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arXiv:1510.05398 · Physical Review B

Self-energy effects in functional renormalization group flows of the two-dimensional t−t′ Hubbard model away from van Hove filling

Andreas Eberlein

Published Dec 28, 2015• 23 citations
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Abstract

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.

DOI
10.1103/physrevb.92.235146

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We study the impact of the fermionic self-energy on one-loop functional renormalization group flows of the two-dimensional t\ensuremath-t\ensuremath′t\text{\ensuremath{-}}{t}^{\ensuremath{'}}t\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.
We study the two-dimensional repulsive Hubbard model by functional renormalization group methods, using our recently proposed channel decomposition of the interaction vertex. The main technical advance of this work is that we calculate the full Matsubara frequency dependence of the self-energy and the interaction vertex in the whole frequency range without simplifying assumptions on its functional form, and that the effects of the self-energy are fully taken into account in the equations for the flow of the two-body vertex function. At Van Hove filling, we find that the Fermi-surface deformations remain small at fixed particle density and have a minor impact on the structure of the interaction vertex. The frequency dependence of the self-energy, however, turns out to be important, especially at a transition from ferromagnetism to ddd-wave superconductivity. We determine non-Fermi-liquid exponents at this transition point.
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 complement previous functional renormalization group (fRG) studies of the two-dimensional Hubbard model by mean-field calculations. The focus falls on Van Hove filling and the hopping amplitude ${t}^{\ensuremath{'}}/t=0.341$. The fRG data suggest a quantum critical point (QCP) in this region and in its vicinity a singular fermionic self-energy $\mathrm{Im}\phantom{\rule{0.16em}{0ex}}\mathrm{\ensuremath{\Sigma}}(\ensuremath{\omega})/\ensuremath{\omega}\ensuremath{\sim}\ensuremath{-}{\left|\ensuremath{\omega}\right|}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}0.26$ [K-U. Giering and M. Salmhofer, Self-energy flows in the two-dimensional repulsive Hubbard model, Phys. Rev. B 86, 245122 (2012)]. Here we start a more detailed investigation of this QCP using a bosonic formulation for the effective action, where the bosons couple to the order parameter fields. To this end, we use the channel decomposition of the fermionic effective action developed in C. Husemann and M. Salmhofer, Efficient Parametrization of the Vertex Function, Omega-Scheme, and the (t, t')-Hubbard Model at Van Hove Filling, Phys. Rev. B 79, 195125 (2009), which allows us to perform Hubbard-Stratonovich transformations for all relevant order parameter fields at any given energy scale $\mathrm{\ensuremath{\Omega}}$. We stop the flow at a scale $\mathrm{\ensuremath{\Omega}}$ where the correlations of the order parameter field are already pronounced, but the flow is still regular, and derive the effective boson theory. It contains ddd-wave superconducting, magnetic, and density-density interactions. We analyze the resulting phase diagram in the mean-field approximation. We show that the singular fermionic self-energy suppresses gap formation both in the superconducting and magnetic channel already at the mean-field level, thus rounding a first-order transition (without self-energy) to a quantum phase transition (with self-energy). We give a simple effective model that shows the generality of this effect. In the two-dimensional Hubbard model, the effective density-density interaction is peaked at a nonzero frequency, so that solving the mean-field equations already involves a functional equation instead of simply a matrix equation (on a technical level, similar to incommensurate phases). Within a certain approximation, we show that such an interaction leads to a short quasiparticle lifetime.
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.
We present the formalism for a two-loop renormalization group (RG) calculation of some order-parameter susceptibilities associated with a two-dimensional (2D) flat Fermi-surface model. In this order of perturbation theory, one must take into account the self-energy effects directly in all RG flow equations. In one-loop order, our calculation reproduces the well-known results obtained previously by other RG schemes. That is, for repulsive interactions all susceptibilities diverge in the low-energy limit and the antiferromagnetic (AF) spin-density-wave correlations produce indeed the leading instability in the system. In contrast, in two-loop order, only the AF susceptibility diverges for this model. However, even this divergence takes place at a much slower rate than in the one-loop RG approach. The purpose of this paper is to show in a very simple setting how to assess the importance of two-loop quantum fluctuations in 2D interacting fermionic models. With some modifications, the present formalism can also be extended to discuss more realistic models such as the paradigmatic 2D Hubbard model.