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International Journal of Theoretical Physics

Complex frequencies, analytic continuation, and equivalent spectra in generalized quantum theory

E. C. G. Sudarshan

Published Jul 1, 1995• 0 citations
Publisher
DOI
10.1007/bf00671360

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The spectral density of a fluctuating light beam may be determined from the knowledge of both the modulus and the phase of the complex degree of self-coherence γ(τ) of the beam. The phase itself may be determined from the modulus and from the location of the zeros of the analytic continuation of γ(τ) in the lower half of the complex τ plane. In the present paper results of an investigation are presented which show that the determination of the zeros is equivalent to the solution of a certain inhomogeneous eigenvalue problem of the Sturm-Liouville type on a semi-infinite frequency range. This eigenvalue problem is found to be equivalent to a certain stability problem in mechanics. Although no general technique for the solution of this type of an eigenvalue problem appears to be known, the new formulation may be used to determine spectral profiles for which the associated degree of self-coherence has zeros at prescribed points in the complex τ plane. Some illustrative examples are given.
The analytic continuation of imaginary-time quantum Monte Carlo data to extract real-frequency spectra remains a key problem in connecting theory with experiment. Here we present a fast and efficient stochastic optimization method (FESOM) as a more accessible variant of the stochastic optimization method introduced by Mishchenko et al. [Phys. Rev. B 62, 6317 (2000)], and we benchmark the resulting spectra with those obtained by the standard maximum entropy method for three representative test cases, including data taken from studies of the two-dimensional Hubbard model. We generally find that our FESOM approach yields spectra similar to the maximum entropy results. In particular, while the maximum entropy method yields superior results when the quality of the data is strong, we find that FESOM is able to resolve fine structure with more detail when the quality of the data is poor. In addition, because of its stochastic nature, the method provides detailed information on the frequency-dependent uncertainty of the resulting spectra, while the maximum entropy method does so only for the spectral weight integrated over a finite frequency region. We therefore believe that this variant of the stochastic optimization approach provides a viable alternative to the routinely used maximum entropy method, especially for data of poor quality.
We report multipronged progress on the stochastic averaging approach to numerical analytic continuation of imaginary-time correlation functions computed by quantum Monte Carlo simulations. After reviewing the conventional maximum-entropy approach and established stochastic analytic continuation methods, we present several new developments in which the configurational entropy of the sampled spectrum plays a key role. Parametrizing the spectrum as a large number of δ-functions in continuous frequency space, an exact calculation of the entropy lends support to a simple goodness-of-fit criterion for the optimal sampling temperature. We also compare spectra sampled in continuous frequency with those from amplitudes sampled on a fixed frequency grid. Insights into the functional form of the entropy in different cases allow us to demonstrate equivalence in a generalized thermodynamic limit (large number of degrees of freedom) of the average spectrum and the maximum-entropy solution, with different parametrizations corresponding to different forms of the entropy in the prior probability. These results 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 quasi-particle 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 spectra 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 in many applications.