Spectral function of a Luttinger liquid coupled to phonons and angle-resolved photoemission measurements in the cuprate superconductors

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by  Ian P. Bindloss

Spectral function of a Luttinger liquid coupled to phonons and angle-resolved
photoemission measurements in the cuprate superconductors
Ian P. Bindloss and Steven A. Kivelson
Department of Physics, University of California, Los Angeles, California 90095–1547, USA
(Dated: February 2, 2008)
Wecomputethefinite-temperaturesingle-particlespectralfunctionofaone-dimensionalLuttinger
5
liquidcoupledtoanopticalphononband. Thecalculationisperformedexactlyforthecaseinwhich
0
electron-phonon coupling is purely forward scattering. We extend the results to include backward
0
scattering with arenormalization group treatment. Thedispersion contains achangein velocity at
2
the phonon energy, qualitatively similar to the case of electron-phonon coupling in a Fermi liquid.

n Ifthebackwardscatteringpartofthetheelectron-phononinteractionisnottoostrongcomparedto
a the forward scattering part, coupling to phonons also produces a pronounced peak in the spectral
J functionatlowenergies. Thecalculatedspectralfunctionisremarkablysimilartotheangle-resolved

5 photoemissionspectraofthehigh-temperaturesuperconductors,includingtheapparentpresenceof
2 “nodalquasiparticles,”thepresenceofa“kink”inthedispersion,andthenon-Fermi-liquidfrequency
and temperature dependencies. Although a microscopic justification has not been established for

] treatingtheelectronicdynamicsofthecupratesasquasi-one-dimensional, attheveryleast wetake
l
e thequalityofthecomparisonasevidenceofthenon-Fermi-liquidcharacterofthemeasuredspectra.
-
r
t
s I. INTRODUCTION liquid–just as many aspects of the Fermi liquid state are
.
t robust, independent of details of material and even di-
a
m Thereisincreasing,althoughcontroversial,experimen- mensionality, we may hope that some features of non-
Fermi-liquidsaresimilarlygeneric,atleastwithinclasses
- talevidencethat,incupratehigh-temperaturesupercon-
d ductors, a strong coupling between electrons and optical ofnon-Fermi-liquids. Inthis case,the LL may serveas a
n paradigmaticmodelforabroaderclassofsystems. (3)It
phonons produces observable features in the single-hole
o spectral function A(k,E) measured by angle-resolved may be the case that the quasi-2D cuprates have signif-
[c photoemissionspectroscopy(ARPES).1,2,3Inattempting icant, self-organized “stripe” structures14 which render
to understand the implications of these measurements, them locally quasi-1D, in which case it may be possi-
5 ble to directly compare the results obtained here with
the experiments have been previously analyzed in the
v experiments5 inthe cupratesandotherhighlycorrelated
contextofthe theoryofthe electron-phonon(el-ph)cou-
7 materials.15
pling in a Fermi liquid. However, the observed spectral
5
4 functionexhibitsamanifestlynon-Fermi-liquidfrequency Let us start with a qualitative summary of the solu-
2 andtemperaturedependence,4,5,6,7sothejustificationfor tionoftheel-phprobleminaFermiliquid16 atzerotem-
0 this mode of analysis is not clear.
perature. Consider the case of an optical phonon with
4
0 In the present paper, we compute and analyze the frequency ω0 and dimensionless el-ph coupling λ′, which
finite-temperature spectral function of a spinful, one-
/ is not too large. For E > ω0, the effect of the el-ph
at dimensional (1D) Luttinger liquid (LL) coupled to a coupling on A(k,E) is|an| E-independent broadening of
m dispersionless optical phonon band (Einstein phonon). the quasiparticle peak. For E ω , the phonons can
0
Such a spectral function was computed previously in | | ≪
- beintegratedouttoproduceneweffectiveinteractionsin
d Ref. 8, but only at T = 0 and in the absence of both theFermiliquid: thelargesteffectisarenormalizationof
n electron-electron (el-el) interactions and el-ph backscat- the Fermi velocity, v v∗ = v /(1+λ′) < v , while
o tering. The LL is a quantum critical point, so the spec- F → F F F
the most dramatic effect is the weak effective attraction
c tral function is a scaling function, and should be com-
: produced between low-energy quasiparticles, which can
v puted at finite temperature; the zero-temperature spec-
lead to a superconducting instability of the Fermi liquid
i tral function reveals only the high-frequency behavior of
X state.
the scaling function. It is also essential to include el-el
r
a interactions,as in realmaterials they are typically much Wehavecomputedthesingle-holespectralfunctionfor
strongerthantheel-phinteractions. Moreover,strongel- a LL coupled to optical phonons for the case when the
el interactions make qualitative changes to the “appear- el-phcouplingdoesnotproduceaspingap. Wefindthat
ance” of the spectral function. The effects of phonon- the same words describe the effects of the el-ph coupling
assisted backward scattering are also important to con- as in the Fermi liquid case, but the results look quite
sider, as we have done below. different because the unperturbed LL spectral function
Our motivation for this study is threefold. (1) There is not a simple Lorentzian. Specifically, if we let g
are a host of interesting quasi-1D materials,9,10 some of represent the set of coupling constants which define{th}e
which are amenable to ARPES studies,11,12,13 to which LL(i.e. the chargeandspinvelocitiesv andv ,andthe
c s
this analysis may be directly applicable. (2) The LL is correspondingLuttingerexponentsK andK ),thenour
c s
thetheoreticallybestunderstoodexampleofanonFermi result can be summarized by
2
A (k,E; g )+ for E ω , no el-ph coupling (K c = 0.15)
A(k,E)∼ ALLLL(k,E;{g}∗ ) ··· for|E|≫ω00, (1) nits) no el-ph coupling (K c = 0.3)
(cid:26) { } | |≪ u with el-ph coupling
and there is a smooth crossover between the two limits arb. of strength = 0.75
wpuhreenL|EL|(∼i.eω.0.inHtehree,aAbLseLnicsetohfeeslp-pechtrcaolufpulninctgi)o,ntohfetehle- E) ( (K c = 0.15, K* c = 0.3)
lipsis represents perturbative corrections to the spectral k,F
A(
function which can be ignored as long as the el-ph cou-
pling isnottoobigorif E ishighenough,and g∗ are
| | { }
renormalizedcoupling constants obtained by integrating
0
out the phonons–we give explicit expressions for these -6 -5 -4 -3 -2 -1 0 1
renormalized couplings below. E/ 0
The principalresults ofthe presentpaper areEqs. (1)
FIG. 1: Comparison of the single-hole spectral functions at
and (9). The latter is an analytic expression for the
k =kF for a LL coupled to optical phonons (solid line), and
space-time spectral function that interpolates between
aLLinabsenceofphononcoupling(dashedanddash-dotted
the two limits in Eq. (1), andis shownto be verynearly
lines). For the solid line, the el-ph coupling strength is λ =
exact in the exactly solvable case of forward scattering 0.75 (vc/vc∗ = Kc∗/Kc = 2). For all curves, vc/vs = 4 and
only. We later generalize it for el-ph couplings that in- ω0/T =10. SeetheplotforthevaluesofKc. Normalizations
clude backscattering interactions. Plots of A(k,E) com- are chosen for graphical clarity.
