Photoinduced High-Frequency Charge Oscillations in Dimerized Systems
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Journal of the Physical Society of Japan
Photoinduced High-Frequency Charge Oscillations in DimerizedSystems
Kenji Yonemitsu ∗ Department of Physics, Chuo University, Bunkyo, Tokyo 112-8551, Japan
Photoinduced charge dynamics in dimerized systems is studied on the basis of the ex-act diagonalization method and the time-dependent Schr¨odinger equation for a one-dimensional spinless-fermion model at half filling and a two-dimensional model for κ -(bis[ethylenedithio]tetrathiafulvalene) X [ κ -(BEDT-TTF) X] at three-quarter filling. Afterthe application of a one-cycle pulse of a specifically polarized electric field, the charge den-sities at half of the sites of the system oscillate in the same phase and those at the other halfoscillate in the opposite phase. For weak fields, the Fourier transform of the time profile of thecharge density at any site after photoexcitation has peaks for finite-sized systems that corre-spond to those of the steady-state optical conductivity spectrum. For strong fields, these peaksare suppressed and a new peak appears on the high-energy side, that is, the charge densitiesmainly oscillate with a single frequency, although the oscillation is eventually damped. In thetwo-dimensional case without intersite repulsion and in the one-dimensional case, this fre-quency corresponds to charge-transfer processes by which all the bonds connecting the twoclasses of sites are exploited. Thus, this oscillation behaves as an electronic breathing mode.The relevance of the new peak to a recently found reflectivity peak in κ -(BEDT-TTF) X afterphotoexcitation is discussed.
1. Introduction
Photoinduced dynamics and phase transitions in itinerant electron systems have re-ceived renewed interest with the observation of strengthened or weakened orders and eventransient suppression of charge motion, enhancing the prospect for controlling the elec-tronic phase.
8, 9)
Here, pictures of photoinduced states are not conventional ones obtainedafter the absorption of photons but rather electrons directly and coherently driven by an os-cillating electric field.
10, 11)
In this context, the concept of dynamical localization may play ∗ E-mail: [email protected] 1 /
21. Phys. Soc. Jpn. an important role, although it is basically applicable to systems that are driven by continuouswaves.
Even after a pulse excitation, states induced by a strong electric field have beenshown to be similar to states expected for dynamical localization.
5, 7, 16, 17)
Dynamical localization describes the long-time behavior obtained by time-averaging.Negative-temperature states and inverted interactions have also been discussed by time-averaging after photoexcitation.
For the long-time behavior, continuous-wave- andpulse-induced phenomena have been compared in a quantitative manner from a broad per-spective.
However, the picture for short-time behavior has not been discussed in a sys-tematic manner. Thus, it is desirable to present a concrete example.Quite recently, a new reflectivity peak has been discovered in photoexcited κ -(bis[ethylenedithio]tetrathiafulvalene) Cu[N(CN) ]Br [ κ -(BEDT-TTF) Cu[N(CN) ]Br] onthe high-energy side of the main reflectivity spectrum. This unprecedented peak is narrow.Its energy is independent of the excitation strength and it survives for a while after photoex-citation; thus, it is not due to the optical Stark e ff ect. The associated charge oscillation hasbeen shown to be enhanced near criticality in the “pressure”-temperature phase diagram. A mechanism of the emergence of such a high-energy peak is theoretically studied using theexact diagonalization method for small clusters in this paper. Thus, the influence of criticalityis beyond the scope of this paper.The object material is one of the κ -(BEDT-TTF) X, which are quasi-two-dimensionalthree-quarter-filled dimerized organic conductors. Photoinduced insulator-metal transitionsare known to take place in these materials.
The intradimer charge degrees of freedomare studied in reference to anomalous dielectric permittivity, which is associated with polarcharge distributions inside dimers.
However, the high-frequency charge oscillation modehad not been discussed before Ref. 25. Thus, the mechanism and condition for the appearanceof this mode are yet to be clarified.Here, we show that such a charge oscillation mode emerges in di ff erent dimerized systemsafter the application of a strong pulse of an oscillating electric field. Numerical results arepresented in a one-dimensional spinless-fermion “ t - t - V ” model at half filling and a two-dimensional extended Hubbard model for κ -(BEDT-TTF) X at three-quarter filling, which isphotoexcited along the a - or c -axis. The high-frequency charge-oscillation mode is shown toappear in a wide parameter space of ground states with a uniform charge distribution. A closeassociation with time-averaged properties is also revealed. /
21. Phys. Soc. Jpn. (a) t t (b)0 1 2 13 (b) b b pp qqb b ac a Fig. 1. (Color online) (a) One-dimensional lattice for spinless-fermion model and (b) two-dimensional latticefor κ -(BEDT-TTF) X.
