A New Type of Quantum Criticality in the Pyrochlore Iridates

A New Type of Quantum Criticality in the Pyrochlore Iridates

Lucile Savary    Eun-Gook Moon Department of Physics, University of California, Santa Barbara, CA 93106-9530, U.S.A.    Leon Balents Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, U.S.A.
February 4, 2014
Abstract

Magnetic fluctuations and electrons couple in intriguing ways in the vicinity of zero temperature phase transitions – quantum critical points – in conducting materials. Quantum criticality is implicated in non-Fermi liquid behavior of diverse materials, and in the formation of unconventional superconductors. Here we uncover an entirely new type of quantum critical point describing the onset of antiferromagnetism in a nodal semimetal engendered by the combination of strong spin-orbit coupling and electron correlations, and which is predicted to occur in the iridium oxide pyrochlores. We formulate and solve a field theory for this quantum critical point by renormalization group techniques, show that electrons and antiferromagnetic fluctuations are strongly coupled, and that both these excitations are modified in an essential way. This quantum critical point has many novel features, including strong emergent spatial anisotropy, a vital role for Coulomb interactions, and highly unconventional critical exponents. Our theory motivates and informs experiments on pyrochlore iridates, and constitutes a singular realistic example of a non-trivial quantum critical point with gapless fermions in three dimensions.

Antiferromagnetic quantum critical points (QCPs) are controlled by the interactions between electrons and magnetic fluctuations Sachdev (2011); Löhneysen et al. (2007). In three dimensional metals with a Fermi surface, it is believed to be sufficient to consider Landau damping of the magnetic order parameter in a purely order parameter theory, which leads, following Hertz Hertz (1976); Millis (1993), to mean field behavior. In two dimensions, the electronic Fermi surface and order parameter are strongly coupled, a fact which may be related to high-temperature superconductivity and associated phenomena. This problem is highly non-trivial and still an active research topic Lee (2009); Metlitski and Sachdev (2010); Mross et al. (2010); Efetov et al. (2013).

In this paper, we uncover a new antiferromagnetic QCP which is strongly coupled in three dimensions, engendered by spin-orbit coupled electronic structure. We consider a quadratic band-touching at the Fermi energy, as in the inverted band gap material HgTe, but having in mind the strongly correlated family of iridium oxide pyrochlores Wan et al. (2011); Moon et al. (2013); Chen et al. (2014); Kondo et al. (2014). The latter have chemical formula AIrO, and an antiferromagnetic phase transition indeed occurs both as a function of temperature and at zero temperature with varying chemical pressure (ionic radius of A) Matsuhira et al. (2011). We show that the replacement of the Fermi surface by a point Fermi node alters the physics in an essential way, suppressing screening of the Coulomb interaction and allowing the order-parameter fluctuations to affect all the low-energy electrons. These two facts lead to a strongly-coupled quantum critical point.

The nodal nature of the Fermi point, happily, also enables a rather complete analysis of the problem, which we present here, using the powerful renormalization group (RG) technique. The complete theory we present is in sharp contrast to the strongly coupled Fermi surface problem in two dimensions, which remains only partially understood and controversial. Finally, the pyrochlore quantum critical point has a remarkable symmetry structure. We find that, unlike at most classical and quantum phase transitions, rotational invariance is strongly broken in the critical theory: the fixed point “remembers” the cubic anisotropy of space (and indeed takes it to an extreme limit, as explained further below). Compensating for the absence of spatial rotational invariance is, however, an emergent invariance of the critical field theory, which is a purely internal symmetry and unrelated to spatial rotations. The anisotropy in real space manifests for example in the formation of “spiky” Fermi surfaces when the system close to the QCP is doped with charge carriers, as seen in Fig. 1.

