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EFFECT OF A LOCALLY REPULSIVE INTERACTION ON S–WAVE SUPERCONDUCTORS

J.-B. Bru, W. de Siqueira Pedra

Published Nov 27, 2013• 13 citations
OpenAlex

Abstract

Abstract. The thermodynamic impact of the Coulomb repulsion on s–wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS–Hubbard Hamiltonian. We show that the one–site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one–site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance near half–filling and at strong repulsion. Our proof of a superconductor–Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of “quantum large deviation ” arguments combined with an adaptation of the proof of Størmer’s theorem [1] to even states on the CAR algebra. 1.

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EFFECT OF A LOCALLY REPULSIVE INTERACTION ON s-WAVE SUPERCONDUCTORS

J.-B. Bru, W. DE SIQUEIRA PEDRA

The thermodynamic impact of the Coulomb repulsion on s-wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one-site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance, near half-filling and at strong repulsion. Our proof of a superconductor-Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of "quantum large deviation" arguments combined with an adaptation of the proof of Størmer's theorem [1] to even states on the CAR algebra.

arXiv:1608.08441•Reviews in Mathematical Physics•Apr 1, 2010•10 citations

Nearly Insulating Strongly Correlated Systems: Gossamer Superconductors and Metals

Bogdan Andrei Bernevig, George Chapline, R. B. Laughlin +2 more

Recently a new phenomenological Hamiltonian was proposed to describe the superconducting cuprates in which correlations and on-site Coulomb repulsion are introduced by partial Gutzwiller projection. This Gossamer Hamiltonian has an exact ground state and differs from the t-J and Hubbard Hamiltonians in possessing a powerful attractive interaction among electrons responsible for Cooper pairing in the d-wave channel. It is a faithful description for a superconductor with strong on-site electronic repulsion. The superconducting tunneling gap remains intact and despite on-site repulsion. Near half-filling the Gossamer superconductor with strong repulsion has suppressed photoemission intensities and superfluid density, is unstable toward an antiferromagnetic insulator and possesses an incipient Mott-Hubbard gap. The Gossamer technique can be applied to metallic ground states thus possibly serving as an apt description of strongly correlated metals. Such a Gossamer metallic phase, just as the Gossamer superconducting one, becomes arbitrarily hard to differentiate from an insulator as one turns the Coulomb correlations up near half-filling. Both the metallic and superconducting states undergo a quantum phase transition to an antiferromagnetic insulator as one increases the on-site Coulomb repulsion. In the Gossamer model we reach the critical point at half-filling by fully projecting out the double occupancy. Such a critical point might be the Anderson Resonating Valence bond state.

arXiv:cond-mat/0312573•Dec 22, 2003•0 citations

A Rigorous Demonstration of Superconductivity in a Repulsive Hubbard Model

Manjinder Kainth, M W Long

We have rigorously shown that a strong Hubbard repulsion can cause superconductivity. The model, which has a particular set of local symmetries, manifests the phase diagram of many unconventional superconductors; anti-ferromagnetism paramagnetism, superconductivity, and even ferromagnetism are all observed. The key technique in the analysis is an exact non-linear fermion transformation, which preserves the correlated motion of electrons while removing the strong interactions. Using resolvent formalism, it is exactly shown that two holes at the Mott point bind to form a localised Cooper pair. As interactions are now weak, we then use BCS mean field theory to calculate the energy, excess pairing, and superconducting gap. These results are compared to exact diagonalisation of finite sized systems and show good agreement. At the Mott point the system is an anti-ferromagnet, and a superconducting phase quickly appears upon doping, and then vanishes.

