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arXiv:0804.2087

Phase transitions on the edge of the ν=5/2 Pfaffian and anti-Pfaffian quantum Hall state

B. J. Overbosch, Xiao-Gang Wen

Published Apr 13, 2008• 6 citations
arXivHTMLPDF

Abstract

Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction $ν={5/2}$ fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from $g=1/4$ to $g=1/2$ for the edge reconstructed Pfaffian state, and changes from $g=1/2$ to $g=0.55-0.75$ for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed $ν={5/2}$ state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.

DOI
10.48550/arxiv.0804.2087

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Starting from the edge reconstructed Pfaffian state and the anti-Pfaffian state for the filling fraction ν=5/2ν={5/2}ν=5/2 fractional quantum Hall (FQH) state with the filled Landau levels included, we find that interactions between counterpropagating edge modes can induce phase transitions on the edge. In the new `Majorana-gapped' phases, a pair of counter propagating neutral Majorana modes becomes gapped. The quasiparticle tunneling exponent changes from g=1/4g=1/4g=1/4 to g=1/2g=1/2g=1/2 for the edge reconstructed Pfaffian state, and changes from g=1/2g=1/2g=1/2 to g=0.55−0.75g=0.55-0.75g=0.55−0.75 for the anti-Pfaffian state, in the new Majorana-gapped phases. The new phases are candidate states for the observed ν=5/2ν={5/2}ν=5/2 state. Furthermore, Majorana-gapped phases provide examples that non-trivial quantum phase transitions can happen on the edge of a FQH state without any change in bulk topological order.
The $\ensuremath{\nu}=5/2$ fractional quantum Hall effect is a system of intense experimental and theoretical interest as its ground state may host non-Abelian excitations, but the exact nature of the ground state is still undetermined. We present the results of an exact diagonalization study of an electron system in the disk configuration, including the effects of Landau level (LL) mixing and the finite thickness of the quantum well confining the electrons. The degeneracy between the two leading candidates for the ground state, the Pfaffian and anti-Pfaffian, is broken by interactions with a neutralizing background, in addition to the inclusion of two- and three-body interactions via LL mixing. As a result of the neutralizing background in the disk configuration, there is a phase transition from the anti-Pfaffian to the Pfaffian as LL mixing is turned on, in stark contrast to what is observed in spherical geometry. This behavior is in agreement with existing experiments, showing the appearance of the Pfaffian state at strong LL mixing before the system enters a compressible phase. The inclusion of LL mixing leads to an increased charge e/4e/4e/4 quasihole size. LL mixing interactions are also shown to overcome the effects of edge reconstruction. Due to finite thickness effects, these properties are enhanced dramatically. We also find that only the Pfaffian and anti-Pfaffian states continue to possess energy gaps at finite width, while gaps for compressible stripe states close, which is in agreement with available experimental data.
We present a comprehensive numerical study of a microscopic model of the fractional quantum Hall system at filling fraction $\ensuremath{\nu}=5∕2$, based on the disk geometry. Our model includes Coulomb interaction and a semirealistic confining potential. We also mix in a three-body interaction in some cases to help elucidate the physics. We obtain a phase diagram, discuss the conditions under which the ground state can be described by the Moore-Read state, and study its competition with neighboring stripe phases. We also study quasihole excitations and edge excitations in the Moore-Read-like state. From the evolution of the edge spectrum, we obtain the velocities of the charge and neutral edge modes, which turn out to be very different. This separation of velocities is a source of decoherence for a non-Abelian quasihole and/or quasiparticle (with charge $\ifmmode\pm\else\textpm\fi{}e∕4$) when propagating at the edge; using numbers obtained from a specific set of parameters, we estimate the decoherence length to be around $4\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{m}$. This sets an upper bound for the separation of the two point contacts in a double point-contact interferometer, designed to detect the non-Abelian nature of such quasiparticles. We also find a state that is a potential candidate for the recently proposed anti-Pfaffian state. We find the speculated anti-Pfaffian state is favored in weak confinement (smooth edge), while the Moore-Read Pfaffian state is favored in strong confinement (sharp edge).
