Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1211.6893 · Physical Review A

Tight-binding models for ultracold atoms in honeycomb optical lattices

Julen Ibañez-Azpiroz, Asier Eiguren, Aitor Bergara, Giulio Pettini, Michele Modugno

Published Jan 9, 2013• 40 citations
arXivHTMLPDF

Abstract

We discuss how to construct tight-binding models for ultracold atoms in honeycomb potentials, by means of the maximally localized Wannier functions (MLWFs) for composite bands introduced by Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)]. In particular, we work out the model with up to third-nearest neighbors, and provide explicit calculations of the MLWFs and of the tunneling coefficients for the graphenelike potential with two degenerate minima per unit cell. Finally, we discuss the degree of accuracy in reproducing the exact Bloch spectrum of different tight-binding approximations, in a range of typical experimental parameters.

DOI
10.1103/physreva.87.011602

Related papers

Maximally localized Wannier functions for ultracold atoms in one-dimensional double-well periodic potentials

Michele Modugno, Giulio Pettini

We discuss a method for constructing generalized Wannier functions that are maximally localized at the minima of a one-dimensional periodic potential with a double well per unit cell. By following the approach of Marzari and Vanderbilt (1997 Phys. Rev. B 56 12847), we consider a set of band-mixing Wannier functions with minimal spread and design a specific two-step gauge transformation of the Bloch functions for a composite two-band system. This method is suited for efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. The behaviour of the tight-binding coefficients is analyzed here as a function of the symmetry properties of the double well (including the possibility of parity-breaking), in a range of feasible experimental parameters.

arXiv:1112.2845•New Journal of Physics•May 1, 2012•22 citations

Self-consistent tight-binding description of Dirac points moving and merging in two-dimensional optical lattices

Julen Ibañez-Azpiroz, Asier Eiguren, Aitor Bergara +2 more

We present an accurate ab initio tight-binding model, capable of describing the dynamics of Dirac points in tunable honeycomb optical lattices following a recent experimental realization [Tarruell et al., Nature (London) 483, 302 (2012)]. Our scheme is based on first-principle maximally localized Wannier functions for composite bands. The tunneling coefficients are calculated for different lattice configurations, and the spectrum properties are well reproduced with high accuracy. In particular, we show which tight-binding description is needed in order to accurately reproduce the position of Dirac points and the dispersion law close to their merging, for different laser intensities.

arXiv:1306.1796•Physical Review A•Sep 30, 2013•17 citations

Ab initio derivation of Hubbard models for cold atoms in optical lattices

R. Walters, G. Cotugno, T. H. Johnson +2 more

We derive ab initio local Hubbard models for several optical-lattice potentials of current interest, including the honeycomb and kagome lattices, verifying their accuracy on each occasion by comparing the interpolated band structures against the originals. To achieve this, we calculate the maximally localized generalized Wannier basis by implementing the steepest-descent algorithm of Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)] directly in one and two dimensions. To avoid local minima we develop an initialization procedure that is both robust and requires no prior knowledge of the optimal Wannier basis.

arXiv:1303.2213•Physical Review A•Apr 10, 2013•58 citations

Ultracold gases in optical lattices: basic concepts

Maciej Lewenstein, Anna Sanpera, V. Ahufinger

This chapter describes basic methods to realise optical potentials and optical lattices, listing in detail what can be controlled in ultracold atomic systems. It revisits the properties of non-interacting particles in periodic lattices: band structure, Bloch functions, and Wannier states. The chapter derives the Hubbard model in the tight binding approximation, and discusses Bose–Einstein condensates in optical lattices in a weak interacting limit, and in a strongly correlated regime.

Oxford University Press eBooks•Mar 8, 2012•3 citations

Ultracold fermions in a graphene-type optical lattice

Kean Loon Lee, Benoît Grémaud, Rui Han +2 more

Some important features of the graphene physics can be reproduced by loading ultracold fermionic atoms in a two-dimensional optical lattice with honeycomb symmetry and we address here its experimental feasibility. We analyze in great detail the optical lattice generated by the coherent superposition of three coplanar running laser waves with respective angles $2\ensuremath{\pi}/3$. The corresponding band structure displays Dirac cones located at the corners of the Brillouin zone and close to half-filling this system is well described by massless Dirac fermions. We characterize their properties by accurately deriving the nearest-neighbor hopping parameter ${t}_{0}$ as a function of the optical lattice parameters. Our semiclassical instanton method proves in excellent agreement with an exact numerical diagonalization of the full Hamilton operator in the tight-binding regime. We conclude that the temperature range needed to access the Dirac fermions regime is within experimental reach. We also analyze imperfections in the laser configuration as they lead to optical lattice distortions which affect the Dirac fermions. We show that the Dirac cones do survive up to some critical intensity or angle mismatches which are easily controlled in actual experiments. In the tight-binding regime, we predict, and numerically confirm, that these critical mismatches are inversely proportional to the square root of the optical potential strength. We also briefly discuss the interesting possibility of fine tuning the mass of the Dirac fermions by controlling the laser phase in an optical lattice generated by the incoherent superposition of three coplanar independent standing waves with respective angles $2\ensuremath{\pi}/3$.

