Geometric Adiabatic Transport in Quantum Hall States

Abstract

We argue that in addition to the Hall conductance and the non-dissipative component of the viscous tensor there exists a third independent transport coefficient, which is precisely quantized, taking on constant values along quantum Hall plateaus. Relying on the holomorphic properties of the quantum Hall states, we show that the new coefficient is the Chern number of a vector bundle over moduli space of surfaces of genus two or higher and therefore can not change continuously along the plateau. As such it does not transpire on a sphere or a torus. In the linear response theory this coefficient determines intensive forces exerted on electronic fluid by adiabatic deformations of geometry and represents the effect of the gravitational anomaly. We also present the method of computing the transport coefficients for QH-states. Precise quantization in materials occurs when the transport is a non-dissipative adiabatic process. QHE is an example of a system where adiabatic conditions are in place. Namely, the low energy states are separated from the rest of the spectrum by a gap and adiabatic changes of parameters produce states with the flux. However, only adiabatic processes with the nontrivial first Chern class yield quantized transport coefficients.In this paper we show that apart from the Hall conductance there exist two more quantized transport coefficients, although at present only the former is experimentally accessible. One of these coefficients is the nondissipative component of viscous tensor introduced in [1][2][3]. Indications for the existence of another precise transport coefficient appeared recently in connection with the gravitational anomaly found in the context of QHE in Refs. [4][5][6][7][8][9][10][11][12], see also [13].Precise quantization on QH plateaus of the nondissipative transport coefficients can be explained from two points of view. The first, topological explanation is through their relation to topological invariants, such as Chern numbers of vector bundles over the appropriate parameter space [14][15][16]. For example, in the case of the Hall conductance, the parameter space is spanned by Aharonov-Bohm fluxes piercing through the handles of the Riemann surface. For the non-dissipative viscosity the relevant parameter space is the moduli space of complex structures on the torus [1][2][3]. In this paper we show that the third coefficient shows up, when the parameter space is the moduli space of complex structures for the surfaces of genus 2 and higher.For this reason, we discuss the precise transport for QH-states on compact Riemann surfaces. We develop a general method to compute all three transport coeffi-

Odd surface waves in two-dimensional incompressible fluids

“…It is clear from the relation (20) that there is a typical length scale δ = ν e /|Ω|. As the frequency Ω of the surface waves remains finite in the limit ν e → 0 we find that the Bcomponent of the solution (16,17) and vorticity of the fluid (18), in particular, is localized near the surface of the fluid within the dynamic boundary layer of thickness δ. For conventional slightly damped gravity waves (without odd viscosity) Ω ≈ ± g|k| and the existence and structure of such a layer is well known [34,43].…”

Multipole expansion in the quantum hall effect

“…Following the discussion of universality in section 3.4, we are lead to conclude that the Laplacian corrections in the third-order W ∞ action (3.40)-(3.42) and the gravitational Wen-Zee term (3.44) are non-universal. We further remark that the curvature correction ω 0 R in (3.43), not obtained in our approach, is believed to be universal because it is also found in the calculation of the Hall viscosity from the Berry phase of the Laughlin wavefunction in curved backgrounds [46,47].…”

Boundary central charge from bulk odd viscosity: Chiral superfluids

“…Analysis of the problem has previously been carried out only within the relativistic limit of the p-wave CSF, where the non-relativistic kinetic energy of the fermions is neglected [12,25,32,[35][36][37]. Within this limit one finds a bulk gravitational Chern-Simons (gCS) term, which implies a c-dependent correction to η (1) o of (1) at small non-zero wave-vector [37][38][39],…”