Collective excitations in unconventional charge-density wave systems

0.42 MB·English

by  A. Greco

Collective excitations in unconventional charge-density wave systems
6 A. Grecoa,b and R. Zeyhera
0 aMax-Planck-Institut fu¨r Festk¨orperforschung,
0 Heisenbergstr.1, 70569 Stuttgart, Germany
2 bPermanent address: Departamento de F´ısica,

n Facultad de Ciencias Exactas e Ingenier´ıa and
a IFIR(UNR-CONICET), Av. Pellegrini 250,
J 2000-Rosario, Argentina

7 (Dated: February 6, 2008)
1
The excitation spectrum of the t-J model is studied on a square lattice in the large N limit in

a doping range where a d-density-wave (DDW) forms below a transition temperature T⋆. Char-
]
l acteristic features of the DDW ground state are circulating currents which fluctuate above and
e condense into a staggered flux state below T⋆ and density fluctuations where the electron and the
-
r holearelocalizedatdifferentsites. Generalexpressionsforthedensityresponsearegivenbothabove
st and below T⋆ and applied to Raman, X-ray, and neutron scattering. Numerical results show that
. the density response is mainly collective in nature consisting of broad, dispersive structures which
t
a transform into well-defined peaks mainly at small momentum transfers. One way to detect these
m excitations is by inelastic neutron scattering at small momentum transfers where the cross section
(typically a few per cents of that for spin scattering) is substantially enhanced, exhibits a strong
-
d dependence on the direction of the transferred momentum and a well-pronounced peak somewhat
n below twice the DDW gap. Scattering from the DDW-induced Bragg peak is found to be weaker
o bytwo orders of magnitude compared with themomentum-integrated inelastic part.
c
[ PACSnumbers: 74.72.-h,74.50.+r,71.10.Hf

1
v
I. INTRODUCTION density-density correlationfunction, whereγ is givenby,
2
8
8iπeµ t
3 Theelectrondensityoperatorρonalatticehasinmo- γ(k,q) = 0 eff
1 ~cq ×
mentum space the general form, | |
0 α
j
( (1 cosq )cosk +sinq sink ) , (2)
06 ρ(q)= N1 γ(k,q)c†σ(k+q)cσ(k). (1) (cid:16)jX=x,y qj − j j j j (cid:17)
t/ c Xkσ with
a
m c†σ(k),cσ(k) are electron creation and annihilation oper- µ q
- atorswithmomentumk andspinprojectionσ, Nc is the α = µ × |q|. (3)
d number of primitive cells, q the transferred momentum, 0
n
and k characterizes the relative motion of electron and TheexpressionEq.(2)appliestoasolidofsquarelayers
o
c hole. If both particles reside always on the same lattice with lattice constant a which we put, together with ~ to
: site γ is independent on k and ρ describes a usual den- oneinthe following. t isaneffectivenearestneighbor
v eff
sitywave. Thiswavemaycondenseforsomewavevector hopping integral, c the velocity of light, µ the magnetic
i
X q different from the reciprocal lattice vectors of the un- µ2
moment of the neutrons, and µ = . We neglect
r modulatedlatticeandaconventionalchargedensitywave 0 q 3
a (CDW)stateisobtained. Iftheelectronandholeexplore hopping between the layers as well as a small contribu-
differentsitesγdependsonkandthecondensationofthis tion to γ from second-nearest neighbor hoppings t′.
∼
wave yields a CDW state with an internal symmetry de- Similarly, inelastic light and non-resonantX-ray scatter-
scribed by the k dependence of γ. We will call this state ingaredeterminedbytheRamanortheX-rayscattering
in the following an unconventional CDW state1. Pro- amplitude which correspondsto the following expression
posals for systems with an unconventional CDW state for γ7,
include organic conductors1,2 and the pseudogap phase
∂2ǫ(k+q/2)
of high-temperature superconductors3,4,5,6. γ(k,q)= es ei. (4)
α ∂k ∂k β
Static and fluctuating density waves of the general X α β
αβ
form of Eq.(1) can be observed by various experimental
probes. The vector potential generated by the magnetic ei andes arethepolarizationvectorsofthe incidentand
moment of neutrons influences the hopping elements of scatteredlight,respectively,andǫ(k)theone-particleen-
electrons via the Peierls substitution. Thus neutrons ergy. Incontrastto Refs.3,7we assumedthatthe Peierls
probe orbitalmagnetic fluctuations associatedwith den- substitution is made in the renormalized Hamiltonian
sityfluctuations. AsshowninRef.3theresultingneutron so that the renormalized hopping t and one-electron
eff
crosssectionisrelatedtotheimaginarypartofaretarded band appear in Eqs.(2) and (4). For Raman scattering
2
the momentum transfer is practically zero whereas no Section III contains numerical results for the formulas
such restriction exists for X-ray scattering. presented in section II. In subsection A the density fluc-
Microscopic models for interacting electrons usually tuationspectruminthed-wavechannelwillbediscussed
contain interactions between local charge densities or asafunctionofmomentum,bothaboveandbelowT∗. In
spin densities, such as, the Coulomb or Heisenberg in- subsectionsBandCresultsforthecrosssectionsfornon-
teraction. k-dependent densities of the form of Eq.(1) resonantinelastic X-rayscattering andfor polarizedand
become important if the exchange terms are competing unpolarizedneutronscatteringwillbegiven,andsubsec-
or dominating the direct, Hartree-like terms. The exci- tion D compares the magnitude of q integrated neutron
tonic insulator is such a case where the Coulombic ex- cross sections for spin and for orbital scattering. Section
change terms not only create excitons but force them to IV contains our conclusions.
