Competition between Interactions and Randomness in Photoinduced Synchronization of Charge Oscillations on a Dimer Lattice

The synchronization of charge oscillations after photoexcitation that has been realized through the emergence of an electronic breathing mode on dimer lattices is studied here from the viewpoint of the competition between interactions and randomness. We employ an extended Hubbard model at three-quarter filling on a simple dimer lattice and add random numbers to all transfer integrals between nearest-neighbor sites. Photoinduced dynamics are calculated using the time-dependent Schr\"odinger equation by the exact diagonalization method. Although the randomness tends to unsynchronize charge oscillations on different bonds during and after photoexcitation, sufficiently strong on-site repulsion $U$ overcomes this effect and synchronizes these charge oscillations some time after strong photoexcitation. The degree of synchronization is evaluated using an order parameter that is derived from the time profiles of the current densities on all bonds. As to the nearest-neighbor interaction $V$, if $V$ is weakly attractive, it increases the order parameter by facilitating the charge oscillations. The relevance of these findings to previously reported experimental and theoretical results for the organic conductor $\kappa$-(bis[ethylenedithio]tetrathiafulvalene)$_2$Cu[N(CN)$_2$]Br is discussed.


Introduction
Photoinduced phase transitions are nonlinear dynamical phenomena, in which events on different timescales are involved. [1][2][3][4][5][6][7] The transient lowering of the symmetry of a manyelectron state is an important issue, and qualitative progress in its observation and understanding has been made by developments in experimental techniques. In equilibrium and continuous phase transitions accompanying symmetry breaking, a long-range order is formed by the spontaneous development of fluctuations. In photoinduced phase transitions, a conventional picture is similar, where fluctuations are produced by photoexcitation. 8) The development of fluctuations is achieved by interactions; thus, photoinduced phase transitions are cooperative phenomena. Their stochastic processes are described with a probability.
As the pulse width becomes small and the amplitude of the optical field becomes large, photoinduced dynamics can be qualitatively changed. The application of an intense optical field transiently and directly lowers the symmetry of a many-electron state, keeping the coherence in many-electron motion 9,10) and leading to transient charge order formation before relaxation becomes significant. 11,12) Such an order is absent before photoexcitation, and it would oscillate and become zero on average.
The excited states that are responsible for ultrafast dynamics inevitably have high energies. The number of such states is large. Dephasing of charge oscillations is often significant especially when electron correlations are strong. In this context, it is nontrivial for electrons to oscillate coherently. In any case, a coherent charge oscillation with a short period is important for the ultrafast lowering of the symmetry. To reduce the effect of dephasing, it is advantageous for charge oscillations to be synchronized. Here, we study competing effects in the photoinduced synchronization of charge oscillations that are previously reported on dimer lattices. 11,13) In the organic superconductor κ-(bis[ethylenedithio]tetrathiafulvalene) 2

Cu[N(CN) 2 ]Br
[κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br] with a dimerized structure, a nonlinear charge oscillation and a resultant stimulated emission have been observed on the high-energy side of the main reflectivity spectrum only after strong photoexcitation. 14) Theoretically, by using an extended Hubbard model at three-quarter filling on a dimer lattice that corresponds to this compound 11) and other models on dimer lattices, 11,13) we can realize a nonlinear charge oscillation only after strong photoexcitation that is characterized as an electronic breathing mode.
It has already been pointed out that sufficiently strong on-site repulsion is necessary for this mode to be dominant over any charge oscillations appearing in the linear conductivity spectra. The synchronization of charge oscillations between charge-rich and charge-poor sites with the help of this interaction is suggested by the fact that time-averaged bond charge densities on different bonds governed by different transfer integrals are similar functions of the amplitude of the optical field (before taking the time average) when the on-site repulsion is sufficiently strong. 11) In this paper, we add random numbers to transfer integrals to intentionally weaken the synchronization of charge oscillations. A synchronization order parameter is defined 12,[15][16][17] using the phases of the oscillating current densities on all bonds connected by transfer integrals. Then, we investigate the competition between interactions and random transfer integrals. Thus, we directly show that the emergence of an electronic breathing mode is caused by the synchronization of charge oscillations through the on-site repulsion.

