Physica C 481 (2012) 132–145

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Physica C

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Stripes in cuprate superconductors: Excitations and dynamic dichotomy

G. Seibold a,⇑, M. Grilli b, J. Lorenzana b

a Institut für Physik, BTU Cottbus, P.O. Box 101344, 03013 Cottbus, Germanyb SMC-INFM-CNR, Dipartimento di Fisica, Università di Roma ‘‘La Sapienza’’, P.le Aldo Moro 5, I-00185 Roma, Italy

a r t i c l e i n f o

Article history:Accepted 25 March 2012Available online 21 April 2012

Keywords:High-temperature superconductorsStripesElectronic structureTransport propertiesCharge excitationsMagnetic excitations

0921-4534/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.physc.2012.03.072

⇑ Corresponding author.E-mail address: goetz@physik.tu-cottbus.de (G. Se

a b s t r a c t

We present a short account of the present experimental situation of stripes in cuprates followed by areview of our present understanding of their ground state and excited state properties. Collective modes,the dynamical structure factor, and the optical conductivity of stripes are computed using the time-dependent Gutzwiller approximation applied to realistic one band and three band Hubbard models,and are found to be in excellent agreement with experiment. On the other hand, experiments likeangle-resolved photoemission and scanning tunneling microscopy show the coexistence of stripes at highenergies with Fermi liquid quasiparticles at low energies. We show that a phenomenological model goingbeyond mean-field can reconcile this dynamic dichotomy.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Since the discovery of high-temperature superconductors(HTSCs) by Bednorz and Müller [1] numerous experiments haveevidenced the existence of electronic inhomogeneities in thesecompounds (cf. e.g. Ref. [2]). While early on these inhomogeneitieswhere believed to be predominantly chemically driven, e.g. due tomaterial imperfections like disorder induced by the dopant ions, itwas subsequently realized that the strongly correlated character ofthe cuprate superconductors and thus the electronic subsystem it-self can favor the formation of inhomogeneities on the nano-scale.According to the analysis by the Rome group [3] the inherentreduction of the quasiparticle kinetic energy together with a shortrange attractive (e.g. electron-lattice) interaction can induce aninstability towards phase separation since competing nestinginstabilities are suppressed due to a residual repulsion of thequasiparticles at large momenta. On the other hand, the long-rangerepulsive Coulomb interaction will spoil the associated zero-momentum instability in the charge sector and instead shift thewave-vector of the ordering transition to finite values. This so-called frustrated phase separation mechanism [4] thus results inan incommensurate charge ordering (CO).

An alternative approach is based on Hartree–Fock (HF) investi-gations of Hubbard (and tJ)-type hamiltonians [5–8] which haverevealed solutions with a combined charge- and spin-densitywave. In contrast to the frustrated phase separation mechanism,the charge order in this case is driven by a spin-density wave insta-

ll rights reserved.

ibold).

bility which via the coupling between longitudinal spin and chargedegrees of freedom results in a concomitant charge density wave.These so-called stripe solutions have been confirmed later on bymore sophisticated numerical methods. Within a density matrixrenormalization group (DMRG) approach White and Scalapino[9–11] have found stable domain wall solutions in the physicallyrelevant doping regime of the tJ-model. The stripe stability in thetJ-model has also been investigated using exact diagonalization[12,13] as well as quantum and variational Monte Carlo techniques[14,15]. Besides the HF approach a variety of methods has alsobeen applied to Hubbard-type models in order to investigate thepossibility of charge and spin order. Fleck and collaborators [16]have found stable stripes using dynamical mean-field theory(DMFT) and a cluster perturbation approach has been applied inRef. [20]. From an ab initio perspective stripe order in lanthanumcuprates has also been shown to be stable in LDA + U calculations[21].

Here we present a short update of the current experimentalsituation (Section 2) followed by an overview over static anddynamical properties of stripes [22,23,25–27,73,74,28–30] ob-tained within the time-dependent Gutzwiller approximation(TDGA) [135–138] and phenomenological models. In Section 3we first derive the parameter set for the extended Hubbard modelwhich in the following sections allows us to make quantitativecomparison with experimental data. We then show in Section 4that the Gutzwiller approximation (GA) of this model in fact leadsto stable stripe ground states which allows for the doping depen-dent evaluation of the corresponding incommensurate spinresponse. In Section 5 we discuss the electronic structure and var-ious transport properties of stripes. Section 6 focusses on the

G. Seibold et al. / Physica C 481 (2012) 132–145 133

momentum and frequency dependent spin excitations where wealso address recent resonant inelastic X-ray scattering experi-ments. Section 7 presents results for the charge excitations fromstripe ground states, focussing on the optical conductivity. Finally,we show in Section 8 how our Gutzwiller variational theory can beextended towards a dynamical description of stripes before wesummarize our discussion in Section 9.

In the following section we start with an update of the experi-mental situation.

2. Brief experimental review

First evidence for stripe order in HTSC’s came from elastic neu-tron scattering experiments by Tranquada and collaborators [31–33]. They observed a splitting of both spin and charge order peaksin La1.48Nd0.4Sr0.12CuO4 (LNSCO) which resembled similar data inthe nickelates where both incommensurate antiferromagnetic(AF) order [34,36] and the ordering of charges [35,36] has beendetected by neutron scattering and electron diffraction, respec-tively. In Ref. [36] it was shown that the magnetic ordering dis-plays itself as an occurence of first and third harmonic Braggpeaks whereas the charge ordering is associated with second har-monic peaks. Meanwhile analogous static stripe order has alsobeen detected in Eu-codoped lanthanum cuprates (LECO) [38]and La2�xBaxCuO4 (LBCO) [37,51] which all display a LTT latticedistortion. That the latter is not necessarily a condition precedentto the formation of stripes has recently been demonstrated byapplying pressure to a La1.875Ba0.125CuO4 sample. This restoresthe structure to fourfold lattice symmetry keeping symmetry bro-ken electronic stripe states [39]. Moreover, charge and spin stripeorder can also be induced in the LTO phase of impurity (Cu, Zn,Fe, Ga) substituted lanthanum cuprates [40] without the needof applying pressure. Whereas charge order in the neutron andhard X-ray scattering [41] experiments mentioned above has onlybeen detected indirectly via the associated lattice modulation,more recent soft resonant X-ray scattering studies on LBCO [42]and LECO [43] have directly revealed the spatial modulation ofcharge in these systems.

The incommensurability d (defined as the shift of the magneticBragg peaks from the antiferromagnetic (AF) wave-vector QAF inreciprocal lattice units) in samples with long range stripe orderfollows the so-called ‘Yamada-Plot’ [45] and depends linearly ondoping d = nh up to nh � 1/8. Here nh is the number of added holesper planar Cu with respect to the parent insulating compound.This behavior is compatible with stripe like modulations of chargeand spin having a linear concentration of added holes m = 1/2 (so-called half-filled stripes) as we will discuss in more detail in Sec-tion 4.

Evidence for some kind of stripe order in other HTSC materialsis based on the doping dependence of the low energy spin re-sponse. In non-codoped lanthanum cuprates (LCO) the incommen-surability d (defined in terms of the shift of the low energymagnetic scattering from QAF) also follows the ‘Yamada-Plot’ ofthe samples with long-range order. This suggests that the staticstripe order is replaced by some kind of ‘fluctuating order’ in theNd- (or Ba-, Eu-) free samples. Above nh � 1/8, the incommensura-bility stays essentially constant but the intensity of the low energyspin fluctuations decreases and vanishes at the same concentrationwhere superconductivity disappears in the overdoped regime [46].This behavior is mirrored at low doping in that superconductivity,in LSCO materials, disappears upon decreasing doping exactly atthe point in which incommensurate scattering parallel to theCuO bond disappears [50]. This strong correlation between super-conductivity and low energy incommensurate scattering parallel tothe CuO bond suggesting an intimate relation between both

phenomena and provides a strong motivation to understand stripephysics in cuprates.

Once superconductivity disappears by decreasing doping belownh � 0.055 in lanthanum cuprates incommensurate scattering doesnot disappear entirely but rotates by 45� [47–50] to the diagonaldirection. Additionally the orthorhombic lattice distortion allowsone to conclude that the elastic diagonal magnetic scattering isone-dimensional with the associated modulation along the ortho-rhombic b⁄-axis, supporting again the picture of stripe formation.When d is measured in units of reciprocal tetragonal lattice unitsin both the vertical and diagonal phase it turns out that the mag-nitude of the incommensurability numerically coincides acrossthe rotation leading to a linear relation d = nh. Upon approachingthe border of the AF phase at nh = 0.02 the incommensurability ap-proaches � = nh [52] where � is measured in units of reciprocalorthorhombic lattice units, thus � ¼

ffiffiffi2p

d. The crossover from diag-onal to vertical incommensurate spin response is so far only ob-served in lanthanum cuprates whereas in all other HTSCcompounds the scattering is always along the Cu-O bond direction.Despite this peculiarity of LCO other experiments point to univer-sal behavior.

