Fingerprints of an Exotic Orbital-Selective Mott Phase in the Block Magnetic State of BaFe$_2$Se$_3$ Ladders
Niravkumar D. Patel Alberto Nocera Gonzalo Alvarez Adriana Moreo Steven Johnston Elbio Dagotto





Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA


Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA


Computer Science & Mathematics 
Division and Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, 
 , USA


Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA


Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
Joint Institute for Advanced Materials at The University of Tennessee, Knoxville, Tennessee 37996, USA


Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996, USA
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA


Resonant Inelastic X-Ray Scattering (RIXS) experiments on the iron-based ladder
BaFe_2Se_3 unveiled an unexpected two-peak structure associated with local
orbital (dd) excitations in a block-type antiferromagnetic phase. A mixed
character between correlated band-like and localized excitations was also
reported. Here, we use the density matrix renormalization group method to
calculate the momentum-resolved charge- and orbital-dynamical response
functions of a multi-orbital Hubbard chain. Remarkably, our results
qualitatively resemble the BaFe_2Se_3 RIXS data, while also capturing the
presence of long-range magnetic order as found in neutron scattering, only when the model is in an exotic orbital-selective Mott phase (OSMP). In
the calculations, the experimentally observed zero-momentum transfer RIXS peaks
correspond to excitations between itinerant and Mott insulating orbitals. We
provide experimentally testable predictions for the momentum-resolved charge
and orbital dynamical structures, which can provide further insight into the
OSMP regime of BaFe_2Se_3. 


Fingerprints of an Exotic Orbital-Selective Mott Phase 
 
in the Block Magnetic State of BaFe_2Se_3 Ladders
    E. Dagotto
    
===========================================================================================================






Introduction — The recent discovery of superconductivity in the two-leg
iron-based ladder compounds BaFe_2S_3 <cit.> and
BaFe_2Se_3 <cit.> opened an exciting new
branch of research: these materials are the first members of the iron
superconductors family
<cit.> without iron layers,
the parent compounds are insulators <cit.>, and their
low-dimensionality allows for accurate theoretical treatment <cit.>.  For this reason, iron-ladder
compounds were the focus of many recent experimental and theoretical studies
<cit.>.  In particular, inelastic neutron
scattering data  <cit.> is compatible with the exciting idea that
BaFe_2Se_3 is in an orbital-selective Mott phase (OSMP) at ambient pressure
<cit.>, where one orbital is localized with a Mott
gap while the others are gapless and itinerant (see Fig. <ref>).
Moreover, this state displays an exotic magnetic order involving 2× 2
ferromagnetic blocks that are antiferromagnetically staggered. 

The intuitive origin of the block magnetic state admits multiple descriptions.
Within Hartree-Fock treatments <cit.>, the “block” structure arises
from magnetic frustration because the system is located in parameter space
between a ferromagnetic state, induced by double-exchange at large Hund
coupling, and a staggered antiferromagnetic state, induced by superexchange at
small Hund coupling. Alternatively, recent efforts based on an itinerant
perspective predicted that the metallic orbitals are the “drivers” while the
localized spins are the “passengers”, with the block structure arising from
Fermi surface nesting <cit.>. From the experimental
perspective, Resonant Inelastic X-ray Scattering (RIXS) and X-ray Photoemission
(XPS) studies report the presence of both localized and itinerant carriers in
BaFe_2Se_3,  also suggesting OSMP physics <cit.>;
however, up to now, no sufficient theoretical study has been carried out to
support these indirect experimental claims of an OSMP ground state in
BaFe_2Se_3. Our primary goal is to fill this gap and provide evidence from
a theoretical perspective that indeed the “123” ladder is in an OSMP state. 

One of the first steps towards unveiling the characteristic  
excitations of an OSMP state is to calculate its single-particle spectral function A(q,) 
and its intra- and inter-orbital dynamical spin S(q,), 
charge N(q,), and orbital L(q,) structure factors. 
Recently, theoretical predictions for S(q,) were presented <cit.> in a block antiferromagnetic (AFM) OSMP. 
Using determinantal quantum Monte Carlo and the maximum entropy method, the low temperature A(q,) was also reported employing an approximation to a multi-orbital Hubbard model <cit.>. 
However,  N(q,) and L(q,), crucial for RIXS experiments, have not been addressed yet. Therefore, here for the first time, we use the density matrix renormalization group (DMRG) technique to compute the momentum-resolved charge and orbital response functions (see Figs. <ref> and Figs. <ref>), as well as the single-particle spectral function (at zero temperature), of the block magnetic OSMP of a multi-orbital Hubbard chain. 