2. Dimerized Models in One and Two Dimensions
In one dimension, we use one of the simplest models, i.e., a spinless fermion model athalf filling, H = t X n ( c † n c n + + c † n + c n ) + t X n ( c † n − c n + c † n c n − ) + V X j n j − ! n j + − ! , (1)where c † j creates a spinless fermion at site j and n j = c † j c j . The parameter V represents thenearest-neighbor repulsion. Large and small transfer integrals, t and t , are alternated, asshown in Fig. 1(a). A 24-site system with a periodic boundary condition is used. The distancebetween neighboring sites is set to be equal and unity. We use t = − . t and V . /
21. Phys. Soc. Jpn.
In two dimensions, we use an extended Hubbard model at three-quarter filling, H = X h i j i σ t i j ( c † i σ c j σ + c † j σ c i σ ) + U X i n i ↑ − ! n i ↓ − ! + X h i j i V i j n i − ! n j − ! , (2)where c † i σ creates an electron with spin σ at site i , n i σ = c † i σ c i σ , and n i = P σ n i σ . The parameter U represents the on-site Coulomb repulsion. The transfer integral t i j and the intersite Coulombrepulsion V i j depend on the bond i j , as shown in Fig. 1(b). A 16-site system with periodicboundary conditions is used. The intermolecular distances and angles are taken from thestructural data for κ -(BEDT-TTF) Cu[N(CN) ]Cl.
36, 37)
Unless stated otherwise, we use, inunits of eV, t b = − . t b = − . t p = − . t q = . U = . V b = . V b = . V p = .
28, and V q = .
24. In Eq. (2), the constant term is subtracted in such a way that the total energybecomes zero in equilibrium at infinite temperature.The initial state is the ground state obtained by the exact diagonalization method. Pho-toexcitation is introduced through the Peierls phase c † i σ c j σ → exp (cid:20) ie ~ c r i j · A ( t ) (cid:21) c † i σ c j σ , (3)which is substituted into Eq. (2) [and its spinless analog is substituted into Eq. (1)] for eachcombination of sites i and j with relative position r i j = r j − r i . We employ symmetric one-cycle electric-field pulses
17, 20, 21) and use the time-dependent vector potential A ( t ) = c F ω c [cos( ω c t ) − θ ( t ) θ πω c − t ! , (4)where F describes the amplitude ( F = | F | ) and polarization of the electric field. Unless statedotherwise, the central frequency ω c is chosen to be ω c = .
7, which is above the main charge-transfer excitations, as shown below. Hereafter, frequencies ω including ω c are also shown inunits of eV (with ~ = The time-dependent Schr¨odinger equation is numerically solved by expanding theexponential evolution operator with a time slice dt =
3. Intradimer and Interdimer Bond Densities in Two-Dimensional Case
In this section, we use the two-dimensional model for κ -(BEDT-TTF) X. Some quantitiestime-averaged after photoexcitation along the c -axis are shown in Fig. 2. In this paper, aver-ages are taken over the interval of 5 T < t < T with T = π/ω c as before. Figure 2(a) /
21. Phys. Soc. Jpn. (a)(b)
Fig. 2. (Color online) (a) Time-averaged bond densities P σ hh (cid:16) c † i σ c j σ + c † j σ c i σ (cid:17) ii for di ff erent combinationsof i j , (b) time-averaged kinetic energy, and total energy after photoexcitation along c -axis as functions of fieldamplitude F . shows the time-averaged bond densities P σ hh (cid:16) c † i σ c j σ + c † j σ c i σ (cid:17) ii for i j along the b , b , p , and q bonds as functions of the field amplitude F . Here, F is shown in units of V / Å. If we takethe length scale a = c -axis of the intradimer inter-molecular relative position in κ -(BEDT-TTF) X, F = eaF / ~ ω c = F increases, the time-averaged bond densities decrease in magnitude, simultaneously vanishat F = ff ective transfer integrals, which are renormalized bythe zeroth-order Bessel functions with bond-dependent arguments and vanish at di ff erentvalues of F . This fact suggests that electrons on di ff erent bonds are transferred in a concertedmanner for large F , which will be discussed in later sections. This behavior is universally /
21. Phys. Soc. Jpn. C ha r ge den s i t y ω c t/ V b = . // c -axis F= . F= . F= . Fig. 3. (Color online) Time evolution of charge density 2 − h n i at zeroth site in Fig. 1(b) during and afterphotoexcitation along c -axis with small F . observed for large F . For ω c = eaF / ~ ω c = ω c = eaF / ~ ω c = F (not shown).Figure 2(b) shows the time-averaged kinetic energy, i.e., the expectation value of thefirst term in Eq. (2), and the total energy after photoexcitation. A negative-temperature stateis realized when the total energy is positive (i.e., larger than that in equilibrium at infinitetemperature). Note that a negative-temperature state is formed after the application of a one-cycle pulse of the electric field in di ff erent models.
