Is graphene on the edge of being a topological insulator?

Is graphene on the edge of being a topological insulator?

J. González Instituto de Estructura de la Materia,
Consejo Superior de Investigaciones Científicas,
Serrano 123, 28006 Madrid, Spain
July 9, 2019
Abstract

We show that, at sufficiently large strength of the long-range Coulomb interaction, a mass term breaking parity (so-called Haldane mass) is dynamically generated in the many-body theory of Dirac fermions describing the graphene layer. While the tendency towards chiral symmetry breaking is stronger than for the dynamical breakdown of parity at spatial dimension , we find that the situation is reversed at . The need to regularize the many-body theory in a gauge-invariant manner (taking the limit ) is what leads to the dominance of the parity-breaking pattern in graphene. We compute the critical coupling for the generation of a parity-breaking mass from the finite radius of convergence of the ladder series supplemented with electron self-energy corrections, finding a value quite close to the effective interaction strength for graphene in vacuum after including Fermi velocity renormalization and static RPA screening of the Coulomb interaction.

I Introduction

During the last years, a great effort has been devoted to the investigation of the so-called graphene, the material made of a one-atom-thick carbon layernovo (). Many of the interesting features of graphene arise from the unconventional band structure of the carbon sheet, where electron quasiparticles behave like massless Dirac fermionsgeim (); kim (). The relativistic-like invariance of the low-energy electron system is at the origin of outstanding transport propertiesrmp (), like the reduced influence of scatterers with size above the C-C distanceando () or the perfect transmission through one-dimensional potential barrierskats ().

In the graphene layer, the Coulomb repulsion between electrons constitutes the dominant interaction. This makes the electron system to be at low energies a variant of Quantum Electrodynamics, but placed here in the strong coupling regime as the ratio of to the Fermi velocity of the electrons is nominally larger than one. There have been already several proposals to observe unconventional signatures of the interacting electrons in graphene, including the anomalous screening of impurities carrying a sufficiently large chargenil (); fog (); shy (); ter (). It has been also found that the own - interaction in the layer should lead to a linear dependence on energy of the quasiparticle decay rateqlt (), as a consequence of the vanishing density of states at the charge neutrality point, and in agreement with measurements carried out in graphiteexp ().

Yet the effects of electron correlations have been quite elusive in graphene (setting aside the observation of the fractional quantum Hall effect). In this respect, a number of theoretical works have studied the dynamical breakdown of the chiral symmetry of the Dirac fermionskhves (); gus (); vafek (); khves2 (); her (); jur (); drut1 (); drut2 (); hands (); hands2 (); gama (); fer (); ggg (); me (); prb (), being remarkable that most part of the approaches have asserted its viability, with a critical coupling below the nominal interaction strength of graphene in vacuum (). Experimentally, it has been observed instead the increase of the Fermi velocity when approaching the charge neutrality pointexp2 (), reflecting the renormalization of the effective interaction strength towards a vanishing value in the low-energy theorynp2 (); prbr ().

It has been proposed that this scaling of the electron system approaching the noninteracting limit could explain the absence of any effect of dynamical gap generation in graphene. Some theoretical studies have indeed incorporated the electron self-energy corrections in the many-body theory to show that they can push the critical coupling for exciton condensation above the largest possible value attained for graphene in vacuumsabio (); prb (). Anyhow, the theoretical analyses have dealt mainly with the static RPA screening of the Coulomb potential, that tends to underestimate the effective strength of the interaction at low energies. The consideration of a more sensible dynamical screening has shown to lead to smaller values of the critical coupling for the excitonic instability, below the value ggg (); prb ().

These preceding investigations have focused on the dynamical generation of a gap from the formation of a staggered charge density in the graphene lattice, while the possible condensation of other order parameters has been mostly overlooked (see however Refs. rag (); herbut ()). Nevertheless, already in the early studies of Quantum Electrodynamics at spatial dimension , it was realized that the dynamical symmetry breaking could develop in two different ways. That is, one could have the condensation of a mass breaking the chiral symmetry but preserving the invariance under parity of the Dirac fermionspis (); qed (); kog (), or otherwise a mass maintaining the chiral symmetry but breaking the invariance under parityparb (); sem1 (). The prevalence of one pattern of symmetry breaking over the other was shown to depend in general on the balance between different local four-fermion interactions in the electron systemsem1 ().

