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arXiv:cond-mat/9905019 · Journal of Low Temperature Physics

Self-Consistent Treatment of Dynamical Correlation Functions Using a Spectral Representation Technique

Martin Letz, F. Marsiglio

Published Oct 1, 1999• 2 citations
arXivHTMLPDF
DOI
10.1023/a:1021822224492

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We theoretically examine the finite-temperature spectral function of Fermi polarons in three dimensions by using a self-consistent many-body TTT-matrix theory in real frequency. In comparison with the previous results from a non-self-consistent many-body TTT-matrix approach, we show that the treatment of self-consistency in the impurity Green's function leads to notable changes in almost all the dynamical quantities, including the vertex function, impurity self-energy, and spectral function. Eventually, it gives rise to quantitatively different predictions for the measurable radio-frequency spectrum and Raman spectrum at finite temperature. Using the recent spectroscopic measurements as a benchmark, we find that the self-consistent many-body TTT-matrix theory somehow provides a better explanation for the experimental data. The notable difference in the predictions from the non-self-consistent and self-consistent theories suggests that more accurate theoretical descriptions are needed in order to fully account for the current spectroscopic observations on Fermi polarons.
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We develop a nonperturbative framework for generic nonintegrable many-body systems that reorganizes the expansion of diagonal Green's functions. Starting from exact projection identities and the spectral representation of the resolvent, we derive a recursive hierarchy for the self-energy in which cross-correlated propagation processes are systematically rewritten in terms of diagonal resolvents. Under a diagonal closure approximation, the hierarchy becomes formally closed yet remains systematically improvable. The framework combines two nonperturbative mechanisms. First, a Lanczos continued-fraction representation provides a recursive single-resolvent structure that naturally produces non-Lorentzian spectral features beyond self-consistent Born approximations. Second, an exact projected multi-resolvent hierarchy introduces nonlocal frequency couplings through products of resolvents and their Hilbert transforms. These contributions mix parity sectors under energy reflection and generate spectral skewness, which is absent in single-resolvent closures. To solve the resulting equations, we employ a hierarchy of Lorentzian, Gaussian, and Voigt-type ansätze together with an effective Faddeeva self-energy representation ensuring analyticity and causality. Spectral broadening, distribution tails, and higher-order fluctuations emerge from the interplay between continued-fraction recursion and multi-resolvent correlations. The framework requires no small expansion parameters or diagrammatic truncations, relying instead on ETH-type statistical assumptions appropriate for dense chaotic spectra. It provides a unified route from microscopic interactions to emergent spectral structure, revealing a progression from single-pole self-consistent dynamics to continued-fraction renormalization and finally to multi-resolvent interference effects.