Stabilization of the Particle-Hole Pfaffian Order by Landau-Level Mixing and Impurities That Break Particle-Hole Symmetry

Abstract

Numerical results suggest that the quantum Hall effect at ν = 5/2 is described by the Pfaffian or anti-Pfaffian state in the absence of disorder and Landau level mixing. Those states are incompatible with the observed transport properties of GaAs heterostructures, where disorder and Landau level mixing are strong. We show that the recent proposal of a PH-Pfaffian topological order by Son is consistent with all experiments. The absence of the particle-hole symmetry at ν = 5/2 is not an obstacle to the existence of the PH-Pfaffian order since the order is robust to symmetry breaking.One of the most interesting features of topological insulators and superconductors is their surface behavior. A great variety of gapless and topologically-ordered gapped surface states have been proposed [1]. Such states are anomalous, that is, they can only exist on the surface of a 3D bulk system and not in a stand-alone film. Finding experimental realizations of exotic surface states proved difficult and most of them have remained theoretical proposals. Thus, it came as a surprise when Son [2] argued that one such exotic state [3], made of Dirac composite fermions with the particle-hole symmetry (PHS), has long been observed experimentally in a two-dimensional system: the electron gas in the quantum Hall effect (QHE) with the filling factor ν = 1/2.At first sight, Son's idea violates the fermion doubling theorem [4]. However, the theorem does not apply to interacting systems such as the one Son considered. Besides, in contrast to the conditions of the doubling theorem, the action of PHS is nonlocal in QHE since it involves filling a Landau level. Interestingly, the picture of composite Dirac fermions sheds light on the geometrical resonance experiments [5] which the classic theory [6] of the 1/2 state could not explain. In this paper we show that a closely related idea [2] provides a natural explanation of the observed phenomenology on the enigmatic QHE plateau at ν = 5/2 in GaAs.Cooper pairing of Dirac composite fermions in the s channel results in a fractional QHE state dubbed PHPfaffian [2,7], where "PH" stands for "particle-hole". We argue that the PH-Pfaffian topological order is present on the QHE plateau at ν = 5/2. This might seem unlikely because the particle-hole symmetry is violated by the Landau level mixing (LLM) in the observed states of the second Landau level in GaAs. Besides, numerics [11][12][13][14][15][16] supports the Pfaffian [8] and anti-Pfaffian [9,10] states at ν = 5/2 even in the presence of PHS. At the same time, the existing numerical work, which always neglects strong disorder and typically neglects strong LLM, is not yet in the position to explain the physics of the 5/2 state, as is evidenced by a large discrepancy between numerical and experimental energy gaps [17]. Note that a recent attempt to incorporate LLM [16] into simulations led to the manifestly wrong conclusion that the 5/2 state does not exist at realistic LLM. Indeed, as discussed in Ref. [16], the existing perturbative methods are justifi...

Onset of quantum criticality in the topological-to-nematic transition in a two-dimensional electron gas at filling factor ν=5/2

“…The existence of a quantum critical point in Fig.3 thus highlights the dichotomy of the two descriptions of the half-filled Landau level: one based on electrons [27][28][29] and another on composite fermions 18,19 . Interest in the half-filled Landau level was recently rekindled by theories according to which the composite fermions are Dirac-like at exact particle-hole symmetry [65][66][67][68] . These theories naturally account for a Fermi sea and for a FQHS at half-filling, but do not accomodate the formation of the stripe phase 65 .…”

Pfaffian state in an electron gas with small Landau level gaps

“…Its particle-hole (PH) conjugate, the anti-Pfaffian state [35] is also likely since the two-body Coulomb interaction can not break the PH symmetry in a half-filled Landau level. Recently, a new Pfaffian-like state with PH symmetry [6,36] was proposed and may be valid for strong LLM or disorder. It is convenient to employ the Haldane pseudopotential to study the overlap between the Pfaffian state and the ground state of an even-electron system in the spherical geometry [37,38].…”

“…This could be due to our choice of planar screened dielectric function. It is also possible that the ground state itself is another Pfaffian-like wave function with PH symmetry [6,36], especially for large κ, the strong LLM region. In the PH symmetric case where the shift is S = −1 and the flux is N P H = 2N e − 1 on a sphere, the ground state is compressible and located at L = 2 for N e = 10 without screening and disorder.…”

Symmetry and Duality in Bosonization of Two-Dimensional Dirac Fermions

“…Since this is a gapped phase, its properties (i.e., its quasiparticle content) should reflect this compatibility regardless of whether this self-duality is microscopically present. (This is analogous to the PH-Pfaffian, which, as a phase of matter, need not be PH symmetric [60].) It might be interesting to explore more generally how the requirement of self-duality in this sense constrains possible phases and their properties.…”