Truncating the memory time in nonequilibrium dynamical mean field theory calculations

Schüler, Michael; Eckstein, Martin; Werner, Philipp

doi:10.1103/physrevb.97.245129

Cited by 27 publications

(35 citation statements)

References 96 publications

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“…For example, if the retarded Green's function decays rapidly in the off-diagonal direction, we do not need to store its entries with moduli smaller than some threshold, and we can ignore them in the history sums. Though this is sometimes the case [50,51], decay in the Green's functions and self energies depends strongly on the parameter regime. On the other hand, we have found that the Green's functions and self energies for many systems of physical interest display a smoothness property which leads to data sparsity; they have numerically low rank off-diagonal blocks.…”

Section: A Fast Memory-efficient Methods Based On Hierarchical Low Rank Compressionmentioning

confidence: 99%

“…51], high-order time stepping and quadrature rules [22], parallelization [49,52], and direct Monte Carlo sampling at long times [38,40]. Of these, only the memory truncation methods succeed in systematically reducing the asymptotic cost and memory complexity, but these are restricted to specific parameter regimes in which the Green's functions are numerically sparse [50,51]. Another alternative is to approximate the full propagation scheme with quantum kinetic equations or their generalizations, like the generalized Kadanoff-Baym ansatz (GKBA) [53][54][55].…”

Section: Introductionmentioning

confidence: 99%

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

Kaye

Golež

2021

SciPost Phys.

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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 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.

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Section: A Fast Memory-efficient Methods Based On Hierarchical Low Rank Compressionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Kaye

Golež

2021

SciPost Phys.

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“…Note that we are using the term "free" in the sense of non-interacting. External fields coupling with the particles are included in the definition of G −1 Q , but interactions are left into the action term S I , equation (11). Our goal here is to find the direct GFs by inverting the matrices G −1 Q .…”

Section: Derivation Of the General Formulamentioning

confidence: 99%

Discrete-time construction of nonequilibrium path integrals on the Kostantinov–Perel’ time contour

Secchi¹,

Polini²

2019

J. Phys. A: Math. Theor.

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Rigorous nonequilibrium actions for the many-body problem are usually derived by means of path integrals combined with a discrete temporal mesh on the Schwinger-Keldysh time contour. The latter suffers from a fundamental limitation: the initial state on this contour cannot be arbitrary, but necessarily needs to be described by a non-interacting density matrix, while interactions are switched on adiabatically. The Kostantinov-Perel' contour overcomes these and other limitations, allowing generic initial-state preparations. In this Article, we apply the technique of the discrete temporal mesh to rigorously build the nonequilibrium path integral on the Kostantinov-Perel' time contour.

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“…For instance, the formation and stability of (Bi-)Polarons is a central problem and considerable effort has been taken for its investigation [6][7][8][9][10][11][12][13][14][15]. A broad class of different methods such as quantum Monte Carlo [16,17], density-functional theory [18], density-matrix embedding theory [19], or dynamical mean-field theory [20][21][22] has been explored to study its various aspects. Evidently, the task to numerically describe such low-dimensional, strongly-correlated quantum systems has been subject to a vast development.…”

Section: Introductionmentioning

confidence: 99%

Efficient and flexible approach to simulate low-dimensional quantum lattice models with large local Hilbert spaces

2021

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Quantum lattice models with large local Hilbert spaces emerge across various fields in quantum many-body physics. Problems such as the interplay between fermions and phonons, the BCS-BEC crossover of interacting bosons, or decoherence in quantum simulators have been extensively studied both theoretically and experimentally. In recent years, tensor network methods have become one of the most successful tools to treat such lattice systems numerically. Nevertheless, systems with large local Hilbert spaces remain challenging. Here, we introduce a mapping that allows to construct artificial U(1)U(1) symmetries for any type of lattice model. Exploiting the generated symmetries, numerical expenses that are related to the local degrees of freedom decrease significantly. This allows for an efficient treatment of systems with large local dimensions. Further exploring this mapping, we reveal an intimate connection between the Schmidt values of the corresponding matrixproductstate representation and the singlesite reduced density matrix. Our findings motivate an intuitive physical picture of the truncations occurring in typical algorithms and we give bounds on the numerical complexity in comparison to standard methods that do not exploit such artificial symmetries. We demonstrate this new mapping, provide an implementation recipe for an existing code, and perform example calculations for the Holstein model at half filling. We studied systems with a very large number of lattice sites up to L=501L=501 while accounting for N_{\rm ph}=63 phonons per site with high precision in the CDW phase.

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Truncating the memory time in nonequilibrium dynamical mean field theory calculations

Cited by 27 publications

References 96 publications

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

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

Discrete-time construction of nonequilibrium path integrals on the Kostantinov–Perel’ time contour

Efficient and flexible approach to simulate low-dimensional quantum lattice models with large local Hilbert spaces

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