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

Physical review. B./Physical review. B

Coupled Bogoliubov equations for electrons and phonons

C. Y. Wang, Sangeeta Sharma, T. Müller, Eberhard K. U. Gross, J. K. Dewhurst

Published May 16, 2022• 3 citations
Publisher

Abstract

The interaction between fermions and bosons is central to modern physics. To describe systems with such interactions, the authors apply Wick's theorem to the square of the Fr\"ohlich Hamiltonian and keep all terms up to quadratic order. What emerges are two coupled Bogoliubov equations: one for electrons and one for phonons. Remarkably, the self-consistent solution to these equations accurately describes both the renormalized gap in insulators and the superconducting gap in metals.

DOI
10.1103/physrevb.105.174509

Related papers

Coupled Kohn-Sham equations for electrons and phonons

C. Y. Wang, Müller, T., Sangeeta Sharma +2 more

This work establishes the algebraic structure of the Kohn-Sham equations to be solved in a density formulation of electron and phonon dynamics, including the superconducting order parameter. A Bogoliubov transform is required to diagonalize both the fermionic and bosonic Kohn-Sham Hamiltonians since they both represent a non-interacting quantum field theory. The Bogoliubov transform for phonons is non-Hermitian in the general case, and the corresponding time-evolution is non-unitary. Several sufficient conditions for ensuring that the bosonic eigenvalues are real are provided and a practical method for solving the system is described. Finally, we produce a set of approximate mean-field potentials which are functionals of the electronic and phononic density matrices and depend on the electron-phonon vertex.

arXiv:1803.07364•Mar 20, 2018•0 citations

Density matrix embedding theory for interacting electron-phonon systems

Barbara Sandhoefer, Garnet Kin‐Lic Chan

We describe the extension of the density matrix embedding theory framework to coupled interacting fermion-boson systems. This provides a frequency-independent, entanglement embedding formalism to treat bulk fermion-boson problems. We illustrate the concepts within the context of the one-dimensional Hubbard-Holstein model, where the phonon bath states are obtained from the Schmidt decomposition of a self-consistently adjusted coherent state. We benchmark our results against accurate density matrix renormalization group calculations.

arXiv:1602.04195•Physical review. B./Physical review. B•Aug 9, 2016•32 citations

Strong interaction of correlated electrons with phonons: A diagrammatic approach

V. A. Moskalenko, P. Entel, D. F. Digor

We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.

Physical review. B, Condensed matter•Jan 1, 1999•45 citations

Electron-electron and electron-phonon interactions in strongly correlated systems

Gerardo Sica

In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]

•Apr 29, 2013•0 citations

The Theory of Superconductivity in the Tamm-Dancoff Approximation

B. P. Nigam

The Tamm-Dancoff approximation is applied to the interaction Hamiltonian used by Bogoliubov in his theory of superconductivity. It is shown that (i) the requirement that there is no virtual production of a single pair of electron and hole without phonons, or (ii) the electron-phonon coupling parameter tends to zero, leads to essentially the same compensating equation as obtained by Bogoliubov.

Progress of Theoretical Physics•Mar 1, 1961•0 citations

Bogoliubov-de Gennes study of nanoscale Hubbard superconductors

César G. Galván, José Manuel Cabrera-Trujillo, Luis Antonio Pérez +1 more

The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov–de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N × N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)2 coupled self-consistent BdG equations for a supercell of (N + s) × (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron–electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site.

