Antiferromagnetic spin fluctuations are a prime suspect for the superconducting glue in cuprates and nickelates. The minimal model reproducing the main physics of these systems is the single-orbital Hubbard model. Diagramatically, the scattering of electrons with these spin fluctuations can be described via a three-point propagator, where the coupling between these two quasiparticles - the propagator with amputated legs - is the Hedin vertex in the magnetic channel, or spin-fermion vertex. It has previously been shown that this coupling shows a rich dependence on fermionic momentum, notably a divergence close to half filling at the antinode. Exact diagonalization calculations for finite clusters in this work reproduce this momentum dependence. Furthermore, the known power law relation from DCA results between the Hedin vertex and the self-energy is also observed. Finally, a physical interpretation of the Hedin vertices' pole at the antinode, due to its established equivalence to the derivative of the self-energy w.r.t to an applied magnetic field is discussed for the 4x4 cluster.
Antiferromagnetic spin fluctuations are a prime suspect for the superconducting glue in cuprates and nickelates. The minimal model reproducing the main physics of these systems is the single-orbital Hubbard model. Diagramatically, the scattering of electrons with these spin fluctuations can be described via a three-point propagator, where the coupling between these two quasiparticles - the propagator with amputated legs - is the Hedin vertex in the magnetic channel, or spin-fermion vertex. It has previously been shown that this coupling shows a rich dependence on fermionic momentum, notably a divergence close to half filling at the antinode. Exact diagonalization calculations for finite clusters in this work reproduce this momentum dependence. Furthermore, the known power law relation from DCA results between the Hedin vertex and the self-energy is also observed. Finally, a physical interpretation of the Hedin vertices' pole at the antinode, due to its established equivalence to the derivative of the self-energy w.r.t to an applied magnetic field is discussed for the 4x4 cluster.
We analyze the three-point vertex function that describes the coupling of fermionic particle-hole pairs in a metal to spin or charge fluctuations at nonzero momentum. We consider Ward identities, which connect two-particle vertex functions to the self-energy, in the framework of a Hubbard model. These are derived using conservation laws following from local symmetries. The generators considered are the spin density and particle density. It is shown that at certain antiferromagnetic critical points, where the quasiparticle effective mass is diverging, the vertex function describing the coupling of particle-hole pairs to the spin density Fourier component at the antiferromagnetic wave vector is also divergent. Then we give an explicit calculation of the irreducible vertex function for the case of three-dimensional antiferromagnetic fluctuations, and show that it is proportional to the diverging quasiparticle effective mass.
Away from half filling and above its Neel temperature, the three-dimensional Hubbard model is a paramagnetic metal with antiferromagnetic spin fluctuations, where controlled momentum-resolved results have been lacking. Using connected-determinant diagrammatic Monte Carlo, we compute the momentum-resolved self-energy for n = 0.4-0.95, T/t = 0.05-0.33 and U/t = 4-8, and the full-zone spin susceptibility at twelve of these points, focusing on U/t = 4. At U/t = 4 and toward half filling the spin fluctuations become strong: for n = 0.7-0.9 at T/t = 0.14 they enhance the susceptibility peak up to sevenfold and raise the momentum-dependent part of the scattering rate to four to five times its second-order value. They are driven by an effective spin interaction screened to about 2U/3 and nearly uniform over the zone. Yet the peak height is set by the slightly smaller value of this interaction at the peak wave vector, and near magnetic order this small difference changes the height by a large factor. Two-particle self-consistent theory reproduces the peak because its sum rule fixes its single constant interaction near this value, but for n = 0.7-0.9 at T/t = 0.14 its self-energy overestimates the local scattering rate and underestimates the momentum-dependent part. Though strong, the fluctuations are short ranged, with a correlation length of at most about two lattice spacings, and for n = 0.7-0.9 and T/t >= 0.14 the scattering rate near the Fermi surface varies by at most about 1% at fixed band energy, without hot spots. At U/t = 4 dynamical mean-field theory (DMFT) reproduces the local scattering rate and quasiparticle weight within a few percent, and the nonlocal part it misses is approximately a function of the band energy. Strong spin fluctuations thus coexist with nearly isotropic quasiparticles, and the results provide a benchmark for extensions of DMFT and for spin-fluctuation theories.
We present a formalism for strongly correlated electron systems which consists in a local approximation of the dynamical three-leg interaction vertex. This vertex is self-consistently computed using a quantum impurity model with dynamical interactions in the charge and spin channels, similar to dynamical mean field theory approaches. The electronic self-energy and the polarization are both frequency and momentum dependent. The method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply the formalism to the Hubbard model on a two-dimensional square lattice and show that as interactions are increased towards the Mott insulating state, the local vertex acquires a strong frequency dependence, driving the system to a Mott transition, while at low enough temperatures the momentum dependence of the self-energy is enhanced due to large spin fluctuations. Upon doping, we find a Fermi arc in the one-particle spectral function, which is one signature of the pseudogap state.
The cluster slave-spin method is introduced to systematically investigate the solutions of the Hubbard model including the symmetry-broken phases. In this method, the electron operator is factorized into a fermionic spinon describing the physical spin and a slave-spin describing the charge fluctuations. Following the U(1) formalism derived by Yu and Si [Phys. Rev. B 86, 085104 (2012)], it is shown that the self-consistent equations to explore various symmetry-broken density wave states can be constructed in general with a cluster of multiple slave-spin sites. We employ this method to study the antiferromagnetic (AFM) state in the single band Hubbard model with the two- and four-site clusters of slave spins. While the Hubbard gap, the charge gap due to the doubly occupied states, scales with the Hubbard interaction U as expected, the AFM gap $\mathrm{\ensuremath{\Delta}}$, the gap in the spinon dispersion in the AFM state, exhibits a crossover from the weak- to strong-coupling behaviors as U increases. Our cluster slave-spin method reproduces not only the traditional mean-field behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}U$ in the weak-coupling limit, but also the behavior of $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{t}^{2}/U$ predicted by the superexchange mechanism in the strong-coupling limit. In addition, the holon-doublon correlator as functions of U and doping x is also computed, which exhibits a strong tendency toward the holon-doublon binding in the strong coupling regime. We further show that the quasiparticle weight obtained by the cluster slave-spin method is in a good agreement with the generalized Gutzwiller approximation in both AFM and paramagnetic states, and the results can be improved beyond the generalized Gutzwiller approximation as the cluster is enlarged from a single site to four sites. Our results demonstrate that the cluster slave-spin method can be a powerful tool to systematically investigate the strongly correlated system.
We have calculated within the fluctuation exchange approximation the full momentum and real-frequency dependence of the quasiparticle self-energy, the gap function, and the dynamic spin susceptibility for the two-dimensional Hubbard model near half filling. The spin-density-wave (SDW) instability is reached only asymptotically in the limit of infinite U or zero temperature. Near the SDW instability we find approximately marginal Fermi-liquid behavior, a pseudogap in the density of states, and a singlet dx2-y2 pairing instability. Below Tc, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The spectral density of the dynamic spin susceptibility agrees qualitatively with the neutron-scattering data on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$SrxCuO4. The quasiparticle scattering rate and spin-lattice relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$ decrease rapidly below Tc. The photoemission intensities along the antinode and node of the gap are in good agreement with the angle-resolved photoemission spectroscopy data on Bi 2212. Including next-nearest-neighbor hopping we obtain for sufficiently large coupling (U/t\ensuremath{\gtrsim}7) a broad commensurate neutron-scattering peak like the observed one in YBa2Cu3O6+x.
A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.