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A time-dependent momentum-resolved scattering approach to core-level spectroscopies

by Krissia Zawadzki, Alberto Nocera, Adrian E. Feiguin

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Submission summary

Authors (as registered SciPost users): Adrian Feiguin
Submission information
Preprint Link: scipost_202302_00015v2  (pdf)
Date accepted: 2023-09-15
Date submitted: 2023-08-31 21:50
Submitted by: Feiguin, Adrian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approach: Computational


While new light sources allow for unprecedented resolution in experiments with X-rays, a theoretical understanding of the scattering cross-section is lacking. In the particular case of strongly correlated electron systems, numerical techniques are quite limited, since conventional approaches rely on calculating a response function (Kramers-Heisenberg formula) that is obtained from a perturbative analysis of scattering processes in the frequency domain. This requires a knowledge of a full set of eigenstates in order to account for all intermediate processes away from equilibrium, limiting the applicability to small tractable systems. In this work, we present an alternative paradigm, recasting the problem in the time domain and explicitly solving the time-dependent Schrödinger equation without the limitations of perturbation theory: a faithful simulation of the scattering processes taking place in actual experiments, including photons and core electrons. We show how this approach can yield the full time and momentum resolved Resonant Inelastic X-Ray Scattering (RIXS) spectrum of strongly interacting many-body systems. We demonstrate the formalism with an application to Mott insulating Hubbard chains using the time-dependent density matrix renormalization group method, which does not require a priory knowledge of the eigenstates and can solve very large systems with dozens of orbitals. This approach can readily be applied to systems out of equilibrium without modification and generalized to other spectroscopies.

Published as SciPost Phys. 15, 166 (2023)

Author comments upon resubmission

We thank the Referees for their insightful and rigorous assessment of our manuscript. We greatly value their attention to detail that has helped us to greatly improve the manuscript. We are also pleased that both consider it worthy of publication.

List of changes

-We have added a clarifying sentence:
"After obtaining the ground state we connect the extended probe at time $t=0$ and measure the momentum distribution function $n_{b,d}(k,t)$ at the detector as a function of time. Since in our tDMRG simulations we use open boundary condition, the proper definition of momenta corresponds to particle in a box states $\sin{(k_jx)}$ with momenta $k_j=j\pi/(L+1)$ with $ (j=1,\cdots,L)$. However, as customary in DMRG calculations, we vary $k$ continuously."

-We have added an entire description or the Hubbard model with additional simulation details directly under "Results".

-We have added the explicit form of the projector in the sentence:

"This can be done by means of a projector:
H_{source}= -|\kin\rangle \langle \kin|+\lambda \sum_{ij}n_{b,s,i}n_{b,s,j},
where $|\kin\rangle \langle \kin|=n_{b,s}(\kk)=\frac{1}{L}\sum_{mn}e^{i\kin(\mathbf{R}_m-\mathbf{R}_n)}b^\dagger_{s,m}b_{s,n}$"

"The full calculation proceeds as follows: The system is first initialized in the ground state of $H_0+H_{source}$...."

-We have added the following discussion in the summary:
"In our calculations accuracy is kept it under control by using a sufficiently large number of DMRG states (bond dimension). Notice that at time $t=0$ the system in in the ground state of $H_0+H_{source}$. The core orbitals are in a product state of double occupied states and do not contribute to the entanglement. As time evolves, one electron will be excited from the core-orbitals, and one photon will eventually be emitted when the core hole recombines. When this occurs, the core orbitals return to a product state. Moreover, there is no hopping for the core degrees of freedom. This means that any additional entanglement will stem from the perturbations left behind in the system (which is a gapped Mott insulator) and the single photon at the detector, which will contribute to the entanglement by a bounded amount $\mathcal{O}(1)$. As a consequence, the entanglement growth will be minimal and simulations can proceed to quite long times. While we have not done a detailed quantitative analysis, we believe that the entanglement growth will be comparable, if not lower, than typical tDMRG simulations of spectral functions, particularly in the case of single-site RIXS."

- We have corrected the notation for Eq.(8)

-We have fixed the notation:
"the system absorbs a photon with energy $\win$ and momentumm $\kin$ and emits another one with energy $\wout$, momentum $\kout$. We hereby focus on the so-called ``direct RIXS'' processes, see Fig.\ref{fig:fig2} and Fig.~1 in Ref.\cite{Kourtis2012-PhysRevB.85.064423}). As a consequence, the photon loses energy $\Delta \omega=|\wout-\win|=\win-\wout$ (from now one referred-to as simply $\omega$) and the electrons in the solid end up in an excited state with momentum $\kout-\kin$. In the following, we consider $\kin=0$ and refer to the momentum transferred simply as $\kk$."

- Modified the sentence: "By an appropriate choice of $\Gamma_s^{\sigma\sigma'}=\Gamma_d^{\tau\tau'}=\Gamma$, and all others set to zero, one evolves the system in time to obtain a wave-function $|\psi_{\sigma\sigma',\tau\tau'}(t)\rangle$. This allows us to resolve the different contributions to the spectrum that split into spin conserving and non-conserving ones:..."

- Modified legends in Fig. 4 and Fig. 5

- We have expanded our description of Fig.5 in the text.

Reports on this Submission

Anonymous Report 1 on 2023-9-2 (Invited Report)


I would like to thank the authors for this revised version, I believe the manuscript is ready for publication.

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Anonymous on 2023-09-02  [id 3946]

The authors answered adequately to my previous comments and modified the manuscript accordingly. Therefore, I recommend the publication of the present manuscript in SciPost.