Stochastic mean-field theory for the disordered Bose-Hubbard model

We investigate the effect of diagonal disorder on bosons in an optical lattice described by an Anderson-Hubbard model at zero temperature. It is known that within Gutzwiller mean-field theory spatially resolved calculations suffer particularly from finite system sizes in the disordered case, while arithmetic averaging of the order parameter cannot describe the Bose glass phase for finite hopping J > 0. Here we present and apply a new stochastic mean-field theory which captures localization due to disorder, includes non-trivial dimensional effects beyond the mean-field scaling level and is applicable in the thermodynamic limit. In contrast to fermionic systems, we find the existence of a critical hopping strength, above which the system remains superfluid for arbitrarily strong disorder. Ever since the seminal paper by Fisher et al. [1], the disordered Bose Hubbard model has been the subject of theoretical and experimental investigation. In particular, the realization of the superfluid-Mott insulator transition in a gas of ultracold bosonic atoms in an optical lattice [2] has sparked a new wave of research on this field. In contrast to typical solid state systems, optical lattices allow the introduction of various types of disorder in a highly controlled manner [3]. Experimentally, several techniques have been implemented, such as speckle laser patterns [4–7], multichromatic lattices with noncommensurate wavelengths [8,9] or an additional species of atoms tunneling at a considerably lower rate [10,11]. In a recent experiment [6] it was possible to generate the first 3D fine-grained disorder potential using a speckle laser, providing an excellent realization of the 3D disordered Bose-Hubbard model. Various theoretical methods have previously been employed to investigate the transitions between Mott insulator (MI), Bose glass (BG) and superfluid (SF), such as Quantum Monte Carlo [12–16], exact diagonalization [9, 17–19], renormalization group [20], density matrix renormalization group [21] and mean-field approximations [1, 22–30]. However, spatially resolved calculations on disordered lattices suffer from finite size effects in the vicinity of phase borders, where the physics is dominated by rare events, while an arithmetically averaged Fig. 1: Within SMFT, the multiple site lattice model is approximated by an effective single-site problem, where a site is coupled to a bath of mean-field parameters (MFPs). Disorderinduced fluctuations of the MFPs are accounted for by a statistical distribution P (ψ). mean-field theory is incapable of describing the Bose glass phase at any finite hopping amplitude and T = 0 [29]. Ultracold bosonic atoms in a sufficiently deep optical lattice at moderate filling are well described by the single band Bose-Hubbard (BH) Hamiltonian [31]