Abstract
We modify a nonlinear $\ensuremath{\sigma}$ model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ for the description of a granular disordered system in the presence of both the Coulomb repulsion and the Cooper pairing. We show that under certain controlled approximations the action of this model is reduced to the Ambegaokar-Eckern-Sch\"on (AES) action, which is further reduced to the Bose-Hubbard (or ``dirty-boson'') model with renormalized coupling constants. We obtain an effective action which is more general than the AES one but still simpler than the full $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ action. This action can be applied in the region of parameters where the reduction to the AES or the Bose-Hubbard model is not justified. This action may lead to a different picture of the superconductor-insulator transition in two-dimensional systems.
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