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.

We study the elastic response of planar black hole (BH) solutions in a simple class of holographic models with broken translational invariance. We compute the transverse quasi-normal mode spectrum and the propagation speed of the lowest energy mode. We find that the speed of the lowest mode relates to the BH rigidity modulus as dictated by elasticity theory. This allows to identify these modes as transverse phonons — the pseudo Goldstone bosons of spontaneously broken translational invariance. In addition, we show that these modes have a mass gap controlled by an explicit source of the translational symmetry breaking. These results provide a new confirmation that the BHs in these models do exhibit solid properties that become more manifest at low temperatures. Also, by the AdS/CFT correspondence, this allows to extend the standard results from the effective field theory for solids to quantum-critical materials.

We provide a complete and unified description of translational symmetry breaking in a simple holographic model. In particular, we focus on the distinction and the interplay between explicit and spontaneous breaking. We consider a class of holographic massive gravity models which allow to range continuously from one situation to the other. We study the collective degrees of freedom, the electric AC conductivity and the shear correlator in function of the explicit and spontaneous scales. We show the possibility of having a sound-to-diffusion crossover for the transverse phonons. Within our model, we verify the validity of the Gell-Mann-Oakes-Renner relation. Despite of strong evidence for the absence of any standard dislocation induced phase relaxation mechanism, we identify a novel relaxation scale controlled by the ratio between the explicit and spontaneous breaking scales. Finally, in the pseudo-spontaneous limit, we prove analytically the relation, which has been discussed in the literature, between this novel relaxation scale, the mass of the pseudo-phonons and the Goldstone diffusivity. Our numerical data confirms this analytic result.

We compute quasinormal modes (QNMs) of the metric and gauge field perturbations about black branes electrically and magnetically charged in the Einstein-Maxwell-Chern-Simons theory. By the gauge/gravity correspondence, this theory is dual to a particular class of field theories with a chiral anomaly, in a thermal charged plasma state subjected to a constant external magnetic field, B. The QNMs are dual to the poles of the two-point functions of the energy-momentum and axial current operators, and they encode information about the dissipation and transport of charges in the plasma. Complementary to the gravity calculation, we work out the hydrodynamic description of the dual field theory in the presence of a chiral anomaly, and a constant external B. We find good agreement with the weak field hydrodynamics, which can extend beyond the weak B regime into intermediate regimes. Furthermore, we provide results that can be tested against thermodynamics and hydrodynamics in the strong B regime. We find QNMs exhibiting Landau level behavior, which become long-lived at large B if the anomaly coefficient exceeds a critical magnitude. Chiral transport is analyzed beyond the hydrodynamic approximation for the five (formerly) hydrodynamic modes, including a chiral magnetic wave.

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.

We construct the general hydrodynamic description of (3+1)-dimensional chiral charged (quantum) fluids subject to a strong external magnetic field with effective field theory methods. We determine the constitutive equations for the energy-momentum tensor and the axial charge current, in part from a generating functional. Furthermore, we derive the Kubo formulas which relate two-point functions of the energy-momentum tensor and charge current to 27 transport coefficients: 8 independent thermodynamic, 4 independent non-dissipative hydrodynamic, and 10 independent dissipative hydrodynamic transport coefficients. Five Onsager relations render 5 more transport coefficients dependent. We uncover four novel transport effects, which are encoded in what we call the shear-induced conductivity, the two expansion-induced longitudinal conductivities and the shear-induced Hall conductivity. Remarkably, the shear-induced Hall conductivity constitutes a novel non-dissipative transport effect. As a demonstration, we compute all transport coefficients explicitly in a strongly coupled quantum fluid via holography.