Ultrafast dynamics in relativistic Mott insulators

The control of material properties with nonequlibrium protocols is one of the most fascinating promises of modern condensed matter physics Basov et al. (2017); Orenstein (2012). One example is the ultrafast manipulation of magnetic order, which can potentially boost major technological advances and reveal indispensable information of the underlying many-body physics Kirilyuk et al. (2010); Orenstein (2012). It has been known for decades that an ultrafast laser pulse can induce a subpicosecond demagnetization Beaurepaire et al. (1996), while a systematic theoretical description of the scenario is still under development. In many experimental studies, multi-temperature models are adopted for the phenomenology, where the laser excitation suddenly injects energy into the electron reservoir in the metal, subsequently heating up and melting the collective magnetic order Kirilyuk et al. (2010). In insulating materials, this mechanism should nevertheless be modified Kimel et al. (2002), since the localized electrons cannot absorb energy effectively from the laser field. One possible mechanism is the delocalization of electrons due to the photoexcitation, termed photodoping. The instantly created photocarriers then transfer their excess energy to the collective order and lead to the ultrafast melting.

Of particular interest are Mott insulators, which are predicted to be metals from band theory but are indeed insulators due to the strong Coulomb repulsion Imada et al. (1998). The family includes various transition metal oxides, such as NiO, V₂O₃, and cuprates in certain parameter regimes Zhang et al. (2019); Held et al. (2001); Cai et al. (2016). A minimal description is given by the one-band Hubbard model, with one orbital per each lattice site and one electron per orbital. The strong onsite Coulomb repulsion $U$ prevents a double occupancy of the same site, leaving no room for electronic conduction. In these materials, the localized spins can order at low temperatures and tend to form an antiferrogmagnetic (AFM) phase on a bipartite crystal lattice, where neighboring spins align oppositely. The mechanism for the photo-induced melting of AFM order in Mott insulators is fundamentally different from the ultrafast demagnetization of ferromagnets. It has been shown that a laser pulse can quickly redistribute the charge among neighboring lattice sites, leading to fast generation of doubly occupied and empty sites (doublons and holons). These photocarriers can modify the electronic structure and the energy gap, as observed, e.g., using angle-resolved photoemission spectroscopy (ARPES) and X-ray spectroscopy Mor et al. (2017); Beaud et al. (2014). The coupling between charge excitations and the spin order has been argued to be the dominant mechanism leading to the ultrafast spin dynamics Lenarčič and Prelovšek (2013); Golez et al. (2014); Balzer et al. (2015); Dal Conte et al. (2015). Specifically, the motion of charge excitations can create trails of defects in the antiferromagnetic background, transferring energy from the hot photocarriers to the AFM order. The scenarios have been examined theoretically through a nonequilibrium generalization Aoki et al. (2014) of the dynamical mean-field theory (DMFT) Georges et al. (1996), which is an established approach to describe the Mott physics in equilibrium. The same spin charge coupling becomes already manifest in the properties of a single hole or electron in the ordered spin background, which transforms into a spin-polaron, i.e., a hybrid of charge and magnon excitationsBrinkman and Rice (1970); Dagotto (1994); Brunner et al. (2000); Sangiovanni et al. (2006); Grusdt et al. (2018).

