Photoinduced Enhancement of Anisotropic Charge Correlationson Triangular Lattices with Trimers
Kenji Yonemitsu+
Department of Physics, Chuo University, Bunkyo, Tokyo 112-8551, Japan
(Received October 18, 2016; accepted December 15, 2016; published online January 26, 2017)
To explore nontrivial photoinduced modulations of charge correlations, we theoretically study photoinduced dynamicsin quarter-filled extended Hubbard models with competing intersite repulsive interactions on triangular lattices withtrimers, where the end points are crystallographically equivalent. The exact diagonalization method is used and the time-dependent Schrödinger equation is numerically solved during and after photoexcitation. Time-averaged doubleoccupancy and intersite density–density correlations can be interpreted as due to effective on-site and intersite repulsiveinteractions, respectively, relative to transfer energies. In the case where the intersite repulsive interactions compete witheach other, the anisotropy of their effective interactions can be enhanced with the help of the trimers, irrespective ofwhether the trimers are linear or bent. In particular, in the case where the arrangement of the trimers is close to that inα-(bis[ethylenedithio]-tetrathiafulvalene)2I3 [α-(BEDT-TTF)2I3] in the metallic phase, the effective on-site repulsion isenhanced relative to the transfer energies. The relevance of this theoretical finding to the experimentally observed opticalfreezing of charge motion is discussed.
1. Introduction
Various photoinduced cooperative phenomena includingphotoinduced phase transitions have been studied intensivelyfor more than a quarter century.1–4) When any order isinvolved, in most cases, the transition is toward destroying ormelting the order. This tendency is natural because theexternal field supplies energy to the system and finallyincreases the temperature of the system. However, in transientstates, the tendency can be the opposite: some order can beconstructed and the electronic motion can be frozen.
In fact, for α-(bis[ethylenedithio]-tetrathiafulvalene)2I3[α-(BEDT-TTF)2I3] in a metallic phase just above thecharge-order5) metal–insulator transition temperature, anintense infrared pulse leads to a reflectivity change as if theelectronic motion is frozen and the system transiently becomesa charge-ordered insulator.6) It is in contrast to the previouslyobserved, photoinduced melting of the charge order in thesame7–11) and similar12,13) compounds. As for frozen motion,dynamical localization is theoretically known in the case ofcontinuous waves.14–16) As a related phenomenon, a negative-temperature state is produced by continuous waves,17)
asymmetric pulses (like half-cycle pulses),18) and symmetricpulses,19,20) although their mechanisms are different. As for thesimilarity and difference between continuous-wave- and pulse-induced phenomena, the photoinduced transition from acharge-ordered-insulator phase to a Mott-insulator phasein the quasi-two-dimensional metal complex Et2Me2Sb[Pd-(dmit)2]2 (dmit = 1,3-dithiol-2-thione-4,5-dithiolate)21–23) canbe controlled in principle by suppressing interdimer electrontransfer in the case of continuous waves, and by suppressingintradimer electron transfer in the case of pulses.24) The issueon the similarity and difference between continuous-wave- andpulse-induced phenomena is also discussed in a one-dimen-sional transverse Ising model.25) In this context, transientlocalization is expected to be caused even by a pulse, asdynamical localization is caused by a continuous wave.
Quite recently, photoinduced changes in the direction ofstrengthening or enhancing the order have been observedexperimentally in different situations. A tendency towards the
transient stabilization of a charge density wave is demon-strated for RTe3 (R = Dy, Ho) after near-infrared excita-tion,26) counteracting the suppression of order in thenonequilibrium state, owing to transiently enhanced Fermisurface nesting. Ultrafast photoexcitation can transientlyenhance the charge-density-wave amplitude in a crystallineCr film,27) resulting from dynamic electron–phonon inter-actions. The quasi-one-dimensional organic conductor(TMTTF)2AsF6 (TMTTF = tetramethyltetrathiafulvalene) isirradiated with 1.5-cycle, 7-fs infrared pulses, which showsan increase in the effective mass, as a strong-light field effectassisted by Coulomb repulsion.28) For the layered semi-conductor Ta2NiSe5, above a critical photoexcitation density,the direct band gap is transiently enhanced,29) owing to itsexotic low-temperature ordered state. As to how intense THzexcitations can be used to resonantly control matter, throughsuperconducting gaps, Josephson plasmons in layered super-conductors, vibrational modes of the crystal lattice, andmagnetic excitations, see Ref. 4.
