Abstract
In this note we study the effects of a magnetic field on transport using holographic models with broken translational invariance. We show that, after carefully subtracting off non-trivial magnetisation currents, it is possible to express the DC transport currents of the boundary theory in terms of properties of a black hole horizon. This allows us to obtain simple analytic expressions for the electrical, thermoelectric and heat conductivity tensors. Our results apply to both isotropic and anisotropic models, including holographic Q-lattices and to certain theories where translational invariance is broken by linear sources for axions.
Highlights
JHEP08(2015)124 for electric Hall transport recently allowed a novel mechanism for obtaining an anomalous scaling of the Hall angle to be identified [18]
Perhaps the most striking aspect of the equations is that, just as in the hydrodynamic analysis of [5], the entire set of DC transport coefficients are described by two parameters
We have defined the timescale τ so that with the above identifications the electric conductivity tensor takes precisely the same form as in [5], where τ −1 corresponded to the momentum dissipation rate the agreement of the electrical conductivity tensor between these two approaches is striking, it does not extend to the thermoelectric response coefficients
Summary
Our goal is to calculate the transport coefficients of simple holographic models in the presence of a magnetic field. As was first realised in [9, 11, 13], the key reason it is possible to calculate transport coefficients in these models is that the currents of the boundary theory can be related to radially-independent quantities in the bulk. I.e. these constant bulk fluxes J i precisely correspond to the transport currents of the boundary theory.. In appendix A, we show that as r → ∞ MQ(r) precisely approaches the heat magnetisation density, MQ = ME − μM , of the dual theory The effect of this additional term is to subtract off the contribution of the magnetisation current from (2.25). The definitions of MQ(r) and M (r) imply that they vanish at r = r+ and so, as for the electrical case, we see that the transport currents can be expressed locally in terms of horizon fields. The heat conductivity, κreads κxx s2T (B2Z + e2V k2Φ) B2ρ2 + (B2Z + e2V k2Φ) r+
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