Bayesian parametric analytic continuation of Green's functions

Rumetshofer, Michael; Bauernfeind, Daniel; Linden, Wolfgang von der

doi:10.1103/physrevb.100.075137

Cited by 15 publications

(9 citation statements)

References 38 publications

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“…Increasing the flexibility beyond some rather small number of parameters leads to problems similar to those encountered with histograms, unless the solution can be sufficiently constrained (regularized) to avoid the anomalies related to positive definiteness, mentioned above. Some progress has been made recently along these lines, by combining rather complex, flexible functional forms with a Bayesian method to discriminate between results of different parametrizations [42]. The spectral functions here are still not completely arbitrary but should be constructed based on prior knowledge.…”

Section: A Data Fitting Based On χmentioning

confidence: 99%

“…We did not optimize ∆ω 1 and ∆ω 2 , and the peaks were much sharper than the Gaussians used in our tests. Spectra with several sharp peaks are also common and have been the subject of recent methods specifically adapted to resolving a number of such peaks [42]. The DPS approach appears to be a competitive approach for such spectra.…”

Section: Default Peak Structures and Profile Default Modelsmentioning

confidence: 99%

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Progress on stochastic analytic continuation of quantum Monte Carlo data

Shao¹,

Sandvik²

2022

Preprint

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We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. With the sampled spectrum parametrized as a large number of δ-functions in continuous frequency space, a calculation of the configurational entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. To further investigate entropic effects, we compare spectra sampled in continuous frequency with results of amplitudes sampled on a fixed frequency grid. We demonstrate equivalences between sampling and optimizing spectral functions with the maximumentropy approach, with different parametrizations corresponding to different forms of the entropy in the prior probability. These insights revise prevailing notions of the maximum-entropy method and its relationship to stochastic analytic continuation. In further developments of the sampling approach, we explore various adjustable (optimized) constraints that allow sharp low-temperature spectral features to be resolved, in particular at the lower frequency edge. The constraints, e.g., the location of the edge or the spectral weight of a quasi-particle peak, are optimized using a statistical criterion based on entropy minimization under the condition of optimal fit. We show with several examples that this method can correctly reproduce both narrow and broad quasi-particle peaks. We next introduce a parametrization for more intricate spectral functions with sharp edges, e.g., power-law singularities. We present tests with synthetic data as well as with real simulation data for the spin-1/2 Heisenberg chain, where a divergent edge of the dynamic structure factor is due to deconfined spinon excitations. Our results demonstrate that distortions of sharp edges or quasiparticle peaks, which arise with other analytic continuation methods, propagate and cause artificial spectral features also at higher energies. The constrained sampling methods overcome this problem and allow analytic continuation of spectral functions with sharp edge features at unprecedented fidelity. We present results for S = 1/2 Heisenberg 2-leg and 3-leg ladders to illustrate the ability of the methods to resolve spectral features arising from both elementary and composite excitations. Finally, we also propose how the methods developed here could be used as "pre processors" for analytic continuation by machine learning. Edge singularities and narrow quasi-particle peaks being ubiquitous in quantum many-body systems, we expect the new methods to be broadly useful and take numerical analytic continuation to a new quantitative level.

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Section: A Data Fitting Based On χmentioning

confidence: 99%

Section: Default Peak Structures and Profile Default Modelsmentioning

confidence: 99%

Progress on stochastic analytic continuation of quantum Monte Carlo data

Shao¹,

Sandvik²

2022

Preprint

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“…Instead, methods that aim to fit the Matsubara data with a physically reasonable (i.e. smooth, positive and normalized) spectral function, such as the maximum entropy analytic continuation [9][10][11][12][13][14][15][16][17][18], the stochastic analytic continuation (SAC) and variants [19][20][21][22][23][24], the sparse modeling method [25,26], or machine learning approaches [27] are used, from which the real and imaginary parts of the retarded Green's functions, self-energies, and susceptibilities are extracted. These methods work well for single-orbital systems and the diagonal components of Green's functions, especially in the presence of data with statistical uncertainties.…”

Section: Introductionmentioning

confidence: 99%

Analytical Continuation of Matrix-Valued Functions: Carathéodory Formalism

Fei,

Yeh,

Zgid

et al. 2021

Preprint

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Finite-temperature quantum field theories are formulated in terms of Green's functions and selfenergies on the Matsubara axis. In multi-orbital systems, these quantities are related to positive semidefinite matrix-valued functions of the Carathéodory and Schur class. Analysis, interpretation and evaluation of derived quantities such as real-frequency response functions requires analytic continuation of the off-diagonal elements to the real axis. We derive the criteria under which such functions exist for given Matsubara data and present an interpolation algorithm that intrinsically respects their mathematical properties. For small systems with precise Matsubara data, we find that the continuation exactly recovers all off-diagonal and diagonal elements. In real-materials systems, we show that the precision of the continuation is sufficient for the analytic continuation to commute with the Dyson equation, and we show that the commonly used truncation of off-diagonal selfenergy elements leads to considerable approximation artifacts. Our method paves the way for the systematic evaluation of Matsubara data with equations of many-body theory on the real-frequency axis.

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“…As the kernel relating G(iω n ) to G R (ω) is ill conditioned [9], a direct inversion of the relation is infeasible in practice. Instead, continuation methods such as a Padé continued fraction fit [10] of Matsubara data [11][12][13][14][15][16], the Maximum Entropy (MaxEnt) method [9,[17][18][19][20][21][22][23][24][25][26], or stochastic analytic continuation (SAC) and variants [27][28][29][30][31][32][33] are employed.…”

mentioning

confidence: 99%

Nevanlinna Analytical Continuation

Fei,

Yeh,

Gull

2020

Preprint

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Simulations of finite temperature quantum systems provide imaginary frequency Green's functions that correspond one-to-one to experimentally measurable real-frequency spectral functions. However, due to the bad conditioning of the continuation transform from imaginary to real frequencies, established methods tend to either wash out spectral features at high frequencies or produce spectral functions with unphysical negative parts. Here, we show that explicitly respecting the analytic 'Nevanlinna' structure of the Green's function leads to intrinsically positive and normalized spectral functions, and we present a continued fraction expansion that yields all possible functions consistent with the analytic structure. Application to synthetic trial data shows that sharp, smooth, and multi-peak data is resolved accurately. Application to the band structure of silicon demonstrates that high energy features are resolved precisely. Continuations in a realistic correlated setup reveal additional features that were previously unresolved. By substantially increasing the resolution of real frequency calculations our work overcomes one of the main limitations of finite-temperature quantum simulations.

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Bayesian parametric analytic continuation of Green's functions

Cited by 15 publications

References 38 publications

Progress on stochastic analytic continuation of quantum Monte Carlo data

Progress on stochastic analytic continuation of quantum Monte Carlo data

Analytical Continuation of Matrix-Valued Functions: Carathéodory Formalism

Nevanlinna Analytical Continuation

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