In this paper, bearing in mind the optical freezing of chargemotion in α-(BEDT-TTF)2I3 in the metallic phase, wetheoretically discuss the feasibility of pulse-induced transientlocalization and modulations of charge correlations byultrafast and intense photoexcitation. Its relationship withthe lattice structure (i.e., the network of transfer integrals) andinteractions will be clarified. For α-(BEDT-TTF)2I3, compet-ing intersite repulsive interactions and the lattice with trimerspossessing crystallographically equivalent end points areshown to be the key ingredients for the observed phenomena.Furthermore, it is shown that the photoinduced enhancementof anisotropic charge correlations is not limited to a latticestructure similar to that of α-(BEDT-TTF)2I3: it is observedregardless of whether the trimers are linear or bent.
2. Extended Hubbard Models on Triangular Latticeswith Trimers
We use quarter-filled extended Hubbard models,
H ¼Xhiji�
tijðcyi�cj� þ cyj�ci�Þ þUXi
ni"ni# þXhiji
Vijninj; ð1Þ
Journal of the Physical Society of Japan 86, 024711 (2017)
where cyi� creates an electron with spin σ at site i, ni� ¼ cyi�ci�,and ni ¼
P� ni�. The parameter U represents the on-site
Coulomb repulsion. We consider the triangular lattice thatconsists of equilateral triangles, where the distance betweenneighboring sites is denoted by a. The location of the ith siteis represented by ri, and the relative position by rij ¼ rj � ri.The nearest-neighbor Coulomb repulsion Vij is assumed tobe Vij ¼ V1 for rij being at a �30 or �150° angle withrespect to the horizontal axis, and Vij ¼ V2 for rij beingparallel to the vertical axis, as shown in Fig. 1. For theinteraction parameters, we use U ¼ 0:8 and V1 ¼ 0:3, andvary V2.
For the transfer integrals tij, we consider the two casesshown in Figs. 1(a) and 1(b). The structure in Fig. 1(a) haslinear trimers and is close to that of α-(BEDT-TTF)2I3 inthe metallic phase, although the latter does not consist ofequilateral triangles. The structure in Fig. 1(b) has benttrimers. The three sites (A, B, and AA in Fig. 1) in a trimer arelinked by the largest transfer integral in magnitude, t1. Thereis another type of trimer, “weak trimer”, where the three sites(A, C, and AA in Fig. 1) are linked by the second largesttransfer integral in magnitude, t2. The end points of a weaktrimer (A and AA) are end points of one [Fig. 1(b)] or two[Fig. 1(a)] “strong trimers”. The structure in Fig. 1(a) hasinversion symmetry with respect to site B, site C, or themidpoint of neighboring sites A and AA. This symmetry ispossessed by α-(BEDT-TTF)2I3 in the metallic phase. Thestructure in Fig. 1(b) has reflection symmetry with respectto a line parallel to the horizontal axis. The notations of sitesA, AA, B, and C are the same as those in Refs. 8 and 9 forα-(BEDT-TTF)2I3. In this paper, we use t1 ¼ �0:14, t2 ¼�0:13, t3 ¼ �0:02, t4 ¼ �0:06, t5 ¼ 0:03, and t05 ¼ �0:03for Fig. 1(a), and t1 ¼ �0:10, t2 ¼ �0:08, t3 ¼ �0:02, and
t4 ¼ �0:01 for Fig. 1(b). The parameters for Fig. 1(a) areclose to those of α-(BEDT-TTF)2I3 in the metallic phase.However, the conclusions of this study do not depend on thedetails in these parameters. Note in Fig. 1(a) that sites A andAA along the vertical axis are alternately linked by t5 and t05,as in α-(BEDT-TTF)2I3, which leads to similar V1 and V2
dependences of the electron density distributions for thenetworks of transfer integrals in Figs. 1(a) and 1(b) and inα-(BEDT-TTF)2I3 in the metallic phase. Note also thatα-(BEDT-TTF)2I3 is a three-quarter-filled system, i.e., aquarter-filled system in the hole picture. The transfer integralsin the hole picture are inverted from those in the electronpicture here.
