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Fermi surface dichotomy in systems with fluctuating order

M. Grilli,1, 2 G. Seibold,3 A. Di Ciolo,1, 2 and J. Lorenzana2,4

1Dipartimento di Fisica, Universita di Roma “La Sapienza”, P.le Aldo Moro 5, I-00185 Roma, Italy2SMC-INFM-CNR, Universita di Roma “La Sapienza”, P.le Aldo Moro 5, I-00185 Roma, Italy

3Institut fur Physik, BTU Cottbus, PBox 101344, 03013 Cottbus, Germany4ISC-CNR, Via dei Taurini 19, I-00185, Roma, Italy

(Dated: January 30, 2009)

We investigate the effect of a dynamical collective mode coupled with quasiparticles at specificwavevectors only. This coupling describes the incipient tendency to order and produces shadowspectral features at high energies, while leaving essentially untouched the low energy quasiparticles.This allows to interpret seemingly contradictory experiments on underdoped cuprates, where manyconverging evidences indicate the presence of charge (stripe or checkerboard) order, which remainsinstead elusive in the Fermi surface obtained from angle-resolved photoemission experiments.

PACS numbers: 74.72.-h, 74.25.Jb, 71.18.+y, 71.45.-d

I. INTRODUCTION

Strongly correlated systems like the heavy fermionsand the superconducting cuprates display excitationsover a variety of energy scales. On short timescales (highenergies) electrons are excited incoherently over energiesranging from the highest local (Hubbard U) repulsion tothe magnetic superexchange interaction. On the otherhand, over long timescales, i.e. at low energies around theFermi level, the excitations can acquire a coherent char-acter typical of the long-lived Fermi-liquid quasiparticles(QPs). These energy scales appear very clearly both intheoretical1 and experimental2 studies of the one particlespectral function of strongly correlated systems. It is usu-ally assumed that the incoherent part has no momentumstructure, an assumption which is emphasized by infinitedimensional studies where the self-energy is momentumindependent and spatial informations on quasi long rangecorrelations close to a phase transition are lost.1

In this paper we discuss how this picture is modified inphysical spatial dimensions. We want to address how thespectral function looks like when the system is close toan ordered phase. This issue is particularly important inthe context of cuprates where it has been proposed thatsome kind of stripe-like order fluctuates in the metallicphase.3,4,5

The scenario that we propose is based on the followingqualitative argument: In physical dimensions a systemmay have long (but finite) ranged order parameter spa-tial correlations which are also long lived close to a quan-tum critical point. This defines a fluctuating frequencyω0 above which the systems appears to be ordered. Weargue that for energies larger than ω0 with respect to theFermi level the spectra should resemble the spectral func-tion of an ordered system. This spectral weight residesin what is usually called the incoherent part, which weargue, can have some important momentum structure.On the other hand at lower energies electrons averageover the order parameter fluctuations and “sense” a dis-ordered system. In this limit we expect Fermi-liquid QPswith all their well known characteristics like a Luttinger

Fermi surface (FS).

To understand the momentum structure of the spectralfunction at energies higher than ω0 is important becauseif the incoherent part carries a memory of the close-byordered phase it should be possible to analyze it to ob-tain informations on what is the underlying fluctuatingorder. Usually ordered systems are well described bymean-field thus one can obtain a first guess of how theincoherent part of the spectral function in the disorderedphase should look like by performing a mean-field com-putation assuming long-range order. Comparison withexperimental data in the absence of long range order canbe useful to identify the fluctuating order parameter.

To fix ideas consider as an example a moderately largeU Hubbard system in a half-filled bipartite lattice in twodimensions at T = 0. In this case an antiferromagneticstate is expected to be a competitive low-energy state.When the system is in the ordered phase the spectralfunction will be reasonably well described by a mean-fieldcomputation and will show two Hubbard bands separatedin energy by mU with m the staggered magnetization.The bands will show some dispersion governed by thescattering of the electrons with the mean-field potential.Suppose that due to some frustrating effect long rangemagnetic order is lost while keeping well formed magneticmoments. We expect that beyond mean field if U is nottoo large (so that the disordered phase is metallic) a QPwill appear at the center of the Hubbard bands with smallspectral weight resembling the dynamical mean field the-ory (DMFT) picture.1 At high energies, however, elec-trons will sense a mean-field-like staggered potential fordistances of the order of the correlation length ξ, whichcan be quite long, and therefore the system will keep sub-stantial memory of the mean-field like bands with theirdispersion. Roughly we expect that the spectral func-tion will look like the superposition of a Fermi-liquid-likespectral function, with a small weight z close to the Fermilevel, plus a blurred mean-field-like spectral function inthe presence of long-range order with a large weight 1−z.This is at first sight similar to the DMFT picture but itdiffers in that in DMFT there are no magnetic correla-

2

tions surviving in the disordered phase and the incoher-ent part becomes momentum independent. We will showthat in finite dimensions the incoherent part carries im-portant informations encoded in the momentum depen-dence which within a DMFT approach would require thecluster extensions developed more recently.6

FIG. 1: Schematic view of heavy-fermion system with aKondo-like resonance arising at the Fermi energy EF from themixing of a deep narrow f level (not shown) and the conduc-tion band (dashed line). Two QP bands arise E1,2(k). Thecorresponding momentum distribution function nk is shownbelow with a true Fermi momentum kQP

F and a “fictitious”Fermi surface at kc

F .

