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Impure AdS/CFT

Sean A. Hartnoll, Christopher P. Herzog

TL;DR

This work develops a holographic framework to quantify momentum relaxation from dilute impurities in a strongly coupled 2+1D CFT, deriving a general impurity-relaxation formula $\frac{1}{\tau_{\text{imp}}} = -\frac{1}{\chi_0} \lim_{\omega\to 0} \frac{\text{Im } G^R_{\mathcal{F}\mathcal{F}}(\omega,0)}{\omega}$ and identifying two impurity channels $\mathcal{O}_\pm$ with $\Delta_{\mathcal{O}}=1$ that couple to magnetic and electric impurities, respectively. In a truncated M2-brane holographic model, the authors compute the corresponding Green's functions via a dyonic AdS$_4$ black hole, revealing that magnetic impurities suppress momentum relaxation with increasing $B$ or $\rho$, while electric impurities drive $1/\tau_{\text{imp}}$ to a divergence at a critical $\mathcal{Q}$, signaling a black-hole instability toward an ordered phase. They also analyze the Nernst response, showing qualitative parallels to experiments in superconductors near quantum criticality and highlighting how $\tau_{\text{imp}}$ controls the magnitude of the Nernst signal. The results illuminate how holographic methods capture impurity-driven transport in strongly correlated quantum critical systems and suggest directions for more complete holographic realizations and experimental connections.

Abstract

We study momentum relaxation due to dilute, weak impurities in a strongly coupled CFT, a truncation of the M2 brane theory. Using the AdS/CFT correspondence, we compute the relaxation time scale as a function of the background magnetic field B and charge density ρ. The theory admits two different types of impurities. We find that for magnetic impurities, momentum relaxation due to the impurity is suppressed by a background B or ρ. For electric impurities, due to an underlying instability in the theory towards an ordered phase, the inverse relaxation time scale increases dramatically near \sqrt{B^2 + ρ^2/σ^2_0} \sim 21 T^2. We compute the Nernst response for the impure theory, and comment on similarities with recent measurements in superconductors.

Impure AdS/CFT

TL;DR

This work develops a holographic framework to quantify momentum relaxation from dilute impurities in a strongly coupled 2+1D CFT, deriving a general impurity-relaxation formula and identifying two impurity channels with that couple to magnetic and electric impurities, respectively. In a truncated M2-brane holographic model, the authors compute the corresponding Green's functions via a dyonic AdS black hole, revealing that magnetic impurities suppress momentum relaxation with increasing or , while electric impurities drive to a divergence at a critical , signaling a black-hole instability toward an ordered phase. They also analyze the Nernst response, showing qualitative parallels to experiments in superconductors near quantum criticality and highlighting how controls the magnitude of the Nernst signal. The results illuminate how holographic methods capture impurity-driven transport in strongly correlated quantum critical systems and suggest directions for more complete holographic realizations and experimental connections.

Abstract

We study momentum relaxation due to dilute, weak impurities in a strongly coupled CFT, a truncation of the M2 brane theory. Using the AdS/CFT correspondence, we compute the relaxation time scale as a function of the background magnetic field B and charge density ρ. The theory admits two different types of impurities. We find that for magnetic impurities, momentum relaxation due to the impurity is suppressed by a background B or ρ. For electric impurities, due to an underlying instability in the theory towards an ordered phase, the inverse relaxation time scale increases dramatically near \sqrt{B^2 + ρ^2/σ^2_0} \sim 21 T^2. We compute the Nernst response for the impure theory, and comment on similarities with recent measurements in superconductors.

Paper Structure

This paper contains 17 sections, 89 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The effect of random disorder
  3. The impurity timescale
  4. Impurity in the absence of a magnetic field
  5. Impurity plus an external magnetic field
  6. Impurity relaxation in the truncated M2 brane theory
  7. Comment on discrete symmetries
  8. Numerical results for $1/\tau_\text{imp}$
  9. A black hole instability
  10. The Nernst effect in the truncated M2 brane theory
  11. Discussion
  12. Summary of results for the impure CFT
  13. Some directions for future work
  14. Relating $G^R_{\mathcal{P}\mathcal{P}}$ to a conductivity
  15. Hydrodynamics and the memory method
  16. ...and 2 more sections

Figures (6)

  • Figure 1: The function $F_\pm$ for the M2 brane theory for a) the scalar field $\psi_+$ and b) $\psi_-$.
  • Figure 2: a) The real part of the cyclotron frequency and b) momentum relaxation due to the cyclotron resonance at $B = T^2$.
  • Figure 3: The location of the smallest pole in $G^R_{\mathcal{O}_- \mathcal{O}_-}$ as a function of $B$ and $\rho$: a) The pole in the complex $k$ plane for $\omega = 0$. Note $k^2$ is real. b) The pole in the complex $\omega$ plane for $k=0$. Note $\hbox{Re}(\omega)=0$.
  • Figure 4: (a) The Nernst signal $N$ for the truncated M2 brane theory as a function of $B$ and $T$, with vanishing charge density $\rho = 0$. The impurity potential is coupled to $\mathcal{O}_+$. Lighter denotes a larger Nernst coefficient. (b) The Nernst signal in the vicinity of a general quantum critical point with $\rho = 0$ and $\Delta_\mathcal{O} = 1$. The shading is logarithmically spaced.
  • Figure 5: The Nernst signal $N$ for the truncated M2 brane theory as a function of $B$ and $T$, with (a) charge density $\rho/\sigma_0 = 8.3 \times 10^{-5}$ and (b) charge density $\rho/\sigma_0 = 10$, and $\bar{V} = 0.1$ in both cases. The impurity potential is coupled to $\mathcal{O}_+$. Lighter denotes a larger Nernst coefficient.
  • ...and 1 more figures