Acoustic wave absorption as a probe of dynamical geometrical response of fractional quantum Hall liquids

Abstract

We show that acoustic crystalline wave gives rise to an effect similar to that of a gravitational wave to an electron gas. Applying this idea to a two-dimensional electron gas in the fractional quantum Hall regime, this allows for experimental study of its intra-Landau level dynamical response in the long-wave length limit. To study such response we generalize Haldane's geometrical description of fractional quantum Hall states to situations where the external metric is time-dependent. We show that such time-dependent metric (generated by acoustic wave) couples to collective modes of the system, including a quadrapolar mode at long wave length, and magneto-roton at finite wave length. Energies of these modes can be revealed in spectroscopic measurements. controlled by strain-induced Fermi velocity anisotropy. We argue that such geometrical probe provides a potentially highly useful alternative probe of quantum Hall liquids, in addition to the usual electromagnetic response. Fractional quantum Hall (FQH) liquid is the prototype topological state of matter. Haldane[1] pointed out recently that description of FQH liquids in terms of topological quantum field theories, while capturing the universal and topological aspect of the physics, is incomplete in the sense that an internal geometrical degree of freedom responsible for the intra-Landau level dynamics of the system is not included. This geometrical degree of freedom, or internal metric, couples to anisotropy in the interaction between electrons[2-4] or the electron band structure [5], and its expectation value is determined by energetics of the system. In a recent work [6] we showed that this internal metric parameter manifests itself as anisotropy of composite fermion Fermi surface, which is measurable. Our quantitative result compares favorably with recent experiments, in which electron mass anisotropy is induced and controlled by an in-plane magnetic field [7][8][9]. This demonstrates the observability of this internal geometry. It has also been argued [1,[10][11][12] that this internal metric may be viewed as a dynamical degree of freedom, whose long-wave length dynamics corresponds to the collective excitations of the system that can be viewed as "gravitons" [13]. In a parallel stream of works, much effort has been devoted to studying FQH liquids in a curved background space [14][15][16][17][18][19][20][21][22][23], following earlier seminal work by Wen and Zee [24].In the existing theoretical studies [2-6, 25, 26], the background geometry (or metric) provided by electronelectron interaction and/or band structure is static. The main purpose of the present work is to generalize this to the case where the background metric is time-dependent, and show that dynamics of the metric couples to the intra-Landau level collective modes of the FQH liquid. In particular, we show that such time-dependent metric can be generated by acoustic waves, that play a role very similar to gravitational wave in this context. Such gravitational wave naturally couples to graviton an...

Geometry of flux attachment in anisotropic fractional quantum Hall states

“…Though alternative ways to test the internal geometry of gapped states have been put forward, e.g. by using acoustic wave absorption 27 , at this stage our knowledge comes almost exclusively from theory and numerics.…”

Very high-energy collective states of partons in fractional quantum Hall liquids

“…The FQH graviton is hidden from conventional probes for collective modes such as inelastic light scattering [28][29][30][31] since these probes are limited to finite momenta. In contrast, recent works [36][37][38][39] have shown that the graviton can be directly excited in a dynamical quench experiment, where the band mass or dielectric tensor of the two-dimensional electron gas is suddenly made anisotropic or the magnetic field is abruptly tilted (see also a recent proposal using surface acoustic waves [40]). demonstrated that the theories of Refs.…”

“…To identify the collective modes more accurately, we perform a finer characterization of the dynamical response by choosing Ô in Eq. ( 24) to be one of the anisotropic pseudopotentials V σ m,s with a given chirality σ=± [36,40,82,83]. This allows us to probe systematically how the system responds to a particular type of metric deformation.…”

The Half-Full Landau Level

“…79 It has been suggested that metric fluctuations produced by an acoustic wave might be used to excite quadrupolar modes of a gapped quantum state. 80 Geometric aspects of states in the lowest Landau level and their response to anisotropy have been emphasized in various investigations. [81][82][83] It is not entirely clear whether there should be unique definition of the shift parameter for a gapless state such as at ν = 1/2, and one must be careful about the proper definition of the Hall viscosity in this case.…”