The two different order parameters characterizing the dynamical symmetry breaking, the parity-invariant mass and the parity-breaking mass, are not related by any fundamental symmetry in the graphene electron system, which means that they can be generated quite independently. The possibility of introducing a mass operator breaking the invariance under parity was considered in the context of a tight-binding model of electrons in the honeycomb lattice in the seminal work of Ref. haldane (). The dynamical generation of the parity-breaking mass establishes in fact a bridge between graphene and the so-called topological insulators, as one of the consequences of the condensation of the parity-breaking order parameter would be the spontaneous development of loop currents in the honeycomb lattice.

In this work we study the dynamical generation of the parity-breaking mass term in the theory of Dirac fermions in graphene, taking the long-range Coulomb repulsion as the relevant interaction between electrons. We will consider the dynamical symmetry breaking analyzing the corresponding order parameter in the many-body theory, comparing it with the more conventional effect of exciton condensation breaking chiral symmetry but preserving parity. We recall that the space of Dirac fermions at is defined by the algebra of matrices () satisfying the anticommutation relations

(1)

In this notation, the order parameter for the parity-invariant mass will correspond to the fermion density averaged with the matrix, while the generation of the parity-breaking mass will be given by a nonvanishing average of the fermion current .

One can check that, at , the product of all the matrices, , commutes with any other matrix in the space of the Dirac fermions, so that its action is equivalent to the identity. This means that, computing for instance the ladder series of many-body corrections, all contributions to the vertex should be formally the same as those to the vertex for the current . The sum of the ladder series provides a particularly sensible approach to dynamical symmetry breaking, as it encodes the most divergent diagrams at each level of the perturbative expansionmis (). It is actually the need to devise a consistent regularization of those divergences which plays a central role in the evaluation of the different order parameters.

We will show that, in the theory regularized with a method chosen specifically to preserve its gauge invariance (such as dimensional regularization), the perfect match between the order parameters for the parity-breaking and the parity-invariant mass is lost. This situation is similar to the well-known case of conventional Quantum Electrodynamics at spatial dimension , where there is no regularization that can simultaneously preserve the gauge and the chiral symmetries, which makes unavoidable to discard the latter in favor of the first.

In the case of graphene, the use of dimensional regularization has shown to be a most suitable choice to preserve the underlying gauge invariance of the many-body theorynp2 (); juricic (); jhep (). Recently, it has been checked that such a method is singled out as the right approach to reproduce the corrections to the DC conductivity calculated from the lattice modelros (). In our problem, the need to regularize the many-body theory while preserving gauge invariance is also what produces the anomalous mismatch between the respective vertices for and the current .

We will see that, at greater than 2, the divergent corrections to the vertex for the parity-invariant mass are in general greater that those to the vertex for the parity-breaking mass, while the situation is reversed for . The diagrams in the Dirac many-body theory are made convergent by analytic continuation to spatial dimension , which explains that the tendency to develop a nonvanishing order parameter for the parity-breaking mass becomes dominant in the physical renormalized theory at . As a consequence, we will see that there is a critical coupling for the dynamical breakdown of parity in graphene which turns out to be smaller than the critical coupling obtained in the more conventional mechanism relying on chiral symmetry breaking.

In the next section, we set up the framework to describe the many-body theory of Dirac fermions, highlighting the classical scale invariance that opens the way to apply the renormalization group approach. We focus on the computation of the vertices for and the current in Sec. III, where we show that it is possible to define cutoff-independent observable quantities from the sum of ladder diagrams. We find that the two ladder series for the renormalized vertices have finite radii of convergence, that we compute in each case to obtain the respective critical couplings signaling the dynamical generation of the parity-invariant and the parity-breaking mass. Finally, we discuss in Sec. IV the physical implications of the dynamical symmetry breaking arising from a parity-breaking mass.

Ii Dirac many-body theory

The low-energy electron quasiparticles are disposed in graphene into conical conduction and valence bands that touch at the six corners of the Brillouin zonermp (). There are two inequivalent classes of electronic states, that can be disposed around a pair of independent Fermi points. Thus, the low-energy electronic excitations can be encoded into a couple of four-dimensional Dirac spinors , which are characterized by having linear energy-momentum dispersion . The index accounts here for the two independent spin projections, but it will be omitted to simplify the notation in what follows. The kinetic term of the hamiltonian in this low-energy theory is given by

(2)

where and is a collection of four-dimensional matrices satisfying the relations (1). It is convenient to represent them in terms of Pauli matrices as

(3)

where the first factor acts on the two sublattice components of the graphene honeycomb lattice and the second factor operates on the set of two independent Fermi points.