arXiv:1904.07138•Apr 15, 2019•1 citation

Resilience of d -wave superconductivity to nearest-neighbor repulsion

David Sénéchal, Alexandre G. R. Day, V. Bouliane +1 more

Many theoretical approaches find $d$-wave superconductivity in the prototypical one-band Hubbard model for high-temperature superconductors. At strong coupling ($U\ensuremath{\ge}W$, where $U$ is the on-site repulsion and $W=8t$ the bandwidth) pairing is controlled by the exchange energy $J=4{t}^{2}/U$. One may then surmise, ignoring retardation effects, that near-neighbor Coulomb repulsion $V$ will destroy superconductivity when it becomes larger than $J$, a condition that is easily satisfied in cuprates, for example. Using cellular dynamical mean-field theory with an exact diagonalization solver for the extended Hubbard model, we show that pairing at strong coupling is preserved, even when $V\ensuremath{\gg}J$, as long as $V\ensuremath{\lesssim}U/2$. While at weak coupling $V$ always reduces the spin fluctuations and hence $d$-wave pairing, at strong coupling, in the underdoped regime, the increase of $J=4{t}^{2}/(U\ensuremath{-}V)$ caused by $V$ increases binding at low frequency while the pair-breaking effect of $V$ is pushed to high frequency. These two effects compensate in the underdoped regime, in the presence of a pseudogap. While the pseudogap competes with superconductivity, the proximity to the Mott transition that leads to the pseudogap, and retardation effects, protect $d$-wave superconductivity from $V$.

arXiv:1212.4503•Physical Review B•Feb 19, 2013•58 citations

Mott localisation, antiferromagnetism and nematic superconductivity in the extended Hubbard model with non-local pairing

ALESSANDRO GORI

High temperature superconductivity in copper oxides (cuprates) represents one of the most striking phenomena in condensed matter physics. These materials host superconducting states with the highest known critical temperatures, emerging upon doping antiferromagnetic insulator compounds. The antiferromagnetic (AF) insulating and superconducting (SC) phases defy standard descriptions in terms of band picture and conventional phonon-mediated superconductivity. In fact, these phenomena are genuine signatures of strong electronic correlations which hinder effective single-particle descriptions of the many-electron system. After 40 years since their discovery, it is widely accepted that the Hubbard model on the square lattice provides a minimal description of the low-energy electronic properties of cuprates. The model describes the competition between the electronic kinetic energy, modelled by nearest-neighbours hopping t, and the Coulomb repulsion, modelled by a purely local interaction potential U. At half-filling, this competition results in the so-called "Mott localisation" phenomenon, which distinguishes the metallic phase for U/t « 1 from an insulating phase for U/t » 1. The latter is characterised by the almost complete localisation of electrons due to the high energetic cost of hopping towards occupied sites. Interestingly, this basic competition is intimately related to AF ordering and, upon doping, to the formation of the SC state with characteristic d-wave symmetry. The fact that a purely repulsive interaction U is responsible for the formation of a SC state is remarkable, being superconductivity typically associated with an effective attractive interaction. In fact, a d-wave SC state can also be obtained, in general, by considering an attractive potential between electrons of neighbouring sites. Motivated by this observation, in this thesis, we study a modified version of the Hubbard model, dubbed as the "extended Hubbard model", in which we include a nearest-neighbours attractive potential -V. The general goal of this work is to explore the interplay between the purely local repulsion and the non-local attraction. This is not only motivated by the fact that both interactions are, in principle, responsible for d-wave superconductivity, but also by recent experimental observations which strongly suggest that the term -V needs to be included in order to accurately describe the magnetic spectra of some specific compounds. In this thesis, we study the extended Hubbard model using the Hartree-Fock method. Namely, we approximate the ground state by implementing a variational search in the subspace of the many-body Hilbert space of gaussian states. Despite its simplicity, the Hartree-Fock method provides a powerful framework to study competition among different phases, originating from different symmetry breaking mechanisms. In particular, we study the competition between three different phases: the normal phase, characterised by no symmetry breaking, the AF phase, characterised by a lowering of SU(2) symmetry, and the SC phase, characterised by the breaking of U(1) charge conservation. This thesis work led to a number of original results, which can be summarised by the general observation that the non-local attraction -V acts cooperatively with the local repulsion U for the stabilisation of the various phases hosted by the Hubbard model. For the normal phase, we find that the non-local attraction acts as a source of Mott localisation, leading to an effective reduction of the electronic bandwidth. In particular, for values of V larger than a critical value and close to half-filling, we find a strong Mott localisation characterised by the complete flattening of the electronic bands. Interestingly, this result shows an example of Mott localisation that can be described at the level of the Hartree-Fock approximation, whereas Mott localisation in the conventional Hubbard model usually requires more sophisticated approximation schemes. For the AF phase, we restrict our analysis to the half-filled case. We find that the pairing-induced Mott localisation leads to a significant enhancement of the magnetisation, primarily originating from the local repulsion U. Finally, we focus on the superconducting phase and derive self-consistency equations in both singlet and triplet Cooper pairing channels, showing how Cooper pairing can be mediated simultaneously by both the local repulsion and the non-local attraction. Therefore, we numerically investigate the singlet channel and find evidence of "symmetry mixing", namely, the formation of a superconducting gap that is neither symmetric nor anti-symmetric under rotations of π/2 in the plane. We show that symmetry mixing is connected to the emergence of a finite nematic order parameter, that signals the breaking of planar rotational symmetry. This nematic symmetry breaking results in the fact that the renormalisation of bands becomes anisotropic, producing a difference in electronic kinetics in the x and y directions. To summarise, the original results obtained in this thesis show that the non-local pairing provides a different route towards the formation of key strongly correlated phenomena, including Mott localisation and nematic superconductivity, and call for future investigation using more sophisticated methods such as, for example, the Dynamical Mean Field Theory.