condenseintoanewgroundstate8. Theseideaswereap-
plied to high-T cuprates by Efetov9 assuming that the
c
barelyscreenedCoulombinteractionbetweenlayerspro- II. DENSITY RESPONSE IN THE D-CDW
duces interlayer excitons and their condensation. More STATE
recently,itwasrecognizedthatthet-J modelinthelarge
N limit (N is the number of spin components) shows a In Refs.16,17 it has been shown that the density re-
transition to a charge-density wave state with internal sponse in the normal state of the t-J model at large N
d-wave symmetry (DDW) at low doping4 with a tran- canbeobtainedbyusingthefollowingtwosetsofdensity
sition temperature T⋆. This model describes electrons operators,
in the CuO planes of high-T cuprates and identifies
2 c
the pseudogap phase at low dopings with a DDW state. ρ′ (q)= 1 E (k,q)c†(k+q)c (k), (5)
One may expect that near the phase boundary to the α Nc X α σ σ
kσ
DDW state or in the DDW state density fluctuations of
the form of Eq.(1) become important and, according to
the abovediscussion,couldbe detectedbyneutronorX- 1
ρ (q)= F (k)c†(k+q)c (k), (6)
ray scattering. It is therefore the purpose of this paper β N β σ σ
c X
to calculate the general density response of this model, kσ
both above and below T⋆, and to make predictions for
with
the magnitude and the momentum and polarization de-
pendence of neutron and X-ray cross sections.
E (k,q)=(1,t(k+q)+J(q),γ (k),γ (k),γ (k),γ (k)),
α 3 4 5 6
The order parameter of a commensurate DDW state (7)
withwavevector(π,π)ispurelyimaginarywhichfollows and
from the hermiticity of the large N Hamiltonian. As a
consequence, circulating, local currents accompagny in F (k)=(t(k),1,2Jγ (k),2Jγ (k),2Jγ (k),2Jγ (k)).
β 3 4 5 6
general density fluctuations in the DDW state produc- (8)
ingorbitalmagneticmomentslocalizedontheplaquettes Compared to Refs.16,17 a slight change in the represen-
of the square lattice. The circulating currents fluctuate tation of the basis functions has been made by using the
above and freeze into a staggered flux phase below T⋆. symmetrized functions
Most previous calculations considered q = 0 quantities
suchasthefrequency-dependentconductivity10,11,12,13or γ = (cosk cosk )/2,
3,5 x y
Ramanscattering10,14. Calculationsforfinitemomentum ±
γ = (sink sink )/2, (9)
4,6 x y
transfers15 to be presentedbelow areinteresting because ±
theyprobethelocalpropertiesofthedensityfluctuations wherethesubscripts3and4refertothe+and5and6to
and the flux lattice. the sign. Here,andinthefollowing,weputthelattice
−
InsectionII we willfirstreformulatethe largeN limit constantofthe squarelatticeto 1. t(k) andJ(k)arethe
of the t-J model in terms of an effective Hamiltonian Fouriertransformsofthehoppingamplitudest andthe
ij
which contains usual electron creation and annihilation Heisenberg term J , respectively. Including nearest and
ij
operators (i.e., no constraint operators), renormalized second nearest neighbor hoppings t and t′, respectively,
bands and an interaction between several charge density we have
waves. Thiseffectivemodelcontainsasaspecialcasethe
models used in Refs.1,10 in discussing anomalous charge t(k)=2t(coskx+cosky)+4t′coskxcosky. (10)
density waves. We then extend previous normal-state
calculations of the density response, given by a 6x6 sus- The corresponding density-density Matsubara Green’s
ceptibility matrixχ,totheDDWstateusingtheNambu function matrix is defined by
formalism. The obtained expressions hold for all mo-
1/T
mentaqandaregeneralenoughtodiscussthedispersion χ (q,iω )= dτe−iωnτ <T ρ′ (qτ)ρ†(q0)>.
of collective modes and the momentum and polarization αβ n −Z τ α β
0
dependencies of experimental cross sections. (11)
3
Thegeneralsolutionforχinthenormalstatehasbeen into parts inside and outside of the RBZ and the two
given in the large N limit and the dispersion of macro- parts then combined using the Nambu vectorsand Pauli
scopic density fluctuations, determined by the element matrices. We obtain then,
χ ,havebeendiscussed16,18. Atlowertemperaturesand
12
dopings the density component E5(k,Q) freezes in for a ρ′ (q)= 1 ′ψ†(k+q)Eˆ (k,q)ψ (k), (15)
wave vector Q approximately equal to (π,π), leading to α NcX σ α σ
kσ
the DDW state.