There are multiple reasons for focusing on a chain geometry as opposed to a ladder. First, a three-orbital Hubbard chain is already equivalent to a three-leg one-orbital ladder; thus a three-orbital Hubbard ladder maps onto a six-leg one-orbital ladder, which is challenging even with DMRG. Second, in the usual “snake” geometry of DMRG, interorbital hoppings mutate into long-distance hoppings in the one-orbital chain analog, here involving sites effectively separated by eight lattice spacings. Such long-range hoppings compromise the accuracy of DMRG. Finally, RIXS in the eV scale usually addresses local excitations, and there should not be much difference between chains and ladders, as recent efforts in neutron scattering using both chains and ladders showed <cit.>.



We compare our results against previously gathered RIXS
data for BaFe_2Se_3 <cit.>, which
measures magnetic, charge, and orbital excitations
simultaneously <cit.>. By calculating the orbital response functions of the competing paramagnetic metal (PM) and ferromagnetic (FM) insulator states, we show that block OSMP has a characteristic two-peak structure that is distinctive and in striking agreement with RIXS results on BaFe_2Se_3 <cit.>. Moreover, we identify the observed dd peaks as orbital excitations between localized and
itinerant orbitals. Our study strongly suggests that the ground state of BaFe_2Se_3 is indeed an OSMP with block magnetic order. 








Results — Figure 1 plots the local charge occupation (panel b) and fluctuations (panel c) of each orbital vs the total electronic
density, which demonstrates that the OSMP
is stable and robust against variations in the hole and electron doping <cit.>. 
First, we focus on the results for 
4 electrons per site (n=4), shown by the vertical dashed line 
in Fig. <ref> (b,c) (note that n=4 in a three-orbital model is the analog of the realistic n=6 in a five-orbital model). At this density, orbital c has one electron per site, and thus becomes Mott localized,
while the two other orbitals remain fractionally occupied, and thus metallic. The reason for this exotic behavior relies on the crystal-field splitting and different bandwidths of the orbitals: as the
Hubbard U interaction opens a gap and shifts energy states “up and down” relative to the gap, it becomes energetically favorable for orbital c 
to have a half-filled lower Hubbard band, which is formed by moving electrons from the other bands into c.  
The corresponding intra-orbital charge fluctuations δ N_γ^2  = ⟨ n_γ^2⟩ - ⟨ n_γ⟩^2 in orbital c are significantly suppressed, compatible with a localized state, while charge fluctuations in  orbitals a and b remain finite, compatible with a metallic state. 
The inset of Fig. <ref> establishes the magnetic order at this filling, and plots the spin structure factor at n=4, displaying the characteristic peak at q = π/2 that is in agreement with the experimentally observed non-trivial block-type AFM order. 
Note that orbital c (representing the d_xy orbital, see methods) has the standard 
characteristics of a Mott phase at n=4, where charge degrees of 
freedom are “frozen” (localized) and accompanied by well-formed local 
magnetic moments, 𝐦_c ∼ 0.98μ_B.  In  contrast, the large local charge fluctuations of the other orbitals suggest a metallic behavior typical of itinerant electrons. 

The total on-site local moment at n=4 is 𝐦_tot≃ 1.97μ_B, a robust value slightly larger than in experiments <cit.>.  
















 The coexistence of localized and itinerant carriers is also evident when examining the electronic density of each orbital vs the global filling near the commensurate value n=4 (Fig. <ref>). 
 The linear behavior of n_a and n_b suggests a band-like picture, 
 while the robust plateau in n_c extending over a wide range of doping (n = [3:4.5]) indicates a Mott gap in the single-particle spectral function of orbital c. Since all these results provide ample evidence that we correctly capture the block-magnetic properties of BaFe_2Se_3, henceforth we fix the filling n=4 to address the features 
of the block OSMP in the RIXS dynamical excitations, the focus of
our publication.  




Dynamical spectra of the block OSMP — Figures <ref> and <ref> show the orbital-resolved single-particle spectral functions. 
Orbitals a and b have a finite quasi-particle weight at the Fermi level E_F, as also shown in the density-of-states in Figure <ref>, confirming once again the itinerant nature of electrons in these orbitals. 





