T c

with t and t' being the hopping amplitudes to nearest neighbor and diagonal next-nearest neighbors on the square lattice. U is the Hubbard onsite repulsion. It serves as initial condition for the coupling function parametrizing the interaction vertex at high scales A0 ~ 4t before any modes have been integrated out, VAo(k1 , k2, k3 = U. Together with the chemical potential the hopping parameters t and t' determine the shape of the Fermi surface. The band structure including ^ is given by e(k) = —2t cos cos — 4t' cos cos —

Any band structure with the inversion symmetry e(k) = e(—k) and nonzero density of states at the Fermi surface exhibits a logarithmically divergent particle-particle diagram for zero incoming wavevector and frequency for T ! 0. These diagrams can be summed up in a ladder summation. For repulsive onsite interactions, this does not cause a divergence, but a strong suppression of scattering processes with zero total wavevector and frequency. On the other hand, particle-hole terms are known to create attractive nonlocal interactions. Inserting these back into the particle-particle ladder into this summation will cause Cooper instabilities in unconventional (possibly very anisotropic) pairing channels if the temperature is low enough. This is the so-called Kohn-Luttinger effect [27,64] that predicts that all inversion-symmetric metals (in at least two dimensions) have a superconducting ground state, unless another instability excludes this. In fact, mathematically it is impossible to guarantee the convergence of perturbation theory for generic systems below a certain energy scale determined by the mentioned particle-particle ladder [17]. Note, however, that the Kohn-Luttinger effect may only occur at very low T and hence may be only of academic interest for many material classes. In particular, impurities could kill many of these low-Tc anisotropic superconducting phases easily. In the 2D Hubbard model on the square lattice, however, particle-hole corrections to the pair scattering have a marked influence and lead to rather high critical scales for Cooper pairing, as described below.

Regarding the possible ground states of the one-band Hubbard model on the square lattice at weak to moderate U, the Fermi surface location and shape plays a dominant role. Let us now consider how the N -patch fRG-flows differ for different Fermi surfaces.

For ^ = 0 and t' = 0, one has a half-filled band with a perfectly nested Fermi surface. All electronic states that are occupied at T = 0, i.e., with e(k) < 0, can be scattered by wavevector Q = (w, w) on unoccupied states, leading to a dominance of particle-hole fluctuations with this nesting wavevector. A random-phase approximation summation of these bubbles would result in a divergent static spin susceptibility at Q for any >0 at sufficiently low temperatures indicating the formation of an antiferromagnetic (AF) spin-density wave. The basic fRG results for low T are shown in the left half of Fig. 12.5. In the upper plot the Fermi surface is shown together with N = 32 discretization points. In the middle plot we show the flow of two families of coupling constants. We can see that some of

1030 20 10

10 20 30 k

27

23

/j

19

3

\V

11

1003

30 20 10

10 20 30 k

10 20 30

Fig. 12.5 N-patch fRG data obtained with the momentum-shell 1PI fRG scheme for the repulsive Hubbard model on the 2D square lattice. Left plots: ^ = 0, t0 = 0 and initial U = 2t, right plots: ^ = 1.2t, t0 = _0.3t, U = 3t. Upper row: Fermi surfaces for the two cases and the N = 32 discretization points for the two incoming k 1, k2 and the first outgoing wavevector k3. Middle row: Solid lines show the flows of components in the coupling function VA (k1, k2, k3) corresponding to Cooper pair scattering with zero total incoming wavevector, k1 C k2 = 0, or |k1 _ k2| = N/2 in terms of patch indices. The dashed lines correspond to processes in the AF-SDW channel with wavevector transfer k2_k3 near ±(w, w). The flow is started at A = 4t and goes to the left toward smaller A. Lower plots: Snapshots of the coupling function VA(k1, k2, k3) near the instability with k1 fixed at point 1 with k1 and k2 moving around the Fermi surface. The colorbars on the right indicate the values of the interactions these lines flow to large values when the scale parameter A is lowered. This is an example for a flow to strong coupling. When the interaction reaches values larger than the band width the flow has to be stopped, as explained in Sect. 12.3. Next to the divergence scale we can also read out from the data which classes of coupling constants grow most strongly. In this case, these are the dashed lines corresponding to interaction processes with wavevector transfer Q = (n, n/ between k2 and k3. The maximal coupling constant that grows most strongly also belongs to this family. All members of this family flow to strong coupling in a rather similar way when A is reduced toward the critical scale Ac ~ 0.161. Other families of couplings constants grow less strongly. Shown as well are as solid lines Cooper processes with zero total incoming wavevector. These processes show some growth as well, but lag behind the leading components. The lowest plot on the left shows a snapshot of the coupling constant when the first outgoing wavevector k3 is fixed at discretization point 1 near (—n, 0/ as function of the incoming wavevectors. We can clearly see two structures: one vertical line with strongly enhanced repulsive interactions at k2 = 24 (corresponding to wavevector transfer Q = (n, n/ between k2 and k3) with very little dependence on k1, and another line at k1 = 24 (corresponding to wavevector transfer Q = (n,n/ between k1 and k3). These values show again only a weak dependence on k2 and are roughly half as large as the vertical feature. Concentrating on these two features we arrive at the following effective interaction near the instability