The Hilbert space of the t-J model does not contain
double occupancies of sites, its operators are therefore 1 ′
ρ (q)= ψ†(k+q)Fˆ (k)ψ (k), (16)
X and not the usualcreationand annihilationoperators β N σ β σ
cX
c† and c of second quantization. In the large N limit kσ
iσ iσ
the t-J model becomes, however, equivalent to the fol- with
lowing effective Hamiltonian in terms of usual creation
and annihilation operators, Fˆ (k) = t(1)(k)P (k+q,k)+t(2)(k)P (k+q,k),
1 o e
Fˆ (k) = P (k+q,k),
6 2 e
N
Heff = ǫ(k)c†σ(k)cσ(k)− 2c ρ′α(q)ρ†α(q). Fˆβ(q) = Fβ(k)Po(k+q,k), (17)
X XX
kσ α=1 q
(12) for β =3...6,
ǫ(k) are the one-particle energies in the large N limit,
Eˆ (k,q) = Fˆ (k),
1 2
δ J(k) 1
ǫ(k)= 2t(k)− 2 · N cos(px)f(ǫ(p)−µ). (13) Eˆ2(k,q) = t(1)(k+q)Po(k+q,k)
c Xp + (t(2)(k+q)+J(q))P (k+q,k),
e
µistherenormalizedchemicalpotentialandf theFermi Eˆ (k,q) = Fˆ (k)/2J, (18)
α α
function. The second term in Eq.(12) represents an ef-
fectiveinteractionoriginatingfromthefunctionalderiva- for α=3...6, and
tive of the self-energy with respect to the Green’s func-
tion. This interaction is hermitean though its present P (k+q,k) = p(k+q,k)σ i(1 p(k+q,k))σ ,
o 3 2
− −
formdoesnotshowthispropertyexplicitlybecauseofthe P (k+q,k) = p(k+q,k)σ +(1 p(k+q,k))σ .
e 0 1
chosen compact form to represent it. Using Eqs.(5) and −
(19)
(6)oneeasilyverifiesthatthe sumofthe termsα=3...6
representtheHeisenberginteraction,writtenasacharge- t(1)(k)andt(2)(k)aretheFouriertransformsofthenear-
charge interaction and as a sum of four separable ker-
est and second-nearest neighbor hopping amplitudes, σ
0
nels. The terms α = 1,2 originate from the constraint
is the 2x2 unit matrix, and σ ,σ ,σ are Pauli matri-
1 2 3
which acts in H as a two-particle interaction. H
eff eff ces. The dash at the summation sign in Eq.(16) means
and the original t-J Hamiltonian in the large N limit
a restriction of the sum to the RBZ. p(k+q,k) with k
become equivalent if only bubble diagrams generated by
in the RBZ is equal to one if k+q, reduced to the BZ,
the second term in H and no self-energy corrections
eff lies in the RBZ and is zero otherwise. Finally, the first
are taken into account. In the DDW state < ρ (Q) >
α and second terms in H can be combined and read in
and < ρ′ (Q) > become non-zero for α = 5. We rewrite eff
α Nambu notation as
H as,
eff
H(0) = ′ψ†(k) (ǫ (k) µ)σ
Heff = ǫ(k)c†σ(k)cσ(k)−Nc <ρ′5(Q)>ρ†5(Q) eff Xkσ σ h + − 0
X
kσ
+ǫ (k)σ +Φγ (k)σ ψ (k). (20)
6 − 3 5 2 σ
Nc ρ˜′ (q)ρ˜†(q). (14) i
− 2 α α
αX=1Xq The energies ǫ± are defined by (ǫ(k) ǫ(k+Q))/2. Φ is
±
anabbreviationfortheamplitude oftheDDWandmust
The tilde at the density operators means that only their be real because of the hermiticity of H .
eff
fluctuating parts should be considered. Using the above Nambu representation the suscepti-
In order to be able to use usual diagrammatic rules in bility matrix χ is obtained by a bubble summation,
αβ
the DDW state we introduce the Nambu notation, i.e.,
therowvectorψ†(k)=(c†(k),c†(k+Q))andthecorre-
σ σ σ
sponding column vector obtained by hermitean conjuga- χ (q)= χ(0)(q)(1+χ(0)(q))−1. (21)
αβ αγ γβ
tion. We also make the convention that the momentum X
γ
in ψ or ψ† are always taken modulo the reduced Bril-
louin zone (RBZ), i.e., lie in the RBZ. The momentum Here we used the abbreviation q = (iν ,q) with the
n
sums over the (large) Brillouin zone (BZ) are then split bosonic Matsubara frequencies ν = 2πTn, where T is
n
4
the temperature. χ(0)(q) stands for a single bubble and 50
is given analytically by the expression δ=0.077 re 0.08
te0.06
χ(α0β)(q)= NTcXkn′TrhG(k+q)Eˆα(k,q)G(k)Fˆβ†(k)i, (22) 3400 q=(π,π) marap redro00..000240 0.01 0.02 0.03 0.04
with the Nambu Green’s function χ"55 temperature T
T=0.032
(iωn+µ ǫ+(k))σ0+ǫ−(k)σ3+Φγ5(k)σ2 20 T=0.030
G(k)= − ,
(iωn E1(k))(iωn E2(k)) T=0.029
− − (23) 10 T=0.028
and the two eigenenergies
0
E (k)=ǫ (k) µ ǫ2(k)+γ2(k)Φ2. (24) 0 0.005 freque0n.01cy ω 0.015 0.02
1,2 + − ±q − 5
After performing the trace over Pauli matrices and the
sum over fermionic Matsubara frequencies in Eq.(22) FIG. 1: Negative imaginary part of χ of the t-J model at
55
only the sum overmomenta is left for a numericalevalu- large N for t′ = −0.3, η = 0.001, q = (π,π), and doping
ation. In the normal state we obtain δ = 0.077. Inset: DDW order parameter Φ as a function of
temperature T.