To proceed with the analysis, we couple the electrons to an Ising magnetic order parameter . This corresponds for the pyrochlore iridates to the translationally-invariant “all-in-all-out” (AIAO) antiferromagnetic state (see “inset” in Fig. 1), for which there is considerable evidence Sagayama et al. (2013); Tomiyasu et al. (2012); Disseler et al. (2012). Due to the time-reversal and inversion symmetries of the paramagnetic state, electron bands are two-fold degenerate, so that band touching necessitates a minimal four-band model. Therefore the Hamiltonian is expressed in terms of four-component fermion operators , , in addition to and the electrostatic field , which mediates the Coulomb interactions. The action is

(1)

where the momentum cutoff () is assumed, and where the Hamiltonian density is . Higher-order terms omitted in Eq. (1) prove irrelevant at the QCP. The ’s (given in the Supp. Mat. sup ()) make a complete basis of the allowed terms quadratic in , chosen such that belong to a three-dimensional representation (often called ) and make a two-dimensional one (commonly referred to as ), the ’s are anticommuting unit matrices, , , and (as they should since they belong to the same representation), and symmetry dictates sup () the order parameter couples via the matrix (). is the magnitude of the electron charge, and parametrizes the coupling strength of the fermions to the order parameter. As discussed in the Supplementary Material sup (), may always be chosen positive, without loss of generality. parametrizes “particle-hole asymmetry”, with denoting a symmetric band structure. Also, when , in the vicinity of the Gamma point, the bands touch at and only at the Gamma point. We assume that the system parameters fall within this range, and find that this is consistent.

The model in Eq. (1) has two phases. For (where is thereby defined), fluctuates around zero, and can be integrated out. This is a magnetically disordered state. The resulting model with Coulomb interactions alone describes a non-Fermi liquid phase, as first discussed by Abrikosov and Beneslavskii Abrikosov and Beneslavskii (1971); Abrikosov (1974) and thoroughly revisited recently Moon et al. (2013). Notably, in this regime, non-trivial scaling exponents arise and the low-energy electronic dispersion renormalizes to become isotropic, i.e. effectively and . For , the expectation value , and replacing causes the two-fold degenerate bands to split, removing the quadratic touching at in favor of eight linearly-dispersing “Weyl points” along the directions: a Weyl semimetal.

Figure 1: Quantum critical point (QCP) and quantum criticality driven by the onset of “all-in-all-out” magnetism. For (in this figure the star indicates ), the “Luttinger-Abrikosov-Beneslavskii” (LAB) phase occurs at , with a quadratic Fermi node, while antiferromagnetic (AFM) “all-in-all-out” ordering occurs for , with the quadratic node replaced by linear Weyl points – a Weyl semimetal. The quantum critical regime occurs at around . Note that the quantum critical-AFM boundary (thick white line) is a true (continuous Ising) phase transition. The axis represents the Fermi energy and parametrizes electron or hole doping. The three-dimensional (orange) surfaces represent the shapes of the corresponding Fermi surfaces at small doping – the increased anisotropy is apparent as one moves towards the QCP. The phase transition denoted by the thick gray line is expected to exhibit critical properties appropriate to a order parameter coupled to a Fermi surface, as in the Hertz formulation Hertz (1976), though subject to the usual uncertainties regarding the theory of that problem Lee (2009); Metlitski and Sachdev (2010); Mross et al. (2010).

We now turn to the critical regime. To proceed, we introduce as a formal device copies of the four fermion fields, replacing (resp. ) and ( is the identity matrix). We organize perturbation theory in powers of , but in the end argue that the results are asymptotically exact for the physical case . To leading order in , we require the two boson self-energies in Fig. 2, and, using the dressed boson propagators including this correction, the fermion self-energy and vertex functions in Fig. 3. These diagrams allow a full calculation of the terms of all critical exponents. The evaluation of the diagrams is complicated by the three mass parameters of the free fermion propagators. Fortunately, a simplification is possible due to the structure of the RG. While the (inverse) mass terms , , and all have identical engineering dimensions, they, in general renormalize differently from loop corrections, and thus their ratios flow in the full RG treatment. We find below that, in the critical regime, under renormalization (arguments why this is the only reasonable choice are given in the Supplementary Material sup ()). This allows technical simplifications in the loop integrals, and also has physical consequences we explore later.

