Introducing a Kondo-type interaction to the model of quantum walkers

Abstract

We introduce a model of discrete-time quantum walkers interacting with a lozalized magnetic impurity. Each quantum walker interacts with an impurity, through which multiple quantum walkers indirectly interact with each other, as in the Kondo model. We first identify a quantum walker as a massless Dirac particle propagating in continuous space via a series of Dirac's delta potentials. Based on the identification, we add a spin-$1/2$ degree of freedom to Dirac's potential at the origin. We derive all scattering matrices for massless Dirac particles arising from the impurity. First, for a simple set of parameter values, we analytically obtain the eigenvalues and eigenvectors of the bound states, in which a quantum walker is bound to the magnetic impurity. Second, we study two quantum walkers indirectly interacting with each other via the magnetic impurity. We numerically simulate the collision dynamics in one dimension when the spin-spin interaction at the origin is of the XX type and the SU(2) Heisenberg type. In the case of the XX interaction, we calculate the entanglement negativity to quantify how much the two quantum walkers are entangled with each other, and find that the negativity increases drastically upon the collision of the two walkers. In the case of the SU(2) Heisenberg interaction, we simulate the dynamics starting from the initial state in which one fermionic walker is in a bound eigenstate around the origin and the other fermionic walker is a delta function colliding with the first walker. We find that a bound eigenstate closest to the singlet state of the first walker and the magnetic impurity is least perturbed by the collision of the second walker. We speculate that this finding may be related to Kondo screening-like behavior at the lowest level of the real-space renormalization-group procedure.