In equilibrium, it is known that the spin-polaron physics is strongly affected by geometric frustration Srivastava and Singh (2005); Tohyama (2006); Hamad et al. (2006, 2008); Läuchli and Poilblanc (2004); Shibata et al. (1999), such as for the $120^{\circ}$ order in a triangular lattice Capriotti et al. (1999); White and Chernyshev (2007); Iida et al. (2019). It should therefore be interesting to see how the mechanism for the melting of AFM order is modified in the presence of geometric frustration Bittner et al. (2020) or spin-orbit coupling (SOC). The latter can also be of interest for a practical reason: spin-orbit coupling can add a ferromagnetic moment to the AFM order, whose dynamics can be probed optically, while a probe of pure antiferromagnetic correlations on the femtosecond timescale is more challenging Afanasiev et al. (2019). Recently, heavy transition metal compounds, including elements such as iridium and ruthenium, have attracted many research interests. Due to the large atomic number, the relativistic effect in these materials is significantly enhanced, and the spin-orbit coupling becomes of similar order of magnitude than other local electronic energy scales (Hund’s coupling, crystal-field splitting, Coulomb interaction), leading to intriguing scenarios. For example, Na₂IrO₃ and $\alpha$-RuCl₃ have been predicted to realize the Kitaev honeycomb model in the low-energy sector, and may contain Majorana excitations promising for topological quantum computation Jackeli and Khaliullin (2009). The layered perovskite Sr₂IrO₄, on other hand, can be well described as a single-band Mott insulator Jin et al. (2009); Wang and Senthil (2011); Kim et al. (2012), but features an anomalously large ferromagnetic (FM) moment and potentially hosts states related to high-temperature superconductivity Yan et al. (2015). The photoinduced ultrafast dynamics in these materials has been under intense scrutiny. With strong SOC, the nonthermal states created with photodoping can exhibit exotic properties that are absent for normal Mott insulators Dean et al. (2016); Versteeg et al. (2020). In particular, the FM moment in Sr₂IrO₄ is rigidly coupled to the AFM order, which provides a unique opportunity for studying the ultrafast spin dynamics and the photodoping process by measuring the evolution of the FM moment Afanasiev et al. (2019). Microscopic theories to describe these observations are scarcely available, partly due to the complex orbital degeneracy and the electron-lattice coupling, which place many challenges for theoretical studies. On the other hand, the strong SOC can split the energy band and sometimes gives rise to a single energy band near the Fermi energy, which can be effectively described by a one-band Hubbard model, as in the case of Sr₂IrO₄. This raises the question of how to understand the photoinduced spin dynamics in a minimal one-band spin-orbit Mott insulators.

In this article, we will concentrate on minimal one-band Hubbard models with SOC on a lattice of square or triangular symmetry, and use nonequilibrium DMFT to solve for the photoinduced dynamics. Two questions are mainly addressed here: (i) How can the presence of SOC affect the photoinduced spin dynamics in Mott insulators? (ii) Is there a general mechanism underpinning the dynamics in the different geometries? We will confirm that the (partial) melting of the spin order parameter is significantly affected by the strength of SOC, giving rise to a different pathway of tuning the ultrafast magnetic dynamics in solids and cold atom systems. The dynamics can be classified into two types, according to the spatial profile of the Dzyaloshinskii-Moriya interaction induced by the SOC. The effect of SOC on the ultrafast spin dynamics can be well understood by the spin polaron physics, i.e., the creation of magnons due to motion of charge excitations, in analogy to the case of normal Mott insulators without SOC.

The article is organized as follows. Sect. II introduces the effective one-band Hubbard model with spin-orbit coupling and its solution within nonequilibrium DMFT. Sec. III discusses the spin order under SOC and the different types of SOC Mott insulators. Sect. IV demonstrates the photoinduced spin dynamics in the prototypical spin-orbit Mott insulator Sr₂IrO₄ and discusses the strong coupling between the AFM order and the canting-induced FM moment. Sec. V considers a triangular lattice with a different SOC pattern from the Sr₂IrO₄ case and shows the photoinduced demagnetization is qualitatively different. Sect. VI summarizes the main results and provides an outlook.

In multiband Mott insulators, the local spin orbit coupling $\bm{L}\cdot\bm{S}$ results in a rotation of the local spin-orbital basis. Up to the leading order, the intersite electron tunneling can mix up different spin and orbital indices when the local electronic Hamiltonian is diagonalized. As a minimal model, we consider the one-band spin-orbit Hubbard model with a spin-mixing hopping matrix,

where $c^{\phantom{\dagger}}_{i\sigma}$ represents the annihilation operator for an electron of spin $\sigma$ at the lattice site $i$ and $U$ is the on-site Coulomb interaction. The parameter $t_{0}$ characterizes the overall strength of electron tunneling between neighboring sites $\langle ij\rangle$, while the hopping matrix $\hat{R}^{\langle ij\rangle}$ determines how spin components are mixed during the tunneling process. $\varphi_{ij}(t)=e\bm{A}(t)\cdot(\bm{r}_{i}-\bm{r}_{j})/\hbar$, with positions $\bm{r}_{i(j)}$ for two neighboring atoms, is the Peierls phase. It takes into account the impact of a laser pulse described by the time-dependent vector potential $\bm{A}(t)$, as in Fig. 1(a).

