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arXiv:1210.0907 · Physical Review B

Quantum phase transition between integer quantum Hall states of bosons

Tarun Grover, Ashvin Vishwanath

Published Jan 31, 2013• 84 citations
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Abstract

Bosonic integer quantum Hall (IQH) phases are a class of symmetry-protected topological (SPT) phases that, similar to the fermionic IQH states, support a quantized Hall conductance. They, however, require interactions for their realization. Here, we study quantum Hall plateau transitions between a trivial insulator and a bosonic IQH phase, in a clean system. Generically, we find an intervening superfluid phase. The presence of additional symmetries, however, can potentially lead to a direct transition between these phases. We employ a fermionic parton description that captures both the insulating phases as well as the transition between them. The critical theory is massless QED-3 with ${N}_{f}=2$ fermion flavors. The fermions have a surprisingly simple interpretation: they are vortices of the superfluid. Therefore, the universal conductivity at the transition, assuming it is continuous, equals the universal resistivity of Dirac fermions in QED-3. We also briefly comment on sigma model descriptions of the transition and the surface states of related three-dimensional SPT phases.

DOI
10.1103/physrevb.87.045129

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Bosonic integer quantum Hall (IQH) phases are a class of symmetry-protected topological (SPT) phases that, similar to the fermionic IQH states, support a quantized Hall conductance. They, however, require interactions for their realization. Here, we study quantum Hall plateau transitions between a trivial insulator and a bosonic IQH phase, in a clean system. Generically, we find an intervening superfluid phase. The presence of additional symmetries, however, can potentially lead to a direct transition between these phases. We employ a fermionic parton description that captures both the insulating phases as well as the transition between them. The critical theory is massless QED-3 with Nf=2{N}_{f}=2Nf​=2 fermion flavors. The fermions have a surprisingly simple interpretation: they are vortices of the superfluid. Therefore, the universal conductivity at the transition, assuming it is continuous, equals the universal resistivity of Dirac fermions in QED-3. We also briefly comment on sigma model descriptions of the transition and the surface states of related three-dimensional SPT phases.
We study a bosonic model with correlated hopping on a honeycomb lattice, and show that its ground state is a bosonic integer quantum Hall (BIQH) phase, a prominent example of a symmetry-protected topological (SPT) phase. By using the infinite density matrix renormalization group method, we establish the existence of the BIQH phase by providing clear numerical evidence: (i) a quantized Hall conductance with |σ_{xy}|=2, (ii) two counterpropagating gapless edge modes. Our simple model is an example of a novel class of systems that can stabilize SPT phases protected by a continuous symmetry on lattices and opens up new possibilities for the experimental realization of these exotic phases.
The deconfined quantum critical point, a prototype Landau-forbidden transition, could exist, in principle, in the phase transitions involving a symmetry-protected topological phase; however, examples of such kinds of transitions in physical systems are rare beyond one-dimensional systems. Here, using a density-matrix renormalization-group calculation, we unveil a bosonic integer quantum Hall phase in a two-dimensional correlated honeycomb lattice, by full identification of its internal structure from the topological K\mathbf{K}K matrix. Moreover, we demonstrate that imbalanced periodic chemical potentials can destroy the bosonic integer quantum Hall state and drive it into a featureless trivial (Mott) insulator, where all physical observables evolve smoothly across the critical point. At the critical point, the entanglement entropy reveals a characteristic scaling behavior, which is consistent with the critical field theory as an emergent QED3{\mathrm{QED}}_{3}QED3​ with two flavors of Dirac fermions.
Topological Mott Insulators and Discontinuous U(1)U(1)U(1) θθθ-Terms
We introduce a lattice field theory that describes the transition between a superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a U(1)U(1)U(1) symmetry protected topological phase labeled by an integer level kkk and possessing an even integer 2ke2h2k\frac{e^2}{h}2khe2​ quantized Hall conductance. Our model differs from the usual 2+12+12+1d XY model by a topological term that vanishes on closed manifolds and in the absence of an applied gauge field, which implies that the critical exponents of the SF-tMI transition are identical to those of the well-studied 2+12+12+1d XY transition. Our formalism predicts a "level-shift" symmetry: in the absence of an applied gauge field, the bulk correlation functions of all local operators are identical for models differing by the topological term. %, for example, near the SF-MI and SF-tMI transitions, and hence the extremely well-studied critical exponents of the 2+12+12+1d XY model apply to the SF-tMI transition. In the presence of a background gauge field, the topological term leads to a quantized Hall response in the tMI phase, and we argue that this quantized Hall effect persists in the vicinity of the phase transition into the SF phase. Our formalism paves the way for other exact lattice descriptions of symmetry-protected-topological (SPT) phases, and mappings of critical exponents between transitions from symmetry-breaking to trivial states and transitions from symmetry-breaking to SPT states. A similar "level shift" symmetry should appear between all group cohomology SPT states protected by the same symmetry.
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.
Inspired by the Chern-Simons effective theory description of symmetry protected topological (SPT) phases in two dimensions, we present a projective construction for many-body wave functions of SPT phases. Using this projective construction, we can systematically write down trial wave functions of SPT phases on a lattice. An explicit example of SPT phase with U(1) symmetry is constructed for two types of bosons with filling ${\ensuremath{\nu}}_{{b}_{1}}={\ensuremath{\nu}}_{{b}_{2}}=\frac{1}{2}$ per site on a square lattice. We study continuous phase transitions between different U(1)-SPT phases based on projective construction. The effective theory around the critical point is an emergent QED3{\mathrm{QED}}_{3}QED3​ with fermion number Nf=2{N}_{f}=2Nf​=2. Such a continuous phase transition, however, needs fine tuning, and in general there are intermediate phases between different U(1)-SPT phases. We show that such an intermediate phase has the same response as an anyon superconductor, and hence dub it ``anyon superfluid.'' A schematic phase diagram of interacting bosons with U(1) symmetry is depicted.