Magnetic End States in a Strongly Interacting One-Dimensional Topological Kondo Insulator

Abstract

Topological Kondo insulators are strongly correlated materials where itinerant electrons hybridize with localized spins, giving rise to a topologically nontrivial band structure. Here, we use nonperturbative bosonization and renormalization-group techniques to study theoretically a one-dimensional topological Kondo insulator, described as a Kondo-Heisenberg model, where the Heisenberg spin-1=2 chain is coupled to a Hubbard chain through a Kondo exchange interaction in the p-wave channel (i.e., a strongly correlated version of the prototypical Tamm-Schockley model). We derive and solve renormalization-group equations at two-loop order in the Kondo parameter, and find that, at half filling, the charge degrees of freedom in the Hubbard chain acquire a Mott gap, even in the case of a noninteracting conduction band (Hubbard parameter U ¼ 0). Furthermore, at low enough temperatures, the system maps onto a spin-1=2 ladder with local ferromagnetic interactions along the rungs, effectively locking the spin degrees of freedom into a spin-1 chain with frozen charge degrees of freedom. This structure behaves as a spin-1 Haldane chain, a prototypical interacting topological spin model, and features two magnetic spin-1=2 end states for chains with open boundary conditions. Our analysis allows us to derive an insightful connection between topological Kondo insulators in one spatial dimension and the well-known physics of the Haldane chain, showing that the ground state of the former is qualitatively different from the predictions of the naive mean-field theory.

Haldane phase in one-dimensional topological Kondo insulators

“…The DMRG method has been used previously to describe the standard 1D Kondo lattice model at half-filling [15][16][17][18][19], where a topologically trivial, fully gapped groundstate was obtained. For the p-KHM, where a topological insulator groundstate was predicted [11,27], there are no DMRG studies to the best of our knowledge. Intuitively, we expect that the charge and spin gaps in this model vanish in the limit J K → 0, as the Hamiltonians H 1 and H 2 are separately gapless in the thermodynamic limit.…”

“…The striking results in Ref. [27] indicate that 1D TKI systems might be much more complex and richer than expected with the naïve mean-field approach, and suggest that they must be reconsidered from the more general perspective of interacting symmetry-protected topological phases [31,32]. In this Letter we study the groundstate properties of the finite-length p-KHM in one dimension using the Den-sity Matrix Renormalization Group (DMRG) [33,34].…”

“…sis, revealing an unexpected connection to the Haldane phase at low temperatures [27]. The Haldane phase is a paradigmatic example of a strongly interacting topological system, with unique features such as topologically protected spin-1/2 end-states, non-vanishing string order parameter, and the breaking of a discrete Z 2 × Z 2 hidden symmetry in the groundstate [28][29][30].…”

“…The Hamiltonian of the p-KHM is H = H 1 + H 2 + H K [11,27], where the conduction band is represented by a L-site tight-binding chain…”

“…The Hamiltonian Fig. 1(a)] corresponds to a spin 1/2 Heisenberg chain, and H K is the Kondo exchange coupling between H 1 and H 2 [11,27] …”