Moreover, we find that the orbital c has a gap of value
Δ_c∼ 1.5 eV <cit.> in c (Figs. <ref> and <ref>), 
compatible with our analysis in Figs. <ref> and <ref>. 
The spectral function demonstrates again the coexistence of gapped Mott (localized) and gapless itinerant carriers and agrees well with an earlier study of a similar model <cit.>. 
Note also the presence of a pseudogap in the itinerant d_xz/yz orbitals (not captured by earlier work <cit.>). 
It is likely that this pseudogap originates from correlations between the block ordered spins of Mott orbital c  
and itinerant electrons in orbitals a and b (Fig. <ref>), since they are not decoupled from one another. 
We also highlight that the Fermi momentum of orbitals a and b 
is approximately q_F ≃π/4, leading to scattering at 2q_F ≃π/2 that is comparable 
to the spin-structure factor peak at q = π/2 corresponding to the 
block AFM order <cit.>. It is conceivable that the block phase of localized spins of the 
Mott orbital c is driven by the Fermi nesting of itinerant orbitals, as
in manganites and heavy-fermion systems <cit.>.


In fact, recent experimental <cit.> and theoretical <cit.> work have
confirmed that the magnetic excitations in BaFe_2Se_3 cannot be 
fully described using an effective Heisenberg model, 
highlighting the role of the other degrees of freedom in this material that we are focusing on here. 









To arrive to our main conclusions, 
we now calculate the charge (Figs. <ref> and <ref>) 
and orbital (Figs. <ref> and <ref>) dynamics of the block-antiferromagnetic OSMP.  
Since the low-energy spin-dynamics have already been reported in the literature, we only focus on the high-energy charge and orbital dynamics that are more relevant to RIXS measurements at energy losses above 1 eV. 
As with a/b, the charge excitations of the itinerant orbitals display a 
gapless continuum (Fig. <ref>) because charge fluctuations can propagate freely in a metallic system (at fixed-energy transfer  many states allow for charge fluctuations without an energy cost). In contrast to  this, the charge excitations of the Mott orbital c display a
gap Δ^N_c∼ 1.5 eV (Fig. <ref>), and c shows incoherent charge excitations, compatible 
with doubly occupied orbitals d_xy (i.e. c) configurations 
with an energy proportional to the Mott gap (Fig. <ref>).
We can contrast our results with the much studied one-orbital two-leg ladder Hubbard model, where charge excitations are gapped in the half-filled Mott phase but display a gapless continuum in the metallic phase away from half-filling <cit.>. Our results show the simultaneous presence of localized and itinerant fermions in our multi-orbital model.






Consider now the q-resolved orbital excitations  in the block OSMP.

Figure <ref> (x/y) shows the excitations 
from (to) the itinerant orbitals a/b to (from) the localized orbital c. 
Figure <ref> displays a gapless continuum in orbital excitations, z, between the itinerant orbitals a and b 
(see illustration Fig. <ref>). 
 These dynamical spectra can be  
 understood using the density-of-states (Fig. <ref>), and 
 poles of the single-particle spectral function of the OSMP (Fig. <ref>). 


Figure <ref> shows that electron scattering from 
P_2 → P_4 requires 
an energy transfer ∼ 0.9 eV (vertical
arrow in Fig. <ref>). 
This scattering creates a gapped response peak in the orbital excitations at q ≃ 0 and 
≃ 0.9 eV (Fig. <ref>). 
The charge gap of
orbital c in c can also be understood via the scattering between points P_1 and P_4 of 
the density-of-states, shown by another vertical arrow in Fig. <ref>. 
Additionally, the gapless continuum in z is very similar to the itinerant charge excitations, 
a/b because only the (almost degenerate) itinerant orbitals contribute in the calculations of z. 







Dynamical spectra of the FM insulator — To identify features of orbital excitations in the block OSMP that are unique, we also show the q-resolved orbital excitations in the competing 
FM insulator known to exist  at U/W=4,<cit.> see
Figs. <ref> and <ref>. 
In this FM phase, which is not yet known to be stable experimentally, 
theoretical calculations show that the single-particle spectral functions of all the orbitals have a Mott gap <cit.>. Figures <ref> and <ref> show that 
the corresponding x/y and z display clearly 
sharper excitations in the FM phase in comparison to the block OSMP. 
Remarkably, z is gapless with a ≃ 0 peak at q = π, implying quasi-long-range antiferro-orbital order in the 
ground state, as discussed before <cit.>. 
In fact, the z of this FM phase resembles 
the  of the spin-1/2 Heisenberg chain with a two-spinon continuum. 
By analogy, we conjecture that the 
z of the FM phase may denote the existence of fractionalized orbital excitations <cit.>
(see illustration Fig. <ref>). 
Orbital and spin excitations were also studied recently in the context of Kugel-Khomskii models <cit.>. 
Here, we show novel results of orbital fractionalization in a FM 
insulator using a general multi-orbital fermionic Hamiltonian.  