χ(0)(q,iω )=
αβ n
1 f(ǫ(k+q)) f(ǫ(k))
N Eα(k,q)Fβ(k)ǫ(k+q) ǫ−(k) iω .(25) χ(000)(q) is given by Eq.(22) where both α and β have
c Xk − − n been replaced by 0. Similarly, the expressions for χ(0)
0β
Two special cases of Eq.(21) should be mentioned. If and χ(0) are obtained from Eq.(22) by replacing only α
α0
q is parallel to the diagonal, i.e., of the form q = (q,q), or β, respectively, by 0. χ is given by Eq.(21).
αβ
the components 5 and 6 in the matrices, which are odd
under reflectionatthe diagonal,decouple fromthe other
components. At the points q = (0,0) and (π,π) they III. NUMERICAL RESULTS
even decouple from each other so that Eq.(21) can be
solved by division, for instance19, A. D-wave susceptibility χ
55
χ (q)=χ(0)(q)/(1+χ(0)(q)), (26)
55 55 55 From now on we will write t = t and use t as the
−| | | |
for q = (0,0) and q = (π,π). In the first case χ energyunit. Furthermore,theHeisenbergconstantJ and
55
describes the B component of Raman scattering14, the dopingδ willbe fixedto the values0.3and0.077,re-
1g
in the second case amplitude fluctuations of the order spectively. The mean-fieldvalue forthe superconducting
parameter15. The other special case refers to a DDW transitiontemperatureisthenpracticallyzerosothatwe
without constraint10,15. Since the components 1 and 2 dealwith a pure DDW system. Fig.1 showsthe negative
were caused by the constraint they can be dropped in imaginary part of the retarded susceptibility χ55(q,ω),
this case and Eq.(21) reduces to a 4x4 matrix equation. denoted by χ′′ in the following, as a function of ω in
55
The four components arise because a nearest neighbor the normal state, using t′ = 0.3 and a small imagi-
−
charge-chargeinteractionleads tofourseparablekernels. nary part η = 0.001. We have chosen the component
If this interaction is furthermore approximated by one χ55 and the wave vector q = (π,π) because the transi-
kernel, associated with the order parameter, Eq.(21) re- tion to the DDW state occurs in this symmetry channel
ducestoonescalarequation,andonearrivesatthemod- and near this wave vector. Decreasing the temperature
els of Refs.1,10. from T = 0.032 to T = 0.028 shifts spectral weight in
The above formulas allow to calculate density correla- the overdamped spectrum from large to small frequen-
tion functions which can be expressed by ρ′ or ρ defined cies until a sharp and intense peak appears very near
in Eqs.(5)-(6). The most general case may, however, in- to the transition point T⋆ 0.028. Decreasing further
∼
volve two additional densities ρ and ρ′ in Eqs.(5) and the temperature the spectrum becomes again broad and
0 0
(6), respectively, where E and F are general functions movestohigherfrequencies. TheinsetofFig.1showsthe
0 0
of k. Going over to the Nambu representation Eˆ and temperaturedependence oftheDDWorderparameterΦ
0
Fˆ are then linear combinations of the Pauli matrices which exhibits the usual mean-field behavior.
0
and general functions of k. Using the diagramatic rules Fig.2 shows a three-dimensional plot of χ′5′5 as a func-
and the effective Hamiltonian one finds for the general tion of ω in the normal state at T = 0.039. The mo-
susceptibility χ (q), mentum runs from the point Γ = (0,0) to the points
00
S =(π,π) andX =(0,π), andback to the point Γ. The
χ (q)=χ(0)(q)+ χ(0)(q)χ (q)χ(0)(q). (27) curvesforΓ,SandX aredrawnasthicklinestofacilitate
00 00 0α αβ β0
X the visiualization. According to Eq.(25), and similar as
αβ
5
2 8
1.6 6
1.2
4
0.8
0.4 2
0 0 Γ
Γ
0.2 X 0.2 X
ω 0.4 0.6 S ω 0.4 0.6 S
0.8 0.8 Γ
Γ
FIG. 2: χ′′ in the normal state at T = 0.039 calculated FIG. 3: χ′′ at T=0. calculated for t′=−0.3, δ=0.077, and
55 55
for t′ = −0.3, δ = 0.077, and along the line Γ-S-X-Γ in the along thelines Γ-S-X-Γ in theBrillouin zone.
