High-Frequency Analysis of Effective Interactions and Bandwidth for Transient States after Monocycle Pulse Excitation of Extended Hubbard Model

Using a high-frequency expansion in periodically driven extended Hubbard models, where the strengths and ranges of density-density interactions are arbitrary, we obtain the effective interactions and bandwidth, which depend sensitively on the polarization of the driving field. Then, we numerically calculate modulations of correlation functions in a quarter-filled extended Hubbard model with nearest-neighbor interactions on a triangular lattice with trimers after monocycle pulse excitation. We discuss how the resultant modulations are compatible with the effective interactions and bandwidth derived above on the basis of their dependence on the polarization of photoexcitation, which is easily accessible by experiments. Some correlation functions after monocycle pulse excitation are consistent with the effective interactions, which are weaker or stronger than the original ones. However, the photoinduced enhancement of anisotropic charge correlations previously discussed for the three-quarter-filled organic conductor $\alpha$-(bis[ethylenedithio]-tetrathiafulvalene)$_2$I$_3$ [$\alpha$-(BEDT-TTF)$_2$I$_3$] in the metallic phase is not fully explained by the effective interactions or bandwidth, which are derived independently of the filling.


Introduction
Nonequilibrium properties of quantum many-body systems have received much attention, which can lead to advances in their real-time and coherent manipulation. Motivated by experiments on ultracold atomic gases, interaction quench has been discussed theoretically in different contexts. [1][2][3][4][5][6][7] For many-electron systems in solids, periodic driving is achievable, including photoexcitation. Electromagnetic fields are often incorporated into the Peierls phase multiplied by transfer integrals. In particular, for continuous waves, long-time dynamics compared with the period of the oscillating field has been discussed to develop the concept of * E-mail: kxy@phys.chuo-u.ac.jp 1/21 dynamical localization. [8][9][10] The corresponding effective Hamiltonian is simply the time average of the time-dependent Hamiltonian and is regarded as the lowest-order (∝ ω 0 ) term in a high-frequency (ω) expansion for an effective Hamiltonian in the framework of quantum Floquet theory. [11][12][13][14][15][16][17] For instance, a Floquet topological insulator can be discussed in the second-lowest order (∝ ω −1 ). 18) In the next order (∝ ω −2 ), local interactions are modulated. [11][12][13][14][15][16][17] Bearing many-electron systems in solids in mind, we first consider extended Hubbard models, where the strengths and ranges of density-density interactions are arbitrary.
Most photoinduced phase transitions are triggered by a pulse of light. [19][20][21][22] As the pulse width decreases, the time resolution is improved and the instantaneous field amplitude increases. At the same time, oscillating electric fields are viewed as coherently driving many electrons. 23,24) Recently, the optical freezing of charge motion 25) and the photoinduced suppression of conductivity 26) have been observed. For the former, the similarity to dynamical localization was pointed out, although dynamical localization is a continuous-wave-induced phenomenon.
Various similarities between continuous-wave-and pulse-induced phenomena are known.
A negative-temperature state is produced in both cases if the electric field amplitude is large and satisfies a certain condition. [27][28][29][30] A sudden application of a continuously oscillating weak electric field to the half-filled Hubbard model immediately decreases the double occupancy. 27) Similar behavior has also been reported in a one-dimensional Bose-Hubbard model. 5) Because this early-stage dynamics does not depend on whether the field continues to be applied or not, in both the continuous-wave and pulse cases the transient state behaves as if the interaction strength were increased relative to the bandwidth. The transition from a chargeordered insulator phase to a Mott insulator phase in the quasi-two-dimensional metal complex Et 2 Me 2 Sb[Pd(dmit) 2 ] 2 (dmit = 1,3-dithiol-2-thione-4,5-dithiolate) 31) can theoretically be controlled by suppressing the effective transfer integrals in both cases. 32) The similarity between continuous-wave-and pulse-induced phenomena has also been discussed for a onedimensional transverse Ising model. 33) In practice, the application of laser pulses is more advantageous than that of continuous waves for ultrafast collective phenomena that become possible only when the electric field amplitude is large. The optical freezing of charge motion 25) is indeed one such phenomenon. Theoretically, for pulse-induced phenomena in quantum many-body systems, only numerical approaches have so far been employed. In this context, an analytic approach will be useful if similarities are empirically found between continuous-wave-and pulse-induced phenomena, even if it is basically developed for continuous waves. In this study, we employ 2/21 a high-frequency expansion to obtain an effective Hamiltonian in the framework of quantum Floquet theory and discuss the behavior generally expected after periodic driving. Then, we tentatively use it to analyze pulse-induced transient states. If transient states including those similar to dynamically localized states survive for a while, they may be described by the effective Hamiltonian, which has renormalized transfer integrals and interactions.
We will compare states expected by the effective Hamiltonian and monocycle-pulseinduced transient states, which are numerically obtained by solving a time-dependent Schrödinger equation. Although the photoinduced enhancement of anisotropic charge correlations previously discussed for the 3/4-filled organic conductor α-(bis[ethylenedithio]tetrathiafulvalene) 2 I 3 [α-(BEDT-TTF) 2 I 3 ] in the metallic phase 34) is not reproduced by the effective Hamiltonian, it is shown to be generally useful when we roughly expect transient states after monocycle pulse excitation.