HASDW/ = Vf X [2S(k2—k3 ± Q/+S(ki—k3 ± Q/] c^yck2yc^y, k1 ;k2 ;k3

s s where k4 is understood to be given by k1 +k2—k3 modulo reciprocal lattice vectors. A simple calculation shows that this interaction is equivalent to a long-ranged AF spin-spin interaction H = —JJ2(i j) elQ'(Ri~Rj/Si ■ S j with spin operators given by the usual fermion bilinears and J / VAF. This effective Hamiltonian exhibits long-ranged AF order at sufficiently low T. In this sense, the fRG flow for these parameters clearly indicate the proximity to an AF-ordered state or an AF spin-density wave (AF-SDW). Strictly speaking, this is the most we can infer at this stage. Whether long-range order actually occurs depends on the subleading terms and on the approximation errors collected on the way to this result. Note that the dynamics and interactions of emergent collective degrees of freedom are not appropriately captured in this approximation. Of course in two dimensions, a spontaneous breaking of a continuous symmetry should not occur at T > 0. On the other hand, most experimental systems described by our model would have additional small couplings in the third direction that change the situation in favor of our result. Moreover, as the leading instability is clearly exposed by this scheme, one could now resort to a bosonized description that allows one to treat the collective infrared physics much better (see e.g., [65] how the continuous symmetry breaking in two dimensions gets healed in the asymptotic flow).

Now let us consider a different Fermi surface with more curvature. We choose t' = —0.3t and / = —1.2, such that the Fermi surface is now curved but still contains the van Hove point (n, 0) and (0, n). The flow is illustrated in the right plots in Fig. 12.5. In the middle plot on the right we now see a dominance of the Cooper scattering processes with zero incoming total wavevector. Here some lines seem to diverge to large positive values, while some other lines take very negative values. The sign structure is also visible quite nicely in the bottom figure on the right. Here the first outgoing wavevector .k)3 is again fixed at point 1 (near (—n, 0)) and the dependence on the incoming wavevectors around the Fermi surface is plotted. One observes diagonal lines of enhanced interactions, corresponding to zero total incoming wavevector k\ + k2 = 0. This pair scattering is attractive when the incoming pair k\, k2 near the same saddle point (±n, 0) as the outgoing pair k3, —k3, and repulsive, when incoming and outgoing pairs are at different saddle points. This is exactly the formfactor d(k) = d0(cos kx —cos ky) of a dx2-y2 -Cooper pairing instability, where the dominant interaction is given by what is obtained by only keeping these diagonal features in VA(k\, k2, k3),

The spin-structure can be understood by forming the antisymmetric vertex again via (12.12). As the dominant features are symmetric with respect to interchange of k1 and k2, the equal-spin vertex vanishes. Again, this effective Hamiltonian can be solved by mean-field theory exactly. It has a d-wave paired ground state. This d -wave pairing instability was obtained by a number of research groups with different variations of fRG approaches (e.g., [45-47, 54-56]). The parameter region for its occurrence is rather wide. It constitutes convincing evidence that the weakly coupled Hubbard model possesses a d-wave superconducting ground state. The pairing mechanism at these weaker couplings is best described as AF-spin-fluctuation induced. This is already visible in the bottom plot on the right-hand side of Fig. 12.5. Here one can see that the d-wave pairing interaction on the diagonal lines with zero total incoming wavevector crosses a region with enhanced repulsive interactions, e.g., near k\ ~ 8,9 and k2 ~ 24,25. This enhancement is the broadened, due to Fermi surface curvature, version of the vertical SDW feature in the left lower plot for the fully nested case. We see that the enhancement to positive values fits perfectly into the sign structure of the d-wave. Studying the flow as a function of the cutoff A, one finds that the SDW features appear first and create the attractive component in the dx2-y2-pairing channel. More drastically, one can put a filter into the fRG scheme that removes these SDW features by hand, e.g., by setting these interactions to their initial values. This way one also looses the d-wave pairing instability in the same way as the AF-SDW instability, showing the strong coupling between the channels. Another way to check this view is to completely drop the particle-hole channels. In this case, no flow to strong coupling is found, as no attractive pairing components are generated, and as the SDW instability is excluded as well. If we only drop the particle-particle channel, we of course destroy the pairing instabilities, and kk the SDW instability takes a wider parameter space, at even higher critical scales comparable to those found in simple RPA summations.

The fRG data discussed so far showed rather clearcut single-channel instabilities, either in the AF-SDW or in the d-wave pairing channel, provided T is low enough. For higher T > Tc the flow remains finite, and the system should be in a metallic state. The critical scale at T = 0 is usually a good estimate for the critical temperatures Tc above which the flow remains finite, up to factors ~1. One can ask how the flow changes when we move from one to the other by changing the parameters. In principle there are two possibilities: One is that the flow to strong coupling changes gradually and the critical scale remains nonzero, with the SDW component getting weaker and the pairing component getting stronger continuously. The second possibility would be a quantum critical point where the critical scale for the run-away flow scale is suppressed to zero. As shown in the upper plots of Fig. 12.6 the change from AF-SDW is of the first type, while for larger t' one finds indications for quantum critical point between d-wave pairing and ferromagnetism, at least in the fermionic fRG flows, as illustrated in the lower plots of Fig. 12.6. The main factor that causes this difference between the phase changes is the overlap of the interaction processes between the two respective channels.