Brillouin zone.
not change much.
inthecaseoftheusualchargesusceptibility,χ exhibits
55
Tounderstandtheoriginandthepropertiesofthelow-
asingularbehavioratsmallfrequenciesandwavevectors
in the limit η 0: Fixing the momentum to q = (0,0) frequency peak in more detail we have plotted in Fig.4
χ (0,ω) is ze→ro for any finite frequency ω. Putting ω χ′′ and χ(0)′′ as a function of frequency for several q
55 55 55
to zero and taking the limit q 0 χ approaches a values between the Γ and S point. The dashed line in
55
finite value. This explains the a→bsence of a thick line the upper diagram represents χ(0)′′ at the wave vector
55
at Γ in Fig.2. Moving the momentum from Γ to S the q =(π,π). It contains only interband transitions across
zero-frequencypeak of measurezeroat Γ aquiresa finite
the DDW gap, mainly between the points X and Y and
width, i.e.,a finite spectralweight,andatthe sametime
betweenthehotspotsontheboundariesoftheRBZ.The
peaksatafinite frequency. Boththe widthandthe peak
intraband contribution is zero by symmetry at S. Since
increase first rapidly with momentum. The peak posi-
Φ is about0.07the interbandtransitionsleadto abroad
tion passes then through a maximum at ω 0.04,starts
peak somewhat higher than twice the DDW gap. The
∼
to decrease, and reaches a minimum at S signalling the solid curve in the upper diagram of Fig.4 represents χ′′
55
transitiontothe DDW state atlowertemperatures. The
at S. It shows a much more pronounced peak than the
width of the peak as well as the extension and intensity
dashed curve which originates from the denominator in
oftheslowlydecayingstructurelessbackgroundincreases
Eq.(26), i.e., which represents a collective excitation. It
uptothepointS. MovingalongthedirectionS-X-Γthe
corresponds to the amplitude mode of the DDW, where
low-frequency peak looses rapidly spectral weight, van-
the order parameter Φ is modulated without changing
ishes practically at X, and recovers somewhat towards
its d-wave symmetry. Its frequency is somewhat smaller
Γ. At the same time a well-pronounced, strong high-
than 2Φ which is expected for a d-wave state. For an
frequency peak develops, its frequency decreases mono-
isotropic s-wave ground state the two frequencies would
tonically along the above path, and its intensity shows
be exactly the same.
a maximum near the point X. The dispersion of this
The lowest panel in Fig.4 shows the same susceptibil-
high-frequency peak is ∼ sinkx between Γ and X, i.e., ities at the Γ point. χ(0)′′ again contains no intraband
it is identical with the sound wave due to usual density 55
fluctuations18. This means that d-wave density fluctua- but only interband contributions consisting of vertical
tions, described by χ′′ , strongly couple to usual density transitions which probe directly the DDW gap. As a
fluctuations along th5e5path S-X-Γ whereas such a cou- result, χ(0)′′ exhibits a well-defined peak at 2Φ and a
55
pling is forbidden by symmetry along Γ-S. long-tail towards smaller frequencies due to the momen-
Fig.3 shows the same 3d plot as Fig.2 but the calcu- tumdependenceoftheorderparameter. Thesolidcurve,
lation is now performed at T=0 in the DDW state. Dif- representing χ′′ , exhibits a well-pronounced peak much
55
ferences in the two figures thus have to be ascribed to below 2Φ. It corresponds to a bound state inside the d-
the formationofthe DDW.Takingthe differentscalesin wavegapcreatedbymultiplescatteringoftheexcitations
thetwofiguresintoaccountoneseesthattheintensityof acrossthe gapdue to the Heisenberginteraction. Due to
thehigh-frequencypeakduetousualdensityfluctuations its large binding energy the density of states inside the
didnotchangemuch. Bigchanges,however,occuratlow gapisrathersmallatthepeakpositionsothatthewidth
frequencies. χ′′ no longer vanishes at finite frequencies of the peak is substantialy smaller than that of the am-
55
at the Γ-point but rather shows there a well-pronounced plitude mode at the momentum (π,π). Previously14, we
and strong peak. Moving away from the Γ point this have associatedthis peak with the B peak seen in Ra-
1g
peak looses rapidly intensity whereas its frequency does man scattering in high-T superconductors. Note that
c
6
0.8 T=0K q=(π , π )
0.4 δ=0.077
0
0.8
0.4
8
0
1 6
0.5
χ"550 4
2 2
1
0 0 Γ
4 0.1 X
2 0.2 S
0 ω 0.3
4 q=(0,0) Γ
2
0
0 0.05 fr0e.1quency ω0.15 0.2 0.25
FIG.5: D-wavecrosssectionfornonresonant,inelasticX-ray
FIG.4: χ′′ (solidlines)andχ(0)′′ (dashedlines)formomenta scattering for the t-J model at large N in the DDW state at
55 55 T = 0. The momentum varies along the points Γ-S-X-Γ in
along thediagonal q=(q,q).
theBrillouin zone.
q=(q,q), we find,
there is roughly a factor 5 difference in the scales of the
top and bottom panels in Fig.4. It means that the am- 2t2 q
plitude mode of the DDW order parameter has a much χX(q,iωn)= eff cos2( )χ55(q,iωn)
J (cid:16) 2
larger spectral weight and also smaller width at the Γ q
than at the S point. +sin2( )χ66(q,iωn) sin(q)χ56(q,iωn) , (28)
2 − (cid:17)
For momenta q between Γ and S both intra- and in-
withtheeffectivenearest-neighborhoppingelementt
terband transitions contribute to χ(505)′′. The intraband determined by the large N dispersion of Eq.(13). Tehffe
part,reflectingtheresidualFermiarcsintheDDWstate, second-nearest hopping element t′ drops out in the con-
turns out to be in general substantially smaller than the sidered symmetry. In Fig.5 we have plotted χ with-
X
interband part and consists of a broad peak which dis- out this prefactor along the points Γ-S-X-Γgeneralizing
perses roughly as sin(q) and which looses its intensity Eq.(28)toanarbitrarykpointintheBrillouinzone. One
when approaching Γ or S. So only the low-frequency feature of χ′′ is that the high-frequency side bands due
X
tails and the structure around the frequency 0.04 in the tousualdensity fluctuationsarepracticallyabsent. This
second panel from above can be attributed to intraband may be explained by the fact that the various contribu-
scattering, the remaining spectral weight is due to inter- tions to χ′′ have both signs causing large cancellation
X
band scattering. Going from Γ to S the solid lines show effects. Another difference between Figs.3 and 5 occurs
the dispersionofthe amplitude mode whichis in general around the point S where additional structure is seen in
wellbelowthemainspectralpeakinχ(505)′′,representinga χ′X′ due to the component χ66.