High-Frequency Approximation for Periodically Driven, Extended Hubbard Models
In this section, we do not specify the dimension or lattice structure (i.e., network of transfer integrals) and generally consider extended Hubbard models, where the strengths and ranges of density-density interactions are arbitrary, where c † iσ creates an electron with spin σ at site i and n iσ =c † iσ c iσ . The parameters t i j and V i j denote the transfer integral and the intersite repulsion, respectively, between sites i and j, and U i denotes the on-site repulsion at site i. Photoexcitation is introduced through the Peierls phase, with r i j = r j − r i . In this section, we consider the time-dependent vector potential for a continuous wave, where ω is the frequency and F describes the amplitude and polarization of the electric field.
When we substitute Eq. (3) into Eq. (2), we obtain the Peierls phase factor, which is expanded with J m (x) being the mth-order Bessel function, a i j =| r i − r j |, and φ i j is the angle between r i − r j (not r j − r i ) and a reference axis. Note that φ ji = φ i j + π (mod 2π); thus, cos For continuous waves, as long as the time evolution is considered in a stroboscopic manner in steps of the period T = 2π/ω, the stroboscopic time evolution is described by a timeindependent effective Hamiltonian. The effective Hamiltonian is approximately derived by a high-frequency expansion. [11][12][13][14][15][16][17] We follow Ref. 14, which employs degenerate perturbation theory in the extended Floquet Hilbert space, and use its notations. In the lowest order, we have where and H int = 1 2 i,σ U i n i,σ n i,−σ + 1 2 i, j( i),σ,τ V i j n i,σ n j,τ .
In the second-lowest order, we have where In the next order, we obtain thus, we define as in Ref. 14.
We decompose the interaction term and to obtain The double commutators in Eq. and Since the modulations δU i and δV i j originate from the double commutators in Eq. (16), they depend sensitively on θ.
So far, we have distinguished on-site (U i ) and intersite (V i j ) repulsions. If we set V ii = U i and sum over sites i and j irrespective of whether they are different or not in the third term and ignore the second term in Eq. (1), the difference between the resultant Hamiltonian and Eq. (1) is a one-body term that becomes a constant when U i is independent of i because the total number of electrons N e is conserved. This fact is useful in checking formulae. In fact, Eqs. (17) and (18)  Equations (17) and (18) satisfy this condition, as easily checked. Note that Eqs. (17) and (18) are independent of the filling or the system size. Therefore, the effective interactions will not be able to describe any phenomena that are sensitive to the filling or the system size.
As is evident from the Appendix, most of the terms in H (3)

F,int
are not interactions between 5/21 site-diagonal densities, but they bring about electron transfers. As a consequence, continuouswave-induced changes in the electronic state will generally tend to homogenize the electron distribution if the charge is initially disproportionated. When we focus on the interaction terms between site-diagonal densities, the modulations of the interaction strengths δU i [Eq. (17)] and δV i j [Eq. (18)] have contributions from themselves with negative signs (thus weakening the effective U i and V i j by themselves) and contributions from other strengths with positive signs (thus are enhanced by all other strengths with the same signs). This implies that large interaction parameters become small while small interaction parameters become large, thus averaging themselves out. Indeed, in the special case where all the interaction parameters are equal (U i = γ and V i j = γ for all i and j), the modulation is absent (all the interaction parameters are already averaged out). It is natural to assume that the modulations approach the special form of the interaction 1 2 γN 2 e , which is actually equivalent to no interaction because N e is a constant.
In realistic cases, interaction strengths are zero or very small between distant sites, large between neighboring sites, and largest on a single site. Therefore, the effective Hamiltonian will possess weaker interactions between neighbors and stronger interactions between distant sites than the original one. If there is anisotropy in intersite interactions between neighboring sites, the anisotropy will be suppressed and the effective Hamiltonian will acquire isotropic interactions. Note that the rates of the modulations are governed by the square of a transfer integral multiplied by a corresponding non-zeroth-order Bessel function divided by ω, as shown in Eqs. (17) and (18), leading to their sensitivity to θ.