Discussion —  Our main result is in Fig. <ref>, where we
compare the calculated orbital dynamical 
response vs RIXS Fe-L_3 edge experimental data for BaFe_2Se_3 at zero-momentum transfer <cit.>. 
Relating the full RIXS intensity to dynamical structure
factors is not straightforward <cit.>. For example, for the single-band Hubbard model, the RIXS intensity can be directly related to dynamical structure factors only in limiting cases. To further complicate matters, 
the experimental effort <cit.> did not report the exact momentum transfer of
the experiment. While the scattering geometry employed nominally probes q = 0 excitations 
along the chain, there were difficulties in aligning the chain orientation during beam time <cit.> 
and some finite q excitations may have been mixed into the spectra.   
Additionally, RIXS measures excitations of many different channels (e.g. spin,
charge, and orbital), making it difficult to differentiate between those
distinct channels. However, comparisons with the dynamical response functions
can still provide at least qualitative insights into
the dominant features and energy scales of RIXS experiments. Moreover, a merit
of our effort is that we calculate excitations for each channel separately, and can, therefore, provide detailed predictions for future
experiments. Until additional data become available
our focus on RIXS experiments at q = 0 momentum transfer <cit.> is
the natural avenue to pursue.











At zero energy transfer, there is an “elastic” peak commonly observed in RIXS experiments. We note that magnon contributions to the RIXS data generally occur at energy losses <0.2 in the Fe-based superconductors, and, 
therefore, are hidden in the elastic contribution in the RIXS experiment <cit.>. For this reason, this low-energy region will not be our focus. 
To identify the unique features of orbital excitations in the block OSMP (U/W=0.8), we also present results for the PM (U/W=0.04) and FM (U/W=4) competing phases.



Overall, we find a close agreement between our dynamical spectra in the block OSMP at q ≃ 0 and the two-peak structure observed in experiments, 
involving localized dd excitations of the iron 3d orbitals. The OSMP two-peak structure 
(Fig. <ref>) is distinct from that in the FM phase, where the two peaks are almost identical in position and width. 
With regards to the PM phase results (Figs. <ref>), there are some similarities with
the spectra shape of the OSMP phase.  The presence of these similarities are natural since
according to Ref. <cit.>, at U/W = 0.04 the population of orbital c is already near 1, although 
the block magnetic phase is still not fully developed.  
One might argue that the PM regime data could also fit the experiments with appropriate broadening, 
particularly considering that high-frequency results involving interorbital excitations do not typically change abruptly across phase transitions; however, neutron scattering experiments tell us that the material is magnetically ordered in a block state. Thus the
PM state cannot be chosen as the correct state even though the RIXS data might be consistent with our predicted dynamical spectra. 

Our model suggests that the experimentally observed peaks represent excitations between localized d_xy and itinerant d_xz/yz orbitals. 
They are gapped with an activation energy 0.7-0.9 eV in the calculations, 
and ∼ 0.35-0.45 eV in the experiments (transitions
involving e_g orbitals, outside our model, should appear at higher energies). 
The difference in absolute gap numbers could be 
fixed by tuning our model parameters still within the OSMP phase 
(or by using ladders instead of chains, which is technically much harder, as discussed before). Since “fine tuning” is not our goal,
but a qualitative understanding of results, we consider our (already costly) simulation results sufficient for our main
qualitative conclusion: the block OSMP is the one that most closely resembles the experimental data. 
In addition, in the real samples, there is broadening due to phonons and non-crystallinity effects 
that could make it challenging to define the magnitude of the gap accurately. Regardless, it is clear from experiments 
that there is a gap in the RIXS q=0 response. 
However, the paramagnetic phase has a gapless q=0 
response in L^x/y(q ≃ 0,) (Fig. <ref>), 
even-though there is a precursor to a two-peak structure, establishing another difference with the OSMP results. 
Moreover, while the ferromagnetic 
insulator shows a gapped L^x/y(q ≃ 0,) response, this is with the presence of a single peak 
at ≃ 1 eV. In summary, the gapped two-peak response is only evident in the block OSMP phase.