resonance mode with a frequency and width determined
mainlybythedenominatorinEq.(26). Fromthedifferent
scales used in each panel it is evident that the intensity C. Inelastic neutron scattering
of the amplitude mode strongly decays away from the Γ
point in agreement with Fig.3. Using the symmetry-adapted functions Eq.(9) the ex-
pression for γ, Eq.(2), can be written as,
6
8πieµ t
γ(k,q)= 0 eff B (q)γ (k), (29)
~cq β β
X
B. Non-resonant inelastic X-ray scattering | | β=3
with
Specifying the polarizationvectors of the incident and αx αy
B = (1 cosq ) (1 cosq ), (30)
scattered light and using Eqs.(13) and (10) for the elec- 3,5 q − x ± q − y
x y
tron dispersion the cross section for inelastic X-ray scat-
teringisdeterminedbyχ′′ andχ isobtainedasalinear
X X αx αy
combination of the susceptibility matrix elements χ . B = sinq sinq , (31)
αβ 4,6 q x± q y
As an example, consider the case ei = (1,1)/√2,es = x y
(1, 1)/√2, which yields for q = 0 the B component where the subscripts 3,4 refer to the + and 5,6 to the
1g
−
of Raman scattering. Choosing the symmetry direction signs, respectively. The charge correlation function,
−
7
0.3
q=1/7(π,π)
0.2
0.1
0
1.5 q=1/8(π,π)
4 1
0.5
0
6
2 u4 q=(0,0)
"N,2
χ
0
0 Γ 2 q=1/5(π,0)
X 1
0.2
ω 0.4 S 0
0.6 q=1/4(π,0)
Γ 0.4
0.2
0
0 0.1 0.2 0.3ω 0.4 0.5
frequency
FIG. 6: χ′′ describing neutron scattering as a function of ω
N
alongthelineΓ-S-X-ΓintheBrillouinzoneintheDDWstate
at T =0 using t′=−0.3, δ=0.077, qz =0.5, and η=0.05. FIG.7: χ′N′,udescribingscatteringwithunpolarizedneutrons
as a function of ω for momenta around the point Γ in the
DDWstateatT =0usingt′ =−0.3,δ=0.077andqz =0.5.
describing neutron scattering, is given by,
1 8πeµ t 2
χN(q,ω)= 2J(cid:16) cq0 eff(cid:17) XBβ(q)Bδ(q)χβδ(q,ω). DχND,Wu(qs,tωa)ted.ivTeorgeexstartacstmiatsllsminogmuleanrtupmarttroannesfemrsayintathkee
| | β,δ
(32) thelimitq 0inthef,g,andχfunctions. Oneobtains
→
then,
Fig. 6 shows χ′′(q,ω) as a function of ω for momenta 1 q2
N χ ( + z )χ (0,iω ). (36)
alongthesymmetrylineΓ-S-X-ΓintheBrillouinzonefor N,u ∼ q2 q4 66 n
the fixed momentuma transfer q =0.5. The spin of the | | | |
z
neutrons was assumed to be polarized along the z-axis. In the normal state χ vanishes at q = 0 for any fi-
66
Thespectrashowessentiallyonemoreorlesswell-defined nitefrequencyascanbeseenfromtheexplicitexpression
peak which disperses roughly between the frequencies Eq.(25). In contrast to that it is finite at q = 0 in the
0.1 0.2. Similarly as in Fig.4 this peak describes a DDW state describing a continuum of particle-hole ex-
−
mainly collective excitation of the system related to a citations with zero total momentum across the gap. As
bound state inside the DDW gap. In spite of the rather a result χ diverges quadratically in the momentum
N,u
large momentum transfer q =0.5 the peak intensity in- q parallel to the layers for q = 0. This divergence is
z k z
creases substantially towards the point Γ. This increase causedby the long-rangepartofthe interactionbetween
becomes more and more pronounced if q is further low- the magnetic moment of the neutron and the electrons
z
ered until it diverges like 1/q2 in the limit q 0. in the layers. It is thus a real effect causing a singular
z
| | →
An important special case of Eq.(32) is scattering by forwardscatteringcontributioninthecrosssection. Inte-
unpolarized neutrons. Averaging over all spin directions grating χ′′ overq diverges logarithmically in a strictly
N,u
for the neutrons yields the charge correlation function 2ddescriptionatq =0whereasitremainsfiniteinthree
z
χ for scattering with unpolarized neutrons, dimensions. The large enhancement of χ′′ near the
N,u N,u
point Γ is illustrated in Fig.7. The curve at Γ shows a
χ (q,ω)= F (q)χ (q,ω), (33) peak atω 0.1wellinside the gapof2∆ 0.14,which
N,u αβ αβ 0
∼ ∼
Xαβ disperses to higher frequencies moving away from Γ, es-
pecially, along the diagonal. More dramatic is, however,
with
the rapid decay of the intensity of the peak away from
1 8πeµ t 2 Γ which can be seen from the different scales used in
F = 0 eff <B (q)B (q)> , (34)
αβ 2J(cid:16) cq (cid:17) α β av plotting the various panels.
