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Quantum anomalies give rise to new transport phenomena. In particular, a magnetic field can induce an anomalous current via the chiral magnetic effect and a vortex in the relativistic fluid can also induce a current via the chiral vortical effect. The related transport coefficients can be calculated via Kubo formulas. We evaluate the Kubo formula for the anomalous vortical conductivity at weak coupling and show that it receives contributions proportional to the gravitational anomaly coefficient. The gravitational anomaly gives rise to an anomalous vortical effect even for an uncharged fluid.

Chiral anomalies give rise to dissipationless transport phenomena such as the chiral magnetic and vortical effects. In these notes I review the theory from a quantum field theoretic, hydrodynamic and holographic perspective. A physical interpretation of the otherwise somewhat obscure concepts of consistent and covariant anomalies will be given. Vanishing of the CME in strict equilibrium will be connected to the boundary conditions in momentum space imposed by the regularization. The role of the gravitational anomaly will be explained. That it contributes to transport in an unexpectedly low order in the derivative expansion can be easiest understood via holography. Anomalous transport is supposed to play also a key role in understanding the electronics of advanced materials, the Dirac-and Weyl (semi)metals. Anomaly related phenomena such as negative magnetoresistivity, anomalous Hall effect, thermal anomalous Hall effect and Fermi arcs can be understood via anomalous transport. Finally I briefly review a holographic model of Weyl semimetal which allows to infer a new phenomenon related to the gravitational anomaly: the presence of odd viscosity.

We analyze a holographic model with a pure gauge and a mixed gaugegravitational Chern-Simons term in the action. These are the holographic implementations of the usual chiral and the mixed gauge-gravitational anomalies in four dimensional field theories with chiral fermions. We discuss the holographic renormalization and show that the gauge-gravitational Chern-Simons term does not induce new divergences. In order to cancel contributions from the extrinsic curvature at a boundary at finite distance a new type of counterterm has to be added however. This counterterm can also serve to make the Dirichlet problem well defined in case the gauge field strength vanishes on the boundary. A charged asymptotically AdS black hole is a solution to the theory and as an application we compute the chiral magnetic and chiral vortical conductivities via Kubo formulas. We find that the characteristic term proportional to T 2 is present also at strong coupling and that its numerical value is not renormalized compared to the weak coupling result.

We show that, as it happens in graphene, elastic deformations couple to the electronic degrees of freedom as pseudo gauge fields in Weyl semimetals. We derive the form of the elastic gauge fields in a tight-binding model hosting Weyl nodes and see that this vector electron-phonon coupling is chiral, providing an example of axial gauge fields in three dimensions. As an example of the new response functions that arise associated to these elastic gauge fields, we derive a non-zero phonon Hall viscosity for the neutral system at zero temperature. The axial nature of the fields provides a test of the chiral anomaly in high energy with three axial vector couplings. PACS numbers:Introduction.-The occurrence of Weyl fermions (massless Dirac fermions of definite chirality) in condensed matter has come always with unexpected phenomena and new physics. Although having a long tradition [1], the best examples so far arose in one spacial dimension (Luttinger liquids) [2] or in two (Graphene [3] and the surface of three dimensional topological insulators [4]). Charged massless fermions are particularly interesting in three dimensions: They do not have counterparts in particle physics and they experience the chiral anomaly [5][6][7][8][9] and its related physical responses.

The conservation laws, such as those of charge, energy and momentum, have a central role in physics. In some special cases, classical conservation laws are broken at the quantum level by quantum fluctuations, in which case the theory is said to have quantum anomalies. One of the most prominent examples is the chiral anomaly, which involves massless chiral fermions. These particles have their spin, or internal angular momentum, aligned either parallel or antiparallel with their linear momentum, labelled as left and right chirality, respectively. In three spatial dimensions, the chiral anomaly is the breakdown (as a result of externally applied parallel electric and magnetic fields) of the classical conservation law that dictates that the number of massless fermions of each chirality are separately conserved. The current that measures the difference between left- and right-handed particles is called the axial current and is not conserved at the quantum level. In addition, an underlying curved space-time provides a distinct contribution to a chiral imbalance, an effect known as the mixed axial-gravitational anomaly, but this anomaly has yet to be confirmed experimentally. However, the presence of a mixed gauge-gravitational anomaly has recently been tied to thermoelectrical transport in a magnetic field, even in flat space-time, suggesting that such types of mixed anomaly could be experimentally probed in condensed matter systems known as Weyl semimetals. Here, using a temperature gradient, we observe experimentally a positive magneto-thermoelectric conductance in the Weyl semimetal niobium phosphide (NbP) for collinear temperature gradients and magnetic fields that vanishes in the ultra-quantum limit, when only a single Landau level is occupied. This observation is consistent with the presence of a mixed axial-gravitational anomaly, providing clear evidence for a theoretical concept that has so far eluded experimental detection.

Experiments on the sensitization of guinea pigs with simple chemical compounds are described. Positive effects were obtained by the administration of small quantities, namely fractions of milligrams, with 1:2:4 chlorodinitrobenzene, p-nitrosodimethylaniline, 1:2:4 trinitrobenzene, picryl chloride, four dichlorodinitrobenzenes, and a number of other aromatic compounds. Several substances chemically similar to those enumerated gave negative results. The first named compound is known to produce hypersensitiveness in human beings, a large number of cases having been observed in factory workers. The mechanism of these effects is discussed.

We study the poles of the retarded Green functions of a holographic superconductor. The model shows a second order phase transition where a charged scalar operator condenses and a U (1) symmetry is spontaneously broken. The poles of the holographic Green functions are the quasinormal modes in an AdS black hole background. We study the spectrum of quasinormal frequencies in the broken phase, where we establish the appearance of a massless or hydrodynamic mode at the critical temperature as expected for a second order phase transition. In the broken phase we find the pole representing second sound. We compute the speed of second sound and its attenuation length as function of the temperature. In addition we find a pseudo diffusion mode, whose frequencies are purely imaginary but with a non-zero gap at zero momentum. This gap goes to zero at the critical temperature. As a technical side result we explain how to calculate holographic Green functions and their quasinormal modes for a set of operators that mix under the RG flow.

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