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/0408246 · Physical Review B

Order-parameter symmetries for magnetic and superconducting instabilities: Bethe-Salpeter analysis of functional renormalization-group solutions

A. A. Katanin, A. P. Kampf

Published Nov 30, 2005• 6 citations
arXivHTMLPDF

Abstract

The Bethe-Salpeter equation is combined with the temperature-cutoff functional renormalization group approach to analyze the order parameter structure for the leading instabilities of the 2D $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model. We find significant deviations from the conventional $s$-, $p$-, or $d$-wave forms, which is due to the frustration of antiferromagnetism at small and intermediate ${t}^{\ensuremath{'}}$. With adding a direct antiferromagnetic spin-exchange coupling the eigenfunctions in the particle-hole channel have extended $s$-wave form, while in the particle-particle singlet pairing channel a higher angular momentum component arises besides the standard ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ wave component, which flattens the angular dependence of the gap. For ${t}^{\ensuremath{'}}$ closer to $t∕2$ we find a delicate competition of ferromagnetism and triplet pairing with a nontrivial pair-wavefunction.

DOI
10.1103/physrevb.72.205128

Related papers

Renormalized mean-field analysis of antiferromagnetism and d -wave superconductivity in the two-dimensional Hubbard model

J. Reiss, D. Rohe, Walter Metzner

We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the $d$-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.

arXiv:cond-mat/0611164•Physical Review B•Feb 20, 2007•92 citations

Temperature-flow renormalization group and the competition between superconductivity and ferromagnetism

Carsten Honerkamp A, Manfred Salmhofer B

We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.

arXiv:cond-mat/0105218•Physical review. B, Condensed matter•Oct 19, 2001•172 citations

Bethe-Salpeter eigenvalues and amplitudes for the half-filled two-dimensional Hubbard model

Niyazi Bulut, Douglas J. Scalapino, Steven R. White

Monte Carlo simulations are used to determine the eigenvalues and eigenfunctions of the particle-hole and particle-particle Bethe-Salpeter equations for 8\ifmmode\times\else\texttimes\fi{}8 half-filled Hubbard lattice with U/t=4 and U/t=8. In the particle-hole channel, the dominant eigenvalue corresponds to the Q=(\ensuremath{\pi},\ensuremath{\pi}) antiferromagnetic correlations. In the particle-particle channel the amplitude of the leading low-temperature eigenvalue is an even-frequency ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$ singlet. Odd-frequency p-wave-singlet and s-wave-triplet amplitudes are also found.

Physical review. B, Condensed matter•Jun 1, 1993•50 citations

Rashba spin-orbit coupling in the square-lattice Hubbard model: A truncated-unity functional renormalization group study

Jacob Beyer, Jonas B. Profe, Lennart Klebl +5 more

The Rashba-Hubbard model on the square lattice is the paradigmatic case for studying the effect of spin-orbit coupling, which breaks spin and inversion symmetry, in a correlated electron system. We employ a truncated-unity variant of the functional renormalization group which allows us to analyze magnetic and superconducting instabilities on equal footing. We derive phase diagrams depending on the strengths of Rasbha spin-orbit coupling, real second-neighbor hopping, and electron filling. We find commensurate and incommensurate magnetic phases which compete with $d$-wave superconductivity. Due to the breaking of inversion symmetry, singlet and triplet components mix; we quantify the mixing of $d$-wave singlet pairing with $f$-wave triplet pairing.

arXiv:2210.09384•Physical review. B./Physical review. B•Mar 7, 2023•23 citations

Renormalization-group analysis of the two-dimensional Hubbard model

Christoph J. Halboth, Walter Metzner

Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofer's renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.

arXiv:cond-mat/9908471•Physical review. B, Condensed matter•Mar 15, 2000•326 citations

Dynamical functional renormalization group computation of order parameters and critical temperatures in the two-dimensional Hubbard model

Demetrio Vilardi, Pietro M. Bonetti, Walter Metzner

We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping $p$ and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the $(p,T)$ phase diagram centered around 15% hole doping.