20, 21)
For ω c = ω c = ω c = F dependences of the time-averaged bond densities are not described bythe corresponding e ff ective transfer integrals, we need to observe the short-time behavior oftransient states. Thus, the time evolution of the charge density 2 − h n i is shown in Fig. 3 forphotoexcitation along the c -axis with small values of F . For photoexcitation along the a -axisor along the c -axis, all sites are classified into two groups according to their charge densi-ties for a reason of symmetry. Because the total charge is conserved, the charge distributionamong sites is determined once the charge density at one site is known. Consequently, underthe condition of Fig. 3, the charge densities at the even-numbered sites in Fig. 1(b) vary withtime as shown here, and those at the odd-numbered sites vary in the opposite phase. /
21. Phys. Soc. Jpn. C ondu c t i v i t y / FT o f den s i t y ω V b = . // c -axis < ω c t/ < Conductivity F= . F= . F= . Fig. 4. (Color online) Absolute values of Fourier transforms of time profiles ( T < t < T ) of charge densityin Fig. 3 compared with optical conductivity spectrum with polarization parallel to c -axis in ground state. A charge distribution deviating from the initial uniform distribution is caused by pho-toexcitation. Thus, the time profile of the charge density above is expected to have similarinformation to the steady-state optical conductivity spectrum. We calculate the absolute val-ues of the Fourier transforms of the time profiles ( T < t < T after photoexcitation) ofthe charge density in Fig. 3 and show them in Fig. 4. For comparison, we also show theoptical conductivity spectrum with polarization parallel to the c -axis in the ground state. Tofacilitate the comparison, we take a rather long time span for the Fourier transforms with asmall frequency slice. It is clearly shown that, for small F , the absolute values of the Fouriertransforms have peaks with large weights at energies where the conductivity spectrum haspeaks.The above result shows how charge densities are modulated by photoexcitation and im-plies that their characteristic time profiles can be observed by optical measurements. In thefollowing sections, we investigate Fourier spectra for large F more systematically in the one-and two-dimensional models (for specifically polarized fields in the two-dimensional case)where there are two inequivalent sites with respect to charge densities.Before discussing specific models, we mention a general fact in noninteracting systems.In any noninteracting system, H NI = P λ ǫ λ c † λ c λ , we have ( id / dt ) c † λ c λ = (cid:0) ǫ λ − ǫ λ (cid:1) c † λ c λ forany λ and λ ; thus, the quantities Q λ λ ≡ c † λ c λ + c † λ c λ and P λ λ ≡ − ic † λ c λ + ic † λ c λ for λ , λ oscillate with ω = | ǫ λ − ǫ λ | similarly to the position and momentum operators of aharmonic oscillator. /
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4. Photoinduced Charge Oscillations in One Dimension
To study photoinduced charge oscillations in dimerized systems from a broad perspective,we here use the one-dimensional spinless fermion model in Eq. (1) at half filling. Here andin the following section, we take a shorter time span of T < t < T after photoexcitationfor Fourier transforms and refer to the absolute value of the Fourier transform of the timeprofile of the charge density simply as a Fourier spectrum. For large F ( F ≥ . F ( F ≥ .
