${}^3\mathrm{He}-R$: A topological $s^{\pm }$ superfluid with triplet pairing

We show that when spin and orbital angular momenta are entangled by spin-orbit coupling, this transforms a topological spin-triplet superfluid/superconductor state, such as ${}^3\text{He-}B$, into a topological $s^{\pm }$ state, with nontrivial gapless edge states. Similar to ${}^3\text{He-}B$, the $s^{\pm }$ state also minimizes on-site Coulomb repulsion for weak to moderate interactions. A phase transition into a topological $d$-wave state occurs for sufficiently strong spin-orbit coupling.

Figure 1

(a) With Rashba coupling, the 2D Fermi surface is split into two with opposing helicities ${\beta }_3=\pm 1$. The relative orientation of the helicity vector $\widehat{z}\times \widehat{k}$ and the triplet pairing $\widehat{d}\left(\mathbf{k}\right)$ vector is $\theta$. (b) In the superfluid ${}^3\mathrm{He}-R$ condensate, the gap is maximized when the helicity and $\widehat{d}\left(\mathbf{k}\right)$ vectors align $\left(\theta =0\right)$, developing an $s^{\pm }$ gap function with opposite signs on the two Fermi surfaces.

Figure 2

Panels (a) and (b) show the direction of rotation of ${\vec{n}}_{\mathbf{k}}$ and ${\vec{d}}_{\mathbf{k}}$ in the low-spin $s^{\pm }$ and high-spin $d_{x^2-y^2}$ state, respectively. The Rashba vector ${\vec{n}}_{\mathbf{k}}=(-k_y,k_x)$ rotates anticlockwise around $\Gamma$ in both states, and is shown only in panel (a) for simplicity. The superconducting vector ${\vec{d}}_{\mathbf{k}}=(-k_y,k_x)\parallel {\vec{n}}_{\mathbf{k}}$ in the $s^{\pm }$ state, while ${\vec{d}}_{\mathbf{k}}=(-k_x,k_y)$ rotates clockwise, opposite to ${\vec{n}}_{\mathbf{k}}$, in the high-spin $d_{x^2-y^2}$ state. The dark red line is the locus marked out by ${\vec{d}}_{\mathbf{k}}$ as it rotates. A phase transition from the $s^{\pm }$ state into a $d$-wave state occurs when the spin-orbit interaction is sufficiently strong to lift one of the helical bands above $E_F$. Panels (c) and (d) show the symmetry of the superconducting gap in the helical quasiparticle basis for the low-spin and high-spin SC state, respectively.

Figure 3

2D helical bands respectively with weak spin-orbit coupling and strong spin-orbit coupling resulting in one of the helical bands pushed above $E_F$.

Figure 4

Strong spin-orbit coupling scenario: in the absence of Coulomb repulsion, the $s^{\pm }$ state is transformed into an $s^{++}$ state on the remaining helical band. In the presence of Coulomb repulsion, the on-site $s$-wave pair amplitude is disfavored, and the system will instead favor a phase transition into a $d$-wave state to minimize the hard-core/Coulomb repulsion. (Technically, the superfluid pairing on the lower helical band has a phase proportional to $\beta =-$, but we follow convention in labeling it as an $s^{++}$ pairing, which is equivalent up to a gauge transformation.)