Fermionic two-loop functional renormalization group for correlated fermions: Method and application to the attractive Hubbard model

Fermionic two-loop functional renormalization group for correlated fermions: Method and application to the attractive Hubbard model

Andreas Eberlein Max Planck Institute for Solid State Research, D-70569 Stuttgart, Germany
July 8, 2019
Abstract

We derive an efficient method for treating renormalization contributions at two-loop level within the functional renormalization group in the one-particle irreducible formalism for fermions. It is based on a decomposition of the two-particle vertex in charge, magnetic and pairing channels. The method treats single-particle and collective excitations in all channels on equal footing, allows for the description of symmetry-breaking and captures collective mode fluctuation physics in the infrared. As a first application, we study the superfluid ground state of the two-dimensional attractive Hubbard model. We obtain superfluid gaps that are reduced by fluctuations in comparison to the one-loop approximation and demonstrate that the method captures the renormalization of the amplitude mode by long-range phase fluctuations.

pacs:
05.10.Cc, 71.10.Fd, 74.20.-z

I Introduction

In the last decade, the functional renormalization group (fRG) proved itself as an excellent tool for the study of competing instabilities in correlated electron systems and a valuable source of new approximation schemes.*[Forareview; see]Metzner2012 It treats charge, magnetic and pairing channels on equal footing and in a scale-separated way. This allows for the study of competing order and fluctuation driven instabilities. One of the major successes of fRG was to provide evidence for -wave superconductivity in the two-dimensional repulsive Hubbard model at weak and intermediate couplings.Zanchi and Schulz (2000); Halboth and Metzner (2000); Honerkamp et al. (2001); Eberlein and Metzner (2014) The method was also successfully applied to quantum wires and dots in and out of equilibrium,Andergassen et al. (2004); Karrasch et al. (2010) multiband systems describing pnictide superconductors Wang et al. (2009); Thomale et al. (2009); Platt et al. (2009) and also spin models.Reuther and Wölfle (2010); Reuther and Thomale (2011)

Most of these fRG studies used the one-particle irreducible (1PI) formalism in a purely fermionic formulation and were restricted to the one-loop level without or with taking the self-energy feedback into account in the flow equation for the vertex as suggested by Katanin.Katanin (2004) The latter modification improves the fulfillment of Ward identities and allows to continue fermionic fRG flows into phases with broken symmetries.Salmhofer et al. (2004); Gersch et al. (2005, 2008); Maier and Honerkamp (2012); Eberlein and Metzner (2013, 2014); Maier et al. (2014) However, in a recent study of superfluidity in the attractive Hubbard model, it was pointed out that the singular infrared behavior of the amplitude mode of the gap due to phase fluctuations Castellani et al. (1997); Pistolesi et al. (2004) is not captured in the Katanin scheme.Eberlein and Metzner (2013); Eberlein (2013)

An alternative route to studying fluctuation physics and symmetry breaking is to decouple the fermionic fields by introducing bosonic auxiliary fields via a Hubbard-Stratonovich transformation. Applying the fRG to the resulting mixed fermion-boson action captures the asymptotic infrared behavior of fermionic superfluids already at one-loop level within a relatively simple truncation.Strack et al. (2008); Obert et al. (2013) Besides fermionic superfluidity,Birse et al. (2005); Strack et al. (2008); Obert et al. (2013) this route was also pursued for studying the BEC-BCS crossover in continuum systems,Diehl et al. (2007); Bartosch et al. (2009); Diehl et al. (2010) and antiferromagnetism Baier et al. (2004) as well as -wave superconductivity Friederich et al. (2011) in the two-dimensional Hubbard model.

