Abstract
We use the Ward identities corresponding to general linear transformations, and derive relations between transport coefficients of (2 + 1)-dimensional systems. Our analysis includes relativistic and Galilean invariant systems, as well as systems without boost invariance such as Lifshitz theories. We consider translation invariant, as well as broken translation invariant cases, and include an external magnetic field. Our results agree with effective theory relations of incompressible Hall fluid, and with holographic calculations in a magnetically charged black hole background.
Highlights
An outline of the paper is the following: starting from general linear transformations, we derive a Ward identity (2.13) in terms of retarded correlators and equal time commutators
After evaluating the equal time commutators, the Ward identity relates the retarded correlators to one point functions (2.18)
When translation invariance is not broken the Ward identity becomes (2.23). This Ward identity can be derived from the conservation equation of the energy-momentum tensor, so it has appeared in other forms in the literature before
Summary
In a Galilean invariant system one can derive general relations between viscosities and conductivities by acting with time derivatives on correlators of the current [2]. Before introducing any explicit form for the correlators, the relations that will be derived from the Ward identities are completely general, and will apply to any field theory, independently of whether it has an effective fluid description or not. They can be used for states with broken translation invariance, that at low energies can behave as elastic media. We will make use of the fact that TB is the density for the generator of translations, and write the following operator identity for the equal time commutators with TB i[TB0i(x), O(x)] = ∂iO(x)δx ,. In order to define the viscosity tensor, these terms are subtracted from the correlator [2, 8]
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