Time-reversal-invariant topological superconductivity induced by repulsive interactions in quantum wires

We consider a model for a one-dimensional quantum wire with Rashba spin-orbit coupling and repulsive interactions, proximity coupled to a conventional $s$-wave superconductor. Using a combination of Hartree-Fock and density matrix renormalization group calculations, we show that for sufficiently strong interactions in the wire, a time-reversal-invariant topological superconducting phase can be stabilized in the absence of an external magnetic field. This phase supports two zero-energy Majorana bound states at each end, which are protected by time-reversal symmetry. The mechanism for the formation of this phase is a reversal of the sign of the effective pair potential in the wire, due to the repulsive interactions. We calculate the differential conductance into the wire and its dependence on an applied magnetic field using the scattering-matrix formalism. The behavior of the zero-bias anomaly as a function of the field direction can serve as a distinct experimental signature of the topological phase.

Figure 1

The proposed system consists of a single quasi-1D wire (modeled by two chains) with SOC, coupled to a conventional $s$-wave superconductor. Integrating out the degrees of freedom of the superconductor generates a pairing potential ${\Delta }_{\mathrm{ind}}$ on the chain adjacent to the superconductor. Repulsive interactions in the wire which resist local pairing of electrons, induce a pairing potential ${\tilde{\Delta }}_b$ on the “$b$” chain with an opposite sign to ${\Delta }_{\mathrm{ind}}$.

Figure 2

Hartree-Fock phase diagram as a function of chemical potential ${\mu }_a={\mu }_b=\mu$, and interaction strength $U_a=U_b=U$, for $t_a=t_b=1,t_{ab}=0.4,{\alpha }_a=0,{\alpha }_b=0.6$, ${\Delta }_{\mathrm{ind}}=1$, and $L=200$. The diagram includes a TRITOPS phase, a trivial superconductor phase, and a region in which the Hartree-Fock solution is locally unstable to the formation of spin-density waves [39].

Figure 3

(a) Phase diagram obtained using DMRG. The system is in the trivial superconducting phase for $U<U_c$, and in the TRITOPS phase for $U>U_c$. The system's parameters are $t_a=t_b=1,t_{ab}=0.4,{\alpha }_a=0,{\alpha }_b=0.6,{\Delta }_{\mathrm{ind}}=1,\mathrm{and}\mu =0.8$. The low-lying energy spectrum of the system vs $1/L$, where $L$ is the length of the wire, for the three marked points is plotted in (b)–(d). Energies plotted in blue (green) correspond to energy states in the even (odd) fermion parity sectors. All energies are plotted with respect to the energy of the ground state, which is in all cases the lowest energy state in the even fermion parity sector, $\Delta E_n=E_n-E_0^{\mathrm{even}}$. (b) $U=0.5<U_c$. The system is in the trivial superconducting phase. The first two states in the even and odd fermion parity sectors are shown. The ground state is unique and the gap tends to a constant as $L\to \infty$, with a quadratic correction in $1/L$ as expected (the red lines are quadratic fits). The first excited state which lies in the odd parity sector is doubly degenerate as expected from Kramers' theorem. (c) $U=U_c=2.4$. Phase transition point. Once again, the first two states in each fermion parity sector are shown. All gaps scale linearly with $1/L$ in agreement with the system being gapless in the infinite size limit (the red lines are linear fits). (d) $U=5.5>U_c$. The system is in the TRITOPS phase. Here, the three lowest states in each fermion parity sector are shown. The result is consistent with a fourfold degenerate ground state in the thermodynamic limit, separated by a finite gap from the rest of the spectrum. The inset shows the energy difference $\Delta E$ between the lowest states in the even and odd fermion parity sectors on a semilogarithmic scale as a function of $L$. The result is consistent with an exponential dependence of $\Delta E$ on the system size.

Figure 4

(a) Phase diagram of the Hartree-Fock Hamiltonian of Eq. (2) as function of chemical potential ${\mu }_a={\mu }_b=\mu$, and Zeeman field along $x$ (perpendicular to the SOC), for $t_a=t_b=1,t_{ab}=0.4,{\alpha }_a=0,{\alpha }_b=0.6,{\tilde{\Delta }}_{\mathrm{ind}}=0.3$, and ${\tilde{\Delta }}_b=-0.15$. The system is in symmetry class BDI, and is characterized by a $\mathbb{Z}$ invariant, $Q$. $Q$ equals the number of MBS at each end of the wire. The TRITOPS phase is marked by a red line. (b) Differential conductance through a single lead connected to the wire as a function of bias voltage and Zeeman field for $\mu =0.15$. A wire of $L=100$ sites was used in the calculation.