The choice of auxiliary fields introduces some bias if only a small number of them and simple truncations are used.Baier et al. (2004) This bias can be reduced by dynamical rebosonization.Gies and Wetterich (2002); Floerchinger and Wetterich (2009) Most fRG studies of partially bosonized actions relied on relatively simple ansätze for the bosonic potential. However, in situations with competition of instabilities, truncating the order parameter potential is not straightforward. It may therefore be advantageous to treat at least the high and intermediate scales in a purely fermionic language. In addition, results for the vertex in the Hubbard model at large couplings indicate that the effective interaction contains contributions that are not well described as boson-mediated interactions.Rohringer et al. (2012); Kinza and Honerkamp (2013) Recently, Veschgini and Salmhofer derived a hierarchy of flow equations for fermionic 1PI vertex functions from Dyson-Schwinger equations and pointed out that including all renormalization contributions at two-loop level may increase the degree of self-consistency.Veschgini and Salmhofer (2013)

All this motivates going beyond the one-loop approximation or Katanin’s scheme within the purely fermionic formalism. One fRG study at two-loop level has been carried out for the repulsive Hubbard model in the symmetric regime by Katanin.Katanin (2009) He used the -patch approximation for the momentum dependence of the vertex and neglected its frequency dependence. For relatively weak couplings, he concluded that fluctuations at two-loop level do not change the flow qualitatively. This is not clear at larger couplings and in symmetry-broken phases. Besides, in such situations the frequency dependence of the vertex should be taken into account, which seems to be beyond the scope of Katanin’s two-loop approach due to the resulting numerical complexity.

In this article, we present a reformulation of the two-loop flow equations that is effectively of one-loop form. It is exact to the third order in the effective interaction and based on a decomposition of the vertex in charge, magnetic and pairing channels.Karrasch et al. (2008); Husemann and Salmhofer (2009) It allows to take the frequency dependence of the vertex into account with a reasonable numerical effort and to continue fRG flows into symmetry-broken phases. We demonstrate that the method captures the singular asymptotic infrared behavior of a fermionic -wave superfluid. Besides the general scheme, we present as a first application a study of the superfluid ground state of the attractive Hubbard model. In the presence of a not too small external pairing field, the one- and two-loop flows are qualitatively similar, albeit with smaller critical scales and gaps at two-loop level. The infrared behavior is different in the two approximations, as expected.

The article is structured as follows. In Sec. II, we describe the reformulated flow equations at two-loop level that allow for an efficient numerical solution. In Sec. III, results for the attractive Hubbard model are presented, including estimates for the infrared behavior and numerical results. Section IV contains a short summary and final remarks.

Ii Method

In this section, we describe the reformulation of the two-loop flow equations as effective one-loop equations, which is at the heart of this article. The result is applicable to all systems whose effective actions contain vertices with an equal number of creation and annihilation operators only (see below) and in which it is meaningful to classify diagrams according to singular dependences on transfer momenta or frequencies.

A general derivation of renormalization group equations in the 1PI framework can be found for example in Ref. Metzner et al., 2012. Flow equations at two-loop level were derived for example in Refs. Salmhofer et al., 2004; Veschgini and Salmhofer, 2013.

ii.1 Flow equations at two-loop level

In this section, we outline the derivation of flow equations at two-loop level in order to make the article self-contained, following the presentations in Refs. Salmhofer et al., 2004; Eberlein, 2013.

Flow equations at two-loop level are based on a truncation of the flowing effective action at the three-particle level,

(1)

where , are anticommuting Grassmann fields and the 1PI -particle vertex functions111 describes the interaction correction to the grand-canonical potential and is not important in the following.. The ellipsis represents four-particle and higher-order terms. The Greek indices , with , collect momenta as well as fermionic Matsubara frequencies and the spin or Nambu index . For describing a system in the symmetric phase, the fermionic fields can be chosen in spinor representation, and , where is the spin orientation. For a fermionic singlet superfluid or superconductor, as considered in the following, the fields are most conveniently chosen in Nambu representation, where is the Nambu index and

(2)

Using this representation in the presence of spin rotation invariance, only vertices that create and annihilate an equal number of Nambu quasiparticles appear in the effective action.Gersch et al. (2008); Eberlein and Metzner (2010)

Inserting the ansatz for the effective action Eq. (1) into the functional flow equation and evaluating appropriate functional derivatives with respect to the fields (see Ref. Metzner et al., 2012) yields flow equations for the self-energy

(3)

and the two-particle vertex

(4)

where is the full fermionic propagator and the so-called single-scale propagator. The self-energy is connected to via a Dyson-Schwinger equation, , where is the regularized bare propagator at scale . The flow equation for the two-particle vertex is illustrated diagrammatically in Fig. 1.