arXiv:0906.4158•Physical Review A•Oct 19, 2009•132 citations

Wannier functions using a discrete variable representation for optical lattices

Saurabh Paul, Eite Tiesinga

We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following S. Kivelson [Phys. Rev. B 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$, and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigenstates using a uniform Fourier grid DVR. We then show, by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion-symmetric and asymmetric potentials to better than 10 significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the $x$ axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard model and show the dependence of these energies on lattice asymmetry.

arXiv:1609.00654•Physical Review A•Sep 7, 2016•10 citations
We discuss how to construct tight-binding models for ultracold atoms in honeycomb potentials, by means of the maximally localized Wannier functions (MLWFs) for composite bands introduced by Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)]. In particular, we work out the model with up to third-nearest neighbors, and provide explicit calculations of the MLWFs and of the tunneling coefficients for the graphenelike potential with two degenerate minima per unit cell. Finally, we discuss the degree of accuracy in reproducing the exact Bloch spectrum of different tight-binding approximations, in a range of typical experimental parameters.
We discuss a method for constructing generalized Wannier functions that are maximally localized at the minima of a one-dimensional periodic potential with a double well per unit cell. By following the approach of Marzari and Vanderbilt (1997 Phys. Rev. B 56 12847), we consider a set of band-mixing Wannier functions with minimal spread and design a specific two-step gauge transformation of the Bloch functions for a composite two-band system. This method is suited for efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. The behaviour of the tight-binding coefficients is analyzed here as a function of the symmetry properties of the double well (including the possibility of parity-breaking), in a range of feasible experimental parameters.
We present an accurate ab initio tight-binding model, capable of describing the dynamics of Dirac points in tunable honeycomb optical lattices following a recent experimental realization [Tarruell et al., Nature (London) 483, 302 (2012)]. Our scheme is based on first-principle maximally localized Wannier functions for composite bands. The tunneling coefficients are calculated for different lattice configurations, and the spectrum properties are well reproduced with high accuracy. In particular, we show which tight-binding description is needed in order to accurately reproduce the position of Dirac points and the dispersion law close to their merging, for different laser intensities.
We derive ab initio local Hubbard models for several optical-lattice potentials of current interest, including the honeycomb and kagome lattices, verifying their accuracy on each occasion by comparing the interpolated band structures against the originals. To achieve this, we calculate the maximally localized generalized Wannier basis by implementing the steepest-descent algorithm of Marzari and Vanderbilt [Phys. Rev. B 56, 12847 (1997)] directly in one and two dimensions. To avoid local minima we develop an initialization procedure that is both robust and requires no prior knowledge of the optimal Wannier basis.
Some important features of the graphene physics can be reproduced by loading ultracold fermionic atoms in a two-dimensional optical lattice with honeycomb symmetry and we address here its experimental feasibility. We analyze in great detail the optical lattice generated by the coherent superposition of three coplanar running laser waves with respective angles $2\ensuremath{\pi}/3$. The corresponding band structure displays Dirac cones located at the corners of the Brillouin zone and close to half-filling this system is well described by massless Dirac fermions. We characterize their properties by accurately deriving the nearest-neighbor hopping parameter t0{t}_{0}t0​ as a function of the optical lattice parameters. Our semiclassical instanton method proves in excellent agreement with an exact numerical diagonalization of the full Hamilton operator in the tight-binding regime. We conclude that the temperature range needed to access the Dirac fermions regime is within experimental reach. We also analyze imperfections in the laser configuration as they lead to optical lattice distortions which affect the Dirac fermions. We show that the Dirac cones do survive up to some critical intensity or angle mismatches which are easily controlled in actual experiments. In the tight-binding regime, we predict, and numerically confirm, that these critical mismatches are inversely proportional to the square root of the optical potential strength. We also briefly discuss the interesting possibility of fine tuning the mass of the Dirac fermions by controlling the laser phase in an optical lattice generated by the incoherent superposition of three coplanar independent standing waves with respective angles $2\ensuremath{\pi}/3$.
We propose a numerical method using the discrete variable representation (DVR) for constructing real-valued Wannier functions localized in a unit cell for both symmetric and asymmetric periodic potentials. We apply these results to finding Wannier functions for ultracold atoms trapped in laser-generated optical lattices. Following S. Kivelson [Phys. Rev. B 26, 4269 (1982)], for a symmetric lattice with inversion symmetry, we construct Wannier functions as eigenstates of the position operators $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x}, \stackrel{\ifmmode \hat{}\else \^{}\fi{}}{y}$, and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ restricted to single-particle Bloch functions belonging to one or more bands. To ensure that the Wannier functions are real-valued, we numerically obtain the band structure and real-valued eigenstates using a uniform Fourier grid DVR. We then show, by a comparison of tunneling energies, that the Wannier functions are accurate for both inversion-symmetric and asymmetric potentials to better than 10 significant digits when using double-precision arithmetic. The calculations are performed for an optical lattice with double-wells per unit cell with tunable asymmetry along the xxx axis and a single sinusoidal potential along the perpendicular directions. Localized functions at the two potential minima within each unit cell are similarly constructed, but using a superposition of single-particle solutions from the two lowest bands. We finally use these localized basis functions to determine the two-body interaction energies in the Bose-Hubbard model and show the dependence of these energies on lattice asymmetry.