The simple model (1) can be implemented with atoms placed in optical lattices Hamner et al. (2015). It has also been argued that the two-dimensional spin-orbit Mott insulator Sr₂IrO₄ is described by a variant of (1). In this case, the strong spin-orbit coupling splits the $t_{2g}$ manifold of the $5d$ atomic shell, which are occupied by 5 electrons, leaving out a single-particle band with effective spin $J_{\rm eff}=1/2$ at the Fermi level Jin et al. (2009). In general, one can assume that the matrix $\hat{R}$ is proportional to a unitary rotation of the local spin-orbital basis. In the remainder of this article, we mainly consider hopping matrices in the following simple form,

With time-reversal symmetry, this is the only possible form if one assumes that the $\hat{R}^{ij}$’s along different bonds can be diagonalized simultaneously. This assumption will not affect the general mechanism which will be discussed later. The spin-dependent hopping matrix (2) has a clear physical meaning: it measures the relative rotation between the local basis at site $i$ and $j$. Specifically, consider an electron at site $i$ with spin-state (a spinor) $\psi=\begin{pmatrix}c_{\uparrow},c_{\downarrow}\end{pmatrix}^{T}$, the hopping process to site $j$ transforms the state into $\hat{R}^{ij}\psi=\begin{pmatrix}e^{{\rm i}\phi_{ij}}c_{\uparrow},e^{-{\rm i}\phi_{ij}}c_{\downarrow}\end{pmatrix}^{T}$.

The SOC angle $\phi_{ij}$ generally depends on the bond $\langle ij\rangle$. In different lattices and with different patterns of $\phi_{ij}$, the physics can differ dramatically, see Fig. 1(b-d) for examples. Here we exemplarily consider a square lattice and a triangular lattice. These lattices are generally related to various transition metal compounds, such as Sr₂IrO₄, the copper-oxide monolayer Bonesteel et al. (1992) and Na₂IrO₃ Shitade et al. (2009). When one electron hops around a loop in the lattice, it captures an overall phase factor $\sigma\sum\phi_{ij}$ accumulating the SOC phases through the trajectory. The phase differs by a sign $\sigma=\pm 1$ for effective spin up and down, respectively. As shown in Fig. 1, the square lattice can have two different SOC patterns, featuring zero or spin-dependent $4\sigma\phi$ flux for each unit cell. Such different patterns imply different magnetic orders, which is best understood in terms of the strong-coupling spin model obtained from the Hubbard model (see Sec. III). Indeed, the single-band effective model of Sr₂IrO₄ Jackeli and Khaliullin (2009) features the zero flux pattern. This leads to a canted antiferromagnetic (AFM) phase with a ferromagnetic (FM) moment. Cuprates, on the other hand, are believed to have the $4\sigma\phi$ SOC flux and frustrated spin canting Bonesteel et al. (1992). Another example of a model with nonzero SOC flux is the triangular lattice shown in Fig. 1(d), which will be discussed in more detail in the following sections. We shall see that the two scenarios with zero or nonzero SOC flux lead to distinct photoinduced spin dynamics.

We first study the spin order in the spin-orbit Mott insulators. For later convenience, we define the spin operator $J^{\mu}=\sum_{\sigma\sigma^{\prime}}c^{\dagger}_{\sigma}\tau^{\mu}_{\sigma\sigma^{\prime}}c_{\sigma}^{\prime}$ in the single-band model. While $\bm{J}$ corresponds to a total angular momentum state and is often referred to as “effective spin” in the literature, we will use the short-hand “spin” for simplicity in the following. However, it is worth noting that the effective spin is generally related to the experimentally observed magnetic moments $\bm{M}$ through a tensor relation Wang and Senthil (2011).

