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

TL;DR

The authors develop a hydrodynamic theory of transport near 2+1D Lorentz-invariant quantum critical points, incorporating weak impurity scattering and a magnetic field, and show that the thermo-electric response, including the Nernst effect, is governed by a universal conductivity $\sigma_Q$, the momentum-relaxation rate $1/\tau_{imp}$, and thermodynamic state variables. They derive explicit frequency-dependent transport coefficients and reveal a hydrodynamic cyclotron mode, with a Wiedemann–Franz–like relation linking thermal and electrical transport. The results are cross-validated by an exact AdS/CFT solution for a dyonic black hole, demonstrating exact agreement in the hydrodynamic limit and exposing a bulk duality that implements particle–vortex duality in the boundary CFT. The work connects quantum-critical transport to experimental observations in cuprates and NbSi, and predicts observable resonances in ultrapure samples, offering a unified framework for understanding Nernst signals in strongly correlated systems.

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, ρ, 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 ρand 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.

Figures (4)

• Figure 1: Zero temperature ($T=0$), zero field ($B=0$) phase diagram in the vicinity of the quantum critical point described by the CFT, represented by the filled circle. The coupling $g$ represents a parameter which tunes between a superfluid and a Mott insulator which is at a density commensurate with the underlying lattice. The chemical potential $\mu$ introduces variations in the density and $\rho$ is difference in the density of pairs of holes in the superfluid from that in the Mott insulator. The thin dashed lines are contours of constant $\rho$. In the application to the cuprate superconductors, the Mott insulator with $\rho=0$ could be, e.g., an insulating state at hole density $\delta_I=1/8$ in a generalized phase diagram; then $\rho = (\delta-\delta_I)/(2a^2)$, where $a$ is the lattice spacing. The thick dotted line represents a possible trajectory of a particular compound as its hole density is decreased; note that the ground state is always a superconductor along this trajectory, even at $\delta=1/8$ (although there will be a dip in $T_c$ near $\delta=1/8$ as is also clear from Fig. \ref{['phasediag2']}). Note that the parent Mott insulator with zero hole density is not shown above. This paper will describe electrical and thermal transport in the above phase diagram perturbed by an applied magnetic field $B$ and a small density of impurities.
• Figure 2: Nonzero temperature ($T$) phase diagram at $B=0$ along three vertical cuts ( i.e. fixed $g$) in Fig. \ref{['phasediag']}. The lines indicate Kosterlitz-Thouless phase transitions at $T=T_{KT}$ associated with the loss of superfluid order as a function of $\mu$ for different values of $g$. At $g=g_c$, $T_{KT}/|\mu|$ is a universal number determined by the CFT at $g=g_c$, $\mu=0$troyer. This paper describes transport properties in the non-superfluid region above $T_{KT}$, in the presence of an applied magnetic field $B$ and a small impurity scattering $\tau_{\rm imp}$. The results of the supersymmetric CFT solvable by AdS/CFT in Section \ref{['sec:dyon']} are limited to $g=g_c$, but allow arbitrary variations in $\mu$ and $B$ away from quantum criticality as long there is no phase transition into a superfluid (or other) state.
• Figure 3: Contour plot (with logarithmic spacing) of the thermoelectric conductivity $\alpha_{xy}$ (Eq. \ref{['alphaplot2']}) as a function of temperature $T$ and magnetic field $B$, for parameters $\hbar v=47$ meV Å, $\delta-\delta_I=0.025$ and $\tau_{\rm imp}=10^{-12}$s estimated for LSCO. In the ordered low temperature regime $T<T_c\approx 30$K, Eq. (\ref{['alphaplot2']}) will receive modifications.
• Figure 4: Contour plot (linear scale) of the Nernst signal $e_N=\vartheta_{yx}$ (Eq. \ref{['nernstplot']}) close to a quantum critical point, as a function of temperature $T$ and magnetic field $B$. The parameters are the same as for Fig. \ref{['alphacontour']}. The signal strength in the plot ranges up to $10\mu$V/K.