Horava-Lifshitz Gravity and Effective Theory of the Fractional Quantum Hall Effect

TL;DR

The paper develops a covariant effective theory for the fractional quantum Hall effect using Horava-Lifshitz gravity as the bulk framework, preserving non-relativistic diffeomorphism and Weyl invariances. A holographic dictionary is constructed to map HL bulk fields to external sources of the boundary field theory, enabling a leading-order Chern-Simons action that captures universal electromagnetic and geometric properties such as the Wen-Zee shift and Hall viscosity. A key result is identifying the shift function with the negative of the guiding center velocity, which decouples guiding-center angular momentum from internal angular momentum; the Hall viscosity is shown to be minus half the internal angular momentum density, while the bulk Hall viscosity is half the guiding-center angular momentum density. The approach provides a consistent, symmetry-guided framework for deriving universal QHE features and offers a route to holographic modeling of geometric responses and quantum-Hall phase structure, with potential extensions to multi-species states and dynamical bulk terms. Overall, the work clarifies geometric and hydrodynamic aspects of FQHE within a covariant, holographically informed effective theory, linking Wen-Zee physics, drift dynamics, and angular-momentum decompositions in a unified formalism.

Abstract

We show that Horava-Lifshitz gravity theory can be employed as a covariant framework to build an effective field theory for the fractional quantum Hall effect that respects all the spacetime symmetries such as non-relativistic diffeomorphism invariance and anisotropic Weyl invariance as well as the gauge symmetry. The key to this formalism is a set of correspondence relations that maps all the field degrees of freedom in the Horava-Lifshitz gravity theory to external background (source) fields among others in the effective action of the quantum Hall effect, according to their symmetry transformation properties. We originally derive the map as a holographic dictionary, but its form is independent of the existence of holographic duality. This paves the way for the application of Horava-Lifshitz holography on fractional quantum Hall effect. Using the simplest holographic Chern-Simons model, we compute the low energy effective action at leading orders and show that it captures universal electromagnetic and geometric properties of quantum Hall states, including the Wen-Zee shift, Hall viscosity, angular momentum density and their relations. We identify the shift function in Horava-Lifshitz gravity theory as minus of guiding center velocity and conjugate to guiding center momentum. This enables us to distinguish guiding center angular momentum density from the internal one, which is the sum of Landau orbit spin and intrinsic (topological) spin of the composite particles. Our effective action shows that Hall viscosity is minus half of the internal angular momentum density and proportional to Wen-Zee shift, and Hall bulk viscosity is half of the guiding center angular momentum density.