Many Body Density of States of a system of non interacting fermions
ORAL
Abstract
Originally considered in the context of nuclear physics by Hans Bethe [1], the problem of calculating a Many Body Density of States (MBDoS) has been eclipsed by the success of mean field theories. Indeed, within this framework, the concept of quasi particle has focused attention on single or few bodies densities of states, in order to explain equilibrium and transport properties of myriads of condensed matter systems.
It is only recently, with the progress in quantum engineering and simulation, that the necessity to calculate Many Body densities of states has came back to light. For instance, a phenomenon like Many Body Localization requires to go beyond the low energy physics and consider excitations from the full energy range of the system spectrum [3,4]. More generally, closed quantum systems exhibiting unconventional stationary states require to enumerate the number of accessible states in order to build their statistical description. Such an enumeration requires the MBDoS whose calculation is challenging due to the difficulty in counting states while enforcing the exchange symmetry [4].
In the present work, we introduce a new approach for calculating the MBDoS of systems of non interacting fermions. The starting point of our method is the principal component analysis of a filling matrix F describing how N fermions can be configured into K single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of universal spectra given by the principal components of the filling matrix. The weighting coefficients involve renormalized energies obtained from the single body spectrum and indicate which components dominate the spectral behavior of the system under consideration [5].
By considering each principal spectrum and their degeneracies, one can classify non interacting many body states and quantify how they will be affected by a subsequent interaction, depending if this interaction is single, two or generally, k-body like. We illustrate the potential of our method on the Anderson localization problem.
[1] H. Bethe, Phys. Rev., (1936)
[2] Schreiber et al, Science, (2015)
[3] Gross et al, Science (2017)
[4] V. Zelevinsky and M. Horoi, Prog. in Part. And Nucl. Phys. (2018)
[5] R. Lefevre et al, arXiv:2208.02236
It is only recently, with the progress in quantum engineering and simulation, that the necessity to calculate Many Body densities of states has came back to light. For instance, a phenomenon like Many Body Localization requires to go beyond the low energy physics and consider excitations from the full energy range of the system spectrum [3,4]. More generally, closed quantum systems exhibiting unconventional stationary states require to enumerate the number of accessible states in order to build their statistical description. Such an enumeration requires the MBDoS whose calculation is challenging due to the difficulty in counting states while enforcing the exchange symmetry [4].
In the present work, we introduce a new approach for calculating the MBDoS of systems of non interacting fermions. The starting point of our method is the principal component analysis of a filling matrix F describing how N fermions can be configured into K single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of universal spectra given by the principal components of the filling matrix. The weighting coefficients involve renormalized energies obtained from the single body spectrum and indicate which components dominate the spectral behavior of the system under consideration [5].
By considering each principal spectrum and their degeneracies, one can classify non interacting many body states and quantify how they will be affected by a subsequent interaction, depending if this interaction is single, two or generally, k-body like. We illustrate the potential of our method on the Anderson localization problem.
[1] H. Bethe, Phys. Rev., (1936)
[2] Schreiber et al, Science, (2015)
[3] Gross et al, Science (2017)
[4] V. Zelevinsky and M. Horoi, Prog. in Part. And Nucl. Phys. (2018)
[5] R. Lefevre et al, arXiv:2208.02236
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Publication: R. Lefevre et al, arXiv:2208.02236
Presenters
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Gregoire Ithier
Royal Holloway University of London
Authors
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Gregoire Ithier
Royal Holloway University of London
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Rémi Lefèvre
Royal Holloway, University of London, Royal Holloway University of London
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Krissia d Zawadzki
Trinity College Dublin