A linear relationship between doping and incommensurabilityis also observed in YBCO6+x although the curve falls below theone of lanthanum cuprates [53–55]. Apart from details, this sug-gests that stripe physics is a universal phenomenon in cuprates.

Upon increasing frequency of spin excitations the dispersiondevelops a spectrum (‘hour glass’) which is similar in many high-Tc cuprates again suggesting universality. At a certain frequencyEs the incommensurate branches merge [56] at the AF wave-vectorand for even higher energies the excitations become incommensu-rate again. The ‘hour glass’ spectrum has been detected in the ver-tical [57,59–61] and diagonal phase [63,65] of lanthanum cuprates,the YBCO compounds [66–70] and Bi2Sr2CaCu2O8+y [71,72].

Magnetic fluctuations which dispersion follows the ‘hour-glass’shape naturally arise from stripe correlations in the ground state[73–84,29] (cf. Section 6). Basically a Goldstone mode arises fromeach of the incommensurate wave-vectors and disperses in acone-shaped structure to higher energies. However, due to disor-der and peculiar magnetic couplings across the stripes (as sup-ported by recent inelastic neutron scattering (INS) experimentson La2�xSrxCoO4 [85]) the weight of the outwards dispersing con-tribution is strongly suppressed so that the excitations which dom-inate the spectral weight disperse towards the AF wave-vector QAF.The corresponding excitation energy Es at QAF is essentially deter-mined by the spin coupling across the antiphase domains. Sinceat high energies the excitations should again resemble those ofthe (undoped) AF one finds an outwards dispersing intensity alsoabove Es.

While apparently the spectrum of spin excitations is compatiblewith stripe correlations the problem is the detection of an associ-ated charge order in the non-codoped lanthanum cuprates, YBCOand bismuthates. Experimental data of local probes like NQR[86–88], NMR [89], EXAFS [184], and X-ray microdiffraction[185] are suggestive of the formation of electronic inhomogenei-ties, possibly induced by magnetic field [90], but cannot provideevidence for long-range order. In this context it is remarkable thatdue to refinements in the experimental technique Haase et al. [89]where able to demonstrate a correlation of charge and density vari-ations on short length scales.

One way to reconcile the ‘stripe-like’ magnetic excitationspectrum with the absence of long-range charge order is the notionof a charge nematic [91] which (in the charge channel) breaksrotational but not translational invariance. This concept is extre-mely useful in detwinned YBCO where both neutron scattering[69,70] and thermoelectric transport [92] data show pronouncedin-plane anisotropies but may also apply to the spin-glass phase

134 G. Seibold et al. / Physica C 481 (2012) 132–145

of lanthanum cuprates where, as mentioned above, the incommen-surate spin scattering is one-dimensional.

Due to the absence of clear signatures of charge ordering inother than co-doped lanthanum cuprates, alternative theories havebeen proposed in order to account for the incommensurate spinscattering and the associated ‘hour glass’ magnetic spectrum[93–101]. Nonetheless there are a number of fingerprints in spec-troscopic probes which support the presence of incommensuratecharge scattering. For example, the anomalous lineshape and tem-perature dependence of the bond-stretching phonons in variouscuprate materials (cf. Ref. [102] and references therein) can beattributed to the influence of stripes. For example, it has been pro-posed that the softening of these phonons at certain momenta isdue to the influence of charge stripe fluctuations on the phononself-energy [103]. In addition, one may also have a Kohn-typeanomaly due to the nesting along the half-filled stripe [104]. Furth-eron, the scattering of charge carriers by incommensurate chargefluctuations should induce a reconstruction of the Fermi surface[20,105,106]. Angle-resolved photoemission experiments (ARPEs)on LSCO [107,108] have indeed shown the appearance of straightFermi surface (FS) segments around the M-points of the Brillouinzone as expected for a striped ground state. However, one peculiarfeature of these experiments concerns the fact that the stripe-likeFS was obtained from the momentum distribution nk, by integrat-ing the spectral function over a broad energy window (�300 meV).On the other hand, upon following the momentum dependence ofthe low-energy part of the energy distribution curves a large FSwas found corresponding to the LDA band-structure and fulfillingLuttinger’s theorem. We will address this dichotomy of stripe-likespectral features at large energies and a ‘protected’ FS at low ener-gies in Section 8.

In LECO, where static charge order has been detected with reso-nant soft X-ray scattering [43], the associated reconstruction of theFS has indeed been resolved by ARPES. [44] Moreover, a recent ser-ies of magnetotransport experiments [111–114] (cf. also the reviewin Ref. [115]) have revealed quantum oscillations suggestive of theformation of electron pockets due to a FS reconstruction. It has beenshown by Millis and Norman [116] that a charge density waveground state with intermediate values of the stripe order parameteris in qualitative agreement with the quantum oscillation data. A re-cent comparative study of thermoelectric properties in YBCO andLECO [117] also provides strong evidence that the Fermi surfacereconstruction in YBCO is associated with stripe correlations. Inthe context of transport experiments one should also mention theNernst effect which in hole doped HTSC’s above Tc usually yields alarge positive signal [118] and which has been considered as indic-ative of fluctuating superconductivity. However, in addition also aFS reconstruction contributes to the Nernst signal and both contri-butions have been resolved in LNCO and LECO [119]. These authorshave attributed the high temperature signal to originate fromchanges in the FS, an interpretation which has recently been chal-lenged in Ref. [120]. A theoretical analysis of the Nernst effect interms of stripe order can be found in Refs. [121,122].

Scanning tunneling microscopy (STM) provides additionalevidence (limited obviously to the surface) for charge order in cup-rate superconductors. Corresponding measurements performed onbismuthate and oxychloride superconductors have revealed a com-plex modulation of the local density of states (LDOSs) both in thesuperconducting (SC) state [123–128] and above Tc [128–130]. Inboth cases one observes peaks in the Fourier transform of the realspace LDOS at wave-vectors Q = 2p/(4a0) . . . 2p/(5a0) suggestive ofcheckerboard or stripe charge order. However, it is important to dis-tinguish between the case where these peaks are non-dispersive inenergy (and thus signature of ‘real’ charge order) or the situationwhere they follow a bias-dependent dispersion due to quasiparticle(QP) interference. In the latter case the spatial LDOS variations can

be understood from the so-called octet model [123,131] which attri-butes the modulations to the elastic scattering between the highdensity regions of the Bogoljubov ‘bananas’ in the superconductingstate. Recent STM investigations [132,133] may resolve this appar-ent conflict since they suggest that both, dispersive and non-disper-sive scattering originates from different regions in momentum andenergy space. The states in the nodal region which are well definedin k-space and undergo a transition to a d-wave SC state below Tc

are then responsible for the low energy QP interference structureof the LDOS, whereas the ill-defined k-space ‘quasiparticle’ statesin the antinodal regions are responsible for the non-dispersive COabove some energy scale x0. As mentioned before we will addressthe issue of high energy electronic order in Section 8.

3. Model and parameters

Moment formation in a spin-density wave solution involves thelarge Hubbard U scale and modifies the electronic structure of thesystem over a large energy range. Therefore, it is not a priori obvi-ous that the reduction of a multiband model to a more simplifiedmodel hamiltonian will retain the relevant features in the spectra,especially when one is concerned with high frequency excitations.A suitable method in order to deal with this problem has to fulfillessentially the following two conflicting criteria: (a) it should besufficiently accurate in order to account for the Hubbard-type cor-relations of the model and (b) it should allow for the investigationsof large systems in order to provide a sufficient resolution for thecorrelation functions in momentum space.

Subsequent to the first stripe calculations [5–8], investigationsof collective excitations have been performed within the Har-tree–Fock (HF) approximation of the one-band Hubbard modelsupplemented with RPA fluctuations [139–143]. However, in caseof the HF approximation [144] it turns out that the stripe solutionsrelevant for cuprates are only favored for unrealistic small valuesof U/t � 3 . . . 5 whereas a ratio of U/t � 8 is required to reproducethe magnon spectrum of LCO (cf. below).