The dominant - interaction is given in graphene by the long-range Coulomb repulsion. This interaction does not have a conventional screening at long distances, given the vanishing density of states at the Fermi points connecting the conduction and valence bands. A suitable description of the Coulomb interaction starts then by considering the unscreened potential . The full hamiltonian including the - interaction can be represented as

(4)

with .

The prevalence of the Coulomb interaction at low energies can be understood from the scaling properties of the system governed by (4). The long-range Coulomb repulsion is the only interaction that is not suppressed, at the classical level, when scaling the many-body theory towards the limit of very large distances or very low energies. From (4), the total action of the system is given by

(5)

This action is invariant under the combined transformation of the space and time variables and the scale of the fields

(6)

This means in particular that the strength of the Coulomb interaction is not diminished when zooming into the low-energy limit . One can check that any other - interaction without the tail, as those that arise effectively from phonon exchange, would be suppressed at least by a power of under the change of variables (6), implying its irrelevance in the low-energy limit.

On the other hand, the many-body theory does not preserve in general the scale invariance of the classical action (5), as a high-energy cutoff has to be introduced to obtain finite results in the computation of many-body corrections to different observables. The analysis of the cutoff dependence of the many-body theory provides deeper insight into the effective low-energy theory. If the theory is renormalizable, it must be possible to absorb all powers of the cutoff dependence into a redefinition of the parameters in the action (5). This should be therefore modified to read

(7)

The assumption is that and (and other renormalization factors for composite operators that do not appear in (5)) can only depend on the cutoff, while they must be precisely chosen to render all electronic correlators cutoff independentamit ().

This possibility of absorbing all the divergent dependences on the cutoff in a finite number of renormalization factors has been checked in the above theory in the limit of a large number of Dirac fermion flavorsprbr (); ale (), as well as in the ladder approximation to vertex diagrams supplemented by electron self-energy correctionsjhep (). We will use this latter approach in the computation of the vertices for and the current . We recall that in the ladder approximation there is no need to renormalize the electron quasiparticle weight, so that , while the renormalization factor for the Fermi velocity coincides with the expression obtained at the first perturbative leveljhep (). The computation of the expectation value of other fermion currents requires however the introduction of new renormalization factors in order to obtain finite, cutoff-independent results, as shown in Ref. jhep (). These renormalization factors encode in general valuable information about the possible condensation of different excitations and the corresponding dynamical symmetry breaking.

The gapless character of the electronic spectrum is in principle protected by two different symmetries of the hamiltonian (4). This is invariant under a chiral symmetry () that consists in the exchange of the two-dimensional spinors attached to the two independent Fermi points

(8)

We could add to the hamiltonian (4) a term breaking the invariance under (8)

(9)

while respecting the space-time symmetriessem (). This term, corresponding to a conventional parity-invariant mass, would certainly open a gap in the electronic spectrum. But this same effect can be obtained with a different perturbation preserving the chiral symmetry (8)

(10)

One can check that (10) breaks instead the invariance under parity ()

(11)

The term (10) corresponds to the parity-breaking mass mentioned beforehaldane (). It can be shown that there are no other operators (not implying the electron spin or the exchange of excitations about the two Fermi points) that may open a gap in the electronic spectrumcham (); mas (). We finally recall that, while (9) or (10) may not be present nominally in the hamiltonian, they can be generated in the many-body theory at sufficiently large interaction strength, leading to the dynamical breakdown of the chiral symmetry or parity in each respective case.

Iii Renormalization of staggered charge density and loop current operators

We focus then on the possible development of a vacuum expectation value for the parity invariant operator

(12)

as well as for the parity-breaking bilinear

(13)

with

(14)

We will characterize such a phenomenon by inspection of the respective vertices

(15)

and

(16)

where 1PI denotes that we take the one-particle irreducible part of the correlator. We are going to see that the above vertices have a finite radius of convergence in perturbation theory, which is the signature of the dynamical breakdown of symmetry driven by the interaction.

