Kubo formulas for viscosity: Hall viscosity, Ward identities, and the relation with conductivity

By Barry Bradlyn, M. Goldstein, N. Read. 2012

223 citations recorded by the source corpus.

We derive from first principles the Kubo formulas for the stress-stress response function at zero wavevector that can be used to define the full complex frequency-dependent viscosity tensor, both with and without a uniform magnetic field. The formulas in the existing literature are frequently incomplete, incorrect, or lack a derivation; in particular, Hall viscosity is overlooked. Our approach begins from the response to a uniform external strain field, which is an active time-dependent coordinate transformation in d space dimensions. These transformations form the group GL(d,R) of invertible matrices, and the infinitesimal generators are called strain generators. These enable us to express the Kubo formula in different ways, related by Ward identities; some of these make contact with the adiabatic transport approach. For Galilean-invariant systems, we derive a relation between the stress response tensor and the conductivity tensor that is valid at all frequencies and in both the presence and absence of a magnetic field. In the presence of a magnetic field and at low frequency, this yields a relation between the Hall viscosity, the q^2 part of the Hall conductivity, the inverse compressibility (suitably defined), and the diverging part of the shear viscosity (if any); this relation generalizes a result found recently. We show that the correct value of the Hall viscosity at zero frequency can be obtained (at least in the absence of low-frequency bulk and shear viscosity) by assuming that there is an orbital spin per particle that couples to a perturbing electromagnetic field as a magnetization per particle. We study several examples as checks on our formulation.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics

View Paper: https://www.semanticscholar.org/paper/541d82521327b4d055036ba13fd18ca73d676afd

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In Atomic and Molecular Physics, and Optics

In Condensed Matter Physics

Citations in the corpus

Torsional anomalies, Hall viscosity, and bulk-boundary correspondence in topological states (2012, background): “QHE with rotation[8, 9] and translation invariance[10], and chiral superconductors[8, 11].”
Quantum Hall physics : Hierarchies and conformal field theory techniques (2016, methodology): “d good agreement with (86). They also checked the stability of the result for the = 1=2 state by adding a perturbation to the exact model Hamiltonian. It was pointed out by Hoyos and Son (2012) and Bradlyn et al. (2012) that { asuming Galilean invariance { the Hall viscosity can be extracted from t”
Novel effects of strains in graphene and other two dimensional materials (2015, methodology): “In a similar way as done by Luttinger with the heat transport, the metric is used more recently to analyse properties such as the Hall viscosity [15] in topological insulators and in quantum Hall fluids [32, 146, 150, 151].”