In order to enable a quantitative comparison with experiment,the results we will discuss in the following sections will be mostlybased on the Gutzwiller approximation (GA) [134] supplementedwith Gaussian fluctuations [135–138] (so-called time-dependentGA [TDGA]).

We consider the one-band Hubbard model

H ¼Xi;j;r

tijcyi;rcj;r þ U

Xi

ni;"ni;# ð1Þ

where ci;rðcyi;rÞ destroys (creates) an electron with spin r at site i,and ni;r ¼ cyi;rci;r. U is the on-site Hubbard repulsion and tij denotesthe hopping parameter between sites i and j. We restrict to hoppingbetween nearest (�t) and next-nearest (�t0) neighbors. For high-Tc

cuprates the value of U can be estimated from the magnon disper-sion of the undoped system [146,147]. In fact, within spin-wavetheory (SWT) applied to the Heisenberg model (corresponding toU/t ?1) the magnon excitations are given by

�hxq /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ½cos qx þ cos qy�

2=4

qð2Þ

and thus the dispersion vanishes along the magnetic Brillouin zone.Including quantum fluctuations beyond SWT [145] leads to excita-tions with lower energy at (p,0) than at (p/2,p/2), contrary to whatis observed in undoped cuprate superconductors [146,147]. It hasbeen argued that in cuprates corrections to the Heisenberg modelarising as higher orders in a t/U expansion are relevant [148–151,146]. The most important of such corrections is a term whichcyclically exchanges four spins on a plaquette. A sizable value forthis term has been revealed by analyzing phonon-assisted multima-gnon infrared absorption [151] and the dispersion relation

(a)

G. Seibold et al. / Physica C 481 (2012) 132–145 135

measured with INS [146,147] shown in Fig. 1. In particular, the dis-persion between (p,0) and (p/2,p/2) is mainly due to this term. Incase of the Hubbard model the magnetic zone edge dispersion isdue to the finiteness of U/t. For moderate values of U/t this leadsto dispersion which decreases from (p, 0) to (p/2,p/2) along themagnetic zone boundary whereas for large U/t the behavior of thespin-1/2 Heisenberg model is recovered [152].

Fig. 1a shows the U/t dependence of the magnetic excitationsfor the half-filled AF as compared to INS data from Ref. [146].The dispersion along the magnetic zone boundary can be accu-rately fitted by a value of U/t = 8 where the absolute energy scaleis fixed by the nearest neighbor hopping t = 360 meV. Panel b ofFig. 1 displays the dependence of the spin excitations on thenext-nearest neighbor hopping t0/t. For small to moderate valuesof t0/t the zone boundary dispersion is only slightly dependent onthis parameter, however, significant softening occurs for t0/t = �0.5. The increasing frustration between nearest and next-nearest neighbor magnetic interactions drives the system towardsan 1D-AF instability the precursor of which is seen as a magneticmode softening at q = (p,0).

Further information on the parameter t0/t can be obtained fromthe optical conductivity which in the inset to Fig. 1b is shown forthe same parameters as used in the main panel. The lowest energyexcitation Xmin for the half-filled system is shifted up with increas-ing jt0/tj and the experimental value of Xmin � 2.1 eV for LCO[153,154] is accurately reproduced for the parameter set U/t = 8,t0/t = �0.2, t = 360 meV. In the following section we will show thata value of t0/t = �0.2 is also appropriate to account for the doping

(π/2,π/2) (π,π) (π,0) (π/2,π/2) (0,0)(q

x,q

y)

0

100

200

300

400

Ene

rgy

(meV

)

U/t=5, t’/t=-0.2, t=300 meVU/t=8, t’/t=-0.2, t=360 meVU/t=10, t’/t=-0.2, t=416 meV

(π/2,π/2) (π,π) (π,0) (π/2,π/2) (0,0)(q

x,q

y)

0

100

200

300

400

Ene

rgy

(meV

)

U/t=8, t’/t=0, t=336 meVU/t=8, t’/t=-0.2, t=360 meVU/t=8, t’/t=-0.5, t=535 meV

1 2 3 4energy [eV]

0

opt.

cond

[a.

u.]

(a)

(b)

Fig. 1. Energy and wave vector dependence of magnetic excitations in the half-filled system as obtained from the TDGA for (a) t0/t = �0.2 and varying U/t and (b) U/t = 8 and varying t0/t. Squares and triangles correspond to data points from INSexperiments on LaCuO4 by Coldea et al. [146]. The inset to (b) shows the opticalconductivity for the same parameters as used in the main panel (b).

dependent incommensurability of lanthanum cuprates [45] basedon a striped ground state.

Note that some few results discussed in the present review(Sections 4.1 and 7) are also obtained within the three-band modelwhich for copper dx2�y2 and oxygen px,y orbitals includes the asso-ciated hopping processes, orbital energies, and intra- and interor-bital Coulomb repulsions. We have taken these parameters eitherfrom first principle computations [169,170] or from more empiri-cal considerations [158].

4. Stripe stability and static properties

4.1. Charge and spin structure

Stripes are characterized by one-dimensional antiphase bound-aries for the AF order which also host the doped holes. These tex-tures can be understood as the strong coupling remnant of aninstability in the spin channel. It can be shown [156] that for dis-persion relations relevant for cuprates the orientation is eitheralong the copper–oxygen (vertical) or the diagonal direction. Thedomain walls can either reside on lattice sites or bonds and thesetwo prototypical stripe solutions for both vertical and diagonal ori-entations are sketched in Fig. 2.

dtet

d tet

(b)

(c)

(d)

Fig. 2. Spin structure of vertical (a,b) and diagonal stripe textures. The doped holesmainly reside on the (shaded) domain walls. In both cases the prototypicalstructures comprise site-centered (a,c) and bond-centered structures (b,d). Thedistance between charge stripes, measured along the copper–oxygen bond direc-tion is denoted as d in units of the tetragonal lattice constant atet.

0.022

0.018 0.018

0.02 0.049 0.036 0.071

0.081

Fig. 3. Charge and spin density for d = 4 BC vertical stripes. Doping nh = 1/8. Opencircles (p orbitals) represent Cu(O) sites. The numbers and the size of symbolsrepresent the excess charge whereas the spin density is proportional to the lengthof arrows.

136 G. Seibold et al. / Physica C 481 (2012) 132–145

By symmetry, for site-centered (SC) stripes the magnetizationvanishes on the domain wall whereas for bond-centered (BC)stripes these are build up from ferromagnetically aligned near-est-neighbor spins. For diagonal BC stripes this leads to the situa-tion that configurations with d = even acquire a macroscopicmagnetization in contrast to vertical BC stripes where the ferro-magnetic bonds alternate along the domain wall. Diagonal BCstripes can be viewed as a realization of the staircase structuresproposed by Granath [157] with step length l = 2.

Fig. 3 shows the charge and spin density for d = 4 BC verticalstripes (doping nh = 1/8) obtained within the 3-band model andparameters taken from Ref. [158]. A similar charge distributionwas predicted in Ref. [25] and found to be in excellent agreementwith a charge sensitive probe by Abbamonte and collaborators[42]. However, these authors could not determine if the stripeswhere Cu centered or O centered. More recently, Davis and collab-orators [159] have imaged glassy stripes which indeed are cen-tered on O as predicted [25]. Taken together, these experimentsgive us amplitude and phase information of stripes in excellentagreement with Ref. [25] thus showing that it is possible to obtainand even predict realistic information on the intermediate scalephysics of cuprates.

0.5 0 0.5qx

0.5

0

0.5

q y

Cu spin 103

0

130.

0

4.9 4.9

0

130.

0

0.50.5

0

0.5

q y

Ox sp

3.3

0

120.

0.5 0 0.5qx

0.5

0

0.5

q y

Cu charge 106

0

33.

0 0

33.

0

0

0.50.5

0

0.5

q y

Ox Char

0

150.

0

Fig. 4. Bragg weights m2Q ;q2

Q at reciprocal lattice vectors (indicated by the dots) for nh

Brillouin zones. We use reciprocal lattice units of the fundamental Brillouin zone. Ox(Oy

The Fourier transform of the charge (/ qQ) and spin distribution(/ mQ) determines Bragg peak weights / q2

Q ;m2Q

� �in scattering

experiments (for the definitions see Ref. [160]) as shown inFig. 4. Disregarding the difference in cross section for different pro-cesses (magnetic neutron scattering vs. X-ray or nuclear neutronscattering) we see that Bragg weights for the charge are practicallythree orders of magnitude smaller than those for the Cu spins. Thisis due to very soft charge distribution shown in Fig. 3 with respectto the spin distribution and explains why it has been so hard to de-tect charge ordering.

