Self-duality of the integer quantum Hall to insulator transition: Composite fermion description

Abstract

The integer quantum Hall to insulator transition (IQHIT) is a paradigmatic quantum critical point. Key aspects of this transition, however, remain mysterious, due to the simultaneous effects of quenched disorder and strong interactions. We study this transition using a composite fermion (CF) representation, which incorporates some of the effects of interactions. As we describe, the transition also marks a IQHIT of CFs: this suggests that the transition may exhibit 'self-duality'. We show the explicit equivalence of the electron and CF Lagrangians at the critical point via the corresponding non-linear sigma models, revealing the self-dual nature of the transition. We show analytically that the resistivity tensor at the critical point is ρ c xx = ρ c xy = h e 2 , which are consistent with the expectations of self-duality, and in rough agreement with experiments.

Composite fermion nonlinear sigma models

“…xx requires the renormalized m 2 = 0. The resulting NSLM reduces to that found in [54] in which a θ = π term appears. We interpret this as an emergent particle-hole symmetry and the irrelevance of m 2 near the diffusive quantum critical point.…”

“…In [54] the θ = π term was extracted by a careful study of the chiral anomaly of a 2D theory closely related to the one in (3.49). In this earlier study, only particle-hole symmetric disorder was considered.…”

“…The Dirac composite fermion mean-field theory (1.1) manifestly has particle-hole symmetry so long as time-reversal invariance is preserved. Significant progress towards an analytical description of this transition was then made in [54], where it was shown that, for particle-hole symmetric disorder and when m 2 = 0, the DC zero-temperature electrical conductivity of composite fermions is encoded in the nonlinear sigma model (NLSM),…”

“…In this paper we generalize [54] to include uniform m 2 and random a 0 (x). First for vanishing a 0 (x) and when particlehole symmetric disorder a i (x) is present, we find that finite σ cf…”

Composite Fermion Nonlinear Sigma Models

“…There has been recent progress towards an analytical description of this quantum criticality [19,20]. Specifically in [19] it was shown that the nonlinear sigma model (NLSM) for the diffusion of Dirac composite fermion [15] charge density fluctuations Q ∈ U(2n)/U(n)×U(n),…”