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arXiv:2509.17113

Third-order quantum phase transitions of bosonic non-Abelian fractional quantum Hall states

Kai‐Wen Huang, Xinggang Hou, Ying-Hai Wu

Published Sep 21, 2025• 0 citations
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Abstract

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 $ν=1/2$ or $1$ 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 $ν=1/2$ Laughlin states to the Moore-Read state in bilayer systems; (2) from one $ν=1/2$ Laughlin state plus one $ν=1$ Moore-Read state to the Read-Rezayi $\mathbb{Z}_{3}$ state in bilayer systems; (3) from three decoupled $ν=1/2$ Laughlin states to the Read-Rezayi $\mathbb{Z}_{3}$ 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.

DOI
10.48550/arxiv.2509.17113

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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.
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.
Bilayer quantum Hall systems, realized either in two separated wells or in the lowest two subbands of a wide quantum well, provide an experimentally realizable way to tune between competing quantum orders at the same filling fraction. Using newly developed density matrix renormalization group techniques combined with exact diagonalization, we return to the problem of quantum Hall bilayers at filling $\ensuremath{\nu}=1/3+1/3$. We first consider the Coulomb interaction at bilayer separation ddd, bilayer tunneling energy ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{SAS}}$, and individual layer width www, where we find a phase diagram which includes three competing Abelian phases: a bilayer Laughlin phase (two nearly decoupled $\ensuremath{\nu}=1/3$ layers), a bilayer spin-singlet phase, and a bilayer symmetric phase. We also study the order of the transitions between these phases. A variety of non-Abelian phases has also been proposed for these systems. While absent in the simplest phase diagram, by slightly modifying the interlayer repulsion we find a robust non-Abelian phase which we identify as the ``interlayer-Pfaffian'' phase. In addition to non-Abelian statistics similar to the Moore-Read state, it exhibits a novel form of bilayer-spin charge separation. Our results suggest that $\ensuremath{\nu}=1/3+1/3$ systems merit further experimental study.
Identifying and understanding interacting systems that can host non-Abelian topological phases with fractionalized quasiparticles have attracted intense attention in the past 20 years. Theoretically, it is possible to realize a rich variety of such states by coupling together two Abelian fractional quantum Hall (FQH) states through gapping out part of the low-energy degrees of freedom. So far, there are some indications, but no robust examples have 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 exact diagonalization and density-matrix renormalization group (DMRG) calculations, which demonstrates a potential regime with the emergence of a non-Abelian bosonic Moore-Read state. Starting from the Abelian phase with fourfold topological degeneracy at weak coupling, with an increase of interlayer tunneling, we find an intermediate regime with a threefold ground-state degeneracy and a finite drag Hall conductance. We find different topological sectors consistent with the Moore-Read state by inserting different fluxes in an adiabatic DMRG study. We also extract the modular S\mathcal{S}S matrix, which supports the emergence of a non-Abelian Ising anyon quasiparticle in this system.
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.
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau-level filling factor 1/2 bilayer problem and make predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong-tunneling regime. In the lowest Landau level, we find that although, in principle, increasing (decreasing) tunneling strength (layer separation) could lead to a transition from the Abelian two-component Halperin 331 to non-Abelian one-component Moore-Read Pfaffian state, the FQHE excitation gap is relatively small in the lowest Landau level Pfaffian regime, and we establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state. In the second Landau level we make the prediction that bilayer structures would manifest two distinct branches of incompressible FQHE corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Moore-Read Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the possible quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Moore-Read Pfaffian state in the second Landau level. We discuss our results in the context of existing experiments and theoretical works.