In particular, in the limit , the interband splitting vanishes along the directions, leading to an extended singularity of the electron Green’s function. In the loop integrals determining the bosonic self-energies, this produces a divergent contribution at non-zero . Technically, with the assumptions and (shown self-consistent below), the low-energy behavior (small ) may be extracted as (see Supplementary Material sup ())

(2)

where , where is an upper momentum cutoff, , , and follows from charge conservation. and the functions are given as integrals in the Supplementary Material sup ().

Note that, at low energy, the dispersive terms in Eq. (2) are much larger than the bare terms they correct, and hence dominate the renormalized Green’s functions. Thus, in the fermion self-energy and vertex correction, the renormalized boson propagator, (note ), must be used.

Figure 2: Boson self energies for the order parameter () and electrostatic () fields.

This renormalized boson propagator corresponds to the result, and already reveals some dramatic features. First, the bosons immediately receive a large anomalous scaling dimension, equal to , and their dynamics becomes damping-dominated, with dynamical critical exponents close to . Second, since the damping terms which dominate are proportional to , it implies that the fermion self-energies, which involve two interaction vertices (see Fig. 3), become independent: this is a sign of universality at the QCP.

Figure 3: diagrams for the fermion self-energy and vertex corrections and (only two-loop diagrams that need be calculated, i.e. that do not vanish or cancel one another, are shown). Double lines indicate the renormalized boson propagators including the self-energies from Fig. 2. Expressions for these diagrams are given in the Supplementary Material sup ().

To confirm the assumed scaling of , , and fully determine the critical behavior, we turn to the renormalization group approach. There, as usual, we apply the following rescaling (applicable in real space)

(3)

where parametrizes the RG flow. The exponents are left allowed to be scale dependent, as is necessary Huh and Sachdev (2008), as we shall see below.

We evaluate the contributions to the fermion propagator and coupling constants due to a small change in the cutoff (which corresponds physically to integrating out modes to keep the rescaled cutoff unchanged). Hence, the RG flow equations are obtained by (i) logarithmically differentiating the fermion self energy and vertex functions with respect to the cutoff (made soft through a rapidly decaying function ) Huh and Sachdev (2008); Vojta et al. (2000), and (ii) identifying the appropriate coefficients of the Taylor expansion (in and ) of the result 111Note that a number of technicalities are involved in this calculation, in particular regarding the convergence of the differentiated functions; All are discussed in the Supplementary Material sup ()..

We leave most details to the Supplementary Material sup (), and only give one example here. To extract the correction to the mass coefficient , we first expand the fermion self-energy as

(4)

and examine the component. The RG equation is then

(5)

(we define sup ()). Similar expressions are obtained for the other parameters of the theory, , , , and . The latter all depend on and through or (expressions are expanded in small , see Supp. Mat. sup ()). Therefore, for the six equations thereby obtained, there are four unknowns (, , and ) which can be chosen to keep four parameter fixed, leaving two left to flow. Here we find it is possible to keep , , and fixed, and thus and will flow. Note that, in doing so, we obtain a critical theory with non-zero coupling of fermions both to order-parameter and Coulomb-potential fluctuations: both effects are crucial and important in stabilizing the QCP. Finally, we obtain

(6)

where , , and .

The flow equations may be solved thanks to that of , which is an analytically-soluble differential equation involving only sup (). Ultimately, we find

(7)

with , and where and are constants which depend on the system’s parameters, namely on . Formally, therefore both the and mass terms are irrelevant in the RG sense, but they can be “dangerously irrelevant” insofar as they control certain physical properties (see below). Note also that not only is irrelevant, but it also flows to zero faster than , so that becomes small at the QCP.