puted from Eq. (9) are shown in the figures.
There are several general features of A(k,E) that are
worth noting. (1) As has been previously emphasized,5
thespectralweightoftheLLisconcentratedinaroughly
triangularregionoftheE-k plane(see Fig. 7), reflecting
by the coupling to phonons. Therefore, for K 1,
thefractionalizedcharacteroftheelementaryexcitations, c ≪
theresulthasthe followingsimilaritywiththe Fermiliq-
incontrasttothe Fermiliquidcaseinwhichthe spectral
uid case: features in A(k,E) appear broader for binding
weightisconcentratedalongthelineE =v (k k ),re-
flectingthequasiparticledispersion. (2)ThFerei−saFrenor- energies above ω0 than for binding energies below. In
the Fermi liquidcase,this comes froma phonon-induced
malizationofthecharge(holon)velocityproducedbythe
broadeningathighenergies,butintheLL itcomesfrom
el-ph coupling, such that v∗ < v , which is analogous to
c c a phonon-induced narrowing at low energies.17
the renormalization of v in the Fermi liquid. This pro-
F
duces a “kink” in the spectrum, as shown in Fig. 8. (3)
The E dependence of A(k,E) at fixed k is referred to A plot of the EDC at k = kF is shown in Fig. 1.
intheARPESliteratureasthe energydistributioncurve The solid line shows A(kF,E) for the case Kc =0.15, in
(EDC), while the k dependence at fixed E is referred the presence of forward scattering el-ph coupling, which
to as the momentum distribution curve (MDC); the ex- produces,atlow energies,arenormalizedKc∗ =0.3. The
tent to which there is a quasiparticle-like peak in the dashed line shows A(kF,E) for the pure LL, but with
EDC of a LL is strongly dependent on the value of Kc. Kc = 0.3, while the dash-dotted line shows it for the
For weakly interacting electrons, Kc 1, and the EDC pure LL with Kc = 0.15. Clearly, for the LL coupled
exhibits a peak, although this peak≈contains a power- to phonons, the renormalized Kc governs the properties
law tail indicating the absence of fermionic quasiparti- at low binding energies, while the bare Kc dictates the
cles. Forstrongrepulsiveinteractionsthataresufficiently behavior at large binding energies.
long range, K 1, in which case the EDC is extremely
c
≪
broad,andcanevenfailto exhibit any welldefined peak In Sec. II, we investigate the exactly solvable model
near the Fermi energy. In contrast, the structure of the (forward scattering only). The results are extended to
MDC is much less variable,5 and remains peaked even general couplings using a renormalization group (RG)
for small K . The presence of extremely broad EDCs at treatmentinSec. III. InSecs. IVandVwespeculateon
c
the same time that the MDCs arenarrowis amanifestly the relevanceofthese resultsto realmaterials,especially
non-Fermi-liquidfeature anda dramaticsignatureofthe thehigh-temperaturesuperconductors. AppendixAcon-
LL. This feature is illustrated in Fig. 13. tains a derivation of the spectral function of the forward
For a model with purely forward scattering el-ph in- scattering only model. There we also present the exact
teractions, and for more generalcouplings as long as the resultfor the frequency- andmomentum-dependent con-
el-ph backscattering is not too strong, at low energies ductivity of this model. The optical conductivity is un-
K is increased by the presence of el-ph interactions, i.e. changedbytheel-phforwardscattering,eventhoughthis
c
K∗ >K . Forthephysicallyrelevantcasewhenthebare interactionhas dramaticeffects onthe spectralfunction.
c c
K <1,thismeansthatfeaturesthataresituatedwithin AppendixBcontainstechnicaldetailsfortheresultspre-
c
ω of the Fermi energy are made sharper (more peaked) sented in Sec. III.
0
3
II. AN EXACTLY SOLVABLE MODEL in the fields. In Appendix A we compute the renormal-
ized couplings that define the g∗ for this model. The
{ }
The model is defined by the Hamiltonian density exact result is
= LL+ ph+ el−ph. (2) v∗ =v √1 λ , K∗ = Kc , (7)
H H H H c c − c √1 λ
Here the purely electronic part of the Hamiltonian −
where
v (∂ φ )2
= α K Π2 + x α (3)
HLL 2 α α K 2K α2
α=c,s (cid:20) α (cid:21) λ= c 2 , (8)
X πv Mω2
c 0
isthefamousspin-charge-separatedTomonaga-Luttinger
liquid model of the interacting one-dimensional electron and the spin couplings are unrenormalized. Note that λ
gas (1DEG) at incommensurate filling. It is expressed depends on both the el-ph and el-el interactions. Also
via bosonization, in terms of bosonic charge (α=c) and note that v /v∗ = K∗/K 1. At low energies, v is
spin (α = s) fields φα, and their canonically conjugate reduced duec toc “phocnoncdr≥ag,” while Kc is increacsed
momenta Πα. Repulsive interactions usually renormal- due to an attractive interaction mediated by phonons.