In the strong-coupling limit $U\gg t_{0}$, the low-energy physics of (1) is captured by a spin exchange model obtained from the Schrieffer-Wolff transformation, while the spin-mixing $\hat{R}$-matrix leads to a Dzyaloshinskii-Moriya (DM) interaction Moriya (1960); Jackeli and Khaliullin (2009). For example, consider $\phi=0$, a half-filled two-site Hubbard chain with site index $i=1,2$ is effectively described by the following Heisenberg model,

where the exchange coupling $I_{\rm ex}=4t_{0}^{2}/U$ as usual. At low temperatures, this Hamiltonian gives rise to an antiferromagnetic ground state. In the presence of SOC, a nonzero phase $\phi$ generally leads to a Dzyaloshinskii-Moriya (DM) interaction. The easiest way to understand the situation is to impose a local basis rotation $c^{\dagger}_{j\sigma}=e^{{\rm i}(-1)^{j}\sigma\phi/2}\tilde{c}^{\dagger}_{j\sigma}$, with $\sigma=\pm 1$ for up/down spin, mapping the model to the $\phi=0$ case. The mapping rotates the local spins in the following way

and transforms the effective hamiltonian (4) to a model with a DM interaction,

Note that, for a two-site model, a general $\hat{R}$ matrix can always be recast to the form in (2) through redefining the $J^{z}$–axis.

The last term in Eq. (6), or the DM interaction, tends to induce a relative rotation between two neighboring spin moments, which is always possible for the two-site model. However, in a lattice with a certain $\phi_{ij}$ pattern, the DM interaction from different bonds adjacent to a given spin can favor inconsistent canting angles, resulting in frustration of the spin canting Moriya (1960); Bonesteel et al. (1992). The diagrams in Fig. 1(b) and (c) show two examples of the unfrustrated (with flux $0$) and frustrated (with flux $4\sigma\phi$) spin canting for a square lattice. The frustration corresponds to a nonzero SOC flux for a closed loop in the lattice. Indeed, if the SOC flux vanishes for all loops in the lattice, one can always impose the above-mentioned local basis rotation to map the full SOC Hubbard model to a non-SOC Hubbard model with $\phi=0$, thus completely eliminating the DM interaction. In this case, the effect of SOC is simply inducing a spin canting, and the resulting spin order is equivalent to the original one up to the local basis rotation. In the following, we consider two examples: Sr₂IrO₄, which corresponds to a lattice of square symmetry with zero SOC flux, and a lattice of triangular symmetry with nonzero flux.

In this section, we consider the photoinduced melting of the spin order in Sr₂IrO₄ using the single-band model. As explained above, the effective model of Sr₂IrO₄ features an SOC pattern with zero SOC flux, leading to a ferromagnetic moment consistent with the DM interaction of bonds in different directions. In this case, the SOC phase factor can be completely gauged away with a unitary transformation, and the photoinduced dynamics should also be unaffected with or without the spin-orbit coupling. The ultrafast dynamics of Sr₂IrO₄ has been experimentally studied in a recent work Afanasiev et al. (2019), which contains a detailed analysis of the evolution of the canting angle.

In a realistic description, the SOC angle $\phi$ is nevertheless controlled by the rotation angle $\theta$ of the IrO₆ octahedra Jin et al. (2009), which simultaneously affects the electron tunneling parameter $t_{0}$ in the following way,

where $\tau^{z}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$, and $\theta=0$ corresponds to the non-SOC case, see Fig. 3(a). In experiments, the angle $\theta$ is controllable through a strain Liu et al. (2015).

In the following, we address the question of how the spin dynamics can be controlled by changing the structural parameter $\theta$. We choose $\bar{t}_{0}=1.0$ in Eq. (7), which can be viewed as the energy unit, and the time unit is $\hbar/\bar{t}_{0}$ with $\hbar=1$. For the interaction parameter, we pick up $U=4.5\bar{t}_{0}$, which is close to the realistic parameters in Sr₂IrO₄. We vary $\theta=5^{\circ},10^{\circ},\ldots,30^{\circ}$ for demonstrating the effect of SOC Afanasiev et al. (2019). $\theta\sim 11^{\circ}$ is reported in experiments. The half-filling condition $n_{\uparrow}+n_{\downarrow}=1$ is imposed with a chemical potential $\mu=U/2$, yielding an insulating ground state with AFM ordering. An electric pulse, as given by Eq. (3), is applied at $t=0$ with parameters $\Sigma=0.5,\tau_{0}=5.0,A_{0}=0.8,T=4.0$. Since we are only concerned with the spin melting following the excitation, the detail of the protocol should not matter too much.

