Momentum dependent light scattering in insulating cuprates

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by  F. H. Vernay

Momentum dependent light scattering in insulating cuprates
F. H. Vernay1,2, M. J. P. Gingras1,3, and T. P. Devereaux1,2
1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
2Pacific Institute for Theoretical Physics, University of British Columbia, Vancouver, B. C. V6T 1Z1, Canada
3Department of Physics and Astronomy, University of Canterbury, Christchurch 8020, New-Zealand
(Dated: February 6, 2008)
7 We investigate the problem of inelastic x-ray scattering in the spin−1 Heisenberg model on the
2
0 squarelattice. WefirstderiveamomentumdependentscatteringoperatorfortheA1gandB1gpolar-
0 izationgeometries. Onthebasisofaspin-waveanalysis,includingmagnon-magnoninteractionsand
2 exact-diagonalizations,wedeterminethequalitativeshapeofthespectra. Wearguethatourresults
may be relevant to help interpret inelastic x-ray scattering experiments in the antiferromagnetic
n
a parent cuprates.
J

2
Theadvancesmadein3rdgenerationlightsourceshave tions in parent insulating cuprate compounds.
1
recently provided new insights into the study of electron We remark at the outset that we neglect specific res-

] dynamics in strongly correlatedsystems via resonantin- onant matrix elements relevant to the RIXS process in-
l
e elasticx-rayscattering(RIXS).Detailedinformationhas cludingtransitionsresultingfromthecreationofthecore
- already been obtained on Mott gap excitations and or- hole, and work in the restricted model of the half-filled
r
t bital transitions as a function of doping in the cuprate single-bandHubbardmodel onthe squarelattice to cap-
s
. families1,2,3,4,5,6. However, due to limitations in resolu- ture scattering via magnon creation. Thus we neglect
t
a tion, ∼300 meV near the elastic line, excitations at low specific pathways of charge excitation, such as the Cu
m energies remain hidden. On the other hand, low energy K-edge 1s-4p transition, and how double-spin flips may
- excitations at small photon momentum transfers, e.g., occur near the site where the core hole is created. Such
d phonons, magnons,andelectron-hole excitations in met- kinematic details are indeed important to determine ac-
n alsandsuperconductors,havebeenwellcharacterizedvia curately the RIXS intensity as well as the proper sym-
o Raman spectroscopy7. Yet, achieving a detailed under- metry and polarizationof spin excitations which may be
c
[ standing of the momentum and polarizationdependence probed by specific x-ray transitions. However, in order
of low energy excitations in strongly correlated matter to obtain a preliminary understanding of how the two-
1 would greatly clarify the interplay between various de- magnon response can be probed and how polarization
v
greesoffreedom,suchasantiferromagnetism,chargeden- may enter, we first simply focus on the evolution of the
9
8 sity wave order, and superconductivity, often present in two-magnon response, far from any specific resonance,
2 system with many competing interactions8. for nonzero photon momentum transfers q.
1 In the energyrangebelow a few hundredmeV lies one Transitions via light scattering can be created via
0 ofthemostprominentfeaturesobservedviaRamanscat- dipole or multipole matrix elements involving states
7 tering in Heisenberg antiferromagnets - the two-magnon within the conduction band or out of the valence band.
0
feature at energy ∼ 2.7J, with J the nearest-neighbor These transition may be selected by orienting incident
/
t magnetic exchange. While the peak frequency of the and scattering polarization light vectors, ˆe and ˆe , re-
a i f
m broad two-magnon peak in Raman scattering is well un- spectively. Thephotonsenteringinthescatteringprocess
derstood, the asymmetry of the lineshape as well as are represented by (ω ,k ,ˆe ) where indices i and
- i,f i,f i,f
d the polarization dependence remains unexplained, even f represent the incoming and outgoing photon, respec-
n though it has been lavished with attention9. Prelimi- tively. Since we are interested in the insulating phase,
o nary RIXS on the Cu K-edge in LaCuO410 and M-edge thescatteringoflightisinducedbytheinterbandpartof
c in CaCuO211 have shown evidence for low energy two the operator. Following Refs. [13,14], we derive a finite
:
v magnon scattering. Since in the near future this low en- momentum transfer q, q ≡ k −k , scattering operator
i f
Xi ergy window will open for inelastic x-ray studies, it will fordifferentpolarizationgeometries. The interbandpart
afford an opportunity to study the dynamics of magnon of the scattering matrix element is given by:
r
excitations via the charge degrees of freedom which will
a
complement neutron and Raman scattering studies.