4.2. Stripe stability and incommensurability

The task is now for a given parameter set and doping to varia-tionally determine the lowest energy state among the variousstripe states and to compare with other possible solutions. Thishas been accomplished in Refs. [27,28,30] for the one-band hamil-tonian Eq. (1) and in Ref. [25] also for the three-band model. Fig. 5displays the binding energy per hole

eh ¼EAFðNh ¼ 0Þ � EtextðNhÞ

Nhð3Þ

for the energetically most competing structures, namely stripes,spirals and spin polarons. Here EAF(Nh = 0) denotes the energy ofthe undoped AF and Etext(Nh) is the energy of a given texture ob-tained for doping the system with Nh holes. We can also definethe filling factor m of stripes which corresponds to the concentrationof holes per unit cell along the stripe. This can be related to the holeconcentration nh and the distance (in atet lattice units) betweencharge stripes d as m = nh � d. From the inset to Fig. 5a it can be seenthat for U/t = 8 and t0/t = �0.2 low doping vertical stripes havemopt � 0.55 whereas the minimum of diagonal SC (BC) domain wallsis at mopt = 1(0.75). We will see below that mopt determines the slopeof the Yamada plot at low doping.

We find that for the one-band model, vertical BC and SC stripesare also practically degenerate in energy in agreement with Refs.[9,16,165]. For definiteness we mainly restrict ourself to the BC casebecause these textures constitute the more stable configuration at

0 0.5qx

in 106

0

120.

0

3.3

0

0.5 0 0.5qx

0.5

0

0.5

q y

Oy spin 106

0

2.6

0

0.78 0.78

0

2.6

0

0 0.5qx

ge 106

0

150.

0

2.3

0.5 0 0.5qx

0.5

0

0.5

q y

Oy charge 106

0

500.

0 0

500.

0

0

= 1/8 and d = 4 stripes. The polygons around each point are the reduced magnetic) are the oxygens in the bonds perpendicular (parallel) to the stripe.

0 0.1 0.2doping

0

0.1

0.2

inco

mm

ensu

rabi

lity

1/(2

d tet) LSCO, Ref. 44

vert spiral.vert. stripeFe-LSCO, Ref. 160

-1.6

-1.5

-1.4

-1.3

-1.2en

ergy

per

hol

e [t

]

diag. spiralvert. spiralPolaron

0.4 0.6 0.8 1ν

-1.6

-1.5

e h [t]

d=10 86

5

4

3

vertical

BC, diag.

SC, diag.

(a)

(b)

vert. stripe

Fig. 5. (a) Binding energy per hole eh as a function of doping for stripes, uniformspirals and spin polaron. The inset shows eh for d = 10 vertical (dashed-dotted),bond centered diagonal (dashed), and site-centered diagonal (solid) stripes as afunction of the filling factor m. (b) Incommensurability 1/(2d) of vertical stripes anduniform spirals as a function of doping. Diamonds are data for LSCO from Ref. [45];full dots are data for Fe-codoped LSCO from Ref. [162]. Results are obtained from theone-band model with parameters: U/t = 8 and t0/t = �0.2.

G. Seibold et al. / Physica C 481 (2012) 132–145 137

nh = 1/8 in the more accurate three-band model [25] and in firstprinciple computations [21]. However, one should keep in mindthat all energetic considerations below hold equally for SC verticalstripes.

From the inset to Fig. 5a it turns out that diagonal BC stripes arepractically (accidentally) degenerate in energy with verticalstripes. The energy of the DSC texture is �0.02t per hole above.These small energy differences should not be significant giventhe simplicity of the model. A precise determination of the relativestability of the different phases would require at least multiorbitaleffects, inclusion of both long-range Coulomb interactions and cou-pling of the holes to the tilts of the CuO4 octahedra. The latter havebeen shown to play a major role in the stabilization of vertical vs.diagonal stripes [161].

In Fig. 5 the dot-dashed line corresponds to the envelope of thestripe binding energy curves for different periodicites d. The solid(dashed) lines show eh for diagonal (vertical) spiral solutions whichapparently are unstable towards phase separation at low doping.The horizontal line is the energy of the phase separated stateobtained from the Maxwell construction. The respective stabilityof stripes and uniform spirals is determined by the ratio betweennext-nearest neighbor and nearest neighbor hopping t0/t [30,163,164]. For t0/t = �0.2 stripes are stable over the whole doping rangeup to nh � 0.28 where their energy becomes degenerate with thatof vertical spirals. In contrast, it is found that with increasing jt0/tjstripes with significant spin canting start to dominate the phasediagram and for even larger jt0/tj also checkerboard textures may be-come the ground state [28,30]. Also the optimum filling mopt changeswith t0/t as expected according to the results of Ref. [27]. The influ-ence of a next-nearest neighbor hopping term on the stripe stabilityhas been previously investigated by various methods. From theDMRG approach applied to the t-t0-J model [11] it turned out thata negative t0/t can suppress both the formation of stripes and pairing

correlations. A weakening of stripe tendencies for t0/t < 0 in the samemodel was also found with exact diagonalization [13] and in theHubbard model with DMFT [16], and the HF approximation [17–19]. All these calculations suggest that static stripes are destabilizedwhen the ratio t0/t becomes negative. This may indicate a moredynamical character of stripes in some systems like Tl and Hg basedcompounds.

With regard to the experimental evidence of stripe correlationsin lanthanum cuprates it turns out that a ratio of t0/t = �0.2 can ac-count for the doping dependent incommensurability d observed inthese compounds [45]. For vertical stripes this quantity is relatedto the modulation of the AF order and thus given by

d ¼ 1=ð2dÞ ¼ nh=ð2mÞ ð4Þ

in units of 2p/atet where atet is the tetragonal lattice constant. Thecharged core of the stripe has a characteristic width n so that whenthe charge periodicity d is larger than n there are negligible inter-stripe interactions and doping proceeds by increasing the numberof stripes. In this regime the number of stripes for a fixed numberof holes is optimized when m = mopt. Thus mopt determines the slopeof the incommensurability as a function of doping, according toEq. (4) and mopt = 0.55 implies d = 0.91nh in good agreement withthe Yamada plot. The regime d > n is characterized by the fact thateh at the minimum is independent of doping or d. From Fig. 5 wefind n = 4 lattice units which coincides with a direct examinationof the charge profile. For d [ n stripes completely cover the planeand increasing further the number of stripes becomes energeticallycostly. Therefore doping proceeds by increasing the charge ofstripes and d becomes constant. The crossover in doping betweenthe two regimes occurs when nh = mopt/d � 1/8 in good agreementwith the Yamada plot in LCO.

As can be seen from Fig. 5b the incommensurability for stripesshows a staircase structure where the steps occur at the crossing ofthe corresponding energy curves (cf. Fig. 5a). For small doping(d > n), since interstripe interactions are negligible, one canproduce a practically continuous curve by considering combina-tions of solutions with periodicity d and d + 1. As explained abovethis is not any more convenient for d [ n and the additional holesare doped into existing stripes producing the plateau in theincommensurability.

Contrary to the stripes, one observes a continuous increase ofd(nh) in case of uniform spirals. Nevertheless, both phases showsimilar evolution of incommensurability with doping and indeedfairly similar values of d(nh) up to nh � 1/8. For larger doping thisagreement is lost due to the tendency of stripes to make wideplateaus.

We have explained the Yamada plot in terms of well formedstripe textures. It is remarkable that the incommensurabilities,both for stripes and spirals, are not far from what one would obtainfrom an analysis of the momentum dependent susceptibility of theFermi liquid phase [155]. In fact, if one adiabatically decreases theinteraction both uniform spirals and stripes originate from thesame magnetic instability [156]. One can call this a ‘‘weak cou-pling’’ instability. On the other hand within the time-dependentGutzwiller approximation, the Hubbard interaction U is stronglyscreened by vertex corrections. Therefore a quite large value of Uis needed in order to make the Fermi liquid unstable and the insta-bility is better characterized as ‘‘intermediate coupling’’.

The time dependent Gutzwiller analysis shows that it is the pla-teau in the spin susceptibility close to QAF = (p,p) (cf. Fig. 1 in Ref.[156]) which drives the system unstable towards spiral or stripeorder for sufficiently large on-site repulsion U/t.