Summary and Conclusions — Our results suggest 
that the experimentally observed gap is not a conventional semi-conducting gap, but instead originates 
from the inter-orbital excitations of a magnetically ordered OSMP. Figure <ref> shows that 
peak A occurs at approximately 0.9 eV, for the zero momentum transfer, as a result of
vertical (Δ q = 0) scattering across E_F from the itinerant d_xz/yz 
orbitals to the d_xy Mott orbital (Fig. <ref>, P_2 → P_4). The 
shoulder/peak labeled as B represents scattering from the localized 
d_xy band below E_F to the itinerant hole pocket bands d_xz/yz above 
E_F, with zero momentum transfer q/π = 0 (Fig. <ref>, P_1 → P_3).
These peaks cannot be described using a simple weak-coupling framework. 






The orbital d_xy charge excitations in our calculations 
generate a response at = 1.4 eV of intensity ∼ 100 times
smaller than the orbital excitations (Figs. <ref>-<ref>). This is
because local charge fluctuations are suppressed 
significantly in the Mott orbital, namely the probabilities associated with a doubly occupied 
or empty site configuration are small compared to configurations where sites are half-filled.
Additionally, the features in a/b and L^z(q,) originate  
from itinerant carriers that are sensitive to the incident x-ray energy (ħω_in) of the RIXS 
experiments. It is known that localized excitations do not shift with ħω_in, 
while itinerant carriers produce a response that shifts linearly with ħω_in, 
becoming part of the fluorescence at large ħω_in <cit.>. 
Therefore, the z peak at the low-energy transfer  (Fig. <ref>, dashed) will shift with
the incident energy. The same is true for itinerant charge excitations a/b. 
In fact, RIXS experiment on BaFe_2Se_3 also find (fluorescence) 
peaks that shift with incident energy and 
merge with the localized excitations (A and B of Fig. <ref>), suggesting the existence 
of both localized and itinerant degrees of freedom at the experimental low temperature, as also found
for the OSMP regime in our calculations. 




To understand better the characteristics of the states 
at peak positions A and B, we also show  
real-space L_x/y orbital correlations vs energy transfer at fixed 
distances (1^st to 4^th nearest neighbor) in Fig. <ref>. 
In the block OSMP the high energy states at peak A have positive L_x/y correlations up to the 4^th nearest neighbor in real-space, while states at peak B have negative L_x/y correlations only up to the nearest neighbor. In Fig. <ref>, we explicitly show that the peak A of the block OSMP represents states with ferro-orbital ordering, 
while B represents states with short-range anti-ferro-orbital fluctuations. 
However, the ground state of block OSMP does not have an orbital ordering unlike the ground-state of the competing
ferromagnetic insulator at U/W = 4, which has uniform magnetic <cit.> and 
staggered L_z orbital order (Fig. <ref>). 
For the U/W = 4 FM case, the corresponding L_x/y is gapped and has excitations 
only at ∼ 1.0 (Fig. <ref>). In real-space, these high-energy states 
show short-range positive correlations, persisting only up to the 2^nd nearest neighbor (not shown). 















Conclusions —
We calculated the momentum-resolved orbital and charge dynamics 
of an orbital-selective Mott phase using the three t_2g orbitals, 
and compared our results to the available RIXS data for 
BaFe_2Se_3 at zero momentum transfer <cit.>. 
We find localized dd excitations that produce particular peaks A and B that are 
very similar 
to those observed experimentally, providing theoretical support that BaFe_2Se_3 is indeed an orbital-selective Mott insulator. 
Moreover, we predict the
q-resolved charge and orbital dynamical spectra that can be measured by RIXS in the future. 
Recent RIXS experiments display a very similar response 
for BaFe_2Se_3 and BaFe_2S_3, implying that an orbital-selective phase is present 
in the  BaFe_2S_3 high-pressure superconductor as well <cit.>. We encourage the measurement of the band structure and density-of-states of these materials using angle-resolved photoemission and scanning tunneling microscopy, to 
confirm our predicted band(s) crossing of the Fermi surface and Mott like band(s) below the 
Fermi surface. We also encourage future q-resolved RIXS measurements 
for BaFe_2Se_3 and BaFe_2S_3 compounds. 
Finally, the novel orbital fractionalization proposed in the FM insulator phase 
defines a new avenue of research that will be pursued soon.