Intuition for the irrelevance of comes from considering the fermion self-energy , which yields the corrections to and to , and is given schematically by (the contributions from each boson field just add up). In the first term, which represents dressing of electrons by magnetic fluctuations, the appearance of , which commutes with but anti-commutes with , portends “opposite” consequences for and . The second term, due to Coulomb effects, tends instead to affect and identically. Our calculation shows that the former tendency prevails, and under RG, as claimed above. Conversely, the fact that should be attributed to the effect of Coulomb forces, which suppress particle-hole asymmetry. Indeed, we have checked that if in the calculations we artificially turn off the long-range Coulomb potential, i.e. take , the QCP is unstable and there is no direct, continuous quantum phase transition from the LAB state to the AIAO one sup ().

Eqs. (6,7) determine the properties at the QCP. We now turn to a discussion of the physical consequences. First we consider some scaling properties. For the correlation length, we need the flow equation for , the deviation from the critical point: , with . This implies, in the usual way, that the correlation length behaves as , up to logarithmic corrections. Also interesting is the order parameter growth in the AIAO phase. By scaling, , with . We also expect the critical temperature of the magnetic state to obey . In asymptopia, i.e. , all the -dependent corrections vanish, and the exponents correspond to those of a saddle-point treatment of . These are still distinct from the usual order parameter mean field theory, as witnessed by the large () anomalous dimensions in this limit, and the unconventional values , . The latter is noteworthy insofar as it implies an unusually wide critical fan at which is controlled by the QCP (see Fig. 1). The RG treatment goes beyond the saddle point in giving the corrections due to finite , which are small only logarithmically, and thus may be significant for physically-realistic situations. For example we find sup (), where is a constant.

The irrelevance of and has other, more direct, physical consequences. Because of the former, the low-energy electronic spectrum becomes approximately particle-hole symmetric. The latter has more implications. Obviously, the electronic spectrum develops pronounced cubic anisotropy, with anomalously low energy excitations along the cubic directions in momentum space. This is in stark contrast to most critical points (for example of Ginzburg-Landau type, or involving Dirac fermions), which typically have emergent spatial isotropy and even conformal symmetry and Lorentz invariance at the fixed point. These low-energy excitations manifest, for example, in the specific heat . Since at the Gaussian level the coefficient of diverges as , we estimate, by using as a cut-off ( is a microscopic energy scale), , with sup (). The emergent anisotropy may also manifest in increasingly-“spiky” Fermi surfaces in lightly doped samples near the QCP, see Fig. 1.

Although rotational symmetry is strongly broken, the vanishing of leads to an emergent internal symmetry, corresponding to rotating the matrices with amongst themselves like a vector. The generator of this symmetry is the pseudo-spin , with

(8)

where . Its integral has commutation relations and commutes with the fixed-point Hamiltonian.

Discussion.— In standard Hertz-Millis theory Hertz (1976); Millis (1993), the inequality implies that the theory is above its critical dimension, and thus has mean field behavior. Although this inequality holds here, taking , the conclusion is false. The Hertz-Millis approach assumes the fermions may be innocuously integrated out, and obtains this inequality by power-counting the term in the Landau action, which is irrelevant. Instead, here we have strong coupling of fermions with the order parameter, and the coupling term is marginal using , , . If one does integrate out the fermions, one obtains a nonanalytic term sup (), which overwhelms the naïve one, and is again marginal by power counting. This dependence was obtained previously in Ref. Kurita et al. (2013), in the context of a mean-field treatment of related transitions. Note, however, that such a mean-field analysis integrating out fermions is not justified and misses important physics.

Our critical theory has some formal similarity to the theory of a two-dimensional nodal nematic QCP in a -wave superconductor Huh and Sachdev (2008), insofar as both theories display “infinite anisotropy”: in our case due to under RG. This suggests that, as in Ref. Huh and Sachdev (2008), at low energy the perturbative expansion parameter is small for all , and that therefore our results apply directly at low energy to the physical case . This conclusion is appealing, though we have not shown it rigorously.

