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0022-2313/$ - se

doi:10.1016/j.jlu

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Journal of Luminescence 128 (2008) 1072–1074

www.elsevier.com/locate/jlumin

Photoinduced Hubbard band mixing of a one-dimensionalMott insulator

J.D. Leea,�, J. Inoueb

aSchool of Materials Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, JapanbQuantum Dot Research Center, National Institute for Materials Science, Tsukuba 305-0044, Japan

Available online 13 November 2007

Abstract

Based on the nonperturbative many-body time-dependent approach, photoinduced Hubbard band mixing by ultrashort laser pumping

is found in a one-dimensional half-filled Hubbard model through the optical conductivity. The photoinduced density of states (DOS)

appears below the gap energy ðEgÞ and is attributed to the metallic weight. In particular, the nonlinear response in the resonant pumping

ðopu � EgÞ and the linear response in the off-resonant pumping ðopubEgÞ imply two qualitatively different processes of the

photoinduced insulator–metal transition, which is confirmed by the photoemission just after the ultrashort laser irradiation, i.e., the

illuminated photoemission.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Mott insulator; Photoinduced phase transition; Illuminated photoemission

Photoinduced phase transitions (PIPTs) in correlatedsystems attract increasing attention and provide a bench-marking cornerstone for a new field of photodynamics [1].One noticeable PIPT is the photoinduced insulator–metaltransition (PIIMT) without structural changes [2]. Ultra-short optical pumping is also a useful doping method asgood as chemical doping. Indeed, the onset of Drude-likeweights just after the photoirradiation, which is acharacteristic of PIIMT, has been reported in stronglycorrelated electron systems (SCES) without structuralchanges; the one-dimensional (1D) halogen-bridged Ni-chain compound (Ni–X chain) ½NiðchxnÞ2Br�Br2 ðchxn ¼cyclohexanediamineÞ [2]. There is, however, a fundamentaldifference between chemical and photo (optical) doping:the former is static and the latter is a dynamic phenom-enon. The explicit dynamical response of SCES during therelevant ultrafast time span after ultrashort optical pump-ing is an unexplored problem, partially because of a lack ofa theoretical framework to tackle the problem. In aconventional theoretical treatment, the Hubbard bandsare considered to be rigid. However, it is not clear at

e front matter r 2007 Elsevier B.V. All rights reserved.

min.2007.11.055

ing author. Tel.: +81761 51 1784; fax: +81 761 51 1149.

ess: jdlee@jaist.ac.jp (J.D. Lee).

all whether the rigid band picture can be really justified ornot. This problem is a highly nontrivial and ratherfundamental issue for a wide range of material scienceresearch fields.For a theoretical study of ultrafast dynamics, a new

nonperturbative formalism called many-body time-depen-dent diagonalization (MTD) is found to be powerful [3].MTD is in fact good at describing the two-body responseswith nonequilibrium states, which naturally reduces to theKubo formula in the limit of a continuous optical wave inits linear order [3]. This method is suitable to study theproblem cited above. We explore electronic reconstructionin SCES by studying the optical conductivity of the half-filled 1D Hubbard system induced by ultrashort opticalpumping within MTD. For this purpose, we first examinehow the metallic states evolve. The qualitative behavior ofthese weights is found to depend strongly on whether theoptical pumping (with laser energy opu) is resonant ðopu �

EgÞ or off-resonant ðopubEgÞ with the gap energy Eg.Finally, the Hubbard band reconstruction is clearlyvisualized through examining the photoemission responsejust after the ultrashort irradiation, i.e., the illuminatedphotoemission spectroscopy (ILPES). ILPES is alsotreated within MTD.

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σ(ω

) (a

rb.

un

its)

ω

ω pu

= 1.75

A = 0.01

A = 0.01

0.0 0.5 1.0 1.5 2.0

A = 0.01

A = 0.01

ωpu

= 1.88

ω 0.0 0.5 1.0 1.5 2.0

A = 0.01

A = 0.01

σ(ω

) (a

rb.

un

its)

ωpu

= 2

ω 0.0 0.5 1.0 1.5 2.0

A = 0.01

A = 0.01

ωpu

= 2.25

ω 0.0 0.5 1.0 1.5 2.0

Fig. 1. Optical conductivity sðoÞ from Eq. (2) with respect to opu and

Apu. U ¼ 3 and t ¼ 1 are taken for intrinsic parameters of the Hubbard

model.

