Symmetry of Photoexcited States and Large-Shift Raman Scattering in Two-Dimensional Mott Insulators

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by  T. Tohyama

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Symmetry of Photoexcited States and Large-Shift Raman Scattering in
Two-Dimensional Mott Insulators
T. Tohyama∗
6 Institute for Materials Research, Tohoku University, Sendai, 980-8577
0 (ReceivedAugust22,2005)
0
Symmetry of photoexcited states with two photoinduced carriers in two-dimensional Mott
2
insulators is examined by applying the numerically exact diagonalization method to finite-
n
size clusters of a half-filled Hubbard model in the strong-coupling limit. The symmetry of
Ja minimum-energy bound state is found to be s-wave, which is different from a dx2−y2 wave
of a two-hole pair in doped Mott insulators. We demonstrate that the difference is originated
3
from an exchange of fermions due to the motion of a doubly occupied site. Correspondingly
1
large-shift Ramanscatteringacross theMott gapexhibitsaminimum-energyexcitation inthe

] A1 (s-wave)channel.WediscussimplicationsoftheresultsfortheRamanscatteringandother
l optical experiments.
e
-
r KEYWORDS: Photoexcited state, Hubbard model, Mott insulator, Raman scattering, Exact diagonaliza-
t
s tion
.
t
a
m 1. Introduction toexcited states and their symmetry in 2D Mott insula-
- The charge gapin Mott insulators is a consequence of tors.12–16 It has been shown analytically13 and numer-
d
n strong electron correlation represented by large on-site ically14 that the bound states with odd-parity (dipole-
o Coulombinteraction.The correlationinduces novelphe- allowedstates)arehigherinenergythanthosewitheven-
c nomena in terms of the interplay of charge and spin de- parity (dipole-forbidden states). However, symmetry of
[ grees of freedom.1 Photoexcitation across the Mott gap the lowest-energybound statehasnotbeenclarifiedyet,
2 induces two carriers, an unoccupied site and a doubly although two possibilities, the A212 or B1,13 have been
v occupied site of electrons. In two dimensions (2D), the proposed. Therefore, it is important to clarify the sym-
9 motionofthetwocarriersisstronglyaffectedbythepres- metry ofthe boundsate inorderto fully understandthe
1
ence of localized spins in the background: The propaga- nature of the photoexcited states. Also it is interesting
5
8 tion of a carrier is known to induce a spin cloud around whetherthesymmetryisrelatedtothatofaboundstate
0 thecarrierasaconsequenceofthemisalignedspinsalong of two-hole pair in doped Mott insulators.
5 the carrier-hoppingpaths, and the two carriersprefer to In this paper, we theoretically examine symmetry of
0 form bound states by minimizing the loss of magnetic photoexcited states and large-shift Raman scattering in
/
t energy due to the spin cloud.2 The formation of such the2DMottinsulators.Weapplyanumericallyexactdi-
a
m boundstatesissimilartothecaseoftwoholesintroduced agonalizationmethodtofinite-sizeclustersofahalf-filled
into 2D Mott insulators. The bound state formed in the Hubbard model in the strong-coupling limit. In the cal-
-
d two-holegroundstatehasdx2−y2 symmetry(equivalently culations,weintroduce variousboundaryconditions.An
n the B representation in the D group), provided that averaging procedure over twisted boundary conditions
1 4
o the exchange interaction between localized spins is not is also used to reduce finite-size effects. The symmetry
c small compared with the nearest-neighbor hopping am- of the lowest-energy bound state is found to be neither
:
v plitude.3 This is considered to be related to dx2−y2 su- A2 nor B1, but A1 (s-wave symmetry). We demonstrate
Xi perconducting symmetry in the high-Tc cuprates. that, if sign changes due to an fermion exchange caused
The nature of photoexcited states in 2D Mott insu- by the motion of a doubly occupied site are not taken
r
a lating cuprates such as Sr2CuO2Cl2 has been examined into account, the symmetry becomes B1. The large-shift
by using linear4,5 and nonlinear optical response exper- Raman scattering exhibits a lowest-energy excitation in
iments6–9 and large-shift Raman scattering experiments the A1 channel. This is different from the experimental