ize the Luttinger parameter Kc below its noninteracting Since this model possesses aninstability at λ=1, where
value of 1 such that 0 < Kc < 1, and renormalize the the charge velocity goes to zero, λ is restricted to the
velocities such that the vs < vF < vc. We will assume range 0 λ < 1. The analog of this instability for the
thesystemisspin-rotationinvariant,whichdictatesthat case of c≤oupling to acoustic phonons has been studied
Ks =1. Expressionsforthevα’sandKα’sintermsofmi- previously.23
croscopicshort-rangeinteractionparametersandreviews In Appendix A we derive an exact expression for the
on the technique of bosonization can be found in many spectralfunction. This quantity is most easily expressed
places in the literature.18,19,20,21,22 Bosonization allows in position space, but even then involves a momentum
the fermionic fields to be expressed directly in terms of integral in the exponent that cannot be performed an-
the bosonic fields: alytically. The integral can be evaluated numerically,
eiηkFx π and fortunately we are able to derive a simple analytic
Ψη,σ = exp i [η(φc+σφs)+θc+σθs] , expression that accurately approximates it, as shown in
√2πa 2
(cid:26) r (cid:27) Appendix A and Fig. 15. We therefore use this analytic
(4)
approximationin subsequent calculations.
whereΨ istheright-orleft-movingfermionicdestruc-
η,σ Specifically, our analytic approximationfor the single-
tion field (η = 1, respectively) for spin up or down
(σ = 1, respect±ively). Here θ (x) = x dx′Π (x′) holeGreen’sfunctionGη(x,t;λ)= Ψ†η,σ(x,t)Ψη,σ(0,0)
± α − −∞ α is
labels the dual bosonic field, and a is a short-distance (cid:10) (cid:11)
R
cutoff corresponding to the lattice parameter. As is well g (x,t;v∗,K∗,v /ω )
known, the Hamiltonian in Eq. (3) describes a line of Gη(x,t;λ) Gη(x,t;0) η c c c 0 , (9)
≈ g (x,t;v ,K ,v /ω )
η c c c 0
fixedpointsforinteractingelectrons,andsocapturesthe
essential low-energy physics of a large class of physical
where the exact Green’s function in the absence of
systems. phononcoupling G (x,t;0) at temperature T =1/β is24
η
The purely vibrational part of the Hamiltonian
Hph = P2+ω02u2 /2M (5) Gη(x,t;0)= e−2iπηkaFx gη(x,t;vα,Kα,a), (10)
describes an Einstein o(cid:2)scillator. H(cid:3)ere u and P are the αY=c,s
phonon field and its canonical conjugate momentum, M
and we have defined the function
is the ion mass, and again ω is the optical phonon fre-
0
quency. Note that we work with units such that Boltz-
g (x,t;v,K,a)
η
mann’s and Planck’s constants are k =h¯ =1.
Inthefollowingsection,wewillconBsiderageneralform ivβ π(vt+jηx ia) −(K−j)2/8K
= sinh − (1.1)
ofthe el-phcoupling, el−ph,butforthe purposesofthe πa vβ
present section, we coHnsider forward scattering interac- jY=±1(cid:26) (cid:20) (cid:21)(cid:27)
tions only,
Equation (9) is the central result of this paper. The
spectralfunctionmeasuredbyARPESisaccuratelygiven
=α u ρˆ=α 2/π u (∂ φ ), (6)
el−ph 2 2 x c
H by its Fourier transform.
where α2 is the el-ph couplingpparameter, ρˆis the long- Specifically, the single-hole spectral function for right-
wavelengthcomponentofthechargedensity,andthesec- moving fermions is25
ond equality makes use of the standard bosonization ex-
∞ ∞
pression for ρˆ. viet The forward scattering model is ex-
A(k,E)= dx dtei(kx−Et)G (x,t;λ). (12)
actly solvable since in its bosonized form it is quadratic 1
Z−∞ Z−∞
4
K =1, K =K =K , K =K∗, (18)
1 2 3 c 4 c
b. units) =a 0d.j6(u5as)ting ca =d 0ju.5s(tibn)g c va11==vas2/=vc,a¯/vβ¯2,=a3v3==a14,=v14/=β¯.vc∗/vc, ((1290))
E) (ar == 00..785 cc == 00..67 iAnbteogvrea,tix˜ona,nδdi3t˜isatrheedKimroennescikoenrledsesltdau,manmdya¯vari1abisleasdoi-f
k,F mensionless cutoff. The spectral function plo≪ts were ob-
A( tained by performing the double integration in Eq. (13)
0 0 numerically. Henceforth, we adopt the notation
-4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1
E E γ (K +K−1 2)/8=γ , (21)
c ≡ c c − 21
s) adjusting vc/vs adjusting 0/T which vanishes in the absence of el-el interactions.
unit (c) (d) A(k,E) depends onlyonk¯, E¯, ω0/T,andonthe three
E) (arb. vvvccc///vvvsss === 345 000///TTT === 51200 phdroimowveintdhseeiotnshpleeescrsterpaaadlrefarumnwecittteihornsaλd,qeupγace,nliadtanstdiovnvect/uhvnesds.2ee6rpsItanarnaomdrdienetgrertosof,
,F inFig. 2weshowhowtheEDC atk =k changeswhen
k F
A( one is varied with the rest held fixed. Henceforth, we
0 0 eitherchoserepresentativeparametersforthefigures,or,
-4 -3 -2 -1 0 1 -4 -3 -2 -1 0 1 forcasesinwhichthetheoryiscomparedtoexperimental
E E data, we fit the parameters. Not surprisingly, for cases
in which we havefitted the parameters,they turn out to
FIG. 2: Dependence of the EDC at k =kF on λ, γc, vc/vs,
and ω0/T [(a), (b), (c), and (d) respectively]. In each panel, be somewhat material dependent.
only one parameter is varied, and the rest are held fixed. Figure3(a)exhibitsthetemperaturedependenceofthe
Unless otherwise labeled, λ =0.75, γc =0.6, vc/vs = 4, and EDC at k = kF. The spectral weight of the low energy
ω0/T = 10. For example, for the curves in (a), λ has the peak is reduced by increasing T, while at the same time
value labeled in the plot legend, while γc = 0.6, vc/vs = 4, thewidthofthepeakincreasesproportionaltoT,dueto
andω0/T =10. In(a),thevaluesofvc/vc∗ =Kc∗/Kcare1.69, thequantumcriticalnatureoftheLL.Thevalueγ =0.6
c
2, 2.24 for λ equal to 0.65, 0.75, 0.8, respectively. In (b), Kc (K 0.15) used here indicates strong el-el repulsion.
is0.172,0.150,0.134forγc equalto0.5,0.6,0.7,respectively. Icn≈Fig. 4 we show the dependence of the EDCs on k¯,
Normalizations arechosenforgraphical clarity(theapparent
withandwithoutforwardscatteringcouplingtophonons,
variation of thetotal spectral weight is an artifact).
for ω /T = 10 and γ = 0.6. Note the “double-peak”
0 c
structure present near the phonon energy for moderate
By combining this with Eq. (9), one obtains values of k¯.
