Emergent Critical Phase and Ricci Flow in a 2D Frustrated Heisenberg Model

We introduce a two-dimensional frustrated Heisenberg antiferromagnet on interpenetrating honeycomb and triangular lattices. Classically the two sublattices decouple, and “order from disorder” drives them into a coplanar state. Applying Friedan’s geometric approach to nonlinear sigma models, we obtain the scaling of the spin stiffnesses governed by the Ricci flow of a four-dimensional metric tensor. At low temperatures, the relative phase between the spins on the two sublattices is described by a six-state clock model with an emergent critical phase.

Figure 1

(a) Heisenberg model on the “windmill” lattice. (b) Definition of angles $\alpha$ and $\beta$ describing the relative orientation of magnetic order on triangular and honeycomb lattice. (c) and (d) Angle dependent free energy correction $\delta F$ from thermal and quantum spin fluctuations for parameters $J_{hh}=J_{tt}=1$, $J_{th}=0.4$, $T=1$. Panel (d) is for fixed $\beta =\pi /2$.

Figure 2

(a) Schematic phase diagram. (b) Coplanar RG flow of the variables $I_{\alpha }^{\prime }$ (blue, increasing), $\overline{I}=(I_1{I}_2{I}_3{)}^{1/3}$ (green dashed), $(I_2-I_1)/\overline{I}$ (red), and $r$ (pink dotted). Curves are normalized to initial values at $l_{\gamma }$. Upper panel is for $J_{tt}\gg J_{hh}$ with $J_{hh}=1$, $J_{tt}=5$, $J_{th}=0.4$, $T=0.6$, and initial values $\overline{I}=5.3$, $(I_2-I_1)/\overline{I}=0.27$, $r=0.82$, $I_{\alpha }^{\prime }=1.2$. Decoupling is due to $(I_1-I_2)/\overline{I}\to 0$. Lower panel is for $J_{hh}\gg J_{tt}$ with $J_{hh}=5$, $J_{tt}=1$, $J_{th}=0.4$, $T=0.5$, and initial values $\overline{I}=4.5$, $(I_2-I_1)/\overline{I}=2.1$, $r=0.11$, $I_{\alpha }^{\prime }=1.1$. Decoupling is due to $r\to 0$.