Entanglement Entropy in a Holographic Kondo Model

TL;DR

The work advances the understanding of how a magnetic impurity embedded in a strongly coupled 1+1D system is encoded holographically by a backreacting brane in BTZ space, showing that impurity degrees of freedom are progressively hidden as a Kondo condensate forms. By computing entanglement entropy with the RT prescription and incorporating Israel junction conditions, the authors demonstrate that brane backreaction shortens bulk geodesics and reduces $S_{imp}$, consistent with the $g$-theorem. Analytically, in the large-distance limit, they derive a simple geometric formula $S_{imp}( ilde{\\ell})=S_{BH}( ilde{\\ell}+D)-S_{BH}( ilde{\\ell})+C(\\\lambda')$, where $D$ plays the role of the Kondo screening length and is related to the IR scale $\\xi_{K}$; numerically, this reproduces the anticipated exponential falloff and matches field-theory results for free electrons in the appropriate regime, highlighting universality of IR perturbations. The results tie holographic impurity physics to boundary RG flows and suggest links to holographic complexity and other entanglement measures, while pointing to future explorations at zero temperature and for more elaborate potentials.

Abstract

We calculate entanglement and impurity entropies in a recent holographic model of a magnetic impurity interacting with a strongly coupled system. There is an RG flow to an IR fixed point where the impurity is screened, leading to a decrease in impurity degrees of freedom. This information loss corresponds to a volume decrease in our dual gravity model, which consists of a codimension one hypersurface embedded in a BTZ black hole background in three dimensions. There are matter fields defined on this hypersurface which are dual to Kondo field theory operators. In the large N limit, the formation of the Kondo cloud corresponds to the condensation of a scalar field. The entropy is calculated according to the Ryu-Takayanagi prescription. This requires to determine the backreaction of the hypersurface on the BTZ geometry, which is achieved by solving the Israel junction conditions. We find that the larger the scalar condensate gets, the more the volume of constant time slices in the bulk is reduced, shortening the bulk geodesics and reducing the impurity entropy. This provides a new non-trivial example of an RG flow satisfying the g-theorem. Moreover, we find explicit expressions for the impurity entropy which are in agreement with previous field theory results for free electrons. This demonstrates the universality of perturbing about an IR fixed point.

Figures (10)

• Figure 1:
• Figure 2: Bulk setup of the holographic Kondo model of Erdmenger:2013dpa. The localised spin impurity on the field theory side is described by a codimension one hypersurface (called 'brane') in the bulk. In this figure, the brane is shown to be trivially embedded, as appropriate for the case without backreaction. We show the boundary interval of length $2\ell$ for which we calculate the entanglement entropy. The corresponding bulk geodesic used in the Ryu-Takayanagi prescription streches into the bulk and crosses the impurity hypersurface.
• Figure 3: Construction underlying the Israel junction conditions. Left: Construction using standard coordinates as in \ref{['BTZlinelement']}. This is the approach folowed in this paper. Right: It is theoretically possible (albeit too complicated in general) to construct Gaussian normal coordinates around the brane.
• Figure 4: Embedding profiles $x_+(z)$ for different values of $T/T_c$, which is adjusted numerically by changing $\mu_T$, see below \ref{['tilde']}. The curves start out at the left for $T=T_c$ and bend to the right as $T/T_c\rightarrow 0$. Compare to figure \ref{['fig::cutnpaste']} for the geometrical construction and the definition of $x_+(z)$. In particular, note that only the part of the spacetime to the right of the embedding curve is physical, while the left part is cut out and replaced by a mirror image of the right part.
• Figure 5: Numerical results for the impurity entropy from the holographic model. For $T/T_c=1$ the normal phase is given be a constant tension solution \ref{['backgroundSolutionGaugeAndEmbedding']} that leads to a constant $S_{imp}(\tilde{\ell})$ which is the uppermost curve in the figure. As the temperature is reduced and the scalar field condenses, due to the formation of the Kondo screening cloud the impurity entanglement for a given $\tilde{\ell}$ is reduced.