The AF-SDW instability and the dx2_y2 both require repulsive interaction processes between the two saddle points region near (^ , 0) and (0 , Hence the growth of one channel also supports the other channel to a large extent. If the Fermi surface remains in the vicinity of the van Hove points for a larger range of there is a rather wide crossover region where both channels, d-wave pairing and AF-SDW, grow strongly in a rather similar fashion. This is the case in the so-called saddle-point regime in the upper right plot of Fig. 12.6. Here, the mutual reinforcement of different interaction processes with initial and final states near the saddle-point regions is reminiscent of the flows to strong coupling in the half-filled two-leg Hubbard model, where d-wave pairing and AF-SDW channel are again driven in parts by the same processes. In this one-dimensional system the ground state does not select one these channels and does not develop any quasi-long-range order, but becomes a short-range correlated spin liquid with a single-particle gap [66,67]. This resemblance motivated interpretations of the flows in two dimensions as indications for a partial truncation of the Fermi surface in the saddle point regions [55,68,69], similar to what is argued to occur in the underdoped high-Tc cuprates. A controlled determination of the resulting strong-coupling state near the saddle-point region for these parameter regions is, however, still missing.

If the Fermi surface is further away from the saddle point regions, the strongest interaction processes with wavevector transfer (^ , occur near the Brillouin zone diagonal where the dx2_y2-formfactor is small. Then d-wave pairing and AF-SDW channels are rather weakly coupled. Correspondingly the energy scales for the two instabilities can be quite different. This is seen in the left upper plot of Fig. 12.6 for band fillings larger than one. Here the high-energy-scale AF-SDW instability gets cut off at a certain critical chemical potential and is replaced by a low-energy-scale dx2_y2 pairing instability. Without further analysis it is difficult to determine the precise nature of this transition. However, as the coupling between the channels

Fig. 12.6 Leading instabilities found with the N-patch fRG in the t-t 0-Hubbard model. Left upper plot: Critical scale Tc below which the fRG flow goes to strong coupling vs. chemical potential ^ for band filling larger than unity, for t0 = _0.3t and U = 3t. There is a high-energy-scale AF SDW instability with a weaker dx2_y2-wave pairing instability when the AF-SDW is cut off. Data from [84]. Right upper plot: Data for the same t0 and U on the "hole-doped" side with band fillings smaller than one, from [55]. Now there is a broad crossover "saddle point regime" between the nesting-driven AF-SDW instability and the dx2_y2-wave pairing regime. Lower left plot: Tc vs. t0 at the van Hove filling where the Fermi surface contains the points (w, 0) and (0, w). For large t0 one finds a ferromagnetic instability. Data from [56] obtained with the T-flow scheme. Right lower plot: Critical scale for the flow to strong coupling, again vs. t0 at van Hove filling, now obtained with the simplified vertex parametrization of [58] and with a smooth frequency cutoff. While the results agree well with the full N-patch flow for t0 ^ 0 and t0 ^ _0.5t, the transition from d-wave pairing to the ferromagnetic regime does not go through a potential quantum critical point as in the left lower plot

Fig. 12.6 Leading instabilities found with the N-patch fRG in the t-t 0-Hubbard model. Left upper plot: Critical scale Tc below which the fRG flow goes to strong coupling vs. chemical potential ^ for band filling larger than unity, for t0 = _0.3t and U = 3t. There is a high-energy-scale AF SDW instability with a weaker dx2_y2-wave pairing instability when the AF-SDW is cut off. Data from [84]. Right upper plot: Data for the same t0 and U on the "hole-doped" side with band fillings smaller than one, from [55]. Now there is a broad crossover "saddle point regime" between the nesting-driven AF-SDW instability and the dx2_y2-wave pairing regime. Lower left plot: Tc vs. t0 at the van Hove filling where the Fermi surface contains the points (w, 0) and (0, w). For large t0 one finds a ferromagnetic instability. Data from [56] obtained with the T-flow scheme. Right lower plot: Critical scale for the flow to strong coupling, again vs. t0 at van Hove filling, now obtained with the simplified vertex parametrization of [58] and with a smooth frequency cutoff. While the results agree well with the full N-patch flow for t0 ^ 0 and t0 ^ _0.5t, the transition from d-wave pairing to the ferromagnetic regime does not go through a potential quantum critical point as in the left lower plot is rather weak, we expect that a more accurate calculation would yield first-order transition.

