Low rank compression in the numerical solution of the nonequilibrium Dyson equation

Abstract

We propose a method to improve the computational and memory efficiency of numerical solvers for the nonequilibrium Dyson equation in the Keldysh formalism. It is based on the empirical observation that the nonequilibrium Green's functions and self energies arising in many problems of physical interest, discretized as matrices, have low rank off-diagonal blocks, and can therefore be compressed using a hierarchical low rank data structure. We describe an efficient algorithm to build this compressed representation on the fly during the course of time stepping, and use the representation to reduce the cost of computing history integrals, which is the main computational bottleneck. For systems with the hierarchical low rank property, our method reduces the computational complexity of solving the nonequilibrium Dyson equation from cubic to near quadratic, and the memory complexity from quadratic to near linear. We demonstrate the full solver for the Falicov-Kimball model exposed to a rapid ramp and Floquet driving of system parameters, and are able to increase feasible propagation times substantially. We present examples with 262 144 time steps, which would require approximately five months of computing time and 2.2 TB of memory using the direct time stepping method, but can be completed in just over a day on a laptop with less than 4 GB of memory using our method. We also confirm the hierarchical low rank property for the driven Hubbard model in the weak coupling regime within the GW approximation, and in the strong coupling regime within dynamical mean-field theory.

Memory truncated Kadanoff-Baym equations

“…The momentum-dependent cutoff could eventually allow for true multi-scale simulations of the condensed matter dynamics, with a consistent treatment of the non-thermal electron dynamics on femtosecond timescales and the order parameter dynamics on picosecond timescales. Further possible future developments include similar memory truncation to higher-order selfenergy approximations within the strong coupling solution of the DMFT impurity model 49 (such as the onecrossing approximation), and the combination of the memory truncation scheme with compact basis representations of the non-equilibrium Green's functions 42 .…”

“…Larger times can be accessed using parallel implementations 40,41 , and high-order time stepping and quadrature rules 27 which reduce the number of time-steps. Beyond this brute force approach, a promising direction are compressed storage representations of the two-time Green's functions which are compact in memory but can nevertheless be incorporated into a time-stepping procedure with little computational overhead 42 .…”

A many-body approach to transport in quantum systems: From the transient regime to the stationary state

“…[196] for large-scale simulations of cold atoms in optical lattices and Ref. [197] of the Falicov-Kimball model).…”

Adaptive Numerical Solution of Kadanoff-Baym Equations

“…The research fields requiring the solution of time-dependent, non-equilibrium many-body problems comprise, among others, cosmology and high-energy particle physics [1][2][3], spin dynamics [4], cold atoms [5][6][7][8], superconductivity [9], the Kondo effect [10,11] and other strongly correlated electronic systems [12][13][14][15]. Due to the great difficulty in addressing such problems with purely analytical methods, their numerical solution is an active area of research [16][17][18][19] and plays a central role in the understanding of time-dependent many-body phenomena.…”

“…The dynamical equations arising from the NE(Q)FT of all of these problems are generally known as Kadanoff-Baym (KB) equations [45], a set of two-time non-linear integro-differential equations. The computational effort to solve the KB equations has led to a number of approximation techniques, among them memory truncation [16,46], the generalised Kadanoff-Baym ansatz [18,47], as well as advanced computational methods such as high-order time-stepping algorithms [17], parallelised programming [48], finite-element representations [18,49], and data compression [16].…”