Paired states at 5/2: Particle-hole Pfaffian and particle-hole symmetry breaking

Abstract

We study Cooper pairing in the Dirac composite fermion (CF) system. The presence of the mass term in the Dirac CF description (which may simulate Landau level mixing) i.e. breaking of particle-hole (PH) symmetry in this system, is a necessary condition for the existence of a PH Pfaffian-like topological state. In the scope of RPA approximation and hydrodynamic approach, we find some signatures of pairing at finite frequencies. Motivated by this insight, we extend our analysis to the case of a different but still Dirac quasi-particle (CF) representation, appropriate in the presence of a mass term, and discuss the likelihood of PH Pfaffian pairing and Pfaffian pairing in general. On the basis of gauge field effects, we find for small Dirac mass anti-Pfaffian or Pfaffian instability depending on the sign of mass, while for large mass (Landau level mixing), irrespective of its sign, PH Pfaffian-like instability.

Model interactions for Pfaffian paired states based on Chern-Simons field theory description

“…2) may be a hallmark of the whole region, −1/2 < λ 1/2, in which the topological pairing at weak coupling of composite fermions in (24) is suppressed under the CS transformation into the electron representation. This system, at 0 < λ 1/2, as we will discuss below, confined to a single LL, also represents a projection of the PH Pfaffian to a fixed LL -any attempt to confine the description of the PH Pfaffian in a fixed LL will produce a gapless state, state close to the CFL [9,12]. But we should note and emphasize that the particular oscillatory behavior of PPs, as described in the lower part of Fig.…”

“…On the basis of the same theory (A) the criticality of the PH Pfaffian (reversed chirality p-wave) for the zero Dirac mass case (i.e. in the presence of the PH symmetry) was predicted [14,9], that is numerically supported (no clear gapped arXiv:1908.03412v1 [cond-mat.str-el]…”

“…Still, after so many years of intensive experimental and theoretical research, we are not sure which paired state can be associated with the observed even-denominator FQHE. A paired state, more precisely the theoretical concept of so-called PH Pfaffian, seems according to the recent experiment [3] and other experimental data and theory [4], and theoretical proposals [4][5][6][7], closest to the solution of the puzzle at 5/2.The PH Pfaffian topological phase may be a result of a disorder dominated physics [5-8], but might be also a result of Landau level (LL) mixing in a system that may be considered uniform [8,9]. It is desirable to understand whether the PH Pfaffian (state) can be supported in a uniform system, as it was done and demonstrated in numerical experiments for Pfaffian and its particle-hole (PH) conjugated partner, anti-Pfaffian [10,11], on the basis of their model interactions.…”

The sixteenfold way and the quantum Hall effect at half-integer filling factors

“…The PH-Pfaffian topological order is consistent with the PH symmetry, but it is not protected by that symmetry, and so the corresponding ground-state wave functions do not have to be particle-hole symmetric. Moreover, it was argued [103] that a PH-symmetric wave function, which would naturally emerge in Son's picture of massless Dirac fermions, must be gapless. A gapped state with the PH-Pfaffian order requires massive Dirac fermions [103].…”

“…Moreover, it was argued [103] that a PH-symmetric wave function, which would naturally emerge in Son's picture of massless Dirac fermions, must be gapless. A gapped state with the PH-Pfaffian order requires massive Dirac fermions [103]. This means the absence of the microscopic PH symmetry.…”