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Excitonic wave-packet evolution in a two-orbital Hubbard model chain: A real-time real-space study

Motivated by experimental developments introducing the concept of spin-orbit separation, we study the real-space time evolution of an excitonic wave-packet using a two-orbital Hubbard model. The exciton is created by exciting an electron from a lower energy half-filled orbital to a higher energy empty orbital. We carry out the real-time dynamics of the resulting excitonic wave-packet, using the time-dependent density matrix renormalization group method. We find clear evidence of charge-spin and spin-orbit separation in real-space, by tracking the time evolution of local observables. We show that the velocity of the orbiton can be tuned by varying the inter-orbital interactions. We also present a comparative study of a hole (in one orbital) and exciton (in two orbitals) dynamics in one-dimensional systems. Moreover, we analyze the dynamics of an exciton with spin-flip excitation, where we observe fractionalized spinons induced by Hund's interaction.

Due to reduced dimensionality and strong correlation effects, quasi one-dimensional (1D) systems display exotic dynamical properties [10][11][12], including the fractionalization of low-energy excitations [1,3] into spin (spinons) and charge (holons) excitations propagating with different velocities [13][14][15]. The existence of spincharge separation was shown experimentally early on in quasi one-dimensional systems [16,17]. Interestingly, the fractionalization of electronic excitations is not limited only to spin and charge, but it can also include the orbital degree of freedom. In fact, spin-orbital separation was observed experimentally in the transition metal compound Sr 2 CuO 3 [18]. Using high-resolution RIXS experiments, spin-orbital separation was also observed in the ladder system CaCu 2 O 3 [19].
Recently, the spin-orbit separation in Mott insulating systems was studied theoretically using the effective Kugel-Khomskii model [20]. In the limit of vanishing Hund's coupling, the propagation of the orbiton in a ferro-orbital and antiferro-magnetic chain was shown to map into a "single hole" moving in an AFM chain with its dynamics described by an effective t − J model [20]. However, in transition metal compounds, the Hund interaction plays an important role and, depending on the ma-terial, it can be strong. Precisely for a strong Hund's coupling, the propagation of an orbiton, with inter-orbital FM and AFM spin alignments (in the excited states of the superexchange process) are not equal, and the simple mapping between orbiton and t − J hole dynamics is no longer valid [21,22]. More recently, using RIXS the impact of the Hund's interaction on the orbiton propagation of the quasi-1D AFM compound Ca 2 CuO 2 was studied [22]. It was observed that robust Hund's interactions are required in the theoretical description to understand the experimental orbital spectrum [22].
In this letter, we provide the first study of spin-orbit separation in a real-time and real-space formalism by creating a finite momentum excitonic wave-packet at time t = 0, using a one-dimensional chain with two orbitals at each site. This wave-packet is created by exciting an electron from a half-filled orbital to an empty higher-energy orbital, as in experiments. Previous studies primarily focused on the spectral properties to study spin-orbit separation and for simplicity relied on Kugel-Khomskii models in the strong coupling limit [20,21]. Here we consider a more general multi-orbital Hubbard Hamiltonian at intermediate coupling strengths, accounting also for charge fluctuations and with focus on the influence of the Hund's coupling. To study the excitonic real-time dynamics, we use the time-dependent density-matrix-renormalization group (t-DMRG) method [23,24]. We have observed that after creating the exciton, the hole (in the half-filled orbital) and the electron (in the empty orbital) always move together, while the spin wave-packet in the halffilled orbital independently evolves from the charge wavepacket. We also compare the dynamics of a hole in a one-orbital chain versus the dynamics of an exciton in a two-orbital chain. Overall, at intermediate coupling and for robust values of the Hund's interaction, we find clear evidence of spin-orbit separation as time grows. Moreover, we quantitatively study the relation between the Hund's coupling and the orbiton velocity, finding that this orbiton's velocity increases with an increase in the Hund's coupling magnitude, while the spinon's velocity remains unaffected. Furthermore, we also present the dynamics of a spin-flip exciton (the previous discussion was for a spin preserving exciton), where we find fractionalized spinons, induced by the strong Hund's coupling.
