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arXiv:1904.01589 · Physical Review Letters

Variational Ansatz for an Abelian to Non-Abelian Topological Phase Transition in ν=1/2+1/2 Bilayers

Valentin Crépel, Benoit Estienne, Nicolas Regnault

Published Sep 20, 2019• 27 citations
arXivHTMLPDF

Abstract

We propose a one-parameter variational ansatz to describe the tunneling-driven Abelian to non-Abelian transition in bosonic ν=1/2+1/2 fractional quantum Hall bilayers. This ansatz, based on exact matrix product states, captures the low-energy physics all along the transition and allows us to probe its characteristic features. The transition is continuous, characterized by the decoupling of antisymmetric degrees of freedom. We futhermore determine the tunneling strength above which non-Abelian statistics should be observed experimentally. Finally, we propose to engineer the interlayer tunneling to create an interface trapping a neutral chiral Majorana fermion. We microscopically characterize such an interface using a slightly modified model wave function.

DOI
10.1103/physrevlett.123.126804

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We propose a one-parameter variational ansatz to describe the tunneling-driven Abelian to non-Abelian transition in bosonic ν=1/2+1/2 fractional quantum Hall bilayers. This ansatz, based on exact matrix product states, captures the low-energy physics all along the transition and allows us to probe its characteristic features. The transition is continuous, characterized by the decoupling of antisymmetric degrees of freedom. We futhermore determine the tunneling strength above which non-Abelian statistics should be observed experimentally. Finally, we propose to engineer the interlayer tunneling to create an interface trapping a neutral chiral Majorana fermion. We microscopically characterize such an interface using a slightly modified model wave function.
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.
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 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 perform exact diagonalization studies for fractional quantum Hall states at filling factor 4/54/54/5 in a bilayer system, on a torus with various aspect ratios and angles. We find that in the absence of tunneling, two weakly coupled 2/52/52/5 layers undergo a phase transition into an interlayer-correlated regime, which is also Abelian with the fivefold degeneracy on the torus. In the limit of zero layer separation, this phase becomes a singlet in the pseudospin variable describing the layer degree of freedom. By studying the Chern-number matrix, we show that the KKK matrix describing the interlayer-correlated regime requires a matrix dimension larger than two and this regime is in particular not described by a Halperin state. A detailed analysis of possible $4\ifmmode\times\else\texttimes\fi{}4\phantom{\rule{4pt}{0ex}}K$ matrices having the requisite symmetries and quantum numbers shows that there is only one equivalence class of such matrices. A model wave function representing this universality class is constructed. The role of separate particle number conservation in both layers is discussed, and it is argued that this additional symmetry allows for the further distinction of two different symmetry-protected Abelian phases in the interlayer correlated regime. Interlayer tunneling breaks this symmetry and can drive the system into a single-layer regime when strong enough. A qualitative phase diagram in the tunneling-layer separation parameter space is proposed based on our numerical results.
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.