•Jun 17, 2026•0 citations

THE EFFECT OF A BOND-SITE INTERACTION ON THE GROUND-STATE INSTABILITIES OF THE ONE-DIMENSIONAL HUBBARD MODEL

Florin D. Buzatu

The ground-state instabilities for a one-dimensional lattice system of electrons with onsite (Hubbard) and bond-site (hopping) interactions are analyzed in a perturbative approach. The zero temperature phase diagram at different band fillings is drawn; an attractive (repulsive) bond-site interaction favors the appearance of a superconductor state at low concentrations of electrons (holes). A comparison with the exact results for the Hubbard model and previous works for particular cases is also discussed.

International Journal of Modern Physics B•May 30, 1995•3 citations
Abstract. The thermodynamic impact of the Coulomb repulsion on s–wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS–Hubbard Hamiltonian. We show that the one–site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one–site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance near half–filling and at strong repulsion. Our proof of a superconductor–Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of “quantum large deviation ” arguments combined with an adaptation of the proof of Størmer’s theorem [1] to even states on the CAR algebra. 1.
The thermodynamic impact of the Coulomb repulsion on s-wave superconductors is analyzed via a rigorous study of equilibrium and ground states of the strong coupling BCS-Hubbard Hamiltonian. We show that the one-site electron repulsion can favor superconductivity at fixed chemical potential by increasing the critical temperature and/or the Cooper pair condensate density. If the one-site repulsion is not too large, a first or a second order superconducting phase transition can appear at low temperatures. The Meißner effect is shown to be rather generic but coexistence of superconducting and ferromagnetic phases is also shown to be feasible, for instance, near half-filling and at strong repulsion. Our proof of a superconductor-Mott insulator phase transition implies a rigorous explanation of the necessity of doping insulators to create superconductors. These mathematical results are consequences of "quantum large deviation" arguments combined with an adaptation of the proof of Størmer's theorem [1] to even states on the CAR algebra.
Recently a new phenomenological Hamiltonian was proposed to describe the superconducting cuprates in which correlations and on-site Coulomb repulsion are introduced by partial Gutzwiller projection. This Gossamer Hamiltonian has an exact ground state and differs from the t-J and Hubbard Hamiltonians in possessing a powerful attractive interaction among electrons responsible for Cooper pairing in the d-wave channel. It is a faithful description for a superconductor with strong on-site electronic repulsion. The superconducting tunneling gap remains intact and despite on-site repulsion. Near half-filling the Gossamer superconductor with strong repulsion has suppressed photoemission intensities and superfluid density, is unstable toward an antiferromagnetic insulator and possesses an incipient Mott-Hubbard gap. The Gossamer technique can be applied to metallic ground states thus possibly serving as an apt description of strongly correlated metals. Such a Gossamer metallic phase, just as the Gossamer superconducting one, becomes arbitrarily hard to differentiate from an insulator as one turns the Coulomb correlations up near half-filling. Both the metallic and superconducting states undergo a quantum phase transition to an antiferromagnetic insulator as one increases the on-site Coulomb repulsion. In the Gossamer model we reach the critical point at half-filling by fully projecting out the double occupancy. Such a critical point might be the Anderson Resonating Valence bond state.
