We describe a generalization of the local density approximation for ultra-cold atomic systems, which takes into account the gradients of the Green’s functions with respect to position when solving the local Dyson equation and, hence, is sensitive to the local variation of the system within the trap. We use this technique along with dynamical mean-field theory to simulate the Hubbard model data of the Swiss group, which measured the Mott transition in a three-dimensional optical lattice with ultra-cold 40 K atoms. We simulate the same size system as used in experiment, which includes about 8 million lattice sites and up to 300,000 atoms. We describe our computational algorithm in detail, along with modifications needed for the communications to make it work on the Cray XE6. We show strong scaling curves up to approximately 43,500 cores. The code performs remarkably well with perfect linear scaling up to the largest systems we ran on. The results that we present are a comparison of the dynamical mean-field theory results within the generalized gradient approximation to the strong-coupling perturbation theory, which is known to be accurate for high-temperature and large interaction strength. This work is part of the Defense Advanced Research Projects Agency (DARPA) optical lattice emulator program.
We describe a generalization of the local density approximation for ultra-cold atomic systems, which takes into account the gradients of the Green’s functions with respect to position when solving the local Dyson equation and, hence, is sensitive to the local variation of the system within the trap. We use this technique along with dynamical mean-field theory to simulate the Hubbard model data of the Swiss group, which measured the Mott transition in a three-dimensional optical lattice with ultra-cold 40 K atoms. We simulate the same size system as used in experiment, which includes about 8 million lattice sites and up to 300,000 atoms. We describe our computational algorithm in detail, along with modifications needed for the communications to make it work on the Cray XE6. We show strong scaling curves up to approximately 43,500 cores. The code performs remarkably well with perfect linear scaling up to the largest systems we ran on. The results that we present are a comparison of the dynamical mean-field theory results within the generalized gradient approximation to the strong-coupling perturbation theory, which is known to be accurate for high-temperature and large interaction strength. This work is part of the Defense Advanced Research Projects Agency (DARPA) optical lattice emulator program.
We present a nonequilibrium strong-coupling approach to inhomogeneous systems of ultracold atoms in optical lattices. We demonstrate its application to the Mott-insulating phase of a two-dimensional Fermi-Hubbard model in the presence of a trap potential. Since the theory is formulated self-consistently, the numerical implementation relies on a massively parallel evaluation of the self-energy and the Green's function at each lattice site, employing thousands of CPUs. While the computation of the self-energy is straightforward to parallelize, the evaluation of the Green's function requires the inversion of a large sparse 10(d) × 10(d) matrix, with d > 6. As a crucial ingredient, our solution heavily relies on the smallness of the hopping as compared to the interaction strength and yields a widely scalable realization of a rapidly converging iterative algorithm which evaluates all elements of the Green's function. Results are validated by comparing with the homogeneous case via the local-density approximation. These calculations also show that the local-density approximation is valid in nonequilibrium setups without mass transport.
Defense Advanced Research Projects Agency (DARPA) is running a program to create an analog quantum-mechanical simulator for strongly interacting quantum particles. It is called an optical lattice emulator, and it involves ultracold neutral atoms moving in a corrugated periodic potential created by an optical lattice. The atoms interact with each other via collisions. In this work, we show how one can apply inhomogenenous dynamical mean-field theory (IDMFT) as an approximate computational tool to study the behavior of just such a system. Ultimately, conventional computation will be employed to benchmark the optical lattice emulator on a numerically tractable problem (a verification and validation study), and then the emulator will be applied to problems that cannot be solved with conventional computation. Here, we demonstrate an efficient massively parallel implementation of the IDMFT algorithm that is transportable, scales to many thousands of processors, and runs at approximately 50% of the theoretical peak speed of the machine.