10) is an intrinsic one, except for in the noninteracting case. A largerfrequency slice (i.e., a shorter time span) than that used for Fig. 4 makes them easy to read.The Fourier spectra thus obtained are shown for di ff erent values of t and V in Fig. 5. If themodel parameters are in units of eV, F is shown in units of V / [intersite distance].In the noninteracting case ( V = | t | > | t | , the one-electron statesof energies from − | t | − | t | to − | t | + | t | are occupied, and those from | t | − | t | to | t | + | t | are unoccupied. Then, the optical conductivity spectrum has weights in the energyrange from 2( | t | − | t | ) to 2( | t | + | t | ) [from 0.2 to 0.6 in Fig. 5(a)]. They form discretepeaks owing to the finite-size e ff ect, and their oscillator strengths become larger as the energyis lowered. For small F ( F = F increases, the weights first increase as a whole ( F = . < F < . F ( F = ω osc = | t | + | t | ) (i.e., with a period of π/ ( | t | + | t | )).In general, on a bipartite lattice (with sublattices “e” and “o”) without interactions, H BL = P i , j t i , j (cid:16) c † e , i c o , j + c † o , j c e , i (cid:17) , where all sites are equivalent in terms of the network oftransfer integrals, the quantity Q ≡ P i , j (cid:16) c † e , i c e , j − c † o , i c o , j (cid:17) behaves as a harmonic oscillator,( id / dt ) Q = ω Q , which is derived through the double commutator. In the ground state, itsexpectation value is zero. If such a system has N sites, we define c e , k = √ / N P j e ik j c e , j and c o , k = √ / N P j e ik j c o , j to have Q = ( N / (cid:16) c † e , k = c e , k = − c † o , k = c o , k = (cid:17) , whose motion isgoverned by the term H BL , k = = ( t + t ) (cid:16) c † e , k = c o , k = + c † o , k = c e , k = (cid:17) in H BL = P k H BL , k in the present case. If t t <
0, we multiply the creation and annihilation operators forsites 2, 3, 6, 7, etc., by ( −
1) to have t t > Q = ( N / (cid:16) c † e , k = π c e , k = π − c † o , k = π c o , k = π (cid:17) and H BL , k = π = ( t − t ) (cid:16) c † e , k = π c o , k = π + c † o , k = π c e , k = π (cid:17) instead ofthe above. Then, it is straightforward to show that ω = ω osc and the charge oscillation isundamped in the noninteracting case. However, with interactions, ( id / dt ) Q , ω Q and the /
21. Phys. Soc. Jpn. (a)(b)(c)(d)
Fig. 5. (Color online) Absolute values of Fourier transforms of time profiles ( T < t < T ) of charge densityat any site in Fig. 1(a) with di ff erent F and optical conductivity spectrum in ground state for (a) t = − V = t = − V = t = − V = t = − V = /
21. Phys. Soc. Jpn. charge oscillation is damped.The above behavior is basically maintained in an interacting case [Fig. 5(b)]. In the op-tical conductivity spectrum, the oscillator strength is concentrated on the low-energy peakowing to the exciton e ff ect. The weight of the Fourier spectrum for small F ( F = F , the low-energypart is decreased and the high-energy part is increased as F increases (0 . < F < . ω osc = | t | + | t | ).Compared with the noninteracting case [Fig. 5(a)], where the charge oscillation is undampedowing to integrability, it is damped; thus, the peak height is smaller.Numerical results for a di ff erent value of t are shown in Figs. 5(c) and 5(d). The F dependence of the Fourier spectrum is similar to above, especially for large F (0 . < F < . ω osc = | t | + | t | ) again, which is 0.8 for Figs. 5(c) and 5(d). For t = − ω osc = V = V = ω osc. Whenwe further increase | t | , we find that it is occasionally necessary to increase the centralfrequency ω c as well (e.g., ω c = t = − t = − ω c to resonate with ω osc) toobtain the strong-field-induced charge oscillation and that it always appears at a frequencynear ω osc = | t | + | t | ). This relation is maintained even when the nearest-neighborrepulsion strengths are alternated. This relation is also maintained even away from half filling.This charge oscillation is not observed in the charge-ordered phase with V > | t | .The above relation between ω osc and the transfer integrals implies that this charge os-cillation is realized by simultaneous charge transfers through the t and t bonds, as shownin Fig. 6. At a site, fermions are simultaneously transferred from the neighboring sites onboth sides to this site and back to these two sites. Because these charge-transfer processes arerealized coherently everywhere in the system, the whole process is e ff ectively described by atwo-site model, H = t (cid:16) c † e c o + c † o c e (cid:17) = t (cid:16) c † b c b − c † a c a (cid:17) , where c b ( a ) = ( c e ± c o ) / √ c † e c e + c † o c o = c † b c b + c † a c a =
1, or equivalently by a one-spin model, H = t σ x , where σ x is the Pauli matrix with eigenvalues ±
1. Its time evolution operator is described (with ~ =
1) by e − itH = e − itt σ x . At t = π/ (2 | t | ), the operator e ∓ i ( π/ σ x = ∓ i σ x gives a completecharge transfer (spin flip). At t = π/ ( | t | ), the operator e ∓ i πσ x = −
39, 40) in many-body-localized driven systems.
