Topological transitions in metals

We investigate if a sharp topological transition in a metal with a large Fermi surface may be detected in transport measurements. In particular, we address if a skew scattering and a side jump on elastic disorder in the bulk of such a metal masks signatures of the topological transition. We conclude that certain transport coefficients exhibit discontinuous changes across the transition. These discontinuities are not smeared or dwarfed by the bulk metallic transport in a broad range of parameters.

Figure 1

[(a)–(d)] Band structure of the model from Ref. [23] at various choices of parameter $Q$ (see Sec. 2 for details of the model). The corresponding phases are: (a) topological insulator/superconductor (TI); (b) topological metal (TM); (c) topological phase transition; (d) normal metal (M). (e) Fermi surfaces, corresponding to cases (b)–(d) above. Blue lines in (a)–(d) represent multiple sub-bands of transversal quantization in quasi-1D geometry.

Figure 2

Schematic system setup. The edge states are in a local equilibrium with the heat reservoirs at temperatures $T_1$ and $T_2$. The temperature gradient is established in the bulk of the system, leading to the quasiparticles drift. The anisotropic (skew) scattering on impurities (blue dots) leads to a heat current in $x$ direction, indicated by the arrows around the blue dot. Notice that in the topological metal state both edges propagate in the same direction, cf. Fig. 1.

Figure 3

The intrinsic contribution to the thermal Hall conductivity in units of ${\kappa }_Q=(\pi k_B^2/12\hslash )T$ as a function of the suppercurrent $Q$ (measured in units of $4m\mathrm{\Delta }$). The calculation is done for $v_F/\mathrm{\Delta }=4.9$.

Figure 4

Particle (orange) and hole (blue) Fermi surfaces close to the topological phase transition; various scattering processes with scattering rates $W_{\mathit{p}{\mathit{p}}^{\prime }}^{\left(s{s}^{\prime }\right)}$ are indicated by arrows; dashed circle of radius $4m\mathrm{\Delta }$ separates two distinct contributions to ${\sum }_{\mathbf{k}}{\widehat{G}}_0$. (Inset) The same Fermi surfaces parameterized by the angle $\phi$.

Figure 5

(a) Longitudinal thermal conductivity ${\kappa }_{yy}^{\mathrm{skew}}$. (b) Thermal Hall conductivity ${\kappa }_{xy}^{\mathrm{skew}}$ around topological phase transition $Q_T$. Different colors correspond to different impurity strengths. Calculations are done with the same parameters as in Fig. 3. Curves cross zero due to cancellation of two time-reversal symmetry breaking mechanisms: the $p+ip$ superconductivity and the applied supercurrent $Q$.

Figure 6

The side jump contribution to the off-diagonal conductance ${\kappa }_{xy}$. Here, $v_F/\mathrm{\Delta }=4.9$. [The apparent discrepancy with Eq. (36) is understood as the consequence of higher-order terms in $\mathrm{\Delta }/v_F$, see Eq. (D11) in Appendix pp4 and the detailed derivation in Appendix pp5.]

Figure 7

The total thermal Hall conductivity ${\kappa }_{xy}$ versus supercurrent strength $Q$ (black); the skew scattering contribution (blue); the Berry phase intrinsic contribution (red); the side jump contribution (brown). The parameters are $g=100$, $v_F/\mathrm{\Delta }=4.9$ and ${\delta }_Q=0.2$.

Figure 8

The side jump velocity for states on the Fermi surfaces, parameterized by the angle $\phi$ as in the inset of Fig. 4. The solid line is plotted from the reduced expression, Eq. (D9), while the dotted line is obtained from the full definition, Eq. (D7). The unit here is chosen to be $v^{*}=\mathrm{\Delta }/\left(k_{F}l\right)$. Both plots are plotted right before topological phase transition $Q<Q_T$. For (b), the red and blue curves should be identical. For visualization, they were being shifted by a small number.