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A timedependent momentumresolved scattering approach to corelevel spectroscopies
by Krissia Zawadzki, Alberto Nocera, Adrian E. Feiguin
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Adrian Feiguin 
Submission information  

Preprint Link:  scipost_202302_00015v2 (pdf) 
Date accepted:  20230915 
Date submitted:  20230831 21:50 
Submitted by:  Feiguin, Adrian 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
While new light sources allow for unprecedented resolution in experiments with Xrays, a theoretical understanding of the scattering crosssection 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 (KramersHeisenberg 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 timedependent SchrÃ¶dinger 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 XRay Scattering (RIXS) spectrum of strongly interacting manybody systems. We demonstrate the formalism with an application to Mott insulating Hubbard chains using the timedependent 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.
Author comments upon resubmission
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:
\begin{eqnarray}
H_{source}= \kin\rangle \langle \kin+\lambda \sum_{ij}n_{b,s,i}n_{b,s,j},
\end{eqnarray}
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 coreorbitals, 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 singlesite 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 socalled ``direct RIXS'' processes, see Fig.\ref{fig:fig2} and Fig.~1 in Ref.\cite{Kourtis2012PhysRevB.85.064423}). As a consequence, the photon loses energy $\Delta \omega=\wout\win=\win\wout$ (from now one referredto 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 wavefunction $\psi_{\sigma\sigma',\tau\tau'}(t)\rangle$. This allows us to resolve the different contributions to the spectrum that split into spin conserving and nonconserving ones:..."
 Modified legends in Fig. 4 and Fig. 5
 We have expanded our description of Fig.5 in the text.
Published as SciPost Phys. 15, 166 (2023)
Anonymous on 20230902 [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.