PHYSICAL REVIEW B 95, 165435 (2017)

Origin of kinks in the energy dispersion of strongly correlated matter

Kazue Matsuyama,1 Edward Perepelitsky,2,3 and B. Sriram Shastry11Physics Department, University of California, Santa Cruz, California 95064, USA

2Centre de Physique Thorique, cole Polytechnique, CNRS, Universit Paris-Saclay, 91128 Palaiseau, France3Collge de France, 11 place Marcelin Berthelot, 75005 Paris, France

(Received 22 January 2017; published 20 April 2017)

We investigate the origin of ubiquitous low-energy kinks found in angle-resolved photoemission experiments ina variety of correlated matter. Such kinks are unexpected from weakly interacting electrons and hence identifyingtheir origin should lead to fundamental insights in strongly correlated matter. We devise a protocol for extractingthe kink momentum and energy from the experimental data which relies solely on the two asymptotic tangentsof each dispersion curve, away from the feature itself. It is thereby insensitive to the different shapes of thekinks as seen in experiments. The body of available data are then analyzed using this method. We proceedto discuss two alternate theoretical explanations of the origin of the kinks. Some theoretical proposals invokelocal bosonic excitations (Einstein phonons or other modes with spin or charge character), located exactly at theenergy of observed kinks, leading to a momentum-independent self-energy of the electrons. A recent alternateis the theory of extremely correlated Fermi liquids (ECFL). This theory predicts kinks in the dispersion arisingfrom a momentum-dependent self-energy of correlated electrons. We present the essential results from bothclasses of theories, and identify experimental features that can help distinguish between the two mechanisms.The ECFL theory is found to be consistent with currently available data on kinks in the nodal direction of cupratesuperconductors, but conclusive tests require higher-resolution energy distribution curve data.

DOI: 10.1103/PhysRevB.95.165435

I. INTRODUCTION

High-precision measurements of electronic spectral dis-persions have been possible in recent years, thanks to theimpressive enhancement of the experimental resolution inthe angle-resolved photoemission spectroscopy (ARPES).This technique measures the single-electron spectral functionA(k,) multiplied by the Fermi occupation function; it canbe scanned at either fixed k as a function of or at fixed as a function of k. These scans produce, respectively, theenergy distribution curves (EDCs) and momentum distributioncurves (MDCs). The line shapes in both these scans are offundamental interest since they provide a direct picture of thequasiparticle and background components of interacting Fermisystems, and thus unravel the roles of various interactions thatare at play in strongly correlated Fermi systems. The dispersionrelation of the electrons can be studied through the location ofthe peaks of A(k,) in constant or constant k scans.

Recent experimental studies have displayed a surprisingubiquity of kinks in the dispersion of strongly correlated matterat low energies 50100 meV. The kinks are bending-typeanomalies (see Fig. 1) of the simple = vF (k kF ), i.e.,linear energy versus momentum dispersion that is expectednear kF from band theory. The special significance of kinkslies in the fact that their existence must signal a departure fromband theory. This departure could be either due to electron-electron interactions or to interaction of the electrons withother bosonic degrees of freedom. Either of them are thereforesignificant enough to leave a direct and observable fingerprintin the spectrum. The goal of this work is to elucidate the originof the observed kinks, and therefore to throw light on thedominant interactions that might presumably lead to high-Tcsuperconductivity.

The purpose of this paper is multifold: We (i) survey theoccurrence of the kinks in a variety of correlated systems of

current interest, (ii) provide a robust protocol for characterizingthe kinks which is insensitive to the detailed shape of the kink,(iii) discuss how these kinks arise in two classes of theories,one based on coupling to a bosonic mode and the other tostrong correlations, and (iv) identify testable predictions thatARPES experiments can use to distinguish between these.

The 15 systems reporting kinks are listed in Table I: theseinclude (1) most high-Tc cuprates in the (nodal) direction11 at various levels of doping from insulating to normalmetallic states in the phase diagram [1,2], (2) charge densitywave systems, (3) cobaltates, and (4) ferromagnetic ironsurfaces. The kinks lose their sharpness as temperature is raised[24], and appear to evolve smoothly between the d-wavesuperconducting state and the normal state.

