Quantum Paramagnet in a $\pi$ Flux Triangular Lattice Hubbard Model

We propose the $\pi$ flux triangular lattice Hubbard model ($\pi$ THM) as a prototypical setup to stabilize magnetically disordered quantum states of matter in the presence of charge fluctuations. The quantum paramagnetic domain of the $\pi$ THM that we identify for intermediate Hubbard $U$ is framed by a Dirac semimetal for weak coupling and by 120° Néel order for strong coupling. Generalizing the Klein duality from spin Hamiltonians to tight-binding models, the $\pi$ THM maps to a Hubbard model which corresponds to the $(J_{\mathrm{H}},J_{\mathrm{K}})=(-1,2)$ Heisenberg-Kitaev model in its strong coupling limit. The $\pi$ THM provides a promising microscopic testing ground for exotic finite-$U$ spin liquid ground states amenable to numerical investigation.

Figure 1

(a) Flux pattern and signs of the real nearest neighbor hoppings on the triangular lattice. The $\pi$’s denote triangles that are threaded by a $\pi$ flux. (b) Contour plot of the spin-degenerate upper band of the semimetallic band structure (blue spots indicate the Dirac nodes at zero energy); the lower band follows from particle-hole symmetry. The Brillouin zone (black) is spanned by the reciprocal vectors ${\mathit{b}}_1=(2\pi /\sqrt{3})(1,\sqrt{3})$ and ${\mathit{b}}_2=(2\pi /\sqrt{3})(-1/2,\sqrt{3}/2)$. (c) Four-sublattice structure of a Klein transformation. “$x$,” “$y$,” and “$z$” indicate the axis around which a spin rotation of angle $\pi$ is performed. The full dot sublattice remains unchanged.

Figure 2

Phase diagram of Eq. (2) as obtained by VCA. $U_{c1}$, $U_{c2}$, ${\mathrm{\Delta }}_{\mathrm{c}}$, and $m_{120°}$ are calculated for a lattice covering with 12-site (mirror) clusters as sketched in the inset (the magnetization $m_{120°}=1$ denotes the saturation value). From single-particle spectra, there are three phases: semimetal (SM), nonmagnetic insulator (NMI), and 120° Néel antiferromagnetic insulator (AFM).

Figure 3

Strength of dimer loop resonances $|E_{12}|$ [see Eqs. (5) and (6)] for the 4-site loop in panel (a) and all 6-site loops in panels (b)–(d) calculated for the $\pi$ THM and the regular THM. The insets illustrate the according resonances. Except for (d), dimer resonances result in a larger energy gain for the $\pi$ THM than the regular THM.