The situation is again different for the transition from the d-wave pairing to the ferromagnetic instability, as illustrated in the lower plots in Fig. 12.6. Here the critical scale for the flow to strong coupling is suppressed strongly in the transition region, and the fermionic fRG flows (left lower plot) even suggest a quantum critical point. While it is difficult to fully understand these coupled nonlinear flows of N3 components of the coupling function using a simple argument, there is now one significant difference compared to the continuous crossovers described above. Let us again consider the interaction process that is common to both, d-wave pairing and ferromagnetic channel. The latter channel is driven by scattering processes with wavevector transfer q (between k2 and k3) going to zero. A closer analysis shows that these processes should be repulsive to cause a ferromagnetic instability, like for the usual Stoner criterion. This, however, means that also the processes with q ! 0 that have zero total momentum and enter the Cooper channel should be repulsive. A positive pair scattering with small wavevector transfer is, however, incompatible with singlet Cooper pairing that needs attractive scattering for these wavevector combinations. In this sense, the overlap between the two ordering channels can either support ferromagnetism or d-wave pairing, but not both tendencies together, and the two channels compete already in the symmetric phase. We note as well that the ferromagnetic instability is rather fragile and limited to the neighborhood of van Hove filling. Upon doping away one finds a smooth crossover to an instability with low critical scale and dominant p-wave pairing interactions [56].

The two different plot in the lower half of Fig. 12.6 are obtained for the same system parameters. The left plot is for the "full" fermionic N-patch fRG in the temperature-flow scheme [56] and shows the quantum critical point between d-wave pairing and ferromagnetic instabilities. The plot on the right-hand side is adapted from [58]. Here the fermionic vertex was decomposed into different functions in charge, spin and pairing channel and with s- and d-wave formfac-tors that depend only on one specific wavevector transfer or the total incoming wavevector (as discussed in Sect. 12.3. While this seems a good approximation in the parameter region further away from the transition between d-wave pairing and ferromagnetism, is seems to overestimate the critical scale in the region near t' & — 1/3 and rather points to a first-order transition. Nevertheless this decomposition of the vertex can be significantly improved by also treating a remainder term capturing the previously ignored higher formfactors [58, 59] and represents a promising direction for an improved wavevector resolution of the interaction vertex.

Summarizing these results on the one-band Hubbard model we see that the fRG is capable of deriving tentative phase diagrams with detailed descriptions of the wavevector-structure (and in principle also frequency-structure) of the effective interactions. Next to establishing broader regime with clearcut instabilities toward phases with unconventional Cooper pairing, the method also shows that the effective interactions at the borders of these regimes are rather complex. To understand the physical meaning of these flows with several strong channels in the effective interaction and to relate them to observable phenomena is an interesting challenge for the future research.

12.4.2 Iron Pnictides

A new field where fRG has contributed already considerably are the newly discovered iron pnictide superconductors [5-7]. Here the applicability of a perturbative technique like the one described here may be even better, as the pncitides are certainly not as strongly correlated as the high-Tc cuprates. This can already be inferred from the experimental phase diagram, where one only finds metallic antiferromagnetic phases (if at all), but never Mott insulating antiferromagnetism. Further, theoretical works that try to assess the value of the iron-d orbital onsite-interaction strengths find values that put the materials into the range of weak to moderate correlations [70, 71]. Regarding the electronic structure, the pnic-tides are, however, more complex than the cuprates. The main reason is that at least three of the five iron d-orbital have non-negligible weight near the Fermi level [72,73], and that these d-orbitals hybridize strongly with the neighboring arsenic p-orbitals. Therefore, even if one is only interested in the vicinity of the Fermi surface, the multiband character has to be kept. The Fermi surface is divided into two hole pockets, centered around the origin of the Brillouin zone at k = 0, and two electron pockets around k = (n; 0) and k = (0; n) in the unfolded zone corresponding to the small unit cell with one iron atom (or k = (n; n) in the folded zone corresponding to the large unit cell with two iron atoms). As pointed out early [72, 74], there is a potential nesting of electron and hole pockets which enhances particle-hole susceptibilities with the wavevector connecting these pockets. In addition, depending on the parameters and approximations [75], there can be a third hole pocket at (n, n) in the unfolded zone.

The first fRG work on the pncitides was performed by Wang et al. [60] for a five-band model. These authors obtained a sign-changing s-wave pairing instability driven by AF fluctuations as the dominant pairing instability. Further they found strongly anisotropic gaps around the electron pockets, with possibility of node formation. The basic structure of the phase diagram with the sign-changing pairing gap between electron- and hole-pockets can be understood already from simplified few-patch RG approaches [76]. This would, however, predict isotropic gaps around these pockets [77]. To understand the gap anisotropy one has to take into account the multiorbital nature of the electronic spectrum in the iron pnictides, as was done by the Berkeley group in their initial study [60]. Let us start with a single-particle Hamiltonian in wavevector-orbital space

k,s,o where the matrices h(k)oor take into account intra- and inter-orbital (density-density interactions and Hund's rule) terms for orbital index o = o' or o = o' respectively. s is the spin quantum number. The energy bands are obtained by a unitary transformation from orbital to band operators (index b), ckbb,s = J2o ubo(k)cko sS. For the various density-functional-theory-based tight-binding parameterizations of the band structure (e.g., [71,78-80]) of the d-dominated bands, three orbitals dxy, dxz and dyz (in the coordinates of the small cell, with x- and y-axis pointing toward the nearest iron neighbor) contribute significantly near the Fermi level. The simplest choice for the interaction between the electrons is to introduce orbital-dependent intra- and inter-orbital onsite repulsions, plus Hund's rule and pair hopping terms. While these local terms lead to wavevector-independent interactions in the orbital basis, parametrized by a tensor Vol,o2,o3,o4, after the transformation to bands one arrives at a wavevector-dependent interaction function

• M¿1,ol(kl)M¿2,o2(k2)«¿3,o3(k3)«¿4,o4(k4).

