Ground state energy of the Bose--Hubbard model with large coordination number with a polaron-type quantum de Finetti theorem
We consider the ground state energy of the Bose--Hubbard model on a graph with large and homogeneous coordination number. In the limit of infinite coordination number, we prove convergence of the ground state energy to the minimizer of a mean-field energy functional. This functional is obtained by averaging the hopping term over the large number of connected sites, while the interaction energy is not averaged. Hence, the resulting mean-field description is in the strong coupling regime, and is expected to provide a qualitatively correct picture of the phase diagram of the Bose--Hubbard model for large enough coordination number. For our proof, we develop a new version of a de Finetti-type theorem, which we call the polaron-type quantum de Finetti theorem, and which we expect to be a more broadly useful extension of existing quantum de Finetti results. Our theorem covers the case where the Hilbert space is a tensor product of some Hilbert space with a bosonic Fock space. This theorem is applied to the convergence of the ground state energy of the Bose--Hubbard model after reducing it to a polaron-type model.