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

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.

Figure 1

Representation of the problem in $\mathbf{k}$ space. (a) The reciprocal-lattice points for a two-dimensional oblique lattice. The black circle has a radius of $G_{\max }$ and is centered on $\Gamma$. Fourier components corresponding to reciprocal lattice points within the circle (red points) are included in the truncated basis set, while those outside (blue points) are not. (b) A mesh of wave vectors $\mathbf{k}$ for a hexagonal two-dimensional lattice used to interpolate values of functions of $\mathbf{k}$ over the Brillouin zone, which is shown by the black hexagon.

Figure 2

Flow diagram of how our software package calculates the maximally localized generalized Wannier states. First, it calculates the band structure, and from this computes the matrix elements $M_{mn}^{(\mathbf{k},\mathbf{b})}={\sum }_{\mathbf{G}}{c}_m^{(\mathbf{k},\mathbf{G})†}{c}_n^{(\mathbf{k}+\mathbf{b},\mathbf{G})}$. Second, it minimizes ${\Omega }_D^n$ for each band as far as possible without mixing the bands. Third, it reduces ${\Omega }_{\mathit{OD}}$. Fourth, it again minimizes ${\Omega }_D^n$ for each band as far as possible without further mixing the bands. Fifth, it minimizes the total $\Omega$ to its global minimum via steepest-descent minimization. Last, we compute the Hubbard parameters and construct the maximally localized generalized Wannier states.

Figure 3

One-dimensional superlattice. (a) The configuration of lasers red detuned from wavelength $\lambda$ to produce the superlattice potential. (b) The potential over the unit cell, for $s=0$ (blue solid line), $s=0.5$ (red dotted line), and $s=1$ (green dashed line). (c) The band structure corresponding to the potentials in (b).

Figure 4

Maximally localized Wannier states for the one-dimensional superlattice with $V_0=20E_R$ and $s=0.999$. (a) The dotted green and dotted-dashed blue lines are the $m=1,2$ generalized Wannier states. (b) The solid red and dashed light blue lines are the $n=1,2$ maximally localized ordinary Wannier states. (c) The spreads of the maximally localized ordinary and generalized Wannier states as a function of $s$ [line type is the same as in (a) and (b)].

Figure 5

Hopping parameters for the one-dimensional superlattice. The lines show the magnitudes $|t_j^{mn}|=|t_{\mathbf{0}\mathbf{R}}^{mn}|$ with $2\left|\mathbf{R}\right|=j\lambda$. The black, blue, red, and green lines correspond to $j=0,1,2,3$, as labeled in the figure. The solid (dashed) lines correspond to the maximally localized generalized (ordinary) Wannier states.

Figure 6

Interaction parameters for the one-dimensional superlattice. The lines show the magnitudes $|U_j^{mn}|=|U_{\mathbf{0}\mathbf{0}\mathbf{R}\mathbf{R}}^{mmnn}|$ of interactions between two particles in bands $m$ and $n$ at sites separated by $2\left|\mathbf{R}\right|=j\lambda$. The black, blue, red, and green lines correspond to $j=0,1,2,3$, as labeled in the figure. The solid (dashed) lines correspond to the maximally localized generalized (ordinary) Wannier states.

Figure 7

Accuracy of the one-dimensional superlattice Hubbard models. (a) Lowest and first excited bands for the superlattice potential with $V_0=20E_R$ and $s=0.999$. The blue and red solid (cyan and magenta dashed) lines show the interpolated lowest and first excited bands, respectively, for a tight-binding model using the maximally localized generalized (ordinary) Wannier states. For maximally localized generalized Wannier states, the interpolated bands are indistinguishable from the exact bands on this scale. (b) The standard deviation $\sigma$ between the exact bands and the interpolated bands as a function of the superlattice parameter $s$. The solid blue (dashed red) line is for maximally localized generalized (ordinary) Wannier states.

