Holographic Hall conductivities from dyonic backgrounds

Abstract

We develop a general framework for computing the holographic 2-point functions and the corresponding conductivities in asymptotically locally AdS backgrounds with an electric charge density, a constant magentic field, and possibly non-trivial scalar profiles, for a broad class of Einstein-Maxwell-Axion-Dilaton theories, including certain Chern-Simons terms. Holographic renormalization is carried out for any theory in this class and the computation of the renormalized AC conductivities at zero spatial momentum is reduced to solving a single decoupled first order Riccati equation. Moreover, we develop a first order fake supergravity formulalism for dyonic renormalization group flows in four dimensions, allowing us to construct analytically infinite families of such backgrounds by specifying a superpotential at will. These RG flows interpolate between AdS 4 in the UV and a hyperscaling violating Lifshitz geometry in the IR with exponents 1 < z < 3 and θ = z + 1. For 1 < z < 2 the spectrum of fluctuations is gapped and discrete. Our hope and intention is that this analysis can serve as a manual for computing the holographic 1-and 2-point functions and the corresponding transport coefficients in any dyonic background, both in the context of AdS/CMT and AdS/QCD. 1 We write µνρσ = e µ a e ν b e ρ c e σ d abcd with ntxy = 1, wheren is the Lorentz frame index corresponding to the direction orthogonal to the boundary of ∂M in M.

Vanishing DC holographic conductivity from a magnetic monopole condensate

“…As discussed before, for mass functions satisfying the near-boundary asymptotics m( ) ∼ a , with a = 1 or a > 3/2, the asymptotic solution of the equation of motion for the 2-form field near the boundary is the same as the asymptotic solution for the Maxwell field, and one can easily show that the on-shell boundary action (2.22) remains finite in the limit → 0, such that one does not need to resort to the holographic renormalization procedure [23][24][25][26][27] in these cases, i.e, the boundary condition for the massive 2-form ensures that the on-shell action is finite at the boundary just like the Maxwell action is in the same dimensionality. Then, we can immediately obtain from (2.22) and (2.9) the expressions for the retarded Green's function and the associated conductivity, respectively:…”

“…This general analytic result will also be useful as a consistency check of the numerical results for the AC conductivity in section 3. In order to compute the AC conductivity (2.24), we substitute the Ansatz (2.21) into (2.19) and define the following quantity 27) which obeys a first order ordinary differential equation [15] …”

Holographic calculation of the QCD crossover temperature in a magnetic field

“…where S GHY is the Gibbons-Hawking-York action [91,92] needed to establish a well-posed variational problem with Dirichlet boundary condition for the metric, and S CT is the counterterm action that can be constructed using the holographic renormalization procedure [93][94][95][96][97]. These two boundary terms contribute to the total on-shell action but not to the equations of motion and, since we shall not need to compute the total on-shell action in the present work, we do not need to worry about their explicit form here.…”

Thermodynamics of spinningAdS4black holes in gauged supergravity

“…We now proceed to the study of the scalar boundary conditions allowed in our system, in light of the results of [53,54]. There it is shown that holographic renormalization acts as a canonical transformation: the boundary counterterms necessary for a well defined variational principle (and for the removal of divergences) are such that the symplectic map φ * of the modes is diagonal.…”

“…We can then can use the procedure of [53] (see also [54]) and add a canonical counterterm S ct (69) which makes the symplectic map diagonal, giving…”