Thermodynamics in topological Josephson junctions

We study the thermodynamic properties of topological Josephson junctions using a quantum spin Hall (QSH) insulator-based junction as an example. In particular, we propose that phase-dependent measurements of the heat capacity offer an alternative to Josephson-current measurements to demonstrate key topological features. Even in an equilibrium situation, where the fermion parity is not conserved, the heat capacity exhibits a pronounced double peak in its phase dependence as a signature of the protected zero-energy crossing in the Andreev spectrum. This double-peak feature is robust against changes of the tunneling barrier and thus allows one to distinguish between topological and trivial junctions. At short time scales, fermion parity is conserved and the heat capacity is $4\pi$ periodic in the superconducting phase difference. We propose a dispersive setup coupling the Josephson junction to a tank LC circuit to measure the heat capacity of the QSH-based Josephson junction sufficiently fast to detect the $4\pi$ periodicity. Although explicitly calculated for a short QSH-based Josephson junction, our results are also applicable to long as well as nanowire-based topological Josephson junctions.

Figure 1

Scheme of a QSH-based topological Josephson junction: $s$-wave superconductors on top of the QSH insulator proximity-induce pairing of amplitude $\mathrm{\Delta }$ into the QSH edge states $|\uparrow \rangle$ and $|\downarrow \rangle$, thus defining the superconducting (S) regions. A superconducting phase difference $\phi$ is applied across the normal QSH weak link. In addition, a magnetic field $B$ in the QSH region induces a finite magnetic flux $B{\mathcal{W}}_S{L}_N$ and a Doppler shift as the Cooper pairs acquire a momentum $p_S$.

Figure 2

Andreev bound states given by Eq. (2) for the top edge with different $p_S$: (a) $v_F{p}_S=0$, (b) $v_F{p}_S=0.5\mathrm{\Delta }$, and (c) $v_F{p}_S=2.5\mathrm{\Delta }$. Here the blue and red lines denote the different spin quantum numbers $s=\uparrow /\downarrow$ and the shaded regions correspond to the continuum states.

Figure 3

Phase dependence of the [(a),(c)] Josephson current $I_t\left({\phi }_t\right)$ and of the [(b),(d)] heat capacity $\delta C_t\left({\phi }_t\right)=C_t\left({\phi }_t\right)-C_t\left(0\right)$ for the top edge, $k_{B}T=0.1\mathrm{\Delta }$ and different $p_S$ [(a),(b)] without and [(c),(d)] with parity constraints. Here $I_0=e\mathrm{\Delta }/2\hslash$.

Figure 4

(a) Setup to experimentally measure the heat capacity: A circuit consisting of a topological Josephson junction (with kinetic inductance ${\mathcal{L}}_K$) inside a superconducting ring (with superconducting gap ${\mathrm{\Delta }}_S$ and inductance ${\mathcal{L}}_S$) threaded by a magnetic flux $\mathrm{\Phi }$ is coupled via the mutual inductance $\mathcal{M}$ to a tank circuit composed of an inductor ${\mathcal{L}}_T$ and a capacitor $C_T$. The tank circuit is connected to an ac current source and shunted to a voltage probe $V_T$. By measuring the dispersive response of the resonant circuit it is possible to monitor the temperature $T_{\mathrm{QSH}}$. The topological Josephson junction is subject to a light pulse with photon energies $\hslash \omega <{\mathrm{\Delta }}_S$. (b) Time evolution of $T_{\mathrm{QSH}}$ after the light pulse at time $t_0$: The pulse generates an increase $\delta T$ in temperature compared to the bath temperature $T$. After the pulse, the temperature decays back to the bath temperature with a characteristic time scale ${\tau }_T$.

Figure 5

Different contributions and full Josephson current of the top edge as a function of $\phi \equiv {\phi }_t$ for $k_{B}T=0.1\mathrm{\Delta }$: [(a),(b)] $v_F{p}_S=0$, [(c),(d)] $v_F{p}_S=0.5\mathrm{\Delta }$, [(e),(f)] $v_F{p}_S=2.5\mathrm{\Delta }$. Here we have used the $\delta$-barrier model and $I_0=e\mathrm{\Delta }/2\hslash$.

Figure 6

Different contributions and full heat capacity of the top edge as a function of $\phi \equiv {\phi }_t$ for $k_{B}T=0.1\mathrm{\Delta }$: [(a),(b)] $v_F{p}_S=0$, [(c),(d)] $v_F{p}_S=0.5\mathrm{\Delta }$, [(e),(f)] $v_F{p}_S=2.5\mathrm{\Delta }$. Here we have used the $\delta$-barrier model. In panels (a), (c), and (e), only the $\phi$-dependent parts of $C_{\mathrm{ABS}}^t(\phi ,T)$ and $C_c^t(\phi ,T)$ are plotted. In panels (b), (d), and (f), $\mathrm{\Delta }{C}_t\left(\phi \right)=C_t\left(\phi \right)-C_t\left(0\right)$.

Figure 7

Phase dependence of [(a),(c)] the relative heat capacity $\delta C^t\left({\phi }_t\right)=C^t\left({\phi }_t\right)-C^t\left(0\right)$ and [(b),(d)] $C_t^{\prime \prime }\left({\phi }_t\right)=d^2{C}^t\left({\phi }_t\right)/d{\phi }_t^2$ for the top edge of a topological Josephson junction and for a trivial Josephson junction with different transmission probabilities ${\tau }_{\mathrm{triv}}$. Here $p_S=0$ and no parity constraints have been imposed on the topological junction. The temperature has been chosen as $k_{B}T=0.1\mathrm{\Delta }$ in panels (a) and (b) and as $k_{B}T=0.02\mathrm{\Delta }$ in panels (c) and (d).