End states in a one-dimensional topological Kondo insulator in large-$N$ limit

To gain further insight into the properties of interacting topological insulators, we study a one-dimensional model of topological Kondo insulators which can be regarded as the strongly interacting limit of the Tamm-Shockley model. Treating the model in a large-$N$ expansion, we find a number of competing ground-state solutions, including topological insulating and valence bond ground states. One of the effects to emerge in our treatment is a reconstruction of the Kondo screening process near the boundary of the material (“Kondo band-bending”). Near the boundary for localization into a valence bond state, we find that the conduction character of the edge state grows substantially, leading to states that extend deeply into the bulk. We speculate that such states are the one-dimensional analog of the light $f$-electron surface states which appear to develop in the putative topological Kondo insulator, ${\mathrm{SmB}}_6$.

Figure 1

Schematic illustration of the 1D $p$-wave Kondo insulator Hamiltonian (1). The model contains a chain of Heisenberg spins coupled by an antiferromagnetic nearest-neighbor Heisenberg coupling, plus a tight-binding chain of conduction electrons (c). Each localized moment is coupled to the conduction sea via a Kondo “cotunneling” term that exchanges spin between a localized moment at site $j$ and a $p$-wave combination of conduction electron states formed between neighboring sites $j-1$ and $j+1$.

Figure 2

Illustrating the tight-binding model. (a) Real-space structure. (b) Dispersion of quasiparticles, showing band inversion at $k=\pi$ [Eq. (13)].

Figure 3

(a) Majorana decomposition of Kitaev model; (b) $t=\Delta =V$ limit of the tight-binding model (13).

Figure 4

Schematic phase diagram of the bulk ground state contains a metallic phase (I) and an insulating phase (II). Phase II can be further divided depending on the properties of its surface states; see Fig. 7.

Figure 5

Cartoon representation of two distinct phases in Fig. 4. We plot spatial dependence of $V_j$ and ${\Delta }_j$ and density of states (DOS). Phase I [(a) and (b)]: Metallic states with suppressed hybridization. It does not support surface states. Phase II [(c) and (d)]: Kondo topological insulator that supports surface states. Inset in (d) is the profile of a typical surface state decaying into the bulk as a function of distance.

Figure 6

Magnetic phase (red color in Fig. 7).

Figure 7

The surface phase diagram. With the addition of a renormalized Kondo coupling at the surface $\alpha =J_K^{\mathrm{boundary}}/J_K$. (a) Red color represents pure $f$ states with no $c$-electron mixing, $n_c=0$. Inset is the cut along $\alpha =2$. (b) is the bulk gap ${\Delta }_g$ in units of $t$ for $\alpha =2$, identifying the “Kitaev point” as the black dot. (c) displays the inferred group velocity measured in units of $at$, derived from the numerically measured penetration depth of the edge states using $v={\Delta }_g\xi$. The mean-field calculations were done on a chain of 70 unit cells.