Effect of non-local correlations on the electronic structure of LiFeAs
Karim Zantout Steffen Backes Roser Valentí

	
	
	
	zantout@itp.uni-frankfurt.de
	Institut für Theoretische Physik, Goethe-Universität 
		Frankfurt, 60438 Frankfurt am Main, Germany		
	
	CPHT, CNRS, Institut Polytechnique de Paris, F-91128 Palaiseau, France
	
	Institut für Theoretische Physik, Goethe-Universität Frankfurt, 60438 Frankfurt am Main, Germany
	

We investigate the role of non-local correlations in LiFeAs by
exploring an ab-initio-derived  multi-orbital Hubbard model 
for LiFeAs via the Two-Particle Self-Consistent (TPSC) approach. The multi-orbital formulation of TPSC 
approximates the irreducible interaction vertex to be an orbital-dependent constant, which is self-consistently
determined from local spin and charge sum rules. Within this approach, we disentangle the contribution of local and non-local correlations in LiFeAs and show that in the local approximation one recovers the dynamical-mean field theory (DMFT) result.
The comparison of our theoretical results to most recent angular-resolved photoemission spectroscopy (ARPES) and de-Haas van Alphen (dHvA)
data shows that non-local correlations in LiFeAs are decisive to describe the measured 
spectral function A(k⃗,ω), Fermi surface and scattering rates.
These findings underline the importance of non-local correlations and benchmark
different theoretical approaches for iron-based superconductors.

	
	Effect of non-local correlations on the electronic structure of LiFeAs
    Roser Valentí
    
======================================================================

Introduction.-
The nature of the electronic structure in 
iron-based superconductors has been intensively scrutinized since
their discovery in 2008<cit.>. 
While ab initio density functional theory (DFT)
calculations can provide a qualitative understanding of their bandstructure and Fermi surface<cit.>, 
it became soon evident that correlation effects originating from the strong local Coulomb
repulsion on the Fe atoms are responsible for many experimental findings  such as large effective masses, Fermi surface
renormalization, finite lifetimes or transfer of spectral weight to high binding energies
<cit.>. 
The combined DFT with Dynamical Mean Field Theory (DFT+DMFT) method, which approximates the electronic self-energy
to be local in space and thus includes frequency- and orbital-dependent local effects of electronic correlations, has been very
successful in capturing many of these observations.  Some examples are orbital-dependent correlations, incoherence properties and
Fermi surface renormalization.<cit.>
However, the single-site DMFT cannot account for possible 
momentum-dependent correlation effects such as relative band shifts in opposite directions of, respectively, 
hole bands centered at Γ and electron bands centered at the Brillouin 
zone edge M  (the so-called
“blue/red shift") in a large class of iron-based superconductors<cit.>, or 
the recently reported<cit.> possible momentum-dependent scattering rates in 
angular-resolved photoemission spectroscopy (ARPES) measurements of LiFeAs.
Some of these effects have been suggested to play an important role in the superconducting pairing mechanism <cit.> as well.

Consideration of momentum dependence in the self-energy in real materials' calculations are scarce
but promising, <cit.> showing, for instance,
effects of bandwidth widening and momentum-dependent bandshifts in the systems studied <cit.>.
Here we explore this dependence by considering an approach where spin fluctuations play the
dominant role and it allows both, a description of local and non-local correlations on an equal footing.


The purpose of this work is twofold: (i) We first introduce the multi-orbital formulation of the Two-Particle Self-Consistent
(TPSC) approach originally conceived for the single-orbital Hubbard model <cit.>, which provides
momentum- and frequency- dependent self-energies in the intermediate coupling regime. 
(ii) We apply the method to the iron-based superconductor LiFeAs.

We find that the momentum-dependence obtained within the TPSC approach introduces drastic 
changes to the LiFeAs Fermi surface and bandstructure with respect to DFT 
results.  First, the innermost hole pocket centered at Γ is shifted below
the Fermi energy E_F as de Haas-van Alphen (dHvA) and ARPES <cit.> 
measurements already suggested. Second, we find a large accumulation of 
incoherent spectral weight around the Γ point as observed in 
ARPES <cit.>. Third, the relative "blue/red shift" of the bands centered at 
Γ and M respectively<cit.>, is properly described and, fourth, the momentum-averaged TPSC results
agree with the results obtained from previous local DFT+DMFT calculations <cit.> pointing to an important relation 
between both approaches in this region of interactions.

Models and methods.- Starting from a DFT calculation of LiFeAs in the tetragonal crystal structure<cit.> 
within the Generalized Gradient Approximation (GGA)<cit.> using the full-potential linear augmented 
plane-wave basis from WIEN2K <cit.>, we derive an effective low-energy
model comprizing the Fe 3d orbitals using maximally localized Wannier functions
as implemented in Wannier90<cit.> (see Supplemental Material<cit.>). 
We effectively then solve a 2-dimensional system by restricting our calculation to the k_z=0 plane,
since the low-energy electronic structure shows only weak dispersion along k_z. 
In this Wannier-projected 2D model we have an electron occupation of 6.
Interaction parameters for the lattice Hubbard model were obtained within the
constrained random-phase approximation (cRPA)<cit.> on the DFT bandstructure (see 
Supplemental Material<cit.>).