Another example can clarify the concept of an inco-herent part with a strong momentum dependence whichcarries physical information on the short range physics.Lets consider the more standard issue of large and smallFSs in heavy fermions represented in Fig. 1. In heavyfermions strongly correlated electrons in a narrow half-filled f level hybridize with electrons in a conductionband and give rise to a Kondo resonance at the Fermilevel formed by coherent QP states. The width (andweight) of this QP band is usually quite small and setsthe scale of the coherence energy in these systems. Nowconsider the momentum distribution function defined by:

nk ≡

∫

dωA(k, ω)f(ω), (1)

where f(ω) is the Fermi function and the spectral density

A(k, ω) ≡1

πImG(k, ω) =

1

π

Σ′′(k, ω)

(ω − Σ′(k, ω))2 + Σ′′2(k, ω)(2)

is proportional to the imaginary part of the electronGreen function with real (imaginary) part of the self-energy Σ′ (Σ′′). It is crucial to recognize that nk involves

all the excitation energies and its features might be dom-inated by the incoherent part of the spectrum if the QPshave a minor weight. Indeed strictly speaking the trueFS at zero temperature is given by the small jump in theFermi distribution function determining the Fermi mo-

menta of the QPs at kQPF . This FS is large and satisfies

the Luttinger theorem with a number of carriers includ-ing the electrons in the f level. This FS would naturallybe determined by following the QP dispersion. On theother hand the shape of nk is substantially determinedby the (incoherent) part of the spectral function, whichhas strong weight at energies corresponding to both thef level and the conduction band. This latter gives rise toa rather sharp decrease of nk at a “fictitious” Fermi mo-mentum kc, corresponding to the FS that the electronsin the conduction band would have in the absence of mix-ing with the f-level. If the hybridization between the flevel and the conduction band is turned off so that theQP weight z is driven to zero one reaches a situation inwhich the decrease at kc becomes a discontinuity and the

small jump at kQPF disappears. It is clear that the sharp

decrease of spectral weight at kc for finite hybridizationhas strong physical content for an observer who ignoresthe underlying model. Mutatis mutandis this shows thata computation in which the fluctuating order is artifi-cially frozen can give some hints on the distribution ofthe incoherent spectral weight in the less trivial case withfluctuations.

The above ideas but with a more complicated order pa-rameter may explain perplexing data on cuprates. Sev-eral years ago underdoped La2−xSrxCuO4 (LSCO) com-pounds were examined and a dichotomy was found inthe Fermi surface (FS) determined by two different treat-ments of the data. On the one hand the momentum de-pendence of the low-energy part of the energy distribu-tion curves was followed, thereby reconstructing the low-energy QP dispersion. In this way a large FS was foundcorresponding to the Fermi-liquid LDA band-structureand fulfilling the Luttinger requirement that the volumeof the FS encircled the whole number of fermionic carri-ers n = 1−x. On the other hand the FS was determinedfrom the momentum distribution nk, obtained by inte-grating the spectral function over a broad energy window(∼ 300 meV). Then the locus of momentum-space pointswhere nk displays a sharp decrease, marked a FS formedby two nearly parallel (weakly modulated) lines alongthe kx direction and crossing two similar lines along theky direction. This crossed FS would naturally arise in asystem with one-dimensional stripes along the x and ydirections.

As a matter of fact the stripe Fermi surface has beenobserved both in systems which show a striped groundstate and in systems where long-range stripe order hasnot been detected.7,8 As in the heavy Fermion case forrelatively small z we expect that nk is dominated by theincoherent spectral weight. According to our scenario nk

should resemble the Fermi surface of stripes in mean-fieldregardless of whether stripe order is static or fluctuat-

3

ing at low frequencies. Indeed LDA computations in thepresence of stripes with magnetic and charge long-rangeorder reproduce this Fermi surface.9

The experimental stripe-like Fermi surface is not flatas could be expected for a perfect one dimensional bandstructure but shows some wavy features, which dependon the details of residual hopping processes perpendicularto the stripes. Remarkably the wavy features are wellreproduced by the LDA computation both with regard toamplitude and periodicity, giving credibility to the ideathat the static LDA computation provides a snapshot ofthe fluctuating order in the disordered phase (cf. alsoRef. 10 for a more phenomenological approach).

The main point is that the formation of QPs due tosome coherence effect is a small perturbation for the over-all distribution of spectral weight. Thus a careful studyof nk can give precious information on the proximity tosome ordered phase. By the same token the presenceof QPs and a Fermi-liquid-like Fermi surface are not in-compatible with two-particle responses (neutrons, opti-cal) which show strong features of ordering (like stripes)at frequencies above ω0. This explains why computationsof fluctuations on top of stripe phases with long rangeorder11,12 explain well optical conductivity and neutronscattering data of systems without long-range order. Italso explains how nodal quasiparticles can coexist withfluctuating stripe order.

In the following section we present a toy model of theKampf-Schrieffer-type13 where the dichotomy between aFermi-liquid-like Fermi surface and a momentum depen-dent incoherent part, reflecting the fluctuating order, canbe illustrated. Although we study self-energy correctionsself-consistently, the lack of vertex corrections makes ourcomputations reliable only in weak coupling where QPweights are close to one. Within this limitation one canshow that the described scenario holds.

In Sec. II we present the model, while in Sec. III wepresent some numerical results. A discussion of the re-sults and our conclusions are reported in Sec. IV.

