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arXiv:1010.4270 · Physical Review B

Bilayer quantum Hall phase transitions and the orbifold non-Abelian fractional quantum Hall states

Maissam Barkeshli, Xiao-Gang Wen

Published Sep 20, 2011• 58 citations
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Abstract

We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the $(ppq)$ Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the ${Z}_{4}$ parafermion (Read-Rezayi) state is a special case. This provides an example in which ${Z}_{2}$ electron fractionalization leads to non-Abelian topological phases. The naive ``ideal'' wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases but, instead, lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern-of-zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the $(ppq)$ bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions $8/3$ and $12/5$ and as a way to tune to non-Abelian states in double-layer or wide quantum wells.

DOI
10.1103/physrevb.84.115121

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We study continuous quantum phase transitions that can occur in bilayer fractional quantum Hall (FQH) systems as the interlayer tunneling and interlayer repulsion are tuned. We introduce a slave-particle gauge theory description of a series of continuous transitions from the (ppq)(ppq)(ppq) Abelian bilayer states to a set of non-Abelian FQH states, which we dub orbifold FQH states, of which the Z4{Z}_{4}Z4​ parafermion (Read-Rezayi) state is a special case. This provides an example in which Z2{Z}_{2}Z2​ electron fractionalization leads to non-Abelian topological phases. The naive ``ideal'' wave functions and ideal Hamiltonians associated with these orbifold states do not in general correspond to incompressible phases but, instead, lie at a nearby critical point. We discuss this unusual situation from the perspective of the pattern-of-zeros/vertex algebra frameworks and discuss implications for the conceptual foundations of these approaches. Due to the proximity in the phase diagram of these non-Abelian states to the (ppq)(ppq)(ppq) bilayer states, they may be experimentally relevant, both as candidates for describing the plateaus in single-layer systems at filling fractions 8/38/38/3 and 12/512/512/5 and as a way to tune to non-Abelian states in double-layer or wide quantum wells.
We find a series of possible continuous quantum phase transitions between fractional quantum Hall states at the same filling fraction in two-component quantum Hall systems. These can be driven by tuning the interlayer tunneling and/or interlayer repulsion. One side of the transition is the Halperin (p,p,p-3) Abelian two-component state, while the other side is the non-Abelian Z4 parafermion (Read-Rezayi) state. We predict that the transition is a continuous transition in the 3D Ising class. The critical point is described by a Z2 gauged Ginzburg-Landau theory. These results have implications for experiments on two-component systems at ν=2/3 and single-component systems at ν=8/3.
There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two-component FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, for integer nnn. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $\ensuremath{\nu}=n+2/3$, including in particular the possibility of the non-Abelian Z4{Z}_{4}Z4​ parafermion state. In $\ensuremath{\nu}=2/3$ bilayers we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the Z4{Z}_{4}Z4​ state. On the other hand, in single-component systems at $\ensuremath{\nu}=8/3$, we find that the Z4{Z}_{4}Z4​ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $\ensuremath{\nu}=8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.
A large class of fractional quantum Hall (FQH) states can be classified according to their pattern of zeros, which describes the way ideal ground-state wave functions go to zero as various clusters of electrons are brought together. In this paper, we generalize this approach to classify multilayer FQH states. Such a classification leads to the construction of a class of non-Abelian multilayer FQH states that are closely related to ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{k}$ parafermion conformal field theories, where ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{g}}_{k}$ is an affine simple Lie algebra. We discuss the possibility of some of the simplest of these non-Abelian states occurring in experiments on bilayer FQH systems at $\ensuremath{\nu}=2/3$, 4/5, 4/7, etc.
We study phase transitions in bilayer and trilayer bosonic quantum Hall systems. In the absence of interlayer tunneling and interaction, each layer is chosen to have filling factor ν=1/2ν=1/2ν=1/2 or 111 to realize the Laughlin state or the Moore-Read state. By tuning interlayer tunneling and/or interaction, multiple phases can be generated. In the absence of interlayer interaction, three phase transitions appear when interlayer tunneling becomes sufficiently strong: (1) from two decoupled ν=1/2ν=1/2ν=1/2 Laughlin states to the Moore-Read state in bilayer systems; (2) from one ν=1/2ν=1/2ν=1/2 Laughlin state plus one ν=1ν=1ν=1 Moore-Read state to the Read-Rezayi Z3\mathbb{Z}_{3}Z3​ state in bilayer systems; (3) from three decoupled ν=1/2ν=1/2ν=1/2 Laughlin states to the Read-Rezayi Z3\mathbb{Z}_{3}Z3​ state in trilayer systems. Numerical calculations suggest that these transitions are third-order ones. We propose non-Abelian Chern-Simons-Higgs theory to describe them. If both interlayer tunneling and interaction are present, one-component or multi-component composite fermion liquids and Jain states can be realized. This leads to intricate phase diagrams that host multiple phase transitions and possibly exotic critical points.
Multicomponent quantum Hall systems with internal degrees of freedom provide a fertile ground for the emergence of exotic quantum liquids. Here, we investigate the possibility of non-Abelian topological order in the half-filled fractional quantum Hall (FQH) bilayer system driven by the tunneling effect between two layers. By means of the state-of-the-art density-matrix renormalization group, we unveil ``fingerprint'' evidence of the non-Abelian Moore-Read Pfaffian state emerging in the intermediate-tunneling regime, including the ground-state degeneracy on the torus geometry and the topological entanglement spectroscopy (entanglement spectrum and topological entanglement entropy) on the spherical geometry, respectively. Remarkably, the phase transition from the previously identified Abelian (331) Halperin state to the non-Abelian Moore-Read Pfaffian state is determined to be continuous, which is signaled by the continuous evolution of the universal part of the entanglement spectrum, and discontinuities in the excitation gap and the derivative of the ground-state energy. Our results not only provide a ``proof-of-principle'' demonstration of realizing a non-Abelian state through coupling different degrees of freedom, but also open up a possibility in FQH bilayer systems for detecting different chiral $p\text{\ensuremath{-}}\text{wave}$ pairing states.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attentions in the past twenty years. Theoretically, it is possible to realize a rich variety of such states by coupling two Abelian fractional quantum Hall (FQH) states together through gapping out part of the low energy degrees of freedom. So far, there are some indications, but no robust example has been established in bilayer systems for realizing the non-Abelian state in the past. Here, we present a phase diagram of a double-layer bosonic FQH system based on the exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrate a potential regime with the emergence of the non-Abelian bosonic Moore-Read state. We start from the Abelian phase with fourfold topological degeneracies on torus geometry when the two layers are weakly coupled. With the increase of interlayer tunneling, we find an intermediate regime with a threefold groundstate degeneracy and a finite fractional drag Hall conductance. We find the different topological sectors in consistent with Moore-Read state by inserting different fluxes in adiabatic DMRG study. We also extract the modular S−\mathcal{S}-S− matrix, which supports the emergence of the non-Abelian Ising anyon quasiparticle in this system.