Journal of Luminescence 94–95 (2001) 499–502

Multi-photon excited states of two-dimensional stronglycorrelated electron systems

Akira Takahashi*, Hiroki Gomi, Masaki Aihara

Graduate School of Materials Science, Nara Institute of Science and Technology, 8916-5 Takayama-cho, Ikoma, Nara 630-0101, Japan

Abstract

The effective Hamiltonian of the Hubbard model for the multi-photon excited states is derived by assuming strongCoulomb repulsion. By diagonalizing the effective Hamiltonian exactly, the lowest energy state in two-photon excitedmanifold is calculated in the two-dimensional Hubbard model at half-filling, and its physical properties are investigated.

The antiferromagnetic spin order characteristic of the ground state is destroyed by two-photon excitation and threephases which have different electronic orders from that of the ground state are photo-generated. The analysis of severalcorrelation functions suggests that they are of spiral spin state and two different d-wave superconducting states.r 2001Elsevier Science B.V. All rights reserved.

Keywords: Two-dimensional strongly correlated electron systems; Multi-photon excited states; Photo-induced phase transition;

Photo-induced superconductivity

1. Introduction

Low-dimensional strongly correlated electronsystems (LDSCES) have antiferromagnetic (AF)spin order at half-filling and carrier dopingweakens and often destroys the spin order. As aresult, physical properties of the LDSCES aredrastically changed and various interesting phasesare generated by carrier doping [1,2]. Since photo-generated carriers will also destroy the AF spinorder, we can expect that optical excitation alsohas drastic effects on LDSCES and these interest-ing phases can be photo-generated. Therefore, wecan expect the possibility of photo-induced phasetransition in which physical properties arechanged and controlled by light in LDSCES.

The physical properties of LDSCES inthe low energy limit have been studied extensively[1,2]. On the other hand, highly photo-excited states of LDSCES will exhibit variousproperties which are not observed in thelow energy states because electronic orderof the ground state is strongly perturbedand the balance in the ground state betweenthe competing interactions is lifted by theexcitation. We can therefore expect a new materialphase in the photo-excited LDSCES whichcannot be generated by doping. Since theexcited state reflects new and rich aspects ofthe Hamiltonian of LDSCES which are signifi-cantly different from those in the ground state,analysis of the multi-photon excited stateswill promote a better understanding of theextraordinary properties of the LDSCES fromnew and general points of view.

*Corresponding author. Fax: +81-743-72-6039.

E-mail address: taka@ms.aist-nara.ac.jp (A. Takahashi).

0022-2313/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 2 3 1 3 ( 0 1 ) 0 0 4 0 8 - 2

2. Model

To describe strongly correlated electron systems,we adopt the Hubbard Hamiltonian given by, H ¼T þ V :Here, T ¼ �t

P/i; jS;s ðc

wiscjs þ cwjscisÞ; V ¼

UP

i cwimcimc

wikcik; and notations are standard.

We consider the strong correlation case Ubt inthis paper. For t ¼ 0; the energy eigenvalue E isdetermined only by the number of doubly occu-pied sites nD; namely, E ¼ nDU: As a result, thereare discrete energy levels with the equidistantenergy separation U and huge number of energyeigenstates are degenerate at each energy level. Forfinite t satisfying the condition Ubt; the degen-eracy is lifted and each discrete energy levelbecomes an energy band. Note that we discusshere the energy bands of many-body Hamiltonian,and not the conventional one-electron bands basedon the independent electron approximation.

We consider resonant excitation by the photonswith the energy of _oEU: In this case, m-photonexcited states belong to the mth excited band. Thestates in the mth excited band mainly consist ofthe states with nD ¼ m and the contributionsfrom the other states can be included as smallperturbations. Taking into account the effects of Tto the second order in t=U; we derive an effectiveHamiltonian for the mth excited band, namely,that for the m-photon excited states, in thefollowing way.