We have rigorously shown that a strong Hubbard repulsion can cause superconductivity. The model, which has a particular set of local symmetries, manifests the phase diagram of many unconventional superconductors; anti-ferromagnetism paramagnetism, superconductivity, and even ferromagnetism are all observed. The key technique in the analysis is an exact non-linear fermion transformation, which preserves the correlated motion of electrons while removing the strong interactions. Using resolvent formalism, it is exactly shown that two holes at the Mott point bind to form a localised Cooper pair. As interactions are now weak, we then use BCS mean field theory to calculate the energy, excess pairing, and superconducting gap. These results are compared to exact diagonalisation of finite sized systems and show good agreement. At the Mott point the system is an anti-ferromagnet, and a superconducting phase quickly appears upon doping, and then vanishes.
Many theoretical approaches find ddd-wave superconductivity in the prototypical one-band Hubbard model for high-temperature superconductors. At strong coupling ($U\ensuremath{\ge}W$, where UUU is the on-site repulsion and W=8tW=8tW=8t the bandwidth) pairing is controlled by the exchange energy J=4t2/UJ=4{t}^{2}/UJ=4t2/U. One may then surmise, ignoring retardation effects, that near-neighbor Coulomb repulsion VVV will destroy superconductivity when it becomes larger than JJJ, a condition that is easily satisfied in cuprates, for example. Using cellular dynamical mean-field theory with an exact diagonalization solver for the extended Hubbard model, we show that pairing at strong coupling is preserved, even when $V\ensuremath{\gg}J$, as long as $V\ensuremath{\lesssim}U/2$. While at weak coupling VVV always reduces the spin fluctuations and hence ddd-wave pairing, at strong coupling, in the underdoped regime, the increase of $J=4{t}^{2}/(U\ensuremath{-}V)$ caused by VVV increases binding at low frequency while the pair-breaking effect of VVV is pushed to high frequency. These two effects compensate in the underdoped regime, in the presence of a pseudogap. While the pseudogap competes with superconductivity, the proximity to the Mott transition that leads to the pseudogap, and retardation effects, protect ddd-wave superconductivity from VVV.
High temperature superconductivity in copper oxides (cuprates) represents one of the most striking phenomena in condensed matter physics. These materials host superconducting states with the highest known critical temperatures, emerging upon doping antiferromagnetic insulator compounds. The antiferromagnetic (AF) insulating and superconducting (SC) phases defy standard descriptions in terms of band picture and conventional phonon-mediated superconductivity. In fact, these phenomena are genuine signatures of strong electronic correlations which hinder effective single-particle descriptions of the many-electron system. After 40 years since their discovery, it is widely accepted that the Hubbard model on the square lattice provides a minimal description of the low-energy electronic properties of cuprates. The model describes the competition between the electronic kinetic energy, modelled by nearest-neighbours hopping t, and the Coulomb repulsion, modelled by a purely local interaction potential U. At half-filling, this competition results in the so-called "Mott localisation" phenomenon, which distinguishes the metallic phase for U/t « 1 from an insulating phase for U/t » 1. The latter is characterised by the almost complete localisation of electrons due to the high energetic cost of hopping towards occupied sites. Interestingly, this basic competition is intimately related to AF ordering and, upon doping, to the formation of the SC state with characteristic d-wave symmetry. The fact that a purely repulsive interaction U is responsible for the formation of a SC state is remarkable, being superconductivity typically associated with an effective attractive interaction. In fact, a d-wave SC state can also be obtained, in general, by considering an attractive potential between electrons of neighbouring sites. Motivated by this observation, in this thesis, we study