We develop a generalized gradient expansion of the inhomogeneous dynamical mean-field theory method for determining properties of ultracold atoms in a trap. This approach goes beyond the well-known local density approximation and at higher temperatures, in the normal phase, it shows why the local density approximation works so well, since the local density and generalized gradient approximations are essentially indistinguishable from each other (and from the exact solution within full inhomogeneous dynamical mean-field theory). But because the generalized gradient expansion only involves nearest-neighbor corrections, it does not work as well at low temperatures, when the systems enter into ordered phases. This is primarily due to the problem that ordered phases often satisfy some global constraints, which determine the spatial ordering pattern, and the local density and generalized gradient approximations are not able to impose those kinds of constraints; they also overestimate the tendency to order. The theory is applied to phase separation of different mass fermionic mixtures represented by the Falicov-Kimball model and to determining the entropy per particle of a fermionic system represented by the Hubbard model. The generalized gradient approximation is a useful diagnostic for the accuracy of the local density approximation---when both methods agree, they are likely accurate, when they disagree, neither is likely to be correct.
With recent advances in the field of ultra cold atoms one can, by trapping atoms at low temperatures by laser beams, simulate systems which can be adequately described by single-band lattice Hamiltonians. Also, due to the high parameter tunability of the experimental setups, lattice disorder can be introduced in a con- trolled fashion in these systems. This thesis considers disordered/ordered interacting fermion lattice systems in equilibrium in one and two dimensions, subject to trap- ping parabolic potentials, where different levels of description, ranging from exact where possible (in 1D, Density Matrix Renormalization Group, DMRG), to approxi- mate (based on several local-density approximations within the framework of Lattice Density Functional Theory) are used. The exchange-correlation potentials consid- ered come from Many-Body Approximations obtained using Green’s functions, as well as from an exact local-density approximation (LDA) based on the Bethe-Ansatz (BALDA) in 1D. Both one- and two-dimensional systems were studied in equilib- rium, essentially looking at the ground state density profiles. Furthermore, in 2D, a pseudo-dynamics representing the trap-opening in the complete adiabatic limit was also studied. For one-dimensional systems, it is found that, in general, BALDA yields very good results compared to DMRG, except for the low density limit, but DMRG can describe features that none of the LDA:s considered can reproduce. It is also found that the strength of the external potential affects the impact that the exchange-correlation potential has on the system. Many of these features translate to the 2D case. However, a new aspect emerges in two dimensions, related to the competition of disorder and interaction. Here an important outcome is that, on opening the trap at an ideally adiabatic rate, different MBA:s (and thus different LDA:s) provide different minimal vs maximal expansion radii of the particle cloud, as a result of the interplay of disorder and interaction, and the underlying square lattice structure. For the 2D results, exact benchmarks were not available, and our findings may thus need further validation, by e.g. considering several disorder configurations or, ideally, by performing full Green’s function calculations. These considerations are summarised in our conclusions and outlook remarks, where possible directions for future investigation are highlighted.
We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean-field theory (DMFT). We benchmark the solver in the two-band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization-group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the antiferromagnetic phase transition in the unfrustrated Bethe lattice. The Neel temperature obtained by the fast impurity solver agrees very well with the QMC results in the large Hubbard U limit. The method is a promising tool to be used in combination with the LDA+DMFT to study Mott insulators starting from first principles.
We present a comprehensive study of the thermodynamic properties of the three-dimensional fermionic Hubbard model, with application to cold fermionic atoms subject to an optical lattice and a trapping potential. Our study is focused on the temperature range of current experimental interest. We employ two theoretical methods---dynamical mean-field theory and high-temperature series---and perform comparative benchmarks to delimit their respective range of validity. Special attention is devoted to understand the implications that thermodynamic properties of this system have on cooling. Considering the distribution function of local occupancies in the inhomogeneous lattice, we show that, under adiabatic evolution, the variation of any observable (e.g., temperature) can be conveniently disentangled into two distinct contributions. The first contribution is due to the redistribution of atoms in the trap during the evolution, while the second one comes from the intrinsic change of the observable. Finally, we provide a simplified picture of a recently proposed cooling procedure, based on spatial entropy separation, by applying this method to an idealized model.