With the above results in hand, we comment on the connection to experiments. In the pyrochlore iridates, the QCP might be tuned by alloying the A-site atoms, e.g. PrYIrO, or by pressurizing stoichiometric compounds nearby. The theory developed here, which relies only on cubic symmetry and strong SOC, may apply to other materials if the bands at the Fermi energy belong to the appropriate irreducible representation, and it would be interesting to search for other examples. Experimentally, the heavily-damped paramagnon could be observed in inelastic neutron or x-ray scattering. An explicit calculation of the fermion spectral function measured in angle resolved photoemission has been made neither here nor for the non-Fermi liquid paramagnetic state Moon et al. (2013), and is an important problem for future theory. However, in general, the weak logarithmic flow of the Hamiltonian parameters signifies large self-energy corrections, and behavior somewhat similar to marginal Fermi-liquid theory may be expected.

We also mention some possible complications in the iridates. Impurity scattering is a relevant perturbation and hence important at low energy close to the band touching. Therefore, our results will apply best in the cleanest samples. Also, an accidental band crossing may occur away from the zone center, thereby shifting the Fermi level a few meV away from the nodal point. This should be addressed by ab initio calculations and experiments. In such a case, our results still hold for energies and/or temperatures above this shift energy. Finally, in many of the pyrochlore iridates, the A-site ion hosts rare-earth moments, which were not included here. They only weakly couple to the Ir electrons and to themselves, so are only important at low energy. On the antiferromagnetic side of the QCP, the Ir spins act as strong local effective magnetic fields, locking the A-site spins. However, when the Ir sites are not ordered, as in PrIrO, A-site ions will have an effect below a few Kelvins. Several authors have proposed scenarios based on RKKY interactions Chen and Hermele (2012); Flint and Senthil (2013); Lee et al. (2013), but the quantum critical theory expounded here should be an apt starting point for a systematic analysis.

Acknowledgements.— We thank Cenke Xu and Yong-Baek Kim for discussions on prior work, Max Metlitski for pointing Ref. Huh and Sachdev (2008) to us, and acknowledge Ru Chen, Satoru Nakatsuji and Takeshi Kondo for sharing unpublished data. The integrals were performed using the Cuba library Hahn (2005), and the Feynman diagrams in Figs. 2 and 3 drawn with JaxoDraw Binosi and Theußl (2004). L.S. and L.B. were supported by the DOE through grant DE-FG02-08ER46524, and E.-G. M. was supported by the MRSEC Program of the National Science Foundation under Award No. DMR 1121053.

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Supplementary Material

In reciprocal space, the action, Eq. (1) in the main text, is

where all the notations are defined in Sec. A of the present Supplementary Material. Throughout the latter, for ease of presentation, we shift the QCP so that .

Appendix A Notations and symmetries

In this section, we provide more information about the notations used in the main text and a more detailed discussion of the symmetries at play.

a.1 Fermion Hamiltonian

The fermionic Hamiltonian density in the disordered (quadratic band touching) phase reads

(10)

where and . The first line uses the conventional Luttinger parameters () in the matrix representation Luttinger (1956), and the second line is the form used in the main text. The Gamma matrices () form a Clifford algebra, , and have been introduced as described in the literature Murakami et al. (2004). Note that quantifies the particle-hole asymmetry, while naturally characterizes the cubic anisotropy. The energy eigenvalues are , where and

It is very important to note that, in the limit , and the energy spectrum become gapless along the directions. When needed, a “regularization” is then possible, for example by introducing higher momentum dependence in , e.g. .

It is straightforward to relate the coefficients used in the main text to the Luttinger parameters. This can be done by expressing the spin operators in terms of the Gamma matrices, using for example the equalities

(11)

where .