The kinks above Tc are smoothed out as one moves awayfrom nodal direction [5]. Recent experiments [6] resolve thismovement of the kinks more finely into two subfeatures. Mostof the studies in Table I focus on MDC kinks; the EDC kinksdata are available for only eight systems so far. Bosonic modeshave been reported in six systems using different probes suchas inelastic x rays or magnetic scattering, with either charge(phonons, plasmons) or spin (magnetic) character, while theremaining nine systems do not report such modes. A fewtheoretical studies of the kinks have implicated the observedlow-energy modes via electron-bosontype calculations; wesummarize this calculation in the Supplemental Material (SM)[7]. We find, in agreement with earlier studies, that the bosoncoupling mechanism yields kinks in the MDC dispersion,provided the electron-boson coupling is taken to be sufficientlylarge. In addition, we find in all cases studied this mechanismalso predicts a jump in the EDC dispersion. It also predicts anextra peak in the spectral function pinned to the kink energyafter the wave vector crosses the kink. These two features areexperimentally testable and differ from the predictions of thecorrelations mechanism discussed next.

2469-9950/2017/95(16)/165435(10) 165435-1 2017 American Physical Society

https://doi.org/10.1103/PhysRevB.95.165435

MATSUYAMA, PEREPELITSKY, AND SHASTRY PHYSICAL REVIEW B 95, 165435 (2017)

VL

(e

V )

K ( -1 )

(EDC) VH*

(MDC)VHFar zone

Far zone

Near zone

Ideal Kink

Low THigh T

(MDC and EDC)

Kkink(-1)

} 0

FIG. 1. A schematic MDC and EDC spectrum displaying typ-ical features of experiments discussed below. Here, k = (k kF ) kF /|kF | is the momentum component normal to the Fermisurface, and we label EDC variables with a star. [The sketch usesparameters VL = 2 eV A, VH = 6 eVA, r = 1.5, kkink = 0.03 A

1,

#0 = 0.03eV, and $0 = 0.01eV in Eqs. (3) and (4).] The tangents inthe far zones identify the asymptotic velocities VL < VH and V L 0, i.e., slightly to the right of the physicalFermi momentum kF , and we consider this implies that theexperimental Fermi momentum determination is subject tosuch a correction, whenever the spectral function Eq. (9)has a momentum-dependent caparison factor (see caption inFig. 2).

TABLE II. Parameter table for ARPES kink analysis for OPT Bi2212 [4] in Fig. 2 presents three essential parameters: VL, VH , and kkink.From the high- and low-temperature MDC dispersions, we measured $0 % 10 meV in Fig. 2(b). With the measured experimental parametersand determining the velocity ratio r in Eq. (2), we are able to estimate the finite-temperature kink energy for EDC and MDC dispersions byEEDCkink = Eidealkink $0 and EMDCkink = Eidealkink $0

r

2r and predict VH by V

H =

3VH VLVH +VL

VL in Eq. (10). The uncertainties for calculated variableswere determined by error propagation, and the uncertainties for experimental variables were given by the half of the instrumental resolution.

MDCs EDCs

OPT Bi2212 ARPES data EMDCkink (meV) EEDCkink (meV) V

H (eV A)

VL (eV A) VH (eV A) kkink (A1

) Calculated Measured Calculated Measured Predicted Measured

1.47 0.07 3.3 0.3 0.037 0.005 67 21 67 8 63 21 65 8 2.60 0.56 2.1 1.1

165435-4

ORIGIN OF KINKS IN THE ENERGY DISPERSION OF . . . PHYSICAL REVIEW B 95, 165435 (2017)

FIG. 2. ARPES kinks data for OPT Bi2212 from Ref. [4] compared to theoretical ECFL curves (solid lines) using parameters listed inTable II. (a) The predicted EDC spectrum (blue) from Eq. (3) versus the experimental EDC data (magenta symbols) at T = 115 K. For referencewe also show the MDC data (red dashed curve) and the corresponding ECFL fit (green solid curve). (b) Experimental MDC spectra at 40 K(below Tc in green dashed line) and 115 K (above Tc in red dashed line) yield common asymptotes shown in black lines from the far zone.These determine the parameters displayed in Table II. (c) At low energy 60 meV, the EDCs spectral function (blue solid line) from Eq. (9) iscontrasted with the corresponding ARPES data from [4]. (d) At = 0 we compare the MDCs spectral function (blue solid line) from Eq. (9)with the corresponding ARPES data from Ref. [4]. The range of validity for the theoretical expansion is kkink(0.037 A

1), the data points in

the range are shown in black circle symbols, while the light gray circle symbols are outside this range. The peak position of the theoretical curvehas been shifted to left by 0.007 A

1, a bit less than the instrumental resolution. A similar shift is made in Fig. 3(l). For analogous reasons, the

EDC peak in A(k,) at kF is shifted to the left, i.e., E(0) ! 0. A small shift to the right is made in Fig. 3(k), in order to compensate for thiseffect. These shift effects are within the resolution with present setups, but should be interesting to look for in future generation experimentssince they give useful insights into the energy momentum dependence of the spectral function.

