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Suppression of scattering from slow to fast subsystems and application to resonantly Floquet-driven impurities in strongly interacting systems

by Friedrich Hübner

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Submission summary

Authors (as registered SciPost users): Friedrich Hübner
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-12-04
Date submitted: 2023-11-29 00:41
Submitted by: Hübner, Friedrich
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Atomic, Molecular and Optical Physics - Theory
  • Condensed Matter Physics - Theory
Approach: Theoretical


We study solutions to the Lippmann-Schwinger equation in systems where a slow subsystem is coupled to a fast subsystem via an impurity. Such situations appear when a high-frequency Floquet-driven impurity is introduced into a low-energy system, but the driving frequency is at resonance with a high-energy band. In contrast to the case of resonant bulk driving, where the particles in the low-energy system are excited into the high-energy band, we surprisingly find that these excitations are suppressed for resonantly driven impurities. Still, the transmission through the impurity is strongly affected by the presence of the high-energy band in a universal way that does not depend on the details of the high-energy band. We apply our general result to two examples and show the suppression of excitations from the low-energy band into the high-energy band: a) bound pairs in a Fermi-Hubbard chain scattering at a driven impurity, which is at resonance with the Hubbard interaction and b) particles in a deep optical lattice described by the tight-binding approximation, which scatter at a driven impurity, whose driving frequency equals the band gap between the two lowest energy bands.

Published as SciPost Phys. 16, 005 (2024)

Author comments upon resubmission

Dear editor,

I took care of the minor revisions asked for by the referee. I added more citations about non-equilibrium Green's function methods. I have also clarified the sentence '... which would be useful to describe impurities which are too complicated to study via non-perturbative methods ...'. and replaced `too complicated' by `too computationally expensive'.

About this I would like to add the following comment: Of course one can in principle study any impurity in any model using NEGF or other non-perturbative methods. What I meant with 'too complicated' was that for sufficiently complicated impurities, performing analytical or numerical computations using exact non-perturbative methods might become infeasible given the available resources. I am not an expert in the theory of NEGF, but to my understanding, in practice, if one would like to apply NEGF to impurity scattering it seems very crucial that the self-energy of the leads is known. For the standard tight-binding lattices without particle interaction these are of course well known.

Please consider the two examples I discuss in appendix G and H. In appendix G I discuss an impurity where the leads are given by semi-infinite Hubbard chains (so particles in the leads are interacting). In appendix H the leads are given by semi-infinite optical lattices (particles are non-interacting, but this model contains infinitely many bands). I do not think the full non-perturbative self-energies of both models are known.

For more general impurities, especially if they are embedded in interacting systems in higher dimensions, an exact numerical analysis using non-perturbative methods might become infeasible due to the exponentially large Hilbert space. This is of course a general problem in interacting many-body quantum systems and there have been many ideas on how to still make good predictions. One of them is the idea of separation of scales, which is the basis for my analysis (but also behind other well-known techniques like the high-frequency expansion or hydrodynamic gradient expansions): The two above mentioned examples can be solved analytically in the regime I am considering, which demonstrates the drastic simplification compared to exact non-perturbative methods. Due to the completely general derivation my technique can also be applied to more general models (in particular interacting ones) given that they are in the correct regime. Even if the resulting simplified expressions are not analytically solvable, they can still add physical intuition and systematically pinpoint the most relevant objects to be computed via other methods. In this paper I only discussed the zeroth order approximation, but by extending to higher order one can systematically gain insight into resonant scattering at impurities even if the system is only approximately in that regime.

So to conclude, in my opinion the method I developed in this paper can help to gain understanding of impurities in physical situations which are difficult to treat with exact non-perturbative methods due to finite availability of resources.

Kind regards,
Friedrich Huebner

List of changes

Changed sentence '... which would be useful to describe impurities which are too complicated to study via non-perturbative methods ...' to 'This could complement exact
non-perturbative methods (like the non-equilibrium Green’s function methods [21,39–45]) in
studying scattering at impurities and could perhaps be used to gain insights into impurities
that are too computationally expensive to treat in practice via these methods.'

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