Figure 8

Hexagonal lattice. (a) The beam configuration for generating the optical lattice. The three beams are blue detuned from the wavelength $\lambda$. (b) The lattice potential, with the white line marking the boundary of the Wigner-Seitz unit cell. (c) The band structure for lattice depth $V_0=10E_R$. The energies are displayed along the path through the Brillouin zone shown in the inset. (d) Similarly for $V_0=30E_R$.

Figure 9

Maximally localized generalized Wannier states for the hexagonal lattice. The two lowest bands are shown for lattice depth $V_0=10E_R$. We have labeled the potential minima with equal hopping and interaction parameters from the “home” minimum by $j=0,1,2,3,4$.

Figure 10

Hopping and interaction parameters for the hexagonal lattice. (a) The magnitudes $|t_j|=|t_{\mathbf{0}\mathbf{R}}^{mn}|$ of the hopping parameters, as a function of lattice depth $V_0$, where the centers of $|\mathbf{0}m\rangle$ and $|\mathbf{R}n\rangle$ are the $j$th smallest distance from each other (cf. Fig. 9). The black solid, blue dotted, red dashed, green dot-dashed, and magenta dot–long-dashed lines are for $j=0,1,2,3,4$ respectively. (b) Similarly for the magnitudes $|U_j|=|U_{\mathbf{0}\mathbf{0}\mathbf{R}\mathbf{R}}^{mmnn}|$ of the interaction parameters. (c) The total standard deviation $\sigma$ between the exact lowest bands and the interpolated tight-binding bands as a function of lattice depth.

Figure 11

The kagome lattice. (a) The beam configuration for generating the optical-lattice potential. The projections of the beam wave vectors are shown in the $x$-$y$ and $x$-$z$ planes. The beams are both red and blue detuned from the same wavelength $\lambda$. (b) The resulting lattice potential, with the white line marking the boundary of the Wigner-Seitz unit cell. (c) The band structure for lattice depth $V_0=2E_R$. The energies are displayed along the path through the Brillouin zone shown in the inset.

Figure 12

Maximally localized generalized Wannier states for the kagome lattice. The three lowest bands are shown for lattice depth $V_0=10E_R$. We have labeled the potential minima with equal hopping parameters from the “home” minimum by $j=0,1,2,3,4$.

Figure 13

Hopping and density-density interaction parameters for the kagome lattice. (a) The magnitudes $|t_j|=|t_{\mathbf{0}\mathbf{R}}^{mn}|$ of the hopping parameters, as a function of lattice depth $V_0$, where the centers of $|\mathbf{0}m\rangle$ and $|\mathbf{R}n\rangle$ are the $j$th smallest distance from each other (cf. Fig. 12). The black solid, blue dotted, red dashed, green dot-dashed, and magenta dot–long-dashed lines are for $j=0,1,2,3,4$, respectively. (b) Similarly for the magnitudes $|U_j|=|U_{\mathbf{0}\mathbf{0}\mathbf{R}\mathbf{R}}^{mmnn}|$ of the interaction parameters. (c) The total standard deviation $\sigma$ between the exact lowest bands and the interpolated tight-binding bands as a function of lattice depth.

Figure 14

Progressive phase update method. Starting from the bottom left corner of the Brillouin zone mesh (blue dots), the phase of the neighbor to the right (connected by the black arrow) is adjusted such that $\text{Im}\left[\text{ln}{M}^{(\mathbf{k},\mathbf{b})}\right]={\vartheta }_{x,1}/\mathtt{M}$ for the pair, where ${\vartheta }_{x,1}$ is the Berry phase in this direction. The same adjustment is made for the next neighbor and so on until the end of the mesh is reached (lower right corner). One then has $\text{Im}\left[\text{ln}{M}^{(\mathbf{k},\mathbf{b})}\right]={\vartheta }_{x,1}/\mathtt{M}$ for all mesh points along this path. The process is repeated for each path in the next reciprocal-lattice direction as shown by the red arrows.