Deconfined quantum criticality in SU(3) antiferromagnets on the triangular lattice

We propose field theories for a deconfined quantum critical point in SU(3) antiferromagnets on the triangular lattice. In particular we consider the continuous transition between a magnetic, three-sublattice color-ordered phase and a trimerized SU(3) singlet phase. Starting from the magnetically ordered state we derive a critical theory in terms of fractional bosonic degrees of freedom, in close analogy to the well-developed description of the SU(2) Néel—valence bond solid (VBS) transition on the square lattice. Our critical theory consists of three coupled $C{P}^2$ models and we study its fixed point structure using a functional renormalization group approach in a suitable large $N$ limit. We find a stable critical fixed point and estimate its critical exponents, thereby providing an example of deconfined criticality beyond the universality class of the $C{P}^N$ model. In addition we present a complementary route towards the critical field theory by studying topological defects of the trimerized SU(3) singlet phase.

Figure 1

A realization of the three-sublattice color-ordered state of SU(3) spins on the triangular lattice.

Figure 2

A particular realization of the trimerized SU(3) singlet state. The arrows represent the ${\mathbb{Z}}_6$-clock order parameter.

Figure 3

Mean field phase diagram corresponding to the potential $V\left(\mathbf{z}\right)$ from Eq. (20). The dashed lines indicate the phase boundaries.

Figure 4

Diagrammatic representation of the flow equations. Full (wiggly) lines denote scalar (gauge) field propagators, while blue square (triangles) denote four-point (three-point) vertices. Vertical lines represent regulator insertions.

Figure 5

Sketch of the four degenerate ground states of the SU(2) VBS. The red lines indicate SU(2) singlets, the blue arrows represent the ${\mathbb{Z}}_4$ order. This picture was adapted from Ref. [11].

Figure 6

Sketch of the vortex and antivortex in the SU(2) case. The black circle indicates the vortex core, which carries a free spin. A translation by a lattice unit vector takes a vortex into an antivortex.

Figure 7

Two out of six degenerate ground states of the trimer state. The red triangles indicate SU(3) singlets; the blue arrows represent the ${\mathbb{Z}}_6$ order.

Figure 8

A vortex $z_{\alpha 1}$ centered on sublattice 1. The black circle represents the free SU(3) spin and the dashed green lines indicate the domain walls.

Figure 9

Two composite vortices centered on sublattice 1, which we represent by the fields $x_{\alpha 1}$ (left) and $y_{\alpha 1}$ (right), respectively. We identify vortices which are obtained from $x_{\alpha i},y_{\alpha i}$ by space reflection along the axis indicated in Fig. 8 with $x_{\alpha i},y_{\alpha i}$, accordingly.

Figure 10

A $z_{\alpha 2}$ vortex centered on sublattice 2, obtained by a shift of a vortex $y_{\alpha 1}$ on sublattice 1 (see Fig. 9) to the right by one unit vector.