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arXiv:cond-mat/0410453

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

Behnam Farid

Published Oct 18, 2004• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.cond-mat/0410453

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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.
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).
Abstract: We illustrate the algorithmic advantages of the recently introduced single-boson exchange (SBE) formulation for the one-loop functional renormalization group (fRG), by applying it to the two-dimensional Hubbard model on a square lattice. We present a detailed analysis of the fermion-boson Yukawa couplings and of the corresponding physical susceptibilities by studying their evolution with temperature and interaction strength, both at half filling and finite doping. The comparison with the conventional fermionic fRG decomposition shows that the rest functions of the SBE algorithm, which describe correlation effects beyond the SBE processes, play a negligible role in the weak-coupling regime above the pseudo-critical temperature, in contrast to the rest functions of the conventional fRG. Remarkably, they remain finite also at the pseudo-critical transition, whereas the corresponding rest functions of the conventional fRG implementation diverge. As a result, the SBE formulation of the fRG flow allows for a substantial reduction of the numerical effort in the treatment of the two-particle vertex function, paving a promising route for future multiboson and multiloop extensions.
We consider the application of the two-loop functional renormalization group (fRG) approach to study the low-dimensional Hubbard model. This approach accounts for both the universal and nonuniversal contributions to the RG flow. While the universal contributions were studied previously within the field-theoretical RG for the one-dimensional Hubbard model with linearized electronic dispersion and the two-dimensional Hubbard model with flat Fermi surface, the nonuniversal contributions to the flow of the vertices and susceptibilities appear to be important at large momenta scales. The two-loop fRG approach is also applied to the two-dimensional Hubbard model with a curved Fermi surface and the van Hove singularities near the Fermi level. The vertices and susceptibilities in the end of the flow of the two-loop approach are suppressed in comparison with both the one-loop approach with vertex projection and the modified one-loop approach with corrected vertex projection errors. The results of the two-loop approach are closer to the results of one-loop approach with the projected vertices, the difference of the results of one- and two-loop fRG approaches decreases when going away from the van Hove band filling. The quasiparticle weight remains finite in two dimensions at not too low temperatures above the paramagnetic ground state.
In this thesis we apply the functional renormalisation group (fRG) to the two-dimensional Hubbard model (2D-HM). We do this by solving approximate renormalisation group equations for the self energy and the two-particle interaction numerically. The purpose of these calculations is two-fold in the sense that we want to learn about the reliability of the method on the one hand, and about the physics of the 2D-HM on the other. We find that different fRG schemes agree in describing the mutual interplay and the resulting competition between different correlations, but also that the results sometimes differ quantitatively. This sharpens our perception when it comes to interpreting fRG results, and tells us that we have to separate the indicative character from the quantitative aspects of the results. The main line of arguments presented in previous fRG studies is confirmed, that in the weak-coupling region particle-hole correlations feed back into the particle-particle channel, triggering strong superconducting correlations in a non-zero parameter range. Concerning the physics of the 2D-HM show new results for the single-particle spectral functions for the case of a Fermi surface which is not perfectly nested and intersects the Umklapp surface at so-called hot spots. These results are obtained from the Wick-ordered fRG scheme, and the calculations are done directly on the real frequency axis. We find that close to a critical temperature TcT_cTc​ where correlations become strong, a dip appears in the single-particle spectral functions for momenta close to and at the hot spots. Away from the van Hove scale, this is traced back to strong correlations in the particle-hole channel and attributed to antiferromagnetic fluctuations, in agreement with numerous previous studies. For a Fermi surface at or very close to the van Hove scale, there exists a degeneracy in the sense that the strongest correlations are found around a point in coupling space which is relevant for both superconductivity and antiferromagnetism, and we can thus not distinguish between the importance of the respective fluctuations.
The functional renormalization group (fRG) provides an unbiased framework to analyze competing instabilities in two-dimensional electron systems and has been used extensively over the past decade [1]. In order to obtain an equally unbiased tool to interprete the flow, we investigate the combination of a many-patch, one-loop calculation with higher dimensional bosonization [2] of the resulting low-energy action. Subsequently a semiclassical approximation [3] can be used to describe the resulting phases. The spinless Hubbard model on a square lattice with nearest neighbor repulsion is investigated as a test case. [1] M. Salmhofer and C. Honerkamp, Prog. Theor. Phys. 105, 1 (2001). [2] A. Houghton, H.-J. Kwon, J. B. Marston, Adv.Phys. 49, 141 (2000); P. Kopietz, Bosonization of interacting fermions in arbitrary dimensions, (Springer, Berlin, 1997). [3] H.-H. Lin, L. Balents, M. P. A. Fisher, Phys. Rev. B 56, 65696593 (1997); J. O. Fjaerestad, J. B. Marston, U. Schollwoeck, Ann. Phys. (N.Y.) 321, 894 (2006).
The functional renormalization group (fRG) is an established tool in the treatment of correlated electron systems, notably for the description of competing instabilities. In recent years, methodological advancements led to the multiloop extension of the fRG, which systematically includes loop corrections beyond the conventional one-loop truncation and yields a quantitatively accurate description of two-dimensional lattice systems. At the same time, the single-boson exchange (SBE) decomposition of the two-particle vertex has been shown to offer both computational and interpretative advantages paving the way to more affordable approximation schemes. We here apply their combination coined as multiloop SBE fRG to the two-dimensional Hubbard model at weak coupling. After providing a detailed account of the underlying formalism in physical channels, we analyze the results for the frequency- and momentum-dependent vertex functions. We find that the SBE approximation, i.e., neglecting the flow of the multi-boson exchange contributions, accurately reproduces the parquet approximation at loop convergence. The presented algorithmic improvement opens the route for the treatment of more challenging parameter regimes and more realistic models.