J.D. Lee, J. Inoue / Journal of Luminescence 128 (2008) 1072–1074 1073

We start with the half-filled 1D Hubbard modelH0. It iscombined with the ultrashort optical pumping V with thepolarization along the chain direction and extended to beH ¼H0 þVðtÞ:

H0 ¼ �tX

ls

ðcy

lþ1scls þ cy

lsclþ1sÞ þUX

l

nl"nl#,

VðtÞ ¼ jApu eioputY0ðtÞ þ h:c:; (1)

where cy

lsðclsÞ is the electron creation (annihilation)

operator with spin s at site l and nls ¼ cy

lscls. When this

model is applied to a 1D Ni–X chain, cy

lsðclsÞ signifies

a 3d outer-shell electron of Ni ðNi3þÞ. The parametert gives the nearest-neighbor hopping and U the on-site

repulsion. j is the current operator defined by j ¼

itP

!sðcy

lþ1scls � cy

lsclþ1sÞ. For parameters of the pumping

laser pulse, Apu is the electric field strength and Y0ðtÞ ¼YðtÞ �Yðt� DÞ with the pulse length D [YðtÞ: Heavisidestep function]. For an implementation of MTD, one firstneeds to set up the many-body Hilbert space spanned by

states ofQ

l jcili, where jc

ili should be the electronic state at

site l, that is, j0i, j "i, j #i,j "#i according to i ¼ 0, 1, 2, 3,respectively. Within the Hilbert space, one can write downthe quantum state of the whole system at time t as

jCðtÞi ¼PfigCfigðtÞ

Ql jc

ili. The ground state for jCð0Þi is

obtained by exact diagonalization with a finite number ofHubbard sites. Dynamics that start by turning on theoptical pumping at t ¼ 0 can then be described by solvingthe coupled differential equations for CfigðtÞ resulting from

the time-dependent Schrodinger equation iq=qtjCðtÞi¼HjCðtÞi. From the solution, one can calculate the

time-dependent optical current JðtÞ ¼ hCðtÞjjjCðtÞi. JðtÞcan be shown to give the usual correlation function (that is,the Kubo formula) under the linear response by the S-matrix expansion [3]. We treat the half-filled N-siteHubbard chain with a periodic boundary conditionimposed. The system size is set to be N ¼ 8 unlessmentioned otherwise.

We consider the Hubbard model with U ¼ 3 and t ¼ 1and the ultrashort pulse with D ¼ 100. To be explicit, fort ¼ 1 eV, we have D ¼ 100 eV�1 � 66 fs. The spectralfunction estimates the gap energy Eg � 1:73. In a givensystem, the optical pulse plays the role of a trigger for thedynamics after t ¼ D. The dynamics can be seen moreclearly from the optical conductivity sðoÞ in the frequencydomain. This can be obtained by the Fourier transforma-tion of JðtÞ=Apu. For the time integral necessary for theFourier transformation, one needs to consider practicallythe finite upper bound tc instead of þ1

sðoÞ ¼Z tc

Ddt eiot½JðtÞ=Apu�. (2)

In Fig. 1, sðoÞ is given with respect to opu and Apu. Wehave taken tc ¼ 1000. In the figure, it is seen that theoptical conductivity consists of two peaks. One is the

strong peak at the position of Egð� 1:73Þ and the other isthe low energy peak at o � 0:22. As opu increases, acontinuous transfer of the optical weight from the gaptransition to the inside of the gap region is found. The lowenergy features would be attributed to the metallictransition by photoexcited carriers [4]. However, we haveno true Drude feature at exactly o ¼ 0 in a finite system [5].Interestingly, those low energy features are shown todevelop nonlinearly with the external optical pumping foropu � Eg, while developing linearly for opubEg.Now let us examine the deeper nature of IMT

incorporating the development of low energy metallicfeatures in sðoÞ with respect to opu and Apu. The lineardevelopment of the metallic weight is one of the featuresof photoconductivity due to photodoping based on therigid band. On the other hand, the physical significanceof the nonlinear development of the metallic weightcould be profound. It is well known that, in a 1D Mottinsulator, the strong enhancement of the nonlinear chargeresponse results due to the large dipole moment fromthe almost degenerate odd and even excited states,especially near resonant optical excitation [6]. We notethat, because of such a strong nonlinear response, V wouldno longer be a small perturbation for H0 in Eq. (1). Thissuggests that the electronic structure undergoes a nontrivialchange in the form of phase transition at this resonantpumping. The most direct way to confirm these twodistinguishable processes of metallic switching would beILPES [7].For ILPES, we extend our model Hamiltonian H to

include photoemission terms and have

H ¼H0 þVðtÞ þXks

ekcy

kscks þVpesðt� DÞ,

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0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

inte

nsity (

arb

. units)

