Abstract
Hall viscosity, also known as the Lorentz shear modulus, has been proposed as a topological property of a quantum Hall fluid. Using a recent formulation of the composite fermion theory on the torus, we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form $\nu=n/(2pn\pm 1)$, where $n$ and $p$ are integers, from the explicit wave functions for these states. The calculated Hall viscosities $\eta^A$ agree with the expression $\eta^A=(\hbar/4) {\cal S}\rho$, where $\rho$ is the density and ${\cal S}=2p\pm n$ is the "shift" in the spherical geometry. We discuss the role of modular invariance of the wave functions, of the center-of-mass momentum, and also of the lowest-Landau-level projection. Finally, we show that the Hall viscosity for $\nu={n\over 2pn+1}$ may be derived analytically from the microscopic wave functions, provided that the overall normalization factor satisfies a certain behavior in the thermodynamic limit. This derivation should be applicable to a class of states in the parton construction, which are products of integer quantum Hall states with magnetic fields pointing in the same direction.
Highlights
The extreme precision of the quantization of the Hall resistance in the integer quantum Hall effect [1] led to a topological interpretation in terms of Chern numbers [2]
Using a recent formulation of the composite fermion theory on the torus, we evaluate the Hall viscosities for a large number of fractional quantum Hall states at filling factors of the form ν = n/(2pn ± 1), where n and p are integers, from the explicit wave functions for these states
We discuss the modular covariance of the Jain states, which is an important requirement from legitimate wave functions in the torus geometry as well as crucial for the evaluation of the Hall viscosity
Summary
The extreme precision of the quantization of the Hall resistance in the integer quantum Hall effect [1] led to a topological interpretation in terms of Chern numbers [2]. The Hall viscosity is by extension believed to be a topological quantum number for a given FQH state. In some sense, it is a manifestation of the orbital spin through a transport coefficient. We discuss the modular covariance of the Jain states, which is an important requirement from legitimate wave functions in the torus geometry as well as crucial for the evaluation of the Hall viscosity. Hall viscosity at ν n 2 pn+1 is given by Eq (10), provided that we assume that the overall normalization factor of the product wave functions has a certain behavior in the thermodynamic limit; this assumption is tested numerically.
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