Non-Equilibrium Instantaneous Approximation and Dipole Forbidden d-d Transitions

Abstract: Time- and angle-resolved photoemission spectroscopy provides direct access to pump-induced changes in the electronic structure of correlated materials, but its theoretical description generally requires computationally demanding two-time non-equilibrium calculations. We introduce an instantaneous approximation for pump-driven correlated systems, based on a separation between the rapid decay of the Green's functions in the relative time variable and their slower evolution in the average time. Combined with a one-shot dynamical mean-field construction at the Hubbard I level, the method incorporates the driven dynamics of the local correlated shell into the lattice Green's function while retaining momentum resolution. We also derive an effective coupling for nominally dipole-forbidden d-d excitations. It arises from a dipole-allowed d-p transition followed by p-d hybridization; beyond the instantaneous limit, the same process produces an energy-dependent vertex correction to the optical response. As a proof of principle, we apply the framework to paramagnetic and antiferromagnetic FePS3, using a density-functional theory derived tight-binding model together with a supercell unfolding procedure. The calculated momentum-resolved spectra reproduce the main qualitative features observed after excitation of the first and second d-d transitions in recent time-resolved photoemission experiments.

• The paper introduces an instantaneous approximation to the Kadanoff–Baym equations combined with one-shot Hubbard I and exact local multiplet dynamics, reducing the cost of simulating pump-driven correlated insulators.
• The effective theory shows that nominally dipole-forbidden d–d transitions gain optical weight through ligand-mediated d–p hopping, producing an instantaneous coupling that can be extended to energy-dependent vertex corrections.
• Applied to FePS₃, the method reproduces the observed contrast between nearly unchanged momentum maps after the spin-allowed transition and strongly altered maps after the spin-forbidden transition, while predicting transient insulator-to-metal behavior.

Motivation and scope

Time- and angle-resolved photoemission spectroscopy (trARPES) provides direct access to pump-induced changes in the momentum-resolved electronic structure of correlated materials, but its first-principles description requires two-time non-equilibrium Green's function calculations that are computationally prohibitive when all five 3d orbitals and their multiplet structure must be retained. This paper introduces an instantaneous approximation (IAP) for the Kadanoff–Baym equations of a pump-driven correlated system, combined with a one-shot DMFT construction at the Hubbard I level, and derives an effective coupling for nominally dipole-forbidden $d$-$d$ transitions. As a proof of principle, the framework is applied to paramagnetic and antiferromagnetic FePS$_3$, reproducing the principal qualitative features of recent trARPES experiments on that material (2608.16219).

The physical setting is the family of layered thiophosphates MPS$_3$, where partially filled transition-metal $d$ shells retain strong atomic character. Although direct $d$-$d$ transitions violate the Laporte rule, they are observed in optical absorption (2608.16219), acquiring weight through ligand-assisted processes combining dipole-allowed $d$-$p$ transitions with $p$-$d$0 hybridization. Recent trARPES measurements on FePS$d$1 have tracked the ultrafast buildup and decay of local $d$2-$d$3 excitations through their momentum-resolved signatures, motivating a theory that connects pump-driven local multiplet dynamics to transient photoemission spectra.

The instantaneous approximation in the Kadanoff–Baym formalism

Starting from the Keldysh contour formulation of the two-time Green's function for a lattice Hamiltonian with local interactions and a pump term, the authors perform a Wigner transform separating the relative time $d$4 from the average time $d$5. The convolution integrals of the KB equations generate gradient operators $d$6. The IAP consists of setting $d$7, justified by the separation between microscopic electronic dynamics (governed by the inverse bandwidth) and macroscopic pump evolution on pulse timescales $d$8 fs. In this limit the adjoint and non-adjoint KB equations coincide, and the problem reduces to solving, at each average time $d$9, Dyson-like equations with an instantaneous frequency-dependent self-energy $_3$0 — structurally identical to equilibrium equations but with $_3$1-dependent ingredients. The lesser Green's function follows from the retarded/advanced components together with the instantaneous self-energy and the equilibrium fluctuation-dissipation theorem applied to the free part.

This is a Markov-type approximation: it assumes fast memory loss of the microscopic dynamics relative to the driving. Its validity rests on the pump being slow compared with electronic timescales, an assumption well satisfied for 100–200 fs pulses.

