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

Abstract

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 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 conductivity͒, 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 coefficients of a supersymmetric conformal field theory ͑CFT͒, which is solvable by the anti-de Sitter ͑AdS͒/CFT correspondence. This correspondence maps the 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.

Nonlinear magnetohydrodynamics from gravity

“…coefficients for the dyonic black brane has already received significant attention in light of its connection to quantum critical points in 2 + 1 dimensional condensed matter systems [11][12][13][14][15]. The approach taken in these works differs from ours, and yields results with a different (but overlapping) regime of validity, as we discuss further below.…”

“…small frequency/wavelength regime, in contrast to [11][12][13][14][15] where full AC results can be found at the level of linear response (at least numerically). Thus, the two approaches are complementary.…”

“…Using the stress tensor conservation equation, we can then read off the current to third order in derivatives. This gives us results for a host of new transport coefficients beyond what was computed in [11][12][13][14][15], including nonlinear terms. On the other hand, for the reasons described above, we are restricted to the…”

“…At intermediate length scales l T ≪ l ≪ l B , we should treat l T ∂ µ as a small parameter but work to all orders in l B ∂ µ . Also, in this regime momentum is approximately conserved, and so the momentum density should be kept as a hydroynamical variable; this is the approach used in [11][12][13][14][15]. At the largest length scales, l ≫ l B , momentum is no longer an independent hydrodynamical variable, but is instead fixed in terms of the energy and electric charge densities.…”

“…In particular, in thermal equilibrium the fluid can be taken to have any constant velocity, and the hydrodynamic expansion then promotes these constants to slowly varying functions. The assumption of a Lorentz invariant background clearly does not hold in the presence of an external magnetic field; nevertheless, (2.15) is taken as the starting point in other treatments of conformal linear magnetohydrodynamics from AdS/CFT [13,15,16]. As we discuss below, it turns out that sensible results can be so obtained provided that one restricts to the regime of linear fluctuations about thermal equilibrium (which is all that is considered in [13,15,16]), but this approach breaks down in general, as it does not respect the principles of hydrodynamic expansions.…”