-5 -4 -3 -2 -1 0 1

binding energy

E

zerobindingenergy

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00in

tensity (

arb

. units)

-5 -4 -3 -2 -1 0 1

binding energy

A = 0.01

A = 0.1

ω = 1.75

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

inte

nsity (

arb

. units)

-5 -4 -3 -2 -1 0 1

binding energy

A = 0.01

A = 0.1

ω = 2

Fig. 2. Photoemission spectra and schematic sketches of the underlying

electronic structure. (a) Photoemission without optical pumping, (b)

illuminated photoemission under resonant optical pumping, i.e.,

opu � Eg, and (c) illuminated photoemission under off-resonant optical

pumping, i.e., opubEg. U ¼ 3 and t ¼ 1 are taken for intrinsic Hubbard

parameters ðEg � 1:73Þ.

J.D. Lee, J. Inoue / Journal of Luminescence 128 (2008) 1072–10741074

VpesðtÞ ¼ Apes

Xkls

cy

kscls eiopestYðtÞ þ h:c:; (3)

where cy

ksðcksÞ is the photoelectron with its kinetic energy

ek ¼ k2=2 and opes and Apes are the energy and the strength

of the photon source adopted for the photoemission,respectively. MTD can also be applied to the calculation ofILPES. It is necessary to extend the Hilbert space to

include all the relevant states spanned byQ

l jcili and

jksiQ

l jcili, where jksi is the state of the photoelectron.

Accordingly, the quantum state of the whole system should

be jCðtÞi ¼PfigCfigðtÞ

Qljc

ili þ

Pks

PfigC

ksfig ðtÞjksi

Ql jc

ili

and ILPES can then be simply obtained asP

s

PfigjC

ksfig ðtÞj

2

from the solution of the time-dependent Schrodingerequation iq=qtjCðtÞi ¼HjCðtÞi in the limit of Apes! 0

when t4D.ILPES and schematic sketches of the corresponding

electronic structures (i.e., density of states) are provided inFig. 2. The occupied (hatched) parts of those sketches can

be directly compared with ILPES. It is most interesting tofind the finite density of states between the original upperand lower Hubbard bands, i.e., in the gap region, in ILPESunder the resonant pumping of Fig. 2(b), which evidentlysignifies nontrivial band mixing. The driving force shouldbe the nonperturbative effect of V due to the stronglyenhanced nonlinearities. On the other hand, we find that inILPES under the off-resonant pumping shown in Fig. 2(c),the overall electronic structure is rigid and the induceddensity of states is accumulated only at the lower edge ofthe upper Hubbard band. To conclude, ILPES shows thereshould be two qualitatively different dynamical processesfor the metallic switching and confirms the conclusion fromthe study of the photoinduced optical conductivity.In summary, we find the photoinduced Hubbard band

mixing of the 1D half-filled Hubbard model from ananalysis of the optical conductivity induced by ultrashortlaser pumping, together with the direct investigation of thechange of electronic structures through ILPES. Thephotoinduced weights evolve below the energy gap andthey are attributed to the Drude weights in the thermo-dynamic limit. The nonlinear behavior of metallic weightsat the resonant pumping directly demonstrates thereconstruction of electronic states, meaning the rigid bandpicture is not valid.

This work was supported by Special CoordinationFunds for Promoting Science and Technology fromMEXT, Japan.

References

[1] Y. Tokura, J. Phys. Soc. Japan 75 (2006) 011008;

K. Yonemitsu, K. Nasu, J. Phys. Soc. Japan 75 (2006) 011008 and

references therein.

[2] S. Iwai, et al., Phys. Rev. Lett. 91 (2003) 057401;

S. Iwai, H. Okamoto, J. Phys. Soc. Japan 75 (2006) 011007.

[3] J.D. Lee, et al., Phys. Rev. Lett. 97 (2006) 157405.

[4] By the matrix element of hnjjjn0i (n; n0: many-body eigenstates), it can

be shown that in the region of jEn � En0 joEg, n 2 1DO and n0 2 1DO

(1DO: subspace with a single double occupancy) is found to be

dominant.

[5] E. Dagotto, Rev. Modern Phys. 66 (1994) 763 and references therein.

[6] M. Ono, et al., Phys. Rev. B 70 (2004) 085101;

M. Ono, et al., Phys. Rev. Lett. 95 (2005) 08741.

[7] J.Y. Son, et al., Phys. Rev. B 72 (2005) 235105.