Bimetric Theory of Fractional Quantum Hall States

We present a bimetric low-energy effective theory of fractional quantum Hall (FQH) states that describes the topological properties and a gapped collective excitation, known as the Girvin-Macdonald-Platzman (GMP) mode. The theory consists of a topological Chern-Simons action, coupled to a symmetric rank-2 tensor, and an action à la bimetric gravity, describing the gapped dynamics of a spin-2 mode. The theory is formulated in curved ambient space and is spatially covariant, which allows us to restrict the form of the effective action and the values of phenomenological coefficients. Using bimetric theory, we calculate the projected static structure factor up to the $k^6$ order in the momentum expansion. To provide further support for the theory, we derive the long-wave limit of the GMP algebra, the dispersion relation of the GMP mode, and the Hall viscosity of FQH states. The particle-hole (PH) transformation of the theory takes a very simple form, making the duality between FQH states and their PH conjugates manifest. We also comment on the possible applications to fractional Chern insulators, where closely related structures arise. It is shown that the familiar FQH observables acquire a curious geometric interpretation within the bimetric formalism.

Popular Summary

A gas of electrons confined to two dimensions and subject to a strong magnetic field and low temperatures exhibits an unusual behavior known as the quantum Hall effect, where the electrical conductance takes on only certain discrete values. When this happens, the interior of the gas acts as an electrical insulator, while the edge behaves like a conductor. It is often assumed that most of the interesting physics occurs at the edge. However, we developed a theory to study the properties of the interior and find a rich behavior that not only provides a unified framework for deriving many of the properties of this state but also has theoretical connections to certain theories of gravity.

Specifically, we studied the bulk properties and dynamics of a 2D electron gas in the fractional quantum Hall (FQH) regime. Our bimetric theory allows one to find the projected static structure factor and the dispersion relation of the lowest excitation, calculate the Hall viscosity, and understand the nematic phase transition—all of which are intrinsically bulk properties of the FQH effect. Some of these properties were previously calculable only from trial wave functions.

Our approach provides a new language and new tools for the analysis of strongly interacting FQH states. It also provides a connection to the bimetric theory of massive gravity and hints at a connection to higher-spin theories. Our theory will have nontrivial extensions when applied to non-Abelian quantum Hall states and quantum Hall liquid crystals.