Absence of Long-Range Order in a Triangular Spin System with Dipolar Interactions

The antiferromagnetic Heisenberg model on the triangular lattice is perhaps the best known example of frustrated magnets, but it orders at low temperatures. Recent density matrix renormalization group (DMRG) calculations find that the next nearest neighbor interaction $J_2$ enhances the frustration, and it leads to a spin liquid for $J_2/J_1\in (0.08,0.15)$. In addition, a DMRG study of a dipolar Heisenberg model with longer range interactions gives evidence for a spin liquid at a small dipole tilting angle $\theta \in [0,10°)$. In both cases, the putative spin liquid region appears to be small. Here, we show that for the triangular lattice dipolar Heisenberg model, a robust quantum paramagnetic phase exists in a surprisingly wide region, $\theta \in [0,54°)$, for dipoles tilted along the lattice diagonal direction. We obtain the phase diagram of the model by functional renormalization group (RG), which treats all magnetic instabilities on equal footing. The quantum paramagnetic phase is characterized by a smooth continuous flow of vertex functions and spin susceptibility down to the lowest RG scale, in contrast to the apparent breakdown of RG flow in phases with stripe or spiral order. Our finding points to a promising direction to search for quantum spin liquids in ultracold dipolar molecules.

Figure 1

A dipolar Heisenberg model on the triangular lattice in the $xy$ plane. (a) The exchange $J_{ij}$ depends on the dipole orientation $\widehat{d}$, with a polar angle $\theta$ and an azimuthal angle $\varphi$. (b) High symmetry points within the Brillouin zone. (c) The competing exchange couplings $J_{ij}$ (site $i$ is at the origin) for $\varphi =30°$ and $\theta =0°$, 45°, and 90°, respectively. The size of the circle indicates the magnitude of $J_{ij}$, with positive (negative) $J_{ij}$ shown in blue (orange). (d) The zero temperature phase diagram at $\varphi =30°$ includes a wide quantum paramagnetic phase, $\theta \in [0,54°)$, and a spiral phase.

Figure 2

A zero temperature phase diagram of the dipolar Heisenberg model on a triangular lattice as a function of the dipole tilting angle ($\theta$, $\varphi$), showing the quantum paramagnetic (PM), stripe, and spiral phases. The solid and dashed white lines are the phase boundaries (see main text). The color contours show estimated critical temperature ${\mathrm{\Lambda }}_s$ from static FRG. Spin susceptibility profiles $\chi (\mathbf{p})$ at scale $\mathrm{\Lambda }<0.5$ are shown in the middle panels for representative points marked with ${\mathcal{P}}_1$, ${\mathcal{M}}_1$ etc. The flows of peak susceptibility ${\chi }_{\max }$ are shown in the bottom panels. For ${\mathcal{M}}_1$ and ${\mathcal{M}}_3$, the flow becomes unstable below some RG scale indicated by the arrows.