The combination of ubos behind the interaction tensor is sometimes called the "orbital make-up" [78, 79]. These prefactors cause in practice that already the initial interaction of the fRG flow exhibits a marked wavevector-structure which is then renormalized during the flow. It turns out that this orbital make-up has an essential influence on the competition between different channels in the flow and is responsible for the gap anisotropies found in the multiband fRG studies by Fa Wang and collaborators [60,81,82] and in subsequent studies [83]. A typical result for the predicted pairing gaps are shown in the right plot of Fig. 12.7.

Summarizing this brief section, the iron superconductors pose an interesting problem to the fRG where the main ordering tendencies have been calculated in good agreement with experiments, at least according to the currently accepted picture. For the future research, one goal should be to make the fRG a useful bridge between ab initio descriptions and experimental observables, in particular regarding materials trends in, e.g., the gap structure or the energy scales of the different systems. Furthermore, the studies should be extended to include the dispersion

-250

cdw'

PI J

f sdw -

d-wavesc

-

st-wave sir

16 32 48 64

Fig. 12.7 N-patch fRG results for the five-pocket scenario for FeAs-compounds. Left plot: fRG flow of the leading ordering tendencies. The sign-changing s-wave channel is competing with SDW and d-wave pairing. Right plot: Leading eigenvector of the Cooper pair scattering around the Fermi pockets, showing the sign change between hole- (patch indices 1-48) and electron-pockets (indices 49-80) and the anisotropy of the suggested gap function. Data taken from [83], interaction parameters U = 3.5eV, U' = 2.0eV, JH = Jpair = 0.7 eV

orthogonal to the iron-pnictide planes, as this would yield additional possibilities for nodes in the gap function [5-7].

12.5 Remarks on the 1PI fRG Scheme

After describing two of the main applications of the fRG for 2D correlated electron systems we discuss some formal issues to clarify the 1PI fRG scheme and to compare with other RG approaches. This will also introduce some important issues for the further development of the method.

12.5.1 Differences to Standard Wilsonian RG

In the formalism described in this chapter, only the first stage of a standard Wilsonian RG procedure, the integrating-out of modes has been performed. This first step may be considered the essential physical ingredient of all Wilson-inspired RG schemes, thus the fRG schemes described here do not fall into any different class. Regarding the other two standard stages of Wilsonian RG (see, e.g., [18,27]), some comments are in place. The second stage usually refers to a momentum-or wavevector-rescaling. This is typically not done in the context described here. The most natural procedure for a many-fermion system would be a rescaling of the wavevector-component perpendicular to the Fermi surface. However, a simple calculation shows that this in general spoils wavevector conservation which we would like to keep, for instance to resolve the anisotropy of the effective interactions. For frequency rescaling there are no such counter-arguments, at least at zero temperature, but all the works described here keep the frequency axis unchanged during the flow. The main purpose for using the renormalization group on the works described here is to sum up the perturbation series in an unbiased way. For this, momentum or frequency-rescaling seems an unnecessary complication. Further, also the field rescaling to keep the quadratic term unchanged, commonly referred to as stage three of Wilsonian RG step, is usually not done in our context. The whole setup of the scheme in general does not depend on whether the system is close to a fixed point or certain parts of the action maintain the same form at all scales such that a field rescaling is not needed. In particular for systems with the Fermi level near a van Hove singularity, the dispersion is not scale-invariant, but the fRG equations still give a valid description of the low-energy effective action - the main target in the present context. It might be that some of these issues have to be changed if one aims at describing fixed points of the flow in more detail, and other flow schemes using all three stages are discussed in the literature [18]. Most of the applications described here concentrate on determining the leading instabilities down to a scale where the truncation error is expected to be unimportant, and for this purpose the present formalism appears to be sufficient.

12.5.2 Higher Loops

Another common question is where higher-loop contributions that one knows from other RG approaches are hidden. Note that in the exact hierarchy of flow equations for the 1PI vertices, no higher-loop diagrams occur on the right-hand side. It is, however, not difficult to see that the solution of the one-loop fRG equations contains contributions of all orders in the bare interactions, and with an arbitrarily high number of loops. To see this for the coupling function VA, one has to insert the solution of the flow equation for VA at scale A, schematically (supressing wavevector and frequency summations) given by

JAo JAQ a with the different particle-hole and particle-particle one-loop diagrams La for least one vertex into the "^-function" for the coupling function on the right-hand side, leading to

The resulting expression corresponds to diagrams with three vertices at different scales and four internal propagators (two coming from each LA). These diagrams are either "chains" of two one-loop diagrams, e.g., when LA and LA' are both particle-particle diagrams, or true two-loop terms, e.g., when LA is a particle-particle loop and LA' is of particle-hole type. Of course, this procedure of reinserting integrated one-loop terms for vertices can be iterated, leading to diagrams of arbitrarily high order in bare interactions and loops. Similar procedures can be used to assess two-loop self-energy effects that otherwise would be lost due to the neglect of the frequency-dependence of the vertices, see e.g., [84-88].