Extended Hubbard model on triangular lattice
In this section, we specify the network of transfer integrals and the strengths and ranges of density-density interactions. We use a quarter-filled extended Hubbard model with on-site and nearest-neighbor repulsions on the triangular lattice with linear trimers shown in Fig. 1, which was previously employed to study the mechanism for the photoinduced tendency toward charge localization. 34) The triangular lattice we consider here consists of equilateral triangles, where the distance between neighboring sites is denoted by a, and has inversion symmetry. The use of this model facilitates a comparison of the high-frequency expansion of the effective Hamiltonian and numerical results.
For the transfer integrals t i j , we use t 1 = −0.14, t 2 = −0.13, t 3 = −0.02, t 4 = −0.06, t 5 = 0.03, and t ′ 5 = −0.03 in Fig. 1, as before. 34) For the on-site repulsion, we consider 6/21 A' A U i = U for all i and use U = 0.8 unless stated otherwise. For the intersite repulsions, we take only nearest-neighbor Coulomb repulsions and set V i j = V 1 for r i j not being parallel to the vertical axis and V i j = V 2 for r i j being parallel to the vertical axis, as shown in Fig. 1, and use The initial state is the ground state obtained by the exact diagonalization method for the 16-site system with periodic boundary conditions. The time-dependent vector potential in the Peierls phase is now set to be 29,30) with F = F(cos θ, sin θ), where F is the amplitude of the electric field and θ is the angle between the field and the horizontal axis. As in the previous paper, 34) we use ω = 0.8. The time-dependent Schrödinger equation is numerically solved by expanding the exponential evolution operator with a time slice dt=0.02 to the 15th order and by checking the conservation of the norm. 35) The time average Q of a quantity Q is calculated by with t s = 5T and t w = 5T , where T is the period T = 2π/ω. Although details are discussed later, here we briefly comment on the fact that the behavior shown in Fig. 2(a)  their ω-dependence is much weaker than that after the application of continuously oscillating fields.

Expectation from effective Hamiltonian
Before showing numerical results for time averages, we discuss what is expected from the effective Hamiltonian obtained in Sect. 2. The bandwidth is a scale of the kinetic energy.
Without interactions and when | t 1 | and | t 2 | are much larger than the other transfer integrals, the bandwidth W is proportional to t 2 1 + t 2 2 . In the lowest order (∝ ω 0 ) of the high-frequency expansion, t 1 is renormalized to be t 1 J 0 (i j ∈ t 1 bond), where "i j ∈ t 1 bond" means that the argument of the Bessel function is that of Eq. (5) with sites i and j being linked by t 1 . In the same manner, t 2 is renormalized to be t 2 J 0 (i j ∈ t 2 bond). The renormalized bandwidth W +δW is proportional to t 2 1 J 2 0 (i j ∈ t 1 bond) + t 2 2 J 2 0 (i j ∈ t 2 bond). Then, in the lowest order, the ratio of any interaction to the bandwidth is increased by a factor of whose dependence on the polarization of photoexcitation θ is shown in Fig. 3  A rough estimation leads to since sites i and j are linked by two large transfer integrals. Depending on the neighboring U i , U j , and V k j in Eqs. (17) and (18), δU i and δV i j can be positive or negative and they have different numerical factors, so that we here show only the relative magnitude and ignore the numerical factor. Thus, the estimation above is very rough. However, these details do not depend on θ, so the dependence of the right-hand side of Eq. (22) on θ, which is shown in 10/21 Fig. 3(b), will be useful for various comparisons.
The quantities in both Eqs. (21) and (22) reach a maximum around θ = 0 (i.e., for polarization nearly parallel to the horizontal axis). When the argument is small, zeroth-order Bessel functions quadratically decrease from unity, so that any quantity with zeroth-order Bessel function(s) in its denominator quadratically increases, and the square of a first-order Bessel function quadratically increases from zero. Thus, the behaviors of the quantities in Eqs. (21) and (22) are similar. The smallness of the quantity shown in Fig. 3(b) suggests that the high-frequency expansion converges rapidly for the field amplitudes used here.

