Limits of Thermal Conductance Quantization in Chiral Topological Josephson Junctions

TL;DR

This work addresses how chiral Majorana modes in multiterminal topological Josephson junctions manifest in thermal versus nonlocal electrical transport. By combining a lattice-regularized Dirac-BdG model with non-equilibrium Green's function transport, it identifies clear conditions under which a single chiral Majorana channel yields half-quantized thermal conductance $\kappa=0.5\kappa_0$ at phase difference $\phi=\pi$, while nonlocal conductance remains suppressed due to particle-hole symmetry. The study shows that in the $C=1$ phase, half-quantized thermal plateaus persist in intermediate-to-long junctions and at finite Zeeman fields, but in the $C=2$ phase the thermal response is not universally quantized and strongly depends on the momentum-space location of Majorana modes. The results emphasize that heat transport signatures hinge on momentum-space structure, finite-size geometry, and sample parameters, offering practical criteria to identify Majorana physics in multiterminal topological superconductors and guiding extensions to other platforms.

Abstract

We investigate thermal and non-local electrical transport in four-terminal Josephson junctions formed by a normal region coupled to two transverse chiral superconducting leads, supporting phases characterized by Chern numbers ${\cal C}=0,\,1$\,and\,2. We identify the conditions under which a single chiral Majorana mode (${\cal C}=1$) produces a robust half-quantized thermal conductance, while non-local electrical conductance remains strongly suppressed by particle-hole symmetry. Thermal conductance quantization occurs near a superconducting phase difference $π$, but only in the low-doping regime of the central region and in the intermediate- to long-junction limits. At finite Zeeman fields, the thermal response broadly follows the topology of the isolated superconducting leads for the $C=1$ phase while, in the ${\cal C}=2$ phase, the thermal conductance generally deviates from quantization, depending on the momentum-space location of the Majorana modes. Our results establish clear criteria for probing chiral Majorana modes in Josephson junctions and highlight the essential role of momentum-space structure, finite-size geometry, and sample parameters in thermal transport.

Figures (12)

• Figure 1: Schematic of the four-terminal Josephson junction. A normal region of size $n_x \times n_y$ and chemical potential $\mu_c$ is coupled to two normal leads (left $L$ and right $R$) and two superconducting leads (top $T$ and bottom $B$). The superconducting terminals are grounded, share a common chemical potential $\mu_s$, and are kept at temperature $T_s$, with pairing potentials $\Delta e^{i\varphi_{T,B}}$. The right normal lead is also grounded ($V_R=0$), while the left normal lead is biased by a voltage $V_L$ and a temperature difference $T_L$, providing the sole source of nonequilibrium. Black bars indicate grounded terminals.
• Figure 2: Bulk energy spectrum of the normal ($\Delta=0$) Hamiltonian, with $\mathcal{Z}=0$, $\mu=0$ and two different values of the Wilson mass, $m_0=0$ (a) and $m_0=0.5$ (b). Energy spectrum for a nanoribbon with $n_y=20a$, also for $m_0=0$ (c) and $m_0=0.5$ (d). All panels are in units of $t$.
• Figure 3: Top panel: Phase diagram of the bulk Hamiltonian as a function of $\mu_s$ and $\mathcal{Z}$, with $m_0=0.8$ and $\Delta=0.35$, delimited by blue curves marking gap closings at high symmetry points. Different colors, white, yellow and blue set $\mathcal{C}=0,\,1,\,2$, respectively. Panels (a)-(h) show exemplary energy spectra of a nanoribbon, with a width of $n_x=60a$. (a) $(\mathcal{Z}, \mu_s)$, (a) $(0.5,0.0)$, (b) $(0.0,0.0)$, (c) $(-0.5,1.5)$, (d) $(-0.75,0.2)$, (e) $(-1.7,0.2)$, (f) $(-2.5,0.2)$, (g) $(-1.5,2.0)$ and (h) $(-2.0,2.0)$.
• Figure 4: Low-doping regime ($\mu_\text{c}=0$): Phase-dependent DOS for a JJ with ${\cal Z}=0$, $\mu_\text{c} = 0$, $\Delta=0.8$, $\mu_s=0.8$ and (a, c) $n_y=2a$ and (b, d) $n_y=20a$ and PBC (a, b) and HWBC (c, d).
• Figure 5: Finite-doping regime ($\mu_\text{c}=0.8$ and $n_y=20a$): (a, b) PBC as a function of $E$ and $k_x$, with (a) $\mu_\text{c}=0$ and (b) $\mu_\text{c}=0.8$ and $\phi=\pi$. (c, d) HWBC DOS as a function of energy and $\phi$ for the intermediate-junction regime, with $\mu_s=0.8$ and $\Delta=0.35$. The rest of the parameters are the same as in Fig. \ref{['fig:ABS-LDOS']}. (d) LDOS as a function of $x$ and $y$ for $E=0$ and $\phi=\pi$.