The initial state is the ground state obtained by the exactdiagonalization method for the 16-site system with periodicboundary conditions. Photoexcitation is introduced throughthe Peierls phase
cyi�cj� ! expie
ħcrij � AðtÞ
� �cyi�cj�; ð2Þ
which is substituted into Eq. (1). We employ symmetricmonocycle electric-field pulses19,20) and use the time-depend-ent vector potential
AðtÞ ¼ cF
!½cosð!tÞ � 1��ðtÞ� 2�
!� t
� �; ð3Þ
with F ¼ Fðcos �; sin �Þ, where F is the amplitude of theelectric field and θ is the angle between the field and thehorizontal axis. The central frequency ω is chosen to be! ¼ 0:8, which is well above the main charge-transferexcitations, as shown below. The optical conductivity spectraare calculated as before.9) The time-dependent Schrödingerequation is numerically solved by expanding the exponentialevolution operator with a time slice dt ¼ 0:02 to the 15thorder and by checking the conservation of the norm.30) Thetime average hhQii of a quantity Q is calculated as
hhQii ¼ 1
tw
Z tsþtw
ts
h�ðtÞjQj�ðtÞi dt; ð4Þ
with ts ¼ 5T, tw ¼ 5T, and T ¼ 2�=! being the period ofthe oscillating electric field. If the model parameters arerepresented in eV, T corresponds to about 5 fs, and ts þ tw ¼10T corresponds to about 52 fs. Qualitative results areunchanged even if different time intervals are used.
3. Triangular Lattice with Linear Trimers
3.1 Ground states with inversion symmetryHere, we consider the triangular lattice with linear trimers
shown in Fig. 1(a). To see how the intersite repulsiveinteractions V1 and V2 govern the ground states, we show theelectron density distribution hnii in Fig. 2(a) and the spatiallyaveraged correlation functions hni"ni#i and hninji, whichcontribute to the interaction energies, in Fig. 2(b), asfunctions of V2. Note that the values and symmetry of thetransfer integrals used here are close to those of α-(BEDT-TTF)2I3 in the metallic phase. In the present finite-sizesystems, the symmetry is not spontaneously broken, so thatthe ground states have the same symmetry as α-(BEDT-TTF)2I3 has in the metallic phase. Sites A and AA arecrystallographically equivalent, so that their electron densitiesare equal in the ground states. The on-site repulsion U is the
(a)
t5 t5 t5t2 t2 t2t3 t3 t3
V2 V1U
V1
t1
t1
t1
t1
t1
t1
t'5t'5
t5
t5 t5
t5 t5
t2 t2 t2t3 t3 t3
t4
t4
t4
t4
t4
t4
V2V1V1
BA’
At2
t1
t1
t'5t4t1
t1
t1
t1
t'5
t5
t5
t5
t5
t5
t5
t2 t2
t t t
t3 t3 t3
t3 t3 t3
t4
t4 t4
t4
t4
Ct5 t5
t2
2 2 2
t2 t2t3 t3 t3
(b)
V2
t4 t4t1
t1
t1t1
t2
t2
t4
t4
t4
t4
t4
t4
t2
t2
t2t3
t3 t3
t3 t3 t3
t3 t3 t3
V1UV2
V1V1
V1
BA
C
t4
t4
t4
t4t1
t1
t1
t1
t1
t4t4t4
t2 t2
t2
t2t3t3 t3
t3 t3 t3
t3 t3 t3
A’t4 t4
t1
t1
t1
t1
t1
t4t4t4t2 t2
t2
t2t3t3 t3
t3 t3 t3
t3 t3 t3
Fig. 1. (Color online) Triangular lattices with (a) linear and (b) benttrimers. The lattice (a) has inversion symmetry, while (b) has reflectionsymmetry. Sites A and AA are crystallographically equivalent.
largest parameter, so that the double occupancy hni"ni#i issmall over the entire region shown in this figure.
When V2 is small, V1 dominates the ground state, and thenearest-neighbor density–density correlations hninji are smallalong the t1, t2, t3, and t4 bonds. In this case, the electrondensity should be distributed mainly to, in principle, eithersites A and AA or sites B and C along the t5 and t05 bonds. Forthe present choice of the transfer integrals, electrons aremainly on sites A and AA. As V2 increases, electrons onneighboring sites A and AA repel each other more strongly,but the symmetry is unchanged, so that their electron
densities remain equal. They simply decrease and the electrondensities increase at sites B and C.