We first define the relevant projection operatorPm onto the Hilbert subspace with nD ¼ m andirrelevant one Qm ¼ 1� Pm: Using these projec-tion operators, the Schr .odinger equation can berewritten as HðPm þQmÞjcS ¼ EjcS: ApplyingPm and Qm to this equation from the left, elimin-ating QmjcS from these two equations, and takingthe terms up to the second order in t=U; theSchr .odinger equation can be deduced as Heff jcS¼ EjcS; where Heff is the effective Hamiltonianfor the m-photon excited states and it is given by

Heff ¼PmHPm �U�1PmTPmþ1TPm

þU�1PmTPm�1TPm: ð1Þ

The first term describes hopping of electrons(holes) at the doubly occupied (empty) sites tothe nearest-neighbor sites. The second term

describes (1) the AF Heisenberg spin–spin inter-action between the nearest-neighbor sites withthe coupling constant J ¼ 4t2=U and also (2) thevirtual three-site hopping, through which theposition of an empty or a doubly occupied site istransferred to the second or the third nearest-neighbor sites. The third term is peculiar to thepresent effective Hamiltonian for the multi-photonexcited states. This part has non-zero matrixelement only when both the bra and the ket stateshave an empty and doubly occupied site pair at thenearest-neighbor, and describes several differenthopping processes of the pair.

The relaxation rate within the excited energyband is usually much faster than the radiativedecay rate in LDSCES [2]. Thus, considering thesituation where the optically excited state quicklyrelaxes within each excited band, we focus on thelowest energy state among the huge number of m-photon excited states and investigate its variousphysical properties. We denote this state simply bythe m-photon excited state in the following.

3. Results and discussion

We adopt the two-dimensional square latticeusing periodic boundary condition with the systemsizes N ¼ 20: Qualitatively, the same results areobtained for N ¼ 18: We have calculated thelowest energy state of Heff with m ¼ 2 using theexact diagonalization method. Here we assumedthat the state has the total momentum K ¼ 0because of the electron–hole symmetry of thepresent Hamiltonian.

We show the t=U dependence of the spincorrelation function ZðrÞ ¼ /Si � SjS; the correla-tion function of the photo-injected same (opposite)charges xðrÞ ¼ /hihjS (zðrÞ ¼ /dihjS), and thepair correlation function for d-wave superconduc-tivity pðrÞ ¼ /Dw

i DjS in Fig. 1. Here, /OS denotesthe expectation value of an operator O for the two-photon excited state, r is the distance between thesites i and j; and S i is a spin operator at the ith site.The positive (negative) charge density operator isdefined by hi ¼ dðniÞ (di ¼ dð2� niÞ), where ni ¼P

s cwiscis: Note that xðrÞ ¼ /didjS because of

the electron–hole symmetry of the present

A. Takahashi et al. / Journal of Luminescence 94–95 (2001) 499–502500

Hamiltonian. The d-wave pair operator is definedby Di ¼

Ps scisðciþx�s þ ci�x�s � ciþy�s � ci�y�sÞ

[1], where i7x (i7y) denotes the nearest-neighborsite of the ith site to the 7x (7y) direction ands ¼ 1ð�1Þ for up (down) spin.

As seen from Fig. 1, these correlation functionschange drastically and very steeply at t=U ¼ 0:017and also at 0.049. Thus, it is natural to considerthat there exist three phases in the two-photonexcited state. We name them as phases I, II and IIIin the increasing order of t=U:

We first discuss Z shown in Fig. 1(a). In phase I,the short-range spin correlations with rp

ffiffiffi2

pare

positive and the long-range ones with rX2 arenegative. This shows that the phase is a spiralspin state [3]. To analyze the spin structures inphases II and III, the staggered spin correlationfunction Z0ðrÞ ¼ t/S i � S jS; where t ¼ 1ð�1Þ if thesites i and j belong to the same (different) sublattice,is plotted as a function of r in Fig. 2. As seen fromthis figure, Z0ðrÞ becomes closer to zero as r isincreased up to the maximum value

ffiffiffiffiffi10

pat

t=U ¼ 0:04 (phase II) and also at t=U ¼ 0:065(phase III). Moreover, the long-range spin correla-tions are altered only a little even when J=t isincreased up to 2 (t=U ¼ 0:5). These results showthat the AF spin order characteristic of theground state is destroyed by two-photon excita-tion unlike the case of one-photon excitation [4].

Next, we discuss x and zshown in Figs. 1(b) and(c). In phase I, xðrÞ increases with increasing r; zðrÞis nearly constant for rX

ffiffiffi2

pand zð1Þ is much

smaller than the other values. In phases II and III,both xðrÞ and zðrÞ are nearly constant for rX

ffiffiffi2

p;

and both xð1Þ and zð1Þ are much smaller than theother values. These results show that (1) thephoto-injected opposite charges interact repul-sively in phase I, (2) except for the case, the

Fig. 1. The t=U dependence of (a) ZðrÞ; (b) xðrÞ; (c) zðrÞ; and(d) pðrÞ with r ¼ 1;

ffiffiffiffi2;

p2;

ffiffiffi5

p; and

ffiffiffiffiffi10

pin the two-photon

excited state.

