Variational study of triangular lattice spin-$1∕2$ model with ring exchanges and spin liquid state in $\kappa \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$

We study triangular lattice spin-$1∕2$ system with antiferromagnetic Heisenberg and ring exchanges using variational approach focusing on possible realization of spin-liquid states. Trial spin liquid wave functions are obtained by Gutzwiller projection of fermionic mean-field states and their energetics is compared against magnetically ordered trial states. We find that in a range of the ring exchange coupling upon destroying the antiferromagnetic order, the best such spin liquid state is essentially a Gutzwiller-projected Fermi sea state. We propose this spin liquid with a spinon Fermi surface as a candidate for the nonmagnetic insulating phase observed in the organic compound $\kappa \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$, and describe some experimental consequences of this proposal.

Figure 1

Phase diagram of the model (1) from the exact diagonalization study of Refs. 1, 2. The magnetic order is destroyed for $J_4\gtrsim 0.07-0.1$; a spin gap is observed in the regime $0.1\lesssim J_4\lesssim 0.25$, but also many singlets below the spin gap. The spin gap is decreasing for $J_4\gtrsim 0.175$.

Figure 2

Variational phase diagram for the Hamiltonian (1). The AF ordered variational state has the lowest energy for small $J_4$, but becomes unstable for $J_4\gtrsim 0.14$ compared with the fermionic spin liquid states. One example of such spin liquid is the projected $d_{x^2-y^2}+i{d}_{xy}$ superconductor Ansatz, with the optimal variational parameter ${(\Delta ∕t)}_{\mathrm{var}}=0.22$, 0.13, 0.05, 0.02 for $J_4=0.15$, 0.20, 0.25, 0.30, respectively. Some other Ansätze give very close optimal energy, and the situation is particularly not clear near the AF state. But for $J_4\gtrsim 0.3$, our best Ansätze become essentially the projected Fermi sea state. We caution that for significantly larger $J_4$ states with more complicated magnetic orders—e.g., with four-sublattice order—may enter the energetics competition (Ref. 8), which is not considered here.

Figure 3

Summary of the large $N$ study of the Hamiltonian ${\widehat{H}}_{\mathrm{SU}\left(N\right)}$. (a) Phase diagram from the mean-field energy optimization over translationally invariant states. (b) Full optimization.

Figure 4

Proposed phase diagram for the triangular lattice Hubbard model. The present study is based on the effective spin Hamiltonian (10) and applies only to the insulating regime expected for $t∕U\lesssim 1∕5$ from Ref. 4. Close to the metal-insulator transition, we propose the spin liquid state with spinon Fermi surface. For smaller $t∕U\lesssim 1∕9$, the best state is AF ordered. The $\kappa \text{−}{\left(\mathrm{ET}\right)}_2{\mathrm{Cu}}_2{\left(\mathrm{CN}\right)}_3$ compound has $t∕U\simeq 1∕8$

Figure 5

Spin correlation in the projected Fermi sea state. Measurements are done on a $24\times 24$ triangular lattice. The mean-field wave function is constructed for periodic boundary conditions and excluding the zero momentum single-particle state in order to avoid Fermi surface points while satisfying the lattice rotation symmetry for the finite system (this does not affect the long-distance properties of the wave function which is our focus here). Note the oscillating character of the correlation [with the period $\approx 2\pi ∕\left(2k_F\right),k_F\approx 2.69$]. Also note that the renormalized mean field roughly reproduces the overall magnitude of the correlations.