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arXiv:hep-ph/0207094 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Flow equations without mean field ambiguity

Joerg Jaeckel, CHRISTOF WETTERICH

Published Jul 28, 2003• 83 citations
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Abstract

We compare different methods used for nonperturbative calculations in strongly interacting fermionic systems. Mean field theory often shows a basic ambiguity related to the possibility to perform Fierz transformations. The results may then depend strongly on an unphysical parameter which reflects the choice of the mean field, thus limiting the reliability. This ambiguity is absent for Schwinger-Dyson equations or fermionic renormalization group equations. Also renormalization group equations in a partially bosonized setting can overcome the Fierz ambiguity if the truncation is chosen appropriately. This is reassuring since the partially bosonized renormalization group approach constitutes a very promising basis for the explicit treatment of condensates and spontaneous symmetry breaking even for situations where the bosonic correlation length is large.

DOI
10.1103/physrevd.68.025020

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We compare different methods used for nonperturbative calculations in strongly interacting fermionic systems. Mean field theory often shows a basic ambiguity related to the possibility to perform Fierz transformations. The results may then depend strongly on an unphysical parameter which reflects the choice of the mean field, thus limiting the reliability. This ambiguity is absent for Schwinger-Dyson equations or fermionic renormalization group equations. Also renormalization group equations in a partially bosonized setting can overcome the Fierz ambiguity if the truncation is chosen appropriately. This is reassuring since the partially bosonized renormalization group approach constitutes a very promising basis for the explicit treatment of condensates and spontaneous symmetry breaking even for situations where the bosonic correlation length is large.
We compare different methods used for non-perturbative calculations in strongly interacting fermionic systems. Mean field theory shows a basic ambiguity related to the possibility to perform Fierz transformations. The results depend strongly on an unphysical parameter which reflects the choice of the mean field. Renormalization group equations in a partially bosonized setting can overcome this problem if the truncation is chosen appropriately. We argue that the renormalization group approach goes beyond the lowest order Schwinger-Dyson equation and constitutes a very promising basis for the explicit treatment of condensates and spontaneous symmetry breaking.
A useful tool in non perturbative studies of fermionic theories is partial bosonization. However, partial bosonization is often connected to an ambiguity due to Fierz rearrangement in the original theory. We discuss two different approximations for the calculation of the effective action ΓΓΓ with respect to a spurious dependence on the choice of Fierz transformation: Mean field theory and the truncated flow of an exact renormalization group equation for the effective average action.
We present a novel scheme for an unbiased, nonperturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, the dynamical mean field theory and the functional renormalization group. Physically, this allows for a systematic inclusion of nonlocal correlations via the functional renormalization group flow equations, after the local correlations are taken into account nonperturbatively by the dynamical mean field theory. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half filling.
We compare different versions of a bosonic description for systems of interacting fermions, with particular emphasis on the free-energy functional. The bosonic effective action makes the issue of symmetries particularly transparent. A systematic expansion for the bosonic effective action starts with a solution to the lowest-order Schwinger-Dyson or gap equation. We propose a two-particle irreducible formulation of an exact functional renormalization group equation for computations beyond leading order. On this basis we suggest a renormalized gap equation. This approach is compared with functional renormalization in a partially bosonized setting.
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 application of the exact renormalisation group to symmetric as well as asymmetric many-fermion systems with a short-range attractive force is studied. Assuming an ansatz for the effective action with effective bosons, describing pairing effects a set of approximate flow equations for the effective coupling including boson and fermionic fluctuations has been derived. The phase transition to a phase with broken symmetry is found at a critical value of the running scale. The mean-field results are recovered if boson-loop effects are omitted. The calculations with two different forms of the regulator are shown to lead to a similar results. We find that, being quite small in the case of the symmetric many-fermion system the corrections to mean field approximation becomes more important with increasing mass asymmetry.