across the Mott gap.10,11 Among these experimental data showing a lowest-energy excitation with A2 sym-
techniques, the large-shift Raman scattering is the best metry10,11 and thus supports a proposalthat the A2 ex-
one to see the symmetry of the photoexcited states di- citation is due to a d-d transition from dx2−y2 to dxy
rectly.However,notonlyexcitationsrelatedtothebound orbitals.11
states in the photoexcitedstates but alsod-d transitions Therestofthispaperisorganizedasfollows.Weintro-
between 3d orbitals contributes to the Raman scatter- duce an effective Hamiltonian of the half-filled 2D Hub-
ing.11 Therefore, it is important to give information on bard model in the strong-coupling limit, and show out-
the contribution of the bound states to the large-shift lines of the procedure for choosing boundary conditions
Raman scattering. in 2. In 3, calculated results of the distribution of
§ §
Thereareseveraltheoreticalstudiesrelatedtothepho- photoexcited states are shown and the symmetry of the
lowest-energy bound state is discussed. In 4, we show
§
calculated spectra of the large-shift Raman scattering
∗E-mailaddress:[email protected]
2 J.Phys.Soc.Jpn. FullPaper T.Tohyama
and compare them with experiments. The summary is We impose periodic boundary conditions for the clus-
given in 5. ters along both the x and y directions. In addition, an-
§
tiperiodicboundaryconditionsandmixedboundarycon-
2. Model and Method
ditions (one direction is periodic and the other is an-
The Hubbard model is given by tiperiodic) are used to check the effect of the boundary
condition on physical quantities. In such small-size clus-
H =H +H (1)
Hub t U ters, we are not free from finite-size effects that some-
with times make the results unreliable. In order to reduce
the finite-size effects, we also introduce various bound-
H = t c† c +H.c. (2)
t − X(cid:16) i,σ i+δ,σ (cid:17) aryconditionswithtwistandaveragephysicalquantities
i,δ,σ
over the twisted boundary conditions. This procedure
and has been applied for various quantities in the t-J19 and
t-t′-t′′-J20 models.
H =U n n , (3)
U Xi i,↑ i,↓ Thetwistinducestheconditionthatci+Ra,σ =eiφaci,σ
where c†i,σ is the creation operator of an electron with Nanodtecit+hRabt,σφa==eiφφbbci=,σ0w(itπh)acrobrirtersaproynpdhsatsoesthφeapaenrdioφdibc.
spin σ at site i, n =c† c , the summation of δ runs (antiperiodic) boundary conditions. The phase φ is
i,σ i,σ i,σ a(b)
over a pair of the nearest-neighbor sites between i and definedasφ =κ R withanarbitrarymomentum
a(b) a(b)
i+δ,tisthenearest-neighborhoppingintegral,andU is κ = κxx+κyy. κ u·sually scans an area surrounded by
tlihmeiton(U-site tC)o,uthloemrebisinntoedraocutbiolyn.ocIncutphieedstsritoenignctohuephlainlfg- ma )sqaunadre wiπth(lf+oumr c,orlne+rsma)t.(Iκnx,oκryd)er=t±o Nπpe(rlf−ormm,tlh+e
filled gro≫und state, resulting in the Heisenberg model: averaging±pNrocedure,−we choose many κ in the square
n n with equal intervals of π/45, π/40, and √2π/8√13 for
H0 =JX(cid:16)Si·Si+δ− i4i+δ(cid:17) , (4) the N =18, 20, and 26 clusters, respectively. The total
i,δ number of κ, N , results in N =450, 320, and 64 for
κ κ
where Si is the spin operator with S = 1/2 at site i, N =18, 20, and 26, respectively. The reason that Nκ
n =n +n , and J =4t2/U. decreaseswithincreasingN isthattimeandmemoryfor
i i,↑ i,↓
Photoexcited states across the Mott gap have both computingincreasewithN.Inparticular,forN =26we
one doubly occupied site and one vacant site. An effec- need to use supercomputing systems and the computing
tive Hamiltonian describing the photoexcited states is time of the optical conductivity for a κ point is about
obtained by restricting the Hilbert spaces to a subspace 130minutesbyusingHitachiSR11000attheInstitutefor
withonedoublyoccupiedsite.Byperformingthesecond Solid State Physics, the University of Tokyo. Therefore,
orderperturbationwithrespecttothehoppingtermHt, itistime-consumingtoincreaseNκ forN =26.Wenote
eq. (2), the effective Hamiltonian is given by14,15 thatimposingthetwistisequivalenttotransformingthe
operator c† c into exp(iκ δ)c† c , δ being the
1 1 i,σ i+δ,σ · i,σ i+δ,σ
H =Π H Π Π H Π H Π + Π H Π H Π +U displacement vector from site i to i+δ. Therefore, the
eff 1 t 1 1 t 2 t 1 1 t 0 t 1
−U U
Heisenberg model, eq. (4), is independent of the choice
(5)
of κ.