12.5.3 Connection to Infinite-Order Single-Channel Summations

The integration of the 1PI equations in the present truncation contains all standard ladder and bubble summations known from perturbation theory. Hence the RG is considered to give a rather unbiased picture of the leading ordering tendencies, even in the case of competing ordering tendencies. The connection to, e.g., ladder summations in the particle-particle channel can be seen if we drop all particle-hole terms on the right-hand side of the flow equation for the interaction. Then the remaining equation has the structure

VA(ki; k2, k3/ = VA(k1, k2, k)LpJ.k, —k + kx + k2)VA(k, —k + kx + k2, IC3).

Again, the sum over k is not written out. For simplicity, let us assume that the initial condition VA0(k\, k2, k3) does not depend on the wavevectors or frequencies, and let us focus on the channel k\ + k2 = 0, i.e., zero total wavevector and zero total momentum. Near the Fermi level and at T = 0, LPJ .k, —k/ typically goes like 1/A, with T J2k LPJ.k, —k/ & p0/A, as the particle-particle bubble at k\ + k2 = 0 diverges logarithmically in the infrared cutoff. These simplifications lead to the equation

If the initial vertex VAo is attractive, this equation has a pole at Ac = A0 exp [—1/(p0VAo\. Hence the RG equation exactly reproduces the well-known Cooper instability. Likewise, all other particle-particle and particle-hole instabilities obtained in ladder or bubble summations can be recovered in suitable simplifications of the full one-loop flow. Some more attention has to be paid for the case of particle-hole instabilities at small wavevector transfer, like ferromagnetic Stoner instabilities. As mentioned in Sect. 12.2.3, the "deficit by construction" in these cases can, however, be remedied by other flow parameters such as the temperature [56], interaction strength [57] or a smooth frequency cutoff [58].

12.5.4 Symmetry-Breaking: Connection to Mean-Field and Eliashberg Theory

Typically, the fRG flows in 2D lattice models lead to strong coupling, at least if the temperature is low enough and if the density of states is nonzero at the Fermi level. This means that at least one class of interaction processes becomes very strong at low scales. Very often, as discussed in Sect. 12.4, this flow to strong coupling suggests a symmetry breaking in a particular channel, accompanied with a gap opening in the single-particle spectrum. Hence, instead of just exploring which type of symmetry breaking is indicated by the flow, an improved flow would also describe the growth of the order parameter and the change in the excitation spectrum. In the fermionic models discussed here, there are various ways to pursue this idea.

One idea is partial bosonization, which means to perform a Hubbard-Stratonovitch transformation of the leading interaction terms (written as a square of appropriate fermionic bilinears) to an appropriate bosonic field. This field couples to the fermionic bilinear and acts like an additional self-energy accounting for the change in the dispersion relation. In a next step, the gapped fermions can be integrated out, and the resulting bosonic theory can be studied. This approach is very useful if the fermionic flow reveals a clearly defined collective boson, and if

one is primarily interested in the collective dynamics at energy scales below the fermionic gap. A more detailed account of this method is beyond the scope of this chapter and the reader is referred to original papers [30,65,89,90]. Interestingly, for the bosonic sector, the full effective potential can be analyzed [91]. The bosonic 1PI flows in vertex expansion were e.g., discussed by Kopietz and collaborators [18,29].

Another less sophisticated way to describe the symmetry-broken phase would be to stop the fRG flow at a certain scale As where the interactions have built up a definite structure. Then one can use a mean-field decoupling of the leading terms and study the self-consistent mean-field solutions for the electronic modes below As. This method has been shown to work reasonably for the repulsive Hubbard model on the square lattice [92]. One might ask whether the outcome depends strongly on the scale As where the transition to the mean-field description is made. Here one is, however, helped by the fact that, e.g., in the simple BCS model with bare attraction VA0, initial bandwidth A0 and constant density of states p0, the scale-dependent attraction grows like VA = VAa/log(A/Ac) toward the critical scale Ac = A0 exp(— 1/VAop0). The BCS gap is then given by A = 2A0 exp(— 1/VAop0) provided that the dimensionless coupling constant VAop0 is smaller than unity. This dependence means that the growth of VAs, when the bandwidth A0 = As is changed to a smaller As , is canceled exactly by the shrinkage of the bandwidth, leaving the gap value unchanged. In more complex models like the repulsive Hubbard model, these dependences are altered by the particle-hole terms and the As-independence is no longer true. Nevertheless, in the study of [92], the As-dependence turned out to be tolerable, and the scheme gave a reasonable phase diagram with gap magnitudes that compare well with other approaches.

