Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0506248 · Physical Review B

Extension of the Fröhlich method to 4-fermion interactions

P. Tarasewicz, Dominik Baran

Published Mar 31, 2006• 19 citations
arXivHTMLPDF

Abstract

Higher-order terms of the transformed electron-phonon Hamiltonian ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$, obtained by performing the Fr\"ohlich transformation, are investigated. The influence of terms discarded by Fr\"ohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Fr\"ohlich-type transformation is performed, which yields, among others, an effective four-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the four-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with the interaction coupling given by $\ensuremath{-}{D}_{{\mathbf{k}}_{F}}^{6}∕{\ensuremath{\omega}}_{{\mathbf{k}}_{F}}^{5}$, where ${D}_{\mathbf{k}}$ is electron-phonon coupling, ${\ensuremath{\omega}}_{\mathbf{k}}$ is the phonon energy, and ${\mathbf{k}}_{F}$ is the Fermi momentum. The form of higher-order terms of the original Fr\"ohlich-transformed ${H}_{e\text{\ensuremath{-}}\mathit{ph}}$ is also found, up to terms proportional to the sixth power of the coupling---that is, up to those which yield the effective four-electron interactions.

DOI
10.1103/physrevb.73.094524

Related papers

Flow Equations for Electron-Phonon Interactions

Peter Lenz, Franz Wegner

A recently [9] proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlichtransformations [5]. The two approaches are compared. Our result will turn out to be less singular than Fröhlich’s. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fröhlich’s transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. 1 New address: Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstraße 55,

•Jan 1, 1996•38 citations

The generalized Lang–Firsov transformation and the three- and the four-fermion interactions in a lattice fermion model

P. Tarasewicz

A new two-band polaronic Hamiltonian is derived from a generalized periodic Anderson model. This is achieved by making use of a new generalized Lang–Firsov transformation on the periodic Anderson model completed with new terms representing the interactions between local phonons and both single electrons and pairs of them with opposite spins residing on the same lattice site. The novelty consists in the appearance of three- and four-fermion terms in the resulting Hamiltonian. The interaction between phonons and pairs of electrons with opposite spins was introduced at first time by Hirsch in order to describe the weakening of the Coulomb repulsion acting between two electrons with opposite spins occupying the same atomic orbital observed in experiments. The three-fermion terms represent the local interactions between a single polarons from one band and a pair of them with opposite spins from the other one. Whether it is attractive or repulsive depends on the signs of couplings of phonons to electron pairs. The four-fermion term stands for the local attraction between pairs of electrons with opposite spins from two bands. In the case of presence of such an interaction, the formation of local fermion quartets is possible. A simplified version of the generalized Hamiltonian is investigated in the end of the paper. The influence of the three- and the four-fermion terms on the properties of the two-band fermion system with two order parameters is examined. Expressions for the order parameters at zero temperatures and finite ones along with the critical temperatures for two-electron subsystems have been derived. This was achieved in the atomic limit by making use of the mean field method.

International Journal of Modern Physics B•Jun 7, 2016•4 citations

Phonon-Induced Renormalization and Interaction: An Improvement on Frohlich Transformation

Han Zheng

Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency $ω_p$ is much smaller than the Fermi energy $E_F$ to that of the small polarons when $ω_p/E_F\ge 1$. (2)The jump of momentum distribution of electron number $n_{\bf k}$ at the Fermi surface goes to zero when the dimensionless coupling constant $λ$ increases to the critical value $λ_c\ge 1$, which means a possible broken down of the Fermi-liquid description.

arXiv:cond-mat/0004388•Apr 24, 2000•0 citations

Similarity renormalization of the electron‐phonon coupling

Andreas Mielke

Abstract We study the problem of the phonon‐induced electron‐electron interaction in a solid. Starting with a Hamiltonian that contains an electron‐phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitly. The effective interactions are calculated to second order in the electronphonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.

arXiv:cond-mat/9609065•Annalen der Physik•Jan 1, 1997•40 citations

A note on the Fröhlich dynamics in the strong coupling limit

David Mitrouskas

Abstract We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state $$\varphi _0 \otimes \xi _\alpha $$ φ0⊗ξα , where $$\varphi _0$$ φ0 is the electron ground state of the Pekar energy functional and $$\xi _\alpha $$ ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to $$\alpha ^2$$ α2 . In the present note we prove that a similar approximation holds for $$t=O(\alpha ^2)$$ t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to $$\alpha ^{-2}$$ α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order $$\alpha ^2$$ α2 , while the phonon fluctuations around the coherent state $$\xi _\alpha $$ ξα can be described by a time-dependent Bogoliubov transformation.

arXiv:2003.11448•Letters in Mathematical Physics•Apr 1, 2021•11 citations

A note on the Fröhlich dynamics in the strong coupling limit

David Mitrouskas

We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ξα, where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.