II. THE MODEL

In order to substantiate the above ideas we consider asystem of electrons coupled to a dynamical order parame-ter which can describe charge ordering (CO) fluctuationsor spin ordering (SO) fluctuations or both. To fix ideaswe consider CO fluctuations described by an effective ac-tion

S = −g2∑

q

∫ β

0

dτ1

∫ β

0

dτ2χq(τ1 − τ2)ρq(τ1)ρ−q(τ2).

(3)In order to simplify the calculations we consider a

Kampf-Schrieffer-type model susceptibility13 which isfactorized into an ω- and q-dependent part, i.e.

χq(iω) = W (iω)J(q) (4)

The (real-frequency)-dependent part is W (ω) =∫

dνFω0,Γ(ν)2ν/(ω2 − ν2), with Fω0,Γ being a normal-ized lorentzian distribution function centered around ω0

with half-width Γ, Fω0,Γ(ω) ∼ Γ/[(ω − ω0)2 + Γ2]. The

momentum-dependent part in D dimensions reads

JD(q) = ND

∑

η=1

∑

±Qη

γ

γ2 + 1 − cos(qη − Qη). (5)

N is a suitable normalization factor introduced to keepthe total scattering strength constant while varying γ ∝ξ−1 (with ξ the CO/SO correlation length). To simplifythe treatment and to make the effect of fluctuations asclear as possible, we will mostly consider the case of spa-tially coherent fluctuations. Although formally the cor-relation length ξ diverges, the ordering is not static andwe will show that this is enough for the spectral functionto converge to the Fermi-liquid Fermi surface at zero fre-quency.

The infinite correlation length case is described in Ddimensions by,

JD(q) =1

4

D∑

η=1

δ(qη − Qη) + δ(qη + Qη). (6)

In previous works the spatially-smeared version of JD(q),Eq. (5), was considered to describe the kink inthe electron dispersions14 and the (still experimentallycontroversial15,16) isotopic dependence of these disper-sions.17 The susceptibility χ contains the charge-chargecorrelations in the case of charge fluctuations and spin-spin correlations in the case of magnetic fluctuations.

If only the dynamical part W (ω) were present inχq(iω), one would have a bosonic spectrum B(ω) =tanh(ω/(kT ))Fω0,Γ(ω) which is a “smeared” version ofthe Holstein phonon considered in Ref. 18. The crucialfeature of the susceptibility (4) is the substantial momen-tum dependence, which describes the (local) order for-mation and reflects the proximity to an instability withbroken translational symmetry.

The static limit Fω0,Γ(ν) = δ(ν) together with an infi-nite charge-charge correlation length (γ → 0) as the oneconsidered in Eq. (6) reproduces mean-field results for along-range phase.13,19 The static limit of (4) has been theobject of an intense activity based on the idea, pioneeredin Ref. 20, that different types of slow order-parameter(OP) fluctuations (SO13 or CO19) can be treated asclassical fluctuations with time-independent correlations.The equivalence of such static degrees of freedoms withquenched impurities has more recently been formalizedand cast in field-theoretical language.21 Although theyallow for (nearly) exact solutions, the main drawback ofthese static approaches is that they do not allow for theaimed separation of energy (i.e. time) scales since tojustify the static character of the OP fluctuations, theyassume that their relaxation time τOP is much longerthan the inelastic scattering rate of the electrons τe,

4

τOP ≫ τe.21 Here we consider instead

Fω0,Γ(ν) = δ(ν − ω0) (7)

representing a dynamical fluctuation oscillating at a fre-quency ω0 and therefore averaging out on timescaleslarger than τOP ∼ 1/ω0.

In the present limit the problem has also a simpleHamiltonian formulation which we introduce for lateruse:

H =∑

kσ

ξkc†kσckσ + ω0(b†QbQ + b†−Qb−Q) (8)

+g∑

kσ

[

c†k−Qσckσ(b†Q + b−Q) + c†k+Qσckσ(b†−Q + bQ)]

where c†kσ creates a free Fermion and b†±Q creates abosonic collective mode excitation and ξk ≡ εk − µ withεk the non interacting dispersion relation and µ the chem-ical potential.

QPs at energies much larger than ω0 see an essen-tially static fluctuation and modify their dispersion asin the mean-field calculation mentioned above. For themthe fluctuations are static and, if the momentum depen-dent part J(q) is strongly peaked around the orderingwavevector Q, they are scattered like in the presence ofa long-range symmetry breaking. Their dispersions aremodified accordingly. On the other hand, for QPs at en-ergies in a shell of width ω0 around the Fermi level theorder parameter fluctuations are far from being staticand have not enough energy to excite them. For theseQPs the system is still in a uniform Fermi-liquid metal-lic state. Therefore, if their dispersion is followed nearthe chemical potential µ, a large FS fully preserving theLuttinger volume is found. In the following we explicitlycalculate the band dispersions and the FS to show thisphysical mechanism at work and the resulting dichotomyof the FS.