Model and Method. We use the two-orbital Hubbard model on a chain. The model can be written as the sum of kinetic and interaction energy terms H = H k + H in [25]. The kinetic (tight-binding) portion contains the nearestneighbor hopping along the chain direction defined as: where c † iσγ creates an electron at the chain site i, with spin z-axis projection σ, and on orbital γ (either orbital a or b). t hop is the hopping integral. For simplicity, we considered only intra-orbital hoppings along the chain and we used identical hopping for both orbitals [t a = t b = t hop = 1]. ∆ γ denotes the crystal field term and n iσγ is the orbital-resolved number operator at site i. We fix the crystal-field parameters as ∆ a = 4.1 and ∆ b = 0. The large crystal field ∆ a = 4.1 4t hop ensures only orbital b is occupied in the non-interacting ground state because the bandwidth W of orbital a is W = 4t hop .
The electronic interaction portion is canonical: The first term is the on-site Hubbard repulsion between ↑ and ↓ electrons in the same orbital. The second term is the electronic repulsion between electrons at different orbitals. The standard relation U = U − 2J H is here assumed. The third term is the ferromagnetic Hund's interaction between electrons occupying the active two orbitals γ = a, b of the same site. S iγ is the total spin of orbital γ at site i. The last term is the pair-hopping between different orbitals, where P iγ =c i↓γ c i↑γ . To obtain the ground state |Ψ 0 of this model, we employed the static DMRG method. For our numerical calculations, we use a system size L = 36 with two orbitals at each site and we kept m = 1200 states. The exciton Gaussian wave-packet is created with spin σ and a crystal momentum k 0 , by applying the operator to the ground state |Ψ 0 . This operator excites an electron from the half-filled orbital b to the empty orbital a, centered at site i 0 = 18 and with width ω r = 2.54. The number 2.54 implies the size of the initial Gaussian is exactly six lattice spacings at half height, a size that we considered adequate for easy visualization. A is the normalization constant of the Gaussian wave-packet. Due to the finite width ω k = 1/2πω r of the Gaussian wavepacket in momentum space, we fix the crystal momentum at k 0 = −0.5π + 4ω k (i. e., close to the highest occupied electronic level with width ω k = 0.06). Because we construct a wave packet with a net nonzero momentum [k 0 (exciton) = k e (electron) + k h (hole) at time t = 0] that points in our case towards the left, the resulting time evolution will not be left-right symmetric. We investigate numerically the time evolution of the one-exciton state |Ψ e = h † σ (k 0 )|Ψ 0 under the influence of H i.e. |Ψ(t) = e −iHt |Ψ e . To perform the time evolution, we have implemented the Krylov space decomposition in the DMRG code [26,27]. For the DMRG calculations, at least 1200 states were kept during the time evolution.
To study the dynamics of the excitonic wave-packet, we measure the following observables at each time step: where n ia (t) and n ib (t) are the orbital-resolved time dependent charge densities of orbitals a and b. S z ia (t) and S z ib (t) are the respective orbital-resolved zcomponent of the time dependent spin densities. τ zi (t)  is the z-component of the time-dependent orbital density. All these quantities are site dependent.
Results. The Hamiltonian ground state, at overall quarter-filling (L = 36 and total number of electrons N e = 36) and parameters U/W = 1.0, J H /U = 0.25 [28], and ∆ a = 4.1, results in a situation where orbital b is a half-filled Mott-insulator with AFM-spin correlations, while orbital a remains empty. At time t = 0, the process previously described leads to an exciton centered in the middle of the chain, i.e. at site i 0 = 18. This results in a hole wave-packet n ib (t) in orbital b [ Fig. 2(a)] and an electron wave-packet n ia (t) in orbital a [ Fig. 2(c)]. The excitation of an electron from orbital b at t = 0 also creates spin-excitations S z ia (t) with up spins in orbital a [ Fig. 2(b)] and down spins in orbital b S z ib (t) [Fig. 2(d)]. As shown in Figs. 2(a) and (b), with increasing time the charge wave-packet n ia (t) and the spin wave-packet S z ia (t) at orbital a (the originally empty orbital) move with similar speeds toward the left from the central site i 0 = 18, indicating no spin-charge separation for or-bital a, as expected for an electron moving in an empty medium. Interestingly, the charge wave-packets n ia (t) and n ib (t) move together, as mirror images of each other [see Figs. 2(a) and (c)]. The reason is that the inter-orbital interaction U = U −2J H acts as an effective attraction between the hole in orbital b and electron in orbital a [29]. Intuitively, when the hole of orbital b and the electron in orbital a are in the same site, the strong inter-orbital repulsion energy U is not active (as compared to the case where two electrons are on the same site). This results in the formation of an electron-hole bound pair exciton which moves together with increasing time t towards the left of site i 0 = 18. The charge wave-packet n ib (t) and spin wave-packet S z ib (t) move in opposite directions with time, providing clear evidence of spin-charge separation [see Fig. 2(c) and (d)], in the half-filled orbital b.