•Jan 1, 2021•14 citations
Higher-order terms of the transformed electron-phonon Hamiltonian He\ensuremath-ph{H}_{e\text{\ensuremath{-}}\mathit{ph}}He\ensuremath-ph​, obtained by performing the Fr"ohlich transformation, are investigated. The influence of terms discarded by Fr"ohlich (in particular those proportional to the third power of electron-phonon coupling) on the effective Hamiltonian is examined. To this end a second Fr"ohlich-type transformation is performed, which yields, among others, an effective four-electron interaction. This interaction is reduced to a form admitting solution of thermodynamics. The form of the coupling of the four-electron interaction is found. By applying standard approximations, it is shown that this interaction is attractive with the interaction coupling given by \ensuremath−DkF6∕\ensuremathωkF5\ensuremath{-}{D}_{{\mathbf{k}}_{F}}^{6}∕{\ensuremath{\omega}}_{{\mathbf{k}}_{F}}^{5}\ensuremath−DkF​6​∕\ensuremathωkF​5​, where Dk{D}_{\mathbf{k}}Dk​ is electron-phonon coupling, \ensuremathωk{\ensuremath{\omega}}_{\mathbf{k}}\ensuremathωk​ is the phonon energy, and kF{\mathbf{k}}_{F}kF​ is the Fermi momentum. The form of higher-order terms of the original Fr"ohlich-transformed He\ensuremath-ph{H}_{e\text{\ensuremath{-}}\mathit{ph}}He\ensuremath-ph​ is also found, up to terms proportional to the sixth power of the coupling---that is, up to those which yield the effective four-electron interactions.
A recently [9] proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlichtransformations [5]. The two approaches are compared. Our result will turn out to be less singular than Fröhlich’s. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fröhlich’s transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. 1 New address: Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstraße 55,
A new two-band polaronic Hamiltonian is derived from a generalized periodic Anderson model. This is achieved by making use of a new generalized Lang–Firsov transformation on the periodic Anderson model completed with new terms representing the interactions between local phonons and both single electrons and pairs of them with opposite spins residing on the same lattice site. The novelty consists in the appearance of three- and four-fermion terms in the resulting Hamiltonian. The interaction between phonons and pairs of electrons with opposite spins was introduced at first time by Hirsch in order to describe the weakening of the Coulomb repulsion acting between two electrons with opposite spins occupying the same atomic orbital observed in experiments. The three-fermion terms represent the local interactions between a single polarons from one band and a pair of them with opposite spins from the other one. Whether it is attractive or repulsive depends on the signs of couplings of phonons to electron pairs. The four-fermion term stands for the local attraction between pairs of electrons with opposite spins from two bands. In the case of presence of such an interaction, the formation of local fermion quartets is possible. A simplified version of the generalized Hamiltonian is investigated in the end of the paper. The influence of the three- and the four-fermion terms on the properties of the two-band fermion system with two order parameters is examined. Expressions for the order parameters at zero temperatures and finite ones along with the critical temperatures for two-electron subsystems have been derived. This was achieved in the atomic limit by making use of the mean field method.
Starting from the Flohlich electron-phonon model, which has a history of 50 years, a new unitary transformation is proposed to implement the perturbation treatment. Our main results are: (1)The phonon-induced interaction shows a crossover from the BCS-like potential when the phonon frequency ωpω_pωp​ is much smaller than the Fermi energy EFE_FEF​ to that of the small polarons when ωp/EF≥1ω_p/E_F\ge 1ωp​/EF​≥1. (2)The jump of momentum distribution of electron number nkn_{\bf k}nk​ at the Fermi surface goes to zero when the dimensionless coupling constant λλλ increases to the critical value λc≥1λ_c\ge 1λc​≥1, which means a possible broken down of the Fermi-liquid description.
Abstract We study the problem of the phonon‐induced electron‐electron interaction in a solid. Starting with a Hamiltonian that contains an electron‐phonon interaction, we perform a similarity renormalization transformation to calculate an effective Hamiltonian. Using this transformation singularities due to degeneracies are avoided explicitly. The effective interactions are calculated to second order in the electronphonon coupling. It is shown that the effective interaction between two electrons forming a Cooper pair is attractive in the whole parameter space. For a simple Einstein model we calculate the renormalization of the electronic energies and the critical temperature of superconductivity.
Abstract We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state φ0⊗ξα\varphi _0 \otimes \xi _\alpha φ0​⊗ξα​ φ0⊗ξα , where φ0\varphi _0φ0​ φ0 is the electron ground state of the Pekar energy functional and ξα\xi _\alpha ξα​ ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2\alpha ^2α2 α2 . In the present note we prove that a similar approximation holds for t=O(α2)t=O(\alpha ^2)t=O(α2) t=O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α−2\alpha ^{-2}α−2 α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2\alpha ^2α2 α2 , while the phonon fluctuations around the coherent state ξα\xi _\alpha ξα​ ξα can be described by a time-dependent Bogoliubov transformation.
We revise a previous result about the Fröhlich dynamics in the strong coupling limit obtained in Griesemer (Rev Math Phys 29(10):1750030, 2017). In the latter it was shown that the Fröhlich time evolution applied to the initial state ϕ0 ⊗ξα, where ϕ0 is the electron ground state of the Pekar energy functional and ξα the associated coherent state of the phonons, can be approximated by a global phase for times small compared to α2. In the present note we prove that a similar approximation holds for t = O(α2) if one includes a nontrivial effective dynamics for the phonons that is generated by an operator proportional to α-2 and quadratic in creation and annihilation operators. Our result implies that the electron ground state remains close to its initial state for times of order α2, while the phonon fluctuations around the coherent state ξα can be described by a time-dependent Bogoliubov transformation.