IV. LSCO LOW-TEMPERATURE DATA

Here, we analyze the LSCO data at low temperature(20 K) and at various doping levels raging from the insulator(x = 0.03) to normal metal (x = 0.3) from Ref. [1]. Theparameters are listed in Table III, where we observe that thevelocity VL is roughly independent of x, and has a somewhatlarger magnitude to that in OPT Bi2212 in Table II. Thekink momentum decreases with decreasing x, roughly askkink = (0.37x 0.77x2) A

1, and the kink energies of EDC

and MDC dispersions are essentially identical. In the regionbeyond the kink, the prediction for V H is interesting sinceit differs measurably from the MDC velocity VH . We findthe ratio VH /V H 1.021.5 is quite spread out at differentdoping.

Our analysis becomes unreliable as lower doping level x |2

|, wherethe momenta

1

and 2

(the low-energy kink momentum)are denoted by the dashed vertical lines in the middlepanel of Fig. (2). In the first region, || < |

1

|, the peaklocation, Ep , disperses according to the equation =Ep

4

!0.4 !0.3 !0.2 !0.1

!0.4

!0.3

!0.2

!0.1

0.1

#

!0.1!0.1!0.2!0.2!0.3!0.3

!0.35

!0.30

!0.25

!0.20

!0.15

!0.10

!0.05

FIG. 3: To explore the eects of raising , we set = 1 while leaving all other parameters unchanged from Fig. (2). As aresult, the kink momentum in the MDC dispersion becomes bigger, the hump in the EDCs is suppressed, the EDC dispersionstays pinned to the phonon frequency over a larger range of momentum, and the magnitude of the jump in the EDC dispersiongrows.

To examine the eects of raising , we set = 1 leavingall other parameters unchanged, and plot the correspond-ing results in Fig. (3). This causes several noticeablechanges to the results in Fig. (2). 1) The kink in thereal part of the self-energy becomes sharper, which leadsto a larger kink momentum,

2

, in the MDC dispersion.2) =m(Eh) becomes bigger, causing the height of thehump to go down. 3) As a direct consequence of 2), therange over which the EDC dispersion stays pinned to thephonon frequency becomes more prolonged in momen-tum space, and therefore the magnitude of the jump in

the EDC dispersion also becomes bigger.

Setting T ! 0 in Eq. (SI-7), and plugging it intoEq. (SI-6), we find that to linear order in ! !

0

,

5

as compared to Fig. (2), but retain the same features. Wesee that the EDC dispersion once again follows the MDCdispersion (closest to zero frequency) in the first two mo-mentum regions, until it (nearly) flattens out in the thirdregion, where the peak is pinned to the phonon frequency,!

0

, in the corresponding EDCs. As the momentum is

increased such that the height of this peak shrinks be-low the height of the hump, the EDC dispersion jumpsdiscontinuously down from the phonon frequency, to theMDC dispersion. Consequently, we see that the veloci-ties of the MDC and EDC dispersion coincide above thekink; i.e. VH = V H .

!0.20.4

0.5

1.0

1.5#

!0.5 1.5E

2 !0.1!0.1!0.2!0.2!0.3!0.3

!0.5

!0.4

!0.3

!0.2

!0.1

FIG. 5: We explore the eects of using the full frequency-dependence of the density of states in Eq. (SI-6), with = 0.84and T = 115 K. Due to the functional form of the density of states (displayed as an inset in the left panel), the MDCdispersion acquires two additional branches which yield large frequency values. Below the low-energy kink momentum, theEDC dispersion follows the lowest-frequency branch of the MDC dispersion. Above the low-energy kink momentum, the EDCdispersion initially stays pinned to the phonon frequency, until it discontinuously jumps onto the highest-frequency branch ofthe MDC dispersion (VH = V

H). A noticeable hump also develops at high-frequencies, in the corresponding EDCs.

