Understanding Cooper Pairs through SU(2)/SO(2) and Generalizing SU(2)/SO(2) to SU(3)/SO(2)

This article explores how a mathematical tool called SU(2)/SO(2) helps us understand the behavior of Cooper pairs, tiny electron couples responsible for superconductivity. Copper pairs consist of two electrons with opposite spins bound together by vibrations in the material. The article explains how SU(2)/SO(2) describes the electron spins and their spatial arrangement within the pairs. It shows how this tool helps predict their distribution, formation process, and magnetic properties. Ultimately, this allows us to understand superconductivity better and design new superconducting materials.This article explores how a mathematical tool called SU(2)/SO(2) helps us understand the behavior of Cooper pairs, tiny electron couples responsible for superconductivity. Copper pairs consist of two electrons with opposite spins bound together by vibrations in the material. The article explains how SU(2)/SO(2) describes the electron spins and their spatial arrangement within the pairs. It shows how this tool helps predict their distribution, formation process, and magnetic properties. Ultimately, this allows us to understand superconductivity better and design new superconducting materials.

Mar 15, 20240 citations

Many-electron characterizations of higher-charge superconductors

Anonymous, Yi Zhang

The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge- 2 e sectors, but with sharp signatures in charge- 4 e or charge- 6 e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory. Furthermore, we demonstrate that at the microscopic level, momentum conservation alone may be vital and sufficient for the robust emergence of higher-charge superconductivity. Our study thus offers a general and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.The theoretical understanding of conventional superconductivity as the phonon-assisted formation and condensation of two-electron Cooper pairs is a significant triumph in condensed matter physics. Here, we propose many-electron characterizations of higher-charge superconductivity with Cooper pairs consisting of more than two electrons, by implementing translation symmetrization on parent pair-density-wave-ordered states. In particular, we demonstrate many-electron constructions with vanishing charge- 2 e sectors, but with sharp signatures in charge- 4 e or charge- 6 e expectation values instead. Such characterizations are consistent with previous phenomenology of vestigial order and Ginzburg-Landau theory. Furthermore, we demonstrate that at the microscopic level, momentum conservation alone may be vital and sufficient for the robust emergence of higher-charge superconductivity. Our study thus offers a general and microscopic route to understand and characterize higher-charge superconductivity, including nontrivial experimental signatures such as fractional magnetic flux and period in interferometry, as well as localized Cooper pairs at lattice topological defects.

arXiv:2512.23801APS Open ScienceJul 21, 20260 citations

The Cooper Pair Problem

Monique Combescot, Shiue-Yuan Shiau

Abstract Chapter 6 introduces the BCS theory of superconductivity, proposed in 1957 by Bardeen, Cooper, and Schrieffer and which is based on the idea that electrons form a coherent state of electron pairs with opposite spins and opposite momenta. It then briefly explains how, along the Fröhlich idea, the electron-phonon interaction can lead to the ion-mediated attractive potential for Cooper pairing. Next, it introduces the reduced BCS potential, which acts between electrons having opposite spins and opposite momenta. Finally, this chapter discusses the drastic consequence of the reduced BCS potential on Cooper pair interaction and explains why Cooper pairs can strongly overlap without dissociating.Abstract Chapter 6 introduces the BCS theory of superconductivity, proposed in 1957 by Bardeen, Cooper, and Schrieffer and which is based on the idea that electrons form a coherent state of electron pairs with opposite spins and opposite momenta. It then briefly explains how, along the Fröhlich idea, the electron-phonon interaction can lead to the ion-mediated attractive potential for Cooper pairing. Next, it introduces the reduced BCS potential, which acts between electrons having opposite spins and opposite momenta. Finally, this chapter discusses the drastic consequence of the reduced BCS potential on Cooper pair interaction and explains why Cooper pairs can strongly overlap without dissociating.

Oxford University Press eBooksDec 1, 20150 citations

Two-Cooper-pair problem and the Pauli exclusion principle

W. V. Pogosov, Monique Combescot, Michel Crouzeix

While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for $N=2$ already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for $N=2$ already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to $N$ pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.

