Haldane phase in one-dimensional topological Kondo insulators

We investigate the ground-state properties of a recently proposed model for a topological Kondo insulator in one dimension (i.e., the $p$-wave Kondo-Heisenberg lattice model) by means of the density-matrix renormalization-group method. The nonstandard Kondo interaction in this model is different from the usual (i.e., local) Kondo interaction in that the localized spins couple to the “$p$-wave” spin density of conduction electrons, inducing a topologically nontrivial insulating ground state. Based on the analysis of the charge- and spin-excitation gaps, the string order parameter, and the spin profile in the ground state, we show that, at half filling and low energies, the system is in the Haldane phase and hosts topologically protected spin-1/2 end states. Beyond its intrinsic interest as a useful “toy model” to understand the effects of strong correlations on topological insulators, we show that the $p$-wave Kondo-Heisenberg model could be experimentally implemented in $p\text{−}\mathrm{band}$ optical lattices loaded with ultracold Fermi gases.

Figure 1

(a) Representation of the $p\text{−}\mathrm{wave}$ Kondo-Heisenberg model $(p$-KHM) in one dimension. Upper leg represents the conduction electron $p$ band, and lower leg corresponds to a spin-1/2 Heisenberg chain. The Kondo exchange $J_K$ couples a spin ${\mathbf{S}}_j$ with the $p$-wave spin density in the conduction band [see Eq. (3)]. (b) Microscopic model effectively realizing the $p$-KHM at low energies, and allowing an experimental implementation in $p\text{−}\mathrm{band}$ optical lattices. The direct hopping across a given rung vanishes due to the different parities of the orbitals.

Figure 2

Charge gap in the thermodynamical limit ${\mathrm{\Delta }}_{\text{c}}\left(\infty \right)$ (shown as black circles) vs $J_K$, obtained after finite-size scaling (see inset). The solid (red) line is a fit ${\mathrm{\Delta }}_{\text{c}}\left(\infty \right)={\alpha }_{\text{c}}{J}_K^2$, valid for small $J_K$, based on the bosonization analysis of Ref. [33]. Dashed lines are a guide to the eye. Inset: Finite-size scaling using the scaling laws ${\mathrm{\Delta }}_{\text{c}}\left(L\right)\approx {\mathrm{\Delta }}_{\text{c}}\left(\infty \right)+{\beta }_{\text{c}}{L}^{-2}$ for $J_K/t>0.3$ [25, 28], and ${\mathrm{\Delta }}_{\text{c}}\left(L\right)\approx \sqrt{{\mathrm{\Delta }}_{\text{c}}^2\left(\infty \right)+{\beta }_{\text{c}}{L}^{-2}}$ for $J_K/t<0.3$.

Figure 3

Extrapolated spin gap ${\mathrm{\Delta }}_{\text{s}}^{\left(2,0\right)}\left(\infty \right)$ (corresponding to the Haldane gap), as a function of $J_K$, obtained after a finite-size scaling analysis (see inset). Dashed lines are a guide to the eye. Inset: Finite-size scaling using the scaling law ${\mathrm{\Delta }}_{\text{s}}\left(L\right)\approx \sqrt{{\mathrm{\Delta }}_{\text{s}}^2\left(\infty \right)+{\beta }_{\text{s}}{L}^{-2}}$.

Figure 4

Spatial profile of $\langle T_j^z\rangle =\langle {\psi }_g^{M^z=1}|T_j^z|{\psi }_g^{M^z=1}\rangle$, i.e., the $z$ component of the spin in the supersite $j$ (left $y$ axis), computed with the ground state of the subspace with total $M^z=1$ for $L=80$. We also show the accumulated magnetization up to a given site $j$: ${M_j}^z\equiv {\sum }_{i=1}^j\langle T_i^z\rangle$ (right $y$ axis). The presence of the topologically protected spin-1/2 states at the ends of the chain is clearly seen.

Figure 5

String order parameter ${\mathcal{O}}_{\text{string}}^{z,\text{bulk}}$ vs $J_K$. Throughout the whole studied regime, ${\mathcal{O}}_{\text{string}}^{z,\text{bulk}}$ remains finite, indicating the presence of a Haldane-insulating phase. Inset: Spatial dependence of ${\mathcal{O}}_{\text{string}}^z\left(d\right)$ vs the distance $d=\left|l-m\right|$, where $l$ and $m$ have been taken symmetrically about the center of the chain.