We now examine how these results are aected by retain-ing the full frequency-dependence of the density of statesin Eq. (SI-6). Just as was done in3, we use the dispersiontb2 from9. In this case, "f = 0 and N("f ) = 0.61 eV

1.Retaining the same values of T = 115 K and = 0.84,we set g = 0.23 eV in Eq. (SI-6). We also set "f = 0.In Fig. (5), we plot !

6

while the real part is as usual given by applying theHilbert transform to Eq. (SI-9). Here, A

el-el,loc

(!) =1

Ns

Pk Ael-el(

~k,!), where Ael-el

(~k,!) is given by Eq. (SI-5) with the substitution (!) !

el-el

(!). Eq. (SI-5)

continues to express A(~k,!) in terms of (~k,!), whereboth objects now include electron-electron and electron-phonon correlations.

In Fig. (6), we plot !

7

We see below that this momentum dependence is essen-tial for describing the low energy kinks in the occupiedpart of the ARPES spectrum.

A. Low energy expansion of the ECFL theory

We start with the ECFL Greens function G expressedin terms of the auxiliary Greens function g and the ca-parison function e Ref. (14) and Ref. (15), we write

G(~k, i!) = g(~k, i!) e(~k, i!), (SI-10)and with the latter expressed in terms of the two selfenergies (~k, i!n), (~k, i!n) as:

e(~k, i!n) = 1 n2+ (~k, i!n) (SI-11)

g1(~k, i!n) = i!n + (1 n2)"k (~k, i!n),

(SI-12)

where n is the electron number per site, !n = (2n+1)/the Matsubara frequency, which we analytically continuei! ! ! + i0+. Let us define k as the normal deviationfrom the Fermi surface i.e. k = (~k ~kF ).~r"kF /|~r"kF |.Our first objective is to Taylor expand these equationsfor small ! and k, as explained above. We carry out alow frequency expansion as follows:

1 n2+ (~k,!) =

0

+ c

(! +

k vf )

+iR/

+O(!3), (SI-13)where the frequently occurring Fermi liquid functionR = {!2 + (kBT )2}, vf = (@k"k)kF is the bare Fermivelocity, and the four parameters

0

, c

,

,

are coef-ficients in the Taylor expansion having suitable dimen-sions. Similarly we expand the auxiliary Greens function

g1(k,!) = (1 + c

)!

k vf

+iR/

+O(!3) , (SI-14)where we have added another three coecients in theTaylor expansion c

,

,

.To carry out this reduction we first trade the two pa-

rameters c

,

in favor of parameters

and s by defin-ing c

= 0

and

= sc , where the dimensionlessparameter 0 s 1. With these expansions and thequasiparticle weight determined in terms of the expan-sion parameters as Z = 0

1+c, we find

G = Z

+ ! +

k vf + iR/(s)!

k vf + iR/. (SI-15)

Using A(k,!) = 1=mG we find the spectral function

A(k,!) =Z

R

(!

k vf )2 + (R

)2 ec(k,!)

(SI-16)

Here the caparison factor, (not to be confused with thecaparison function in Eq. (SI-10)), is found as

ec(k,!) = 1 (k,!)(k,!) =

1

0

(! 0

k vf ) (SI-17)

In Eq. (SI-17) we have introduced two composite param-eters

0

=s

1 s , and 0 =1

1 s +s

1 s .(SI-18)

This procedure eliminates the three old parameters s,

and

in favor of the two emergent energy scale 0

andvelocity

0

.It is interesting to count the reduction in the number

of free parameters from the starting value of seven inEq. (SI-13) and Eq. (SI-14). Already in Eq. (SI-15) wehave a reduction to six, since the quasiparticle weight Zcombines two of the original parameters. Since Eq. (SI-18) subsumes three parameters into two, the spectralfunction in Eq. (SI-16) contains only five parameters: thetwo velocities

0

vf , vf , and the two energies ,0,in addition to the overall scale factor Z.We will see below that the parameters that are measur-

able from energy dispersions are best expressed in termsof certain combinations of the velocities. In order to makethe connection with the experiments close, we will rede-fine the two velocities in terms of an important dispersionvelocity at the lowest energies VL and a dimensionless ra-tio r, on using the definitions:

vf = VL0

vf = r VL. (SI-19)