arXiv:0911.1688Physical Review BMay 13, 201022 citations

New Theory of Superconductivity

В. V. Bondarev

This book contains detailed insights on the calculus of variations that studies the equilibrium density matrix for many-particle Fermi systems. There are two approximations taken into account in the book. The simplest one is the mean field approximation. The second approximation applies not only to the particle distribution pattern but also to the correlation function. The variational principle for electron distribution function among wave vectors has been denoted in the work. The method that makes using the density matrix for finding the valence electron energy in metal has been proposed. It has been proved that the Coulomb interaction of electrons in a crystal lattice results in production of the model Hamiltonian consisting of two components. One component describes attraction of electrons with the equal wave vectors inducing formation of specific electron pairs. The other component describes repulsion of electrons with the wave vectors being equal by size and opposite by direction. This component denotes the anisotropy regarding distribution of electrons among wave vectors thus indicating the superconducting substance ability. The Fermi-Dirac distribution function, as considered together with the model Hamiltonian, explains all the properties exhibited by superconductors. Thus, the effect of a magnetic field on the superconducting states is studied in the book. The book can be interesting for senior schoolchildren, students of higher educational institutions, postgraduates and teachers.This book contains detailed insights on the calculus of variations that studies the equilibrium density matrix for many-particle Fermi systems. There are two approximations taken into account in the book. The simplest one is the mean field approximation. The second approximation applies not only to the particle distribution pattern but also to the correlation function. The variational principle for electron distribution function among wave vectors has been denoted in the work. The method that makes using the density matrix for finding the valence electron energy in metal has been proposed. It has been proved that the Coulomb interaction of electrons in a crystal lattice results in production of the model Hamiltonian consisting of two components. One component describes attraction of electrons with the equal wave vectors inducing formation of specific electron pairs. The other component describes repulsion of electrons with the wave vectors being equal by size and opposite by direction. This component denotes the anisotropy regarding distribution of electrons among wave vectors thus indicating the superconducting substance ability. The Fermi-Dirac distribution function, as considered together with the model Hamiltonian, explains all the properties exhibited by superconductors. Thus, the effect of a magnetic field on the superconducting states is studied in the book. The book can be interesting for senior schoolchildren, students of higher educational institutions, postgraduates and teachers.

Jan 1, 20170 citations

Correlated electrons in high-temperature superconductors

Elbio R. Dagotto

Theoretical ideas and experimental results concerning high-temperature superconductors are reviewed. Special emphasis is given to calculations performed with the help of computers applied to models of strongly correlated electrons proposed to describe the two-dimensional Cu ${\mathrm{O}}_{2}$ planes. The review also includes results using several analytical techniques. The one- and three-band Hubbard models and the $t\ensuremath{-}J$ model are discussed, and their behavior compared against experiments when available. The author found, among the conclusions of the review, that some experimentally observed unusual properties of the cuprates have a natural explanation through Hubbard-like models. In particular, abnormal features like the mid-infrared band of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$, the new states observed in the gap in photoemission experiments, the behavior of the spin correlations with doping, and the presence of phase separation in the copper oxide superconductors may be explained, at least in part, by these models. Finally, the existence of superconductivity in Hubbard-like models is analyzed. Some aspects of the recently proposed ideas to describe the cuprates as having a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting condensate at low temperatures are discussed. Numerical results favor this scenario over others. It is concluded that computational techniques provide a useful, unbiased tool for studying the difficult regime where electrons are strongly interacting, and that considerable progress can be achieved by comparing numerical results against analytical predictions for the properties of these models. Future directions of the active field of computational studies of correlated electrons are briefly discussed.Theoretical ideas and experimental results concerning high-temperature superconductors are reviewed. Special emphasis is given to calculations performed with the help of computers applied to models of strongly correlated electrons proposed to describe the two-dimensional Cu${\mathrm{O}}_{2}$ planes. The review also includes results using several analytical techniques. The one- and three-band Hubbard models and the $t\ensuremath{-}J$ model are discussed, and their behavior compared against experiments when available. The author found, among the conclusions of the review, that some experimentally observed unusual properties of the cuprates have a natural explanation through Hubbard-like models. In particular, abnormal features like the mid-infrared band of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$, the new states observed in the gap in photoemission experiments, the behavior of the spin correlations with doping, and the presence of phase separation in the copper oxide superconductors may be explained, at least in part, by these models. Finally, the existence of superconductivity in Hubbard-like models is analyzed. Some aspects of the recently proposed ideas to describe the cuprates as having a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting condensate at low temperatures are discussed. Numerical results favor this scenario over others. It is concluded that computational techniques provide a useful, unbiased tool for studying the difficult regime where electrons are strongly interacting, and that considerable progress can be achieved by comparing numerical results against analytical predictions for the properties of these models. Future directions of the active field of computational studies of correlated electrons are briefly discussed.

arXiv:cond-mat/9311013Reviews of Modern PhysicsJul 1, 19943,118 citations

N=2

already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to

N

pairs but no precise description of this continuous change has been given so far. Using Richardson's procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs---the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.

{\mathrm{O}}_{2}

planes. The review also includes results using several analytical techniques. The one- and three-band Hubbard models and the $t\ensuremath{-}J$ model are discussed, and their behavior compared against experiments when available. The author found, among the conclusions of the review, that some experimentally observed unusual properties of the cuprates have a natural explanation through Hubbard-like models. In particular, abnormal features like the mid-infrared band of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$, the new states observed in the gap in photoemission experiments, the behavior of the spin correlations with doping, and the presence of phase separation in the copper oxide superconductors may be explained, at least in part, by these models. Finally, the existence of superconductivity in Hubbard-like models is analyzed. Some aspects of the recently proposed ideas to describe the cuprates as having a ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconducting condensate at low temperatures are discussed. Numerical results favor this scenario over others. It is concluded that computational techniques provide a useful, unbiased tool for studying the difficult regime where electrons are strongly interacting, and that considerable progress can be achieved by comparing numerical results against analytical predictions for the properties of these models. Future directions of the active field of computational studies of correlated electrons are briefly discussed.