Figure 5 shows MDCs at various values of E¯ for the
A(k,E) β¯2 ∞ dx˜ ∞dt˜χ(x˜,t˜) same parameters as Fig. 4. A similar plot is shown in
≈ πa¯ω Fig. 6 at a lower temperature (ω /T = 40). Because
0 Z−∞ Z0 0
cos β¯(k¯x˜ E¯t˜) Θ(x˜,t˜) , (13) ofthelowertemperature,hereonecanresolvethreelocal
× − − maximaformoderatevaluesofE¯. AsshowninFig. 6(c),
where we defined the(cid:2) dimensionless quan(cid:3)tities β¯ = which is an enlargement of the E¯ = 1 curve, the cen-
−
ω β/π, a¯=aω /v , ter local maximum disperses at the renormalized charge
0 0 c
velocity v∗. Since this peak is by far the dominant one
k¯=vc(k kF)/ω0, E¯ =E/ω0, (14) for E¯ c1, the low-energy dispersion is characterized
− | | ≪
by what appears to be a single peak dispersing with ve-
and the functions
locity v∗. At higher E¯ , the v∗ peak disappears. If the
4 (a /v )2 γij/2 temperactureisincrea|sed|sufficiecntly,thethreelocalmax-
χ(x˜,t˜)= i i
sinh2(t˜+jx˜/v )+sin2(a /v ) ima can no longer be resolved and appear as one peak,
i=1j=±1(cid:20) i i i (cid:21) as seen in Fig. 5(a). Also note that phonon coupling
Y Y
(15)
creates larger spectral weight at the spinon velocity v ,
s
and
due to the highereffective K (this featuredisappearsat
c
4 tanh(t˜+jx˜/v ) high binding energies).
Θ(x˜,t˜)= γ arctan i , (16) In Fig. 7 we present contour plots in the E¯-k¯ plane,
ij
tan(a /v )
i=1j=±1 (cid:20) i i (cid:21) with and without phonon coupling, for the same param-
X X
eters as Fig. 4. Note the pronounced peak (red spot) at
with
low binding energies in Fig. 7(a), due to the increase in
(K j)2 the effective K . The reductionin chargevelocity at low
i c
γ =(1 2δ ) − , (17)
ij − i3 8K binding energies can also be seen here.
i
5
(a) (b) λ λ
0.5 = 0.72 (a) = 0 (b) 0.5
Luttingλer liquid s)
λ = 0.75 nit 0.0 0.0
u
= 0 b.
E) (arb. units) TT==7400KK nsity (arb. units) A(k,E) (ar---110...505 ---110...505
A(k,F Inte -2.0 -6 -4 -2 0 2 -6 -4 -2 0 2 -2.0
k k
T=100K
FIG.5: MDCs(a)withand(b)withoutcouplingtophonons
of strength λ = 0.72 for ω0/T = 10. All parameters are the
T=125K same as Fig. 4. The curvesare offset by E¯.
-0.20 0.00 Figure 8 shows the dispersion in the E¯-k¯ plane, de-
Energy [eV]
termined by fitting MDC curves to Lorentzian functions
FIG. 3: Comparison of the temperature dependence of the (the same method used in the ARPES literature). A
EDCs at k = kF for (a) a LL coupled to phonons and (b) change in the slope occurs at E¯ 1. The ratio of the
ARPES data of optimally doped Bi2212 (Tc = 89 K) at the slopeat E¯ 1totheslopeat E¯≈ − 1isapproximately
Fermi surface crossing along k = (0,0) to (π,π) (the nodal v∗/v =|√1|≪λ. Note that the| d|is≫persion, when deter-
direction) (Ref. 34). In (a), ω0 = 70 meV, γc = 0.6, and mcinecd by fitt−ing to Lorentzians, is weakly T dependent.
vc/vs =4. The solid line is for λ=0.75, and thedashed line
is the result for thepureLL.
III. GENERAL ELECTRON-PHONON
λ
COUPLING
λ = 0.72
= 0
k = 0
In general,both forwardand backwardscattering (i.e.
with momentum transfer near 2k ) are possible. Thus,
F
k = -0.4 in general, we should consider both processes:
units) k = -0.8 Hel−ph =u"α2 ρˆ+α1 Ψ†1,σΨ−1,σ+H.c. #. (22)
b. Xσ (cid:16) (cid:17)
ar
E) ( k = -1.2 Ibnostohniisccfiaesled,s,btehcaeupsreobΨlemis aisnnoontlienxeaacrtlfyunsoctlvioanbleo.f Wthee
A(k, thereforetreatthebackscatteringtermα1 withapertur-
k = -1.6 bative renormalization group scheme. It is important to
notethatifα issufficientlystrong,agapopensupinthe
1
spin sector, and the system is a Luther-Emery liquid27
k = -2
(LEL) instead of a LL (see Appendix B). We refer the
readertoRef. 28foradetailedstudyofthephasebound-
k = -2.4 ary separating the LL and LEL phases. Here we limit
ourselves to the case in which the spectrum is gapless.