Lorenzana and Sawatzky12 investigated the proper- hf|Mr|ii = Pνhhf|Jkf·ˆeǫfν|−νiǫhiν−|ωJ−iki·ˆei|ii (1)
tfeiersroomfpaghnoentosn,-aanssdisitnevdeastbisgoartpetdiotnheinb2i-DmHagenisoennbsperegctarnutmi- + hf|J−kiǫ·ˆeνi−|νǫii+hνω|Jfkf·ˆef|iii
probedbyinfraredconductivity. Inthispaperweextend
where ν represents states out of the lower
these calculations to present a theory of inelastic x-ray
Hubbard band, and J is the current op-
scattering of the two-magnon response in a Heisenberg k
antiferromagnet. Inparticularweshowthatthe momen- erator Jk = Pp,σ ∂∂ǫpp c†p+k2,σcp−k2,σ, with
tum and polarization dependence can provide detailed ǫ = −2t[cos(p a)+cos(p a)] for a square lattice
p x y
information on the nature of magnon-magnon interac- with lattice constant a, and nearest-neighbor hopping t.
2
The current may be expressed as: operatorwhich,beingproportionaltoH,commuteswith
H, giving no Raman scattering in that channel.
Jkf ·ˆef = itPr,δ,σeˆδfe−ikf·(r+δ/2) One alsofinds OA1g,B1g vanishfor q=Q=(π,π) and
× hc†r+δ,σcr,σ−c†r,σcr+δ,σi (2) symmetryrelatedpoints forbothA1g andB1g. This is a
consequence of including only nearest neighbor spin in-
≡ eˆδJ (r,δ),
Pr,δ,σ f kf teractions in the Heisenberg model21. While in our case
theresponsevanishesfortheantiferromagneticreciprocal
δ
whereeˆ istheprojectionofthepolarizationvectoralong
f lattice vectors, if one includes longer range interactions,
the neighbor direction δ. In the insulating state, |ii and then this restrictionmay be lifted19. Thus the x-rayRa-
|fi are single occupied states, whereas |νi are excited man response for these wavevectors may provide a win-
statesconsistingofa doublyoccupiedstateatsite r+δ. dow to sample the role of longer-rangespin interactions.
Thus the energy of the excited state is roughly given by Thescatteringintensityisproportionaltohf|O |ii2,
ǫν ≈U+ǫi,andtheintermediatestatesmaybecollapsed while satisfying energy conservation ω =ωk+ωk+Bq1g. We
using the identity 14 −Si·Sj = 21c†i,σcj,σ c†j,σ′ci,σ′, which first focus on the B1g channel, where the two-magnon
simply means that the exchange process (and therefore scatteringis mostprominentfor q=0. The leftpanelof
a doubly occupied site) is allowed only if spins σ and σ Fig. 1showstheB barescatteringintensity,neglecting
′ 1g
are opposite14. Finally, we recast hf|M |ii in Eq. 1 via magnon-magnon interactions: I (q,ω) = 1 δ(ω +
r 0 2N Pk k
the following scattering operator: ω −ω)b2 ,whereb isthetermincurlybracketsin
k+q k,q k,q
Eq. 5. The I (q = 0,ω) intensity recovers the standard
O(q) = 8t2Pr,δeiq·(r+δ/2)(δˆ·ˆef)(δˆ·ˆei) (3) Raman respon0se, which for the non-interacting case has
× (cid:0)41 −Sr·Sr+δ(cid:1)hU+1ωf + U−1ωii. astapteeaskatatkω==(π4,J0)dpureojteocttehdeoluartgbeymtahgenBon1gdoepnesritaytoorf.