A more ambitious, fully fermionic approach is to allow for a tiny symmetry-breaking in the initial condition for the self-energy. Then the flow of this self-energy needs to be treated together with the flow of the interactions. The symmetry breaking might create new types of vertices. Then, at the critical scale Ac, the anomalous self-energy will grow rapidly and alter the flow of the interactions, preventing a true divergence. Therefore all modes can be integrated out down to A = 0, and the scheme provides a renormalized single-particle excitation spectrum in the symmetry-broken phase. These schemes were first tried out using momentum-shell cutoffs for simplified BCS-pairing [28] and charge-density-wave models [93] with strongly restricted interactions (e.g., only particle pairs with total momentum zero interact) for which mean-field theory and also the truncation of the fRG hierarchy is exact. It turned out that a reordering of the hierarchy for the flow of the 1PI vertex functions is essential to obtain correct results. More precisely, in the so-called Katanin-modification [94] one replaces the product of a scale-dependent Green's function and a single-scale propagator, GASA by GaGA. While the single-scale-propagator only contains a scale-derivative of the cutoff-function, the latter term also contains in addition the derivative of the self-energy / £A. This term represents a part of the feedback of the formerly neglected six-point vertex on the flow of the four-point vertex [28]. Katanin showed that keeping these additional contributions is important to fulfill certain Ward identities better, or even exactly in the case of simplified mean-field models. Physically, the role of these additional contributions is as follows. The flow is started with a very small initial symmetry breaking or gap parameter. This means that high-energy modes get integrated out with a gap that is different from the final one at the end of the flow. The additional EA -term contributions from all scales above A and corrects the contribution these of higher modes when the anomalous self-energy changes during the flow. With this Katanin-modification the "gap-flow" produces the exact BCS results when all scales are integrated out and in the limit of initial symmetry breaking going to zero. One can also nicely see diagrammatically that the fRG solution for this model is fully equivalent to solving the BCS gap equation [28]. For models with less restricted interactions, mean-field theory is no longer exact, and the fRG includes a number of corrections to mean-field theory, such as the suppression of the pairing strength through one-loop particle-hole diagrams [99]. One can also show how standard Eliashberg theory [100-102] is contained as a specific approximation in the fRG scheme (see [103] for the connection to Eliashberg theory above and at Tc, and [104] for a general argument including the symmetry-broken phase). Further, the approach in the symmetry-broken phase can be adapted to search for first-order transitions [105] that are not detected by run-away flows in the symmetric phase.

The fate of the running interactions depends on the type of symmetry-breaking. For breaking of a discrete symmetry, the interactions become smaller again at scales below the rapid growth of the self-energy (the gap opening) [93]. For continuous symmetry breaking as in the case of the BCS model, one can distinguish between an "amplitude mode" in the vertex that also flows back to smaller values below the gap opening, and a "Goldstone mode" that for A ! 0 basically saturates at a value inversely proportional to the initial symmetry breaking [28]. Hence, for realistic situations, these flows always produce large interactions with rather sharp momentum and frequency structures. It is by no means clear that the simple patching schemes described in this chapter are able to capture this behavior correctly, plus the truncation error might become more important. These issues have slowed down the application of these fermionic flows into the symmetry-broken phase for models where mean-field theory is not exact. Nevertheless the scheme has been tried in the attractive Hubbard model [95] and for competing orders in one dimension [96]. In both cases the results were rather promising, such that future work using this approach might provide more results.

12.5.5 Normal-State Self-Energy

In most applications of the N -patch fRG, the self-energy flow has been ignored completely. By now several aspects of the normal state self-energy EA(£,&>„) have been studied. EA(&,!„) can in principle be computed from (12.13). The information contained in the result depends, however, on the approximation employed for the vertex or coupling functions appearing on the right-hand side. In most N-patch studies, the frequency dependence of the vertex is neglected and hence EA(£,&>„)

comes out frequency-independent and usually real. This level of approximation is still enough to obtain information on the flow of the Fermi surface. A first study was presented in [55] for the one-band Hubbard model, where the flow of the Fermi surface turned out to be rather mild with a tendency toward a flatter Fermi surface down to a scale where the flow to strong coupling occurred. While this gave some first insights, it also emphasized some formal problems that arise. First of all, the particle number usually changes when the self-energy flow is included, and for flows with fixed particle number one has to cope with a scale-dependent chemical potential. Then, in momentum-shell schemes, it might happen that the renormalized Fermi surface flows into regions that have been integrated out already. This can lead to divergences. These problems have been addressed so far on a conceptual level [97,98], but have not yet led to a clear picture on the Fermi surface renormalization in the t-t' Hubbard model. Based on the experience in [55] we do not believe that this aspect changes the obtained phase diagrams in the one-band model considerably. It might become more important in multiband models, where the orbital mixture of the bands enters the interaction functions, as explained in Sect. 12.4, and the interplay between orbital weights and Fermi surface effects is more subtle.

To obtain the imaginary part and also the frequency dependence of the self-energy, one can reinsert the solution of the one-loop flow equation for the interaction into the flow equation of the self-energy, similar to what is described in Sect. 12.5.2. This leads to a frequency-dependent two-loop contribution with nonzero imaginary part on the real frequency axis. From this, quasiparticle scattering rates, quasiparti-cle weights as frequency derivatives and the whole renormalization of the spectral function can be computed.

The scattering rates in the one-band Hubbard model have been addressed to some extent in [84], where the anisotropy was described as function of the Fermi surface shape and of the temperature. If the Fermi surface is located near the (n, 0), (0, n) points of the Brillouin zone, the scattering rate is usually strongly anisotropic and higher near (n, 0), (0, n) than near the Brillouin zone diagonal. Interestingly, some experiments [106,107] point to a positive correlation between the anisotropic component of the transport scattering rate and the superconducting Tc. This trend comes out of the fRG calculation of the quasiparticle scattering rates as well [86]. The reason for this trend are again the scattering processes between the saddle points that are responsible for both the d-wave pairing and the enhanced self-energy in these regions.