Fig. 5b also shows data from more recent neutron scatteringexperiments [162] on Fe-LSCO. In these overdoped samples anelastic incommensurate spin response was found close to the dom-inant nesting vectors as extracted from ARPES experiments. It was

-1 -0.5 0 0.5 1kx [π]

-1

-0.5

0

0.5

1

k y[π

]

-1 -0.5 0 0.5 1kx [π]

-1

-0.5

0

0.5

1

k y[π

]

(a)

(b)

Fig. 7. Fermi surface for d = 4 bond centered stripes at nh = 0.125. (a) y-Axisoriented stripes and (b) superposition of x- and y-axis oriented stripes.

138 G. Seibold et al. / Physica C 481 (2012) 132–145

thus concluded that the induced incommensurate response signalsan inherent instability of the itinerant charge carriers, being differ-ent from the low doping stripes arising from localized Cu spins.However, from Fig. 5b it turns out that the data are rather closeto the incommensurability curve of stripes obtained for the LSCOparameter set. Remarkably this behavior was predicted in Ref.[25] much before the measurement became available. Since thesestripe calculations are based on an itinerant approach, there is no‘dichotomy’ between low doping ‘localized’ spin stripes and largedoping itinerant ones, but both appear as different limits of thesame model.

Finally we briefly address the incommensurability of diagonalstripes. In tetragonal units the incommensurability is a factor offfiffiffi

2p

larger than that for vertical stripes given in Eq. (4). The morestable BC diagonal textures have mopt � 0.75 so that ddiag ¼nh=

ffiffiffi2p

mopt

� �� nh=1:06. Therefore the doping dependence of d for

BC diagonal and vertical stripes is almost identical in agreementwith neutron scattering data [47–50]. One should keep in mindthe effect of disorder which in the diagonal phase reduces the corre-lation to the same order or even smaller than the periodicity [48]whereas for vertical stripes the correlation length can reach 150atet

or around 20 times the magnetic stripe periodicity. This becomesespecially important in context of spin excitations from stripes inthe spin glass phase (cf. Section 6).

5. Electronic structure and transport properties

Fig. 6 shows the bandstructure for half-filled d = 4 bond-cen-tered stripes which are oriented along the y-direction. The elec-tronic states of the undoped AF ordered regions (magnetizationper spin m) give rise to lower and upper Hubbard bands (LHB/UHB) which are separated by D � U �m. Stripes induce the forma-tion of additional bands in the gap and for partially filled stripes(i.e. 0 < m < 1) one of them (‘active band’) crosses the Fermi energy.For the present case of ‘half-filled’ stripes the crossing occurs atkF

y ¼ �p=4 for kx = 0 (atet = 1). Perpendicular to the domain wallsthe bands are rather flat with the residual small dispersion dueto the overlap between adjacent stripes.

The ‘active band’ implies a quasi one-dimensional Fermi surfaceat momentum ky = p/4 (cf. Fig. 7a). The corresponding weight inthe full Brillouin zone is concentrated around the (p,0) point whichresults in the two parallel FS segments. There is also a crossing atky = 3p/4 with less spectral weight and the associated segment is

(0,0)(0,−π/2) (0,−π/4) (0,π/4) (0,π/2)(k

x,k

y)

-2

0

2

4

6

E-E

F [t]

UHB

LHB

AB

Fig. 6. Band structure for d = 4 bond-centered stripes oriented along the y-direction. The dispersion is shown for a cut along the stripe and kx = 0. Theelectronically relevant active band (AB) crosses the Fermi energy (dopingnh = 0.125) and is in the gap between lower (LHB) and upper (UHB) Hubbard band.The AF modulation in the direction of the stripes leads to a doubling of theperiodicity so that �p/2 6 ky < p/2.

shifted more towards kx = 0. In LSCO one expects a superpositionfrom x- and y-oriented stripes which results in the FS shown inFig. 7b. Such a prediction is in good agreement with ARPES data[107,108] and even fine details as the small curvature of the seg-ments due to interstripe hopping are reproduced. It should bementioned that the experimental data [107,108] strongly dependon the energy window which is chosen to extract the FS. We willaddress this point in Section 7.

As discussed above, stripes are approximately half-filled fordoping nh [ 1/8 and for nh J 1/8 additional holes are doped intothe active band. Therefore the chemical potential is expected tobe approximately constant for nh [ 1/8 and decreases fornh J 1/8 in qualitative agreement with the observed behavior[109,110]. The rate of change of l with doping, being a high deriv-ative of the energy, is very sensitive to finite size effects and, more-over, few experimental points are available in this doping range inorder to allow for a precise comparison. A rough estimate indicatesthat the theoretical rate of change of l with doping for nh > 1/8 isapproximately a factor of 2 larger than the experimental one[109,110]. This may be attributed to an underestimation of themass renormalization in mean-field. Another possibility whichgoes in the right direction is phase separation among the d = 4stripe solution and the paramagnetic overdoped Fermi liquidwhich is also suggested by the doping evolution of magnetic exci-tations (cf. Section 6).

We finally note that also anomalies in the Hall and Nernst coef-ficient can be attributed to the FS reconstruction due to stripedground states [25,121,122].

6. Spin excitations

Fig. 8 reports constant frequency scans of the imaginary part ofthe transverse magnetic susceptibility v00qðxÞ ¼ �Im 1

N

Reixt T SþqD

ðtÞSþ�qð0Þi for y-axis oriented d = 4 BC stripes in the magnetic Brillou-in zone (cf. Fig. 9).

The low frequency intensity is concentrated around the wavevectors Qs = p(1 ± 1/4,1) corresponding to the spin periodicity ofthe underlying d = 4 stripe structure. In fact, it turns out that theweight of the elastic magnetic peaks is much larger at Qs than atthe higher harmonics (cf. Fig. 4 where the corresponding weights

0.5

1

1.50.5

1.5

1.50.50.5

1

1.5

1.510.50.5

1.5

1

1

1

55 meV

105 meV

5 meV

Fig. 8. Constant frequency cuts of the magnetic excitation spectrum at x = 5, 55,and 105 meV for the d = 4, nh = 0.125 BC stripe structure. The momentum spacecorresponds to the magnetic Brillouin zone which is rotated by 45� with respect tothe stripe direction (cf. Fig. 9).

Fig. 9. Subdivision of the full Brillouin zone into reduced zones defined from theelementary cell of y-axis oriented d = 4 BC stripes. Arrows (a) . . . (d) indicate thescan direction for the magnetic dispersions shown in Figs. 10 and 12. The intensityplots of Fig. 8 are shown within the magnetic Brillouin zone indicated by the dashedsquare.

G. Seibold et al. / Physica C 481 (2012) 132–145 139

obtained from the 3-band model calculations are shown). Note alsothat the magnetic weight vanishes identically on the ‘nuclear’Bragg points Qn = n(p/2,0) which only contribute in the chargechannel. As a result the dispersion along the cut (qx,p) (cf.Fig. 10, nh = 0.125) shows the expected behavior for spin-waves

in a striped system [74,82–84]. Starting from the Goldstone modeat Qs one observes two branches of spin waves where the largeintensity one disperses towards QAF = (p,p). The other branch rap-idly looses intensity which is analogous to what is found withinlinear spin-wave theory of the Heisenberg model [74,82–84] forsmall ratios of J\/Jk. Here J\ denotes the (ferromagnetic) exchangecoupling across a stripe whereas Jk denotes the coupling within theAF ordered legs. In addition the intensity of the outwards dispers-ing branch is strongly reduced by disorder [85].

At E � 55 meV the spin excitations have finally shifted towardsthe AF wave-vector producing an intense feature at QAF (middle pa-nel of Fig. 8) which is associated with a saddle-point in the magneticdispersion [Fig. 10 (nh = 0.125)]. At the same time one observes theappearance of intensity in the stripe direction starting from QAF.From Fig. 10 (nh = 0.125) it turns out that this feature is due to aslight softening of the magnetic excitations upon dispersing awayfrom the saddle-point in the direction of the stripe. This softeningis absent at lower doping [cf. Fig. 10 (nh = 0.0625,nh = 0.083)] andindicates the susceptibility towards a magnetic instability alongthe stripe at higher doping. It has been argued [74] that this rotonlike minima explains INS in untwinned samples of YBCO by Hinkovand co-workers [62].

Finally, at higher energies the excitations again spread out andform a ring-shaped feature around QAF with a small anisotropicintensity modulation (Fig. 8, upper panel). Note that our spectraare for one-dimensional stripes without any orientational averageso that C4 symmetry is broken. The resulting quasi two-dimen-sional excitation spectra above the saddle-point energy are inmarked contrast to the one-dimensional characteristics obtainedfrom localized spin models [75,76]. This has profound implicationson the interpretation of INS data from detwinned samples [69,70]which also show pronounced 1-D characteristics at low but a 2-Dmagnetic distribution of magnetic energies at high energies in ac-cord with our result.