a modified version of the Hubbard model, dubbed as the "extended Hubbard model", in which we include a nearest-neighbours attractive potential -V. The general goal of this work is to explore the interplay between the purely local repulsion and the non-local attraction. This is not only motivated by the fact that both interactions are, in principle, responsible for d-wave superconductivity, but also by recent experimental observations which strongly suggest that the term -V needs to be included in order to accurately describe the magnetic spectra of some specific compounds. In this thesis, we study the extended Hubbard model using the Hartree-Fock method. Namely, we approximate the ground state by implementing a variational search in the subspace of the many-body Hilbert space of gaussian states. Despite its simplicity, the Hartree-Fock method provides a powerful framework to study competition among different phases, originating from different symmetry breaking mechanisms. In particular, we study the competition between three different phases: the normal phase, characterised by no symmetry breaking, the AF phase, characterised by a lowering of SU(2) symmetry, and the SC phase, characterised by the breaking of U(1) charge conservation. This thesis work led to a number of original results, which can be summarised by the general observation that the non-local attraction -V acts cooperatively with the local repulsion U for the stabilisation of the various phases hosted by the Hubbard model. For the normal phase, we find that the non-local attraction acts as a source of Mott localisation, leading to an effective reduction of the electronic bandwidth. In particular, for values of V larger than a critical value and close to half-filling, we find a strong Mott localisation characterised by the complete flattening of the electronic bands. Interestingly, this result shows an example of Mott localisation that can be described at the level of the Hartree-Fock approximation, whereas Mott localisation in the conventional Hubbard model usually requires more sophisticated approximation schemes. For the AF phase, we restrict our analysis to the half-filled case. We find that the pairing-induced Mott localisation leads to a significant enhancement of the magnetisation, primarily originating from the local repulsion U. Finally, we focus on the superconducting phase and derive self-consistency equations in both singlet and triplet Cooper pairing channels, showing how Cooper pairing can be mediated simultaneously by both the local repulsion and the non-local attraction. Therefore, we numerically investigate the singlet channel and find evidence of "symmetry mixing", namely, the formation of a superconducting gap that is neither symmetric nor anti-symmetric under rotations of π/2 in the plane. We show that symmetry mixing is connected to the emergence of a finite nematic order parameter, that signals the breaking of planar rotational symmetry. This nematic symmetry breaking results in the fact that the renormalisation of bands becomes anisotropic, producing a difference in electronic kinetics in the x and y directions. To summarise, the original results obtained in this thesis show that the non-local pairing provides a different route towards the formation of key strongly correlated phenomena, including Mott localisation and nematic superconductivity, and call for future investigation using more sophisticated methods such as, for example, the Dynamical Mean Field Theory.
The ground-state instabilities for a one-dimensional lattice system of electrons with onsite (Hubbard) and bond-site (hopping) interactions are analyzed in a perturbative approach. The zero temperature phase diagram at different band fillings is drawn; an attractive (repulsive) bond-site interaction favors the appearance of a superconductor state at low concentrations of electrons (holes). A comparison with the exact results for the Hubbard model and previous works for particular cases is also discussed.