In the electron-hole pair exciton, the electron promoted from the half-filled orbital b to the unoccupied orbital a is also equivalent to creating an orbiton [21,30]. In Fig. 2(e), we show the orbiton dynamics via τ zi (t) evolving with time t. The orbital wave-packet τ zi (t) moves similarly to n ib (t) and n ia (t) , towards the left form the central site i 0 = 18, while the spin wave-packet S z ib (t) moves toward the right. Thus, our result can be reinterpreted as a signature of spin-orbit separation in real-space with increasing time t. To determine the velocities of the orbital and spin excitations, we monitored the positions of the peak values of τ zi (t) and S z ib (t) vs. time. Using simple linear fits to extract the orbiton (v τ ) and spinon (v s ) velocities [ Fig. 2   (a) Charge densities ni(t) (one-orbital) and n ib (t) (twoorbital) (b) Spin densities S z i (t) (one-orbital) and S z ib (t) (two-orbital).
Next, to study the role of the inter-orbital repulsion U and Hund's coupling J H in the dynamics of the exciton wave-packet, we calculate τ zi (t) and S z ib (t) for different values of J H /U . Figure 3(a) displays the orbital wave-packet at time t = 5 but for three different values of J H /U . We find that for the smaller coupling J H /U = 0.05 the wave-packet τ zi (t) traveled only a very short distance from the central site i 0 = 18. However, increasing the Hund coupling to J H /U = 0.25, still at time t = 5, τ zi (t) traveled a larger distance (five lattice spacings) from the central site i 0 = 18. The inset shows the orbiton and spinon speeds |v τ | and |v s |, respectively, vs. J H /U . We find that the orbital velocity increases significantly with increasing J H /U . We believe this is because increasing J H /U reduces the interorbital interaction U , which results in a less-tightly bound electron-hole pair and, thus, the exciton becomes less heavier and can move at a faster rate |v τ |. On the other hand, at small J H /U the inter-orbital interaction U increases and results in heavier excitons, which naturally are more localized [7,29]. The larger value of orbiton velocity was observed in RIXS experiments because of the large Hund's coupling in Ca 2 CuO 3 [22]. The increase in orbiton velocity was explained in terms of the superexchange process [21,22], where they showed that the energy of the intermediate state during the movement of the orbiton depends on Hund's coupling J H . For completeness, note that we find the spin speed |v s | (inset) does not change much with increasing J H /U , and remains unaffected by the concomitant modifications in U , which is intuitively reasonable.
In Figs. 4(a) and (b), we show a comparison of the dynamics of a hole in the one-orbital Hubbard model (halffilled chain, U/W = 1.0) and in the two-orbital Hubbard model (quarter-filled, U/W = 1.0, U /W = 1.0, and J H /U = 0) chain system. At t = 0, a hole was created at the central site i 0 = 18, either by removing an electron at site i 0 = 18 for the one-orbital case or, for two orbitals, removing an electron in orbital b and exciting this electron to orbital a at the same site i 0 = 18. The results for the charge wave-packets are remarkably different. While the charge wave-packet in the one-orbital system moves quite fast and splits into left and right moving wave-packets, the charge wave-packet in the two-orbital system moves very slowly due to the formation of the strong electron (in orbital a)-hole (in orbital b) bound state. The heaviness of the bound state electron-hole is natural because to propagate to the next site, it involves two hoppings t a and t b and an intermediate state with energy proportional to U [scale as t a t b /U ], while the bare hole in one orbital propagates easily with just a hopping t hop . Interestingly, the spin wave-packets in both systems move with a similar speed and towards the right from the central site i 0 = 18 [see Fig. 4(b)]. This is expected because after the separation of spin and charge wave packets in the two-orbital system (at quarter-filling), the spin moves approximately guided by the scale t 2 hop /U , the same as the spinon follows in the one-orbital half-filled system [14].
