The Kondo effect in the quantum $XX$ spin chain

We investigate the boundary phenomena that arise in a finite-size $XX$ spin chain interacting through an $XX$ interaction with a spin$-\frac{1}{2}$ impurity located at its edge. Upon Jordan-Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. Using both \textit{Bethe Ansatz} and \textit{exact diagonalization} techniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling. When the ratio falls below the critical value $\sqrt{2}$, the impurity is screened via the multiparticle Kondo effect. However, when the ratio between the coupling exceeds the critical value , a bound mode is formed at the impurity site which screens the spin of the impurity. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling $J_{\mathrm{imp}}$ exceeds the critical value $\sqrt{2}J$, the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.