NEQ-IAP Hubbard I embedding

Within DMFT, the self-energy is taken local. Rather than iterating the self-consistency loop out of equilibrium — the major bottleneck of non-equilibrium DMFT — the authors stop after the first iteration, which is equivalent to Hubbard I. The isolated driven impurity problem,

$_3$2

is solved exactly by propagating the density matrix via the von-Neumann equation within the finite Fock space (with the option of adding Lindblad dissipators). Wigner-transformed lesser and greater impurity Green's functions yield the instantaneous self-energy, which is then inserted into the lattice Dyson equation. Hubbard I fails for metals and intermediate coupling (it misses quasiparticle bands), but is accurate deep in the Mott insulating phase — precisely the regime of FePS$_3$3, which sits near the Mott-Hubbard/charge-transfer crossover.

A key simplification deployed in the application: rather than tracking the full time propagation during the pulse, the calculation evaluates the self-energy only at the instant of population inversion between ground and laser-matched excited multiplets, replacing the ground-state projector in the density matrix by the excited-state projector. This "one-instant" evaluation is a computational restriction beyond the IAP itself; full time-resolved tracking is deferred to future work.

Effective theory of dipole-forbidden d-d transitions

The paper derives, via a Grassmann path integral with the ligand $_3$4 sector integrated out, an effective action for the correlated $_3$5 shell. Integrating out the uncorrelated $_3$6 orbitals generates a self-energy containing the vector potential:

$_3$7

using the dipole selection rules $_3$8. Physically, light drives a dipole-allowed $_3$9 transition (or the reverse), the electron propagates in the $_3$0 manifold, and a $_3$1-$_3$2 hop restores the charge of the correlated shell while leaving it in an excited multiplet. Because the $_3$3-$_3$4 process is fast relative to both the pulse duration and the local dynamics, this reduces to an effective instantaneous $_3$5-$_3$6 matrix element. The process is perturbative in $_3$7, consistent with the weak experimental intensity of these lines. Beyond the instantaneous limit, the same mechanism produces an energy-dependent vertex correction to the optical conductivity, connecting the present construction to earlier vertex-function treatments of forbidden transitions. This derivation supplies the microscopic justification for restricting the explicitly driven dynamics to the $_3$8 shell alone.

Downfolding, upfolding, and supercell unfolding

To connect to real materials, Kohn-Sham states are mapped onto localized Wannier functions via disentanglement and localization unitaries, with explicit downfolding/upfolding transformations for Green's functions. For the antiferromagnetic phase, a supercell related to the primitive cell by an epitaxy matrix $_3$9 is required; the paper formulates an unfolding scheme based on overlap amplitudes $d$0 carrying Kronecker constraints $d$1, allowing supercell spectral quantities to be represented in the primitive-cell Brillouin zone for direct comparison with experimental momentum maps.

Application to FePS₃

Parameterization and its compromises

DFT calculations (QUANTUM ESPRESSO, vdW-DF-C09, experimental lattice parameters) followed by Wannierisation on Fe $d$2 and S $d$3 manifolds produce the tight-binding model. Several tensions are acknowledged plainly. LDA predicts a metal with roughly 7 electrons in the Fe $d$4 shell, whereas experiment indicates an insulator with Fe$d$5 character; recent RIXS work reports a mixed-valent $d$6/$d$7 superposition corresponding to $d$8 $d$9 electrons. The extended nature of the constructed Wannier functions ($d$0 Ų) yields overly large $d$1 hoppings, preventing a paramagnetic insulator at literature interaction values. The authors respond by treating $d$2 and $d$3 as adjustable, using $d$4 eV and $d$5 eV — above literature values — chosen so that both magnetic phases are insulating, with the DFT crystal-field splitting $d$6 eV retained. They note that using the spectroscopic splitting of $d$7 eV gives analogous results. A double-counting correction is tuned to reproduce $d$8 $d$9 electrons locally. These choices mean the insulating gap is engineered through parameter adjustment rather than predicted, a limitation inherent to single-shot Hubbard I without the self-consistent feedback that would narrow the bands.

