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

Abstract

We synthesize and partly review recent developments relating the physics of the half-filled Landau level in two dimensions to correlated surface states of topological insulators in three dimensions. The latter are in turn related to the physics of certain three-dimensional quantum spin liquid states. The resulting insights provide an interesting answer to the old question of how particle-hole symmetry is realized in composite fermion liquids. Specifically the metallic state at filling ν = 1 2 -described originally in pioneering work by Halperin, Lee, and Read as a liquid of composite fermions-was proposed recently by Son to be described by a particle-hole symmetric effective field theory distinct from that in the prior literature. We show how the relation to topological insulator surface states leads to a physical understanding of the correctness of this proposal. We develop a simple picture of the particle-hole symmetric composite fermion through a modification of older pictures as electrically neutral "dipolar" particles. We revisit the phenomenology of composite fermi liquids (with or without particle-hole symmetry), and show that their heat/electrical transport dramatically violates the conventional Wiedemann-Franz law but satisfies a modified one. We also discuss the implications of these insights for finding physical realizations of correlated topological insulator surfaces.

Dual Dirac Liquid on the Surface of the Electron Topological Insulator

“…We now briefly discuss how the present work is connected to Son's proposal. A more elaborate discussion of the half-filled Landau level will be presented in a separate work [34].…”

Symmetry enriched U(1) quantum spin liquids

“…This line of thinking has proven to be very powerful in studying difficult problems in strong correlation physics in two space dimensions. Examples include quantum Hall systems, especially the half-filled Landau level [31,[36][37][38][39][40], interacting topological insulator surfaces [32,34,35], quantum electrodynamics in 2 + 1 dimensions [33,41], and a class of Landau-forbidden quantum phase transitions known as deconfined quantum criticality [42]. The lower dimensional dualities are also interesting on their own as nontrivial results in 2 + 1 dimensional quantum field theory [43][44][45][46].…”

Pairing instabilities of Dirac composite fermions

“…At filling factor ν = 1/2, essentially they absorb the external flux, and make a metallic state [2] with its own Fermi surface -Fermi surface of CF's. By slightly modifying Read's dipole construction of composite (neutral) fermions in the half-filled lowest Landau level [3], an argument can be given for the accumulation of Berry phase equal to π as a CF encircles its own Fermi surface [4]. This has motivated a description of the CF's in this setting in terms of Dirac fermions, which has been recently introduced in Ref.…”

“…[5], and have attracted some interest [4,[6][7][8][9][10][11][12][13]. The PH symmetric description of the half-filled LL is given in terms of a Dirac system of composite quasiparticlesDirac CF's at a finite chemical potential [4] and in the presence of a gauge field. However, the implied existence of singularity at zero momentum in the CF spectrum was criticized [14,15].…”