Fractional quantum Hall (FQH) system at Landau level filling fraction $\ensuremath{\nu}=5/2$ has long been suggested to be non-Abelian, either Pfaffian (Pf) or antiPfaffian (APf) states by numerical studies, both with quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=5{e}^{2}/2h$. Thermal Hall conductances of the Pf and APf states are quantized at ${\ensuremath{\kappa}}_{xy}=7/2$ and ${\ensuremath{\kappa}}_{xy}=3/2$, respectively, in a proper unit. However, a recent experiment shows the thermal Hall conductance of $\ensuremath{\nu}=5/2$ FQH state is ${\ensuremath{\kappa}}_{xy}=5/2$. It has been speculated that the system contains random Pf and APf domains driven by disorders, and the neutral chiral Majorana modes on the domain walls may undergo a percolation transition to a ${\ensuremath{\kappa}}_{xy}=5/2$ phase. In this paper, we do perturbative and nonperturbative analyses on the domain walls between Pf and APf. We show the domain wall theory possesses an emergent SO(4) symmetry at energy scales below a threshold ${\mathrm{\ensuremath{\Lambda}}}_{1}$, which is lowered to an emergent $U(1)\ifmmode\times\else\texttimes\fi{}U$(1) symmetry at energy scales between ${\mathrm{\ensuremath{\Lambda}}}_{1}$ and a higher value ${\mathrm{\ensuremath{\Lambda}}}_{2}$, and is finally lowered to the composite fermion parity symmetry Z2F{\mathbb{Z}}_{2}^{F}Z2F​ above ${\mathrm{\ensuremath{\Lambda}}}_{2}$. Based on the emergent symmetries, we propose a phase diagram of the disordered $\ensuremath{\nu}=5/2$ FQH system and show that a ${\ensuremath{\kappa}}_{xy}=5/2$ phase arises at disorder energy scales $\mathrm{\ensuremath{\Lambda}}>{\mathrm{\ensuremath{\Lambda}}}_{1}$. Furthermore, we show the gapped double-semion sector of ND{N}_{D}ND​ compact domain walls contributes nonlocal topological degeneracy ${2}^{{N}_{D}\ensuremath{-}1}$, causing a low-temperature peak in the heat capacity. We implement a nonperturbative method to bootstrap generic topological 1+1D1+1\mathrm{D}1+1D domain walls (two-surface defects) applicable to any 2+1D2+1\mathrm{D}2+1D non-Abelian topological order. We also identify potentially relevant spin topological quantum field theories (TQFTs) for various $\ensuremath{\nu}=5/2$ FQH states in terms of fermionic version of $U{(1)}_{\ifmmode\pm\else\textpm\fi{}8}$ Chern-Simons $\mathrm{theory}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{8}$-class TQFTs.
The fractional quantum Hall (FQH) effect at the filling number ν=5/2 is a primary candidate for non-Abelian topological order, while the fate of such a state in the presence of random disorder has not been resolved. We address this open question by implementing an unbiased diagnosis based on numerical exact diagonalization. We calculate the disorder averaged Hall conductance and the associated statistical distribution of the topological invariant Chern number, which unambiguously characterize the disorder-driven collapse of the FQH state. As the disorder strength increases towards a critical value, a continuous phase transition is detected based on the disorder configuration averaged wave function fidelity and the entanglement entropy. In the strong disorder regime, we identify a composite Fermi liquid phase with fluctuating Chern numbers, in striking contrast to the well-known ν=1/3 case where an Anderson insulator appears. Interestingly, the lowest Landau level projected a local density profile, the wave function overlap, and the entanglement entropy as a function of disorder strength simultaneously signal an intermediate phase, which may be relevant to the recent proposal of a particle-hole Pfaffian state or Pfaffian-anti-Pfaffian puddle state.
We show that in quantum Hall systems at half filling, edge potentials alone can drive transitions between the Pfaffian and anti-Pfaffian topological phases. We conjecture this is true in realistic systems even in the presence of bulk interactions that weakly break the particle-hole symmetry. The strong effects of edge potentials could be understood from different topological shifts of competing phases at half filling, manifested on the disk geometry as the variation of orbital numbers at a fixed number of particles. In particular, we show analytically that the particle-hole conjugation of Hamiltonians on the disk is equivalent to the tuning of edge potentials, which allows us to explicitly demonstrate the phase transition numerically. The importance of edge potentials in various experimental contexts, including the recently discovered particle-hole symmetric phase, is also discussed.