Abstract:The holographic Weyl semimetal is a model of a strongly coupled topological semi-metal. A topological quantum phase transition separates a topological phase with non-vanishing anomalous Hall conductivity from a trivial state. We investigate how this phase transition depends on the parameters of the scalar potential (mass and quartic self coupling) finding that the quantum phase transition persists for a large region in parameter space. We then compute the axial Hall conductivity. The algebraic structure of the axial anomaly predicts it to be 1/3 of the electric Hall conductivity. We find that this holds once a non-trivial renormalization effect on the external axial gauge fields is taken into account. Finally we show that the phase transition also occurs in a top-down model based on a consistent truncation of type IIB supergravity.

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We show that 3D Lifshitz fermions arising as the critical theory at the Weyl semimetal/insulator transition naturally develop an anomalous Hall viscosity at finite temperature. We discuss how to couple the system to non-relativistic background sources for stress-tensor and momentum currents via a form of Newton-Cartan geometry with torsion and derive the Kubo formulas for the Hall viscosities. While the Lifshitz system that arises most naturally has scaling exponent z = 2 we also generalize the theory for arbitrary Lifshitz scaling z and show that, in the limit z → 0, it may be given a Chern-Simons interpretation by dimensionally reducing along the anisotropic direction.The Hall viscosities are expressed in terms of zeta functions and their temperature dependence is dictated by the scaling exponent.

We consider toy models of holography arising from 3d Chern-Simons theory. In this context a duality to an ensemble average over 2d CFTs has been recently proposed. We put forward an alternative approach in which, rather than summing over bulk geometries, one gauges a one-form global symmetry of the bulk theory. This accomplishes two tasks: it ensures that the bulk theory has no global symmetries, as expected for a theory of quantum gravity, and it makes the partition function on spacetimes with boundaries coincide with that of a modular-invariant 2d CFT on the boundary. In particular, on wormhole geometries one finds a factorized answer for the partition function. In the case of non-Abelian Chern-Simons theories, the relevant one-form symmetry is non-invertible, and its "gauging" corresponds to the condensation of a Lagrangian anyon.the lack of factorization and lead to well-defined quantum mechanical systems with discrete spectrum.3 Besides, in both cases there usually are 0-form symmetries as well.gauging for non-invertible symmetries. In both cases, after gauging, the bulk Chern-Simons theory acquires the following pleasant properties:1) The theory becomes trivial in the bulk, in the sense that the Euclidean partition function on any (oriented) closed 3-manifold equals 1. After all, this is what we expect from a holographic theory: the degrees of freedom only live at the boundary.2) Given a (possibly disconnected) 2d boundary and a boundary condition, the Euclidean partition function on any 3-manifold with that boundary gives the same result (this follows from point 1). Chern-Simons theory is a generally-covariant theory [37,38], in the sense that its partition function on closed 3-manifolds does not depend on a choice of metric -but it does depend on the topology. After gauging, the theory becomes independent from the bulk topology as well.3) The partition function of the gravitational theory is defined as the one on an arbitrary 3-manifold with the given boundary conditions (not as a sum over all possible geometries, or some subset thereof), because of point 2). Factorization in the case of disconnected boundaries immediately follows.4) The partition function with boundary conditions equals the partition function of a single and well-defined boundary CFT. The details of such a CFT are encoded in the bulk gauge group and in the specific chosen gauging of the 1-form symmetry.5) What we described extends to correlation functions of the boundary theory, and hinges on the fact that all lines are transparent in the bulk (because of point 1).