Neglecting vertex corrections we iteratively solve atzero temperature the following coupled set of equationsfor the self-energy and the Green’s function

Σ(k, iω) = −g2

β

∑

q,ip

χq(ip)G(k − q, iω − ip) (9)

G(k, iω) =1

iω − ξk − Σ(k, iω). (10)

To the best of our knowledge our self-consistent treat-ment goes beyond previous solutions that remained atthe perturbative level. The latter can be obtained atlowest order by replacing the Green’s function in Eq. (9)by the non-interacting G0. For our specific model of thesusceptibility the self-energy becomes:

Σ(1)(k, iω) = g2

f(ξk+Q)

iω + ω0 − ξk+Q

+1 − f(ξk+Q)

iω − ω0 − ξk+Q

(11)where the superscript (1) denotes the order of the iter-ation and f(ξk+Q) = Θ(−ξk+Q) is the occupation num-ber. Inserting this self-energy into Eq. (10) leads to

a Green’s function G(1)(k, iω) which displays now twopoles and which can be used to compute Σ(2)(k + Q, iω)(and thus G(2)(k + Q, iω)) and so on. The detailed pro-cedure is described in the Appendix.

For the sake of simplicity, we first implement our nu-merical analysis in one dimension and then we describethe case of two dimensions, which is more relevant forlayered materials.

III. NUMERICAL RESULTS OF

FLUCTUATING ORDER

A. Numerical analysis in one dimension

Coherent order parameter fluctuations

We first present the results for a one-dimensionalmodel with a commensurate ordering wavevector Q = πcorresponding to CO/SO with a doubling of the unit cell.In particular we consider a band of QPs εk = − cos(k)(we choose a unit hopping parameter t = 1 and a uni-tary lattice spacing) coupled via a CO/SO mode givenby Eqs. (4) and (6). Fig. 2 reports the simplest case inwhich only two poles are considered in the Green’s func-tion given by G(1) for a generic filling (n = 0.67: hereand in the following densities are defined as total num-ber of particles per site). This is the standard mean-field

FIG. 2: (Color online) Upper panel: Two-pole band structurein one dimension for g = 0.5t, ω0 = 0 and particle densityn = 0.67. The width of the curve is proportional to theweight of the state and energies are measured with respect toEF . Lower panel: Momentum distribution curve.

solution, with a doubling of the unit cell and a foldingof the bands resulting in a double FS. In the absence

5

of interaction the Fermi momentum would be given bykF = nπ/2 = 1.05. The dispersion relation instead dis-plays two Fermi points and a very different Fermi surfacevolume, “violating” the Luttinger theorem.

The situation is very different for a dynamical fluctu-ation. In Fig. 3 we report the spectral function withω0 = 0.3 and moderately weak coupling g2/ω0 = 0.5.Also in this case shadow bands appear so that the elec-tronic structure has bands close to the location of thebands in the broken symmetry state. The shadow bands,however, do not reach the Fermi level and one has onlyone FS point at k = kF . Therefore the Luttinger theo-rem is satisfied since the divergent pole in G(k2

F , ω = 0)from the static solution now turns into a step so thatG(k, ω = 0) < 0 for k > kF .

The spectral function can be written in the Lehmannrepresentation on the real axis as:

A(k, ω > µ) =∑

ν

|〈φN+1ν |c†kσ|φ

N0 〉|2δ(ω − EN+1

ν + EN0 )

(12)

A(k, ω < µ) =∑

ν

|〈φN−1ν |ckσ|φ

N0 〉|2δ(ω − EN

0 + EN−1ν )

A(k, ω) has structure at the energies of the excitationof the system with one added particle (ω > µ) or oneremoved particle (ω < µ). The result of Fig. 3 can beunderstood from the low energy excitation of the systemwhen g = 0. These are listed in Table I and the cor-responding excitation energies are shown in Fig. 4. Forg = 0 all the weight is in the main band labeled ξk be-cause the matrix elements in Eq. (12) vanish when oneboson is present. The effect of a finite g is to give somespectral weight to the “shadow” band at ξk−Q±ω0 and tointroduce some level repulsion when the bands cross. Theimportant point is that the shadow band never touchesthe Fermi level but it is separated from it by the energyto create the bosonic excitation. We associate the mainband with the QP band and the shadow band with theincoherent spectral weight. Clearly close enough to theFermi level only the QP band exists.

Addition ω > 0State EN+1

ν − EN0 Momentum

c†kσ|φN0 〉 ξk k

c†k−Qσb†Q|φN0 〉 ξk−Q + ω0 k

Removal ω < 0State EN

0 − EN−1ν Momentum

c−kσ|φN0 〉 ξ−k k

c−k+Qσb†−Q|φN0 〉 ξ−k+Q − ω0 k

TABLE I: Low energy excitations in the g = 0 limit. Thecentral column shows the excitation energy. Notice thatξ−k = ξk.

Notice that the shadow bands are quite similar to thecase of a symmetry-broken state (but for a shift in energy

of order ω0 = 0.3, which is small on the scale of thehopping t = 1).

Because the QP states at the Fermi level are negligiblyaffected by the scattering with the dynamical orderingmode, the FS (here represented by points) is preservedand no shadow branches appear at low energy. There-fore already this simple weak-coupling case shows thatan ordered state would be inferred from the presence ofshadow bands at high energies, while the low-energy QPare characteristic of a uniform state.

FIG. 3: (Color online) Upper panel: Two-pole band structurein one dimension for g2/ω0 = 0.5, ω0 = 0.3 and electrondensity n = 0.57. The width of the curve is proportionalto the weight of the state and energies are measured withrespect to EF . Lower panel: Momentum distribution curve.For k > 2 the occupation number nk has been scaled by afactor 5 to enhance the visibility.

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-0.5

0.0

0.5

1.0

1.5

k

Ω

ΞkΞk-Q +Ω0

Ξk-Q -Ω0

FIG. 4: (Color online) Schematic electronic structure for g =0.

