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arXiv:1411.1342 · Physical Review B

Correlated starting points for the functional renormalization group

Nils Wentzell, Ciro Taranto, A. A. Katanin, A. Toschi, Sabine Andergassen

Published Jan 16, 2015• 47 citations
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Abstract

We present a general frame to extend functional renormalization group (fRG) based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a nonperturbative inclusion of correlations. Introducing auxiliary fermionic fields by means of a Hubbard-Stratonovich transformation, we derive the flow equations for the auxiliary fields and determine the relation to the conventional weak-coupling truncation of the hierarchy of flow equations. As a specific example we consider the dynamical mean field theory (DMFT) solution as reference system, and discuss the relation to the recently introduced ${\mathrm{DMF}}^{2}\mathrm{RG}$ and the dual-fermion formalism.

DOI
10.1103/physrevb.91.045120

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We present a general frame to extend functional renormalization group (fRG) based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a nonperturbative inclusion of correlations. Introducing auxiliary fermionic fields by means of a Hubbard-Stratonovich transformation, we derive the flow equations for the auxiliary fields and determine the relation to the conventional weak-coupling truncation of the hierarchy of flow equations. As a specific example we consider the dynamical mean field theory (DMFT) solution as reference system, and discuss the relation to the recently introduced DMF2RG{\mathrm{DMF}}^{2}\mathrm{RG}DMF2RG and the dual-fermion formalism.
We consider formulations of the functional renormalization-group (fRG) flow for correlated electronic systems with the dynamical mean-field theory as a starting point. We classify the corresponding renormalization-group schemes into those neglecting one-particle irreducible six-point vertices (with respect to the local Green’s functions) and neglecting one-particle reducible six-point vertices. The former class is represented by the recently introduced DMF 2 RG approach [31], but also by the scale-dependent generalization of the one-particle irreducible representation (with respect to local Green’s functions, 1PI-LGF) of the generating functional [20]. The second class is represented by the fRG flow within the dual fermion approach [16, 32]. We compare formulations of the fRG approach in each of these cases and suggest their further application to study 2D systems within the Hubbard model.
We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model (i.e., one-dimensional fermions with linear energy dispersion and interactions involving only small momentum transfers) emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integrodifferential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.
This thesis aims at developing new schemes for the treatment of correlation effects in condensed matter systems using quantum field theoretical approaches. In particular, our goal is to extend the description of correlation physics at the two-particle level. This is necessary for an unbiased treatment of condensed matter systems that exhibit electronic correlations and competing ordering tendencies. In this respect, the functional renormalization group (fRG) approaches have surely contributed substantially over the last years, as they account for all scattering channels and their mutual feedback effects in an unbiased way. In spite of its flexibility, the application of the fRG is limited by its inherent perturbative nature. To go beyond the conventional weak-coupling implementations, we discuss the general idea to extend fRG based computational schemes by using an exactly solvable interacting reference problem as starting point for the RG flow. The systematic expansion around this solution accounts for a non-perturbative inclusion of correlations at both, the one-particle (self-energy) and two-particle (vertex functions) level. The full treatment of the two-particle vertex functions, however, poses a huge limitation to the numerical performance, not only in the fRG, but in several forefront many-body algorithms. In this perspective, we provide a detailed diagrammatic analysis of the frequency and momentum structures of the vertex functions, together with their physical interpretation. This constitutes the basis for sophisticated parametrization schemes. We then explain the technical details necessary for cutting-edge numerical implementations, and further benchmark our ideas using refined implementations of both, the fRG and the parquet approximation (PA).
We derive functional renormalization group (fRG) schemes for Fermi systems which are based on the two-particle irreducible approach to the quantum many-body problem. In a first step, the cutoff is introduced in the non-interacting propagator as it is commonly done in fRG based on one-particle irreducible vertex functions. The most natural truncation of the resulting infinite hierarchy of flow equations is shown to be fully equivalent to self-consistent perturbation theory. An earlier suggested alternative truncation strategy is considered as well. In a second step, the cutoff is introduced in the two-particle interaction. Again two truncation procedures are investigated, one of which was derived before. In the latter, the mean-field solution of the many-body problem is considered as the starting point of the renormalization group flow. We compare the performance and the required numerical resources for solving the coupled flow equations for all the approximate schemes by applying them to the problem of the quantum anharmonic oscillator. In a functional integral representation, this model has a formal similarity to the quantum many-body problem. The perspectives for applying the derived two-particle irreducible fRG approaches to zero- and one-dimensional systems of correlated fermions are discussed.
We derive an expansion of the functional renormalization (fRG) equations that treats the frequency and momentum dependencies of the vertices in a systematic manner. The scheme extends the channel-decomposed fRG equations to the frequency domain and reformulates them as a series of linear integral equations in the particle-particle, particle-hole and particle-hole exchange channels. We show that the linearity of the equations offers numerous computational advantages and leads to converged, stable solutions for a variety of Hamiltonians. As the expansion is in the coupling between channels, the truncations that are necessary to making the scheme computationally viable still lead to equations that treat contributions from all channels equally. As a first benchmark we apply the two-loop fRG equations to the single impurity Anderson model. We consider the sources of error within the fRG, the computational cost associated with each, and how the choice of regulator affects the flow of the fRG. We then use the optimal truncation scheme to study the Extended Hubbard Hamiltonian in one and two dimensions. We find that in many cases of interest the fRG flow converges to a stable vertex and self-energy from which we can extract the various correlation functions and susceptibilities of interest.
We construct exact functional renormalization group (RG) flow equations for nonrelativistic fermions in arbitrary dimensions, taking into account not only mode elimination, but also the rescaling of the momenta, frequencies, and fermionic fields. The complete RG flow of all relevant, marginal, and irrelevant couplings can be described by a system of coupled flow equations for the irreducible n-point vertices. Introducing suitable dimensionless variables, we obtain flow equations for generalized scaling functions which are continuous functions of the flow parameter, even if we consider quantities which are dominated by momenta close to the Fermi surface, such as the density-density correlation function at long wavelengths. We also show how the problem of constructing the renormalized Fermi surface can be reduced to the problem of finding the RG fixed point of the irreducible two-point vertex at vanishing momentum and frequency. We argue that only if the degrees of freedom are properly rescaled is it possible to reach scale-invariant non-Fermi-liquid fixed points within a truncation of the exact RG flow equations.