physica status solidi (b)•Apr 23, 2016•1 citation
The interaction between fermions and bosons is central to modern physics. To describe systems with such interactions, the authors apply Wick's theorem to the square of the Fr"ohlich Hamiltonian and keep all terms up to quadratic order. What emerges are two coupled Bogoliubov equations: one for electrons and one for phonons. Remarkably, the self-consistent solution to these equations accurately describes both the renormalized gap in insulators and the superconducting gap in metals.
This work establishes the algebraic structure of the Kohn-Sham equations to be solved in a density formulation of electron and phonon dynamics, including the superconducting order parameter. A Bogoliubov transform is required to diagonalize both the fermionic and bosonic Kohn-Sham Hamiltonians since they both represent a non-interacting quantum field theory. The Bogoliubov transform for phonons is non-Hermitian in the general case, and the corresponding time-evolution is non-unitary. Several sufficient conditions for ensuring that the bosonic eigenvalues are real are provided and a practical method for solving the system is described. Finally, we produce a set of approximate mean-field potentials which are functionals of the electronic and phononic density matrices and depend on the electron-phonon vertex.
We describe the extension of the density matrix embedding theory framework to coupled interacting fermion-boson systems. This provides a frequency-independent, entanglement embedding formalism to treat bulk fermion-boson problems. We illustrate the concepts within the context of the one-dimensional Hubbard-Holstein model, where the phonon bath states are obtained from the Schmidt decomposition of a self-consistently adjusted coherent state. We benchmark our results against accurate density matrix renormalization group calculations.
We investigate the interaction of strongly correlated electrons with phonons in the frame of the Hubbard-Holstein model. The electron-phonon interaction is considered to be strong and is an important parameter of the model besides the Coulomb repulsion of electrons and band filling. This interaction with the nondispersive optical phonons has been transformed to the problem of mobile polarons by using the canonical transformation of Lang and Firsov. A new diagram technique is used in order to handle the strong Coulomb repulsion of the electrons, which leads to a closed set of integral equations for the irreducible on-site many-particle Green's functions. The coupling to the phonons leads to an essential renormalization of the one- and two-particle irreducible Green's functions. It also influences the parameters which control the metal-insulator or superconducting phase transition. In addition we find a collective mode of phonon clouds which surround the polarons. The physics of the emission and absorption of this mode by the polarons has been investigated.
In this work we investigate some aspects of the physics of strongly correlated systems by taking into account both electron-electron and electron-phonon interactions as basic mechanisms for reproducing electronic correlations in real materials. The relevance of the electron-electron interactions is discussed in the first part of this thesis in the framework of a self-consistent theoretical approach, named Composite Operator Method (COM), which accounts for the relevant quasi-particle excitations in terms of a set of composite operators that appear as a result of the modification imposed by the interactions on the canonical electronic fields. We show that the COM allows the calculation of all the relevant Green’s and correlation functions in terms of a number of unknown internal parameters to be determined self-consistently. Therefore, depending on the balance between unknown parameters and self-consistent equations, exact and approximate solutions can be obtained. By way of example, we discuss the application of the COM to the extended t-U- J-h model in the atomic limit, and to the two-dimensional single-band Hubbard model. In the former case, we show that the COM provides the exact solution of the model in one dimension. We study the effects of electronic correlations as responsible for the formation of a plethora of different charge and/or spin orderings. We report the phase diagram of the model, as well as a detailed analysis of both zero and finite temperature single-particle and thermodynamic properties. As far as the single-band Hubbard model is concerned, we illustrate an approximated selfconsistent scheme based on the choice of a two-field basis. We report a detailed analysis of many unconventional features that arise in single-particle properties, thermodynamics and system’s response functions. We emphasize that the accuracy of the COM in describing the effects of electronic correlations strongly relies on the choice of the basis, paving the way for possible multi-pole extensions to the twofield theory. To this purpose, we also study a three-field approach to the single-band Hubbard model, showing a significant step forward in the agreements with numerical data with respect to the two-pole results. The role of the electron-phonon interaction in the physics of strongly correlated systems is discussed in the second part of this thesis. We show that in highly polarizable lattices the competition between unscreened Coulomb and Fröhlich interactions results in a short-range polaronic exchange term Jp that favours the formation of local and light pairs of bosonic nature, named bipolarons, which condense with a critical temperature well in excess of hundred kelvins. These findings, discussed in the framework of the so-called polaronic t-Jp model, are further investigated in the presence of a finite on-site potential ~U , coming from the competition between on-site Coulomb and Fröhlich interactions. We discuss the role of ~U as the driving parameter for a small-to-large bipolaron transition, providing a possible explanation of the BEC-BCS crossover in terms of the properties of the bipolaronic ground state. Finally, we show that a hard-core bipolarons gas, studied as a charged Bose-Fermi mixture, allows for the description of many non Fermi liquid behaviours, allowing also for a microscopic explanation of pseudogap features in terms of a thermal-induced recombination of polarons and bipolarons, without any assumption on preexisting order or broken symmetries. [edited by Author]
The effects of quantum confinement on the superconducting ground state are studied within the Bogoliubov–de Gennes (BdG) formalism and an attractive Hubbard model. We consider a periodic arrangement of two-dimensional superconducting grains composed by N × N atoms surrounded by insulating, metallic or superconducting stripes with a thickness of s atoms, leading to 2(N + s)2 coupled self-consistent BdG equations for a supercell of (N + s) × (N + s) atoms. These equations determine the spatial variation of superconducting gap as functions of temperature, electron–electron interaction, and hopping integrals analyzing three types of boundary stripes. The results show a clear enhancement of the superconducting gap and critical temperature induced by the electron confinement in the grain, being larger for the insulating boundary case. Finally, the numerical solutions of BdG equations are compared with those obtained by applying the BCS theory to each grain site.