where Π , Π , and Π are projection operators onto
0 1 2
the Hilbert space with zero, one, and two doubly occu-
3. Symmetry of Photoexcited States
piedsites,respectively.Wenotethattheeffectivemodel,
The regular part of the optical conductivity detects
eq. (5), can reproduce well the optical conductivity of
optical-allowedphotoexcited states. The real part of the
the Hubbard model with large U under periodic bound-
ary conditions for the N = 18 and 20 clusters17 and conductivity for a given κ is expressed as
under antiperiodic boundary conditions for the N = 16 σ (ω)= π Ψκ jκ 0 2δ(ω Eκ +E ), (6)
cluster.18 Hereafter ~=e=c=1,e andc being the ele- κ Nω X|h m| x| i| − m 0
m
mentary charge and the speed of light, respectively, and
where 0 is the ground state of the Heisenberg model
the distance between the nearest-neighbor sites in the | i κ
withenergyE ,and Ψ representsaphotoexcitedstate
two-dimensional lattice is set to be unity. Throughout 0κ κ| mi
with energy E . j is the x component of the current
this paper, we take U/t=10. m x
operator under the twist up to second order of t:
Weusetheexactdiagonalizationmethodbasedonthe
Lanczosalgorithmtocalculatelow-lyingeigenstateofthe t2
jκ = it δ e−iκ·δc˜† c˜ +i (δ δ′)
Heisenberg and effective Hamiltonians. In 2D systems, x X x i+δ,σ i,σ U X x− x
i,δ,σ i,δ,δ′,σ
one uses a N-site square lattice with the translational
vNec=tolr2s+Rma2=wlixth+inmteygearsndl,mRb =0.−Hmexre+,xlyan,dbeyinagreththaet ×(cid:16)eiκ·(δ′−δ)c˜†i+δ,σc˜†i,−σc˜i,−σc˜i+δ′,σ
≥
dveircetcotrisoncos,nnreescptiencgtivneelayr.eIsnt-ntheiigshsbtourdysi,tewseintatkheeNx a=nd1y8 −eiκ·δ′c˜†i+δ,σc˜†i,−σc˜i+δ,−σc˜i+δ′,σ(cid:17). (7)
(l = 3,m = 3), N = 20 (l = 4,m = 2), and N = 26 The creation and annihilation operators, c˜† and c˜ ,
i,σ i,σ
(l =5,m=1).
are projected onto the subspace with either zero or one
doubly occupied site. The δ is the x components of δ.
x
J.Phys.Soc.Jpn. FullPaper T.Tohyama 3
N=20 N=26 N=20 N=26
0.3 0.3
(a) (b) (a) (b)
0.2 0.2
) )
( (
0.1 0.1
0.0 0.0
4 6 8 10 124 6 8 10 12 4 6 8 10 124 6 8 10 12
/t /t /t /t
1.2 1.2
(c) A1 (d) A1 (c) A1 (d) A1
B1 B1 B1 B1
) 0.8 ) 0.8
( (
S S
0.4 0.4
0.0 0.0
4 6 8 10 124 6 8 10 12 4 6 8 10 124 6 8 10 12
/t /t /t /t
Fig. 1. (Color online) Optical conductivity σκ(ω) for half-filled Fig. 2. (Coloronline)ThesameasFig.1,butunderantiperiodic
Hubbard clusters in the strong coupling limit with the size of boundaryconditions.