Figure 1: Simplified diagrammatic representation of the one-loop renormalization group equation for the two-particle vertex.
Figure 2: Simplified diagrammatic representation of the one-loop renormalization group equation for the three-particle vertex.

The flow equation for the three-particle vertex can be written schematically as222The flow equation for including all indices and contributions in follows from Eqs. (9) and (10) by applying the derivative to the right hand sides.

(5)

and is illustrated diagrammatically in Fig. 2. A solution for in is obtained by neglecting the contributions in the first line of Eq. (5), because these are at least of fourth order in the effective interaction. At the same level of approximation, the scale-derivative in the second line of Eq. (5) can be replaced by a full scale-derivative that acts also on the self-energy and the vertex. The latter modification generates terms at least of . The resulting flow equation Salmhofer et al. (2004)

(6)

can straightforwardly be integrated because the right hand side is a total derivative, yielding

(7)

in a system with only two-particle interactions at the microscopic level. In more detail, the three-particle vertex in reads

(8)
(9)
(10)

In Eq. (9) all internal propagators run into the same direction, while this is not the case in Eq. (10). Inserting these expressions in Eq. (4) yields the flow equation for the two-particle vertex at two-loop level, which is illustrated schematically in Fig. 3.

Figure 3: Simplified diagrammatic representation of the two-loop flow equation for the two-particle vertex. The second and third diagrams on the right hand side can be classified as two-loop contributions with non-overlapping and overlapping loops, respectively.

The two-loop diagrams can be classified in contributions with non-overlapping and overlapping loops, which have the topological structure of the second and third diagram on the right hand side of Fig. 3, respectively. In two-loop diagrams with non-overlapping loops, the single-scale propagator in Eq. (4) contracts two indices at the same vertex. As discussed by Katanin,Katanin (2004) these contributions can be rewritten as one-loop diagrams with an insertion of . Exploiting the relation , they allow to replace the single-scale propagators in the one-loop contributions in Eq. (4) by scale-differentiated full propagators.

The two-loop contributions with overlapping loops can be treated in a similar fashion after decomposing the vertex in interaction channels, which is discussed in the next section.

ii.2 Channel-decomposed flow equations at two-loop level: scheme

In this section, we discuss how the two-loop contributions with overlapping loops can be reformulated effectively as one-loop contributions, and derive a channel-decomposition scheme for the vertex and the flow equations at two-loop level. The latter allows to extend the one-loop schemes by Karrasch et al. Karrasch et al. (2008) and Husemann and Salmhofer Husemann and Salmhofer (2009) for the symmetric phase, by the author and Metzner for singlet superconductors,Eberlein and Metzner (2010) and by Maier and Honerkamp for antiferromagnets.Maier and Honerkamp (2012)

The basic idea is to rewrite the insertions of two vertices that are connected by a full and a single-scale propagator in the two-loop contributions with overlapping loops (see third diagram on the right hand side of Fig. 3) as one-loop contribution (see first diagram on right hand side of Fig. 3). In order to make use of this idea, the vertex has to be decomposed in interaction channels and the diagrams in the flow equation have to be assigned to channels according to their leading singular dependence on external momenta and frequencies.

For a decomposition in interaction channels, the two-particle vertex is written as a sum of several terms, where each describes a possibly singular dependence of the vertex on momenta and frequencies. Assuming translation invariance, the multiindex notation of section II.1 is specialized to

(11)

where is nonzero only for . For the sake of compactness of the presentation, we stick to the multiindex notation in the major part of this section. The channel-decomposition of the vertex reads

(12)

where is the microscopic interaction and

(13)
(14)

describe fluctuation corrections in the (Nambu) particle-hole and (Nambu) particle-particle channel, respectively. The first argument of and describes the possibly singular dependence on the transfer momentum, while the last two momentum arguments describe dependences on fermionic relative momenta. The decomposition of the vertex is illustrated diagrammatically in Fig. 4.

Figure 4: Diagrammatic representation of the decomposition of the (Nambu) two-particle vertex in bare interaction, particle-hole channels and particle-particle channel.