It is convenient to define the dimensionless el-ph cou-
-6 -5 -4 -3 -2 -1 0 1 plings
E
α2 α2
FIG. 4: EDCs with (solid lines) and without (dashed lines) λ = 1 , λ = 2 . (23)
forward scattering coupling to phonons of strength λ = 0.72 1 πv Mω2 2 πv Mω2
(vc/vc∗ = Kc∗/Kc = 1.9), with γc = 0.6, vc/vs = 3, and F 0 F 0
ω0/T =10. The vertical dashed line is drawn at the phonon In Appendix B we derive the following relations, which
energy. The notation is k¯ = vc(k−kF)/ω0 and E¯ = E/ω0, are generalizations of Eq. (7) to the case of nonzero λ1:
where E is measured with respect to the Fermi energy. The
curveshavevertical offsets proportional to k¯. v∗ = v (1 Λ)(1 λ+K2Λ), (24)
c c − − c
p
6
λ λ 0.0
= 0.72 (a) = 0 (b)
s) 0.0 0.0 -0.5
nit
u
b. -0.5 -0.5 -1.0
ar E
E) (-1.0 -1.0 -1.5
k, 0/T = 10
A(-1.5 -1.5 -2.0 0/T = 40
-2.5
-3 -2 -1 0
-2.0 -2.0 k
-6 -4 -2 0 2 -6 -4 -2 0 2
FIG.8: Dispersionforλ=0.72,determinedbyfittingMDCs
k k
to Lorentzians, for ω0/T = 10 (solid line) and ω0/T = 40
vc* vc (dashed line). Interaction strengths are the same as in Fig.
ωk,-)0 vs (c) -vc 4d.ispTerhseiodnos.tted lines are fits to the high-|E¯| portion of the
A(
0
-4 -2 0 2 5
k
= 0.75
lFf(aocIbr)Gesωl.hs06o/t:wThseM=avDne4lCo0ecsnail(tanaired)gswetamhitteehwnshatanimocdhfe(tvibhna)etreiwEor¯uiat=cshtofi−eouant1tucscotrureurespvndlegiintsfhgrposetmroassep(.ahF)Ioi,ngna.o(nna4d)s. K , v/v*ccc34 = 0 vc/vc*
and (b) the curvesare offset by E¯. /* 2
Kc
Kc*/Kc
1
λ λ A(k,E)
(a) = 0.72 (b) = 0 (arb. units) 0
0.0 0.2 0.4 0.6 0.8 1.0
0 0
0.0
FIG. 9: Dependence of the magnitude of the kink (vc/vc∗)
0.5
-1 -1 and the degree to which the spectral function is peaked at
E 1.0 lowenergiescomparedtohighenergies(Kc∗/Kc)ontheeffec-
tivebackscatteringel-phinteractionparameterΛ,forthecase
-2 -2 Kc =0.15. The dashed lines are the result in the absence of
forward scattering el-ph interaction (λ = 0), while the solid
lines are for λ=0.75.
-3 -3
-10 -8 -6 -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 4
k k
ically interesting case E >ω , λ∗ depends on the ratio
FIG. 7: Contour plot for ω0/T =10 of the single-hole spec- F 0 1
E /ω and on the strength of the el-el interactions (see
tral function (a) with and (b) without forward scattering to F 0
phonons of strength λ = 0.72. All parameters are the same Appendix B for an explicit relation). As before, the for-
as in Fig. 4. ward scattering el-ph parameter is λ = (2KcvF/vc)λ2;
note that λ contains no asterisk because it remains un-
2
renormalized regardless of E /ω .29
F 0
1 Λ Equations (24) and (25) are actually completely gen-
K∗ = K − , (25)
c cs1−λ+Kc2Λ ewraayl,onforneplaetritnugrbtahteiveeffeecxtpivreespsiaornasm. etHerowΛevteor,thoeumr iocnroly-
wherewehaveintroducedtheeffectivebackscatteringel- scopic parameter λ is through Eq. (B6), which was ob-
1
ph parameter tainedfromone-loopRG,andisthereforevalidonlywhen
theel-elinteractionsareweakandλ 1. ButEqs. (24)
Λ= vF λ∗, (26) and(25) remainvalidevenif the RG1 ≪is carriedoutto an
2K v 1
c c infinite number of loops.
which lies in the range 0 Λ< 1. Here λ∗ is the renor- If fitting to particular ARPES data, the parameters
malized value of the bare≤el-ph backscatt1ering parame- K , K∗, and v∗/v are easily obtained. Then, λ and
c c c c
ter λ . For the case in which the el-ph interaction is Λ can be determined by inverting Eqs. (24) and (25).
1
unretarded (when the Fermi energy E < ω ), λ re- Therefore, for quasi-1D systems, ARPES is an effective
F 0 1
mains unrenormalized(λ∗ =λ ). However,for the phys- probe of the relative amounts of forward and (renormal-
1 1
7
ized) backwardscattering el-ph interactions.
Note that for nonzero Λ, the relation K∗/K = v /v∗ (a) (b)
c c c c
no longer holds; instead K∗/K = (1 Λ)v /v∗. For
K <1,the relationv∗ <v choldcs regard−lessocfΛcandλ.
c c c k = -0.4
However, K∗ > K holds only if the ratio Λ/λ is small
c c
wenhoiuchghm. Seapnescifithcaaltlyt,hiefΛlo/wλ->en1e/rg(1y+spKec2c)t,rathlefneaKtuc∗re<sKarce, nits) k = -1.2 units)
cvnilacolu/sesvlotc∗rKn,agictseers=inmtch0rae.ed1ae5rse.emdsuToblrthyeeotpfumeriannakigcnerngdeiatuudspiduneeΛgto,ΛofwtphahhtileoefinkKxoiennc∗dksi.,sλg,Freiifvdgoeuurncrteebhdy9e. k,E) (arb. u kk == --21.6 ensity (arb.
For all the spectral function plots in this paper, we have A( nt
set Λ=0. k = -2.4 I
k = -4
IV. COMPARISONS WITH ARPES
EXPERIMENTS IN THE CUPRATES
-400 0
E (meV) E (meV)
Inthissection,wewishtoillustratetheextenttowhich
theobservedARPESspectrainthecupratesuperconduc- FIG. 10: Comparison of EDCs for (a) a LL coupled to
phononsand (b)ARPESdataof slightly underdopedBi2212
tors resemble those of a LL coupled to optical phonons.
along the nodal direction (Tc = 84 K) at various momenta
At a gross level, the character of the resemblance be-
(Ref. 1). For(a),λ=0.72,γc =0.6,vc/vs =3,ω0 =70meV,
tween the observed spectrum and a pure LL was estab-
T =80K,and thedashed line is drawn at |E|=ω0.