Fig. 2. The r dependence of Z0ðrÞ at t=U ¼ 0:04; 0.065, and 0.5

in the two-photon excited state. That of the ground state is also

shown.

A. Takahashi et al. / Journal of Luminescence 94–95 (2001) 499–502 501

photo-injected same charges and also oppositecharges interact very weakly for rX

ffiffiffi2

p; and (3)

the possibility that they are situated at the nearestneighbors is minimal. There is no region wherephoto-injected charges become bound, and theunbound–bound crossover observed in the one-photon excited state [4] does not occur in the two-photon excited state. This is because AFspin order, which induces attractive interactionsbetween the charges, does not exist in thetwo-photon excited state. Moreover, we havecalculated the weights of all the charge arrange-ments and have found that the weights are broadlydistributed and there is no dominant arrangement.Consequently, photo-injected charges have nospecific order in any of the phases. The significantreduction of zð1Þ is observed even in the one-photon excited state and it is due to the third termof Heff [4]. The similar behavior of xðrÞ can beunderstood as a result of hard core repulsionbetween the same charges.

Finally, we discuss p shown Fig. 1 (d). The d-wave superconducting order is photo-generated inphase II, although pðrÞ for large r are very small inphase I. In phase III, long-range pðrÞ are very smallin the region shown in Fig. 1 but they increaseslowly with increasing t=U and become compar-able to those in phase II around t=U ¼ 0:2: Toobserve this in detail, we show the r dependence ofpðrÞ in Fig. 3. As seen from this figure, pðrÞD0 forrX

ffiffiffi5

pat t=U ¼ 0:003 (phase I) and t=U ¼ 0:065

(phase III), but pðrÞ remains finite and almostconstant for rX

ffiffiffi5

pat t=U ¼ 0:04 (phase II) and

t=U ¼ 0:3 (phase III). We cannot decisively con-

clude whether the long-range superconductivityorder exists or not in the present study on small sizesystems. This, however, suggests that pðNÞa0 andphases II and III are two different d-wave super-conducting states. As mentioned earlier, photo-injected opposite charges are unbound in the t=Uregions of the phases, which are consistent with thepresence of superconducting order. We should notethat pðrÞ of phase II (III) is larger (smaller) than thatin the four-hole doping case for a given t=U for allthe distances. In particular, pð

ffiffiffiffiffi10

pÞ of the two-

photon excited state in phase II is about twice largerthan that in the four-hole doping case.

Wehave also calculated a few lowest energy statesin the second excited band, and have found that (1)the second lowest energy state of the band in the t=Uregion of phase III (II) has the correlation functionssimilar to those of phase II (III), and (2) thedifference in energy between the lowest and thesecond lowest energy states is very small.

In conclusion, we have calculated two-photonexcited states in the two-dimensional Hubbardmodel in the strong correlation case, and havefound that there exist three phases which havedifferent electronic orders from that of the groundstate. The present results strongly suggest that theyare of spiral spin state and two different d-wavesuperconducting states. Although the results ob-tained in this paper are preliminary, their analysismay shed new light on the remarkable phenomenain optically excited LDSCES.

Acknowledgements

This work is supported by a Grant-in-Aid forScientific Research on Priority Areas, ‘Photo-induced Phase Transition and Their Dynamics’,from the Ministry of Education, Science, Sportsand Culture of Japan.

References

[1] E. Dagotto, Rev. Mod. Phys. 66 (1994) 763.

[2] M. Imada, A. Fujimori, Y. Tokura, Rev. Mod. Phys. 70

(1998) 1039.

[3] B. Doucot, X.G. Wen, Phys. Rev. B 40 (1989) 2719.

[4] A. Takahashi, S. Yoshikawa, M. Aihara, unpublished.Fig. 3. The r dependence of pðrÞ at t=U ¼ 0:003; 0.04, 0.065,and 0.3 in the two-photon excited state.

A. Takahashi et al. / Journal of Luminescence 94–95 (2001) 499–502502