Flow equations for the effective interactions in the particle-hole and particle-particle channels are derived by inserting the ansatz Eq. (12) into the flow equation (4) and assigning diagrams to interaction channels according to their leading singular dependence on external momenta. The one-loop contributions (i. e., Eq. (4) without the last term) are assigned to interaction channels as in the channel-decomposition schemes at one-loop level in such a way that the transfer momentum is transported through the diagrams by the fermionic propagators. The first and second contributions in the square bracket in Eq. (4) are assigned to the direct and crossed particle-hole channel, respectively. The third contribution is assigned to the particle-particle channel. This yields

(15)
(16)

After rewriting the two-loop diagrams with non-overlapping loops as one-loop diagrams with -insertions, they are assigned similarly according to the transfer momentum in the fermionic loops,

(17)
(18)

where is a shorthand for a -derivative that acts on the fermionic self-energy only, .

Figure 5: Illustration of the reorganization of two-loop diagrams with overlapping loops in a one-loop diagram with a scale-differentiated effective interaction.

Before assigning the two-loop contributions with overlapping loops to the interaction channels, it is convenient to rewrite them effectively as one-loop diagrams. This is possible after expressing insertions of two vertices that are connected by a full and a single-scale propagator as one-loop scale-derivatives of effective interactions. Consider as an example the renormalization contribution to the two-particle vertex arising from the first two terms in the first line of Eq. (10). After insertion into Eq. (4), these contributions read

(19)

Renaming summation indices and exploiting the antisymmetry of the vertex under particle exchange, this expression can be rewritten as

(20)

where Eq. (15) was exploited. This reorganization is illustrated diagrammatically in Fig. 5 and works similarly for all contributions in Eq. (4) after inserting Eqs. (9) and (10). Collecting terms yields the two-loop flow equation for the vertex expressed effectively as a one-loop equation,

(21)
Figure 6: Examples for (a) vertex correction and (b,c) box diagrams. At two-loop level, the scale-derivative acts on the effective interactions that transport the loop momentum . The transfer momentum is transported through the diagram by the fermionic propagators in (a) and (b), while it is transported by the effective interactions in (c).

In order to make use of this reorganization, we insert the decomposition of the vertex in interaction channels Eq. (12) on both sides of Eq. (21) and assign diagrams to interaction channels according to their leading singular dependence on external momenta. The assignment of the contributions in the first line of Eq. (21) was discussed above. After inserting the decomposition of the vertex, the second and third lines contain diagrams that can be classified in two-loop vertex correction diagrams (for an example see Fig. 6) and two-loop box diagrams (for examples see Figs. 6 and \subreffig6c:TBE). No two-loop propagator renormalization diagrams appear as a consequence of the topological structure of the two-loop diagrams with overlapping loops.

Like at one-loop level, the vertex correction diagrams are assigned to interaction channels according to the transfer momentum in the fermionic loop and in one effective interaction. The singular dependence on momentum in the other effective interaction is integrated and does not give rise to singular renormalization contributions in the ground state of a fermionic -wave superfluid (see section III.2). Differently from one-loop level, the two-loop box diagrams are assigned to interaction channels in such a way that the transfer momentum is transported through the diagram by the effective interactions (as shown in the example in Fig. 6). The reason is that these diagrams become important close to and below the critical scale,Salmhofer and Honerkamp (2001) where the effective interactions and their scale-derivatives already developed a strong dependence on momentum and frequency. The assignment according to the “bosonic” singularity yields a better treatment of the strong momentum and frequency dependence of the scale-differentiated effective interactions. In section III.2, we demonstrate that this assignment allows to capture the singular renormalization of the amplitude mode by long-range phase fluctuations in a fermionic -wave superfluid. The two-loop renormalization contributions to the effective interaction in the (Nambu) particle-hole and particle-particle channels read

(22)
(23)

Taking into account the one-loop contributions and the two-loop contributions with non-overlapping loops yields the channel-decomposed flow equations at two-loop level

(24)
(25)

These flow equations are illustrated diagrammatically in Figs. 7 and 8. The two-loop contributions are computed using and .

