Theory of Nematic Fractional Quantum Hall States

We derive an effective field theory for the isotropic-nematic quantum phase transition of fractional quantum Hall states. We demonstrate that for a system with an isotropic background the low-energy effective theory of the nematic order parameter has $z=2$ dynamical scaling exponent, due to a Berry phase term of the order parameter, which is related to the nondissipative Hall viscosity. Employing the composite fermion theory with a quadrupolar interaction between electrons, we show that a sufficiently attractive quadrupolar interaction triggers a phase transition from the isotropic fractional quantum Hall fluid into a nematic fractional quantum Hall phase. By investigating the spectrum of collective excitations, we demonstrate that the mass gap of the Girvin-MacDonald-Platzman mode collapses at the isotropic-nematic quantum phase transition. On the other hand, Laughlin quasiparticles and the Kohn collective mode remain gapped at this quantum phase transition, and Kohn’s theorem is satisfied. The leading couplings between the nematic order parameter and the gauge fields include a term of the same form as the Wen-Zee term. A disclination of the nematic order parameter carries an unquantized electric charge. We also discuss the relation between nematic degrees of freedom and the geometrical response of the fractional quantum Hall fluid.

Popular Summary

Quantum Hall fluids are fascinating states of quantum matter observed in two-dimensional electron gases in high magnetic fields and at low temperatures. In a fractional quantum Hall state, the electron fluid is in a topological phase characterized by a fractional Hall conductivity (in units of ${\mathrm{e}}^2/\mathrm{h}$), and its excitations are vortices that carry a fraction of the charge of an electron and fractional exchange statistics. Experiments have shown that electronic transport in tilted magnetic fields in these topological fluids becomes anisotropic at low temperatures and that the anisotropy has a pronounced temperature dependence. This effect suggests that the electron fluid inside this fractional quantum Hall phase may be close to a phase transition to a state in which the rotational invariance of the fluid is broken spontaneously, a state that is known as an electronic nematic phase. A nematic electronic fluid state has been found earlier in regimes in which the two-dimensional electron gas is not topological, and in some cuprate and pnictide superconductors, but it had not been seen before in topological phases of matter.

We describe this new state of matter by means of an effective field theory. This theory allows us to study the nematic transition and the nematic phase inside a fractional quantum Hall fluid. We show that the quantum phase transition is triggered by effective attractive interactions in the quadrupolar channel, which cause the lowest collective mode of the fractional quantum Hall fluid to condense. The resulting nematic state is characterized by an order parameter that represents these quadrupolar fluctuations, which play the role of fluctuations of the local geometry of the quantum fluid. In other terms, the nematic fluctuations behave as a fluctuating metric for the electrons in the fluid. The result is that they mimic a gravitational interaction among these degrees of freedom. An interesting feature of the nematic phase is that it has topological defects known as disclinations that act as local centers of spatial curvature for the electronic degrees of freedom. The effective field theory provides a full description of the response of the quantum fluid to external electromagnetic probes and to local deformations of the underlying crystal.

Although the theory that we derived is specific for fractional quantum Hall states, these ideas and mechanisms are of general interest to understanding the behavior of geometric fluctuations in other topological phases in condensed matter.

Figure 1

Correlator of the nematic order parameters.

Figure 2

Diagrams contributing to the Wen-Zee term. Here, the wiggly line represents the gauge field $a_{\mu }$ and the dotted line represents the nematic field $M_i$. The thick line represents the composite fermion propagator.