Particle-hole symmetry and the composite Fermi liquid

Abstract

The half-filled Landau level is widely believed to be described by the Halperin-Lee-Read theory of the composite Fermi liquid (CFL). In this paper, we develop a theory for the particle-hole conjugate of the CFL, the Anti-CFL, which we argue to be a distinct phase of matter as compared with the CFL. The Anti-CFL provides a possible explanation of a recent experiment [Kamburov et. al., Phys. Rev. Lett. 113, 196801 (2014)] demonstrating that the density of composite fermions in GaAs quantum wells corresponds to the electron density when the filling fraction ν < 1/2 and to the hole density when ν > 1/2. We introduce a local field theory for the CFL and Anti-CFL in the presence of a boundary, which we use to study CFL -Insulator -CFL junctions, and the interface between the Anti-CFL and CFL. We show that the CFL -Anti-CFL interface allows partially fused boundary phases in which "composite electrons" can directly tunnel into "composite holes," providing a nontrivial example of transmutation between topologically distinct quasiparticles. We discuss several observable consequences of the Anti-CFL, including a predicted resistivity jump at a first order transition between uniform CFL and Anti-CFL phases. We also present a theory of a continuous quantum phase transition between the CFL and Anti-CFL. We conclude that particle-hole symmetry requires a modified view of the half-filled Landau level, in the presence of strong electron-electron interactions and weak disorder, as a critical point between the CFL and the Anti-CFL.

Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids

“…With a two-body interaction (e.g., Coulomb) the resulting Hamiltonian has a "particle-hole" symmetry at ν = 1 2 . This symmetry is not manifest in the HLR description of the composite fermi liquid, and is possibly even violated by it [17,18]. A lowest Landau level description is often routinely used in theoretical discussions and numerical calculations of quantum hall states, including at ν = thus important to understand how the particle-hole symmetry should be incorporated into the theory of the composite Fermi liquid.…”

“…It has been appreciated for some time [17,18] that the effective field theory proposed by HLR for the half-filled Landau level is not manifestly particle-hole symmetric, and is perhaps even inconsistent with it. On the other hand, numerical calculations performed in the lowest Landau level show that with the projected two-body Coulomb interaction the Fermi-liquid-like state at half-filling preserves particlehole symmetry (see, for instance, Ref.…”

“…An alternate recent discussion [18] of particle-hole symmetry in the half-filled Landau level proposes an "anti-HLR" state as a particle-hole conjugate of the HLR state. We will, however, not describe it here.…”

Nonperturbative emergence of the Dirac fermion in a strongly correlated composite Fermi liquid

“…With a CFL, the Moore-Read Pfaffian state 8 becomes natural because it is nothing but a p x + ip y -wave pairing states of CFs 7 . However, the lacking of the PHS leads to the anti-Pfaffian, the particle-hole conjugation of Pfaffian, is not simply defined in HLR's framework 19,37,38 .…”

“…Recently, the enthusiasm of research for reexamining the CFL theory is aroused by experiments [15][16][17] . A careful experiment of the composite fermion (CF) Fermi wave vector measurement through commensurability effects in the presence of a periodic grating suggests the breakdown of the PHS in FQHE 15 while theoretical explanation for the experiment must arise from PHS models 18,19 . Furthermore, the PHS breaking CFL state is not energetically favored as shown by numerical simulations 20 .…”