Numerical results for time-averaged correlations
Now we show numerical results for time averages of correlation functions after monocycle pulse excitation. For V 2 = 0.35, whose case was investigated in detail in the previous paper, 34) the spatially and temporally averaged double occupancy n i↑ n i↓ is shown in Fig. 4(a) as a function of the polarization of photoexcitation θ. It decreases as if the on-site repulsion U were transiently increased relative to the bandwidth after photoexcitation, as previously reported. Its θ dependence is similar to that in Fig. 3(a). To investigate whether the decrease in n i↑ n i↓ can be explained quantitatively by the increased ratio of the on-site repulsion U to the renormalized bandwidth, we vary U and calculate the spatially averaged double occupancy of the ground state n i↑ n i↓ , as shown in Fig. 4(b). For θ = 0 and eaF/( ω) = 0.4 in Fig. 4(a), n i↑ n i↓ is about 0.030. To reproduce this value in the ground state, we need to increase U by 4%, as shown in Fig. 4(b). Figure 3(a) shows that the ratio of the on-site repulsion U to the renormalized bandwidth is increased by about 3% for θ = 0 and eaF/( ω) = 0.4. These values are comparable.
Next we show the spatially and temporally averaged nearest-neighbor density-density correlation n i n j . For V 2 = 0.35, which is slightly larger than V 1 = 0.3, we know that the anisotropy in the effective intersite repulsive interactions is enhanced by photoexcitation, 34) which contradicts Eq. (18). Then, we use V 2 = 0.25 and show n i n j for non-vertical bonds in Fig. 5(a) as a function of θ. It increases as if the intersite repulsion V 1 were transiently decreased relative to the bandwidth after photoexcitation. Then, we vary V 1 and calculate the spatially averaged nearest-neighbor density-density correlation of the ground state n i n j for non-vertical bonds, as shown in Fig. 5(b). For θ = 0 and eaF/( ω) = 0.4 in Fig. 5(a), n i n j is about 0.17. To reproduce this value in the ground state, we need to decrease V 1 by 8%, as shown in Fig. 5(b). This is not explained by the lowest-order effect shown in Fig. 3(a). Spatially averaged double occupancy of ground state n i↑ n i↓ as a function of U for V 1 =0.3 and V 2 =0.35. and 5 and in Fig. 6 later. We need at least a second-order effect because in the high-frequency expansion different interactions are modulated differently from the second order. Figure 3(b) shows the right-hand side of Eq. (22), which implies that the modulation is about 1.5%.
Because the actual modulation depends on the other interaction parameters in Eq. (18) and there are effective interactions that are not in the form of density-density interactions, the discrepancy appears to be not so large.
Finally we show n i n j for vertical bonds as a function of θ for V 2 = 0.25 in Fig. 6(a), Fig. 6(b), and for V 2 = 0.4 in Fig. 6(c). It decreases for V 2 = 0.25 < V 1 as if the intersite repulsion V 2 were increased, increases for V 2 = 0.4 > V 1 as if V 2 were decreased, and therefore behaves as if the difference between V 1 and V 2 were suppressed. For V 2 = 0.35, as previously reported, 34) the anisotropy in the effective V i j is enhanced by photoexcitation, so that it is not described by the present effective Hamiltonian and its θ dependence is different from that in Figs. 3(a) and 3(b). We vary V 2 and calculate the ground state n i n j for vertical bonds, as shown in Fig. 6(d). For θ = 0 and eaF/( ω) = 0.4, n i n j is about 0.222 in Fig. 6(a), which corresponds to an increase in V 2 in the ground state by 8% (from V 2 = 0.25 to Fig. 6(d). At the same photoexcitation, n i n j is about 0.13 in Fig. 6(c), which corresponds to a decrease in V 2 in the ground state by 5% (from V 2 = 0.4 to V 2 = 0.38) in Fig. 6(d). These values (8% and 5%) are close to that for the required modulation of V 1 (8%) above. The discrepancy from Fig. 3(b) is not very large. More importantly, the anisotropy in the effective V i j is reduced by photoexcitation in Figs. 6(a) and 6(c), which is consistent with Eq. (18), and its θ dependence is similar to that in Fig. 3(b).

14/21
We have compared a continuous-wave-induced phenomenon with a pulse-induced one.
For continuous waves, we employ a high-frequency expansion in the framework of quantum From all these comparisons, we find that the effective Hamiltonian is useful in roughly predicting tendencies in correlation functions after monocycle pulse excitation. However, the effective Hamiltonian is independent of the filling or the system size, so that it is not directly be applicable to phenomena particular to a special filling or highly nonlinear phenomena such as the photoinduced enhancement of anisotropic charge correlations in α-(BEDT-TTF) 2 I 3 in the metallic phase.

Acknowledgments
The author is grateful to S. Iwai and Y. Tanaka for various discussions.