In this section, we turn to the spin-orbit Hubbard model with the triangular geometry. In this case, we set $t_{0}=1.0$ as an energy unit and directly vary the SOC angle $\phi$. With the DMFT calculations, the $120^{\circ}$ AFM has been found in the ground state at inverse temperature $\beta=50$. The spin canting is completely frustrated in this case, unless $\phi=120^{\circ}$, which amounts to a spatial reflection and is not important in the present study. We will be mostly interested in $\phi\neq 120^{\circ}$, in particular smaller values of $\phi$.

As mentioned above, both the $120^{\circ}$ AFM order and the SOC pattern can have two distinct chiralities. In the following discussion, we will fix the chirality of the order to be counterclockwise when traversing sublattices in the order 1 to 2 to 3 (see Fig. 1(d)), and change $\phi$ for positive and negative values.

In this article, we studied the ultrafast spin dynamics of the Mott insulators with spin-orbit coupling and demonstrated the possibility of tuning ultrafast demagnetization with changing SOC in two prototypical examples, including the one-band Hubbard model on a square lattice and on a triangular lattice. The models are generally relevant to transition metal compounds and cold atom systems Struck et al. (2011, 2014); Hamner et al. (2015). We find that the light-induced melting of effective spin moments in Sr₂IrO₄ strongly relies on its crystal structure parameter, i.e. the rotation angle $\theta$ of the IrO₆ octahedra, which determines the SOC strength and controls the bandwidth. The SOC-induced canting angle is nevertheless intact during the demagnetization. In the triangular lattice, the SOC strongly affects the melting of the $120^{\circ}$ order. Specifically, an SOC angle $\phi>0$ suppresses the melting, whereas $\phi<0$ enhances the melting, allowing for flexible tunability of the dynamics. This trend can be explained by the modulation of spin exchange energy by the SOC, which increases (decreases) the energy cost for a charge excitation to hop in the AFM background when the SOC angle is consistent (inconsistent) with the chirality of the spin order. This effect can be observed in terms of the spin polaron peaks in the spectrum.

The square and triangular geometries can be identified as representative of two more general scenarios how the SOC influences the spin dynamics. The first type gives rise to unfrustrated spin canting, such as in Sr₂IrO₄, and the Hamiltonian can be mapped to a non-SOC Hubbard model. The dynamics of the canted AFM is then related to that of the collinear AFM in the non-SOC case with a time-independent spin canting, leading to a fixed canting angle. The second type, considered in the triangular lattice, results in frustrated spin canting. In this case, the spin exchange energy is effectively modified by the SOC, modulating the coupling between the AFM order and charge excitations and controlling the spin dynamics.

We have studied the SOC physics in minimal models with a single band. The underpinning mechanisms can be readily generalized to more complex situations, in which the SOC can result in canting of the combined spin and orbital orders and modify the exchange coupling in the compass model Jackeli and Khaliullin (2009). In the future, further studies can be carried out for the spin-orbit multi-band Hubbard models with, for example, $t_{2g}$ orbital degeneracy and Hund’s coupling Khomskii (2014). The SOC engineering of the order-parameter dynamics can be generalized to intertwined spin, orbital, and charge orders and may have crucial consequences on photoinduced hidden phases Li et al. (2018). For this prospect, the effects of nonlocal fluctuations and electron-lattice coupling can be studied with newly developed methods, such as $GW$+DMFT Sun and Kotliar (2002); Biermann et al. (2003) in the nonequilibrium regime Golez et al. (2019) and an exact treatment of the phonon degree of freedom Grandi et al. (2020). These methods can be combined with the steady-state theory of photodoping Li and Eckstein (2020) to assess the long-time behavior of the order parameters after the excitation.