Dimerized Model with Disorder in Two Dimensions
In previous studies, the electronic breathing mode is analyzed on a two-dimensional lattice with one type 13) or two types 11) of dimers. Here, we use a square lattice that is similar to but simpler than that used in Ref. 13, and configure dimers, "t 1 bonds," as shown in Fig. 1.
We employ an extended Hubbard model at three-quarter filling, where c † iσ creates an electron with spin σ at site i, n iσ =c † iσ c iσ , and n i = σ n iσ . The parameters U and V represent the on-site and nearest-neighbor Coulomb repulsion strengths, respectively. The transfer integral is t i j = t 1 (1 + δ i j ) inside a dimer along the x-axis, t i j = t 2 (1 + δ i j ) outside a dimer along the x-axis, or t i j = t y (1 + δ i j ) along the y-axis, as shown in Fig. 1 4×4-site system with periodic boundary conditions is used unless stated otherwise. We use t 1 =−0.3, t 2 =−0.1, and t y =−0.1. If we regard these values as given in units of eV, they roughly correspond to intradimer and interdimer transfer integrals in dimerized organic conductors κ-(BEDT-TTF) 2 X. 18,19) In Eq. (1), the constant term is subtracted in such a way that the total energy becomes zero in equilibrium at infinite temperature.
The initial state is the ground state obtained by the exact diagonalization method. Photoexcitation is introduced through the Peierls phase which is substituted into Eq. (1) for each combination of sites i and j with relative position r i j = r j − r i . Hereafter, we use e=a= =c=1. We employ symmetric one-cycle electric-field pulses 11,13,[20][21][22] and use the time-dependent vector potential where the central frequency ω c is chosen to be ω c = 0.7 throughout the paper because the qualitative results are independent of its value as in previous studies. 11,13) The electric field is polarized along (1, 1) and its maximum is

Effect of Randomness on Electronic Breathing Mode
In our previous paper where we did not consider randomness on dimer lattices, 11 11,13) becomes dominant. In the present dimer lattice of a small size, the peak below 0.6 is also noticeable, but mean-field calculations indicate that this peak becomes less noticeable for larger sizes and merged into a continuum spectrum in the thermodynamic limit. The frequency of the electronic breathing mode is almost independent of the interval [−ǫ, ǫ] of random numbers, indicating that this mode does not lose its identity even if its charge oscillations are inhomogeneous. Later, we will demonstrate that charge oscillations on different bonds are indeed synchronized for sufficiently large U to maintain its identity.

Definition of Synchronization Order Parameter
To define a measure of how synchronized the charge oscillations are, we refer to the synchronization order parameter that is used in the Kuramoto model. 12,[15][16][17] In the electronic breathing mode, the current distribution alternates between the patterns shown in the left and right panels of Fig. 3. Then, we regard the current density on each bond ( σ −ic † iσ c jσ +ic † jσ c iσ between sites i and j if A(t)=0) as taking a positive (negative) value if the current flows as in the left (right) panel. We assign the argument φ of a complex number e iφ to the current density on each bond by following its time evolution as follows. Note that we assign φ only when the current density changes in time. When it evolves between local maxima A, A ′ , · · · and local minima B, B ′ , · · · as shown in the upper panel of Fig. 4, we divide the time evolution into , and so on. When the current density j m (t) on bond m at time t decreases from A to B, we describe its evolution as j m (t) = A+B 2 + A−B 2 sin φ m (t) to obtain φ m (t) in the interval π/2 ≤ φ m (t) ≤ 3π/2. When it increases from B to A ′ , we describe its evolution as j m (t)

Competition with on-site repulsion
The are linear and do not decay without dephasing, so that the order parameter is smaller than the repulsive case even when F is large. Therefore, around F=0.8, the order parameter is described by an increasing function of U.
For even larger F values, the total energy after photoexcitation becomes smaller than that at F=0.7, but it does not reach the ground-state energy at F=0, as shown in Fig. 6(b). Around F=0.7, the total energy goes beyond that in equilibrium at infinite temperature, indicating that it becomes a negative-temperature state with a suppressed rise in the entropy. 21  For F > 0.8, however, the rise in the entropy would become nonnegligible and disturb the synchronization. Similar results are obtained in our previous paper, 11) where the electronic breathing mode becomes less dominant for very large F. Thus, the averaged synchronization order parameter is suppressed for such large F values.
To see the competition between the effect of on-site repulsion U and that of randomness ǫ, we plot the averaged synchronization order parameter in Fig. 7(a) as their function for F=0.6 and V=0. Its contour lines are plotted in Fig. 7(b). Particularly for small U, it is apparent that the randomness ǫ in transfer integrals reduces the order parameter by inhomogeneously for larger systems and find that it is smaller than that in the present 4×4-site system. The overestimation in the present system is due to the fact that a very small number of charge oscillations, as suggested in Fig. 2, contribute to the order parameter without decaying. The time span of 3T < t < 6T is too short to distinguish the frequency of the electronic breathing mode from twice the frequency of the low-energy mode. This situation should disappear for larger systems. Therefore, the averaged order parameter is expected to be overestimated in the present small system at least in the small-U-small-ǫ region of Fig. 7(b). To see such a finite size effect, we calculate the averaged order parameter for an even smaller 4×3-site system with 20 random number distributions and otherwise the same parameters and show it in Fig. 7(c). The averaged order parameter is indeed overestimated for a smaller system, but the general behavior is essentially the same as that observed in Fig. 7(b). Thus, we expect that the present finding is valid for larger systems. 11

