Restoration of topological properties at finite temperatures in a heavy-fermion system

We study how topological phases evolve in the Kane-Mele-Kondo lattice at finite temperatures and obtain the topological Doniach phase diagram. In particular, we find an intriguing crossover behavior induced by the interplay between the topological structure and electron correlations; the topological properties are restored by temperature effects. This restoration can be observed in the behavior of the bulk as well as the edge. In the bulk, we observe an increase of the spin-Hall conductivity at finite temperatures, while it is zero in the low-temperature region. At the edge, we observe gapless edge modes emerging with increasing temperature.

Figure 1

Sketch of the lattice. The electron of the up-spin state hops from site $j$ to $i$ with $i{t}_{\mathrm{so}}\mathrm{sgn}(i,j)$, where $\mathrm{sgn}(i,j)=({\widehat{\mathit{d}}}_i\times {\widehat{\mathit{d}}}_j)/|{\widehat{\mathit{d}}}_i\times {\widehat{\mathit{d}}}_j|$.

Figure 2

Phase diagram of temperature ($T$) vs the antiferromagnetic coupling ($J$). Because of the presence of the bulk gap, a first order transition (solid orange line) is observed in the weak coupling region, while it changes to second order (dashed green line) with increasing coupling. In this figure, one can observe the topological phase transition. For $J<0.683t$ a topological antiferromagnetic phase (AFTI) emerges. This phase changes into a trivial antifferomagnetic phase and an ordinary Kondo insulator as the interaction is increased. The topological structure in Eq. (3) is well defined only at zero temperature.

Figure 3

The spin Chern number and magnetic moments. In panels (a) and (b), temperature dependence of the antiferromagntic moments is plotted for $J=t$ and $J=0.1t$, respectively. $m_c$ and $m_{\mathrm{spin}}$ denote the magnetic moment of the itinerant electron and localized spin. (c) The antiferromagntic moment as functions of the interaction strength. (d) The spin Chern number as a function of the interaction strength $J$. In panels (c) and (d), the temperature is set to $T=0.0001t$.

Figure 4

Momentum-resolved spectral function $A(\omega ,\mathit{k})=-\mathrm{Im}{\sum }_{\sigma }\mathrm{tr}{\widehat{G}}_{\sigma }^R(\omega ,\mathit{k})/\pi$ for several values of the interaction strength $J$. In this figure the momentum $\mathit{k}$ sweeps the high symmetry points in the Brillouin zone: $(k_x,k_y)=(0,0)\to (\pi /\sqrt{3},\pi )\to (0,4\pi /3)\to (0,0)$. Here we have taken the distance of neighboring sites as the length unit. The data for $J=0.6t$, $0.683t$, and $0.8t$ are plotted from bottom to top. Right panels are the magnified version around $\omega =0$. The temperature is set to $T=0.0001t$.

Figure 5

(a) Temperature dependence of the spin-Hall conductivity. (b) [(c)] Self-energies of up-spin state in the paramagnetic [antiferromagnetic] phase. Here, we plot them along the imaginary axis. In the antiferromagnetic phase, we plot the self-energy at the $A$ site where the electron state for up-spin is dominant. For $t_{\mathrm{so}}=0$, the spin-Hall conductivity remains zero even in the high temperature region.

Figure 6

(a) and (b) Momentum-resolved spectral function for up-spin state $A_{\uparrow }(\omega ,\mathit{k})=-\mathrm{Im}{\left[{\widehat{G}}_{\uparrow }^R(\omega ,\mathit{k})\right]}_{2i,2i}/\pi$ at the $A$ site in the bulk (i.e., $i=L/2$ with $L=20$) for $T=0.0001t$ and $T=0.1t$, respectively. (c) and (d) The spectrum at the edge (i.e., $i=0$ and $A$ site) for $T=0.0001t$ and $T=0.1t$. Here, the data for $T=0.1t$ are scaled by $0.5$. The R-DMFT calculation is performed under the open (periodic) boundary condition for the $x\left(y\right)$ direction. We have zig-zag edges at $x=0$ and $L-1$. In the spectral weight for down-spin state, we can find edge modes propagating in opposite direction. (e) Spectral function $A_{\uparrow }\left(\omega \right)=-\mathrm{Im}{\sum }_{k_y}{\left[{\widehat{G}}_{\uparrow }^R\right]}_{0,0}(\omega ,k_y)/2{\pi }^2$. For $T=0.0001t$ edge modes are gapped due to the Kondo effect, while the gap disappears with increasing temperature. For $T=0.1t$, a tiny pseudogap is observed. We conclude that this is due to the finite size effect because the gap width should be the same as that for $T=0.0001$ if the Kondo effect induces it.

Figure 7

Temperature dependence of the spin-Hall conductivity for several values of coupling strength $J$. The error bars in the low-temperature region arise from extrapolation to the $\omega \to 0$ limit. At high temperatures these error bars are smaller than symbols. Arrows denote antiferromagnetic transition points.

Figure 8

(a) Self-energy at $J=0.8t$. (b) the real part of the self-energy at $i{\omega }_0=i\pi T$. Here, the temperature is set to $T=0.0001t$.

Figure 9

Spectral weight for $t_{\mathrm{so}}=t$ and $J=1.5t$. (a) and (b) Momentum-resolved spectral function for up-spin state $A_{\uparrow }(\omega ,\mathit{k})=-\mathrm{Im}{\left[{\widehat{G}}_{\uparrow }^R(\omega ,\mathit{k})\right]}_{2i,2i}/\pi$ at the $A$ site in the bulk (i.e., $i=L/2$ with $L=20$) for $T=0.0001t$ and $T=0.1t$, respectively. (c) and (d) The spectrum at the edge (i.e., $i=0$ and $A$ site) for $T=0.0001t$ and $T=0.1t$. The R-DMFT calculation is performed under the open (periodic) boundary condition for the $x\left(y\right)$ direction. We have zig-zag edges at $x=0$ and $L-1$. (e) Spectral function $A_{\uparrow }\left(\omega \right)=-\mathrm{Im}{\sum }_{k_y}{\left[{\widehat{G}}_{\uparrow }^R\right]}_{0,0}(\omega ,k_y)/2{\pi }^2$.