A Holographic Model of the Kondo Effect

TL;DR

The paper introduces a holographic Kondo model that merges a CS gauge field in AdS$_3$ with a holographic superconductor on AdS$_2$, providing a controlled large-$N$ framework to study impurity screening and RG flows. It demonstrates a dynamically generated Kondo scale $T_K$, a second-order mean-field phase transition at $T_c$, and power-law low-temperature behavior controlled by a nontrivial IR fixed point with a leading irrelevant operator of non-integer dimension. The model reproduces key Kondo signatures, including impurity screening, a phase shift, and a spectral flow, while offering a versatile platform to explore extensions such as Kondo lattices, multiple channels, and entanglement properties in strongly coupled impurity systems. Overall, it links large-$N$ Kondo physics to holographic superconductivity, providing analytic and numerical tools to study impurity dynamics in strongly correlated environments.

Abstract

We propose a model of the Kondo effect based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. The Kondo effect is the screening of a magnetic impurity coupled anti-ferromagnetically to a bath of conduction electrons at low temperatures. In a (1+1)-dimensional CFT description, the Kondo effect is a renormalization group flow triggered by a marginally relevant (0+1)-dimensional operator between two fixed points with the same Kac-Moody current algebra. In the large-N limit, with spin SU(N) and charge U(1) symmetries, the Kondo effect appears as a (0+1)-dimensional second-order mean-field transition in which the U(1) charge symmetry is spontaneously broken. Our holographic model, which combines the CFT and large-N descriptions, is a Chern-Simons gauge field in (2+1)-dimensional AdS space, AdS3, dual to the Kac-Moody current, coupled to a holographic superconductor along an AdS2 subspace. Our model exhibits several characteristic features of the Kondo effect, including a dynamically generated scale, a resistivity with power-law behavior in temperature at low temperatures, and a spectral flow producing a phase shift. Our holographic Kondo model may be useful for studying many open problems involving impurities, including for example the Kondo lattice problem.

Figures (6)

• Figure 1: The value of $\kappa_T$ in eq. \ref{['eq:betaTkappaT']} as a function of $(2\pi T)/\Lambda$ for a representative negative (anti-ferromagnetic) value of the double-trace coupling $\kappa$. For the plot we chose $\kappa=-1$, leading to the divergence of $\kappa_T$ at $(2\pi T)/\Lambda = e^{-1} \approx 0.368$, which allows us to identify the Kondo temperature $T_K = \frac{1}{2\pi}\Lambda \, e^{1/\kappa} = \frac{1}{2 \pi} \Lambda \, e^{-1}$.
• Figure 2: The value of $\kappa_T$ in eq. \ref{['kappatomega']} as a function of the imaginary part of the frequency, $\omega_I \equiv \textrm{Im}\,\omega$, with vanishing real part, $\omega_R \equiv \textrm{Re}\,\omega=0$, for (a.) $Q=0.1$, (b.) $Q=0.35$, and (c.) $Q=0.5$.
• Figure 3: Log-linear plot of our numerical results for the free energy difference $\Delta {\mathcal{F}}$ between the condensed ($\langle {\mathcal{O}}\rangle\neq0$) and uncondensed ($\langle {\mathcal{O}}\rangle=0$) phases, in units of $(2\pi NT)$, as a function of $T/T_c$. We find $\Delta {\mathcal{F}} <0$, indicating that the condensed phase is thermodynamically favored for $T\leq T_c$.
• Figure 4: Plots of our numerical results for $\kappa \beta/\sqrt{T_c} \propto \langle {\mathcal{O}}\rangle/(N\sqrt{T_c})$ as a function of $T/T_c$. (a.) Log-linear plot for $T$ just below $T_c$. The solid red curve is $0.30(1-T/T_c)^{1/2}$, where we obtained the number $0.30$ from a fit to the data. The exponent $1/2$ reveals a mean-field transition. (b.) Log-log plot over a larger range of $T/T_c$, revealing that $\langle {\mathcal{O}} \rangle$ approaches a finite constant as $T/T_c \to 0$.
• Figure 5: Log-log plot of our numerical results for the value of the scalar at the horizon, $\phi(z=1)$, as a function of $T/T_c$, down to $T/T_c = 0.012$, for $Q=-1/2$. We find that $\phi(z=1)$ appears to approach a non-zero constant as $T/T_c \to 0$, namely $\phi(z=1)\approx 0.2$.