Fornonzeroq,twomagnonsarecreatedwithwavevectors
For crossed polarizations, ˆe = 1 (xˆ±yˆ), transform-
i,f √2 k of different magnitude and direction, leaving behind a
ing as B , and for parallel polarizations along xˆ and yˆ,
1g reorganizedspinconfiguration,andthe response is given
transformingasA ,weobtainthefollowingexpression:
1g by convolving two magnon density of states at different
k separated by q. We note that while the response is
×Pi,δPAO1,BA11g(,δB)1[gS(qi·)S=i+−δ8]tc2ohs(Uq+1·ωrfi++qU−·1ω2δi)i, (4) idssteihnllisgigthoylvyoefrsnustepadptrebesysbse∼edina4gsJlaqfrogarepspfitrnaoittaeckhq=esd(Quπe.,0t)o,tthheeinmtaengnsiotny
withP (δ)=1,forδ =axˆ,andequals1,-1forδ =ayˆ
A1,B1
forA andB ,respectively. Thisisafinitemomentum
1g 1g
generalization of the usual Loudon-Fleury light scatter-
ing operator, and for q = 0 the above expressions give
the standard Raman results13,14. We note that in gen-
eralsubtractions ofspectra for different polarizationori-
0
entations are needed in order to fully extract symmetry
2.0
deconvolved spectra7.
4.0
Henceforth,werestrictourselvestothehalf-filledHub-
6.0
bard model with t/U small, and focus on its spin-1 an-
2 8.0
tiferromagnetic Heisenberg representation with Hamil-
10
tonian H. We first investigate the q-dependent in-
elastic response via the spin-wave (SW) approxima-
tion. We proceed as usual9,14,15,16,17,18,19,20 and ex-
press H in its SW representation, HSW = Cst + 1 2 3 4 1 2 3 4 5
Pkωk(cid:16)α†kαk+βk†βk(cid:17), with ωk = JSZp1−γk2 the /J
magnon dispersion, and 2u2 −1=2v2+1=1/ 1−γ2
and γ = (1/2)[cos(k ak)+cos(kka)]. Peprforminkg FIG.1: Spin-waveintensityspectrumwithout (I0(q,ω),left)
k x y and with (IRPA(q,ω), right) magnon-magnon interaction.
the same transformations to the scattering operators
The color intensity scale is 2.2 times smaller for the right
O we obtain:
A1g,B1g panel than theleft.
OA1g,B1g ∝Pkβk†α†k+q(cid:8)−(cid:2)cos(qx2a)±cos(qy2a)(cid:3)
×(ukvk+q+vkuk+q)+(ukuk+q+vkvk+q) Magnon-magnon interactions need to be in-
× cos(k a)cos(qxa)±cos(k a)cos(qya) , cluded to describe more realistically the local spin
(cid:2) x 2 y 2 (cid:3)(cid:9)
(5) rearrangement9,14,15,16,17,18,19,20. In particular, interac-
where we have neglected the prefactor in Eq. 4. For A tions lead to a reduction of the peak frequency for B
1g 1g
and q = 0, we recover the familiar form of the Raman Raman to 2.78J, where J, as shown by Singh et al.22,
3
is also renormalized by quantum corrections. Many for the results presented here, favoring q = 0 for bi-
diagrams contribute to the magnon-magnon vertex magnonin B channel, anda growthof A component
1g 1g
corrections,andasubsethas beeninvestigatedfor q=0 withincreasingq. Thus the A responseis moresimilar
1g
Raman scattering19. In that case, magnon-magnon to the infrared one, as it weights out similar regions in
interactions lower the relevant energy scale from 4J to the magnetic Brillouinzone (BZ). We note that phonon-
3J due to the local breaking of six exchange bonds for assistedbi-magnonresponseforx-rayRaman,considered
two neighboring spin flips. For finite q considered here, in the same vein as Ref. [12], would also have similar
the two magnons are created with net momentum q polarization dependent form factors as those considered
which distribute the spin flips over longer length scales. here, coming from the symmetry classification of the ac-
Since magnon-magnon vertex corrections are expected tive phonon modes involved in the scattering.
