Magnetism and superconductivity of strongly correlated electrons on the triangular lattice

We investigate the phase diagram of the $t\text{−}J$ model on a triangular lattice using a variational Monte Carlo approach. We use an extended set of Gutzwiller projected fermionic trial wave functions allowing for simultaneous magnetic and superconducting order parameters. We obtain energies at zero doping for the spin-$1∕2$ Heisenberg model in very good agreement with the best estimates. Upon electron doping (with a hopping integral $t<0$) this phase is surprisingly stable variationally up to $n\approx 1.4$, while the $d_{x^2-y^2}+i{d}_{xy}$ order parameter is rather weak and disappears at $n\approx 1.1$. For hole doping, however, the coplanar magnetic state is almost immediately destroyed and $d_{x^2-y^2}+i{d}_{xy}$ superconductivity survives down to $n\approx 0.8$. For lower $n$, between $0.2$ and $0.8$, we find saturated ferromagnetism. Moreover, there is evidence for a narrow spin density wave phase around $n\approx 0.8$. Commensurate flux phases were also considered, but these turned out not to be competitive at finite doping.

Figure 1

Three-site supercell of the triangular lattice. The onsite magnetic variational parameters can vary independently on each of the site $A$, $B$, and $C$ of the supercell. The BCS pairing as well as the flux vary independently on each of the different dashed bonds.

Figure 2

The variational parameters ${\mathbf{h}}_i$ for the coplanar $120°$ antiferromagnetic order. The spins lie in the $x$-$y$ plane. The $z$ component of the vector chirality $(\pm 1)$ on each triangular plaquette forms a staggered pattern.

Figure 3

Energy per site $e=3⟨{\mathbf{S}}_i\bullet {\mathbf{S}}_j⟩$ of the different variational wave functions for the Heisenberg model versus the system size $N^{-3∕2}$, with $N=36,48,108$ sites. Ordered by increasing condensation energy we find $d^{+}$ (open squares), $\mathrm{AF}+d^{+}∕J$ (open triangles), $\mathrm{AF}+d^{+}+\mathrm{SFL}∕J$ (open diamonds), $\mathrm{AF}+d^{+}∕J∕\mathrm{Ls}$ (full triangles) and the $\mathrm{AF}+d^{+}+\mathrm{SFL}∕J∕\mathrm{Ls}$ (full diamonds). The stars are the best estimates of the ground-state energy available in the literature (Refs. 12, 33).

Figure 4

Condensation energy per site versus the electron doping for a 12 site cluster for the different variational wave functions. We have done exact diagonalization (ED) for a 12 sites cluster with same periodic boundary conditions.

Figure 5

Condensation energy per site $e_c$ versus electron doping for the 108 sites lattice. We show different wave functions and also the best estimate in the literature (Ref. 12 at half filling (open diamond).

Figure 6

Superconducting order parameter $\Delta$ for a 108 site triangular cluster in our best wave function (full triangles), in the $d^{+}$ wave function (open squares). For comparison we show the amplitude of the $d$-wave gap in a $10\times 10$ square lattice (dashed line).

Figure 7

Amplitude of the ${120}^{\circ }$ magnetic order $M_{\mathrm{AF}}$ measured in the $\mathrm{AF}+d^{+}∕J$ for a 108 site cluster. Inset: the amplitude of the staggered spin-current $\chi$ in the same wave function.

Figure 8

Condensation energy per site $e_c$ for different wave functions in a 108 site cluster.

Figure 9

Amplitude of the ${120}^{\circ }$ magnetic order measured in our best wavefunction for the 108 site cluster (left scale, full triangles) and the ratio of the polarization $M_F$ on the saturated polarization $M_{\mathrm{sat}}$ in our best wave function (right scale, full circles). We show also the absolute magnetization $M_{\mathrm{SDW}}$ for the spin density wave wave function (left scale, see also Fig. 10).

Figure 10

On-site magnetization for each site of a 108 site lattice for the spin density wave function. Open (filled) circles denotes down (up) spins. The size of each circle is proportional to the respective amplitude of the on-site magnetization. We find that the spins forms a stripelike pattern, alternating ferromagnetic bonds in the ${\mathbf{a}}_2$ direction, and antiferromagnetic bond in the two other directions. The average on the lattice sites of the absolute value of the local magnetization is shown in Fig. 9.

Figure 11

(Color online) Cartoon picture of the phase diagram of the $t\text{−}J$ model we get with $t<0$. Here we sketch on an arbitrary scale the order parameter amplitude of the $120°$ magnetic phase, the ferromagnetic phase $\left(F\right)$, the superconducting $d_{x^2-y^2}+i{d}_{xy}$ phase (SC), and the commensurate spin density wave (SDW). Note that for electron density $n>1.04$ and $n<0.96$, the energy is degenerated, within the error bars due to the Monte Carlo sampling, with the pairings $d_{x^2-y^2}$, $d_{xy}$ and $d_{x^2-y^2}-i{d}_{xy}$. The pitch vector of the commensurate spin density wave is ${\mathbf{Q}}_N=(\pi ,-\pi ∕\sqrt{3})$, and this phase is depicted more in details in Fig. 10.