Analytic exposition of the graviton modes in fractional quantum Hall effects and its physical implications

Abstract

Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. In this work, we derived a set of analytical results for the energy gap of the graviton modes with two-body and three-body Hamiltonians in both the long-wavelength and thermodynamic limit. These allow us to construct model Hamiltonians for the graviton modes in different FQH phases, and to elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) with respect to the graviton modes and their corresponding ground states. Using the analytical tools developed, we perform numerical analysis with a particular focus on the Laughlin ν = 1/5 and the Gaffnian ν = 2/5 phases. Our calculation shows that for gapped phases, low-lying neutral excitations can undergo a "phase transition" even when the ground state is invariant. We discuss about the compressibility of the Gaffnian phase, the possibility of multiple graviton modes, and the transition from the graviton modes to the "hollow-core" modes, as well as their experimental consequences.

Parametrization and thermodynamic scaling of pair correlation functions for the Fractional Quantum Hall Effect

“…The magneto-roton branch is often the lowest-lying neutral excitation and therefore plays a vital role in the stability of the fractional quantum Hall liquid. It gained much attention in the early literature [43,44], and has more recently seen a revival due to the connection with gravity modes and holography [45][46][47][48][49]. The magneto-roton gaps have been studied in detail using the composite fermion theory [50,51], finite thickness calculations [52], and recently also using DMRG [53].…”

Graviton modes in fractional quantum hall liquids

“…showing that multiple graviton modes (gravitons for brevity) are a generic feature for FQH fluids, from the splitting of the long wavelength limit of the GMP mode in different subspaces in a single LL [52]. Using the analytic tools we developed earlier [354], we demonstrate that the number of gravitons is dynamical in nature, and is only meaningful when referring to specific interaction Hamiltonians. Each graviton can be interpreted as the metric fluctuation of a CHS within a single LL.…”

“…We have introduced the microscopic theory about the collective neutral excitations with spin-2 in FQH phases, called the graviton modes as a typical manifestation of the interplay between topology and geometry. While Lorentz invariance is absent in this (2 + 1)-dimensional space-time, these 2D graviton modes encode topological information about their respective FQH phases, and their dynamics lead to rich physics ranging from ground state incompressibility to the dynamical phase transitions of the low-lying excitations [140,354]. The effective field theory studying the graviton modes has been proposed by using the Newton-Carton metric, and various experimental proposals for the observation of these modes have been put forward [187,236,238,318,321,322,[324][325][326]355].…”

“…Meanwhile, with the microscopic theory of the graviton modes established in the last chapter, one can construct model Hamiltonians for these modes in the FQH fluids at different filling factors [333,354]. Recently numerical results have implied the signature of multiple graviton modes in FQH states [335,356].…”

“…In Fig. (5.1), we illustrate a hierarchical structure of different H α in the LLL [354]. For a given H α , finding another H β ⊂ H α is possible.…”