Figure 7: Diagrammatic representation of the two-loop renormalization group equation for the effective interaction in the (Nambu) particle-hole channel . The first term on the right hand side represents the one-loop contributions where acts on fermionic propagators. The other terms in the first line represent two-loop box diagrams where is a shorthand for a scale-derivative acting on effective interactions and yielding the one-loop result. The other terms represent two-loop vertex correction diagrams where the effective interactions with dots represent the one-loop contribution.
Figure 8: Diagrammatic representation of the two-loop renormalization group equation for the effective interaction in the (Nambu) particle-particle channel . The first term on the right hand side represents the one-loop contribution. The last term in the second line and the contributions in the third line represent two-loop vertex correction diagrams. The other contributions are two-loop box diagrams. The notation is the same as in Fig. 7.

In the next section, we give a few remarks about the assignment of diagrams in the two-loop channel-decomposition scheme.

ii.3 Discussion of assignment of diagrams

The channel-decomposition scheme derived in section II.2 is exact to the third order in the effective interaction. Due to the use of an approximation for the three-particle vertex of that order for its derivation, no -insertions appear in the scale-differentiated effective interactions in the two-loop contributions. However, within the same order of approximation the -insertions can be added to the scale-differentiated effective interactions in the two-loop contributions, because this introduces terms of . The resulting channel-decomposition scheme would correspond to the two-loop flow equations for the two-particle vertex proposed by Veschgini and Salmhofer.Veschgini and Salmhofer (2013) For the attractive Hubbard model, we checked such an assignment and found only minor differences in the results in the presence of a not too small external pairing field. Note however that this may change in other contexts. For pairing field flows, including the -insertions in the two-loop contributions leads to a somewhat stronger renormalization of exchange propagators, but qualitatively similar results. In the numerical study of the attractive Hubbard model, we therefore did not include the -insertions in the two-loop contributions.

The scheme presented in the last section is relatively compact, because we only distinguished between singularities arising from fermionic propagators or effective interactions when assigning diagrams to interaction channels. For very small external pairing fields, it may be advantageous to further distinguish between one-loop box diagrams with normal or anomalous fermionic propagators. The reason is that the anomalous self-energy receives strong renormalizations in the pairing field flow due to the singular behavior of the amplitude mode. Insertions of the resulting in the one-loop box diagrams may lead to artificial logarithmic singularities in non-Cooper channels when assigned as described above. These can be avoided by assigning the one-loop box diagrams with anomalous fermionic propagators in such a way that the transfer momentum is transported through the diagrams by the effective interactions. The one-loop box diagrams with normal fermionic propagators do not cause difficulties and should be assigned as described above. These subtleties matter only for very small external pairing fields (well beyond those that were accessible in the numerics). Therefore we decided to present the two-loop channel-decomposition scheme in the simpler form as above, but consider the more sophisticated version in the estimates in section III.2. We checked that the alternative assignment of one-loop box diagrams with anomalous propagators indeed yields very similar results in the numerically accessible range of external pairing fields.

Iii Attractive Hubbard model

In this section, we study the ground state of the attractive Hubbard model on the square lattice as a prototype for a singlet superfluid *[Forareview; see][]Micnas1990 using the two-loop channel-decomposition scheme. Progress in experiments with cold atoms in optical lattices sparked renewed research interest in this model in the last decade, because it can be simulated in such systems.Bloch et al. (2008); Esslinger (2010)

The attractive Hubbard model describes spin- fermions with a local attractive interaction on a lattice. Its Hamiltonian is

(26)

where and are creation and annihilation operators for fermions with spin orientation on lattice site . The interaction parameter is negative. The hopping of fermions is restricted to nearest- and next-nearest-neighbour sites with amplitudes and , respectively, yielding the dispersion relation

(27)

In the following, we use as the unit of energy.

The ground state of this model is an -wave spin-singlet superfluid at any fermionic density. For and (half-filling), superfluidity is degenerate with charge-density wave order. The model has been studied in its ground state and at finite temperatures with a variety of methods, including resummations of perturbation theory,Fresard et al. (1992); Pedersen et al. (1997); Letz and Gooding (1998); Keller et al. (1999) quantum Randeria et al. (1992); Trivedi and Randeria (1995); Santos (1994); Singer et al. (1996) and variational Tamura and Yokoyama (2012) Monte Carlo methods, dynamical mean-field theory,Keller et al. (2001, 2002); Capone et al. (2002) and functional renormalization group.Gersch et al. (2008); Strack et al. (2008); Eberlein and Metzner (2013); Obert et al. (2013)

iii.1 Parametrization and approximations

We now describe the approximate parametrization of the interaction vertex and self-energy, which allow for a numerical solution of the two-loop flow equations. The parametrizations are similar to those of Ref. Eberlein and Metzner, 2013.