1 cm
Methods
Model —  
We use a multi-orbital Hubbard model composed of kinetic energy 
and interaction terms as H = H_K + H_I. The kinetic energy part of the 
Hamiltonian is 



    H_K   = ∑_i,σ,γ,γ^' t^γγ^' 
    ( i γσi̧+̧1̧ ̧γ̧^̧'̧σ̧^ + H.c. ) + ∑_i,σ,γϵ_γ n_i σγ,



where i γσ (i̧ ̧γ̧σ̧) creates (destroys) 
an electron at site i, orbital γ, 
and spin σ. The first term represents nearest-neighbor electron hopping 
from orbital γ' to γ with a hopping amplitude 
t^γγ'. We denote the relevant 
orbitals d_xz, d_yz, and d_xy as a, b, and c, respectively. 
The second term contains the orbital-dependent crystal field splitting. 
The parameters are (eV units) ϵ_a =-0.1, ϵ_b=0.0,  ϵ_c=0.8, 
t^aa = t^bb = 0.5, t^cc = 0.15, and t^ac = t^bc = t^ca = t^cb = -0.10.
The non-interacting bandwidth is W = 4.9t^aa. 
This set of parameters is known <cit.> to produce bands that emulate 
iron-based superconductors, with hole pockets 
at q = 0 and an electron pocket at q=±π <cit.>. Our reported results are all at zero temperature.

The interaction term, in standard notation, is 



    H_I   = U ∑_i, γ n_i γ↑ n_i γ↓ 
    		+ (U^'- J_H/2) ∑_i, γ < γ' n_i γ n_i γ'
       - 2J_H∑_i, γ < γ'𝐒_i γ·𝐒_i γ' 
         + J_H∑_i, γ < γ' ( P^†_iγP^_iγ' + H.c.).



The first term is the intraorbital Hubbard repulsion U. 
The second is the interorbital repulsion between electrons at different orbitals, with U^' = U - 2J_H. 
The third term is the Hund's coupling J_H, 
and the last term represents the on-site interorbital hopping of electron pairs
(P_i γ' = c_i γ' ↑ c_i γ' ↓). 
Explicit definitions for all the operators in the model are in 
the supplementary material sections I and II.
The rich physical properties realized by this model were discussed extensively in
Ref. <cit.>. 
Most of the data presented here was gathered at J_H/U = 0.25 
(as used extensively before <cit.>) and
U/W = 0.8 where the block-type AFM OSMP is known to be stable <cit.>. 
However, we also show results for the 
paramagnetic metal (PM, small U) and ferromagnetic insulator (FM, large U) 
phases to highlight unique features of the block OSMP by contrast.

Observables — To characterize the OSMP, we calculate the 
dynamical response functions



    O(i,j,ω)    = -1/πIm[ ⟨ψ_0 | O^†_i1/ω - H + E_g + iη O_j | ψ_0 ⟩]



using DMRG within the correction-vector formulation in Krylov 
space <cit.>.
The single-particle photoemission spectral function is obtained using 
O_i=i̧ ̧σ̧γ̧. 
The intraorbital particle-hole (charge) excitations arise using 
O_i = ∑_σ n_i σγ -  n_i σγ, 
where we explicitly subtract the ground-state contribution to measure only the fluctuations. 
Finally, the orbital excitations are obtained using O_i = {L^x_i, L^y_i, L^z_i}, where 



    L^x_i = i∑_σ (iaσi̧çσ̧^ - icσi̧a̧σ̧^), 
    
    L^y_i = i∑_σ (ibσi̧çσ̧^ - icσi̧b̧σ̧^), 
    
    L^z_i = i∑_σ (iaσi̧b̧σ̧^ - ibσi̧a̧σ̧^).



These operators are derived from the L=2 angular momentum operators 
in the t_2g orbital basis, see supplemental material section II. 

















Code Availability — 
Computer codes used in this study are available at <https://g1257.github.io/dmrgPlusPlus/>. 





Data Availability — 
The data that support the findings of this study are available from the corresponding
author upon request. 








naturemag







Acknowledgments — 
We thank C. Monney and T. Schmitt for providing a copy of the
experimental data shown in Fig. <ref>.
N. D. P., A. N., A. M., and E. D. were supported by the 
U.S. Department of Energy (DOE),
Office of Science, Basic Energy Sciences (BES), Materials Science and
Engineering Division. G. A. and S. J. were supported by 
the Scientific Discovery through Advanced Computing (SciDAC) program funded by
the U.S. Department of Energy, Office of Science, Advanced Scientific Computing
Research and Basic Energy Sciences, Division of Materials Sciences and
Engineering. 
N. D. P. was also partially supported by the National
Science Foundation Grant No. DMR-1404375.
Part of this work was conducted at the
Center for Nanophase Materials Sciences, sponsored by the
Scientific User Facilities Division (SUFD), BES, DOE, under
contract with UT-Battelle. 
Computer time provided in part by resources supported
by the University of Tennessee and Oak Ridge National
Laboratory Joint Institute for Computational Sciences.



