Generalized hard-core dimer model approach to low-energy Heisenberg frustrated antiferromagnets: General properties and application to the kagome antiferromagnet

We propose a general nonperturbative scheme that quantitatively maps the low-energy sector of spin-1/2 frustrated Heisenberg antiferromagnets to effective generalized quantum dimer models. We develop the formal lattice-independent frame and establish some important results on (i) the locality of the generated Hamiltonians, (ii) how full resummations can be performed in this renormalization scheme. The method is then applied to the much debated kagome antiferromagnet for which a fully resummed effective Hamiltonian—shown to capture the essential properties and provide deep insights on the microscopic model [D. Poilblanc, M. Mambrini, and D. Schwandt, Phys. Rev. B 81, 180402(R) (2010)]—is derived.

Figure 1

(Color online) Overlap $⟨\phi |\psi ⟩$ between two VB wave functions $|\phi ⟩$ and $|\psi ⟩$. The amplitude of $⟨\phi |\psi ⟩$ is driven by the number of loops appearing in the overlap graph (bottom frame).

Figure 2

(Color online) Application of ${\widehat{P}}_l=2{\widehat{\mathbf{S}}}_i.{\widehat{\mathbf{S}}}_j+1/2$ on a loop diagram $⟨\phi |\psi ⟩$ for the three types of bonds $l=\left(i,j\right)$ (dashed lines) described in the text: (a) “internal odd,” (b) “external,” and (c) “internal even.” The state $|\psi ⟩$ (respectively, $|\phi ⟩$) is represented with blue thick (respectively, red thin) bonds. Note that $P_l$ is applied on $|\psi ⟩$. Figures nearby the diagrams are the ratios $⟨\phi |{\widehat{P}}_l|\psi ⟩/⟨\phi |\psi ⟩$ (bold) and ${\epsilon }_{i,j}^{\phi ,\psi }=⟨\phi |{\widehat{\mathbf{S}}}_i.{\widehat{\mathbf{S}}}_j|\psi ⟩/⟨\phi |\psi ⟩$ (italic).

Figure 3

(Color online). Semiquantitative phase diagram of the extended GQDM for the kagome lattice as a function of the pinwheel resonance amplitude $J_{12}$ and the parameter $\gamma$ [see Eq. (43)]. The two dashed blue lines represent (i) an interpolation $\lambda {\widehat{\mathcal{H}}}_{\text{eff}}+\left(1-\lambda \right){\widehat{\mathcal{H}}}_{\text{RK}}$ between the RK model and the effective model for the Heisenberg quantum antiferromagnet and (ii) a cut at $\gamma =1$ along $J_{12}$, and correspond to the simulations performed in Ref. 7. Red lines are qualitative phase transitions. Note that for $\gamma >0.9$, a finite $J_{12}$ lifts the degeneracy between two VBC’s with identical 36-site unit cells but opposite parities $P_{\pm }$ of their resonating pinwheels (yellow stars).

Figure 4

(Color online) Explicit bosonic $\left(\epsilon =+1\right)$ and fermionic $\left(\epsilon =-1\right)$ conventions for the 15 VB states on six sites.