6

In Fig. 3 we also report the momentum distributionnk. From this quantity one can see that some weightis indeed present at high energy via the shadow bandappearing at momenta above k∗ ≈ 2.3. This is a sig-nature of some ordering in the system. However, sincethe CO/SO is dynamical the dispersion discontinuouslyjumps from above to below the Fermi level without pro-ducing an additional branch to the FS and with no vio-lation of the Luttinger theorem. In this last respect weexplicitly checked that the Green function at zero fre-quency is positive for k < kF , it changes sign through apole at kF and stays negative all the way up to π. TheFermi momentum kF is the same that one would havein a non-interacting tight-binding system with the samenumber of particles kF = nπ/2 with n = 0.57 in thiscase.

The above picture remains valid if additional polesare considered in the Green function provided the QP-CO/SO-mode coupling is in a weak-to-moderate regime.Fig. 5 reports the case of three poles for the same param-eters of Fig. 3. The increased allowed number of poles

FIG. 5: (Color online) Upper panel: Six-pole band structurein one dimension for g2/ω0 = 0.5 and ω0 = 0.2. The electrondensity is n = 0.55. The width of the curve is proportionalto the weight of the state and energies are measured withrespect to EF . Lower panel: Momentum distribution curve.For k > 2 the occupation number nk has been scaled by afactor 5 to enhance the visibility.

better represents the shift of spectral weight at high en-ergies induced by multiple shadow bands. Nevertheless,the finite frequency of the mode protects the low-energyQPs from being scattered and leaves the low-energy spec-trum unaffected: The FS is still formed by just two pointsdespite the appearance of several shadow bands at highenergy typical of a CO/SO state. The effect of the selfconsistency is to “blur” the shadow bands but still the

incoherent spectral weight retains a strong momentumdependence.

Upon increasing the QP-CO/SO mode coupling g oneeventually enters a regime where our non-crossing pertur-bative scheme neglecting vertex corrections breaks downand the Luttinger theorem is violated. This situationis reported in Fig. 6, where a second branch of the FSappears due to the strong bending down of the band atmomenta k ≈ k∗. In this case the Green function at zerofrequency G(ω = 0) changes sign twice (and diverges) atthe two Fermi momenta kF1 and kF2 while it changessign passing through zero at k = k∗, where the jump inthe dispersion below µ signals a divergence of the self-energy (11). One can check explicitly that the volumecorresponding to a positive G(ω = 0) is larger than theone given by occupied states in the non-interacting sys-tem thus Luttinger theorem is not satisfied. We believethis is an artifact due to the lack of vertex correctionswhich become important as the coupling is increased.

FIG. 6: (Color online) Two-pole band structure in one di-mension for g2/ω0 = 2. and ω0 = 0.3. The electron densityis n = 0.83. The width of the curve is proportional to theweight of the state and energies are measured with respect toEF . Lower panel: Momentum distribution curve.

The apparent success of this simple theory on illus-trating the spectral function of a system with fluctuatingorder is encouraging. However, a different problem arisesif one considers the half-filled case. Without long rangeorder we would expect a metallic state if we were able tosolve the model exactly. On the contrary commensuratescattering in the perturbative solution produces the un-physical result of a gaped FS in contrast with the factthat no true broken symmetry is present in the system.On the other hand it is not surprising that our perturba-tive approach fails when our singular interaction χ(q, ω)connects two degenerate states at the FS as it occurs in

7

this case. We believe this failure is due to the lack ofvertex corrections which we expect to suppress the scat-tering at the Fermi level. We have tested this idea phe-nomenologically by assuming that the QPs at the Fermilevel are protected against this singular scattering by amomentum dependence of the coupling of the form

g(εk, εk+q) = g tanh(εk

ω0) tanh(

εk+q

ω0) (13)

One technical remark is in order here. Perturbatively onecould introduce on the r.h.s. of Eq. (13) the bare QP dis-persion εk. However, at moderate-strong couplings theQP dispersions are substantially modified by the cou-pling with the modes to dressed QP dispersions εk. Tosuppress the scattering of the dressed QPs near the trueFS, one must insert in (13) the renormalized dispersionsεk. This need of a self-consistency scheme considerablycomplicates the calculations and led us to consider onlysimple symmetry breakings (cell doublings).

b)

a)

FIG. 7: Two-pole band structure in one dimension forg2/ω0 = 0.5 and ω0 = 0.3 at half-filling n = 1. In (a) thebare vertex g is used, while in (b) the phenomenological ver-tex g of Eq. (13) is used.

Although the form (13) is adopted on a purely phe-nomenological basis, we like to remind that several mi-croscopic calculations22,23 show that in strongly corre-lated systems the coupling between QPs and phonons isseverely suppressed at low energies. This suppression is

mostly effective when the exchanged momenta vF q arelarger than the typical exchanged energy ω0. This is thecase here, where the exchanged momenta are peaked atsizable Q’s.

Figs. 7(a) and (b) illustrate the effect of this vertexcorrection in the half-filled case n = 1, where it playsa crucial role to restore the metallicity of the system.For the previously considered cases of generic filling wefind that the vertex corrections play a minor role and theresults obtained with g differ little at moderate couplingfrom those reported above. However, for strong couplingthe phenomenological form of the vertex Eq. (13) againprevents the system to violate the Luttinger theorem ase.g. in case of the result shown in Fig. 6.