The flow of the quasiparticle weight, the Z-factor, determined by the frequency derivative of the self-energy at zero frequency, has also been studied with similar techniques in at least two works [85, 109]. Here the suppression of the quasiparticle weight is strongest near the saddle points, but is not strong enough to change the leading flows to strong coupling. More drastic effects are found when the full frequency dependence of the self-energy on the real axis is considered. This has been studied in the one-band Hubbard model in [88,108], again using two-loop self-energies. These works showed that the real part of the retarded self-energy develops an additional kink at low frequencies near the instability scale and near the saddle points. This leads, if strong enough, to an anisotropic split-up of the quasiparticle peak. These works also showed that a parametrization of the self-energy to a Z-factor is in principle insufficient. The trends toward an anisotropic destruction of the quasiparticles are quite promising in connection with the phenomenology of high-Tc cuprates, but to date no firm connection between these weak coupling trends and observable phenomena could be established. In particular more work is needed that allows one to assess the precision of the fRG results very close to the instability. Note also that in all these works on the frequency dependence and on the imaginary part of the self-energy, the self-energy feedback on the flow of the vertex function was ignored beyond the inclusion of Z-factors in a few works [85,87,109]. In this regard and on a technical level, some fRG studies performed in impurity systems [61] and for quantum spin [38] are more advanced, as they include the self-energy feedback at least on some approximate level into the flow of the vertices. It is mainly due to the rich wavevector structure of the interactions that for 2D many-electron lattice systems the full self-energy feedback has not been tackled yet.

12.5.6 Refined Studies

Here we mention two refinements of the usual N -patch approach that give additional support to the picture established in the simplest approximation. The first issue is the frequency dependence of the vertex. Here, a simple patching of the Matsubara frequency axis into a number of segments has been introduced in [110,111]. This was employed in [110] to study the impact of various phonon-mediated electron-electron interactions on the flow to strong coupling in the t-t'-Hubbard model. Regarding the basic trends toward AF-SDW and d-wave pairing instabilities, the refined treatment did not result in qualitative changes. However, it provided some information on the frequency structure of the pairing interaction. Its width as function of the frequency transfer between incoming and outgoing legs can be interpreted as characteristic frequency of the "pairing boson." This turned out to very similar to the frequency width of the spin susceptibility at wavevector {n, n), again corroborating the picture of d-wave pairing mediated by AF spin fluctuations.

Another important and very useful step to improve the previous N-patch results for the t-t'-Hubbard model was undertaken by Katanin [87]. He included into the flow of the four-point vertex the integrated solution of a term in third-order of the four-point vertex that in the usual 1PI flow hierarchy contributes to the flow of the six-point vertex. This goes beyond the usual truncation. Katanin also computed real and imaginary part of the self-energy at the Fermi surface and included the self-energy into the flow via a Z-factor. Summarizing his results, Katanin found that these corrections do not change the leading instability but may lead to a slight shift of the phase boundaries in comparison with the previous one-loop analysis. In agreement with earlier studies [85], for curved Fermi surface and not too low temperatures the quasiparticle weight was only mildly reduced so that the quasiparticles remain well defined during the fRG flow. The quasiparticle scattering and Fermi surface shifts came out numerically small. In this sense, for most of the parameter space, the commonly used one-loop fRG approach without self-energy corrections should be expected to give a correct physical picture about the types and characteristic scales of the leading instabilities.

12.6 Conclusions and Outlook

The fRG for interacting fermions is a widely applicable and flexible method based on the safe and powerful foundation of exact flow equations for generating function-als. In this chapter we have focussed on the use of this method for investigating the leading low-temperature ordering tendencies of correlated electrons on 2D lattices. For the one-band Hubbard model, one main result of the method has been to show that there is a wide parameter regime for a d-wave Cooper-pairing instability, and to map out the other main instabilities depending on the model parameters. While most of the studies showing these instabilities still involve many approximations like the neglect of self-energy effects, all available information indicates that the obtained picture is qualitatively correct. The next steps will bring more detailed investigations of the effective interactions and of self-energy effects. Another important extension would be a detailed treatment of the symmetry-broken state for relevant models like the 2D repulsive Hubbard model, where to the flow is continued down to A = 0. Then the fRG can not only give information on the leading instabilities, but also provide renormalized excitation spectra and correlation functions. For pnictide superconductors, the fRG has proven to be a method that allows one study how details of the microscopic model, like the orbital compositions of the bands, lead to changes in the effective interactions and the predicted pairing gaps. Here the task will be to describe experimentally observed material-specific properties (e.g., the presence or absence of magnetically ordered phases or superconducting gap nodes and their location on the Fermi surfaces) in a reliable but still transparent way. This will again require some further developments of the fRG method and its implementation, as in these complex multiband systems the observables of interest obtained so far might still depend on the degree of approximation.

Besides these two fields of applications described here, there are many other activities in condensed matter physics using fRG methods, some of them were mentioned in the Introduction. In particular the large freedom to choose the RG flow trajectory will lead to many other occasions where the same fundamental flow equations will lead to interesting and relevant results.

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