The fermion bare Green’s function is

where the sum over is implicit and is a projection operator, .

a.2 Symmetries

It is useful to recap the symmetries of the system in the absence of all-in-all-out order, and detail the remaining symmetries in its presence.

As defined above and in Refs. Murakami et al. (2004) and Moon et al. (2013), the matrices are even under time-reversal and inversion symmetry, while the are even under inversion, but odd under time-reversal.

As is well-known for some semiconductors, like HgTe, the touching of four bands at the Gamma point is protected by cubic symmetries (the bands at the Gamma point belong to a four-dimensional representation of the cubic group ), and the absence of a linear term follows from time-reversal and cubic (inversion) symmetries. Moreover, thanks to inversion and time-reversal symmetries, all bands are doubly-degenerate away from the Gamma point.

The magnetic order parameter field transforms as follows under the symmetries of the “disordered” system. It is odd under time-reversal symmetry (since the spins under time-reversal), and so only the (time-reversal-odd) can couple to it. It is even under inversion (since under inversion), unchanged under three-fold rotations, and odd under the allowed reflections of the pyrochlore lattice. A single Gamma matrix, namely Murakami et al. (2004); Moon et al. (2013) (see below), transforms identically.

The Hamiltonian at fixed , i.e. , together with the coupling to the order parameter with , which we call , have the following transformation properties. For , is invariant under three-fold rotations about , and reflections with respect to planes that contain . For , there is additionally inversion symmetry. The symmetry group at then decomposes the four bands of interest into two two-dimensional representations. For , symmetries do not impose bands to cross, hence making any crossings “accidental.” However, it is noteworthy that the purely quadratic Hamiltonian we study, with and , in the presence of the linear coupling to the order parameter leads inevitably to band crossings along the directions.

Note that the system in the presence of an external applied magnetic field, discussed in Ref. Moon et al. (2013), is less symmetric. The system’s Hamiltonian at fixed , which we call , is only invariant under three-fold rotations about if both the magnetic field and point along the same direction. For the system has additionally inversion symmetry, but all the representations of the symmetry group are one-dimensional anyway, and there is a priori no degeneracy at . Away from , any band crossing is, again, accidental.

It is important to note that, although no crossings are required by symmetry, once the crossings are found to happen, their properties are “stable” in the sense that (i) no symmetry-preserving perturbation will remove them, (ii) the dispersion along the crossings will remain linear, (iii) they will not move away from the axes.

a.3 Couplings

The long-range Coulomb interaction is described by introducing the Hubbard-Stratonovich field, , which couples to the density of fermions.

The all-in all-out operator is represented by the time-reversal symmetry breaking Ising field () corresponding to in Luttinger’s notation Luttinger (1956). In terms of the Gamma matrices, the order parameter is . Thus, finally, the interaction part of the action is the “vertex term” given, in real space and imaginary time, by

(12)

where is the four-component spinor field. Upon extending the field space to flavors of fermions, this term becomes

(13)

By appropriately transforming the Gamma matrices with transformations not belonging to the cubic group, one may show that the signs of may always be taken positive. Therefore, throughout the paper we assume . We also assume , i.e. we assume the two sets of bands have opposite curvatures in all directions at the Gamma point, or, in other words that the Fermi energy goes through the band touching point.

a.4 Green’s function and self-energy conventions

We use the following conventions for the boson Green’s functions, with , fermion Green’s function, , boson self-energies, and fermion self-energy, :

where (or ) but are omitted throughout. The “bare propagators” are denoted with the subscript or superscript “.”


Appendix B Asymptotic limits of the bosonic self-energies

We first evaluate the boson self-energies in the large- limit. They are given by

(14)

where , , and ( is the identity matrix). Here the subscript in the integral indicates that an ultraviolet cutoff is required to keep finite. This determines the (non-universal) location of the QCP. However, we seek the corrections to this term for non-zero frequency and momenta, which are cutoff independent, and will be therefore obtained below without further discussion of . We will return later to the role of the cutoff when considering fermionic self-energy terms, and treat it in more detail. The explicit expression for at is

where and , with and (resp. ) for (resp. ). Note that is (and not ); mathematically this is because of the trace, which yields a factor of .

