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Theory of the Nernst effect near quantum phase transitions in condensed matter, and in dyonic black holes


We present a general hydrodynamic theory of transport in the vicinity of  superfluid-insulator transitions in two spatial dimensions described by
``Lorentz''-invariant quantum critical points. We allow for a weak  impurity scattering rate, a magnetic field B , and a deviation in the
density, $\rho$, from that of the insulator. We show that the  frequency-dependent thermal and electric linear response functions,
including the Nernst coefficient, are fully determined by a single  transport coefficient (a universal electrical conductivity), the impurity
scattering rate, and a few thermodynamic state variables. With reasonable  estimates for the parameters, our results predict a magnetic field and
temperature dependence of the Nernst signal which resembles measurements  in the cuprates, including the overall magnitude. Our theory predicts a
``hydrodynamic cyclotron mode'' which could be observable in ultrapure  samples. We also present exact results for the zero frequency transport
co-efficients of a supersymmetric conformal field theory (CFT), which is  solvable by the AdS/CFT correspondence. This correspondence maps the
$\rho$ and B perturbations of the 2+1 dimensional CFT to electric and magnetic charges of a black hole in the 3+1 dimensional anti-de Sitter
space. These exact results are found to be in full agreement with the  general predictions of our hydrodynamic analysis in the appropriate
limiting regime. The mapping of the hydrodynamic and AdS/CFT results under  particle-vortex duality is also described.