Abstract
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 Girvin-Macdonald-Platzman (GMP) mode. The theory consist of a topological Chern-Simons action, coupled to a symmetric rank two tensor, and an action \`a la bimetric gravity, describing the gapped dynamics of the spin-$2$ GMP mode. The theory is formulated in curved ambient space and is spatially covariant, which allows to restrict the form of the effective action and the values of phenomenological coefficients. Using the 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. We also comment on the possible applications to fractional Chern insulators, where closely related structures arise. Finally, it is shown that the familiar FQH observables acquire a curious geometric interpretation within the bimetric formalism.
Highlights
During the last two decades, topological quantum field theory (TQFT) has firmly established itself as a useful lowenergy theory of fractional quantum Hall states [1,2]
We evaluate the projected static structure factor (SSF), which is given by the equal-time correlation function of the Ricci scalar within bimetric theory, and we match it to the microscopic result of Ref. [20]
We have formulated a bimetric theory for the gapped collective excitations in fractional quantum Hall states
Summary
During the last two decades, topological quantum field theory (TQFT) has firmly established itself as a useful lowenergy theory of fractional quantum Hall states [1,2] (and, more generally, of topological phases in two spatial dimensions). A quantum Hall state such as the Wen-Zee shift [6,7,8], the Hall viscosity [9,10,11,12], and the central charge [13,14,15,16,17,18,19] Another remarkable feature of fractional quantum Hall states, not shared generically by other topological phases of matter, is the presence of a gapped collective excitation first proposed by Girvin, Macdonald, and Platzman (GMP) [20]. One remarkable property of the GMP mode is that it carries angular momentum L 1⁄4 2 at zero linear momentum This property of the GMP mode is one of the motivations that led Haldane to propose that fractional quantum Hall states have a hidden sector described by a gapped effective theory of a geometric nature [12,24,25].
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