Study of the triangular lattice $tV$ model near $x=\frac{1}{3}$

Study of the triangular lattice $t V$ model near $x = \frac{1}{3}$

O. I. Motrunich and Patrick A. Lee

Phys. Rev. B 70, 024514 – Published 22 July 2004

Abstract

We study the extended Hubbard model on a triangular lattice near doping $x = \frac{1}{3}$ , which may be relevant for the recently discovered superconductor ${Na}_{x} {CoO}_{2} \cdot y H_{2} O$ . By generalizing this model to $N$ fermionic species, we formulate a meanfield description in the limit of large $N$ . In meanfield, we find two possible phases: a renormalized Fermi liquid and a $\sqrt{3} \times \sqrt{3}$ charge density wave state. The transition between the two phases is driven by increasing the nearest-neighbor repulsion and is found to be first order for doping $x = \frac{1}{3}$ , but occurs close to the point of the local instability of the uniform liquid. We also study fluctuations about the uniform meanfield state in a systematic $1 ∕ N$ expansion, focusing on the residual interaction of quasiparticles and possible superconducting instabilities due to this interaction. Upon moving towards the charge density wave instability, the increasing charge fluctuations favor a particular $f$ -wave triplet state. (This state was recently discussed by Tanaka et al., cond-mat/0311266.) We also report a direct Gutzwiller wave function study of the spin- $\frac{1}{2}$ model.

Received 5 February 2004

DOI:https://doi.org/10.1103/PhysRevB.70.024514

Authors & Affiliations

O. I. Motrunich and Patrick A. Lee

Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

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Vol. 70, Iss. 2 — 1 July 2004

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Images

Figure 1
Schematics of the nesting argument for doping $x = \frac{1}{3}$ . External hexagon shows triangular lattice Brillouin zone, while internal hexagon shows reduced BZ for the $\sqrt{3} \times \sqrt{3}$ order. Flat surfaces of the reduced BZ boundary are connected by the $\sqrt{3} \times \sqrt{3}$ ordering wave vectors perpendicular to the surfaces. Near these momenta transfer, $V_{eff} (θ, θ^{'})$ becomes attractive close to the CDW instability. We also show the $f$ -wave triplet gap function that we find to be the dominant superconducting channel from the residual interaction; this channel utilizes the near nesting of $θ_{M}$ points indicated with black diamonds.Reuse & Permissions
Figure 2
Meanfield phase diagram of the $t V$ Hamiltonian. For $x < 0.14$ , the uniform state is unstable at $q = 0$ (towards phase separation) for small repulsion $V$ , but is stabilized for moderate $V$ . For large $V$ , the instability occurs at the $\sqrt{3} \times \sqrt{3}$ ordering wave vector and the critical ${(V ∕ t)}_{c}$ is shown with the dotted line. In the region between the dotted lines, the uniform state is locally stable at all wave vectors. The circles show the actual meanfield transitions with general $\sqrt{3} \times \sqrt{3}$ ansatz; filled circles indicate first-order phase transitions, while open circles indicate second-order transitions.Reuse & Permissions
Figure 3
Residual quasiparticle interaction relevant for the pairing instability [cf. Eq. (29)] plotted for the momenta $k_{1}$ and $k_{2}$ on the Fermi surface. One angle is kept fixed while the other is varied over the Fermi Surface as indicated in the insets, which also show the triangular lattice BZ and the reduced BZ for the $\sqrt{3} \times \sqrt{3}$ CDW order (see Fig. 1). The data are for $x = \frac{1}{3}$ , and the uniform state becomes unstable at $V_{c} = 2.11$ . The vertical scale is in units of $t$ . Top panel: The fixed angle is in the direction of the $K$ point of the full BZ boundary. As we approach the CDW instability, $180 °$ scattering amplitude becomes attractive, which is associated with the fact that the corresponding Fermi surface points (black circles) are nearly connected by one of the $\sqrt{3} \times \sqrt{3}$ ordering wave vectors. Bottom panel: The fixed angle is in the direction of the $M$ point; the three nearly nested points occur at $120 °$ from each other.Reuse & Permissions
Figure 4
Coupling constants for the triangular lattice harmonics in Table (the $s$ -wave ansatz is below the bottom of the plot). The doping is $x = \frac{1}{3}$ , and the critical $V_{c} = 2.11$ sets the right-hand plot boundary. We also show the maximal and the second eigenvalue; the NNN $f$ -wave coupling constant coincides with the maximal eigenvalue for all $V$ shown.Reuse & Permissions
Figure 5
Maximal eigenvalue for dopings $x = 0.20$ , $0.30$ , and $0.40$ . The critical $V_{c}$ from the quadratic fluctuation analysis is indicated by vertical line in each case, while the first-order transition is indicated with the corresponding arrow near the bottom of the graph (cf. Fig. 2).Reuse & Permissions

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