Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and nontopological superconductors (TRITOPS-S). We focus on wide junctions for which several channels intervene in the tunneling coupling. We derive effective Hamiltonians to describe the topological edge modes for different TRITOPS models, including Hamiltonians with $p$-wave pairing and Hamiltonians combining $s$-wave pairing with spin-orbit coupling. We also derive effective low-energy Hamiltonians to describe the Josephson junction. These can be solved analytically and explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge modes yield peculiar features to the CPR for both junction types. The primary effects occur for the response of the Majorana zero modes at half-flux quantum phase $\varphi \approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\varphi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong for two-component nematic superconductors. The second effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence. We analyze in this case the role of the phase fluctuations. For weakly coupled junctions, we show that time-reversal symmetry is restored for large enough stiffness in these fluctuations.

Figure 1

Edge states and spectra for the topological phase of Hamiltonians with $p$-wave pairing. (a) and (b) correspond to the Hamiltonian of Eq. (1), with ${\mathrm{\Delta }}_{\mathbf{k}}$ belonging to the one-dimensional irreducible representations of Eq. (3), while (c) and (d) to the two-dimensional representation ${\mathrm{E}}_u$. Only ${\mathrm{A}}_{1u}$ (equal to ${\mathrm{A}}_{2u}$) is shown. ${\mathrm{B}}_{1u}$ (equal to ${\mathrm{B}}_{2u}$) has the same spin structure with opposite chiralities. The spectra are calculated for a system with open boundaries along the $x$ direction and periodic boundary conditions along $y$ (only $k_y\ge 0$ is shown). The edge states are indicated in light blue. These are twofold and fourfold degenerate in (b) and (d), respectively.

Figure 2

(a) Bands of the Hamiltonian $H^{\mathrm{ZKM}}$ without pairing for $k_y=0$. The yellow region indicates the range of values of $\mu$ within which the topological phase develops. The Fermi points indicated in dots are $-k_{F1},-k_{F2},k_{F2},k_{F1}$ (from left to right). (b) Sketch of the edge states for the continuum Hamiltonian. (c) Spectrum of $H^{\mathrm{ZKM}}$ with periodic boundary conditions along $y$ and open boundary conditions along $x$ in a system with $N_x=200$. Parameters are ${\mathrm{\Delta }}_0=-2{\mathrm{\Delta }}_1=-0.4t, \lambda =0.5t$, and $\mu =-2t$. The (doubly degenerate) edge states are indicated in light blue.

Figure 3

$k$-resolved Josephson current (in units of $e/\hslash$) as a function of the phase difference in the topological phase for a junction of 2D topological superconductors with $p$-wave pairing calculated with numerical exact diagonalization. The upper and lower panels correspond to the representations ${\mathrm{A}}_{1u}$ (identical results are obtained for the ${\mathrm{A}}_{2u}$ and ${\mathrm{B}}_{ju},j=1,2$) and ${\text{E}}_u$, respectively. The plots in thick lines correspond to $k=0$. Plots in violet correspond to the edge states, while the other $k$ values are shown in red. The parameters are $t_J=\lambda =0.5t, \mathrm{\Delta }=t$, and $\mu =-3t$. A similar behavior is observed for other parameters within the topological phase ($-4t\le \mu \le 4t$)

Figure 4

Josephson current (in units of $e/\hslash$) as a function of the phase difference in the topological phase for a junction of 2D topological superconductors with $p$-wave pairing calculated by numerically diagonalizing the coupled lattice Hamiltonians. Solid and dashed lines correspond to the representation $B_{1u}$ and ${\text{E}}_u$ for $t_J=\lambda =0.5t, \mathrm{\Delta }=t$, and $\mu =-3t$.

Figure 5

(a) $k$-resolved Josephson current as a function of the phase difference in the topological phase for a junction of 2D topological superconductors with $t_J=t/2, {\mathrm{\Delta }}_0=2{\mathrm{\Delta }}_1$, and $\mu ={\varepsilon }_0$. The plot in thick lines corresponds to $k=\pi$. Plots in violet correspond to the edge states, while the other $k$ values are shown in red. The upper insets show the total Josephson current and the effective Josephson current. The lower inset shows the comparison of the exact numerical solution with the prediction based on Eq. (35) with the effective parameters calculated with the exact solution as explained in Appendix pp3 (dashed lines). (b) Same as top panel for ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_1$ and $t_{\mathrm{J}}=t$.

Figure 6

$k$-resolved Josephson current for a junction tilted an angle $\beta =\pi /4$ respect to the $xz$ plane with $t_J=t/2, {\mathrm{\Delta }}_0=2{\mathrm{\Delta }}_1, \mu ={\epsilon }_0$. Inset: Comparison with the effective model. Dashed lines correspond to Eq. (35), with the parameters defined in Eq. (36).

Figure 7

$k$-resolved Josephson current as a function of the phase difference in the topological phase for a junction between a 2D topological superconductor with $p$-wave pairing (upper and middle panels) and a nontopological one with $s$-wave pairing. These panels correspond to the representation ${\text{A}}_{1u}$ and ${\text{E}}_u$, respectively. The lower panel corresponds to a junction between the ZKM model and an ordinary superconductor (corresponding to ${\mathrm{\Delta }}_1=\lambda =0$). The parameters are $t_J=\lambda =0.5t, \mathrm{\Delta }={\mathrm{\Delta }}_0=2{\mathrm{\Delta }}_1=0.4t, \mu ={\mathrm{\Delta }}_0/{\mathrm{\Delta }}_{1}t$.

Figure 8

Total Josephson current as a function of the phase difference in the topological phase for a junction between a 2D topological superconductor and a nontopological one with $s$-wave pairing. Light blue, violet, and red colors correspond to the ${\text{A}}_{1u}, {\text{E}}_u$, and ZKM models, respectively.