N = 20 (a) and N = 26 (b), and dynamical correlation func-
tionSκ(ω)ofA1-symmetryoperator(thethickline)andofB1-
symmetry operator (the thin line) for N = 20 (c) and N = 26
(d). U/t = 10. Periodic boundary conditions are employed and
N=20 N=26
delta-functions are broadened by a Lorentzian with a width of
0.3
0.05t.
(a) (b)
0.2
)
Fortheperiodicboundaryconditions,κischosentosat- (
isfy κ Ra = κ Rb = 0 (π). For the mixed boundary 0.1
conditi·ons, we ta·ke an average of σ (ω) with κ R =0
κ a
andκ R =π andthat with κ R =π and κ ·R =0.
b a b 0.0
· · ·
The optical conductivity under the averaging procedure 4 6 8 10 124 6 8 10 12
is given by /t /t
1.2
1 (c) A1 (d) A1
σ (ω)= σ (ω). (8)
ave N X κ B1 B1
κ κ ) 0.8
We show the optical conductivity under various (
S
boundaryconditionsfortheN =20andN =26clusters 0.4
in the panels (a)and (b), respectively,ofFigs. 1 - 4:Re-
sults for the periodic, antiperiodic, mixed, and averaged 0.0
boundary conditions are shown in Fig. 1, Fig. 2, Fig. 3, 4 6 8 10 124 6 8 10 12
/t /t
Fig. 4, respectively. Both the N =20 and N =26 cases
inFigs.1(a)and1(b)showagloballysimilardistribution Fig. 3. (Color online) The same as Fig. 1, but under mixed
boundary conditions, where the average of two cases [periodic
of the spectral weight, exhibiting two prominent struc-
(antiperiodic) along the x (y) direction and antiperiodic (peri-
tures: One is a broad-peak structure centered at around
odic)alongthex(y)direction]istaken.
ω =10t,andtheotherisanabsorption-edgestructureat
aroundω =6tseparatedfromthebroadpeak.Thelatter
structureissensitivetotheexchangeinteractionJ,2 and
peak appears at ω = 6.2t for N = 20, while for N = 26
thus originates from magnetically induced bound states.
there are four peaks with similar magnitude within the
This global distribution of the spectral weight does not
regionof6t.ω .7t.After takingthe averagingoverκ,
change under the antiperiodic and mixed boundary con-
the edge structure becomes wider for N = 20 but nar-
ditions as shownin Figs. 2 and 3, though fine structures
rowerfor N =26.Therefore, the difference appearing in
in the spectra depend on the boundary conditions.
the cases of the periodic boundary conditions is reduced
The similarity of the spectral-weight distribution be-
in Fig.4,althougha two-peakstructure still remainsfor
tween N =20 and N =26 is also seen in Figs. 4(a) and
N =26.Thereductiondemonstratesefficiencyoftheav-
4(b), where we take the averaging procedure, eq. (8).
eraging procedure.We note that σ (ω) of N =18 (not
The averagingis foundto reduceadifference ofthe edge ave
shown) is similar to that of N =20.
structure for N = 20 and N = 26 under the periodic
In order to examine whether optical-forbidden states
boundaryconditions21 [see Figs.1(a)and1(b)]:A single
4 J.Phys.Soc.Jpn. FullPaper T.Tohyama

N=20 N=26 1.2
A1 A1
0.3 (a) V/t=1 (b)
B1 B1
(a) (b)
) 0.8
) 0.2 (ave
(e S 0.4
av
0.1
0.0
4 6 8 10 12 4 6 8 10 12
0.0
4 6 8 10 124 6 8 10 12 /t /t
/t /t Fig. 5. (Color online)Dynamical correlationfunction Save(ω)of
1.2 A1-symmetryoperator(the thickline)andofB1-symmetryop-
(c) A1 (d) A1 erator (the thin line) for a N = 20 half-filled Hubbard cluster
B1 B1 in the strong coupling limit. (a) U/t = 10 and V/t = 1. (b)
) 0.8
U/t = 10, but sign changes due to the exchange of fermions
(
e
av caused by the motion of a doubly occupied site are neglected.