A sensible approach to determine the critical interaction strengths at which (15) and (16) diverge consists in performing the sum of ladder diagrams for the two vertices. The ladder contributions contain indeed the most divergent part of the vertex at each level of the perturbative expansionmis (). The ladder series built in that way can be easily encoded in the self-consistent diagrammatic equation represented in Fig. 1. In what follows, we will see how to obtain the respective critical couplings in a manner that they do not depend on the high-energy cutoff needed to regularize the theory, relying only on observable quantities of the electron system.

Figure 1: Self-consistent diagrammatic equation for a generic vertex in the ladder approximation.

iii.1 Loop current vertex

We study first the vertex for the loop current operator. In order to make the approach covariant and valid in any spatial dimension, we consider the correlations of the loop operator

(17)

This is now a second-rank tensor, leading to the vertex

(18)

In the ladder approximation, the vertex has to satisfy in the limit and the self-consistent equation

(19)

where we have regularized the momentum integrals by continuing the number of spatial dimensions to . Accordingly, is now a dimensionful coupling which demands the introduction of an auxiliary momentum scale through the relation

(20)

At this point we may resort to a perturbative resolution for the vertex, noticing that it can only depend on the dimensionless ratio . It is easily realized that the operator with the tensor can mix with the other antisymmetric second-rank tensor structure one can build, corresponding to . Thus, the most general solution of Eq. (19) can be expressed as a power series in the coupling

(21)

where we have called .

Each term in the series (21) can be obtained by means of an iterative procedure from the previous orders. By inserting a term proportional to at the right-hand-side of Eq. (19), we get for instance

(22)

with

(23)
(24)

Moreover, when a term proportional to is introduced at the right-hand-side of the integral equation, we obtain

(25)

The above formulas can be summarized in the recursion relations

(26)
(27)

In compact form, one can also write the relation

(28)

We observe that the result of computing the integral in (19) always diverges in the limit , leading to a sequence of higher-order poles in the parameter as we look at higher perturbative levels in the solution of the self-consistent equation. Good news are however that all these divergences can be reabsorbed after a multiplicative renormalization by a single factor , ending up with a finite vertex in the limit

(29)

The renormalization factor can have in general the structure

(30)

The important point is that it can be shown that can be made finite at with a set of functions that do not depend on the external momenta of the vertex. This is the hallmark of renormalizability, by which one can reabsorb the cutoff dependences into the renormalization factors for a few local operators of the theory.

Most interestingly, this procedure of renormalization works in the present case even when self-energy corrections are considered in the internal electron and hole states of the vertex. These supplementary contributions to the ladder series can be taken systematically into account by replacing the constant in Eq. (19) by the effective Fermi velocity found after dressing the Dirac propagator with the electron self-energy correction

(31)

Recalling the expression of the electron self-energy in the same ladder approximationjhep (), we get

(32)

The effective is in principle singular in the limit , but the divergence can be absorbed by a simple renormalization of the Fermi velocity

(33)

Taking the simple pole structure

(34)

one needs to set in order to render finite at as a function of .

The remarkable point is that the vertex , supplemented with the electron self-energy corrections, can be made also finite in the limit with a modified set of coefficients in (30), expressed now as functions of the renormalized coupling

(35)

The first orders in the analytic computation of the residues , including the effect of the electron self-energy corrections, turn out to be

(36)
(37)
(38)
(39)

The function has particular significance since it can be used to determine the anomalous dimension of the vertex, which comes from the dependence on the dimensionful parameter of the renormalized theoryamit ()

(40)

The exponent can be defined in terms of the renormalization factor by

(41)

When applying Eq. (41), it is a highly nontrivial result that all the pole contributions coming from (30) may cancel out to give a finite answer for at . This happens whenever one can enforce the set of conditionsram ()

(42)

In that case, the anomalous exponent is given by

(43)

One can see that the analytic expressions (36)-(39) satisfy the conditions (42), and we have checked that these hold also for the numerical solution of the functions that we have obtained up to order in the perturbative expansion. The characterization of dynamical symmetry breaking from turns out to be then very convenient since, according to (43), the exponent is a scale invariant quantity depending only on the renormalized coupling . It is actually observed in the numerical resolution that the perturbative orders of have a geometric growth, as shown in Fig. 2. We can then use the same scaling technique as in Ref. jhep (), which allowed to reproduce there the value of the critical coupling obtained for chiral symmetry breaking from the resolution of the gap equationgama (). From the expansion up to order , we get now the result for the finite radius of convergence of

(44)

Figure 2: Plot of the absolute value of the coefficients (upper points) and (lower points) in the respective expansions of and as power series of the renormalized coupling .