41, 42)
Here, we do not use acontinuous wave but a pulse; thus, we do not need a many-body-localized system to avoidthermalization, which is similar to the situation in Refs. 24 and 43. Of course, the pulse- /
21. Phys. Soc. Jpn. Fig. 6. (Color online) Schematic view of photoinduced charge oscillation. induced charge oscillation decays with time and the Fourier spectra show a broad peak; thus,only the short-time behavior is approximately described by this two-site one-fermion model.
5. Photoinduced Charge Oscillations in Two Dimensions
Now, we return to the two-dimensional model for κ -(BEDT-TTF) X in Eq. (2) at three-quarter filling and take the time span of T < t < T again for Fourier transforms. The Fourierspectra thus obtained are shown for di ff erent strengths of intersite repulsive interactions andpolarizations of photoexcitation in Fig. 7.In the case without intersite repulsion ( V i j = a - ( c -)axis. For small F ( F = ω < .
8, similarly to the corresponding conductivityspectra. As F increases, the weights first increase as a whole ( F = //
8, similarly to the corresponding conductivityspectra. As F increases, the weights first increase as a whole ( F = //
21. Phys. Soc. Jpn. (a)(b)(c)(d)
Fig. 7. (Color online) Absolute values of Fourier transforms of time profiles ( T < t < T ) of charge den-sity at any site in Fig. 1(b) with di ff erent F and optical conductivity spectrum in ground state for (a), (b) V b = V b = V p = V q = a - and (b) c -axes, and (c), (d) V b = V b = V p = V q = a - and (d) c -axes. In all cases, U = /
21. Phys. Soc. Jpn. part ( ω < .
8) is then decreased, while the high-energy part is further increased (0 . < F < . F ( F = ω = a -axis [Fig. 7(a)], which is slightly lower than ω osc = | t b | + | t b | + | t p | ), amountingto 1.27 here, and around ω = c -axis [Fig. 7(b)], which is closeto ω osc = | t b | + | t b | + | t q | ), amounting to 1.00 here. The peak energy for large F generally well matches the equation for ω osc above for each polarization in cases withdi ff erent values of t i j and U as long as the intersite interactions are absent, V i j = ω osc and the transfer integrals is similar to that in the one-dimensional spinless fermion model. The strong-field-induced charge oscillation is realizedby simultaneous charge transfers through the t b , t b , and two t p bonds for polarization alongthe a -axis, as shown in Figs. 8(a) and 8(b), and those through the t b , t b , and two t q bondsfor polarization along the c -axis, as shown in Figs. 8(c) and 8(d). Through these bonds, thecharge-rich sites are connected to the charge-poor sites. At a charge-poor site, holes are si-multaneously transferred from all of the four neighboring charge-rich sites to this site andthen back to these four sites. Thus, this oscillation looks like an electronic breathing mode ora checkerboard pattern where black and white are time-periodically exchanged.The fact that ω osc is independent of the sign of t p or that of t q is due to the particularstructure shown in Fig. 1(b). If we multiply the creation and annihilation operators for reddimers (sites 2, 3, 6, 7, 10, 11, 14, and 15) by ( − t p and t q . In general, the equation for ω osc is sensitive to the sign of the transfer integral.For instance, if we add a third-neighbor hopping term, t P j (cid:16) c † j c j + + c † j + c j (cid:17) , to the spinlessmodel of Eq. (1), ω osc depends on the relative signs of the transfer integrals and it doesnot always correspond to the highest-energy peak of the optical conductivity spectrum in thenoninteracting case: ω osc = | t + t + t | if t t > ω osc = | t − t | if t t < | t | , | t |≥ | t | .The above behaviors for large F are partly changed by intersite repulsive interactions.Examples are shown in Fig. 7(c) [Fig. 7(d)] for polarization along the a - ( c -)axis. The pa-rameters used for Fig. 7(d) are the same as those used for Fig. 2. For small F ( F = ω < .
