gauge field which directly couples to the spin density and spin current of SzS^zSz components, we also calculate the quantum spin Hall conductance in each SPT state. The projective ground states can be further studied numerically in the future by variational Monte Carlo etc.