When V2 becomes comparable to V1 (¼ 0:3), the spatialvariation of hni"ni#i and that of hninji become small. In thiscase, their distributions are governed by the transfer integrals,i.e., they are determined in such a way that the kinetic energyis lowered: the electron densities at sites A, B, and AA ontrimers are slightly larger than that at site C, and hninji alongthe t1 bond is slightly larger than hninji along the t2 bond.This situation is realized in α-(BEDT-TTF)2I3 in the metallicphase: the charge disproportionation at sites B and C is due tothis network of transfer integrals. As V2 further increases, theelectron densities at sites A, B, and AA on a trimer approach0.5, 1, and 0.5, respectively, maintaining the symmetry withrespect to the exchange of sites A and AA.
Hereafter, we mainly use V2 ¼ 0:35 for photoinduceddynamics. Then, we show the optical conductivity spectra forV2 ¼ 0:35 in Fig. 2(c). The main charge-transfer excitationsare seen below 0.4, so that ! ¼ 0:8 for photoexcitation usedbelow is well above them.
3.2 Densities and correlations after pulse excitationDensities and correlation functions that are time-averaged
after the monocycle pulse excitation as explained in Sect. 2are shown below, as functions of the dimensionless quantityequal to the ratio of the electric field amplitude to the centralfrequency eaF=ðħ!Þ. Unless stated otherwise, we use V2 ¼0:35 and � ¼ 0 in Sect. 3.2.
The time-averaged electron densities hhniii are shownin Fig. 3(a). In the ground state before photoexcitation,electrons are mainly on trimers (i.e., sites A–B–AA). Afterphotoexcitation, as the field amplitude increases, the chargedisproportionation becomes weak, and the time-averagedelectron density distribution approaches the homogeneousone, as expected. Here, the equivalence of sites A and AAis broken by the field. When the field is inverted (� !� þ 180°), hhniii at site A and hhniii at site AA are exchanged.Because the charge disproportionation at sites B and C is ofkinetic origin, as mentioned above, and because the 0.5–0.5density distribution at sites A–AA can be a linear combinationof the 1–0 and 0–1 density distributions, we need to see thecorrelation functions.
The spatially and temporally averaged double occupancyhhni"ni#ii once decreases and then increases as the fieldamplitude increases, as shown in Fig. 3(b). The decreasedhhni"ni#ii means that electrons with opposite spins avoidbeing on the same site more strongly as if the on-siterepulsion U were increased relative to the transfer integralstransiently after photoexcitation. The increased hhni"ni#ii forlarger field amplitudes is natural since the total energyincreases and the interaction energy due to the doubleoccupancy contributes to the total energy.
The spatially and temporally averaged nearest-neighbordensity–density correlations hhninjii show anisotropic behav-ior, as shown in Fig. 3(c). Because of V2 > V1, hhninjii alongthe t5 and t05 bonds is smaller than the others already beforephotoexcitation (at F ¼ 0), and it is further decreased by not-too-strong fields [eaF=ðħ!Þ � 0:5]. This behavior means thatelectrons avoid neighboring along the t5 and t05 bonds morestrongly as if the intersite repulsive interaction V2 wereincreased relative to the transfer integrals transiently after
(a)
(c)
(b)
Fig. 2. (Color online) (a) Electron density hnii at sites A, AA, B, and C,and (b) spatially averaged, double occupancy hni"ni#i, nearest-neighbordensity–density correlation hninji for rij at a −30° angle with respect tohorizontal axis (i.e., parallel to t1 and t3 bonds), hninji for rij at a 30° angle(i.e., parallel to t2 and t4 bonds), and hninji for rij parallel to vertical axis(i.e., parallel to t5 and t05 bonds), as functions of V2. (c) Optical conductivityspectra for V2 ¼ 0:35 with different polarizations as indicated. The peak-broadening parameter is set at 0.005.
photoexcitation. Meanwhile, hhninjii along the t1 and t3bonds and hhninjii along the t2 and t4 bonds increase withthe field amplitude. Thus, the anisotropy in hhninjii isenhanced by electric-field pulses, which is interpreted as thephotoinduced enhancement of the anisotropy in the effectiveintersite repulsive interactions.