Author contributions — 
N. D. Patel and E. Dagotto planned the project. N. D. Patel performed all the DMRG calculations for multi-orbital Hubbard model. A. Nocera and G. Alvarez developed the DMRG++ computer program. E. Dagotto, A. Moreo and S. Johnston provided important insight into the understanding of orbital excitations, and comparison with RIXS experiment. 



Competing interests — 
The authors declare no competing interests. 
















Supplemental: Fingerprints of an Exotic Orbital-Selective Mott Phase 
 
in the Block Magnetic State of BaFe_2Se_3 Ladders























§ OPERATORS AND OBSERVABLES


In this section, we define all the operators that are used in the main text:




    n_iγσ   = c^†_i γσ c_i γσ,  
    
    n_iγ   = n_i γ + n_i γ, 
    
    S^κ_i γ   =1/2∑_σ,σ' c^†_i γσσ^κ_σσ' c_i γσ',  
    
    P_i γ   = c_i γ c_i γ,



where γ is the orbital index, σ and σ' are the spin index, and 
σ^κ are the Pauli matrices with κ={x,y,z} being Cartesian components.
The average occupation and charge fluctuations (Fig.1 of main text) are
defined as



    n_γ   = 1/L∑_i, σ n_iγσ, 
    δ N_γ^2    = 1/L∑_i n_i γ n_i γ -  n_i γ n_i γ,



where L is the number of sites in the lattice.
The static spin-spin correlations (inset of Fig. 1) is calculated using


    S(k) = 1/L^2∑_i,j e^-i k (i - j)⟨𝐒_i·𝐒_j⟩,


where 𝐒_i = ∑_γ𝐒_i γ.
In this supplemental, we also show results for n^γ_k defined as 


    c_j γ   = c_j γ + c_j γ, 
    
    n^γ(k)    = n^γ_k = 1/L^2∑_i,j e^-i k (i - j)⟨ c^†_i γ c_j γ⟩.








 §.§ Spectral functions and sum rules

To characterize the OSMP, the dynamical response functions (shown below) are calculated 
using DMRG 



    O(i,j,ω)    = -1/πIm[ ⟨ψ_0 | O^†_i1/ω - H + E_g + iη O_j | ψ_0 ⟩],



where the local operator O_i can represent any degree of freedom of the model. 
In general, these 
functions are Fourier transformed into the crystal momentum domain to calculate the 
momentum-energy resolved spectra that is relevant to experiments: 



    O(k,ω)    =  1/L^2∑_i,j e^-ik(i-j) O(i,j,ω).



Note that within DMRG, the site j is fixed to the center of the 
lattice (d=L/2-1) to reduce the edge effects and computational cost, and therefore the modified 
Fourier transform becomes  



    O(k,ω)    =  1/L∑_i e^-ik(i-d) O(i,d,ω).



Additionally, we use open boundary conditions in the DMRG simulation and therefore 
the quasi crystal-momenta are defined as 



    k = π n/L+1   where   n = 1,2 ...  L.







§ ORBITAL OPERATORS


The dominant orbitals of an iron atom in iron-based superconductors 
are the five 3d orbitals, 
corresponding to well-known L=2 orbital angular momenta. 
The corresponding operators {L_x,L_y,L_z} are written in the basis of L_z = {-2,-1,0,1,2}, forming 
5×5 matrices:



    L_x = 1/2[    0    2    0    0    0;    2    0 √(6)    0    0;    0 √(6)    0 √(6)    0;    0    0 √(6)    0    2;    0    0    0    2    0;      ]


    L_y = -i/2[     0     2     0     0     0;    -2     0  √(6)     0     0;     0 -√(6)     0  √(6)     0;     0     0 -√(6)     0     2;     0     0     0    -2     0;       ]


    L_z = 
    [  2  0  0  0  0;  0  1  0  0  0;  0  0  0  0  0;  0  0  0 -1  0;  0  0  0  0 -2;    ]



DMRG is a real-space algorithm, and therefore we must write 
these matrices in the orbital basis using the transformation



    |-2   = 1/√(2)( |x^2 - y^2 - i |xy), 
    
    |-1   = 1/√(2)( |xz - i |yz), 
    
    |0   = |z^2, 
    
    |1   = -1/√(2)( |xz + i |yz), 
    
    |2   = 1/√(2)( |x^2 - y^2 + i |xy),



where { |x^2 - y^2, |z^2, |xz, |yz, |xy} are the 
five iron 3d orbitals. The operators {L_x,L_y,L_z} represented in 
the orbital basis are:



