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In this paper, we analyze a proposed gravity dual to a $SU(N)$ Bose-Hubbardmodel, as well as construct a holographic dual of a $SU(N)$ Fermi-Hubbard modelfrom D-branes in string theory. In both cases, the $SU(N)$ is dynamical, i.e.the hopping degrees of freedom are strongly coupled to $SU(N)$ gauge bosonswhich themselves are strongly interacting. The vacuum expectation value (VEV)of the hopping term (i.e. the hopping energy) is analyzed in the gravity dualas a function of the bulk mass of the field dual to the hopping term, as wellas of the coupling constants of the model. The bulk mass controls the anomalousdimension (i.e. the critical exponent) of the hopping term in the $SU(N)$Bose-Hubbard model. We compare the hopping energy to the corresponding resultin a numerical simulation of the ungauged $SU(N)$ Bose-Hubbard model. We findagreement when the hopping parameter is smaller than the other couplings. Ouranalysis shows that the kinetic energy increases as the bulk mass increases,due to increased contributions from the IR. The holographic Bose-Hubbard modelis then compared with the string theory construction of a $SU(N)$ Fermi-Hubbardmodel. The string theory construction makes it possible to describefluctuations around a half-filled state in the supergravity limit, which map to${\cal O}(1)$ occupation number fluctuations in the Fermi-Hubbard model at halffilling. Finally, the VEV of the Bose-Hubbard model is shown to agree with theone of the fermionic Hubbard model with the help of a two-site version of theJordan-Wigner transformation.

Effective Hopping in Holographic Bose and Fermi-Hubbard Models