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DoD High Performance Computing Modernization Program Users Group Conference (HPCMP-UGC)

Efficiently Generalizing Ultra-Cold Atomic Simulations via Inhomogeneous Dynamical Mean-Field Theory from Two- to Three-Dimensions

James K. Freericks, H. R. Krishnamurthy, Pierre Carrier, Yousef Saad

Published Jun 1, 2010• 2 citations
Publisher

Abstract

We describe techniques that we are implementing to move inhomogeneous dynamical mean-field theory simulations from two-to three-dimensions. Two-dimensional simulations typically run on 2,000 - 10,000 lattice sites, while three-dimensional simulations typically need to run on 1,000,000 or more lattice sites. The inhomogeneous dynamical mean-field theory requires the diagonal of the inverse of many sparse matrices with the same sparsity pattern, and a dimension equal to the number of lattice-sites. For two-dimensional systems, we have employed general dense LAPACK routines since the matrices are small enough. For three-dimensional systems, we need to employ sparse matrix techniques. Here, we present one possible strategy for the sparse matrix routine, based on the well-known Lanczos technique, with a long run of the algorithm and (partial) reorthogonalization. This approach is about two-times faster than the LAPACK routines with identical accuracy, and hence will become the standard we use on the two-dimensional problems. We illustrate this approach on the problem of increasing the efficiency for pre-forming dipolar molecules in K-Rb mixtures on a lattice. We compare the local density approximation to inhomogeneous dynamical mean-field theory to illustrate how the local density approximation fails at low-temperature, and to illustrate the benefits of the new algorithms. For a three-dimensional problem, a speed-up of 1,000 or more times is needed. We end by discussing some options that are promising toward reaching this goal.

DOI
10.1109/hpcmp-ugc.2010.17

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We describe techniques that we are implementing to move inhomogeneous dynamical mean-field theory simulations from two-to three-dimensions. Two-dimensional simulations typically run on 2,000 - 10,000 lattice sites, while three-dimensional simulations typically need to run on 1,000,000 or more lattice sites. The inhomogeneous dynamical mean-field theory requires the diagonal of the inverse of many sparse matrices with the same sparsity pattern, and a dimension equal to the number of lattice-sites. For two-dimensional systems, we have employed general dense LAPACK routines since the matrices are small enough. For three-dimensional systems, we need to employ sparse matrix techniques. Here, we present one possible strategy for the sparse matrix routine, based on the well-known Lanczos technique, with a long run of the algorithm and (partial) reorthogonalization. This approach is about two-times faster than the LAPACK routines with identical accuracy, and hence will become the standard we use on the two-dimensional problems. We illustrate this approach on the problem of increasing the efficiency for pre-forming dipolar molecules in K-Rb mixtures on a lattice. We compare the local density approximation to inhomogeneous dynamical mean-field theory to illustrate how the local density approximation fails at low-temperature, and to illustrate the benefits of the new algorithms. For a three-dimensional problem, a speed-up of 1,000 or more times is needed. We end by discussing some options that are promising toward reaching this goal.
Inhomogeneous dynamical mean-field theory has been employed to solve many interesting strongly interacting problems from transport in multilayered devices to the properties of ultracold atoms in a trap. The main computational step, especially for large systems, is the problem of calculating the inverse of a large sparse matrix to solve Dyson's equation and determine the local Green's function at each lattice site from the corresponding local self-energy. We present a new e_cient algorithm, the Lanczos-based low-rank algorithm, for the calculation of the inverse of a large sparse matrix which yields this local (imaginary time) Green's function. The Lanczos-based low-rank algorithm is based on a domain decomposition viewpoint, but avoids explicit calculation of Schur complements and relies instead on low-rank matrix approximations derived from the Lanczos algorithm, for solving the Dyson equation. We report at least a 25-fold improvement of performance compared to explicit decomposition (such as sparse LU) of the matrix inverse. We also report that scaling relative to matrix sizes, of the low-rank correction method on the one hand and domain decomposition methods on the other, are comparable.
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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 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.
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.
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.