to weaken as the spin arrangement occurs at larger Inordertoexplorethesemi-quantitativevalidityofthe
lengthscales,we approximatethe renormalizedresponse SW/RPA results, we investigate the q-dependent inelas-
by a generalized random-phase approximation (RPA) tic x-ray scattering for the Heisenberg spin Hamiltonian
form I (q,ω) ∼ I0(q,ω) , where the function H, using an exact diagonalization approach, consider-
RPA 1+J(q)I0(q,ω) ing both the A and B channels. Although exact-
J(q) = [cos(q a/2)+cos(q a/2)]/2 is taken to recover 1g 1g
x y
diagonalization is limited because of the prohibitive size
the Raman form at q=0 and properties of the solution
of the Hilbert space, Gagliano and Balseiro23 demon-
to the Bethe-Salpeter equation for finite q. A proper
strated that it is a powerful technique that allows to
treatmentofmagnon-magnoninteractionsforallqinthe
compute the dynamical quantities easily. Noteworthy
SW framework is a topic of future research. As shown
in the context of numerical investigations of Raman
in the right panel of Fig. 1, the magnon interactions
spectra, Sandvik et al.15 showed that even though the
for larger q bring the peak down to ω = 2.78J for B
1g
spectra obtained by the Lanczos method are extremely
q=0 Raman, but the general weakening of the magnon
size-dependent, they are of direct relevance to test the
interactions at larger q gives back the unrenormalized
response calculated from a spin-wave analysis. As in
responseatlargerqwith,inparticular,theresponsestill
Ref. [24], the spectra are evaluated by computing the
vanishing at q = Q. On the other hand, the dispersion
continued fraction:
of the peak changes dramatically when magnon-magnon
interactions are included, where the peak hardens at
finite q from the Γ-point, q=0. 1 1
I(q,ω)=− ImhΨ |O (−q) O(q)|Ψ i (6)
g.s. † g.s.
π z−H
withz =ω+iǫ+E ,whereE istheground-stateenergy,
0 0
ǫ is a damping factor.
Wefirstcomputetheground-stateenergy. ForN =16
sites we have E /(NJ) ≈ −0.70178020 and wave-vector
0
|Ψ i. We then begin evaluating the continued fraction
g.s.
with the starting state:
O(q)|Ψ i
|Φ(q)i= g.s. (7)
hΨ |O (−q)O(q)|Ψ i
p g.s. † g.s.
Here, compared to Raman scattering, care has to be
taken in computing Eq. (6): the state |Φ(q)i being in
a different subspace at nonzero q than the ground-state,
the matrixelementsofH tocompute the continuedfrac-
FIG. 2: 16-site cluster : exact diagonalizations for A1g and tionhaveto be expressedinthe correspondingsubspace.
B1g polarizations, to be compared with spin-wave results. The results are summarized in Figs. 2-3.
The spectra are shown for the independent momenta q al-
For the caseof B scattering,we recoverprior results
lowed for the N =16 site cluster (given on the side of right- 1g
forq=014,whileweseethatthepeakdispersestohigher
most panel). The relative intensity of the curves is given by
hO(q)2i. energies,approaching4J both for q along the BZ diago-
nal as well as along the axes. In light of the RPA results
in the right panel of Fig. 1, we attribute this dispersion
The results above differ slightly from those obtained as a weakening of magnon interactions at larger q. At
in Ref. [12] for infrared absorption from multi-magnons. the same time, the overallintensity diminishes for larger
Whilethedispersionofthepeaksissimilartoourresults, q due to the form factor appearing in Eq. (4).