In order to account for the dierence between laserARPES and synchrotron AREPS having dierent inci-dent photon energies, we will make two phenomenologicalmodifications in Eq. (SI-16) following Ref. (16)

R(!)/

! R(0)/

= {kBT}2/ + 0(SI-20)where represents an elastic energy from impurity scat-tering, dependent upon the energy of the incident photonin the ARPES experiments. In the spirit of a low energyexpansionR is evaluated at ! = 0. Thus

0

is a T depen-dent constant, which subsumes the two parameters and

, and thus the total parameter count is still five. Sec-ondly for extension to higher energies, we renormalizethe parameter in Eq. (SI-17) according to a recentlydiscussed prescription following from a theoretical calcu-lation Ref. (17) as fc ! {1 p

1+ca2}, where ca 5.4

near optimum doping 0.15 as estimated recently.This correction ensures that the caparison factor exhibitsthe correct linear behavior for small , and remains posi-tive definite at high energies. Thus we write the spectral

8

function in terms of the new variables as

A(~k,!) =Z

0

(! VL k)2 + 20

{1 p1 + ca2

},

(SI-21)

with = 10

(! r VL k). We should keep in mind thatthese expressions follow from a low energy expansion, andis limited to small k and !, so that the dimensionless vari-able ||max O(1). Microscopic calculations of all theseparameters is possible in the ECFL theory. One impor-tant parameter is the energy scale

0

which is foundto be much reduced from the band width, due to ex-tremely strong correlations. A related energy is the eec-tive Fermi liquid temperature scale where the T 2 depen-dence of the resistivity gives way to a linear dependence.This scale is estimated in the limit of large dimensionsfrom Ref. (17) to be as low as 45 K near optimum doping,i.e. much reduced from naive expectations.

For the present purposes we take a dierent track, wenote that the ARPES fits are overdetermined, so thatwe can determine the few parameters of the low energytheory from a fairly small subset of measurements. Thefive final (composite) parameters defining the spectralfunction Eq. (SI-21) are Z, VL, r,0,0, where ca 5.4.Of these Z is multiplicative, it is only needed for gettingthe absolute scale of the spectral function, and ca doesnot play a significant role near zero energy, it is requiredonly at high energies. Thus the spectra relevant to EDCand MDC will require only four parameters VL, r,0,0.These suce to determine the low energy theory andthus to make a large number of predictions; i.e. implyingnon trivial relationships amongst observables. Many ofthe predictions rely only on the overall structure of thetheory and not its details.

B. The EDC and MDC dispersion relations andkinks

Starting from Eq. (SI-21), we can compute the energydispersions for MDC (varying k while keeping ! fixed)and the EDC spectra (varying ! while keeping k fixed).In terms of a momentum type variable

Q(k) = 0

+ (r 1)k VL (SI-22)we can locate the peaks of Eq. (SI-21) using elementarycalculus since ca only plays a role at high energies, we setca ! 0 when performing the extremization and find theMDC dispersion

E(k) =1

2 rk VL +0

qr(2 r)2

0

+Q2,

(SI-23)

and the EDC dispersion

E(k) =

r k VL +0

q20

+Q2. (SI-24)

Using these two dispersions and expanding them in dif-ferent regimes, we can extract all the parameters of thekinks.

1. Kink momentum

As explained in the main paper, when we set T = 0 = so that

0

= 0, both the EDC and MDC dispersionscontain an ideal kink at the kink momentum. Therefore,using Eqs. (SI-23) and (SI-24), the condition Q = 0locates the kink momentum for both dispersions:

kkink =

0

(1 r)VL , (SI-25)

it corresponds to occupied momenta, i.e. kkinkvf < 0,provided that r > 1. We thus can express

0

=kkink VL(1 r), enabling us to usefully rewrite

Q = (r 1)VL (k kkink) = 0 {1 kkkink

}.(SI-26)

As required by the ideal kink, Q changes sign at the kinkmomentum,

sign(Q) = sign(k kkink). (SI-27)

2. Ideal Kink energies: T=0

Using Eq. (SI-23) and Eq. (SI-24), in conjunction withEq. (SI-25), the ideal kink energy is the same for bothdispersions, and is given by

Eidealkink = 1

r 10. (SI-28)

We can also usefully estimate this ideal kink energy fromthe asymptotic velocities in the far zone, as explained inthe main paper.