The present ground state corresponds to α-(BEDT-TTF)2I3in the metallic phase once electrons in the former areregarded as holes in the latter. Note that, in equilibrium, thecharge order in the latter below the transition temperature ismainly caused by the intersite repulsive interaction V2 (> V1)and is assisted by the coupling to the lattice degrees of
freedom.31,32) An infrared pulse causes a reflectivity changeas if the system becomes a charge-ordered insulator for50 fs.6) The short lifetime is possibly due to a short-rangenature of the order. This experimental fact is consistent withthe present theoretical result in the sense that the enhancedeffective interactions U and V2 relative to the transferintegrals can support a short-range charge order since theobserved state is close to the metal–insulator phase boundary.The numerical calculations are performed for exact many-electron wave functions on finite-size systems that are notcoupled to lattice degrees of freedom, so that the symmetrycannot be broken spontaneously but can be broken by theexternal field. If we could approach the thermodynamic limit,the modulations of correlation functions would be larger thanthe present ones.
The dependence on the polarization of photoexcitation θ isshown in Fig. 4 for hhninjii along the t5 and t05 bonds. Notethat � ¼ 0 corresponds to the polarization along the b-axis inα-(BEDT-TTF)2I3. For small field amplitudes, eaF=ðħ!Þ ¼0:2 and 0.3, hhninjii becomes smallest at approximately� ¼ 30°. For larger field amplitudes, eaF=ðħ!Þ ¼ 0:4, 0.5,and 0.6, hhninjii becomes smallest at approximately � ¼ 60°.The polarization � ¼ 30° is along the t2 bonds in Fig. 1(a),where the transfer integral is the second largest in magnitude.The corresponding electron transfer process efficiently usessite C where the electron density is lowest in the groundstate. Sites A (AA) and B linked by t1 are already almostsingly occupied and the electron transfer through this bond(� ¼ 150°) is hindered by the strong on-site repulsion U, sothat the t1 process is less efficient. The t3 process (� ¼ 150°)can efficiently use site C in principle, but t3 is much smallerin magnitude. Then, the t2 process efficiently makes hhniii atsites A and AA approach 0.5, enhancing the 1–0=0–1 densitycorrelation at sites A–AA on the t5 and t05 bonds. The precisepolarization dependence is not so simple because the networkof transfer integrals is rather complex.
To see how general=special the photoinduced enhance-ment of the anisotropy in hhninjii is, hhninjii for V2 ¼ 0:25and 0.4 are shown in Figs. 5(a) and 5(b), respectively. ForV2 ¼ 0:25, hhninjii along the t1, t2, t3, and t4 bonds, which
(a)
(c)
(b)
Fig. 3. (Color online) (a) Time-averaged electron densities hhniii at sitesA, AA, B, and C, (b) spatially and temporally averaged double occupancyhhni"ni#ii, and (c) spatially and temporally averaged nearest-neighbordensity–density correlations hhninjii for rij in different directions asindicated, for V2 ¼ 0:35 and � ¼ 0, as functions of eaF=ðħ!Þ.
0.15
0.16
0.17
0.18
0.19
0.2
0 0.2 0.4 0.6 0.8 1Tim
e-av
erag
ed n
.n. c
orr.
// t 5
bon
ds
eaF/ ω
θ= 0 degθ= 30 degθ= 60 degθ= 90 deg
θ=120 degθ=150 deg
Fig. 4. (Color online) Spatially and temporally averaged nearest-neighbordensity–density correlation hhninjii for rij parallel to vertical axis (i.e.,parallel to t5 and t05 bonds), for V2 ¼ 0:35 and different polarizations ofphotoexcitation θ, as functions of eaF=ðħ!Þ.
are suppressed by V1 (> V2) before photoexcitation, increasewith the field amplitude, while hhninjii along the t5 and t05bonds decreases. The anisotropy in hhninjii is weakened byphotoexcitation. For V2 ¼ 0:4, the roles of the interactions V1
and V2 are only exchanged, so that the anisotropy in hhninjiiis also weakened here. These behaviors are in contrast to thephotoinduced enhancement of the anisotropy in hhninjii forV2 ¼ 0:35 shown in Fig. 3(c). The latter behavior is observedonly when V2 is slightly larger than V1 and hhninjii indifferent directions compete with each other. The differencebetween the cases of V2 > V1 and V1 > V2 is due to thepresence of trimers since the difference is already seen in theground states: Fig. 2(b) is not symmetric with respect to theV2 ¼ V1 point, owing to the structure with trimers. Then, thephotoinduced enhancement of anisotropic charge correlationsis presumed to be caused by the competing intersite repulsiveinteractions on a lattice with trimers. The next issue to beresolved is how generally this occurs on different lattices withtrimers.