The high energy distribution of magnetic intensity can beunderstood from the energy weighted sum rule

M1q �

Z 1

0dx xv00qðxÞ ¼ �

p2N

Xk;r

ðek � ekþqÞnk;r ð5Þ

where ek denotes the single-particle dispersion of the underlyingitinerant model and nk,r is the number operator for particles withmomentum k and spin r. In the presence of time reversal symmetryand inversion (nk,r = n�k,�r) this can be simplified to

M1q ¼

pN

Xd

sin2 qd2

� �Td ð6Þ

and d runs over the vectors connecting a specific lattice site to itsrespective neighbors with only one representative of the pair(d,�d). Thus in case of a square lattice d ¼ x; y for nearest neighbors,d ¼ xþ y; x� y for next nearest neighbors, etc. and Td denotes thekinetic energy in d-direction. Since M1

q weights more the excitationsat higher energies it turns out from Eq. (6) that with increasing xthe intensity in the transverse magnetic susceptibility should beconfined to wave-vectors around QAF = (p,p).

Consider now the case of stripes oriented along the y-direction.Although we are dealing with a perfectly ordered quasi 1D struc-ture we find from our calculations that e.g. for d = 4 BC stripesthe kinetic energy along the y-direction is only �10% larger thanthat perpendicular to the stripes. Neglecting the contribution fromnext-nearest neighbor terms Eq. (6) becomes

M1q ¼

pT x

Nsin2 qxd

2

� �þ pT y

Nsin2 qyd

2

� �ð7Þ

so that even in the presence of stripes the high energy magnetic re-sponse is essentially two-dimensional but slightly anisotropic

ener

gy [m

eV]

(π,0) <− (π,π) −> (0,π)

qxqy

qxqy

qxqy

qxqy

qxqy

qxqy

qxqy

qxqy

50

100

150

200

δ=0.0625 d=8 (π,0) <− (π,π) −> (0,π) d=6δ=0.083 (π,0) <− (π,π) −> (0,π)δ=0.125 d=4

ener

gy [m

eV]

(π,0) <− (π,π) −> (0,π)

50

100

150

200

ener

gy [m

eV]

50

100

150

200

ener

gy [m

eV]

50

100

150

200

ener

gy [m

eV]

50

100

150

200

ener

gy [m

eV]

50

100

150

200

ener

gy [m

eV]

50

100

150

200

ener

gy [m

eV]

50

100

150

200

δ=0.132 d=4 (π,0) <− (π,π) −> (0,π)δ=0.149 d=4 (π,0) <− (π,π) −> (0,π) d=4δ=0.16 (π,0) <− (π,π) −> (0,π)δ=0.167 d=3

(π,0) <− (π,π) −> (0,π)

π/2 3π/4 π/4(π,π) π/20 0 π/2 3π/4 π/4(π,π) π/20 0 π/2 3π/4 π/4(π,π) π/20 0 π/2 3π/4 π/4(π,π) π/20 0

π/2 3π/4 π/4(π,π) π/20 0π/2 3π/4 π/4(π,π) π/20 0π/2 3π/4 π/4(π,π) π/20 0π/2 3π/4 π/4(π,π) π/20 0

δ=0.1 d=5

d=4

d=8 d=6 d=5 d=4

d=4 d=4 d=3

x=0.0625 x=0.083 x=0.1

x=0.132 x=0.149 x=0.16

x=0.125

x=0.167

Fig. 10. Dispersion of spin excitations perpendicular and along y-axis oriented BC stripes for various dopings. Experimental data are from Ref. [61] (nh = 0.1, large circles), Ref.[57] (nh = 0.125, error bars), and Ref. [60] (nh = 0.16, large circles).

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Doping

0

20

40

60

80

100

ωs [

meV

]

Experimentvertical BCdiagonal BCdiagonal site centered

Fig. 11. Doping dependence of the saddle-point excitation xs at QAF for diagonalsite- and bond-centered stripes and vertical bond-centered stripes. Circles witherror bars are experimental data from Refs. [57,58,60,61,63,65]. The horizontaldashed line starting from nh = 0.125 is the expected behavior for a phase separationscenario for stripes.

140 G. Seibold et al. / Physica C 481 (2012) 132–145

around QAF. In Fig. 8 this is exactly the distribution of the opticalmagnetic excitations one finds above the saddle-point energy. Weanticipate that this feature can also be observed in the spin glass-phase of LSCO where due to the imbalance of twin domains theone-dimensional character of low energy spin excitations has beenresolved by inelastic neutron scattering experiments [65]. Above xs

the measurements seem to indicate that the excitations acquireagain the (two-dimensional) character of the magnons in the un-doped system in agreement with the prediction for a striped groundstate [29].

Another interesting aspect of the sum rule Eq. (5), that haspassed unnoticed so far, is that it allows to measure the changein kinetic energy on entering the superconducting state in a waythat is alternative to the proposal in Ref. [64]. The present proposalneeds accurate measurements in the higher part of the spectrawhich should become available with the new pulsed neutronsources.

The overall doping evolution of the vertical stripe magneticexcitations together with available experimental data is depictedin Fig. 10. Here we show the excitations along scans (a,b) sketchedin Fig. 9. It should be noted that the magnetic excitations for SCstripes are almost identical with only slight differences regardingthe intensity distribution and the gap structure of optical branches[73]. It turns out from Fig. 10 that for doping nh > 0.125 the saddle-point excitation xs at QAF rapidly shifts to higher energies togetherwith an increasing softening of the magnetic excitations along thestripe. In fact, the value of xs is determined by the magnetic cou-pling across the domain wall. Because this coupling is mediated byvirtual hopping processes of holes between stripe sites and adja-cent AF regions it strongly depends on the stripe filling factor mas defined above. As we have shown above for nh < 1/8 the fillingof the core of the stripe remains practically constant and xs hasa weak dependence on doping. On the other hand for nh > 1/8 dop-ing proceeds by changing the filling of the stripes at constantincommensurability. This produces a rapid renormalization of theeffective exchange interaction across the core and a concomitantrapid increase in xs.

In Fig. 11 we compare the doping dependence of xs with avail-able experimental data [57,58,60,61,63,65]. The latter seem to indi-cate [60] that in LSCO the saddle-point excitation at QAF stays at leastconstant beyond nh = 1/8 in contrast to what is obtained from the

spectra of doped stripes shown in Fig. 10. It is conceivable that in-stead of doping additional holes into the domain walls these (fluctu-ating) textures in LSCO stay half-filled beyond nh = 1/8 leading tophase separation between free carriers and those which are in-volved in the stripe correlations [25]. On the other hand Wakimotoand collaborators [24] find that at large doping nh = 0.3 scatteringstarts around 80 meV which is compatible with a strong increaseof xs as we predict for the doped stripes. Thus more experimentalwork is needed at intermediate dopings to clarify the situation. Withunderdoping the computations increasingly overestimate the en-ergy of the experimentally determined xs, especially in the diagonalphase. The main reason for the disagreement in this doping range isprobably the increasing disorder character of the stripes. To supportthis idea we have performed [29] linear spin wave theory calcula-tion of disordered diagonal bond centered stripes. Is is found thatalready a small amount of disorder leads to a softening and broad-ening of xs which would bring the theoretical data in Fig. 11 closerto the experimental ones.

Recent experimental effort has been made to measure magneticexcitations in LSCO with resonant inelastic X-ray scattering [166].

(0,0) (π,0)

100

200

50

150

ener

gy [m

eV]

(0,π)

Fig. 12. Magnetic excitations in the nuclear Brillouin zone for d = 4 y-orientedstripes. The spectra correspond to the scans labeled (c,d) in Fig. 9.

0 0.5 1 1.5 2 2.5 3ω [eV]

0

1

2

3

4

5

6

σ(ω

) [a

.u.]

nh=0nh=0.028 (polaron)nh=0.071; D=7nh=0.167; D=4nh=0.222; D=3

0 0.25 0.5 0.75ω[eV]

0

1

σ(ω

)

Fig. 13. Optical conductivity of the three-band Hubbard model evaluated withinthe time-dependent GA for various hole concentrations. In the inset: The evolutionof the mid-infrared peak as a function of doping. The Drude contribution has set tozero in this case.