8, similarly to the cor-responding conductivity spectra. As F increases, the weights first increase as a whole (witha maximum around F = F further increases, the low-energy part is significantly de-creased, and the high-energy part is also decreased to some degree (0 . < F < . /
21. Phys. Soc. Jpn. (a) (b)9 1011 1413 15 9 10 141311 15 F F Fig. 8. (Color online) Schematic view of charge oscillation between (a) and (b) after photoexcitation along a -axis and that between (c) and (d) after photoexcitation along c -axis. energies are significantly lower than ω osc in the case without intersite repulsion. The intersiterepulsive interactions (attractive interactions between charge-rich and charge-poor sites) slowdown this charge oscillation. Even in one dimension, if we add a second-neighbor hoppingterm to the spinless model of Eq. (1) to allow an exchange of fermions, the frequency of thestrong-field-induced charge oscillation is altered by interactions (not shown).Thus, immediately after the application of a strong pulse of an oscillating electric field,charge densities coherently oscillate with a finite lifetime. This fact is consistent with the /
21. Phys. Soc. Jpn. finding in Fig. 2(a), that is, the time-averaged bond densities are synchronized with eachother as functions of F . During the charge oscillation, holes are transferred through di ff erentbonds simultaneously with a common frequency; thus, the time-averaged bond densities ondi ff erent bonds behave similarly as functions of F . These simultaneous charge transfers arenecessary to suppress the rise in the entropy and to realize a negative-temperature state inFig. 2(b).Now, we consider what is necessary for this synchronized behavior. Each of the e ff ec-tive transfer integrals is renormalized by the zeroth-order Bessel function with the bond-dependent argument proportional to the inner product of F and r i j . If the e ff ective transferintegrals governed the long-time behavior, the synchronized behavior would not be obtained.The picture based on the e ff ective transfer integrals should basically be applied to noninter-acting and weakly interacting systems. Then, in the case without intersite repulsion and withdi ff erent values of U , we show Fourier spectra for large F ( F = c -axis [as used in Fig. 7(b)] in Fig. 9(a). In general, charge oscillations are undamped in thenoninteracting case; thus, their lifetimes are long for weakly interacting systems. That is why,as U decreases, the overall weights in the Fourier spectra increase. Roughly speaking, thesespectra consist of a low-energy part ( ω < .
8) corresponding to the conductivity spectrum anda high-energy peak around ω = U de-creases, the suppression of the low-energy part becomes weak; thus, the strong-field-inducedcharge oscillation becomes relatively weaker. This implies that the tendency for simultaneouscharge transfers becomes weak as U decreases.Then, we show the time-averaged bond densities P σ hh (cid:16) c † i σ c j σ + c † j σ c i σ (cid:17) ii in Figs. 9(b) and9(c) for the small- U cases of Fig. 9(a). Indeed, for small U ( U = F [Fig. 9(b)]. As F increases, the e ff ectivetransfer integral for the b bond first vanishes, then that for the p bond vanishes, and soon. Thus, the behaviors of the time-averaged bond densities are not simply described by thecorresponding e ff ective transfer integrals. For larger U , the time-averaged bond densities arealmost [Fig. 9(c) for U = U = F . Thus, interactions are found to be essential for the simultaneous chargetransfers: the interactions cause damping of charge oscillations and synchronization of thecharge transfers at the same time, which result in the strong damping of low-frequency chargeoscillations and the relatively weak damping of the high-frequency charge oscillation. Thisfact is reminiscent of the situation in a discrete time crystal, where interactions are essentialfor collective synchronization in strongly disordered systems. /
21. Phys. Soc. Jpn. (a)(b)(c)
Fig. 9. (Color online) (a) Fourier spectra similar to Fig. 7 but for F = ff erent U , and (b),(c) time-averaged bond densities similar to Fig. 2(a) but for (b) U = U = V b = V b = V p = V q = c -axis.16 //
Fig. 9. (Color online) (a) Fourier spectra similar to Fig. 7 but for F = ff erent U , and (b),(c) time-averaged bond densities similar to Fig. 2(a) but for (b) U = U = V b = V b = V p = V q = c -axis.16 //
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The emergence of the strong-field-induced charge oscillation is basically limited to thepolarizations along the a - and c -axes, which guarantee that there are only two groups of sitesaccording to their charge densities and that the Fourier spectra are common to all sites. Whenthe polarization deviates from these axes, the charge densities at the four sites in the unit cellbecome nonequivalent after photoexcitation; thus, the time profiles depend on the site andtheir Fourier spectra become di ff erent. Even for large F , substantially large weights remainat low energies.Thus, the presence of only two nonequivalent sites with respect to charge density isimportant. Dimerized structures are favorable in this sense. Indeed, similar results are ob-tained in a one-dimensional spin-1 / ω osc = | t | + | t | ) for di ff erent values of U and V in the uniform-charge-density phase.