4. Triangular Lattice with Bent Trimers
4.1 Ground states with reflection symmetryNow we consider the triangular lattice with bent trimers
shown in Fig. 1(b). To see how the intersite repulsiveinteractions V1 and V2 govern the ground states, we showthe electron density distribution hnii in Fig. 6(a) and thespatially averaged correlation functions hni"ni#i and hninjiin Fig. 6(b) as functions of V2. Sites A and AA are again
crystallographically equivalent [although the symmetry isdifferent from that in Fig. 1(a)], so that their electrondensities are equal in the ground states. Most of the behaviorsare similar to those in the previous section once the bonds areclassified as vertical and nonvertical. Figure 6(a) is similarto Fig. 2(a). In Fig. 6(b), hninji for rij at a 30° angle withrespect to the horizontal axis and that at a −30° angle areidentical because of the symmetry with respect to this axis.Otherwise, Fig. 6(b) is similar to Fig. 2(b). Since we mainlyuse V2 ¼ 0:35 for photoinduced dynamics below, we showthe optical conductivity spectra for V2 ¼ 0:35 in Fig. 6(c).The main charge-transfer excitations are seen below 0.4, so
(a)
(b)
Fig. 5. (Color online) Spatially and temporally averaged nearest-neighbordensity–density correlations hhninjii for rij in different directions as indicated,for � ¼ 0, (a) V2 ¼ 0:25 and (b) V2 ¼ 0:4, as functions of eaF=ðħ!Þ.
(a)
(c)
(b)
Fig. 6. (Color online) (a) Electron density hnii at sites A, AA, B, and C,and (b) spatially averaged, double occupancy hni"ni#i, nearest-neighbordensity–density correlation hninji for rij at �30° angles with respect tohorizontal axis (i.e., parallel to t1, t2, and t3 bonds), and hninji for rij parallelto vertical axis (i.e., parallel to t4 bonds), as functions of V2. (c) Opticalconductivity spectra for V2 ¼ 0:35 with different polarizations as indicated.The peak-broadening parameter is set at 0.005.
that ! ¼ 0:8 for photoexcitation used below is well abovethem.
4.2 Densities and correlations after pulse excitationUnless stated otherwise, we use � ¼ �90° in Sect. 4.2,
which corresponds to the direction parallel to the lineconnecting the end points of a bent trimer A–B–AA (and alsoof a weak trimer A–C–AA). The time-averaged electrondensity distribution hhniii is shown in Fig. 7(a). The behaviorin the range of 0 < eaF=ðħ!Þ < 1 in Fig. 7(a) is roughlysimilar to that in the range of 0 < eaF=ðħ!Þ < 0:5 inFig. 3(a). The equivalence of sites A and AA is broken by thefield. In the present case, hhniii at site A and hhniii at site AA
are exchanged for � ! ��. The averaged double occupancyhhni"ni#ii shown in Fig. 7(b) behaves differently from that inFig. 3(b): it monotonically increases with the field amplitude.For small field amplitudes, however, the change rate is quitesmall. There exist polarizations for which hhni"ni#ii decreasesfor small field amplitudes, but the change rate in that case isalso quite small (not shown). Note that the range of theordinate in Fig. 7(b) is one order of magnitude narrower thanthat in Fig. 3(b).
The behavior of the averaged nearest-neighbor density–density correlations hhninjii shown in Fig. 7(c) is similar tothat in Fig. 3(c). Because of V2 > V1, hhninjii along the t4bonds is smaller than the others already before photo-excitation (at F ¼ 0), and it is further decreased by electric-field pulses. The anisotropy in hhninjii is enhanced by them,which is interpreted as the photoinduced enhancement of theanisotropy in the effective intersite repulsive interactions,even for the present structure with bent trimers.