Finite range fluctuations

Since in realistic systems the dynamical fluctuationshave a finite coherence length and a finite lifetime, wealso investigated the case of a one-dimensional QP bandexposed to fluctuations having finite Γ and γ in Fω0,Γ

and in Eq. (5). Obviously the finite extension in spaceand time of the fluctuations produces broadening in thespectra. As it is natural, one finds that the broadeningin the momentum direction is ruled by γ, while Γ rulesthe broadening along the energy axis. As it clearly ap-pears in Fig. 8, the broadening of the spectra does notspoil the essential feature of the coupling with a dynam-ical mode (cf. Fig. 3 for γ = Γ = 0 but same parametersotherwise): The shadow bands persist as high-energy sig-natures of a (local) order, while the FS stays unchangedand Luttinger theorem is obeyed provided the mode issufficiently narrow in energy (γ < ω0) so that one canneglect the “leakage” of weight down to the Fermi level.

FIG. 8: Two-pole band structure in one dimension forg2/ω0 = 0.5 and ω0 = 0.3 at generic filling n = 0.57. Here afinite inverse coherence length γ = 0.02 and time Γ = 0.1 areused.

8

Pairing effects

We further explore our one-dimensional model to in-vestigate the effects of a particle-particle pairing. To thispurpose we introduce in our bare QP band structure a fi-nite gap ∆, which, as it is usual for this type of pairing, istight to the FS. Accordingly the Bogoliubov particle-holemixing occurs near the FS and the QP band opens a gapcentered at the Fermi level. The branch below the Fermilevel bends down giving rise to a maximum occurring ata momentum coinciding with the Fermi momentum ofthe unpaired QPs. Fig. 9 displays this effect for our typ-ical parameter set at a generic filling. In particular onecan see that the particle-particle pairing only modifiesthe low-energy states over scales of order ∆, while leav-ing unchanged the high-energy states, which still displaythe clear effect of the dynamical ordering. This find-ing shares close resemblance with recent scanning tun-nelling experiments in cuprates29, where an energy scale∆0 (analogous to our ω0) separates the low-energy Bo-goliubov coherent QPs from the high-energy excitationscarrying information of a charge ordering tendency. Theone-dimensional character of our toy model helps in sepa-rating these different excitations in momentum space. Amore detailed analysis of this issue would require mod-eling specific features of the cuprates and is beyond thescope of the present paper, but we find this preliminaryone dimensional finding rather suggestive.

FIG. 9: Two-pole band structure in one dimension forg2/ω0 = 0.5 and ω0 = 0.3 at generic filling n = 0.4. Herea finite inverse coherence length γ = 0.02 and time Γ = 0.1are used as well as a finite pairing gap ∆ = 0.1.

B. Numerical analysis in two dimensions

Once the main effects of the dynamical mode exchangehave been presented in one dimension, we illustrate thetwo-dimensional case. For the bare electron dispersion

we use:

εk = −2t(cos(kx) + cos(ky)) − 4t′ cos(kx) cos(ky)

For concreteness the parameters have been chosen to re-produce the FS of LSCO, taking t = 342 meV, t′/t =−0.2, and µ = −0.1t. For the mode frequency and cou-pling, we choose ω0 = 50 meV and g2 = 1.5ω0, while weconsider only coherent fluctuations with vanishing broad-enings (Γ = γ = 0). To avoid too many band foldings,which could complicate the analysis, we choose a ordergiven by a simple cell doubling in the x direction. Thisgives rise to Q = (±π, 0). Of course this does not corre-spond to the charge/spin modulation observed in LSCOcuprates, but it better fits our simplicity and illustrativepurposes (on the other hand it corresponds to the spinordering in FeAs stoichiometric compounds24,25).

Figs. 10 (a) and (b) report the two-dimensional FSobtained by integrating the spectral density in Eq. (1)over a small (Wl = 10 meV) and a large (Wh = 100meV) energy range respectively. Only two poles havebeen considered in the recursive Green’s function andthe QP-mode coupling is dressed via the ’self-consistent’vertex corrections [i.e. corrected by the interaction it-self, see the remark after Eq. (13)]. These correctionsare required here because at the typical fillings we con-sider, the two-dimensional FS has branches connected bythe critical wavevectors (the so-called “hot spots” in theframework of superconducting cuprates). The states con-nected by Q would display a gap opening similar to the(spurious) one obtained in one dimension at n = 1. Forthis reason we decided to phenomenologically suppressthis exceedingly strong effect which we attribute to thesimple perturbative treatment of the singular interactionsin the model.

In the case of integration over the low-energy statesonly, one obtains a FS closely resembling the one of un-perturbed free QPs’. On the contrary, upon integratingover a broad energy range, the “FS” appears folded andclosely tracks the one expected for a system with a staticlong-ranged broken symmetry. Therefore, also in two di-mensions we see that coupling the QPs with dynamical

modes preserves the QPs from a strong rearrangement ofall the states (including those near the FS). Notice thatthe vertex corrections are relevant in this regard onlyaround the hot spots, but do not prevent the generic for-mation of shadow bands or related features appearing innk.

