Nematic quantum phase transition of composite Fermi liquids in half-filled Landau levels and their geometric response

We present a theory of the isotropic-nematic quantum phase transition in the composite Fermi liquid arising in half-filled Landau levels. We show that the quantum phase transition between the isotropic and the nematic phase is triggered by an attractive quadrupolar interaction between electrons, as in the case of conventional Fermi liquids. We derive the theory of the nematic state and of the phase transition. This theory is based on the flux attachment procedure, which maps an electron liquid in half-filled Landau levels into the composite Fermi liquid close to a nematic transition. We show that the local fluctuations of the nematic order parameters act as an effective dynamical metric interplaying with the underlying Chern-Simons gauge fields associated with the flux attachment. Both the fluctuations of the Chern-Simons gauge field and the nematic order parameter can destroy the composite fermion quasiparticles and drive the system into a non-Fermi liquid state. The effective-field theory for the isotropic-nematic phase transition is shown to have $z=3$ dynamical exponent due to the Landau damping of the dense Fermi system. We show that there is a Berry-phase-type term that governs the effective dynamics of the nematic order parameter fluctuations, which can be interpreted as a nonuniversal “Hall viscosity” of the dynamical metric. We also show that the effective-field theory of this compressible fluid has a Wen-Zee-type term. Both terms originate from the time-reversal breaking fluctuation of the Chern-Simons gauge fields. We present a perturbative (one-loop) computation of the Hall viscosity and also show that this term is also obtained by a Ward identity. We show that the topological excitation of the nematic fluid, the disclination, carries an electric charge. We show that a resonance observed in radio-frequency conductivity experiments can be interpreted as a Goldstone nematic mode gapped by lattice effects.

Figure 1

Nematic correlator: the full lines represent the composite fermion propagator and the broken lines represent the nematic order parameters.

Figure 2

Berry phase term. Here the bubble is the fermion loop corrected by the fluctuating gauge field.

Figure 3

Leading parity-even contribution to the Wen-Zee term. The wavy lines are gauge field propagators and the broken line is a nematic propagator. The blobs are composite fermion loops.

Figure 4

The gauge boson correction to the fermion propagator.

Figure 5

Parity-odd contribution to the Wen-Zee term at cubic level. The wavy lines are gauge field propagators and the broken line is a nematic propagator.

Figure 6

Leading corrections from gauge field fluctuations to the nematic correlators: (a) and (b) composite fermion self-energy corrections and (c) vertex correction.