Functional renormalization group for the anisotropic triangular antiferromagnet

We present a functional renormalization group scheme that allows us to calculate frustrated magnetic systems of arbitrary lattice geometry beyond $O(200)$ sites from first principles. We study the magnetic susceptibility of the antiferromagnetic (AFM) spin-$1/2$ Heisenberg model ground state on the spatially anisotropic triangular lattice, where $J^{\prime }$ denotes the coupling strength of the intrachain bonds along one lattice direction and $J$ the coupling strength of the interchain bonds. We identify three distinct phases of the Heisenberg model. Increasing $\xi =J^{\prime }/J$ from the effective square lattice $\xi =0$, we find an AFM Néel order to spiral order transition at ${\xi }_{c1}~0.6–0.7$, with an indication that it is of second order. In addition, above the isotropic point at ${\xi }_{c2}~1.1$, we find a first-order transition to a magnetically disordered phase with collinear AFM stripe fluctuations.

Figure 1

(a) Triangular lattice structure. The horizontal bonds correspond to coupling $J^{\prime }$ and the others to coupling $J$. (b) Schematic plot of the hexagonal Brillouin zone. Different magnetic order resides at different points [AFM Néel order, ${120}^{°}$ Néel order, collinear AFM order (c-AFM)]. The open circles relate to the filled circles by reciprocal lattice vectors, i.e., AFM order corresponds to two points in the Brillouin zone, collinear AFM to four points, and ${120}^{°}$ Néel order to six points.

Figure 2

Graphic representation of the FRG differential equations. Bare lines denote the (renormalized and scale-dependent) Green’s functions and slashed lines the single scale propagators. Equations (a) and (b) are the FRG equations for the self-energy and the two-particle vertex, respectively [without distinguishing between the different pairing channels on the right side of (b)]. The Katanin scheme is given by the replacement (c). By fusing the external legs of the two-particle vertex, the magnetic susceptibility is obtained, as shown in (d).

Figure 3

Static magnetic susceptibility resolved for the whole Brillouin zone, by varying $\xi =J^{\prime }/J$ from the effective square lattice $\xi =0.0$ close to the triangular lattice $\xi =0.9$. Bottom Left: Two-particle vertex flow for the AFM Néel channel (blue) versus the ${120}^{°}$ Néel channel (red) at $\xi =0$ with respect to the IR cutoff flow parameter $\Lambda$. We observe that the AFM vertices start to diverge, signaling a magnetic instability. Susceptibilities are always given in units of $1/J$. The respective types of order for the distinct peaks positions are shown in Fig. 1b.

Figure 4

Static magnetic susceptibility, by varying $\xi$ from the AFM Néel side $\xi =0.55$ across the phase transition to spiral ordering. The already broadened AFM Néel peak splits into two incommensurate peaks that successively shift toward the edges of the Brillouin zone where the ${120}^{°}$ Néel order peaks reside.

Figure 5

Static magnetic susceptibility, by varying $\xi$ from the isotropic triangular lattice $\xi =1.0$ toward the one-dimensional chain limit $\xi \to \infty$. Bottom left: Vertex flow at the $\xi =1$ point. The rise of the ${120}^{°}$ Néel channel shows the ordering instability at the isotropic triangle point. Compared to the flow at $\xi =0$ (Fig. 3), the rise takes place at a much lower scale of $\Lambda$, indicating a lower ordering scale for the triangular lattice.

Figure 6

Static magnetic susceptibility, by varying $\xi$ in the strongly anisotropic regime from $\xi =2$ to $\xi =4$. While the spectral structure remains rather invariant of c-AFM type, which is strongly smeared along the $k_y$ direction, the total spectral weight slightly decreases with increasing $\xi$.