| | Another special case of Eq.(32) is obtained for an ar-
where <...>av denotes an average over all directions of bitrary polarization for the neutron moment and a van-
the moment of the neutron. We obtain, ishing momentum transfer in the z direction, i.e., for
q = (q ,q ,0). The functions B assume in this case
1 x y i
<Bα(q)Bβ(q)>av= q2[(qy2+qz2)fαfβ the form for i=3,4,
| | µ
−qxqy(fαgβ +gαfβ)+(qx2+qz2)gαgβ], (35) Bi = µ zq (−qyfi+qxgi),
0
| |
with f = f = (1 cosq )/q , g = g = (1 µ
cosqy)/3qy, f45= f6 =−sinqx/xqx, xg4 =3 −g6−=5sinqy/q−y. Bi+2 = µ0|zq|(−qyfi−qxgi). (37)
8
ThepolarizationdependenceofχN issimplycos2θwhere Pdyn(q ) = 2 ∞ dω(1+n(ω))χ˜′′ (q,ω)
θistheanglebetweentheneutronspinandthenormalto orb z Nc2/3 Xq Z−∞ N,u
thelayers. χ assumesitslargestvalueifthespinisper- k
N
2
pendicularandiszeroifthe spinis paralleltothe layers. = F (q) ρ˜′ (q )ρ˜†(q ) . (40)
In this case χN diverges as 1/|q|2 for all polarizations of Nc2/3 q|X|,α,β αβ h α k β k i
the neutron and the singular contribution is isotropic in
the qz =0 plane. For qz 6= 0 the dependencies of χN on InEq.(38)Porb hasbeensplittedintoastaticBraggcon-
the polarization and the momentum no longer factorize tributionPBragg andadynamicpartPdyn. Correspond-
orb orb
but can be worked out for the most general case using
ingly,thetildeinEq.(40)indicatesthatonlythefluctuat-
Eq.(32).
ingpartofthedensitiesshouldbeusedinthecorrelation
functions.
Similar considerations apply to unpolarized neutron
D. Comparison of scattering efficiencies from spin scattering from spin fluctuations. The relevant suscep-
and orbital fluctuations tibility in the paramagnetic state is22,
µ2 8πeF(q) 2
Asdiscussedintheprevioussectionsthetotalmagnetic χ (q,ω). (41)
neutron cross section contains, in addition to the domi- 6 (cid:16) mca3 (cid:17) zz
nating spin flipcontribution, anorbitalpart. Above and
F(q) is the atomic structure factor and χ (q,ω) the zz
nottoofarawayfromT⋆totheDDWstatethescattering zz
componentoftheelectronicspinsusceptibility. Usingthe
from fluctuating orbital moments produced by circulat-
fluctuation-dissipation theorem, approximating F(q) by
ing currentsarounda plaquette are mainly concentrated
F(0) = 1, and assuming no magnetic coupling between
aroundthewavevectorQ=(π,π). BelowT⋆ thereisan
the layers, the total scattering probability P by spin
elasticBraggcomponentatwavevectorQandafluctuat- s
fluctuations becomes23,
ingpartthroughouttheBrillouinzone. Thoughmagnetic
scatteringfromspinflipandorbitalscatteringhavemany µ2 8πe 2
features in common there are three main differences: a) Ps = 12(cid:16)mca3(cid:17) π(1−δ). (42)
In spin scattering the form factor reflects the electron
density distribution within an atom whereas in orbital Most interesting are the ratios of integrated orbital to
scattering the form factor is related to the plaquette ge- spin scattering, i.e., RBragg = PBragg/P and Rdyn =
orb s
ometrywhichgivesrisetothe trigonometricfunctions in Pdyn/P . Using the above results one obtains,
the functions B ; b) The long-range dipole-dipole inter- orb s
β
action between neutrons and the electrons dominates in 1 B (q)B (q)
oprebaiktableslocawttTecriwngheyrieealdsitnhgeainstienrgauctlaiornfobrewtwaredenscnaetutterroinngs Rdyn(qz) = ANc2/3 qXk,α,β h α (aqβ)2 iav
andelectronsforspinscatteringreducestoalocalone;c) ∞
Theintensityoforbitalscatteringissubstantiallysmaller ·Z d(~)ω (1+n(ω))χ′α′β(q/|t|,ω), (43)
−∞
thanthatforspinscattering. Forinstance,aratioof1:70
has been obtained in Ref.3 for Bragg scattering. In the with the dimensionless constant,
following we present results on frequency and in-plane
momentumintegratedscatteringprobabilitiesinthe two 4 teff 2
A= , (44)
cases. Inthiswaytheanomalousforwardscatteringcon- π(1 δ)(J/t)(cid:16)~2/(ma2)(cid:17)
− | |
tribution in orbital scattering can be taken into account
in this comparison. We also recalculate the intensity for and
Braggscatteringandfinditmuchsmallerthanthe value
(B (Q))2
calculated in Ref.3. RBragg(q )=Ah 5 iav ρ′(Q) ρ†(Q) . (45)
The expression 2χ′′ (q,ω)(1 + n(ω)) describes the z a2(Q2+qz2) h 5 ih 5 i
N,u
scattering probability of an unpolarized neutron with
Using t = 0.5 eV we obtain from Eq.(13) t
momentum and frequency transfers q and ω, respec- | | | eff| ∼
0.07eV,whichyieldstogtherwitha=3.856˚A A 0.09
tively, from orbital fluctuations. Integrating over fre- ∼
for our doping δ = 0.077. Comparison of Eq.(14) with
quency and the in-plane momentum q the integrated