    L_x = i/2[     0     0     0     1     0;     0     0     0  √(3)     0;     0     0     0     0     1;    -1 -√(3)     0     0     0;     0     0    -1     0     0;       ]



    L_y = i/2[     0     0    -1     0     0;     0     0  √(3)     0     0;     1 -√(3)     0     0     0;     0     0     0     0     1;     0     0     0    -1     0;       ]



    L_z = i[  0  0  0  0  2;  0  0  0  0  0;  0  0  0  1  0;  0  0 -1  0  0; -2  0  0  0  0;    ].



In the main text, we employ the iron t_2g orbitals, 
i.e., we drop the contribution from |x^2 - y^2, |z^2 
orbitals. After this approximation, we obtain the local orbital angular momentum 
operators used in the main text (site index understood)



    L^x   = i (xzx̧y̧^ - xyx̧z̧^), 
    
    L^y   = i (yzx̧y̧^ - xyy̧z̧^), 
    
    L^z   = i (xzy̧z̧^ - yzx̧z̧^).








§ ADDITIONAL RESULTS


In this section, we show tests performed to ensure the quality of the
presented results. We first show n^γ(k) that is in 
agreement with previous studies with a similar model <cit.>.
A significant change in n^a/b(k) indicates metallic behavior 
of these orbitals. On the contrary, n^c(k) shows little variation in 
k that is associated with a gapped orbital c. 







Furthermore, all spectral quantities defined by 
equations <ref> and <ref> must satisfy a sum rule. 
In general, it can be shown that integrating the spectral function over momentum k and energy  
gives a quantity related to a unique static observable. As an example, 
we use the single-particle spectral function A^c(k,) of orbital c below (above) the 
Fermi energy (E_F) representing the filled (unfilled) electron (hole) states. 
The electron and hole components of A^c(k,) are defined as 



    A_e^c(k,ω)    = -1/π L^2×
    ∑_i,jIm[    ⟨ψ_0 | c^†_ic1/ω - H + E_g + iη c_jc | ψ_0 ⟩] e^-ik(i-j), 
    
    
    A_h^c(k,ω)    = -1/π L^2×
    ∑_i,jIm[    ⟨ψ_0 | c_ic1/ω + H + E_g + iη c^†_jc | ψ_0 ⟩]  e^-ik(i-j),



where c is the orbital index and c_jc = c_jc + c_jc. 
To obtain the sum rule of this quantity, we first sum over the momentum k. 
This sum simply results in L δ_i,j, giving us the local response that is the
single-particle density of states,








    A_e^c(ω)    =  -1/π L∑_iIm[ ⟨ψ_0 | c^†_ic1/ω - H + E_g + iη c_ic | ψ_0 ⟩], 
    
    A_h^c(ω)    =  -1/π L∑_iIm[ ⟨ψ_0 | c_ic1/ω + H + E_g + iη c^†_ic | ψ_0 ⟩],



shown in Figure <ref>. Further integration over  of the imaginary part 
(Lorentzian poles) of the electron (hole) part simply gives the total electron (hole) 
density of orbital c:



    n_e^c    =  1/L∑_i⟨ψ_0 | c^†_i c_i | ψ_0 ⟩ , 
    
    n_h^c    =  1/L∑_i⟨ψ_0 | c_i c^†_i | ψ_0 ⟩ .



This is explicitly done within our calculations by integrating over the density of 
states (Fig. <ref>). The integration of the electron (blue curve) 
and hole (red curve) portions lead to approximately 1.0 that is consistent, within the accuracy
of our results, with calculations 
of the local density from the ground-state. Note that a singly occupied orbital is 
one of the characteristics of a Mott insulator. 


We also performed finite-size scaling on the density of states at the Fermi energy E_F ≃ 4.3 eV. 
Figure <ref> shows that the L →∞ extrapolated quasi-particle weight, 
A(=E_F), of orbital c is an order of magnitude smaller than the
almost degenerate orbitals a and b. The near zero weight of orbital c at the Fermi 
energy is consistent with a Mott phase, providing further evidence of the presence of an OSMP 
as the ground-state.










§ REPRODUCING DATA USING DMRG++

The full open source code, sample inputs, and corresponding
computational details can be found at
<https://g1257.github.io/papers/86/>.