S
Anaveraging procedure for twisted boundary conditions is em-
0.4
ployed and delta-functions are broadened by a Lorentzian with
awidthof0.05t.
0.0
4 6 8 10 124 6 8 10 12
/t /t
the A dominatedstates.Theseresults indicate that the
Fig. 4. (Coloronline)ThesameasFig.1,butanaveragingpro- 1
cedurefortwistedboundaryconditionsisemployed. lowest-energy bound state in the photoexcited states of
the Hubbard model has the A symmetry, i.e., s-wave
1
symmetry.
This conclusion is not altered even if we include an
are lower or higher in energy than allowed ones, we in-
attractive interaction, V, between neighboring doubly
troduce operators with A and B symmetry composed
1 1
occupied and vacant sites, which induces an excitonic
of the nearest-neighbor hoppings:
boundstate.TheresultingeffectiveHamiltonianisgiven
Cκ = e−iκxc˜† c˜ e−iκyc˜† c˜ +h.c. , by
± X(cid:16) i+x,σ i,σ ± i+y,σ i,σ (cid:17)
i,σ
H = H V n n (1 n )(1 n )
(9) V eff − X(cid:2) i,↑ i,↓ − i+δ,↑ − i+δ,↓
where the plus (+) and minus ( ) signs correspond to i,δ
−
A1 and B1, respectively, and κα is the α component of +(1 ni,↑)(1 ni,↓)ni+δ,↑ni+δ,↓ . (12)
κ. The dynamical correlation function of the operators − − (cid:3)
for a given κ is given by Figure 5(a) exhibits Save(ω) for the N =20 cluster with
V =t.Wefindnoqualitativechangeoftheboundstates
S (ω)= 1 Ψκ Cκ 0 2δ(ω Eκ +E ), (10) ascomparedwithFig.4(c):The excitationenergyofthe
κ N X(cid:12)(cid:10) m(cid:12) ±(cid:12) (cid:11)(cid:12) − m 0 A bound state is lower than that of B .
m (cid:12) (cid:12) (cid:12) (cid:12) 1 1
In the two-hole doped Mott insulator, the hole pair
and an averagedcorrelation is expressed as
forms a dx2−y2 wave in the ground state.22 This is in
1
contrast with the present results that the two-carrier
S (ω)= S (ω). (11)
ave Nκ Xκ κ pair produced by photoexcitation forms an s wave. It
is important to clarify what is the origin of this dif-
κ
Wenotethatthefinalstates Ψ cannotbedividedinto
| mi ference. We can easily notice a remarkably difference
subspacesoftheserepresentationsbecauseofthetwisted
between the two-hole and two-photoexcited-carrier sys-
boundary conditions.
tems. That is the difference of the electric charge of
S (ω) for N = 20 and N = 26 is shown in the pan-
κ carriers. In the photoexcited states, one of the carriers
els (c) and (d), respectively, of Figs. 1 - 3, and S (ω)
ave is not a hole but contains two electrons. Therefore, in
is shown in Figs. 4(c) and 4(d). For the cases of the
a basis representation where electrons are sorted in or-
periodic, antiperiodic, and mixed boundary conditions
der of site, the motion of this carrier inevitably induces
(Figs. 1 - 3), we find edge-structures in the A correla-
1 an exchange of fermions and gives an extra sign. For
tionseparatedfrombroadspectrum,indicatingthepres-
instance, in the case of a two-site cluster with a dou-
enceofboundstates.Wealsofindthatintheseboundary
bly occupied site and a singly occupied site, the hop-
conditions the energy of the edge position of the A cor-
1 ping of the doubly occupied site induces an extra sign:
relation is lower than those of the B1 correlation and ( tc† c )c† c† c† vac = +tc† c† c† vac , vac
σκ(ω).Asisthecaseofσave(ω),theaveragingprocedure b−eing2,↑th1e,↑vac2,u↓u1m,↓s1t,a↑t|e. i 2,↓ 2,↑ 1,↓| i | i
makes differences between N = 20 and N = 26 smaller.