IV. DISCUSSION AND CONCLUSIONS

The results of the previous sections clearly indicatethat dynamical CO/SO fluctuations, at least in a nottoo strong coupling regime, do not produce substantialeffects on QPs around the FS. Therefore the absence oflow-energy signatures of CO/SO is compatible with a dy-namical type of order. Formally we have used a Kampf-Schrieffer-type13 approach extended by a self-consistent

9

FIG. 10: Two dimensional FS obtained from the momentumdistribution function nk in Eq. (1). The spectral function hasbeen integrated over: (a) a low-energy range Wl = 10 meV;(b) a high-energy range Wl = 100 meV.

evaluation of self-energy corrections. However, due tothe lack of vertex corrections our computations are re-stricted to weak coupling therefore QP weights are closeto one and the incoherent weights are small. To havethe reverse ratios of weight we need to go to strong cou-pling, which is unfortunately not feasible. Therefore wecan only speculate on how the spectral function will looklike.

We expect that as the coupling is increased some fea-tures of our computations will persist. For example thefact that the momentum dependent incoherent part ofthe spectral function is separated by a bosonic excitationfrom the Fermi level is expected to be quite robust instrong coupling. Indeed one can construct approximated

excitations as in Table I but with heavily dressed QPsrather than with particles which suggest that a similarphysics will be at play. More complicated excitationsmay give spectral weight closer to the Fermi level butfor those the momentum dependence will be completelywashed out providing a structureless background.

The fact that the incoherent part should resemble theelectronic structure of the ordered state follows from con-tinuity arguments. For example, as one crosses a transi-tion where the spin gets disordered at low energies onedoes not expect dramatic changes in the overall distribu-tions of weight in the spectral functions. Those will bedetermined to a large extent by the short range correla-tions which may change very little across the order dis-order transition. In this sense a faint QP peak emergingat low energies, which is certainly a dramatic perturba-tion from the point of view of the Fermi liquid propertiesbecomes a small perturbation from the point of view ofthe spectral weight distribution. Thus we expect thisresemblance to persist in the strong coupling limit.

The scenario we propose is of obvious pertinence incuprates, where standard ARPES experiments usuallyreport a Fermi-liquid Fermi surface or at least well de-fined nodal particles which at first sight are incompatiblewith vertical stripes. On the other hand other experi-ments display the signatures of dynamic order like the fa-mous hourglass dispersion relation observed in cuprates26

even when CO/SO is not detected.27,28 This hourglassdispersion has been explained computing the fluctuationson top of an ordered ground state which will not displaynodal quasiparticles.11 Our analysis reconcile, at leastqualitatively, this apparent contradiction: fluctuating or-der can coexist with a Fermi-liquid like Fermi surface (in-cluding nodal states) and still behave at energies largerthan ω0 as a stripe ordered state.

It is also interesting that recent scanning tunnelingmicroscopy data29 reveal the simultaneous existence oflow energy (Bogoliubov) QPs and high energy excitationswhich break translational invariance which thus closelyagrees with the prediction of our analysis. In particular,Fig. 9 clearly shows the simultaneous presence of a su-perconducting gap with Bogoliubov QPs at low-energyand of high-energy spectral features, which clearly carryinformation on a spatial order. For these high-energystates the construction of a local density of states natu-rally displays the structure of the staggered order relatedto the scattering wavevector Q.

As we mention in the introduction Zhou7,8 and collab-orators find different Fermi surfaces when analyzing thespectral function integrated in a narrow window aroundthe Fermi level or in a large energy window. Fig. 10 il-lustrates the weak coupling version of these effect withthe small window Fermi surface being Fermi liquid likeand the large window pseudo-Fermi surface showing theeffects of fluctuating order.

The dichotomy between low and high energy spectralfeatures as discussed in the present paper is closely re-lated to another dichotomy observed in ARPES experi-

10

ments, namely the one between nodal and antinodal QPsin underdoped cuprates.30 For stripe-like charge order pa-rameter fluctuations the characteristic scattering vectorsare of a comparable magnitude than the separation ofthe Fermi surface segments at the antinodal regions. Asa consequence (and quantitatively analyzed in Ref. 14within a similar formalism) the scattering is much moreeffective for antinodal states than for nodal ones sincenodal quasiparticles get protected by momentum conser-vation in addition to the dynamical protection discussedhere. Antinodal states will be affected by the vertex cor-rections for which we have only a weak coupling guessand therefore require further study but it is conceivablethat spectral weight at the Fermi level will be suppressedin these regions due to transfer of spectral weight to theshadows.

The ascription of the dichotomy between nodal andantinodal QPs to low energy charge order parameter fluc-tuations is also discussed in Ref. 19 within a frustratedphase separation scenario. For charge scattering alongthe Cu-O bond direction one also obtains a FS where thespectral weight is confined to the nodal regions and thusis compatible with a momentum dependent pseudogap inthe sense of an anisotropy in nk along the FS.

One difficulty in interpreting data in the cuprates is thepresence of disorder. It is possible that the system hasan ordered ground state but, due to quenched disorder ora complex energy landscape31, it never becomes ordered.As for a structural glass, the system has long range intime positional order but a structural factor characteris-tic of a disordered state. In this case scattering probeswill fail to detect order even though the charge is notfluctuating and is inhomogeneous. Local probes, how-ever, should be able to detect the ordering but those aremore scarce and more difficult to interpret. Both staticdisorder and our picture will predict similar results athigh energies but will strongly differ at low energies wereonly within our picture one recovers a uniform Fermi liq-uid. It is not clear at the moment which effect prevailsin different systems.