One has to keep in mind that this singularity at implies the divergence of the correlators of the loop current operator , as they all need to be renormalized by factors of . This implies that their anomalous dimensions are given in general by multiples of . We find therefore at the onset of a new phase of the electron system that must correspond to dynamical symmetry breaking, as it is characterized by the appearance of a nonvanishing expectation value of the operator at that critical coupling.

iii.2 Staggered charge density vertex

We review the computation of the vertex (15), which has been addressed in Ref. prb (), in order to establish here the comparison with the loop current vertex. In the ladder approximation, is proportional to the unit matrix, and the resolution of the equation in Fig. 1 proceeds by noticing that

(45)

We arrive then at the self-consistent equation

(46)

The solution of Eq. (46) does not depend on the frequency , and the self-consistent equation becomes

(47)

We resort as before to a perturbative resolution for the vertex, expressing the solution of (47) as a power series in the coupling

(48)

Successive terms in (48) can be obtained by means of an iterative procedure, using the formula

(49)

where

(50)

We get therefore the relation

(51)

It is remarkable that the recursion (51) almost coincides with (28), the deviation being a term that vanishes in the limit . We have in fact the result

(52)

The agreement between the two recursion for and would be perfect if we had a factor multiplying the right-hand-side of (26), which is instead missing due to the second-rank character of the tensor renormalized there. It can be shown that the precise consideration of this tensor character is a must for the correct implementation of a gauge-invariant renormalization, as observed for instance in the case of the electron current vertexjhep (). We are going to see that the mismatch in the two ladder series, when combined with the effect of the electron self-energy corrections, leads to significant differences in the renormalization factors of the vertices and .

It can be checked that a multiplicative renormalization by a factor is enough to absorb the divergences in (48). Thus, we define the renormalized vertex, finite at , by

(53)

The pole structure of the renormalization factor takes the form

(54)

At this point, we can consider again the effect of the electron self-energy corrections on the vertex by simply replacing the constant inside the integral in Eq. (47) by the effective Fermi velocity (32). After renormalizing by the factor , it becomes possible to absorb all the poles at in the vertex by an appropriate choice of coefficients depending only on the renormalized coupling . The first terms in the perturbative expansion can be computed analytically, with the result that

(55)
(56)
(57)
(58)

The most important effect of the renormalization is again the anomalous scaling of the vertex , arising from its dependence on the momentum scale amit (),

(59)

The anomalous dimension can be obtained from the renormalization factor as

(60)

This equation turns out to have a finite limit provided the set of conditions

(61)

are fulfilledram (). In that case, the anomalous exponent is given by

(62)

therefore depending solely on the renormalized coupling .

Quite nicely, it can be seen that the analytic expressions in (55)-(58) satisfy the equations (61). We have also checked by numerical computation that the perturbative expansions of the functions fulfill that set of conditions, at least up to the order we carried out the calculation. We can therefore rely on Eq. (62) to obtain a sensible result for the anomalous dimension . The important point is that, as well as in the case of , the power series in turns out to have a finite radius of convergence. This can be already noticed in the representation of the first orders of the expansion for , plotted in Fig. 2. The accurate fit achieved in Ref. jhep () (from the expansion to order ) allows to get the value for the critical coupling

(63)

The critical coupling should mark the onset of a new phase in the electron system, characterized by a nonvanishing expectation value of the operator . We observe however that the critical value (63) is larger than the result (44) obtained for the coupling . This implies that the susceptibility to the development of loop currents should be stronger than that for the formation of a staggered charge density in graphene. Consequently, the phase with condensation of circular currents should take place first, as long as the system has a relative interaction strength above the critical coupling given by (44).

Iv Discussion

We have seen that, at sufficiently large interaction strength, it is possible to dynamically generate a mass term breaking parity (the so-called Haldane mass) in the many-body theory of Dirac fermions describing the graphene layer. We have shown that the critical coupling for the development of such a mass is below that corresponding to the usual excitonic instability, so that the dynamical breakdown of parity should prevail in graphene over the opening of a gap in the electronic spectrum from chiral symmetry breaking.