6. Conclusions and Discussion
Instead of the long-time behavior often explained by concepts such as dynamical local-ization and modified e ff ective interactions, we pay attention to a short-time behavior that isnonlinear with respect to the field amplitude F and varies on a time scale of the period of theexternal field. This study is motivated by a recent experiment suggesting the importance ofshort-time behavior. Because of the dimerized structures and specifically polarized fieldsconsidered in this study, all sites are classified into two groups according to their charge den-sities. Charge densities take a common value within a group after photoexcitation. Numericalresults are presented for a one-dimensional spinless-fermion model at half filling and a two-dimensional model for κ -(BEDT-TTF) X at three-quarter filling, but the main conclusions arenot limited to these models as long as all sites are classified into two groups and the groundstate is in the uniform-charge-density phase.For small F , Fourier spectra for charge-density time profiles after photoexcitation havepeaks at energies where the corresponding conductivity spectra have peaks. For large F , thespectral distribution is changed and has a peak at a single energy on the high-energy side.For the models we use in this paper and without intersite repulsion in two dimensions, thepeak energy is given by twice the sum of the absolute values of the transfer integrals betweena site and all neighboring sites with di ff erent charge density. In two dimensions, this peakenergy is lowered by intersite repulsion. However, we can construct a model where the long-range hopping increases, decreases, or maintains the peak energy depending on the relativesigns of transfer integrals. This field-induced charge oscillation appears only when F is large; /
21. Phys. Soc. Jpn. thus, it is a nonlinear phenomenon that emerges when a strong pulse of an oscillating electricfield is applied to dimerized systems and charge densities are shaken coherently. This strong-field-induced charge oscillation is considered to be closely related to the newly observedreflectivity peak in photoexcited κ -(BEDT-TTF) Cu[N(CN) ]Br on the high-energy side ofthe main reflectivity spectrum. The coherence associated with the strong-field-induced charge oscillation is responsiblefor the behaviors of the time-averaged bond densities, which decrease in magnitude, simulta-neously vanish at a particular value of F , and invert their signs as F increases. After the signinversion, a negative-temperature state is realized, which implies that the rise in the entropyis suppressed. This suppression is enabled by coherently shaking charge densities. Note thata negative-temperature state is more generally realized even without dimerization. For continuous waves, thermalization is suppressed in a many-body-localized system,which is necessary to realize a discrete time crystal.
39, 40)
The time evolution operators forthe two-site one-fermion model that is referred to for the discussion of the frequency ofthe strong-field-induced charge oscillation in Sect. 4 are similar to local unitary operatorsdiscussed for a discrete time crystal. This similarity may be helpful when considering thepossibility of emergent charge oscillations in di ff erent situations. Acknowledgment
The author is grateful to S. Iwai and Y. Tanaka for various discussions. This work wassupported by Grants-in-Aid for Scientific Research (C) (Grant No. 16K05459) and ScientificResearch (A) (Grant No. 15H02100) from the Ministry of Education, Culture, Sports, Scienceand Technology of Japan. //
The author is grateful to S. Iwai and Y. Tanaka for various discussions. This work wassupported by Grants-in-Aid for Scientific Research (C) (Grant No. 16K05459) and ScientificResearch (A) (Grant No. 15H02100) from the Ministry of Education, Culture, Sports, Scienceand Technology of Japan. //
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