Weak-pairing higher order topological superconductors

Conventional topological superconductors are fully gapped in the bulk but host gapless Majorana modes on their boundaries. We instead focus on a new class of superconductors, second-order topological superconductors, that have gapped, topological surfaces and gapless Majorana modes instead on lower-dimensional boundaries, i.e., corners of a two-dimensional system or hinges for a three-dimensional system. Here, we propose two general scenarios in which second-order topological superconductivity can be realized spontaneously with weak-pairing instabilities. First, we show that $\left(p_x+i{p}_y\right)$-wave pairing in a (doped) Dirac semimetal in two dimensions with four mirror-symmetric Dirac nodes realizes second-order topological superconductivity. Second, we show that $p+id$ pairing on an ordinary spin-degenerate Fermi surface realizes second-order topological superconductivity as well. In the latter case, we find that the topological invariants describing the system can be written using simple formulas involving only the low-energy properties of the Fermi surfaces and superconducting pairing. In both cases, we show that these exotic superconducting states can be intrinsically realized in a metallic system with electronic interactions. For the latter case, we also show it can be induced by proximity effect in a heterostructure of cuprate and topological superconductors.

Figure 1

Lattice representation of 2D second-order topological superconductor Hamiltonian in a Majorana basis as in Eq. (1). Each unit cell has four Majorana fermions represented by black dots as labeled. The tunneling strength $-2t$ is indicated by a solid line, while $2t$ is represented by a dashed line. As a result, there is a $\pi$ flux in each plaquette.

Figure 2

The Fermi surface for the BdG Hamiltonian in Eq. (3) with $(t,\mathrm{\Delta })=(1,\frac{1}{2})$, and (a) $\mu =0.90$, (b) $\mu =1.11$, and (c) $\mu =1.20$. The phase of the $p$-wave order parameter and the spin texture of each pocket are indicated around each Fermi surface. Upon increasing the chemical potential $\mu$, the pockets will increase in sizes, and merge, and vanish at the high-symmetry points in the BZ.

Figure 3

(a) The normal-state band structure of a cuprate. With the $d$-wave order nodes are generated, as indicated by the four Dirac cones along the nodal directions. The phase of the $d$-wave order parameter is indicated by the sign structure in orange. In the presence of $p$-wave order, the Dirac cones in the nodal directions are coupled, as shown in blue, which results in a completely gapped superconducting order. (b) A ${\text{TSC}}_2$ in 2D with gapped edges. The Majorana gaps have opposite signs on neighboring edges such that there is a MBS residing at the corner. (c) A ${\text{TSC}}_2$ in 3D with gapped surfaces, and gapless Majorana modes along the hinges. The colors red and green indicate that the neighboring hinges have different chiralities.

Figure 4

Topological phase transitions for (a) $\mathcal{T}$-invariant TSC and (b) $C_4\mathcal{T}$-invariant second-order TSC. In the trivial phase of a $\mathcal{T}$-invariant TSC, the winding number is $\nu =0$ as the SC gaps on the two FS's have the same sign. A topological transition occurs when a pair of nodal lines are created on one of the FS's, and the SC gap changes from positive to negative on this FS as the nodal lines nucleate, sweep over the FS, and eventually annihilate. The TSC enters the topological phase with $\nu =1$, after the nodal lines are annihilated. In the presence of $C_4\mathcal{T}$, however, a $d$-wave order can gap most out the nodal lines except at eight Weyl points. When the Weyl points are annihilated, a second-rder TSC is formed.

Figure 5

Two experimental setups for a proximity-induced ${\text{TSC}}_2$ phase. (a) A cuprate superconductor with $d$-wave symmetry is placed on top of a $\mathcal{T}$-invariant $p$-wave superconductor. The Josephson coupling induces a quartic term in the free energy, forcing the two order parameters to differ by a $\pm \pi /2$ phase. Due to the $C_4$ symmetry of the cuprate system, it develops a $C_4\mathcal{T}$-invariant $\left(p+id\right)$-wave order with MBS's at the corners. (b) A cuprate superconductor is sandwiched between two iron-based superconductors ${\text{FeTe}}_{0.55}{\text{Se}}_{0.45}$. The cuprate layer, together with its interfaces with ${\text{FeTe}}_{0.55}{\text{Se}}_{0.45}$ layers, realizes a 2D ${\mathrm{TSC}}_2$, with four MBS's at the corners.

Figure 6

The form factors of the superconducting pairing functions for the two leading instabilities. We find the two leading instabilities are $p$-wave (red) and $d$-wave order. For $p$-wave order, the two red curves are for the gap functions ${\mathrm{\Delta }}_{\pm }\left(\theta \right)$ separately, and we can see the relative sign change that is characteristic of $p$-wave order. The $d$-wave pairing has the characteristic four-node structure at $\theta =\pm \pi /4,\pm 3\pi /4$. To ensure that the Fermi surface is large enough to allow for a $\mathbf{Q}$ momentum transfer, we chose $k_F=5\pi /\left(6a_0\right)$. We have set the antiferromagnetic correlation length $\xi =1,$ and we have verified that choosing other values does not change our results.