As already mentioned, the mismatch between the critical couplings for the dynamical breakdown of parity and chiral symmetry may be paradoxical, given that the commutation properties of the vertices for and the current are the same for Dirac fermions at . We have clarified that the difference between the respective many-body corrections to the two vertices arises from the need to regularize high-energy contributions in a gauge-invariant manner, by analytic continuation of the momentum integrals below dimension . The suitability of this method to produce precise physical results for the carbon layer has been recently checked in Ref. ros, . In this regard, the different strength of parity and chiral symmetry breaking is a quantum field theory “anomaly” arising as a remnant of high-energy contributions to the low-energy effective theory of graphene, similar to the anomalous non-conservation of the chiral current in conventional Quantum Electrodynamics at when a gauge invariant method is used to regularize the theory.

It is interesting to observe that the above computation of anomalous exponents can be also applied to investigate the dynamical symmetry breaking of the many-body theory of Dirac fermions at any spatial dimension . This is so as the classical scale invariance (6) can be extended to hold for the action of the Dirac fermions interacting with the same Coulomb potential at arbitrary . It can be shown then that all cutoff dependences can be always absorbed into a finite number of renormalization factors, at least in the ladder approximation supplemented with electron self-energy corrections. The anomalous exponents of the order parameters corresponding to and the current display in general finite radii of convergence in our ladder approach, which we have computed for spatial dimension . The results are shown in Fig. 3. We observe for instance that the critical coupling for the conventional excitonic instability is the lowest at , while we find the reverse situation for . At , it is actually the need to regularize the many-body theory by computing in the limit which makes the dynamical breakdown of parity to prevail over chiral symmetry breaking at the integer physical dimension.

Figure 3: Plot of the critical couplings for the dynamical symmetry breaking of parity (points on the dashed line) and for the breakdown of chiral symmetry (points on the full line) depending on the spatial dimension of the interacting electron system.

Regarding the actual value of the critical coupling (44), we may ask whether there is any chance to observe the effects of a parity-breaking mass in real graphene samples. At this point, we note that the coupling cannot be compared directly to nominal values of the graphene fine structure constant, as should correspond in our approach to the effective coupling after screening of the interaction. That is, the value of must be referred to the relative strength of the dressed Coulomb potential. Under the assumption of static RPA screening of the interaction, we get for instance the relation between the effective coupling and the nominal value of the graphene fine structure constant

(64)

The above formula takes into account the effect of two fermion flavors with different spin projection. In the case of graphene isolated in vacuum, the bare value leads to , which is very close to the computed critical value (44). We recall anyhow that the use of the static RPA tends to underestimate appreciably the relative strength of the dressed interaction. The value of obtained after dynamical screening of the interaction has been shown to be (starting from ) above the critical coupling in (63) ggg (); prb (). Therefore, we are led to conclude that the electronic state of graphene in vacuum should correspond to the phase with dynamical breakdown of parity above .

We note finally that the phase with the dynamical breaking of parity may not have necessarily insulating character, as it corresponds to the spontaneous development of an order parameter (a vector) with two possible orientations. We may think of situations in which the space is divided into different domains where the order parameter alternates between opposite directions. In these configurations, electronic transport may be possible along the interface between neighboring domains. The study of the dynamics of this kind of disorder at a microscopic level is beyond the scope of this work, but it is plausible that, for suitably small domains, the phase with the dynamical breaking of parity may still have metallic properties. The development of an ordered phase over macroscopic regions may require some external source contributing to the complete alignment of the order parameter. This can be achieved for instance by introducing a transverse magnetic field, setting in this way a preferred orientation in the parity-breaking mechanism. Signatures of the onset of the ordered phase may have been already seen in the metal-insulator transition observed in graphene at relatively low magnetic fieldscheck (); zhang (). Further efforts should be devoted to the detection of the broken-symmetry phase, which should anyhow appear in the close proximity (about 1 meV or below) around the charge neutrality point.

V Acknowledgments

We thank F. de Juan, F. Guinea, V. Juričić and C. P. Martín for useful insights into the subject. The financial support from MICINN (Spain) through grants FIS2008-00124/FIS and FIS2011-23713 is gratefully acknowledged.

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