Viscoelastic response of quantum Hall fluids in a tilted field

In this paper, we examine the viscoelastic properties of integer quantum Hall (IQH) states in a tilted magnetic field. In particular, we explore to what extent the tilted-field system behaves like a two-dimensional electron gas with anisotropic mass in the presence of strain deformations. We first review the Kubo formalism for viscosity in an external magnetic field, paying particular attention to the role of rotational symmetry and contact terms. Next, we compute the conductivity, stress, and viscosity tensors for IQH states in the presence of a tilted field and vertical confining potential. By comparing our results with the recently developed bimetric formalism, we show that, at the level of the contracted Hall viscosity tensor, the mapping between tilted field and effective mass anisotropy holds only if we simultaneously modify the background perpendicular magnetic field; in other words, a simultaneous measurement of the density, contracted Hall viscosity, and Hall conductivity at fixed particle number can distinguish between tilted field and effective mass anisotropy. Additionally, we show that in the presence of a tilted magnetic field, the stress tensor acquires an unusual anisotropic ground state average, leading to anomalous elastic response functions. We develop a formalism for projecting a three-dimensional Hamiltonian with confining potential and magnetic field to a two-dimensional Hamiltonian in order to further address the phenomenology of the tilted-field IQH fluid. We find that the projected fluid couples nonminimally to geometric deformations, indicating the presence of internal geometric degrees of freedom.

Figure 1

Setup for the tilted-field system. We consider electrons (black circles) in a quantum well (blue region) moving in the background of a perpendicular magnetic field $B_z$, along with an in-plane field $B_x$. The electrons are confined to the quantum well by a harmonic potential $V=1/2m{\omega }_0^2{z}^2$, depicted on the left.

Figure 2

Schematic of the results of the projection procedure for the tilted-field system. The effective two-dimensional system is composed of electrons (black ovals) in a true two-dimensional system (blue region), moving in background of a perpendicular magnetic field $B_z$. Each electron has a tower of “internal” states corresponding to the energy levels of the $Y$ harmonic oscillator of Sec. 4, which we project into the ground state.