lished in Ref. 5. The strength of this analogy is further
supported13 by direct comparison between the experi-
mentallymeasuredA(k,E)inaquasi-1Dbronzeandthe
( a) (b)
cuprates. However, as more and better data have be- 0
come available, it has become clear that there are fea-
tures in the cuprate data–especially the widely reported
“kinks” in the dispersion of the MDC peaks1,2,30,31,32–
tLhLatisaraelsqouaulnitaabtilveeltyoarbespernotdfurcoemtthhee“pnuordeaLlLq.uTahsieppaurtrie- E (meV)-0100 E (meV)
cles”seeninexperimentswhilesimultaneouslyproducing
broad features at high energy. Here, therefore, we pro-
pose to make a comparison between the measured spec-
tral functions, and the spectral function of a LL coupled -200
0 1 2 3
to an optical phonon.33 Since the LL describes a gapless -k
state, we only compare our results to data taken in the
FIG.11: Comparison ofthedispersionof(a)theLLcoupled
gapless “nodal” direction, which is defined as k = (0,0)
tophononswithω0 =70meV at T =30Kto(b)thedisper-
to (π,π). In other directions, the ARPES spectrum of
sioninBi2212(Tc =84K)atT =30Kinthenodaldirection
the cuprates develops a gap below a certain tempera- (Ref. 1). For (a), all interaction strengths are the same as
ture, whereas the spectrum remains gapless in the nodal in Fig. 10. For both plots, the dispersion was obtained by
direction even in the superconducting state. fittingMDCcurvestoLorentzians. In(b),theauthorsdefine
In Fig. 3 we have fitted the theoretical EDCs to the rescaled momentum k′, by normalizing to 1 the value of
ARPES data7 in optimally doped Bi Sr CaCu O kF−k atE =−170meV. Thedashedlinesareguidestothe
2 2 2 8+δ
(Bi2212)34 at the Fermi surface crossing in the nodal di- eye.
rection, shown at various temperatures, both above and
belowT . Theexperimentaltemperaturedependencehas
c
beenpreviouslyinterpretedasevidenceforquantumcrit- Figure 11(a) shows the theoretical dispersion for the
ical behavior. Since the LL is a quantum critical state, same parameters as Fig. 10(a), except with T =
the resemblance between theory and experiment in the 30 K, determined by fitting the theoretical MDCs to
present paper supports this interpretation. Lorentzians. Fig. 11(b) shows the experimental disper-
Figure 10 presents a fit to normal state EDCs for sion for slightly underdoped1 Bi2212 in the nodal direc-
slightly underdoped1 Bi2212 along the nodal direction, tion at T = 30 K, which the authors obtained with the
at various momenta. The theoretical curves contain a same fitting procedure. The bizarre feature of the ex-
similar double-peak structure as the experiment. The perimental data in which the dispersion at high binding
line shapes of both the theory and experiment are char- energiesdoesnot extrapolateto the originis alsoseenin
acterized by extremely long high energy tails. the theory.35
8
(a) (b) EDC
0.0 0.0 nits) (a) nits) (b)
u u
b. b.
ω--10..05E/0 -0.1E (eV) k,E) (arF nsity (ar
A( nte
I
-1.5
-1 ω 0 -0.2 -0.1 0.0
-0.2 E/ 0 E (eV)
-4 -2 ω 0 -0.4 -0.2 0.π0 0.2 MDC
A(a(rkb,.E -u)0nits)ω: E/0--100...v050c(k(c -) kF)/ 0 I(natrebn. 0suintyits): E (eV)-00k..10 - k(Fd ) ( /a) A(k,0) (arb. units) (c) ntensity (arb. units) (d)
I
1 -1.5 1 -3 0 ω 3 -0.3 0.0 π 0.3
-0.2 vc(k - kF)/ 0 k - kF ( /a)
-2 -1 ω 0 -0.08 -0.04π 0.00
vc(k - kF)/ 0 k - kF ( /a) FIG.13: EDCat k=kF for (a)LL coupledtophononsand
(b) underdoped nonsuperconducting LSCO (Ref. 36) (nodal
FIG. 12: Comparison of intensity plot of (a) the single-hole
direction);andMDCatE =0for(c)LLcoupledtophonons
spectral function of a LL coupled to phonons with γc = 1.5, and (d) underdopedLSCO (nodal direction). Panels (a) and
λ = 0.93, ω0/T = 20, and vc/vs = 3 to (b) the ARPES (c)areslicesfromtheplotinFig. 12(a);(b)and(d)areslices
spectrum of nonsuperconducting LSCO (3% doping) at T =
from the datain Fig. 12(b).
30K(Ref. 36). In(a)and(b),theredlineandpointsarethe
dispersion, obtained by fitting MDCs to Lorentzians. Panels
(c) and (d)show enlargements of thedispersions; thedashed
In Refs. 37 and 38, an effort is made to fit the ex-
lines are fitsto thehigh-binding-energy portion.
perimentaldispersionofthecupratestotheconventional
theory of el-ph coupling in a Fermi liquid. Both papers
reportthatgettingagoodfittotheexperimentalkinkre-
For the theory plots, the manner by which the disper-
quires coupling between electrons and a broad spectrum
sion is extracted (least-squares fitting to Lorentzians),
of phonon modes that extends all the way up to two or
yieldsa weightedaverageofthe fourvelocitieslabeledin
three hundred meV. As the authors point out, this is
Fig. 6(c). It is the contribution from the velocity v
c
− unphysical, since from neutron scattering, the highest-
thatis responsiblefor the plotted high-energydispersion
energy phonon mode is less than 100 meV in all of the
not extrapolating to the origin at E = 0 and k = k .
