Systematic low-energy effective field theory for magnons and holes in an antiferromagnet on the honeycomb lattice

Based on a symmetry analysis of the microscopic Hubbard and $t$-$J$ models, a systematic low-energy effective field theory is constructed for hole-doped antiferromagnets on the honeycomb lattice. In the antiferromagnetic phase, doped holes are massive due to the spontaneous breakdown of the $\text{SU}{\left(2\right)}_s$ symmetry, just as nucleons in Quantum Chromodynamics (QCD) pick up their mass from spontaneous chiral symmetry breaking. In the broken phase, the effective action contains a single-derivative term, similar to the Shraiman-Siggia term in the square lattice case. Interestingly, an accidental continuous spatial rotation symmetry arises at leading order. As an application of the effective field theory, we consider one-magnon exchange between two holes and the formation of two-hole bound states. As an unambiguous prediction of the effective theory, the wave function for the ground state of two holes bound by magnon exchange exhibits $f$-wave symmetry.

Figure 1

Bipartite non-Bravais honeycomb lattice consisting of two triangular Bravais sublattices. The translation vectors are $a_1$ and $a_2$.

Figure 2

The dispersion relation $E\left(k\right)/t$ for a single hole in an antiferromagnet on the honeycomb lattice simulated in the $t$-$J$ model for $J/t=2$ (see Ref. 26).

Figure 3

$\{A_1,A_2,A_3\}$ and $\{B_1,B_2,B_3\}$ sublattice structure and the corresponding primitive lattice vectors.

Figure 4

Sublattice vectors from Eq. (4.13).

Figure 5

Tree-level Feynman diagram for one-magnon exchange between two holes.

Figure 6

Probability distribution for the ground state of two holes of flavors $\alpha$ and $\beta$.