Parameters of the Bose-Hubbard model require the knowledge of the Wannier functions from the lowets band of the optical lattice potential $V(x) = V_{0}sin^{2}(kx)$ according to equations:
$$J = int dx w(x)left[-frac{hbar^2}{2m}frac{d^2}{dx^2} + V(x) right]w(x-d),$$
$$U propto int dx |w(x)|^4.$$
Here $d$ – lattice spacing and $k = pi/d$. Wannier function can be constructed from the Bloch functions associated with the lowest band:
$$w(x) = frac{1}{sqrt{N}}sumlimits_{q in rm BZ}psi_{q}(x),$$
where $N$ – number of lattice points, $q = frac{p}{N}frac{2pi}{d}$, $0 le p le N-1$. Using Bloch’s theorem $psi_{q}(x) = e^{iqx}u_{q}(x)$, one can write equation for periodic function $u_{q}(x)$:
$$left[ -left(frac{d}{dy} + ifrac{q}{k} right)^2 + frac{V_{0}}{2 E_{R}}left(1 - cos(2y) right)right]u_{q}(y) = frac{E}{E_{R}}u_{q}(y),$$
with $E_{R} = frac{hbar^2 k^2}{2m}$ and $y = kx$. This is a Mathieu-like equation. I solved it numerically using truncated Fourier expansion for function $u_{q}(y)$:
$$u_{q}(y) approx sumlimits_{l=-M_{rm cutoff}}^{M_{cutoff}}C^{q}_{l}e^{i2ly}.$$
With this procedure I can find Bloch functions $psi_{q}(y)$ and then construct wannier function. The problem is that is not localized at all. I read here that it may be a problem of ‘gauge freedom’. Basically, bloch functions can be multiplied by the phase factor $e^{iphi(q)}psi_{q}(x)$ and still be a good solution. Of course this operation influences wannier function and may result in better localization.
Is there a simple method to construct maximally localized wannier function without going through the theory of Marzari et al.? I only want wannier function for 1D optical lattice potential in the case of isolated band (so $V_{0}$ large enough).