The polarization (θ) dependence of hhninjii along the t4bonds is shown in Fig. 8. In the present structure with benttrimers, the inversion symmetry does not exist, in contrast tothe previous structure with linear trimers possessing thissymmetry. Then, the data for θ and � þ 180°, which wouldbe difficult to distinguish experimentally, are averaged andmade easy to see. At � ¼ �90° (the data for � ¼ 90 and −90°are actually identical), hhninjii becomes smallest for anyeaF=ðħ!Þ. This polarization is parallel to the line connectingthe end points of two types of trimers (linked by t1 and by t2).In this sense, the present result is similar to the previous onefor linear trimers: the t2 process efficiently uses site C wherethe electron density is lowest in the ground state. For� ¼ 60°, hhninjii is also decreased by the field in the entirerange displayed. For � ¼ 30°, hhninjii slightly decreases forvery small field amplitudes; it increases otherwise. For� ¼ 0°, hhninjii monotonically increases with the fieldamplitude. Note that photoexcitation with � ¼ 0° does notbreak the reflection symmetry, and that sites A and AA remaincrystallographically equivalent for � ¼ 0°. This implies thatthe photoinduced enhancement of the anisotropy requires an
(a)
(c)
(b)
Fig. 7. (Color online) (a) Time-averaged electron densities hhniii at sitesA, AA, B, and C, (b) spatially and temporally averaged double occupancyhhni"ni#ii, and (c) spatially and temporally averaged nearest-neighbordensity–density correlations hhninjii for rij in different directions asindicated, for V2 ¼ 0:35 and � ¼ �90°, as functions of eaF=ðħ!Þ.
Fig. 8. (Color online) Spatially and temporally averaged nearest-neighbordensity–density correlation hhninjii for rij parallel to vertical axis (i.e.,parallel to t4 bonds), for V2 ¼ 0:35 and different polarizations ofphotoexcitation θ, as functions of eaF=ðħ!Þ. Here, the data for θ and� þ 180° are averaged.
escape from the crystallographic equivalence. If directionallyaveraged over θ, hhninjii is decreased by not-too-strongfields.
Finally, hhninjii for V2 ¼ 0:25 and 0.4 are shown inFigs. 9(a) and 9(b), respectively. In both cases, hhninjii that islarger before photoexcitation basically decreases, and hhninjiithat is smaller before photoexcitation increases, as the fieldamplitude increases. Thus, the anisotropy in hhninjii isweakened by photoexcitation, in a manner similar to theprevious one for linear trimers. These behaviors are incontrast to the photoinduced enhancement of the anisotropyin hhninjii for V2 ¼ 0:35 shown in Fig. 7(c). In any case, afact that is more important is the photoinduced enhancementof anisotropic charge correlations, which is caused by V2
being slightly larger than V1 on different lattices with trimers.
5. Conclusions and Discussion
To explore the feasibility of photoinduced charge local-ization and nontrivial modulations of charge correlations, wetreat quarter-filled extended Hubbard models on triangularlattices with trimers, where the end points are crystallo-graphically equivalent. To clarify the generality of thefindings, we employ a triangular lattice with linear trimerslike that of α-(BEDT-TTF)2I3 in the metallic phase andanother lattice with bent trimers. For different relativestrengths of anisotropic intersite interactions, we calculatephotoinduced dynamics using the exact diagonalizationmethod and the time-dependent Schrödinger equation.
For the triangular lattice with linear trimers, the spatiallyand temporally averaged double occupancy decreases afternot-too-strong electric-field pulses (even for other polar-izations θ and for V2 ¼ 0:25 and 0.4 in addition to the datapresented above, not shown) are applied as if the on-siterepulsion U were increased relative to the transfer integralstransiently after photoexcitation. In α-(BEDT-TTF)2I3, theintersite repulsion V2 is known to be slightly larger than V1
owing to the difference in the intermolecular distance. In thiscase, the spatially and temporally averaged nearest-neighbordensity–density correlations behave as if the intersiterepulsion V2 were increased and V1 were decreased relativeto the transfer integrals after photoexcitation.