In order to check the effect of the fermion exchange,
Afteraveraging,thefactthattheedgeofA islowerthan
1 we introduce an effective Hamiltonian that is the same
that of B is still preserved, indicating an intrinsic na-
1 aseq.(5)butreversingthesignofhoppingofthedoubly
tureofthephotoexcitedstatesofthehalf-filledHubbard
occupiedsite.Figure5(b)showsS (ω)forthe effective
model with large U. As will be shown below, the states ave
model. We find that the B correlation exhibits bound
havingA andB components arehigherinenergythan 1
2 2 states whose energies are lower than those of A , indi-
1
J.Phys.Soc.Jpn. FullPaper T.Tohyama 5

(a) (c) (a) (c)
0.8 0.8
A1 B1 ) A1 B1
)
(
( e
R 0.4 Rav0.4
0.0 0.0
(b) Periodic (d) (b) (d)
0.8 0.8
Antiperiodic
) A2 B2 ) A2 B2
(
( e
R av
0.4 R 0.4
0.0 0.0
4 6 8 10 124 6 8 10 12 4 6 8 10 124 6 8 10 12
/t /t /t /t
Fig. 6. (Color online) Large-shiftRaman scattering Rκ(ω) for a Fig. 7. ThesameasFig.6butanaveragingprocedurefortwisted
N =20half-filledHubbardcluster inthestrong couplinglimit. boundaryconditions isemployed.
U/t = 10. (a) A1, (b) A2, (c) B1, and (d) B2 scattering. The
thickandthinlinesrepresenttheresultsunderperiodicandan-
tiperiodicboundaryconditions,respectively.Delta-functionsare
broadenedbyaLorentzianwithawidthof0.05t. order to reduce the dependence, we use the averaging
procedure, eq. (15). R (ω) is shown in Fig. 7. Among
ave
allpossiblesymmetry,the A Ramanscatteringexhibits
1
the lowest-energy excitation, being consistent with the
cating the lowest-energy state to be dx2−y2-wave. From
A correlationfunction discussed above. For the B and
this result, we can conclude that the fermion-exchange 1 1
B Raman scattering, spectral weights appear slightly
process for the doubly occupied site plays a crucial role 2
above the A scattering.In the A scattering, no weight
in making the A bound state lowest in energy. 1 2
1
is observed at the edge region around ω = 6t. These
4. Large-Shift Raman Scattering resultsdemonstrateagainthattheA stateisthelowest-
1
energy state in the photoexcited states of the 2D Mott
Large-shift Raman scattering for the half-filled Hub-
insulator.
bardmodelisagoodquantitytostudythephotoexcited
Since the N =20cluster is not a simple squarelattice
statesinthesymmetry-resolvedform.TheRamaninten-
sity for a given κ is given by but a tilted one, it does not have the D4 point group.
In order to check whether the data in Figs. 6 and 7 are
Rκ(ω)= N1 X(cid:12)(cid:10)Ψκf |MR|0(cid:11)(cid:12)2δ(ω−Efκ+E0) (13) artifacts of tilted lattice or not, we examine Rκ(ω) and
f (cid:12) (cid:12) Rave(ω) for the N = 18 cluster that has the D4 group.
TheresultsareshowninFigs.8and9.Inthe caseofthe
with symmetry-resolved Raman operators M : M =
R A1 periodic boundary conditions where the D point group
M +M ,M =M M ,M =M M ,and 4
Mxx = Myy +AM2 . Txhye−opeyrxatorBM1 (αxx,β−=yxy,y) is is fully held, Rκ(ω) exhibits the same behaviors as the
B2 xy yx αβ case of N =20 under the periodic boundary conditions:
expressed as
TheA scatteringshowsthelowest-energyexcitationand
1
κ κ κ κ
M = Jβ |ΨmihΨm|jα , (14) the A2 scattering has no weight at around ω = 6t. Al-
αβ Xm Emκ −E0−ωi+iη though the difference of Rκ(ω) under the periodic and
antiperiodic boundary conditions is larger in the case of
κ
where J connects two subspaces in the photoexcited
β N =18thanthat inthe caseofN =20 shownin Fig.6,
states and is given by Jβκ =jβκ+i[U−1Π1HtΠ0HtΠ1,βˆ], Rave(ω)obtainedaftertheaveraginginFig.9showsspec-
where βˆ is the β component of the total position opera- tralfeaturessimilartothecaseofN =20inFig.7.This
tor.TheRamanintensityundertheaveragingprocedure indicates that the behaviors of Raman intensities shown
is given by in Figs. 6 and 7 are inherent in the square-lattice Hub-
bard model.