Local multiplet physics

Exact diagonalization of the local problem yields the spin-allowed $d$0 transition at $d$1 eV (set by the crystal-field splitting) and the spin-forbidden $d$2 transition at $d$3 eV (set by $d$4, $d$5, and partly $d$6), close to the experimental values of 1.1 eV and 1.8 eV. The ground state is a Hund's-rule $d$7, $d$8 configuration expressible as a single Slater determinant; the second excitation is a genuine multi-determinantal many-body state involving pair-fluctuation couplings. This structural difference between the two excited states turns out to govern their distinct spectral fingerprints.

Equilibrium validation

Hubbard I spectra reproduce the qualitative $d$9/$d$0 orbital disposition in both phases. Notably, while the Fe $d$1 spectral weight shifts to lower energy upon entering the AF phase — a red shift of the $d$2 contribution, in contrast to blue shifts of Mott gaps reported in effective single-band models — the physical valence-conduction gap red-shifts as well, with calculated gaps of $d$3 eV (AF) versus $d$4 eV (PM). The AF phase shows broken $d$5 symmetry in the calculated k-maps at several binding energies, whereas the paramagnetic k-maps agree significantly with experimental ARPES data. The small AF gap again reflects the overestimated $d$6-$d$7 hybridization.

Non-equilibrium trARPES k-maps

The central result concerns the transient lesser Green's function under population of the first versus second excited multiplet:

• First excitation ($d$8, $d$9 eV): the atomic lesser Green's function equals the ground-state one shifted by the excitation energy, and the resulting momentum maps closely mirror the equilibrium valence maps — matching the experimental observation that off-resonant excitation of the spin-allowed transition leaves the k-maps largely intact. Deviations from exact mirroring are attributed to $p$0-$p$1 hybridization, which modifies the effective bands rather than merely translating them.
• Second excitation ($p$2, $p$3 eV): because this state is a superposition of several Slater determinants, its atomic Green's function differs qualitatively, and the calculated k-maps change substantially — peaks shift to different momenta and the $p$4-point peak disappears at $p$5 and $p$6 eV — again in line with experiment.

Both excitations induce an insulator-to-metal transition in the calculated spectra, with populated conduction states and redistributed spectral weight; the first excitation energy falls inside the equilibrium gap, underscoring its many-body origin beyond any single-particle picture. The robustness of these conclusions was checked against an alternative parameterization (literature $p$7 eV, $p$8 eV with a hopping rescaling factor $p$9), which yields very good agreement of equilibrium k-maps with experiment and analogous excited-state results.

Limitations and open questions

The paper is explicit about its approximations. The IAP neglects gradient corrections and memory effects; the deployed scheme further collapses the time propagation to a single instant (population inversion), so no time-resolved buildup or decay curves are computed here. The one-shot Hubbard I construction omits the self-consistent feedback of the lattice on the local non-equilibrium dynamics, surface sensitivity, finite probe penetration, relaxation, and scattering processes — hence quantitative one-to-one agreement with measured maps is not expected, and the AF gap is substantially underestimated. The parameter dependence (adjusted $p$0, $p$1; hopping rescaling) reflects the absence of the DMFT loop that would renormalize the $p$2-$p$3 hybridization self-consistently. Open questions left by the work include whether full time tracking under the von-Neumann propagation within IAP reproduces the measured femtosecond decay constants, how the energy-dependent vertex correction beyond the instantaneous limit affects quantitative optical weights, and whether self-consistent impurity schemes can cure the gap underestimation without empirical parameter tuning.

Conclusion

This work provides a computationally economical route to momentum-resolved pump-probe spectra of correlated insulators: an instantaneous approximation to the Kadanoff–Baym equations, a one-shot Hubbard I embedding of exactly solved local multiplet dynamics, a path-integral derivation of effective ligand-mediated $p$4-$p$5 couplings, and a supercell unfolding procedure. Applied to FePS$p$6, the framework captures the essential experimental dichotomy — near-invariant momentum maps for the first $p$7-$p$8 excitation versus strongly modified maps for the second — supporting an interpretation in which transient trARPES features are governed by local multiplet populations embedded in the dispersive band structure, while leaving quantitative refinement to self-consistent and fully time-resolved extensions.