G. Seibold et al. / Physica C 481 (2012) 132–145 141

Because this technique essentially measures in the nuclear Brillou-in zone (i.e. close to momentum q = 0) we show in Fig. 12 the cor-responding spectra for d = 4 y-axis oriented bond centered stripes.Consider first the scan starting from momentum q = (0,0) [labeled(d) in Fig. 9] in the direction along the stripe. Because the system isstill AF ordered along the stripe (cf. Fig. 2) one has a doubling of theunit cell in this direction. At the same time the doping of the stripeis approximately m � 0.5 so that the low energy structure of thespectra reflects the continuum of spinon like spin-flip particle-holeexcitations for the active band shown in Fig. 6. At higher energy,part of the (high intensity) optical magnon band overlaps withthe spin-flip continuum and dominates the spectrum. In the direc-tion perpendicular to the stripe scan (c) is connected to scan (a) bythe addition of a reciprocal lattice vector (3p/4,p). Therefore thelow energy magnon band which disperses from (0,0) to (p,0) isthe replica of the Goldstone mode which has been discussed above.The same holds for the higher energy branches which, however, inthe nuclear and magnetic zones strongly differ in weight.

7. Optical conductivity

In the previous sections our considerations where mostly basedon the extended one-band model, which should be appropriate aslong as interband transitions (as in the case of spin excitations) donot play a significant role. In the present section we discuss theoptical conductivity of striped systems which involves also higherenergy excitations. We therefore investigate the correspondingspectra in the framework of the three-band model which is againtreated within the time-dependent Gutzwiller approach [25,26].

It is also possible to obtain an optical conductivity that in themain features resembles experiment, using a one-band Hubbardmodel. Indeed, one could certainly argue from the inset to Fig. 1bthat the transition between LHB and UHB mimics the charge-trans-fer excitation of cuprates and therefore should provide a reasonablestarting point. However, the point is that the charge excitations willalso be strongly influenced by the respective stabilitity of bond- andsite-centered stripes which are energetically degenerate in the one-band model. This leads to soft lateral fluctuations of the stripes(phason mode) which are optically active. Because of the degener-acy between the two textures, the phason is soft in the one-bandmodel at low doping. In contrast, in the more realistic three-bandmodel the two textures become only degenerate at higher dopingwhich, as shown below, is in good agreement with experiment.

Based on the three-band hamiltonian Fig. 13 reports the opticalconductivity for AF and inhomogeneous ground states at variousdopings. The parameter set has been taken from the first principlecomputations of Ref. [169] so we have no adjustable parameters.

At very dilute doping (nh [ 0.03) due to the long-range Cou-lomb interaction (not included in our calculations) each hole willbe close to an acceptor preventing the formation of stripes. TheRPA optical conductivity for the corresponding single-hole solution(nh = 0.028) leads to the formation of a doping induced MIR band

close to 0.5 eV and doping induced transfer of spectral weight fromthe charge transfer (CT) band to the MIR region in agreement withexperiment in this doping range [153].

For distant stripes (d = 7) the single-hole MIR band now splitsinto two bands. The one at higher energy is a band of incoherentparticle-hole excitations close to 1.3 eV which provides a theoret-ical explanation for the shoulder to the CT excitation seen at thesame energy in optical absorption through LSCO thin films [167],and electron energy loss spectroscopy [168].

The low energy MIR peak is a collective mode associated withthe soft lateral displacements of the stripe mentioned above. It islocated at 0.3 eV at low doing and becomes soft only at optimumdoping where it merges with the Drude response in good agree-ment with experiment. This soft collective modes are low energyexcitations which have no counterpart in a noninteracting systembreaking the paradigm of Fermi liquid theory [26]. It is also possi-ble that coupling to these modes contributes significantly to thepairing. It is worth to remark that new developments in time re-solved spectroscopy show [171] that the high energy excitationsbelow the CT band do not participate in the pairing but excitationsat the CT band do. The region below 1.6 eV has not been measuredyet.

It is interesting to remark that also in nickelates the MIR peakhas been attributed to the formation of mid-gap states due to thestripe ground state [172].

8. Dynamical stripes

The investigation of stripe ‘dynamical properties’ in the previ-ous sections concerned the excitations on top of a stripe groundstate with long range order. However, as discussed in Sections 2and 6 in most cuprate materials long range static spin correlationsare not observed and also direct evidence for static charge order isquite elusive.

Local probes like STM often point to a broken translational sym-metry in real space specially when the high energy part of thespectra is observed. Momentum space probes like ARPES and INSresemble those of an ordered state at high energy but interpolateto a state without broken symmetry at low energy. This points to-wards a dynamical or a glassy nature of the stripe correlations.

There are two ways in which this dichotomy between low andhigh energy quasiparticle excitations can be solved: (1) proximityto a quantum/classical critical point in the disordered side (Section8.1). (2) Stripes with long range order but with protected low en-ergy quasiparticles (Section 8.2). For the last possibility there are

142 G. Seibold et al. / Physica C 481 (2012) 132–145

also two variants: (2a) Long range order in spin and charge and(2b) Long range order only in the charge.

8.1. Proximity to a quantum/classical critical point

The classical version of scenario (1) consists of thermally disor-dered stripes at finite temperature. This has been the route fol-lowed by Vojta and collaborators [79] who have investigated aLandau model for coupled charge and spin fluctuations in theBorn–Oppenheimer approximation. Although not mentionedexplicitly, this computation also describes a glassy stripe state.

The quantum version of scenario (1) follows from this qualita-tive argument: In physical dimensions a system may have long(but finite) ranged order parameter spatial correlations which arealso long lived close to a quantum critical point. This defines a fluc-tuating frequency x0 above which the systems appears to be or-dered (or at least critical). Then, for energies larger than x0 withrespect to the Fermi level, the spectral function should resemblethe spectral function of an ordered system. This high energy spec-tral weight which normally one would term ‘‘incoherent’’ may inreality carry important information on the momentum structureof the close-by ordered phase. This momentum structure may bedetermined with ARPES from the momentum distribution nk byintegrating the spectral function over a broad energy window orby STM and INS experiments by direct examination of the high en-ergy spectra.

On the other hand, electrons at lower energies average over theorder parameter fluctuations and ‘‘sense’’ a disordered system. Inthis limit we expect Fermi liquid quasiparticles with all their wellknown characteristics like a Luttinger Fermi surface. This dichot-omy in the momentum structure of low and high energy quasipar-ticles is consistent with ARPES data [107,108] on lanthanumcuprates.

In a series of papers [173–176] which were based on a Kampf–Schrieffer type approach [178] we have worked out the scenario offermionic quasiparticles coupled to dynamical charge order fluctu-ations with regard to several experimental observations. These in-clude the angular dependence of the quasiparticle weight inBi2201 [179], the isotope shift of the high-energy electronic disper-sion [180], and the dichotomy in the Fermi surface of high-Tc cup-rates [107,108] as mentioned before.

The Kampf–Schrieffer approach describes quite successfully sit-uations in which the quasiparticles are coupled to locally wellformed order-parameter fluctuations but cannot be derived froma microscopic approach. On the other hand a formal derivationclose to a quantum critical point leads to quasiparticles coupledto diffusive collective modes (CMs). This leads to a form which iscustomary in quantum critical phenomena and has often beenused for spin fluctuations [181]

Dkðq;xÞ ¼ �1

mk þ mkðq� qkÞ2 � ix�x2=Xk

ð8Þ

where k = c, s refers to charge or spin CMs substantially peaked atcharacteristic wavevectors qs � (p,p) and qc � (±p/2,0), (0, ±p/2)respectively. Here the mass mk is the minimum energy requiredto excite the CM and mk is a fermion scale setting the CM momen-tum dispersion. This dispersion is limited by an energy cutoff Kk.The dimensionless quantities mk/Kk are the inverse square correla-tion lengths (in units of the lattice spacing), which measure thetypical size of ordered domains. The ix term establishes the low-energy diffusive character of these fluctuations due to decay intoparticle-hole pairs, whereas above the scale set by Xk the CM hasa more propagating character. This approach has been used toinvestigate the critical behavior of the low-frequency optical con-ductivity [182] and, more recently, to calculate the Raman response

function including self-energy and vertex corrections [183]. In par-ticular, assuming as mediators spin and charge fluctuations withdifferent characteristic wavevectors, it was possible to selectivelyindividuate the charge and spin contributions in the Raman re-sponse function. By fitting the experimental spectra in La2�xSrxCuO4

single crystals within the above theoretical framework, it was foundthat both charge and spin fluctuations were contributing to the qua-siparticle scattering (a distinctive feature of stripe fluctuations).However, it was also found that, upon increasing doping, chargeCMs progressively acquire more relevance while spin fluctuationsbecome weaker and weaker. This indicates that upon increasingdoping into the overdoped regime, stripes naturally evolve into har-monic damped charge-density-wave fluctuations, while spin modeseventually loose weight.