1
R (ω)= R (ω). (15) The large-shift Raman scattering experiments for
ave N X κ
κ κ parent compounds of high-T superconductors such as
c
Wesettheincidentphotonenergyωi tobeωi =U+6t= Nd2CuO4 have shown that the A2 scattering has the
16t and focus on the Raman shift ω of order of U. We largest spectral weight whose excitation energy is lower
take the relaxation energy η to be 0.2t. than the absorption-peak position.11 This is completely
Figure 6 shows R (ω) for the N =20 cluster with the different from the present results shown in Figs. 7 and
κ
periodicandantiperiodicboundaryconditions.Although 9.We considerthat,as discussedin ref.11,the observed
global features up to the high-energy region are similar large A2 scattering would come from a d-d transition
between the two boundary conditions, edge structures fromdx2−y2 to dxy orbitalsthroughphotoexcitedstates.
of the spectrum depend on the boundary conditions. In
6 J.Phys.Soc.Jpn. FullPaper T.Tohyama

we need further theoreticalstudies of, for instance,TPA
(a) (c) spectra under the presence of various d orbitals.
0.8
Insummary,we haveexaminedsymmetry ofphotoex-
A1 B1
) cited states and symmetry-resolved large-shift Raman
(
R 0.4 scattering in the 2D Mott insulators by using an effec-
tive Hamiltonian of a half-filled Hubbard model in the
strong-coupling limit and a numerically exact diagonal-
0.0 ization method on finite-size clusters. The symmetry of
the lowest-energy bound state is found to be neither
(b) Periodic (d)
0.8 A nor B , but A . Therefore, the symmetry is differ-
Antiperiodic 2 1 1
) A2 B2 ent from that of a hole pair in doped Mott insulators.
(R We demonstrate that the difference is originated from
0.4
fermionexchange inducedby the motionofa doubly oc-
cupied site. The large-shift Raman scattering exhibits a
lowest-energyexcitationintheA channel.Thisisdiffer-
0.0 1
4 6 8 10 124 6 8 10 12 ent from the experiments showing a lowest-energy exci-
/t /t
tationwithA symmetry,whichisprobablyduetoad-d
2
Fig. 8. (Coloronline)ThesameasFig.6butforN =18. transitionfromdx2−y2 todxy orbitals.Itseemstobenec-
essary to include other d orbitals into the photoexcited
states for a complete understanding of the experimental

data. This remains as a future problem.
(a) (c)
0.8 Acknowledgment
) A1 B1
The author is very grateful to H. Kishida and H.
(
Rave0.4 Okamoto for their insightful comments on this work.
ValuablediscussionswithK.TsutsuiandS.Maekawaare
also appreciated. This work was supported by a Grant-
0.0 in-AidforscientificResearchfromtheMinistryofEduca-
tion, Culture, Sports, Science and Technology of Japan,
(b) (d)
0.8 CREST, and NAREGI project. The numerical calcula-
) A2 B2 tions were partly performed in the supercomputing fa-
(
ave cilities in ISSP, University of Tokyo, and IMR, Tohoku
R 0.4
University.
0.0
4 6 8 10 124 6 8 10 12 1) See, for example, S. Maekawa and T. Tohyama: Rep. Prog.
/t /t Phys.64(2001)383;andreferencestherein.
2) T.Tohyama,H.Onodera,K.Tsutsui,andS.Maekawa:Phys.
Fig. 9. The same as Fig. 6 but for N = 18 and an averaging
Rev.Lett. 89(2002) 257405.
procedurefortwistedboundaryconditionsisemployed.
