In this paper, we analyze a proposed gravity dual to a SU(N) Bose-Hubbard model, as well as construct a holographic dual of a SU(N) Fermi-Hubbard model from 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 bosons which themselves are strongly interacting. The vacuum expectation value (VEV) of the hopping term (i.e. the hopping energy) is analyzed in the gravity dual as a function of the bulk mass of the field dual to the hopping term, as well as of the coupling constants of the model. The bulk mass controls the anomalous dimension (i.e. the critical exponent) of the hopping term in the SU(N) Bose-Hubbard model. We compare the hopping energy to the corresponding result in a numerical simulation of the ungauged SU(N ) Bose-Hubbard model. We find agreement when the hopping parameter is smaller than the other couplings. Our analysis shows that the kinetic energy increases as the bulk mass increases, due to increased contributions from the IR. The holographic Bose-Hubbard model is then compared with the string theory construction of a SU(N) Fermi-Hubbard model. The string theory construction makes it possible to describe fluctuations around a half-filled state in the supergravity limit, which map to O(1) occupation number fluctuations in the Fermi-Hubbard model at half filling. Finally, the VEV of the Bose-Hubbard model is shown to agree with the one of the fermionic Hubbard model with the help of a two-site version of the Jordan-Wigner transformation.

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Published in Journal of High Energy Physics, v. 2019, issue 1, article 45, p. 1-49.

© The Author(s) 2019

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Article funded by SCOAP3.

This work was in part supported by JSPS KAKENHI Grants No. 26870284 and JP17K17822 (M.T.). The work of R.M. was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, by the U.S. Department of Energy under Contract No. DE-FG-88ER40388, by the Alexander-von- Humboldt Foundation through a Feodor Lynen postdoctoral fellowship, and by the DFG via SFB 1170 "Tocotronics". S.P. was supported by NSF grant DMR-1056536, and thanks the hospitality of the TIFR for the final stage of this work.

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