Tight-binding models for ultracold atoms in optical lattices: general formulation and applications

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SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 59, Issue 6: 660001(2016) https://doi.org/10.1007/s11433-015-0514-5

Tight-binding models for ultracold atoms in optical lattices: general formulation and applications

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  • ReceivedDec 18, 2015
  • AcceptedFeb 3, 2016
  • PublishedApr 8, 2016
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Abstract

Tight-binding models for ultracold atoms in optical lattices can be properly defined by using the concept of maximally localized Wannier functions for composite bands. The basic principles of this approach are reviewed here, along with different applications to lattice potentials with two minima per unit cell, in one and two spatial dimensions. Two independent methods for computing the tight-binding coefficients---one ab initio, based on the maximally localized Wannier functions, the other through analytic expressions in terms of the energy spectrum---are considered. In the one dimensional case, where the tight-binding coefficients can be obtained by designing a specific gauge transformation, we consider both the case of quasi resonance between the two lowest bands, and that between $s$ and $p$ orbitals. In the latter case, the role of the Wannier functions in the derivation of an effective Dirac equation is also reviewed. Then, we consider the case of a two dimensional honeycomb potential, with particular emphasis on the Haldane model, its phase diagram, and the breakdown of the Peierls substitution. Tunable honeycomb lattices, characterized by movable Dirac points, are also considered. Finally, general considerations for dealing with the interaction terms are presented.


Acknowledgment

Acknowledgments

Most of the material presented in this review is the result of previous collaborations with A. Bergara, A. Eiguren, X. Lopez-Gonzalez, J. Sisti, J. Zakrzewski. We would like to thank them all. This work was supported by the Universidad del Pais Vasco/Euskal Herriko Unibertsitatea (Grant No. UFI 11/55), the Ministerio de Economia y Competitividad (Grant No. FIS2012-36673-C03-03), the Basque Government (Grant No. IT-472-10), the Helmholtz Gemeinschaft Deutscher-Young Investigators Group (Grant No. VH-NG-717, Functional Nanoscale Structure and Probe Simulation Laboratory) and the Impuls und Vernetzungsfonds der Helmholtz-Gemeinschaft Postdoc Programme.


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