3) A.L.Chernyshev,P.W.Leung,andR.J.Gooding:Phys.Rev.
B58(1998)13594.
4) H.S.Choi,Y.S.Lee,T.W.Noh,E.J.Choi,Y.Bang,andY.
5. Summary J.Kim:Phys.Rev.B60(1999)4646.
5) R. Lo¨venich, A. B. Schumacher, J. S. Dodge, D. S. Chemla,
Before summarizing our results, we discuss implica- andL.L.Miller:Phys.Rev.B63(2001)235104.
tionsofthepresentresultsforrecenttwo-photonabsorp- 6) A.B.Schumacher,J.S.Dodge,M.A.Carnahan,R.A.Kaindl,
tion(TPA)experimentonNd CuO .9Intheexperiment, D. S. Chemla, and L. L. Miller: Phys. Rev. Lett. 87 (2001)
2 4
127006.
there is a small absorption hump whose energy is about
7) M.Ashida,Y.Taguchi,Y.Tokura,R.T.Clay,S.Mazumdar,
0.3 eV lower than the optical (linear)absorptionenergy.
Yu. P. Svirko, and M. Kuwata-Gonokami: Euro. Phys. Lett.
Combining this experimental data with third-harmonic 58(2002) 455.
generationdata8,23 andanalyzingthedatabyusingaset 8) H. Kishida,M.Ono, A.Sawa, M.Kawasaki, Y.Tokura, and
ofdiscreteexcitation-energylevels,Kishidaetal.havear- H.Okamoto: Phys.Rev.B68(2003) 075101.
9) A. Maeda, M. Ono, H. Kishida, T. Manako, A. Sawa, M.
guedthatthehumpstructureinTPAcomesfromapho-
Kawasaki,Y.Tokura,andH.Okamoto:Phys.Rev.B70(2004)
toexcited state with B symmetry and thus the lowest-
1 125117.
energyphotoexcitationisduetotheB1symmetry.23This 10) R.Liu,D.Salamon,M.V.Klein,S.L.Cooper,W.C.Lee,S-
isdifferentfromthepresentresultsthattheminimumex- W.Cheong,andD.M.Ginsberg:Phys.Rev.Lett.71(1993)
citation comes from the A symmetry. According to the 3709.
1
11) D.Salamon,R.Liu,M.V.Klein,M.A.Karlow,S.L.Cooper,
Kishida’s analysis, it is difficult to assign the hump as
S-W.Cheong,W.C.Lee,andD.M.Ginsberg:Phys.Rev.B
the A1 excitation. Therefore,a possible interpretationis 51(1995) 6617.
that the hump structure in TPA would originate from a 12) D.V.KhveshchenkoandP.B.Wiegmann:Phys.Rev.Lett.73
d-d excitation between, for instance, dx2−y2 and d3z2−r2 (1994) 500.
orbitals. However, in order to come to final conclusions, 13) P.Wro´belandR.Eder:Phys.Rev.B66(2002)035111.
14) M.Takahashi,T.Tohyama,andS.Maekawa:Phys.Rev.B66
J.Phys.Soc.Jpn. FullPaper T.Tohyama 7
(2002) 125102. 18) H.NakanoandM.Imada:J.Phys.Soc.Jpn.68(1999)1458.
15) A.Takahashi,S.Yoshikawa,andM.Aihara:Phys.Rev.B65 19) D.Poilblanc:Phys.Rev.B44(1991) 9562.
(2002) 085103. 20) T.Tohyama:Phys.Rev.B70(2004) 174517.
16) A.Takahashi,H.Gomi,andM.Aihara:Phys.Rev.B66(2002) 21) H.Onodera,T.Tohyama, andS.Maekawa: Phys.Rev.B69
115103. (2004) 245117.
17) T. Tohyama, Y. Inoue, K. Tsutsui, and S. Maekawa: Phys. 22) S.R.WhiteandD.J.Scalapino:Phys.Rev.B55(1997)6504.
Rev.B72(2005) 045113. 23) H.KishidaandH.Okamoto:privatecommunication.