In order to describe the superfluid state, we use Nambu fields (see Eq. (2)). In this representation, the fermionic propagator is a matrix and reads

(28)

where and are its normal and anomalous components, respectively. The propagator is connected with the regularized bare propagator and the Nambu self-energy by the Dyson equation . reads

(29)

where and is a counterterm. The regulator function

(30)

regularizes the fermionic singularities by replacing small frequencies with by . The external pairing field appears in the bare propagator, because it serves as a regulator for pairing field flows,Eberlein and Metzner (2013) in which is eliminated after integrating out the fermionic modes at . The Nambu self-energy is given by

(31)

where the off-diagonal entries are chosen in such a way that the fermionic gap is . The counterterm is related to the normal component of the self-energy via

(32)

for on the Fermi surface at all , such that the Fermi surface remains fixed during the flow. Similar to Ref. Eberlein and Metzner, 2013, we neglect the momentum dependence of the self-energy but keep its dependence on frequency,

(33)

The momentum dependence of the self-energy is not expected to be important at low fermionic densities, where the Fermi surface is almost circular. Furthermore, it lead only to minor changes of fRG flows for the weakly-coupled repulsive Hubbard model at van Hove filling.Giering and Salmhofer (2012) By choosing the external pairing field to be real, we fix the phase of the anomalous self-energy so that is also real. The frequency dependence of the self-energy is discretized on a grid of 30 points that is denser near and becomes sparser towards higher frequencies, with a maximal frequency around 300. Cubic spline interpolation is used to determine the self-energy at intermediate frequencies.

The interaction vertex is fully described by several coupling functions:Eberlein and Metzner (2010, 2013) and describe charge and spin fluctuations, respectively. The amplitude and phase mode of the superfluid gap are described by and . The imaginary part of the normal interaction in the Cooper channel is denoted as . The real and imaginary part of the anomalous effective interaction is described by and , respectively. The coupling functions are expanded in exchange propagators that describe the singular dependence on the transfer momentum and fermion-boson vertices for the more regular dependences on the fermionic relative momenta and . As in Ref. Eberlein and Metzner, 2013, we restrict this expansion to the -wave channel and approximate the coupling functions by the following ansatz:

(34)

In comparison to Ref. Eberlein and Metzner, 2013, we neglect the renormalization of the fermion-boson vertices in the particle-hole channel, for the anomalous effective interactions and for the imaginary part of the normal interaction in the Cooper channel. and are kept in order to obtain a meaningful frequency dependence of the gap function and to improve the fulfillment of the Ward identity for the global charge symmetry. These simplifications are justified because the neglected frequency-dependences of fermion-boson vertices had only a minor influence on the flow at one-loop level. The effective interactions in the Nambu particle-hole and Nambu particle-particle channel then read

(35)
(36)

where are Pauli matrices () and the unit matrix (). All functions in Eqs. (35) and (36) are even functions of momentum. and are odd functions of frequency, while all other functions are even. The frequency dependence of the fermion-boson vertices is discretized like for the self-energy. For the exchange propagators, we discretize the dependence on momenta and frequencies on a three-dimensional grid and trilinear interpolation is used at intermediate momenta and frequencies. The frequency dependence is discretized with 40 frequencies between and 300, with grid points denser at small frequencies. The momentum dependence is discretized with cylindrical coordinates around and , similar to Ref. Husemann and Salmhofer, 2009. The angular dependences are resolved with three angles between and . At quarter-filling, the radial dependence of the singular exchange propagators , , and around is discretized with 25 points between radius 0 and , with denser distribution of points near . All other exchange propagators have a weaker dependence on momentum and are thus described with only 10 points in the radial direction.