In the transient state experimentally observed immediatelyafter photoexcitation,6) the lattice is not yet expected to besufficiently distorted to stabilize long-range charge order, andthe optical freezing of charge motion lasts for only 50 fs,so that the order, if any, would be short-ranged. In this sense,the numerical calculations performed on finite-size systemsfor ultrafast dynamics without coupling to a heat bath wouldbe suitable. The theoretically found, enhanced effectiveinteractions U and V2 relative to the transfer integrals areconsistent with the experimentally observed optical freezingof charge motion.6)
The lattice structures with trimers studied in this paperallow the presence of a charge-rich site B at the center andthe crystallographically equivalent end points A and AA of atrimer. The intersite repulsion V2 is substantial between thesesites [A and AA belong to neighboring trimers in Fig. 1(a) andto a single trimer in Fig. 1(b)]. In both cases of linear andbent trimers, the photoinduced enhancement of anisotropiccharge correlations is observed only when V2 is slightlylarger than V1.
The condition on V1 and V2 is understood from thefollowing points. (i) Before photoexcitation, the end points ofa trimer are equivalent, so that their electron densities areequal. (ii) The intersite repulsion V2 is substantial betweenthem, so that they are ready to disproportionate. Charge ordermay emerge as a consequence of freedom from oppression,i.e., crystallographic equivalence. (iii) The emergence istriggered by a rather small perturbation such as coupling tolattice degrees of freedom (as realized in equilibrium belowthe charge-order metal–insulator transition temperature,31–33)
which is an example of electronic ferroelectricity34)) or anelectric field. Ultrafast photoexcitation does not allow thelattice to be substantially distorted, but it can preventthermalization at an early stage. If V2 is small, the condition(ii) is not fulfilled. If V2 is too large (but substantially smallerthan U), the strong density–density correlation between sitesA and AA, which is already developed before photoexcitation,is merely weakened by photoexcitation. Therefore, it is anissue of balance. A trimer works indeed as a balancing toythat accepts largely inclined states. To coherently inclinebalancing toys, we need interactions between them andexternal stimuli.
From the above discussions, the existence of crystallo-graphically equivalent sites is found to be essential. Indeed,even if we distort the triangular lattices to make square latticeswith diagonal lines in one direction, keeping the crystallo-graphic equivalence of sites A and AA, we find very similarphenomena (not shown). Therefore, this is different from a
(a)
(b)
Fig. 9. (Color online) Spatially and temporally averaged nearest-neighbordensity–density correlations hhninjii for rij in different directions as indicated,for � ¼ �90°, (a) V2 ¼ 0:25 and (b) V2 ¼ 0:4, as functions of eaF=ðħ!Þ.
charge-ordered liquid state for spinless fermions on triangularlattices,35,36) which has been discussed in the context ofgeometrical frustration in triangular lattices as realized inθ-(BEDT-TTF)2X compounds in the metallic phase.
From these discussions, the optical freezing of chargemotion observed in α-(BEDT-TTF)2I36) is presumablyattributed to the competing intersite repulsive interactionsand the lattice structure with trimers possessing crystallo-graphically equivalent end points. It is not as simple asdynamical localization. Although the importance of inter-actions has been suggested,6,28) a mechanism is made clearerafter considering lattice structures and the competition ofinteractions. For V2 ¼ 0:35 and linear trimers, the decrease inthe averaged double occupancy for � ¼ 0 and eaF=ðħ!Þ ¼0:4 is reproduced in the ground state by the increase in Urelative to the bandwidth by 4%, while the theory fordynamical localization leads to the increase in U relative tothe renormalized bandwidth by 3% at the same parameters forθ and eaF=ðħ!Þ. However, the photoinduced enhancement ofanisotropic charge correlations is not explained by the theoryfor dynamical localization or an extension of it within theframework of the quantum Floquet theory, which will bereported elsewhere. A similar behavior regarding the photo-induced enhancement of the anisotropy is realized even iftrimers are bent as long as the end points are crystallo-graphically equivalent. It would be experimentally difficultto verify the mechanism, but it would be useful to see thedependence of the phenomenon on the polarization ofphotoexcitation. For the present triangular lattices consistingof equilateral triangles, the photoinduced enhancement ofanisotropic charge correlations is almost maximized whenpolarized parallel to the line connecting the end points of aweak trimer. For specific materials, we would need a detailedstudy of the polarization dependence.
Acknowledgments
The author is grateful to S. Iwai for continual discussionsfrom the discovery of the phenomenon and for Y. Tanakafor various theoretical discussions. This work was supportedby Grants-in-Aid for Scientific Research (C) (Grant No.16K05459) and Scientific Research (A) (Grant No.15H02100) from the Ministry of Education, Culture, Sports,Science and Technology of Japan.
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