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arXiv:1111.6802 · Physical Review B

Frequency-dependent vertex functions of the ( t,t′ ) Hubbard model at weak coupling

Christoph Husemann, Kay-Uwe Giering, Manfred Salmhofer B

Published Feb 21, 2012• 62 citations
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Abstract

We present a functional renormalization group calculation for the two-dimensional $(t,{t}^{\ensuremath{'}})$-Hubbard model at Van Hove filling. Using a channel decomposition we describe the momentum and frequency dependence of the vertex function in the normal phase. Compared to previous studies that neglect frequency dependencies we find higher pseudocritical scales and a smaller region of $d$-wave superconductivity. A large contribution to the effective interaction is given by a forward-scattering process with finite frequency exchange. We test different frequency parametrizations and in a second calculation include the frequency dependence of the imaginary self-energy. We also generalize the channel decomposition to frequency-dependent fermion-boson vertex functions.

DOI
10.1103/physrevb.85.075121

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We present a functional renormalization group calculation for the two-dimensional (t,t\ensuremath′)(t,{t}^{\ensuremath{'}})(t,t\ensuremath′)-Hubbard model at Van Hove filling. Using a channel decomposition we describe the momentum and frequency dependence of the vertex function in the normal phase. Compared to previous studies that neglect frequency dependencies we find higher pseudocritical scales and a smaller region of ddd-wave superconductivity. A large contribution to the effective interaction is given by a forward-scattering process with finite frequency exchange. We test different frequency parametrizations and in a second calculation include the frequency dependence of the imaginary self-energy. We also generalize the channel decomposition to frequency-dependent fermion-boson vertex functions.
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.
We study the impact of including self-energy feedback and frequency-dependent interactions on functional renormalization group flows for the two-dimensional Hubbard model on the square lattice at weak to moderate coupling strength. Previous studies using the functional renormalization group had ignored these two ingredients to a large extent, and the question is how much the flows to strong coupling analyzed by this method depend on these approximations. Here we include the imaginary part of the self-energy on the imaginary axis and the frequency dependence of the running interactions on a frequency mesh of 10 frequencies on the Matsubara axis. We find that (i) the critical scales for the flows to strong coupling are shifted downward by a factor that is usually of order 1 but can get larger in specific parameter regions, and (ii) that the leading channel in this flow does not depend strongly on whether self-energies and frequency dependence is included or not. We also discuss the main features of the self-energies developing during the flows.
The functional renormalization group has become a widely used tool for the analysis of the leading low-temperature correlations in weakly to moderately coupled many-fermion lattice systems. A bottleneck for quantitatively more precise results is that the treatment of the frequency dependence of the flowing interactions is numerically quite demanding. Yet the frequency dependence is needed to compute relevant self-energies and hence for controlled results on the energy scales for ordering or for the quasiparticle properties. Here we explore an approximate parametrization of the frequency dependence of the interaction vertex that is inspired by established simplifications in the theory of superconductivity and that keeps the numerical effort bounded. We demonstrate the validity of the approximation for Cooper pairing problems and apply it to the two-dimensional Hubbard model.
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.
The treatment of strongly interacting two-dimensional Fermi systems constitutes one of the most challenging problems in the field of condensed matter physics. Many theoretical works focused on the strongly coupled Hubbard model, since it is expected to capture the most important physics of cuprate superconductors. Due to our methodological improvements and understanding of the frequency dependence of the two-particle vertex function, the functional renormalization group combined with the dynamical mean field theory can now be used to study competing correlations in the strongly interacting regime. While limited to moderate interaction strengths, the functional renormalization group describes efficiently systems with a hierarchy of different energy scales and competing correlations. For instance, it provides definite evidence for d-wave superconductivity in the two-dimensional Hubbard model at moderate coupling. In a first project, we study the frequency dependence of the one-particle irreducible vertex function generated by the functional renormalization group flow. The frequency dependence, which becomes singular for strong interactions, appears to be important already for moderate couplings, and it cannot be represented by separate channels each depending on a single linear combination of frequencies. For strongly interacting systems, the dynamical mean field theory captures strong local correlation effects nonperturbatively. With this approximation, we study the impact of local correlations on the magnetic susceptibility. The local dynamics strongly affects the spin response function and its momentum dependence. In contrast to the widely used random-phase approximation with self-energy corrections that predicts Néel antiferromagnetic order, the local vertex corrections favor an incommensurate order similar to the ordering instability predicted by the Fermi surface geometry, as for weakly interacting systems. The dynamical mean field theory is also used as starting point for the functional renormalization group flow. We demonstrate that, due to our improvements for the parametrization of the vertex function, this approach is actually able to access the strong coupling physics. Moreover, we derive a flow scheme that conserves the local contributions and reduces the truncation error of the flow equations. In the strongly interacting regime, we capture strong d-wave pairing correlations driven by magnetic fluctuations with a mechanism similar to the one observed in the weakly interacting system.
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.