arXiv:2010.06267•Physical review. B./Physical review. B•Dec 18, 2020•33 citations
The Bethe-Salpeter equation is combined with the temperature-cutoff functional renormalization group approach to analyze the order parameter structure for the leading instabilities of the 2D t\ensuremath-t\ensuremath′t\text{\ensuremath{-}}{t}^{\ensuremath{'}}t\ensuremath-t\ensuremath′ Hubbard model. We find significant deviations from the conventional sss-, ppp-, or ddd-wave forms, which is due to the frustration of antiferromagnetism at small and intermediate t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′. With adding a direct antiferromagnetic spin-exchange coupling the eigenfunctions in the particle-hole channel have extended sss-wave form, while in the particle-particle singlet pairing channel a higher angular momentum component arises besides the standard dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ wave component, which flattens the angular dependence of the gap. For t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ closer to t∕2t∕2t∕2 we find a delicate competition of ferromagnetism and triplet pairing with a nontrivial pair-wavefunction.
We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the ddd-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.
Monte Carlo simulations are used to determine the eigenvalues and eigenfunctions of the particle-hole and particle-particle Bethe-Salpeter equations for 8\ifmmode\times\else\texttimes\fi{}8 half-filled Hubbard lattice with U/t=4 and U/t=8. In the particle-hole channel, the dominant eigenvalue corresponds to the Q=(\ensuremath{\pi},\ensuremath{\pi}) antiferromagnetic correlations. In the particle-particle channel the amplitude of the leading low-temperature eigenvalue is an even-frequency dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 singlet. Odd-frequency p-wave-singlet and s-wave-triplet amplitudes are also found.
The Rashba-Hubbard model on the square lattice is the paradigmatic case for studying the effect of spin-orbit coupling, which breaks spin and inversion symmetry, in a correlated electron system. We employ a truncated-unity variant of the functional renormalization group which allows us to analyze magnetic and superconducting instabilities on equal footing. We derive phase diagrams depending on the strengths of Rasbha spin-orbit coupling, real second-neighbor hopping, and electron filling. We find commensurate and incommensurate magnetic phases which compete with ddd-wave superconductivity. Due to the breaking of inversion symmetry, singlet and triplet components mix; we quantify the mixing of ddd-wave singlet pairing with fff-wave triplet pairing.
Salmhofer [Commun. Math. Phys. 194, 249 (1998)] recently developed a new renormalization-group method for interacting Fermi systems, where the complete flow from the bare action of a microscopic model to the effective low-energy action, as a function of a continuously decreasing infrared cutoff, is given by a differential flow equation which is local in the flow parameter. We apply this approach to the repulsive two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping amplitudes. The flow equation for the effective interaction is evaluated numerically on a one-loop level. The effective interactions diverge at a finite-energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions we extend Salmhofer's renormalization group for the calculation of susceptibilities. We compute the singlet superconducting susceptibilities for various pairing symmetries, and also charge- and spin-density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength, and band filling) we find commensurate and incommensurate antiferromagnetic instabilities or d-wave superconductivity as leading instability. We present the resulting phase diagram in the vicinity of half-filling, and also results for the density dependence of the critical energy scale.
We analyze the interplay of antiferromagnetism and pairing in the two-dimensional Hubbard model with a moderate repulsive interaction. Coupled charge, magnetic, and pairing fluctuations above the energy scale of spontaneous symmetry breaking are treated by a functional renormalization group flow, while the formation of gaps and order below that scale is treated in mean-field theory. The full frequency dependences of the interaction vertices and gap functions are taken into account. We compute the magnetic and pairing gap functions as a function of doping ppp and compare with results from a static approximation. In spite of the strong frequency dependences of the effective interactions and of the pairing gap, important physical results from previous static functional renormalization group calculations are confirmed. In particular, there is a sizable doping regime with robust pairing coexisting with N\'eel or incommensurate antiferromagnetism. The critical temperature for magnetic order is interpreted as the pseudogap crossover temperature. Computing the Kosterlitz-Thouless temperature from the superfluid phase stiffness, we obtain a superconducting dome in the (p,T)(p,T)(p,T) phase diagram centered around 15% hole doping.