There are cases in which CO has been detected32,33,34

and still the stripe Fermi surface computed in LDA9 andmeasured from the nk does not show up close to theFermi level. For example no visible shadow FS’s appearsin Ref. 35, where the FS of a CO LBCO sample onlydisplayed Fermi arcs due to a particle-particle pseudo-gap. It is possible that in this case the system chargeorders because the commensuration with the underly-ing lattice provides a pinning potential, which helps tostabilize the charge but the spin is still quantum disor-dered. Indeed spin order breaks a continuous symme-try whereas commensurate CO, as usually observed incuprates, only breaks a discrete symmetry. Thereforethe former is much more fragile than the latter. Thus wepropose a state in which the charge is ordered but for thespin our dynamical picture applies. One can still wonderwhy one does not see shadows due to the CO. One shouldkeep in mind that the CO is a minor perturbation to the

electrons compared to SO. In a SO state the effective po-tential seen by the electrons oscillates on a scale of U .In order to check the effect of CO alone without SO wehave computed the FS in the presence of a charge mod-ulation similar to the one observed. The result is a largeFermi liquid like FS with weak distortions36. These maybe related to features that only very recently have beenresolved for stripe-ordered LSCO codoped with Eu.37 Itis only at moderate energies, above ω0 for the magneticexcitations, that the mixed CO-SO fluctuations produceshadow features in the spectra and give rise to “crossedFS’s” like those observed in LSCO in Refs. 7,8.

A momentum dependence of the so-called “hump” fea-tures at relatively high energies (of order 0.1 eV) has alsobeen detected in Bi2Sr2CaCu2O8+δ

38. One can speculatethat scattering with a magnetic mode related to SO asdescribed here is responsible for this effect. (The authorsoffer a related explanation involving the (π, π) spin res-onance). An analysis of nk similar to the one carriedout for LSCO in Refs. 7,8 could also help in discerningwhether this is just a remnant effect of the proximityto the insulating antiferromagnet or due to fluctuatingstripes.

In conclusion, in the light of the analysis carried outin this paper, we propose that cuprates are affected byfinite-energy spin/charge fluctuations related to proxim-ity to ordered phases. Whether CO is actually realized ina static way is rather immaterial from the point of viewof low-energy ARPES spectra due to the weakness of thecharge modulations. Only when the energy is above thespin and charge fluctuating scale the electronic spectrumis sizably affected. At these energies, however, the detec-tion of CO-SO-related pseudogaps (not tied to the FS) isquite hard due to the largely incoherent character of thespectral lines. Only passing through nk this dynamicaltendency to order becomes visible.

We acknowledge interesting discussions with C. Castel-lani, C. Di Castro. M. G. acknowledges financial supportfrom MIUR Cofin 2005 n. 2005022492 and M.G and G.S. from the Vigoni foundation.

APPENDIX A: ITERATIVE SOLUTION OF THE

MODEL

In this Appendix we give a detailed description of theiterative solution of the two coupled equations (9,10). In-serting the lowest-order self-energy Σ(1)(k, iω) (Eq. (11))into Eq. (10) leads to a G(1)(k, iω) which displays twopoles and which can be used to compute Σ(2)(k + Q, iω)(and thus G(2)(k + Q, iω)) and so on. This proceduretherefore generates the series

G(1)(k, iω) → G(2)(k + Q, iω) → G(3)(k, iω)

→ G(4)(k + Q, iω) → G(5)(k, iω) · · · → G(n−1)(k + Q, iω)

11

where G(n−1)(k + Q, iω) has a n-pole structure that canbe represented as

G(n−1)(k + Q, iω) =

n∑

s=1

[α(n)s (k)]2

iω − E(n)s (k)

. (A1)

Consequently the n-th order for the self-energy is givenby

Σ(n)(k, iω) = g2n

∑

s=1

[α(n)s (k)]2

iω ± ω0 − E(n)s (k)

(A2)

and the sign of ω0 in the denominator depends on

whether f(E(n)s (k)) = 0, 1 (cf. Eq. (11)).

Examining the equation for the n-order Green’s func-tion (iω − εk − Σ(n+1))G(n+1) = 1 it turns out that thesolution can be conveniently obtained by solving the ma-trix equation:

[

iω1 − M]

K = 1 (A3)

with

M =

εk gα(n)1 (k) gα

(n)2 (k) · · · gα

(n)n (k)

gα(n)1 (k) E

(n)1 (k) ∓ ω0 0 · · · 0

gα(n)2 (k) 0 E

(n)2 (k) ∓ ω0 · · · 0

......

......

.... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......

......

...

gα(n)n (k) 0 0 · · · E

(n)n (k) ∓ ω0

. (A4)

and we can identify the (n)-order Green’s function as the(11)-element of the matrix K. Denoting by T the trans-formation which diagonalizes M , thereby yielding (n+1)

eigenvalues E(n+1)s (k), the (n)-order Green’s function is

thus obtained as

G(n)(k, iω) =

n+1∑

s=1

[T1s]2

iω − E(n+1)s (k)

(A5)

which also yields the new weights α(n+1)s (k) ≡ T1s.

This scheme allows for a systematic evaluation of the

Green’s function up to some given order (n). In the first

step, the matrix M is of order 2× 2 with α(1)1 (k) = 1 (cf.

Eq. (9)) and E(1)n (k) = εk+Q. Diagonalization yields the

weights and energies of G(1)(k, iω) which can be used toconstruct the matrix for G(2)(k +Q, iω) (where the (11)-element of M in Eq. (A4) is εk+Q) and so on. Thisprocedure creates spectral functions with more and morepoles, however, in case of a not too strong coupling, itwill converge in the sense that the weight of newly createdpoles becomes smaller and smaller so that the series canbe cut at the desired accuracy.

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