F
cuprates. This suggests that either phonons are not re-
Thisisbecause,whenthedispersionisdeterminedbyfits
sponsible for the kink, or if they are, that the correct
toLorentzians,thepresenceofspectralweightatk >k
F
theory is very different from the conventional Fermi liq-
[near the right side of the triangle in Fig. 12(a)] “pulls”
uid one. In the present paper, the phonon density of
the high-binding-energy dispersion to higher velocities.
states that couples to electrons is a single δ function at
Figure 12(a) shows a contour plot of the theoreti-
cal spectral function in the E¯-k¯ plane and compares it ω0(Einsteinphonon). Buteventreatingthephononden-
sity of states as adjustable, we suspect that a Fermi liq-
to ARPES data36 for underdoped, nonsuperconducting
uidtreatmentwouldnotbe ableto fit the frequencyand
La2−xSrxCuO4 (LSCO) with x=0.03 [Fig. 12(b)]. The
value of λ was chosen to give v /v∗ = K∗/K = 3.8, temperaturedependenceofthespectralfunction,northe
c c c c linear dependence of the MDC width on energy.7
which is the same as the ratio of the high to low binding
energyvelocitiesinthe experimentalplot. InFigs. 12(c)
and12(d) we show the dispersions,obtainedinthe same
manner as in Fig. 11. The EDCs and MDCs from these V. CONCLUSIONS
plots,attheFermimomentumandFermienergy,respec-
tively, are shown explicitly in Fig. 13. For both the the- Wehaveanalyzedtheeffectoftheel-phcouplingonthe
ory and experiment, the contrast between the sharpness single-particle spectral function of the theoretically best
of the MDCs and the breadth of the EDCs is dramatic. understood (and exactly solvable) non-Fermi-liquid, the
Note that the large values of γ and large values of LL. Since, by definition, a non-Fermi-liquid is a state in
c
λ used for all the theoretical plots above indicate the which the elementary excitations are not simply dressed
presence of very strong el-el and el-ph interactions, with electrons or holes, A(k,E) should be the measurable
aneffectiveel-phinteractionthatispeakedintheforward quantity in which non-Fermi-liquid effects are most dra-
scattering direction (since we used Λ=0). matic. Thus,itisimportanttohaveaclearideaofwhich
9
features of this function best distinguish a Fermi liquid a path integral over the bosonic fields Π , φ , Π , φ , P,
c c s s
from a non-Fermi-liquid. It has been argued previously5 and u. Since the Lagrangian is quadratic in all of these
that the most dramatic signature of a LL is the appear- fields, we can integrate out all fields except φ and φ .
c s
anceofextremelybroadtailsintheEDC,althoughpeaks This results in the Lagrangian = [ϕ ]+ [ϕ ] with
c c s s
L L L
in the MDC remain relatively sharp. (This signature
is particularly dramatic when the el-el interactions are 1 ω2v2q2
[ϕ ]= ω2 +v2q2 λ 0 c ϕ 2, (A1)
strong;whilethesamedistinctionappliesin principlefor Lc c 2K v n c − ω2 +ω2 | c|
c c (cid:20) n 0(cid:21)
weak interactions, in practice the LL is harder to distin-
1
guish from a Fermi liquid when the couplings are weak.) Ls[ϕs]= 2K v ωn2 +vs2q2 |ϕs|2. (A2)
s s
We have shown that some of the gross characteristic
(cid:2) (cid:3)
features of el-ph coupling in a Fermi liquid–an appar- Here the bosonic Matsubara frequency is ω = 2πn/β,
n
ent kink in the dispersion relations and EDCs that are the field ϕ = ϕ (q,ω ) is defined by φ (x,τ) =
α α n α
more peaked at |E| < ω0 compared to |E| > ω0–are (2πβL)−1 n dq e−iωnτ+iqxϕα(q,ωn), and L is the
also present in a LL with strong el-el repulsion and for- length of the system. The partition functional is then
wardscatteringcouplingtophonons. ForaLLwithmore givenby thPe paRth integralZ = ϕc ϕse−Sc[ϕc]−Ss[ϕs]
general el-ph couplings, the kink is still present, but the with S [ϕ ] = (2πβL)−1 Ddq D[ϕ ]. The zero-
tendency of the EDC to be more peaked for E < ω0 temperaαturαe dispersion relantioRns Lfoαr tαhe hybridized
is eliminated by sufficiently strong el-ph backsc|at|tering. charge-phononbosonic collePctivRe modes are thus
However, in either case, the Fermi liquid analogy can-
not be taken too far: The basic discrepancy between the 1/2
ω2+v2q2 (ω2 v2q2)2+4λω2v2q2
sharpness of the MDCs and the breadth of the EDCs, ω = 0 c ± 0 − c 0 c .
±
2
and the “triangular” confinement of spectral weight in " p #
the E-k plane, remain striking aspects of the LL which (A3)
differentiate it from the Fermi liquid.
We have also shown examples of measured spectral Note that for λ = 0, ω = v q is the long-wavelength
− c
functions in the cuprates, and drawn attention to the density mode of the LL, and ω = ω is the Einstein
+ 0
similaritiesbetweenthemandourtheoreticalresults. We phonon mode. In Fig. 14(a), we plot ω and ω for
− +
believe that the similarities are dramatic. Exactly why various values of λ.
the measured spectral functions look so much like the Itisageneralfeatureofthebosonizationapproachthat
LL coupled to phonons is a deep question, which we will the fermionic correlation functions are most simply ex-
not explore further, here. However,at the very least, we pressed as a function of space and time. The Matsubara
feelthatthiscomparisonreinforcestheconclusion,which space-timeGreen’sfunctionwaspreviouslycomputedfor
has certainly been reached4,5,6,7 on the basis of a variety a LL with forward scattering interactions with acoustic
of other experimental observations in the cuprates, that (instead of optical) phonons.39,40 The analytic structure
these materials are not well described in terms of the of Eq. (A1) differs from the acoustic phonon model ac-
conventional electronic quasiparticles of simple metals. cordingtoω cq,wherecisthevelocityoftheacoustic
0
→
phonon. The initial steps for computing the space-time
Green’sfunctioninvolvingfunctionalintegrationoverthe
Acknowledgments fieldsandMatsubarafrequencysummationsarethesame
for our model, except for this “parameter” change. We
We wish to thank J. Allen, N. P. Armitage, P. John- will therefore not reproduce these steps here, but rather
son, T. Valla, S. Sachdev, G. H. Gweon, E. Arrigoni, D. refer the reader to Ref. 39. The final momentum inte-
Orgad,Z.-X. Shen, V. Oganesyan,J.Tranquada,andA. gral done in the exponent, however, is different for the
Lanzara for useful discussions, and X. J. Zhou for sup- two models.