k Eq.(20) gives
scattering probability P becomes, using Eq.(33) and
orb
reinserting a and ~ for clarity,
Φ2
ρ′(Q) ρ†(Q) = 0.082, (46)
P (q )=PBragg(q )+Pdyn(q ), (38) h 5 ih 5 i 2J ∼
orb z orb z orb z
for T = 0 and δ = 0.077. Furthermore, from Eq.(35) we
get
PBragg(q )=2F (Q,q ) ρ′(Q) ρ†(Q) , (39) (B (Q))2 =8/π2, (47)
orb z 55 z h 5 ih 5 i h 5 iav
9
0.03 a transition temperature T⋆. The density response in-
dyn cludes “conventional” local density fluctuations, charac-
R
terized by the energy scale t, where practically all spec-
Bragg
R x500 tral weight is concentrated in a collective sound wave
0.02
with approximately a sinusoidal dispersion. In addition,
and this was the main topic of this investigation, there
is a “unconventional”density response from orbitalfluc-
0.01 tuationscausedbyfluctuatingcirculatingcurrentsabove
T⋆ andfluctuations arounda staggeredflux phase below
T⋆. The corresponding spectra are again mainly collec-
tive, their energy scale is J or a fraction of it, and they
0 0.2 q 0.4 0.6 0.8 canbeviewedasorderparametermodesoftheDDW.At
z
smallmomentumtransfersthe spectraaredominatedby
onewell-definedpeakcorrespondingtoamplitudefluctu-
FIG. 8: Ratios RBragg and Rdyn for elastic and inelastic ations of the DDW. For larger momentum transfers, un-
conventional density fluctuations tend to become broad
orbital tospin scattering, respectively.
and their intensity suppressed. This is especially true
nearthepointXwherepracticallynoweightisleftforor-
which yields bitalfluctuationsatlowfrequencies,butinsteadarather
sharp peak due to sound waves appear at large frequen-
RBragg(0)=3 10−5. (48) cies reflecting the coupling of the two kinds of density
·
fluctuations.
Unlike in the case of spin scattering the momentum and
frequency sums in Eq.(43) are non-trivial and we have
In principle the above orbital fluctuations show up
carried them out by numerical methods.
in the Raman, X-ray and neutron scattering spectra.
Fig. 8 shows the results for Rdyn and RBragg as a
The predicted scattering intensities are, however, rather
function of q at zero temperature. As expected Rdyn
z small. The reason for this is that, on the one hand, a
increases strongly at small q due to its logarithmic sin-
z rather small effective nearest-neighbor hopping element
gularity atq =0. Integrating Rdyn also overq yields a
z z t is needed to provide a sufficiently large density of
valueofabout10−2whichismorethan2ordersofmagni- eff
statesandtheinstabilitytotheDDW,ontheotherhand,
tudelargerthanRBragg. Thereasonforthislargediffer-
the square of t appears as a prefactor in the various
eff
enceisduetothefactthatinthemomentumintegration
cross sections due to the Peierls substitution. As a re-
inRdynmainlythesmallmomentacontributewhere1/q2
sult, the q integratedinelastic crosssectionis only a few
is larger than 1. In contrast to that, RBragg probes the
per cents of that for spin scattering and the elastic scat-
integrand of Rdyn at the large value Q where 1/q2 is
verysmall. TheextremesmallvalueRB|ra|gg 10−5casts tering from the Bragg peak is even much weaker. From
∼ our calculations it is also evident that a DDW state in
doubts on the identification of the observed Bragg peak
the unconstrained, weak-coupling case would have very
at Q with the Bragg peak due a DDW24. On the other
similar properties as in the t-J model. In particular, the
hand, if a DDW state is realized in some material we curvesforRdyn andRBragg wouldcloselyresemblethose
think that the resulting strong forward scattering peak
in Fig.8. An experimental verification of a DDW state
in neutron scattering would be observable. According to
viainelastic neutronsseemsto be notunfeasible because
ourpredictionsitwouldexhaustroughly3percentofthe
a good deal of the scattering intensity is concentratedin
sumruleforspinscatteringandthusbeofthesameorder
awell-pronounced,inelasticpeaksomewhatsmallerthan
of magnitude as the recently detected elastic scattering
the DDW gapandinsmallregionaroundthe originin k
from weak magnetic moments directed at least partially
space.
perpendiculartothelayers25inhigh-T superconductors.
c
Acknowledgements The authors are grateful to M.
IV. CONCLUSIONS Geisler andT. Enss for help with the figures,H. Yamase
for useful discussions, and D. Manske for a critical read-
ing of the manuscript.
The excitation spectrum of the t-J model at large
N was studied for dopings where a DDW forms below
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