As mentioned above, the boson self-energy is finite but depends upon the cutoff ( is proportional to ). Again, this determines the location of the QCP at , and when we focus on the critical theory, this zero-frequency zero-momentum contribution is exactly cancelled by the bare value of . Hence we are left with the corrections at non-zero frequency and momenta, which we isolate by considering the self-energy difference (for the second term is zero by charge conservation). This difference is finite and cutoff independent. In the limit, which will be the case in the critical theory, the self-energy differences show logarithmic divergences, i.e. contain . Conveniently, as mentioned in the main text, the latter will act as a control parameter Huh and Sachdev (2008), in addition to , in the critical theory.

In the following, we thereby obtain the one-loop bosonic self energy,

(15)

For future convenience, we take henceforth and denote . It is straightforward to obtain the coefficients of the frequency dependences, . Because is larger than the bare term at , which goes as , throughout this work, we take , where is a full boson Green’s function. Finally, note that we used an expansion in small of , i.e. of the inverse of Eq. (15), in some of the calculations.

By evaluating , we find and taking , and . Note that in the limit, the frequency dependence is subdominant and the bosonic propagator becomes static.

We now extract the non-trivial logarithmic momentum dependence, .

b.1 Coefficient of the logarithm

As mentioned above, when , to which the theory flows at the QCP, the energy and spectrum vanish for any , which renders the self-energy difference, , divergent. The appearance of a divergence is subtle: for general , the denominator in Eq. (B) appears relatively well-behaved since the singularity occurs only when both and lie along a axis. The singularity actually arises from the regions of integration at large along these directions, where , so that both energies are small. We analyze it below. In the limit (i.e. with nonzero and small), which is the actual behavior in the RG flows, the divergence is removed, and the result is large in . In this subsection, we extract the leading result in this limit. Notably, in this limit, the result is independent of , and can be approximated by taking simply .

To extract the coefficient of the logarithm, , we rotate to bases whose -axes point along one of the directions, and make a change of variables such that

(16)

where (allows to span the eight directions). This rewriting is chosen so that for of , the region near the ray is singled out. The Jacobian of this coordinate transformation is . Now, we rewrite the functions involved in the integrand of the self-energies, Eq. (B), in these new coordinates, and we obtain the leading asymptotic behavior of each such function at small .

For example, we find

(17)

where the and () are functions of (and of course of the ’s) only. We are then in a position to take the logarithmic derivatives of the boson self-energies. A major simplification thereby occurs: the frequency dependence drops out of . We find

(18)
(19)

where

In the above formula, we introduced several expressions:

(21)
(22)
(23)
(24)
(25)
(26)

where all the functions defined above, namely , , , , and (), are taken at (and are also functions of the ’s although we have written the latter explicitly for only). Note that the integrations over and are taken all the way from to although the sum over the eight directions, is also taken. This is because, for non-zero , the integrations have a priori upper bounds of order , which is taken to infinity. In the present order of limits, all contributions arise from regions of angular width of order from the rays.

The integrals, Eq. (18), are evaluated thanks to the Cuba library, using the “Cuhre” routine Hahn (2005).

b.2 Approximation

Figure 4: Plot of . The line represents a direction. The whole surface can be obtained from the plotted points by applying cubic symmetries (note that the set of plotted points is larger than the minimal set of points). The yellow surface is a sphere of radius .

Since is very smooth (see Fig. 4), we approximate it by a low-order polynomial of in order to be able to take accurate derivatives of as required to compute the flow of (and ) – see Sec. C. Imposing cubic symmetry, the most general polynomial to order six can take the form

(27)

and fits with , and and , and