Nodal $d+id$ Pairing and Topological Phases on the Triangular Lattice of ${\mathrm{Na}}_x{\mathrm{CoO}}_2·y{\mathrm{H}}_2\mathrm{O}$: Evidence for an Unconventional Superconducting State

We show that finite angular momentum pairing chiral superconductors on the triangular lattice have point zeroes in the complex gap function. A topological quantum phase transition takes place through a nodal superconducting state at a specific carrier density $x_c$ where the normal state Fermi surface crosses the isolated zeros. For spin-singlet pairing, we show that the second-nearest-neighbor ($d+id$)-wave pairing can be the dominant pairing channel. The gapless critical state at $x_c\simeq 0.25$ has six Dirac points and is topologically nontrivial with a $T^3$ spin relaxation rate below $T_c$. This picture provides a possible explanation for the unconventional superconducting state of ${\mathrm{Na}}_x{\mathrm{CoO}}_2·y{\mathrm{H}}_2\mathrm{O}$. Analyzing a pairing model with strong correlation using the Gutzwiller projection and symmetry arguments, we study these topological phases and phase transitions as a function of Na doping.

Figure 1

Nodes of chiral ($\ell ,n$)-wave pairing where the gap function ${\Delta }_{\ell ,n}(k)$ vanishes. (a) $p$ wave and (b) $d$ wave with normal state FS at $x=0.25$ (dashed line).

Figure 2

(a) Comparison of $d+id$ pairing order parameters at several dopings for $W_{1,2,3}=50\mathrm{meV}$. (b)–(d) Properties of seconnd-NN ($d+id$)-wave pairing at $x=0.19=x_c$ for $W_2=50\mathrm{meV}$ and $x=0.4>x_c$ for $W_2=73\mathrm{meV}$: Tunneling DOS (b), temperature dependence of Knight shift $K_s$ (c), and NMR relaxation rate $1/T_1$ (d) normalized by their values at $T_c$.

Figure 3

Topological winder number as a function of doping in the second-NN ($d+id$)-wave superconducting state. Insets show the locations of the nodes in the complex gap function and the normal state FS.