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The $\mathbb{Z}_2$ topological phase in the quantum dimer model on theKagom\'e-lattice is a candidate for the description of the low-energy physicsof the anti-ferromagnetic Heisenberg model on the same lattice. We study theextend of the topological phase by interpolating between the exactly solvableparent Hamiltonian of the topological phase and an effective low-energydescription of the Heisenberg model in terms of a quantum-dimer Hamiltonian.Therefore, we perform a perturbative treatment of the low-energy excitations inthe topological phase including free and interacting quasi-particles. We find aphase transition out of the topological phase far from the Heisenberg point.The resulting phase is characterized by a spontaneously broken rotationalsymmetry and a unit cell involving six sites.

Extend of the $\mathbb{Z}_2$-spin liquid phase on the Kagom\'e-lattice