Renormalization group analysis of dipolar Heisenberg model on square lattice

We present a detailed functional renormalization group analysis of spin-$\frac{1}{2}$ dipolar Heisenberg model on square lattice. This model is similar to the well-known $J_1\text{−}{J}_2$ model and describes the pseudospin degrees of freedom of polar molecules confined in deep optical lattice with long-range anisotropic dipole-dipole interactions. Previous study of this model based on tensor network ansatz indicates a paramagnetic ground state for certain dipole tilting angles which can be tuned in experiments to control the exchange couplings. The tensor ansatz formulated on a small cluster unit cell is inadequate to describe the spiral order, and therefore the phase diagram at high azimuthal tilting angles remains undetermined. Here, we obtain the full phase diagram of the model from numerical pseudofermion functional renormalization group calculations. We show that an extended quantum paramagnetic phase is realized between the Néel and stripe/spiral phases. In this region, the spin susceptibility flows smoothly down to the lowest numerical renormalization group scales with no sign of divergence or breakdown of the flow, in sharp contrast to the flow towards the long-range-ordered phases. Our results provide further evidence that the dipolar Heisenberg model is a fertile ground for quantum spin liquids.

Figure 1

Geometry of the dipolar Heisenberg model on square lattice. There is one polar molecule localized on each site of the square lattice within the $xy$ plane (b). Each molecule carries pseudo-spin-$\frac{1}{2}$. Their dipole moments are aligned along a common direction $\widehat{d}$ specified by the polar angle $\theta$ and azimuthal angle $\varphi$ (a). The Brillouin zone and selected high-symmetry points are shown in (c).

Figure 2

Spin configurations for the Néel, stripe, and spiral orders. These are the three competing classical long-range orders for the dipolar Heisenberg model on square lattice.

Figure 3

Phase diagram of dipolar Heisenberg model on square lattice as function of dipole tilting angles $\theta$ and $\varphi$. Three long-range orders are Néel, stripe, and spiral phases. Color contours are critical RG scales obtained from frequency-independent FRG. For three selected points (black diamonds) in the phase diagram, the flows of maximum susceptibility are shown in the lower panel. The corresponding susceptibility profiles throughout the Brillouin zone at a small RG scale are shown as the insets of the lower panel. The orange arrows denoted by ${\mathcal{C}}_1$, ${\mathcal{C}}_2$, and ${\mathcal{C}}_3$ are cuts near the phase boundaries. The dashed line indicates a qualitative boundary of the quantum paramagnetic (PM) phase.

Figure 4

Renormalization group flows of maximum susceptibility ${\chi }_{\max }$ along three cuts in the vicinity of the phase boundaries. Top row is the ${\mathcal{C}}_1$ cut from the Néel phase to stripe phase, middle row is the ${\mathcal{C}}_2$ cut from the Néel phase to spiral phase, and the bottom row is the ${\mathcal{C}}_3$ cut from the stripe to spiral phase. Insets show the susceptibility profiles near the end of each flow. Within an extended window of angle $\theta$ (top and middle row), the susceptibility flows smoothly down to $\mathrm{\Lambda }\to 0$ and shows no sign of divergence or breakdown. This points to a paramagnetic ground state.