In the RIXS experiment during the creation of orbital excitations spin-flip processes are also allowed [21,22]. Figure 5 presents a comparison of orbiton dynamics with and without spin-flip during the exciton generation, for different values of J H /U . At t = 0, for the spin-flip process the exciton wave-packet was created by the operator A j e −(j−j0) 2 /2ωr 2 e −ik0j c † j↓a c j↑b acting on the ground state wave-function |Ψ 0 . We found that the orbital velocity of τ zi (t) f when spin flip occurs is only slightly reduced compared to the previously described spin nonflip case τ zi (t) at J H /U = 0.25 [see Fig. 5(a)]. The slower speed of the spin-flip orbiton, compared to the spin-non-flip orbiton, is not expected from the superexchange picture [22] (where the spin-flip orbiton moves regulated by t a t b /(U − 3J H ) and for spin non-flip case by t a t b /(U − 2J H )) [31]. For a smaller J H /U = 0.05, the results are almost identical and the orbital wave-packet moves very slowly in both cases [see Fig. 5 The spin-flip excitonic process leads to the creation of spin wave-packets S z ib (t) f and S z ia (t) f in the spindown state [ Fig. 5(c)]. Interestingly, at large J H /U the spin-wave packet splits into two wave-packets with time (t 3), travelling in opposite directions (starting at the central site i 0 = 18). This curious splitting of the spin wave-packet S z ib (t) f indicates the presence of two fractionalized spinons [32] The left moving wave-packet S z ib (t) f travels with similar speed as S z ia (t) f of orbital a and τ zi (t) f . This could be due to the strong Hund's interaction between spin wave-packets of orbital a and b, which favors parallel alignment (spin-down state) of spin wave-packets S z ib (t) f and S z ia (t) f . At large J H /U , the creation of additional spinons was suggested in the spin-orbital spectrum [21]. In the case of spin-flip excitation, a strong J H /U also leads to attraction between orbiton and spinon [21], which may be related to the slight slowdown of the orbiton velocity [ Fig. 5(a)] compared to the without-spin-flip case (where spinon and orbiton repel each other [21]). The right moving spinwave packet S z ib (t) f moves with speed similar to that of S z ib (t) (without spin-flip case) [ Fig. 5(c)]. On the other hand, for smaller J H /U , the spin wave-packet S z ib (t) f does not split into two parts. S z ib (t) f (spin-flip case) and S z ib (t) (without spin-flip) move with similar speeds (see Fig. 5(d)).
Conclusions. Using the Krylov-space t-DMRG method we studied the real-time dynamics of an excitonic wavepacket evolving via a two-orbital Hubbard model on a chain, at intermediate coupling U/W . We observed the real-space spin-orbit and spin-charge separation by monitoring the dynamics of spin, charge, and orbital wavepackets. We find that the charge and spin wave-packets of the higher energy orbital a move together, whereas the charge and spin wave-packets of the lower energy orbital b moves in opposite direction (Fig. 6). The electron in the higher energy orbital and hole in the lower energy orbital always moves together. The inter-orbital interactions (U and J H ) play a crucial role in orbiton dynamics. For example, the orbiton velocity increases significantly by increasing J H /U , whereas the spinon velocity remains unchanged. Interestingly, we found that a hole in a one-orbital chain moves much faster than a hole in a two-orbital chain, because the hole in the lower energy orbital forms a (heavy) bound pair with the electron in the higher energy orbital. Moreover, we presented the dynamics of the spin-flipped exciton, where we found evidence of fractional spinons at large Hund's coupling. Our calculations will be extended in future work in various directions. For example, into other chain multiorbital systems with exotic states [33], including spin-orbit coupling [34], into ladder geometries [35], in materials with orbital order [36], ruthenates [37], and into generic t − J models [38].