We present a class of holographic massive gravity models that realize a spontaneous breaking of translational symmetry-they exhibit transverse phonon modes whose speed relates to the elastic shear modulus according to elasticity theory. Massive gravity theories thus emerge as versatile and convenient theories to model generic types of translational symmetry breaking: explicit, spontaneous, and a mixture of both. The nature of the breaking is encoded in the radial dependence of the graviton mass. As an application of the model, we compute the temperature dependence of the shear modulus and find that it features a glasslike melting transition.

The Goldstone theorem implies the appearance of an ungapped mode whenever a continuous global symmetry is spontaneously broken. In general it does not say anything about the precise form of the dispersion relation nor does it imply that there is one massless mode for each broken symmetry generator. It is a well-established fact that even for relativistic field theories in the presence of a chemical potential Goldstone modes with quadratic dispersion relation, the type II Goldstone bosons, appear in the spectrum. We develop two holographic models that feature type II Goldstone modes as part of the quasinormal mode spectrum. The models are based on simple generalizations with U (2) symmetry of the well-studied holographic s-wave superfluid. Our results include Goldstone modes without broken generators but with unusual realization of symmetries and a frequency dependent conductivity of striking resemblance to the one of Graphene.

We study the magnetoresponse with nonconserved currents in holography. Nonconserved currents are dual to massive vector fields in anti-de Sitter (AdS). We introduce the mass in a gauge invariant way via the Stückelberg mechanism. In particular we find generalizations of the chiral magnetic effect, the chiral separation effect and the chiral magnetic wave. Since the associated charge is not conserved we need to source it explicitly by a coupling, the generalization of the chemical potential. In this setup we find that in general the anomalous transport phenomena are still realized. The values we find for nonzero mass connect continuously to the values of the anomalous conductivities of the consistent currents, i.e. the proper chiral magnetic effect vanishes for all masses (as it does for the consistent current in the zero mass case) whereas the chiral separation effect is fully present. The generalization of the chiral magnetic wave shows that for small momenta there is no propagating wave but two purely absorptive modes (one of them diffusive). At higher momenta we recover the chiral magnetic wave as a combination of the two absorptive modes. We also study the negative magneto resistivity and find that it grows quadratically with the magnetic field. The chiral magnetic wave and the negative magneto resistivity are manifestations of the chiral magnetic effect that takes place when the (nonconserved) charge is allowed to fluctuate freely in contrast to the case where the charge is fixed by an explicit source. Since the (classical) Uð1Þ A symmetry of QCD is not at all a symmetry at the quantum level we also argue that using massive vectors in AdS to model the axial singlet current might result in a more realistic holographic model of QCD and should be a good starting point to investigate the dynamics of anomalous transport in the strongly coupled quark gluon plasma.

Abstract:We study the response of the chiral magnetic effect due to continuous quenches induced by time dependent electric fields within holography. Concretely, we consider a holographic model with dual chiral anomaly and compute the electric current parallel to a constant, homogeneous magnetic field and a time dependent electric field in the probe approximation. We explicitly solve the PDEs by means of pseudospectral methods in spatial and time directions and study the transition to an universal "fast" quench response. Moreover, we compute the amplitudes, i.e., residues of the quasi normal modes, by solving the (ODE) Laplace transformed equations. We investigate the possibility of considering the asymptotic growth rate of the amplitudes as a well defined notion of initial time scale for linearized systems. Finally, we highlight the existence of Landau level resonances in the electrical conductivity parallel to a magnetic field at finite frequency and show explicitly that these only appear in presence of the anomaly. We show that the existence of these resonances induces, among others, a long-lived AC electric current once the electric field is switched off.

Abstract:We revisit the question of stability of holographic superfluids with finite superfluid velocity. Our method is based on applying the Landau criterion to the Quasinormal Mode (QNM) spectrum. In particular we study the QNMs related to the Goldstone modes of spontaneous symmetry breaking with linear and quadratic dispersions. In the linear case we show that the sound velocity becomes negative for large enough superfluid velocity and that the imaginary part of the quasinormal frequency moves to the upper half plane. Since the instability is strongest at finite wavelength, we take this as an indication for the existence of an inhomogeneous or striped condensed phase for large superfluid velocity. In the quadratic case the instability is present for arbitrarily small superfluid velocity.

Abstract:We study the magnetoconductivity induced by the axial anomaly via the chiral magnetic effect in strongly coupled holographic models. An important ingredient in our models is that the axial charge is non-conserved beyond the axial anomaly. We achieve this either by explicit symmetry breaking via a non-vanishing non-normalisable mode of an axially charged scalar or using a Stückelberg field to make the AdS-bulk gauge field massive. The DC magnetoconductivites can be calculated analytically. They take a universal form in terms of gauge field mass at the horizon and quadratic dependence on the magnetic field. The axial charge relaxation time grows linearly with magnetic field in the large B regime. Most strikingly positive magnetoconductivity is still present even when the relaxation times are short τ 5 ≈ 1/(πT ) and the axial charge can not be thought of as an approximate symmetry. In the U(1) A explicit breaking model, we also observe that the chiral separation conductivity and the axial magnetic conductivity for the consistent axial current vanish in the limit of strong symmetry breaking.

We study the phase diagram of a holographic model realizing a U (2) global symmetry on the boundary and show that at low temperature a phase with both scalar s and vector p condensates exists. This is the s+p-wave phase where the global U (2) symmetry and also the spatial rotational symmetry are spontaneously broken. By studying the free energy we show that this phase is preferred when it exists. We also consider unbalanced configurations where a second chemical potential is turned on. They present a rich phase diagram characterized by the competition and coexistence of the s and p order parameters.

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