Jason Kaye, Sophie Beck, Alex Barnett, Lorenzo Van Muñoz, Olivier Parcollet
SciPost Phys. 15, 062 (2023) ·
published 15 August 2023
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We present efficient methods for Brillouin zone integration with a non-zero but possibly very small broadening factor $\eta$, focusing on cases in which downfolded Hamiltonians can be evaluated efficiently using Wannier interpolation. We describe robust, high-order accurate algorithms automating convergence to a user-specified error tolerance $\varepsilon$, emphasizing an efficient computational scaling with respect to $\eta$. After analyzing the standard equispaced integration method, applicable in the case of large broadening, we describe a simple iterated adaptive integration algorithm effective in the small $\eta$ regime. Its computational cost scales as $\mathcal{O}(\log^3(\eta^{-1}))$ as $\eta \to 0^+$ in three dimensions, as opposed to $\mathcal{O}(\eta^{-3})$ for equispaced integration. We argue that, by contrast, tree-based adaptive integration methods scale only as $\mathcal{O}(\log(\eta^{-1})/\eta^{2})$ for typical Brillouin zone integrals. In addition to its favorable scaling, the iterated adaptive algorithm is straightforward to implement, particularly for integration on the irreducible Brillouin zone, for which it avoids the tetrahedral meshes required for tree-based schemes. We illustrate the algorithms by calculating the spectral function of SrVO$_3$ with broadening on the meV scale.
SciPost Phys. 10, 091 (2021) ·
published 26 April 2021
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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
262144 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.
Dr Kaye: "We thank the referee for a tho..."
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