Ref.[12]foundabi-magnonresponsewhichisdominated Sinceexactdiagonalizationsdealwithsmallclusters,it
by a sharppeak with a small bi-magnonlifetime for mo- is very difficult to make a quantitative finite-size scaling
mentum transfers (π,0). Since the form factors are dif- analysisforthespectralshape15. Thereforewerepeatthe
ferent for infrared and polarized x-ray Raman measure- calculation for the B polarization with a 20-site clus-
1g
ments, the bi-magnonspectrumhas differentprojections ter, and plot the results in Fig. 3. It is well known that
4
nated by the B channel for small q, and a mixture of
1g
A and B for larger q 26. The main results are thus
1g 1g
(1)forsmallq,thetwo-magnonRamanpeakrapidlydis-
persesupwardinenergy,bothforqalongtheBZdiagonal
and BZ axes. At the same time, a contribution from the
A channel develops, as the intensity of the B falls
1g 1g
with increasing q. We note, however, that the problems
ofdescribingthetwo-magnonprofileforq=0,whichin-
cludes the differences between A and B intensities,
1g 1g
as well as the width of the lineshapes, remains an issue
hereaswell(see,e.g.,Ref. 9). Thus moredetailedtreat-
ments including further spin-spin interactions, ring ex-
change, and possibly spin-phonon coupling, would need
tobeincorporatedinlowenergyinelasticx-rayscattering
FIG. 3: 20-site cluster : exact diagonalizations in the B1g as well. Besides affecting the overall lineshape, the be-
channel. Theinsetisaschematicoftheinvestigatedq-points. havioratdifferentmomentumpoints, suchasthe special
point at (π,π) may change. Thus, the detailed momen-
tum dependence of the spectra may be able to provide
importantinformationofthetypesandextensionsofspin
the16-siteclusterhasadditionalsymmetries(hypercube-
like)25 and it therefore important to check for similar q- interactions in antiferromagnets.
points in the 20-site cluster. Due to the different shape
Finally we remark on our results in terms of the cur-
and boundary conditions of this cluster, we are not able
rentcapabilitiesofRIXS experiments. Empirically,from
to explore the same momenta in the BZ. However direct
fitting the two-magnon position and spin wave velocity,
inspection of the q points (0,0), (π,0) and (π,π) reveal
J in the cuprates is estimated to J ≈0.13 eV. Thus the
that many of the features are qualitatively similar: the
two-magnonRamanpeakwouldliegenerallyobscuredin
q=0 peak is at lower frequency than (π,0), and the in-
the elastic line at currently available resolution, which is
tensity weakens at finite q ≡ |q| as in the 16-site case
severalordersofmagnitudelargerthantheinelasticcon-
(compare with the middle panel of Fig. 2). In addition,
tribution. It is a possibility that the two-magnoncontri-
the dispersion along the path in the q-space is of the
bution would emerge from underneath the elastic line at
same order ∼ J. In the RPA calculation, the minimum
largermomentumtransfersqasthepeakinx-rayRaman
frequency of the main peak is at (0,0). For the com-
dispersesto4J. Iftheenergyresolutioncanbeenhanced,
patible q-points,the resultsareconsistentwiththe RPA
weproposethatperhapsacleartwo-magnoncontribution
approach as well (same qualitative dispersion and mini-
will become visible. However, a proper treatment of the
mum). We therefore believe that exact diagonalizations
resonantmatrix elements needs to be considered, which,
give a correct qualitative description of the response.
while not affecting the dispersion, may change the rela-
For A , a peak appears for finite q21. Generally the
1g tive intensity of the spectra at specific q. This remains
peak disperses towards 4J for larger q, following essen- a topic for future research.27
tially the B spectrum. The B intensity falls with
1g 1g
increasing q while at the same time the A spectral in- Note added in proof: After completionand submission
1g
tensity grows. In fact, we note that the intensity for A ofourworkwebecameawareofacloselyrelatedpreprint
1g
approaches that of the B spectrum for q = (π,π/2), by Donkov and Chubukov (cond-mat/0609002) The re-
1g
where the scattering operators in Eq. (4) are the same. sults are similar at small momenta but they differ for
Thus, the overallintensity in an unpolarized measure- q = (π,π) due to a slight difference in the form of their
ment, given by I +I as shown in Fig. 2, is domi- scattering operator.
A1g B1g
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