Abstract:
We study the properties of the Schr\"odinger-type non-relativistic holography for general dynamical exponent z with and without hyperscaling violation exponent \theta. The scalar correlation function has a more general form due to general z as well as the presence of \theta, whose effects also modify the scaling dimension of the scalar operator. We propose a prescription for minimal surfaces of this "codimension 2 holography," and demonstrate the (d-1) dimensional area law for the entanglement entropy from (d+3) dimensional Schr\"odinger backgrounds. Surprisingly, the area law is violated for d+1 < z < d+2, even without hyperscaling violation, which interpolates between the logarithmic violation and extensive volume dependence of entanglement entropy. Similar violations are also found in the presence of the hyperscaling violation. Their dual field theories are expected to have novel phases for the parameter range, including Fermi surface. We also analyze string theory embeddings using non-relativistic branes.

Abstract:
We investigate systematic classifications of low energy and lower dimensional effective holographic theories with Lifshitz and Schr\"odinger scaling symmetries only using metrics in terms of hyperscaling violation ($\theta$) and dynamical ($z$) exponents. Their consistent parameter spaces are constrained by null energy and positive specific heat conditions, whose validity is explicitly checked against a previously known result. From dimensional reductions of many microscopic string solutions, we observe the classifications are tied with the number of scales in the original microscopic theories. Conformal theories do not generate a nontrivial $\theta$ for a simple sphere reduction. Theories with Lifshitz scaling with one scale are completely fixed by $\theta$ and $z$, and have a universal emblackening factor at finite temperature. Dimensional reduction of intersecting M2-M5 requires, we call, spatial anisotropic exponents ($\sharp$), along with $z$=1, $\theta$=0, because of another scale. Theories with Schr\"odinger scaling show similar simple classifications at zero temperature, while require more care due to an additional parameter being a thermodynamic variable at finite temperature.

Abstract:
Properties of Schroedinger black holes are derived from AdS black holes expressed in light-cone coordinates with a particular normalization. The advantages of this method over the direct analysis of the Schroedinger geometry are the simplicity and the well-defined Brown-York procedure with the standard counterterms. The method is demonstrated for several physical interests, including the computation of the DC conductivity and the derivation of the R-charged black hole thermodynamic properties.

Abstract:
We briefly explain the consistency conditions imposed on the effective holographic theories, which are parameterized by two real exponents $(\gamma,\delta)$ that control the IR dynamics. The general scaling of DC resistivity with temperature at low temperature and AC conductivity at low frequency across the whole $(\gamma,\delta)$ plane are explained. There is a codimension-one region where the DC resistivity is linear in the temperature. For massive carriers, it is shown that when the scalar operator is not the dilaton, the DC resistivity scales as the heat capacity (and entropy) for $(2+1)$-dimensional systems. Regions are identified where the theory at finite density is a Mott-like insulator. This contribution is based on arXiv:1005.4690 with emphasis on the transport properties of charged dilatonic black holes with potential.

Abstract:
We review a recent holographic analysis arXiv:1005.4690 of charged black holes with scalar hair in view of their applications to the cuprate high temperature superconductors. We show in particular that these black holes show an interesting phase structure including critical behaviour at zero temperature or charge, describe both conductors and insulators (including holographic Mott-like insulators), generically have no residual entropy and exhibit experimentally observed scaling relations between electronic entropy, specific heat and (linear) DC resistivity. Transport properties are discussed in the companion contribution to these proceedings.

Abstract:
We consider a covariant formulation of field theories with Lifshitz scaling, and analyze the energy-momentum tensor and the scale symmetry Ward identity. We derive the equation of state and the ideal Lifshitz hydrodynamics in agreement with arXiv:1304.7481, where they were determined by using thermodynamics and symmetry properties. We construct the charged ideal Lifshitz hydrodynamics in the generating functional framework as well as in the gravitational holographic dual description. At the first viscous order, an analysis of the entropy current reveals two additional transport coefficients (one dissipative and one dissipationless) compared to the neutral case, contributing to the charge current and to the asymmetric part of the energy-momentum tensor.

Abstract:
We use the Ward identities corresponding to general linear transformations, and derive relations between transport coefficients of $(2+1)$-dimensional systems. Our analysis includes relativistic and Galilean invariant systems, as well as systems without boost invariance such as Lifshitz theories. We consider translation invariant, as well as broken translation invariant cases, and include an external magnetic field. Our results agree with effective theory relations of incompressible Hall fluid, and with holographic calculations in a magnetically charged black hole background.

Abstract:
We derive a generalized set of Ward identities that captures the effects of topological charge on 2+1 dimensional Hall transport. The Ward identities follow from the momentum algebra, which includes a central extension proportional to the topological charge density. In the presence of topological objects like baby Skyrmions, the central term leads to a direct relation between the thermal Hall conductivity and the charge density. This relation is further extended in the presence of a magnetic field and a conserved current. The topological charge density produces a distinct signature in the electric Hall conductivity, which is identified in existing experimental data, and yields further novel predictions. For isolated Skyrmions in insulating materials, the Hall viscosity can be expressed in terms of the charge density, thermal Hall conductivity and angular momentum, and could be determined by measuring those quantities.

Abstract:
We compute the bulk viscosity in holographic models dual to theories with Lifshitz scaling and/or hyperscaling violation, using a generalization of the bulk viscosity formula derived in arXiv:1103.1657 from the null focusing equation. We find that only a class of models with massive vector fields are truly Lifshitz scale invariant, and have a vanishing bulk viscosity. For other holographic models with scalars and/or massless vector fields we find a universal formula in terms of the dynamical exponent and the hyperscaling violation exponent.

Abstract:
We derive quantum field theory Ward identities based on linear area preserving and conformal transformations in 2+1 dimensions. The identities relate Hall viscosities, Hall conductivities and the angular momentum. They apply both for relativistic and non relativistic systems, at zero and at finite temperature. We consider systems with or without translation invariance, and introduce an external magnetic field and viscous drag terms. A special case of the identities yields the well known relation between the Hall conductivity and half the angular momentum density.