We study a holographic superconductor model with momentum relaxation due to massless scalar fields linear to spatial coordinates(ψ I = βδ Ii x i ), where β is the strength of momentum relaxation. In addition to the original superconductor induced by the chemical potential(µ) at β = 0, there exists a new type of superconductor induced by β even at µ = 0. It may imply a new 'pairing' mechanism of particles and antiparticles interacting with β, which may be interpreted as 'impurity'. Two parameters µ and β compete in forming superconducting phase. As a result, the critical temperature behaves differently depending on β/µ. It decreases when β/µ is small and increases when β/µ is large, which is a novel feature compared to other models. After analysing ground states and phase diagrams for various β/µ, we study optical electric(σ), thermoelectric(α), and thermal(κ) conductivities. When the system undergoes a phase transition from normal to a superconducting phase, 1/ω pole appears in the imaginary part of the electric conductivity, implying infinite DC conductivity. If β/µ < 1, at small ω, a two-fluid model with an imaginary 1/ω pole and the Drude peak works for σ, α, andκ, but If β/µ > 1 a non-Drude peak replaces the Drude peak. It is consistent with the coherent/incoherent metal transition in its metal phase. The Ferrell-Glover-Tinkham (FGT) sum rule is satisfied for all cases even when µ = 0.

We extend the recent D = 5 results of Dias, Horowitz and Santos by finding asymptotically AdS rotating black hole and boson star solutions with scalar hair in arbitrary odd spacetime dimension. Both the black holes and the boson stars are invariant under a single Killing vector field which corotates with the scalar field and, in the black hole case, is tangent to the generator of the horizon. Furthermore, we explicitly construct boson star and small black hole (r + ≪ ℓ) solutions perturbatively assuming a small amplitude for the scalar field, resulting in solutions valid for low energies and angular momenta. We find that just as in D = 5, the angular momentum is primarily carried by the scalar field in D > 5, whereas unlike D = 5 the energy is also primarily carried by the scalar field in D > 5; the thermodynamics in D = 5 are governed by both the black hole and scalar field whereas in D > 5 they are governed primarily by the scalar field alone. We focus on cataloguing these solutions for the spacetime dimensions of interest in string theory, namely D = 5, 7, 9, 11.

Abstract:We study three properties of a holographic superconductor related to conductivities, where momentum relaxation plays an important role. First, we find that there are constraints between electric, thermoelectric and thermal conductivities. The constraints are analytically derived by the Ward identities regarding diffeomorphism from field theory perspective. We confirm them by numerically computing all two-point functions from holographic perspective. Second, we investigate Homes' law and Uemura's law for various high-temperature and conventional superconductors. They are empirical and (material independent) universal relations between the superfluid density at zero temperature, the transition temperature, and the electric DC conductivity right above the transition temperature. In our model, it turns out that the Homes' law does not hold but the Uemura's law holds at small momentum relaxation related to coherent metal regime. Third, we explicitly show that the DC electric conductivity is finite for a neutral scalar instability while it is infinite for a complex scalar instability. This shows that the neutral scalar instability has nothing to do with superconductivity as expected.

We examine the concept of black hole thermodynamic volume and its consistency with thermodynamic mass in spacetimes that are not asymptotically flat but instead have anisotropic Lifshitz scaling symmetry. We find that the generalized Smarr relation in anti de Sitter spaceextended to include a pressure-volume term -holds here as well, and that there exists a definition of thermodynamic mass and thermodynamic volume that satisfy both this relation and the 1 st law of thermodynamics. We compare the thermodynamic mass with other known quantities such as Arnowitt-Deser-Misner, Brown-York and Hollands-Ishibashi-Marolf masses. We also conjecture methods for obtaining a thermodynamic mass where there is ambiguity due to the cosmological constant lengthscale depending on the horizon radius lengthscale. I. BACKGROUNDGauge-gravity duality remains a subject of considerable interest, in large part because of the insights it yields into quantum gravity. Asymptotically anti de Sitter (AdS) spacetime admits a strongly coupled gauge theory description at its boundary via a holographic dictionary. It is straightforward to define thermodynamic equilibrium in this case, in turn giving rise to thermal radiation/large AdS black hole phase transitions [1].An interesting development in this subject that has been the subject of much current interest is the proposal that the mass of an AdS black hole can be understood as the enthalpy of spacetime [2]. This notion emerges from regarding the cosmological constant Λ as a thermodynamic variable [3] analogous to pressure in the first law [2,[4][5][6][7][8][9][10][11][12][13][14][15][16][17][18], along with a notion of conjugate volume [2,12]. A complete analogy between 4-dimensional Reissner-Nordström AdS black holes and the Van der Waals liquid-gas system can be shown to hold [13]. The critical exponents are the same as those in the Van der Waals system, modifying previous considerations that emerged from earlier studies [19][20][21][22] of the duality description. Intensive investigation in a broad variety of contexts [23-37] has led to the discovery of a variety of new thermodynamic phenomena for both AdS and de Sitter [38] black holes, including the existence of reentrant phase transitions in Born-Infeld [14] and rotating [39] black holes, the existence of a tricritical point in rotating black holes analogous to the triple point in water [39], a new type of thermodynamic criticality in the highercurvature case [40], and the notion of a holographic heat engine [37, 41]. Indeed, the thermal radiation/large AdS black hole phase transition [1] can be understood as a solid/liquid phase transition from this perspective [34].Here we begin the first study of extended thermodynamics in the context of Lifshitz duality. Motivated by * Electronic address: wbrenna@uwaterloo.ca † Electronic address: rbmann@uwaterloo.ca ‡ Electronic address: miokpark@kias.re.kr the hope of obtaining a duality between condensed matter physics with quantum criticality, the anisotropic scaling properties of these systems im...

We study dynamical condensation process in a holographic superconductor model with anisotropy. The time-dependent numerical solution is constructed for the Einstein-Maxwell-dilaton theory with complex scalar in asymptotic AdS spacetime. The introduction of dilaton field generates the anisotropy in boundary spatial directions. In analogy of isotropic case, we have two black hole solutions below certain critical temperature T c , the anisotropic charged black hole with and without scalar hair, corresponding respectively to the supercooled normal phase and superconducting phase in the boundary theory. We observe a nonlinear evolution from a supercooled anisotropic black hole without scalar hair to a anisotropic hairy black hole. Via AdS/CFT correspondence, we extract time evolution of the condensate operator, which shows an exponential growth and subsequent saturation, similar to the isotropic case. Furthermore, we obtain a nontrivial time evolution of the boundary pressure, while in isotropic case it remains a constant. We also generalize quasinormal modes calculation to anisotropic black holes and shows scalar quasinormal modes match with relaxation time scale of the condensate operator. In addition, we present the final temperature and anisotropic pressure as functions of initial temperature and background anisotropy.