The flow of the exchange propagators and fermion-boson vertices is extracted from the flow equations for the coupling functions as described in Ref. Eberlein and Metzner, 2013 by averaging the external fermionic momenta and over the Fermi surface for suitable choices of the transfer momentum and the fermionic frequencies. The flow of the exchange propagators is evaluated for . The renormalization contributions to and are obtained after setting and . The flow of the fermionic self-energy is evaluated similarly by averaging the external fermionic momentum over the Fermi surface.

By discretizing the dependences on momenta and frequencies, the functional flow equations were transformed into a system of around 20000 non-linear ordinary differential equations with three-dimensional loop integrals on the right-hand sides. These loop integrals were performed with an adaptive integration algorithm. The system of differential equations was integrated using an adaptive third-order Runge-Kutta routine. Depending on the parameters, the numerical integration of a flow on 32 CPU cores required around three days at one-loop level and between two and four weeks at two-loop level. Due to the large number of flowing couplings and the fact that the computations of their renormalization contributions are independent at a given scale, the flow equations are well suited for parallelization.

The numerical integration of the flow equations was started at a large finite scale , which is of the order of several times the band width. The fermionic modes for were treated in second-order perturbation theory, yielding exchange propagators of the order of for , so that the contributions from are small compared to . Treating the high energy scales in perturbation theory also provides a well defined starting point for the flow of the fermion-boson vertices, which were set to one at . The normal self-energy receives a sizeable contribution from the tadpole diagram at any finite , yielding . The anomalous self-energy is determined self-consistently from the gap equation at scale , but the corrections to are small (of order ).

Due to the truncation of the hierarchy of flow equations and the approximations for the coupling functions, the Ward identity for the global charge symmetry is violated in the two-loop flows.Eberlein (2013) We did not systematically study how large these violation are quantitatively. For computing the results in section III.3, the Ward identity was enforced by a projection of the coupling constants as in Ref. Eberlein and Metzner, 2013. At low scales, this procedure effectively amounts to determining the Goldstone mass from the Ward identity instead of the flow equation.

iii.2 Analytical estimates for infrared behavior

Before presenting results from numerical solutions of the flow equations, we discuss the infrared behavior of the vertex in a fermionic -wave superfluid in the BCS regime at zero temperature. We assume that the fermionic modes at have been integrated out in the presence of an external pairing field . The latter regularizes the phase mode of the superfluid gap and is treated in a pairing field flow Eberlein and Metzner (2013) in this section. For simplicity, we assume a circular Fermi surface, but its shape is not expected to influence the conclusions. The infrared behavior at one-loop level was discussed in Ref. Eberlein and Metzner, 2013. In this section, we discuss only estimates for the most singular contributions to the flow at two-loop level. The flow equations including all terms are rather lengthy and can be found in Ref. Eberlein, 2013.

At small transfer momenta and frequencies, the exchange propagators in the Cooper channel and for the imaginary part of the anomalous (3+1) effective interaction are well described by

(37)

where the superscripts indicate that is the flow parameter. The ansätze for , and are consistent with a resummation of all chains of Nambu particle-hole diagrams Eberlein and Metzner (2013) and can be justified non-perturbatively for and using Ward identities.Castellani et al. (1997); Pistolesi et al. (2004) The ansatz for is consistent with the expected singular infrared behavior of the amplitude mode in an interacting Bose gas and reproduces the singular infrared scaling that was described in Refs. Strack et al., 2008; Obert et al., 2013 in terms of divergent wave function renormalization factors. The above-mentioned works indicate that the coefficients and in the ansätze remain finite when defined as above. We assume that all other exchange propagators are less singular in the limit where the external pairing field vanishes. Below we show that these assumptions are justified and the ansätze consistent with the infrared behavior of the flow. The fermion-boson vertices are set to one in this section, as they are not expected to influence the singular behavior.

In the presence of a superfluid gap and close to the Fermi surface, the fermionic propagator behaves like

(38)

for small and , together with appropriate ultraviolet cutoffs, where is the Fermi velocity and a unit vector pointing in the direction of . In this section, we neglect the normal self-energy and assume that it can be subsumed into Fermi liquid like renormalization factors, which do not influence the singular infrared behavior as they remain finite (see below).

The most singular contributions at two-loop level arise from diagrams involving scale-derivatives of the phase mode. The two-loop vertex correction diagrams of this kind involve an integral of the form