8.2. Protected low energy quasiparticles

STM [132,133] experiments provide strong evidence for thesimultaneous existence of low energy (Bogoljubov) quasiparticlesand manifestations of truly broken translational symmetry at highenergy.

In another recent approach [186,176] we have developed a phe-nomenological description with broken translational symmetrybut a dynamical order parameter which applies to this situation.The theory also accounts for the contrast reversal in the STM spec-tra between positive and negative bias [177], in contrast to meanfield like static order where this effect in general occurs away fromthe Fermi level. Here we show that this theory can be combinedwith the GA computations discussed in Section 4 in order to phe-nomenologically account for the electronic structure of a fluctuat-ing stripe state.

The GA is a renormalized mean-field theory which allows forthe definition of an effective Gutzwiller hamiltonian (cf. e.g. Ref.[136])

HGA ¼ bT þ bV �Xijr

etij cyircjr þ

Xir

kircyircir ð9Þ

where the cðyÞir are now (creation) annihilation operator of the Gut-zwiller quasiparticles. The etij are renormalized hopping amplitudesand kir correspond to local chemical potentials. Both parameter setsdepend on the value of the onsite repulsion but also on the chargeand spin structure of the inhomogeneous ground state.

We want to construct an Ansatz for the self energy due to elec-tron–electron interactions. In order to ensure that the self-energyis physical (i.e. that obeys all analytical properties of an electronicself energy) we map the problem to that of electrons coupled to aset of auxiliary states. The idea is to enlarge the Hilbert space byintroducing the coupling to an additional set of localized states(described by creation and destruction operators f ðyÞnr which thus re-sults in the following (auxiliary) Fano–Anderson Hamiltonian,

Haux ¼ bT þXi;n;r

v i;n cyirfin;r þ h:c:� �

þXi;n;r

einf yin;rfin;r:

As a result the Greens function for the Gutzwiller quasiparticlesdepends on an additional self-energy which is determined by theelectronic structure of the f-states via

RiðxÞ ¼X

n

v2i;n

x� �fin

þ Di � fiðxÞ þ Di ð10Þ

with the constant Di controlling the high frequency limit.Note that this kind of approach automatically maintains the

correct analytical properties of the (frequency dependent) self-en-ergy which replaces the static single-particle potential in Eq. (9).

Fig. 14. Constant frequency cuts of the electronic dispersion for dynamical d = 4bond centered Gutzwiller stripes. (a) x = EF and (b) x = �0.6t. Parameters: U/t = 8,t0/t = �0.2. The frequency structure is obtained from Eq. (12) with x0 = 0.2t andv2 = 0.02.

G. Seibold et al. / Physica C 481 (2012) 132–145 143

For the frequency structure in Eq. (10) we choose a two-poleAnsatz

fiðxÞ ¼ai

x�x0þ bi

xþx0ð11Þ

Di ¼ai � bi

x0ð12Þ

with ai + bi = v2 and v2 controls the strength of the scattering. Thusat each site an asymmetry is introduced in the spectral distributionwhich below is taken to be proportional to the charge modulation.The constant Di guarantees the limit Ri(x = 0) = 0 at each site.Denoting by �k the average local chemical potential the weights ofthe poles are choose as ai ¼ 1=2½1� tanhð1� ki=�kÞ�.

Fig. 14 reports the corresponding scans of the electronic struc-ture at the Fermi energy (a) and at an energy of 0.6t below EF.Clearly the low energy electronic structure resembles that of ahomogeneous system due to the fact that the Ansatz Eq. (12) leadsto Ri(x = 0) = 0. Instead at high energies x x0 the scattering iscontrolled by Ri(x ? 1) = Di which according to the above defini-tions is governed by the local charge and spin structure. One canclearly see the similarities between the electronic structure atx = �0.6t (cf. Fig. 14b) and the reconstructed Fermi surface fromthe static stripe solution shown in Fig. 7a.

Therefore this phenomenological theory reproduces our sce-nario of low energy protected quasiparticles but emerging spectralproperties of charge and spin order at large energies. In contrast tothe Kampf–Schrieffer approach [178] which is based on homoge-neous ground states the present theory explicitly describes sys-tems with broken symmetry, however, this broken symmetrymanifests only at high energies.

9. Conclusions

In this paper we have reviewed evidence for stripe correlationsin cuprate superconductors based on Gutzwiller variational calcu-lations supplemented with Gaussian fluctuations. Because our aimis to provide also a quantitative analysis of static and dynamical

properties, the appropriate parameter set for lanthanum cupratesbased on the extended Hubbard model was derived in Section 3in some detail. Our value of U/t = 8 agrees with estimates fromRef. [146] and the next-nearest neighbor hopping t0/t = �0.2 is inthe range of values as derived from LDA computations in Ref. [187].

The main emphasis of this review is put on the spin excitationson top of stripe ground states. As we have discussed in Section 6the time-dependent Gutzwiller approach yields an excellent quan-titative agreement with the spectra observed by INS when thestripes are static, i.e. in the LBCO compounds. Upon underdopingand in the non-codoped system our calculations in general overes-timate the excitation energies. We attribute this discrepancy to themuch reduced correlation length as extracted from the low energyspectra due to the dynamics of the stripes and disorder effects [29].In this regard it would also be interesting to compute the spin exci-tations on top of the dynamical stripe solutions which have beendiscussed in Section 8.

As one approaches the overdoped regime the nature of stripecorrelations in the ground state is not so clear since one has to takeinto account the melting of the stripes and non-Gaussian fluctua-tion effects will become important. Also it is conceivable thatphase separation between the underdoped stripes and an overd-oped Fermi liquid occurs as discussed in Ref. [25]. Evidence for thisscenario was presented in Section 6 from the doping dependenceof the saddle point energy. Due to the fact that in the phase sepa-ration scenario the population of the domain walls by holes satu-rates a further increase of the saddle point energy with doping ishampered in agreement with the INS experiments in Ref. [60].On the other hand, Ref. [24] is in accord with a substantial increaseof the saddle point energy.

Another possibility is that the anharmonic structure of thestripes is gradually lost and stripes become harmonic dynamicaland damped charge-density wave fluctuations. This leads to themore standard scenario of a second order transition ending into aquantum critical point slightly above optimal doping [3] drivenby a frustrated phase separation mechanism [4]. In this case thecombined effect of charge and spin fluctuations can simply be trea-ted within perturbation theory [188].

The natural question which arises is whether the results of Sec-tion 6 may then also be relevant for an understanding of spin exci-tations in YBCO. In this compounds the acoustic dispersion shows alow-energy spin gap (D � 30 meV at optimal doping) and thusdoes not reach the incommensurate wave-vector at x = 0. There-fore it is more likely that the system is in a quantum disorderedspin phase as suggested by the ladder theories [75–77]. Alterna-tively a scenario of fluctuating stripes where also the charge losesits long-range order can capture the effect of a spin-gap [79]. Oneexpects that the systems show short range order with a correlationlength of the order of D/(⁄c) � 5 a and that for energies larger thanD the system resembles that of an ordered phase. For this reasonone can expect that a computation as the one discussed in Section6 is suitable for a description of the universal high energy spin re-sponse [57,59–61,63,65–68,70]. Again, to capture the excitationsover the full energy range would require an approach which incor-porates the dynamical nature of stripes as the one proposed in Sec-tion 8.

Concluding, we have shown that the analysis of numerousexperiments provides strong evidence for stripe correlations inhigh-Tc superconductors. This more or less hidden electronic ordercan account for both normal-state anomalies and high-tempera-ture superconductivity in the cuprates. In fact, the proximity toan instability (particularly if it is a second-order ‘‘critical line’’ end-ing at zero temperature into a quantum critical point) marking theonset of order naturally brings along abundant fluctuations andleads to strongly temperature- and doping-dependent features,which account for the non-Fermi liquid properties and for a strong

144 G. Seibold et al. / Physica C 481 (2012) 132–145

pairing interaction. The participation of such stripe fluctuations inthe ‘pairing glue’ is suggested by the recent analysis of Raman scat-tering data [188] but in any case additional work has to be done inorder to elucidate the source of strong scattering/pairing betweenthe charge carriers leading to high-Tc.

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