Effect of nearest-neighbor interaction
In our previous paper, 11) we showed that repulsive intersite interactions (with different strengths for different intersite distances on the two-dimensional lattice for κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br) enhance dephasing and broaden the Fourier spectrum of the chargedensity difference. Thus, a repulsive nearest-neighbor interaction V is expected to reduce the synchronization. Here, we see the effect of the nearest-neighbor interaction V in the disordered case of ǫ > 0. In reality, V should be positive, but we investigate its effect for both It is known in the ground state that a positive V makes charge-density correlations n i n j ter photoexcitation with F ≤ 0.5. The nearest-neighbor charge-density correlations n i n j averaged over the time span of 3T < t < 6T , random number distributions, and different types of bonds or all bonds are shown in Fig. 8(a) for F=0.5. All of them increase with −V indeed. Note that after photoexcitation with F=0.6, the transient state is almost a negativetemperature state as suggested in Fig. 6(b). In negative-temperature states, the correlation functions generally behave as if the interactions were inverted. 21,22,24,25) Indeed, as shown in Fig. 8(b) for F=0.6, the averaged n i n j for any type of bonds decreases as −V increases as if the attractive nearest-neighbor interaction V were replaced by a repulsive one.
If the magnitude of V is large, it enhances dephasing and makes charge oscillations decay faster. If the magnitude of V is small, a negative V contributes to an increase in the magnitude of the current density on each bond, as shown later in Fig. 10(a). Thus, a weakly attractive nearest-neighbor interaction V increases the amplitudes of charge oscillations. This fact is consistent with the behavior shown in Fig. 9(a), which reveals that the Fourier spectrum of the charge-density difference averaged over random number distributions is heightened by the weakly attractive nearest-neighbor interaction V. The peak due to the electronic breathing This finding with respect to V is consistent with recent experimental results. Although the Coulomb interaction in real systems is repulsive (V > 0), the stimulated emission, which is supposed to be due to the electronic breathing mode, 11) is enhanced by superconducting fluctuations in κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Br. 14) In this organic superconductor, the super-  conducting gap is considered to have the d-wave symmetry. [26][27][28] Theoretically, in models with on-site repulsion U > 0 and nearest-neighbor attraction V < 0, a d-wave pairing correlation is generally enhanced [29][30][31][32][33] unless phase separation is realized. 34) Here, we consider a pairing of the form  nearest-neighbor attraction. Although the realistic nearest-neighbor interaction is repulsive, the effect of increasing d-wave superconducting fluctuations with decreasing temperature in the experiment might be simulated to some extent by the attractive nearest-neighbor interaction in the present calculations. We suspect that superconducting fluctuations enhance an electron's transfer correlated with another electron's transfer, which would facilitate the synchronized charge oscillations and consequently the stimulated emission. However, the present system is too small to judge even whether it is metallic or insulating, which prevents quantitative discussions.

Conclusions
On the basis of the previously reported emergence of an electronic breathing mode and synchronization of charge oscillations after strong photoexcitation on dimer lattices, 11,14) we theoretically study the competition between the effect of interactions U and V and that of randomness ǫ introduced into transfer integrals in an extended Hubbard model at three-quarter filling on a simple dimer lattice. For the definition of a synchronization order parameter, we use only current densities on bonds, derive phases φ from their time profiles, and average e iφ over all bonds: it is defined only when current densities change in time. Owing to the randomness, current densities for U=V=0 on different bonds oscillate with different phases, so that the synchronization order parameter is small.
When the optical field amplitude F is large (but not too large to raise the entropy significantly), the on-site repulsion U assists the charge oscillations to be synchronized and increases the order parameter. A sufficiently strong interaction U overcomes the effect of randomness; thus, the order parameter almost reaches the maximum value. An even larger U makes the charge oscillations decay faster through dephasing, so that it becomes difficult to observe the synchronization. As to the nearest-neighbor interaction V, a weakly attractive one enhances the synchronization by enhancing current flows. It is reminiscent of enhanced stimulated emission above the superconducting transition temperature where superconducting fluctuations are expected to assist it, 14) in view of the fact that the stimulated emission is caused by an electronic breathing mode. 11) However, the interaction V is repulsive in real materials, and a repulsive interaction V in small systems that can be treated by the exact diagonalization method decreases the synchronization order parameter, which is consistent with the previous result. 11) The effect of superconducting fluctuations is beyond the scope of this study and left for future studies. 16/19