Exact Overlaps in the Kondo Problem

It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz. Using a strategy based on the integrable structure of the corresponding quantum field theory, we propose an exact formula for this overlap, which we check against extensive density matrix renormalization group calculations.

Figure 1

The overlap $|⟨{\mathrm{\Omega }}_2|{\mathrm{\Omega }}_1⟩|$ can be extracted from the partition function of the system with the insertion of a Kondo impurity and two different values of the coupling. In this picture, the boundary condition corresponds to the spin impurity while the bulk describes a critical statistical mechanics problem associated with the spin mode ${\varphi }_s$.

Figure 2

Theoretical result for the overlap for various anisotropies as a function of the ratio $T_K^{(2)}/T_K^{(1)}$. The isotropic Kondo problem then corresponds to $\xi =\infty$. Note the extreme values on the $x$ axis.

Figure 3

Numerical checks of the main result (8). Note that the comparison between numerical and field theory results does not involve any free parameter. (a) Measures of the overlap in the free-fermion Toulouse case $\xi =1$, modeled by a spinless noninteracting resonant level, tunnel coupled with parameter $J_{\ell }$ to two metallic reservoirs with $N=4096$ sites each. The dashed line is the analytical result, with $T_K\propto J_{\ell }^2$. Inset: finite size scaling. (b) DMRG results for the overlap in the interacting case $\xi =1/3$, modeled by an $XXZ$ spin chain with anisotropy $\mathrm{\Delta }=-1/\sqrt{2}$ and $N$ sites, with an extra site at the edge characterized by a weak link $J_{\ell }$, corresponding to the impurity. The numerical results for $N=200,\dots ,800$ are extrapolated in the thermodynamic limit $N\to \infty$. The red line is the analytical result, with $T_K\propto J_{\ell }^{4/3}$, and the black symbols correspond to the extrapolation $J_{\ell }\to 0$. Inset: examples of finite size extrapolations for fixed $J_{\ell }\equiv J_{\ell }^{(1)}$ and different values of $J_{\ell }^{(2)}$.