Tri-component-pairing chiral superconductivity on the honeycomb lattice with mixed $s$- and $d$-wave symmetries

TL;DR

This work shows that coexisting $s$- and two $d$-wave components on a honeycomb lattice can realize chiral topological superconductivity with a tri-component order parameter $s+d_{x^2-y^2}e^{i\phi_1}+d_{xy}e^{i\phi_2}$. A Ginzburg–Landau free energy analysis reveals spontaneous breaking of time-reversal and $C_{6}$ symmetry to $C_2$, yielding 12 degenerate ground states, while a microscopic BdG model confirms a fully gapped spectrum with nonzero Chern numbers and chiral Majorana edge modes. The system also exhibits an anisotropic anomalous AC Hall conductivity $\sigma_H(\omega)$, vanishing under certain symmetries in the chiral $d+id$ case but remaining nonzero here, enabling experimental distinctions via Kerr-type probes. In addition, the multi-component order parameter supports exotic topological excitations, including fractional magnetic vortices, highlighting potential platforms for nematic superconductivity and topological quantum phenomena on two-dimensional lattices.

Abstract

In this work, we investigate chiral topological superconductors on a two-dimensional honeycomb lattice with coexisting $d_{x^2-y^2}$, $d_{xy}$, and $s$-wave pairing symmetries. Using a Ginzburg-Landau free energy analysis, the pairing gap function is shown to exhibit a tri-component form $s+d_{x^2-y^2}e^{iφ_1}+d_{xy}e^{iφ_2}$, where $φ_1$ and $φ_2$ are phase differences between the $d$- and $s$-wave pairing components, which spontaneously breaks both time reversal and $C_6$ rotational symmetries. Chern numbers of the energy bands are calculated to be nonzero, demonstrating the topologically nontrivial nature of the system. The anomalous AC Hall conductivity is computed, which is not invariant under $C_6$ rotations, reflecting the anisotropic nature of the pairing gap function. Fractional magnetic vortices are also discussed, arising from the multi-component nature of the pairing gap function.

Figures (11)

• Figure 1: One of the twelve degenerate configurations of the tri-component pairing gap function on honeycomb lattice, in which $\psi_s$, $\psi_1$, and $\psi_2$ represent the $s$-, $d_{x^2-y^2}$-, and $d_{xy}$-pairing components, respectively. The phase of $s$-wave order parameter is fixed to zero, i.e., $|\psi_s| + |\psi_1| e^{i\phi_1} + |\psi_2| e^{i\phi_2}$. The choices of the values of parameters in the free energy are included in the text. It is worth emphasizing that the two components $\psi_1$ and $\psi_2$ in the figure have a phase difference of $0.452\pi$, not perpendicular with each other.
• Figure 2: Schematic plot of a two-dimensional honeycomb lattice, where $A$ and $B$ denote sites in the two inequivalent sublattices. The three nearest-neighbour vectors for the sublattice site $A$ are shown as the black arrows as $\boldsymbol{a_1}=(a,0)$, $\boldsymbol{a_2}=(-a/2,\sqrt{3}a/2)$, and $\boldsymbol{a_3}=(-a/2,-\sqrt{3}a/2)$, in which the lattice constant of the honeycomb lattice is $a$. The $x$-direction is taken as the direction pointing from sublattice site $A$ to $B$, and the $y$-direction is in the perpendicular direction.
• Figure 3: Schematic plot of the twelve symmetry elements of the $C_{6v}$ group consisting of six rotation and six reflection operations. The rotation operations are represented as $C_6$, $C_3$, $C_2$, $C_3^2$ and $C_6^5$ in the text, corresponding to rotations around $z$-axis by angles $\pi/3$, $2\pi/3$, $\pi$, $4\pi/3$, and $5\pi/3$. The six reflection planes of the reflection operations are determined by the planes spanned by $z$-axis and the dashed lines $l_1$, $l_2$, $l_3$, $l_4$, $l_5$, $l_6$.
• Figure 4: Degenerate configurations of the tri-component pairing $s + d_{x^2-y^2} e^{i\phi_1} + d_{xy} e^{i\phi_2}$. Symmetry operations which can generate the configuration from the one in (a$_1$) are indicated on top of each figure, where $E$ is identity operation; $T$ is time reversal; $C_6$, $C_3$, ..., $C_6^5$ are rotations; and $l_1$, $l_2$, ..., $l_6$ are reflections. In all panels, $\phi_1$ and $\phi_2$ are not perpendicular with each other. In panels (a$_3$), (a$_5$), (b$_3$), and (b$_5$), $\psi_1$ and $\psi_s$ are not exactly collinear, and $\psi_2$ is not precisely equal to $\pm \pi/2$. The parameters in Eq. \ref{['eq:F']} are chosen as $\alpha_s=-N_F$, $\alpha_d=-3.179N_F$, $\beta_s=2.635N_F/T_c^2$, $\beta_d=0.790N_F/T_c^2$, $\gamma=0$, $g_{dd}=2.640N_F/T_c^2$, $g_{sd}=0.275N_F/T_c^2$, and $g'_{sd}=-1.525N_F/T_c^2$, where $N_F$ is the density of states at the Fermi level and $T_c$ is the superconducting transition temperature.
• Figure 5: Eigenvalues of the BdG Hamiltonian in Eq. \ref{['eq:BdGhk']}. Panel (a) presents three-dimensional plots of the positive $E_1$ and $E_2$ as functions of $\boldsymbol{k}$, where a gap opens at the Dirac points. Panel (b) provides a top view of Panel (a). Panel (c) illustrates the variation of the four eigenvalues with respect to $k_x$, with the $k_y$ coordinate fixed to be the $y$-component of the $K$ point. In all plots, we set $|\psi_s|=0.0604t$, $|\psi_1|=0.1029t$, $|\psi_2|=0.0962t$, $\phi_1=0.383\pi$, $\phi_2=0.835\pi$, and $\mu=-0.4t$.