Topology and Quantum-Spin-Classical-Spin Crossover of the Gapped Kondo Effect

Abstract

The gapped Kondo effect describes the screening of an $S=\frac12$ impurity spin locally coupled via an antiferromagnetic exchange interaction to a conduction-electron system exhibiting a finite hard gap. Using a combination of a Lanczos transformation and a self-consistent configuration-interaction scheme, we numerically investigate the local phase diagram. Furthermore, we show that the different phases can be characterized by several topological invariants: the conventional momentum-space Chern number of the underlying two-dimensional host system, corresponding to a Chern insulator; the B-space Chern number, defined by coupling the impurity spin to a fictitious local magnetic field $\boldsymbol B$, in the limit $B \to 0$; and the S-space Chern number, defined for a classical impurity spin, i.e., a vector of fixed length. The classical-spin limit is obtained for $B \to \infty$. By varying the field strength, we can therefore continuously interpolate between quantum-impurity-spin and classical-impurity-spin Hamiltonians and investigate whether the corresponding phase diagrams are likewise continuously connected. The gapped underscreened Kondo effect is studied for an impurity spin $S>\frac12$ as well as in the classical-spin limit approached via $B \to \infty$ or $S \to \infty$. Different variants of scattering theory are employed to interpret the resulting phases. Finally, the gapped two-channel overscreened Kondo effect, realized by coupling a quantum impurity spin equally to the local electron spins of both orbitals within a unit cell, is shown to be characterized by spontaneous particle-hole symmetry breaking. This leads to a highly nontrivial quantum-classical phase diagram.