Probing a topological quantum critical point in semiconductor-superconductor heterostructures

Quantum ground states on the nontrivial side of a topological quantum critical point (TQCP) have unique properties that make them attractive candidates for quantum information applications. A recent example is provided by $s$-wave superconductivity on a semiconductor platform, which is tuned through a TQCP to a topological superconducting (TS) state by an external Zeeman field. Despite many attractive features of TS states, TQCPs themselves do not break any symmetries, making it impossible to distinguish the TS state from a regular superconductor in conventional bulk measurements. Here we show that for the semiconductor TQCP this problem can be overcome by tracking suitable bulk transport properties across the topological quantum critical regime itself. The universal low-energy effective theory and the scaling form of the relevant susceptibilities also provide a useful theoretical framework in which to understand the topological transitions in semiconductor heterostructures. Based on our theory, specific bulk measurements are proposed here in order to characterize the novel TQCP in semiconductor heterostructures.

Figure 1

(a) Phase diagram associated with a conventional QCP in the ($T-g$) plane, where $g$ is the tuning parameter in the Hamiltonian. Solid curves denote true phase transitions, while dashed curves denote only a crossover. (b) Finite-temperature phase diagram associated with the TQCP in a spin-orbit coupled semiconductor. The tuning parameter $\Gamma$ represents a suitably directed Zeeman splitting. The superconducting pair potential $\Delta$ is perfectly continuous and nonzero at, and on either side of, the TQCP ($T=0,\Gamma ={\Gamma }_c$). Consequently, all lines on this diagram are crossover lines defined by $k_{B}T=E_0$ where $E_0$ is the zero-temperature energy gap.

Figure 2

Nanowire geometry for identifying the topological phase transition with increasing Zeeman splitting $\Gamma$. The nanowire (shown in green) is contacted by two superconducting leads (blue) which produce the proximity effect. The leads are placed with a finite potential difference $V$ and the current $I$ is measured. All quantities $V\left(\omega \right)$, $I\left(\omega \right)$, and $\Phi \left(\omega \right)$ are frequency dependent. The Josephson phase is $\Phi \left(\omega \right)=-iV\left(\omega \right)/\omega$.

Figure 3

Dissipative susceptibility ${\chi }_2$ ($G_0=2e^2/h$) as a function of frequency $\omega$. If experimentally $\omega \ll T$ is satisfied, ${\chi }_2$ should be plotted with the Zeeman splitting, see Fig. 4 and the discussion following it. ${\chi }_2$ increases at lower frequencies as the Zeeman splitting $\Gamma$ is tuned through the TQCP ($\Gamma ={\Gamma }_c=0.5$ meV). For $\Gamma$ away from ${\Gamma }_c$, in both the NTS and TS states the low-frequency [$\omega \ll E_0\left(\Gamma \right)$] dissipative response is negligible. As the single-particle gap $E_0\left(\Gamma \right)$ closes at $k=0$, the threshold frequency for the dissipative susceptibility approaches zero at the TQCP. Parameters are $\Delta =0.5$ meV, $\alpha =0.3$ meV, $\mu =0$, and $T=0.1$ meV corresponding to 1 K. For reference we have used 1 meV $=$ 250 GHz.

Figure 4

Dissipative susceptibility ${\chi }_2(T,\omega ,\delta )$ shows scaling in the vicinity of the topological critical point $\delta =\Gamma -{\Gamma }_c=0$. Appropriately scaled susceptibility data for $\omega /T=0.5$, calculated using Eq. (9) for different values of $\omega$ and $T$, coincide and collapse on the scaling function Eq. (11) in the region near the critical point ($\delta \sim 0$). The slight asymmetry of the data about $\delta =0$ is due to the asymmetry of the zero-temperature gap $E_0\left(\Gamma \right)$ about $\Gamma ={\Gamma }_c$ ($\delta =0$).

Figure 5

Spectral decomposition of current $\langle {\left|I\left(\omega \right)\right|}^2\rangle$ as a function of rescaled frequency $\omega /\left|\delta \right|$ where $\delta =\Gamma -{\Gamma }_c$ at several values of $\delta$ characterized by $\delta /T$. The applied dc voltage is taken to be $V=\delta /2$ so that the ac Josephson current in the nontopological phase (i.e., $\delta <0$) shows a peak at $\omega =\delta /2$ characteristic of the conventional ac Josephson effect. At and above $\delta /T\sim 1$ (i.e., the crossover curve $k_{B}T\sim E_0$ for $\delta >0$) a peak appears at $\omega =\delta /4$ which signals the appearance of the fractional Josephson effect and an underlying $T=0$ TQCP at $\delta =0$.