3. The non-ideal i.e. T > 0 kink energy

The EDC and MDC kink energies for the non-idealcase can be viewed in a couple of ways. We have arguedin the main paper that these are best defined by fixingthe momentum k = kkink and reading o the energy atthis value. This is an unambiguous method independentof the detailed shape of the kink, since it only requiresknowledge of kkink, which can be found from an asymp-totic measurement as we have argued in the main paper.We can put Q = 0 and k ! kkink in Eq. (SI-24) andEq. (SI-23) and read o the kink energies:

EEDCkink = Eidealkink 0, (SI-29)

EMDCkink = Eidealkink 0

rr

2 r . (SI-30)

9

We observe that the MDC kink energy is real provided2 r 1. Note also that at T = 0 and = 0, the twoenergies both reduce to the ideal kink energy.

4. The ideal energy dispersions

At T = 0 or for |Q| 0

, the two dispersionsEq. (SI-24) and Eq. (SI-23) become:

E(k) hr (r 1) sign(k kkink)

ik VL

+ 20

(kkink k) (SI-31)and

E(k) 12 r

h1 (r 1) sign(k kkink)

ik VL

+2

0

2 r(kkink k). (SI-32)

The velocities in the asymptotic regime |k| kkinkcan be found from the slopes of these, and are there-fore temperature-independent. For k kkink we get thelow velocities

dE(k)

dk= VL

dE(k)

dk= V L = VL (SI-33)

and thus the EDC and MDC velocities are identical. Fork kkink we get the high velocities

VH =dE(k)

dk=

r

2 rVL, (SI-34)

V H =dE(k)

dk= (2r 1)VL. (SI-35)

We may cast Eq. (SI-35) into an interesting form

V H =

3VH VLVH + VL

VL, (SI-36)

it is significant since the EDC spectrum velocity is ex-actly determined in terms of the two MDC spectrumvelocities. It is also a testable result, we show else-where in the paper how this compares with known data.Note that the four independent parameters VL, r,0,0alluded to in the discussion below Eq. (SI-21), canbe determined from the directly measurable parametersVL, VH , kkink,0 (SI-34,SI-25,SI-3). Therefore, either setof parameters gives complete knowledge of the EDC andMDC dispersions, as well as the spectral function (up toan overall scale).

5. Near Zone: Corrections to Energy dispersion due to

finite T.

In the regime dominated by finite T and eects of the elastic scattering parameter, we can also perform an

expansion in the limit when |Q| 0

, using Eq. (SI-23)and Eq. (SI-24). The the first few terms are

E(k) =

0

1 r r

r

2 r 0 +VL

2 r (k kkink)

(1 r)2

2pr(2 r)3

V 2L0

(k kkink)2 + . . .(SI-37)

Similarly for the EDC dispersion

E(k) =

0

1 r 0 + rVL(k kkink)

(1 r)2

2

V 2L0

(k kkink)2 + . . . (SI-38)

These formulas display a shift in the energies due to 0

and also a 0

dependent curvature. Since the regime ofthis expansion, |Q| <

0

is dierent from that of theexpansion in Eq. (SI-35) and Eq. (SI-33), we note thatvelocities are dierent as well. Thus one must be carefulabout specifying the regime for using the velocity formu-lae.Let us note that in this regime |Q| <

0

the two dis-persions dier, with the EDC higher.

E(k) E(k) = {r

r

2 r 1}0

(1 r)2

2 r VL(k kkink) + . . . (SI-39)

This equation gives a prescription for estimating 0

incases where the other parameters are known. Alterna-tively in the MDC dispersion we expect to see a curva-ture only near the location of the kink, this is sucientto fix

0

: from Eq. (SI-37)

d2E(k)

dk2= (r 1)

2

pr(2 r)3

V 2L0

. (SI-40)

The curvature d2E(k)

dk2can be estimated from the experi-

mental data to provide an estimate of 0

.

C. The Dyson self energy

For completeness we present the low energy expansionof the Dyson self energy, which gives rise to the spectralfunction in Eq. (SI-21). We may define the Dyson selfenergy from

D = ! + "k G1 (SI-41)Using Eq. (SI-15) we obtain

=mD = 1Z

R

1 10

(! 0

k vf )

{1 + (! +

k vf )/ }2 + R2s222 (SI-42)

10

The corresponding real part is given by

1. From Eq. (SI-44) we see that thisrequires a finite

(so that 1 > s > 0). We also need

0

> 0 and1 +

> 0.

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