Majorana ensembles with fractional entropy and conductance in nanoscopic systems

Quantum thermodynamics is a promising route to unambiguous detections of Majorana bound states. Being fundamentally different from quantum transport, this approach reveals unique Majorana thermodynamic behavior and deepens our insight into Majorana quantum transport itself. Here, we demonstrate that a nanoscopic system with topological superconductors produces a remarkable accumulation of Majorana thermodynamic states in wide ranges of Majorana tunneling phases by means of increasing its temperature $T$. Revealing this physical behavior is twofold beneficial. First, it significantly reduces the dependence of the entropy on the tunneling phases which become almost irrelevant in experiments. Second, the fractional Majorana entropy $S_M^{\left(2\right)}=k_{B}ln\left(2^{\frac{3}{2}}\right)$ may be observed at high temperatures, substantially facilitating experiments. Analyzing quantum transport, we predict that when the temperature increases, the above thermodynamic behavior will induce an anomalous increase of the linear conductance from vanishing values up to the unitary fractional Majorana plateau $G_M=e^2/2h$ extending to high temperatures.

Figure 1

Outline of a quantum device revealing unique Majorana thermodynamic and transport behavior.

Figure 2

Low-temperature behavior of the entropy $S$ as a function of the tunneling phase difference $\mathrm{\Delta }{\varphi }_{13}$ in polar coordinates. Here, $k_{B}T/\mathrm{\Gamma }={10}^{-5}, {\epsilon }_d/\mathrm{\Gamma }={10}^{-1}, |{\eta }_1|/\mathrm{\Gamma }={10}^{-2}, |{\eta }_2|/\mathrm{\Gamma }=5\times {10}^{-3}, |{\eta }_3|/\mathrm{\Gamma }=1, |{\eta }_4|/\mathrm{\Gamma }=4\times {10}^{-2}, {\xi }_{12}/\mathrm{\Gamma }={10}^{-7}, {\xi }_{34}/\mathrm{\Gamma }={10}^{-7}$. The solid curve is for $\mathrm{\Delta }{\varphi }_{14}=0$ while the dashed curve is for $\mathrm{\Delta }{\varphi }_{14}=\pi /18$.

Figure 3

High-temperature behavior of the entropy $S$ as a function of the tunneling phase difference $\mathrm{\Delta }{\varphi }_{13}$ in polar coordinates. Here, $k_{B}T/\mathrm{\Gamma }={10}^{-2}$. The other parameters have the same values as in Fig. 2. The solid curve is for $\mathrm{\Delta }{\varphi }_{14}=0$ while the dashed curve is for $\mathrm{\Delta }{\varphi }_{14}=\pi /4$.

Figure 4

The entropy $S$ (upper panel) and linear conductance $G$ (lower panel) as functions of the gate voltage controlling the location of the energy level ${\epsilon }_d$. Here, $k_{B}T/\mathrm{\Gamma }={10}^{-2}, \mathrm{\Delta }{\varphi }_{13}=\pi /18, \mathrm{\Delta }{\varphi }_{14}=0, \mathrm{\Delta }{\varphi }_{12}=0$. The other parameters have the same values as in Fig. 2.

Figure 5

The entropy $S$ (upper panel) and linear conductance $G$ (lower panel) as functions of the temperature $T$ for various values of the tunneling phase difference $\mathrm{\Delta }{\varphi }_{13}$. Here, $\mathrm{\Delta }{\varphi }_{14}=0, \mathrm{\Delta }{\varphi }_{12}=0$, and the values of the other parameters are the same as in Fig. 2.