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arXiv:cond-mat/0310112 · Physical Review Letters

Quasiparticle Anisotropy and Pseudogap Formation from the Weak-Coupling Renormalization Group Point of View

A. A. Katanin, A. P. Kampf

Published Sep 3, 2004• 80 citations
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Abstract

Using the one-loop functional renormalization group technique, we evaluate the self-energy in the weak-coupling regime of the 2D t-t(') Hubbard model. At van Hove (vH) band fillings and at low temperatures, the quasiparticle weight along the Fermi surface (FS) continuously vanishes on approaching the (pi,0) point where the quasiparticle concept is invalid. Away from vH band fillings the quasiparticle peak is formed inside an anisotropic pseudogap and the self-energy has the conventional Fermi-liquid characteristics near the Fermi level. The spectral weight of the quasiparticle features is reduced on parts of the FS between the near vicinity of hot spots and the FS points closest to (pi,0) and (0,pi).

DOI
10.1103/physrevlett.93.106406

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Comment on ``Quasiparticle Anisotropy and Pseudogap Formation from the Weak-Coupling Renormalization Group Point of View''

Behnam Farid

In their recent Letter (Phys. Rev. Lett., Vol. 93, 106406 (2004)), Katanin and Kampf reported numerical results for the self-energy Sigma(k;e), at real values of e, of the single-band Hubbard Hamiltonian in two space dimensions, obtained through employing the functional renormalization-group (fRG) formalism at the one-loop level. Several of the results by Katanin and Kampf are in full conformity with the exact formal results reported earlier by the present author. This, as we shall elaborate in this contribution, strengthens one's confidence in the reliability of the fRG in dealing with models of strongly-correlated fermions.

arXiv:cond-mat/0410453•Oct 18, 2004•2 citations
Using the one-loop functional renormalization group technique, we evaluate the self-energy in the weak-coupling regime of the 2D t-t(') Hubbard model. At van Hove (vH) band fillings and at low temperatures, the quasiparticle weight along the Fermi surface (FS) continuously vanishes on approaching the (pi,0) point where the quasiparticle concept is invalid. Away from vH band fillings the quasiparticle peak is formed inside an anisotropic pseudogap and the self-energy has the conventional Fermi-liquid characteristics near the Fermi level. The spectral weight of the quasiparticle features is reduced on parts of the FS between the near vicinity of hot spots and the FS points closest to (pi,0) and (0,pi).
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperature. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the imaginary part of the self-energy. Concerning the energy dependence of the spectral function and the self-energy, two theorems are proved. They give a necessary condition in the self-energy to produce the pseudogap at the Fermi level. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. It opens initially in the (±π,0)(\pm π,0)(±π,0), (0,±π)(0,\pm π)(0,±π) regions as the normal state pseudogap observed in the high-TcT_cTc​ superconductors and if the interaction is increased, it spreads to other Fermi surface sectors. The anisotropy of the pseudogap is produced by the low-energy enhancement of the spin excitation around Q=(π,π){\bf Q}=(π,π)Q=(π,π) and the flatness of the band dispersion around the saddle point.
We analyze the interaction-induced renormalization of single-particle excitations in the two-dimensional Hubbard model at weak coupling using the Wick-ordered version of the functional renormalization group. The self-energy is computed for real frequencies by integrating a flow equation with renormalized two-particle interactions. In the vicinity of hot spots, that is, points where the Fermi surface intersects the umklapp surface, self-energy effects beyond the usual quasiparticle renormalizations and damping occur near instabilities of the normal, metallic phase. Strongly enhanced renormalized interactions between particles at different hot spots generate a pronounced low-energy peak in the imaginary part of the self-energy, leading to a pseudogaplike double-peak structure in the spectral function for single-particle excitations.
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperatures. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the self-energy imaginary part. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. This anisotropy of the pseudogap is produced by the flatness of the band dispersion around the saddle point rather than the nesting condition on the Fermi level.
We examine the quasiparticle lifetime and spectral weight near the Fermi surface in the two-dimensional Hubbard model. We use the FLEX approximation to self-consistently generate the Matsubara Green's functions and then we analytically continue to the real axis to obtain the quasiparticle spectral functions. We compare the spectral functions found in the nearest neighbor hopping only Hubbard model with those found when second neighbor hopping is included. This separates the effects of nesting, the Van Hove singularity and the short-ranged antiferromagnetic correlations. The quasiparticle scattering rate is enhanced along the (0,pi) to (pi,pi) Brillouin zone diagonal. When the density is close to half-filling these 'hot-spots' lie on the Fermi surface and the scattering rate increases with decreasing temperature. For the next-nearest neighbor hopping scenario we observe a large range of doping where there is no antiferromagnetism but the scattering rate has a linear temperature dependence. On decreasing the interaction this non-Fermi liquid behavior is confined to doping levels where the Fermi energy lies near to the Van Hove singularity. We conclude that the 'hot-spots' are associated with the antiferromagnetic phase transition while the linear temperature dependence of the scattering rate is associated with the Van Hove singularity.
We study the impact of the fermionic self-energy on one-loop functional renormalization group flows of the two-dimensional $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}$ Hubbard model, with emphasis on electronic densities away from van Hove filling. In the presence of antiferromagnetic hot spots, antiferromagnetic fluctuations lead to a flattening of the Fermi surface, shift magnetic phase boundaries, and significantly enhance critical scales. We trace back this effect to the presence of a magnetic first-order transition. For some parameters, the first-order character of the latter is reduced by self-energy effects. For reliably determining phase diagrams, the fermionic self-energy should be taken into account in functional renormalization group studies if scattering between hot spots is important.
In their recent Letter (Phys. Rev. Lett., Vol. 93, 106406 (2004)), Katanin and Kampf reported numerical results for the self-energy Sigma(k;e), at real values of e, of the single-band Hubbard Hamiltonian in two space dimensions, obtained through employing the functional renormalization-group (fRG) formalism at the one-loop level. Several of the results by Katanin and Kampf are in full conformity with the exact formal results reported earlier by the present author. This, as we shall elaborate in this contribution, strengthens one's confidence in the reliability of the fRG in dealing with models of strongly-correlated fermions.