We will first outline the state of the art of experiments on cold atoms in an optical lattice. We will then show how numerics are useful for these systems, and finally discuss how successful numerical studies have been, which is the keystone of this proposal.
Since the seminal paper by Greiner et al. , showing the transition from the superfluid to the Mott-insulating phase in the Bose-Hubbard model, physicists have realized that modeling strongly-interacting systems in the atomic physics lab is feasible.
The field had started with the theoretical prediction by Jaksch et al., showing theoretically that this model could indeed be analyzed realistically in experiments . The key observation was that the atom-photon interactions could be used and controlled in such a way that the Bose-Hubbard model can be realized for a sufficiently long time and at sufficiently low temperature.
Systems consisting of atoms in an optical lattice have the advantage over strongly-interacting systems of (1) being ultra-clean, (2) the interaction couplings are controllable, and (3) the type, mass and density of the atoms are controllable. Having the possibility to change the parameters of the Hamiltonian at will makes these systems prime candidates for building a quantum analog computer. On the negative side, the detection of (neutral) atoms is more difficult than detection in condensed matter physics, the temperature is relatively high (e.g., in units of the Fermi temperature for fermionic systems present experiments operate on the scale of 0.1 ), there are no phonons since the lattice is built by laser light, and there is an external harmonic trapping potential.
Experimental progress has been rapid. The most recent advances include the observation of the Mott-instulating phase in the Fermi-Hubbard model  and the observation of the superexchange mechanism in a two-well system .
These experiments have mostly been explained qualitatively so far. Further progress is only possible when lower temperatures are reached and better detection schemes developed, but also when current experiments are better understood quantitatively, and this is where numerics become useful.
In particular for the bosons, the theory is well understood and the numerical algorithms (Quantum Monte Carlo) very powerful. The so-called 'worm algorithm' is at present able to address the Bose-Hubbard model from first principles and for realistic system sizes . Previous studies addressed individual aspects of the experiments, such as the role of the trapping potential, the role of entropy in fixing the temperature and the influence of the time-of-flight duration . Fully ab-initio studies took off approximately a year ago and are still going on, with promising first results. The overall agreement is close to excellent, but it turns out that there are deviations which are not completely negligible though controllable. They hint at experimental imperfections, such as density variations, lattice laser calibration uncertainties, heating, and losses of atoms. Estimating the importance of such effects accurately has now become available thanks to the large-scale simulations.
For the fermions, such numerical methods as DMFT  and high-temperature series expansions  addressed the recent observation of the Mott-insulating state. Fermions are intrinsically harder to handle experimentally than bosons. The Mott-insulating state has only been observed experimentally over the last few months . Numerical studies using Dynamical Mean Field Theory (DMFT) and high-temperature series expansions have already appeared [8, 2], but finding a quantity that clearly indicated the transition is hard to find. Temperature and out-of-equilibrium effects are more serious concerns here than for the bosons. It is clear that much more numerical and experimental work is needed before consensus can be reached in the community. Numerical studies have also looked at lower temperatures, trying to see where anti-ferromagnetism sets in . At these temperatures the numerical methods are no longer exact, but the approximations need further assessment.
Experimental studies have also been undertaken to measure critical exponents. The critical exponent of the correlation length of the U(1) transition between a normal and a condensed three-dimensional Bose gas was measured . The Berezinskii-Kosterlitz-Thouless for a two-dimensional Bose gas was also successfully studied experimentally . A full numerical study taking the trapping potential, temperature, heating, and detection schemes into account is numerically challenging and largely open.
(Collège de France, Paris and Flatiron Institute, New York)
(University of Stuttgart)
(Swiss Federal Institute of Technology Zurich (ETHZ))