plying the data in Fig. 12(b). This work was supported, We write the single-particle imaginary-time Green’s
in part, by the National Science Foundation Grant No. function for right moving fermions (x,τ;λ) =
G
DMR 01-10329 (S.A.K.) and by the Department of En- T Ψ (x,τ)Ψ† (0,0) , where T is the imaginary
ergy Contract No. DE-FG03-00ER45798(I.P.B.). − τ 1,σ 1,σ τ
timDe ordering operator, asE
eikFx
APPENDIX A: THE SPECTRAL FUNCTION (x,τ;λ)= exp[ fc(x¯,τ¯;λ) fs(x¯,τ¯)] (A4)
G − 2πa − −
AND CONDUCTIVITY OF THE EXACTLY
SOLVABLE MODEL (results for left-moving fermions are obtained by chang-
ingx x). Heref andf arethechargeandspincon-
c s
→−
We nowderivethe spectralfunctionofthe modelwith tributions to the exponent, which we will write in terms
purely forward scattering interactions. We work in the of the dimensionless variables x¯ = xω /v and τ¯ = ω τ.
0 c 0
Matsubara representation, in which τ = it is the imagi- To simplify notation, for now we express the result in
nary time, and the partition functional is represented as the limit T 0. For τ > 0 and infinitely large system
→
10
length, the exact result is where A (k,E;v ,K ,v ,K ) = A(k,E;0) is the spec-
LL c c s s
tral function of the pure Luttinger liquid.
∞
f (x¯,τ¯;λ) = dq¯e−a¯q¯ [1 cos(q¯x¯)e−ω¯ντ¯] For the physically interesting case a¯ 1, we were
c Z0 ν=±{Aν − abletoderiveanaccurateanalyticapproxi≪mationforEq.
X
i sin(q¯x¯)e−ω¯ντ¯ , (A5) (A5), which is the following:
ν
− B }
f (x¯,τ¯;λ) f (x¯,τ¯;0)
c c
fs(x¯,τ¯) = F(x¯,τ¯;vs/vc,Ks,a¯), (A6) +F(x¯,τ¯;v≈∗/v ,K∗,1) F(x¯,τ¯;1,K ,1), (A16)
c c c − c
where a¯=aω /v and we defined the functions
0 c
which gives for the Green’s function
(K2+1)(ω¯2 1)+λ
A± = 4cKcω¯±(ω¯±±2 −−ω¯∓2) , (A7) G(x,τ;λ) ≈G(x,τ;0)HH((xx,,ττ;;vvc∗,,KKc∗,,vvc//ωω0))
ω¯2 1 c c c 0
= ±− , (A8) fora¯ 1, (A17)
B± 2q¯(ω¯2 ω¯2) ≪
±− ∓
ω¯±2 = 1+q¯2± (12−q¯2)2+4λq¯2 , (A9) wsinitghlev-hc∗oalendreaKl-c∗timgieveGnraegenai’ns fbuyncEtiqo.n (i7s)w. rTittheen,Tu6=sin0g
p this approximation,in Eq. (9).
and We demonstrate the accuracy of this analytic approx-
1 1 x¯2+(a+vτ¯)2 imation by comparing Eq. (A16) with the exact result,
F(x¯,τ¯;v,K,a) = K+ ln obtained by performing the integration in Eq. (A5) nu-
8 K a2
(cid:18) (cid:19) (cid:20) (cid:21) merically. From Figs. 15(c) and 15(d) we see that for
i x¯
arctan . (A10) λ = 0.75 and γc = 0.6, the approximation gives within
− 2 a+vτ¯ 9%ofthe exactresultfor the realpartoff (x¯,τ¯;λ). For
(cid:20) (cid:21) c
λ = 0.25 [Figs. 15(a) and 15(b)], the error is less than
Forλ=0,thechargeandspinpartsarethesameafter
3%. The agreement with the imaginary part is similarly
an appropriate change of parameters:
excellent.
f (x¯,τ¯;0)=F(x¯,τ¯;1,K ,a¯), (A11) We have also computed the frequency- and
c c
momentum-dependent conductivity for the LL with
yielding the known24 Green’s function of a pure LL forward scattering off phonons. The exact result for the
real part of the conductivity at T =0 is
eikFx
(x,τ;0)= H(x,τ;v ,K ,a), (A12)
α α
G − 2πa αY=c,s σ1(q,ω;λ) = e2Kcvc [Wδ(ω−νω+)
ν=±1
where X
+ (1 W)δ(ω νω )]. (A18)
−
− −
H(x,τ;v,K,a)
a2 (K−1)2/8K a Hanedreteheisstpheeccthraalrgweeoigfhttheofeltehcetrωon,mδiosdteheisdeltafunction,
= .(A13) +
(a+vτ)2+x2 a+vτ ix
(cid:20) (cid:21) r − ω2 ω2
For arbitrary values of the parameters, although it is W = +− 0 , (A19)
ω2 ω2
straightforwardtoevaluateEq. (A5)numerically,itdoes +− −
not appear possible to perform the integration analyti-
which depends only on v q/ω and λ. We plot W in
cally. We were, however, able to perform it analytically c 0
Fig. 14(b). Note that the conductivity at q =0 (optical
for the case of arbitrary λ but a¯ 1:
≫ conductivity) remains unchanged by forward scattering
eikFx off phonons:
(x,τ;λ)= H(x,τ;v∗,K∗,a)H(x,τ;v ,K ,a)
G − 2πa c c s s
σ (0,ω;λ)=σ (0,ω;0)=2e2K v δ(ω), (A20)
fora¯ 1, (A14) 1 1 c c
≫
with v∗ and K∗ given by Eq. (7). The limit a¯ 1 yet this same interaction produces dramatic changes to
is typiccally notcsatisfied in real materials, but since≫the the spectral function.
spectral function is independent of a¯ for k¯ , E¯ 1/a¯,
| | | | ≪
we can use Eq. (A14) to deduce the exact behavior of
the spectral function in the limit k¯ , E¯ 1, regardless APPENDIX B: RG TREATMENT OF GENERAL
| | | |≪ ELECTRON-PHONON COUPLING
of the value of a¯:
A(k,E;λ)∝ALL(k,E;vc∗,Kc∗,vs,Ks) for|E|≪ω0, In the case in which the el-el couplings and the
(A15) backscattering el-ph coupling λ [Eq. (23)] are weak,
1