Effective Hopping in Holographic Bose and Fermi-Hubbard Models

TL;DR

This paper develops holographic dual models for $SU(N)$ Bose-Hubbard and Fermi-Hubbard systems, analyzing their hopping energies and comparing with numerical and string theory results, revealing insights into strongly coupled gauge interactions.

Contribution

It constructs and analyzes holographic duals for $SU(N)$ Bose-Hubbard and Fermi-Hubbard models, linking gravity dual parameters to lattice model properties and comparing with numerical simulations.

Findings

Hopping energy depends on bulk mass and coupling constants.

Agreement with numerical simulations when hopping is small.

Holographic models capture fluctuations around half-filling.

Abstract

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 ${\cal O}(1)$ 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.