The TPSC method considers the Luttinger-Ward functional Φ[G]<cit.>,
which is a functional of the interacting Green's function G
and yields the self-energy Σ and two-particle irreducible four-point vertex Γ as functional
derivatives

    Σ = δΦ/δ G,     Γ   = δ^2 Φ/δ G^2.

Within the TPSC approach one approximates the vertex Γ to be static and momentum independent<cit.> (but fully orbital dependent).
One obtains a set of self-consistent and conserving equations that satisfy the Pauli principle 
and Mermin-Wagner theorem. 
The range of validity of TPSC is the regime of 
weak to intermediate couplings where the local and static approximation of the vertex is valid, i.e. away from any phase transition.
This method has been extended to 
multi-site,<cit.> nearest-neighbor interaction,<cit.> and 
multi-orbital<cit.> generalizations of the Hubbard model and has 
provided valuable insights on the pseudogap physics in the 
cuprates <cit.> and unconventional 
superconductivity. <cit.>

In the multi-orbital generalization of the TPSC method 
similar to the original formulation<cit.>
we first introduce the non-interacting susceptibility χ^0
given by 

    [ χ^0_λμνξ(q⃗, iq_m) = [G_νλ^0⋆ G^0_μξ](q⃗,iq_m) ]

where G^0 denotes the non-interacting Green's function in orbital-space, ⋆ denotes a convolution over frequency
and momentum and q_m = 2mπ T the m-th bosonic Matsubara frequency.
The interacting susceptibility χ is decomposed into the spin and charge channel ( χ^sp and χ^ch respectively) and reads

    [                          χ^sp(q⃗, iq_m)  = [𝕀-χ^0(q⃗,iq_m)U^sp]^-12χ^0(q⃗,iq_m);                          χ^ch(q⃗, iq_m) = [𝕀+χ^0(q⃗,iq_m)U^ch]^-12χ^0(q⃗,iq_m), ]

where the inversion of a 4-index tensor is given as the matrix inverse
after combining the first and last two indices of λμνξ 
into a superindex (λμ)(νξ).

We only consider the U^ch/sp_ααββ 
and U^ch/sp_αβαβ=U^ch/sp_αββα
matrix elements of the renormalized irreducible vertices in the spin U^sp and 
the charge channel U^ch to be nonzero, corresponding to the atomic symmetry of 3d orbitals.
Those elements are determined by enforcing the following local spin and charge sum rules

    [                                                                     T/N_q⃗∑_q⃗,mχ^sp_μνμν(q⃗, iq_m) = ⟨ n_μ↑⟩+⟨ n_ν↑⟩ - 2⟨ n_μ↑n_ν↓⟩,;                                                                     T/N_q⃗∑_q⃗,mχ^sp_μμνν(q⃗, iq_m) μ≠ν= 2⟨ n_μ↑n_ν↑⟩ - 2⟨ n_μ↑n_ν↓⟩,;                                                                      T/N_q⃗∑_q⃗,mχ^ch_μμνν(q⃗, iq_m) = 2⟨ (n_μ↑+n_μ↓)n_ν↑⟩ - n_μ n_ν,;                                                                    T/N_q⃗∑_q⃗,mχ^ch_μνμν(q⃗,iq_m) μ≠ν=n_μ+n_ν/2-⟨ (4n_μ↑-2n_μ↓)n_ν↑⟩. ]

In order to solve this underdetermined set of equations we employ an ansatz for the spin 
vertex U^sp that is motivated by the Kanamori-Brueckner 
screening <cit.> and introduce an additional particle-hole symmetrization to keep all equations within TPSC particle-hole symmetric: 

    [                                                                     U^sp_μμμμ                         = 1/2(⟨ n_μ↑n_μ↓⟩/⟨ n_μ↑⟩⟨ n_μ↓⟩ + particle↔hole)U_μμ;                                                                     U^sp_μνμν = 1/2[⟨ n_μ↑n_ν↓⟩/⟨ n_μ↑⟩⟨ n_ν↓⟩U_μν + ⟨ n_μ↑n_ν↑⟩/⟨ n_μ↑⟩⟨ n_ν↑⟩(U_μν-J_μν).;                                                                                                                 .     + particle↔hole]=U^sp_μμνν=U^sp_μννμ. ]

The local spin vertex U^sp can be obtained by iterating the 
equations above.
For the charge channel we optimize U^ch in order to fulfill
the corresponding charge sum rules, restricting to positive values of U^ch
because in certain cases negative values can lead to non-causal self-energies.
Due to the constraint we search for values of U^ch that minimize (see Supplemental Material <cit.>) the difference between left-hand side and
right-hand side of the charge sume rules (Eq. (<ref>)).


After the determination of the spin and charge vertices the self-energy Σ and interacting Green's function G are then
given as

    Σ_μν =   1/4∑_α,β[U^spχ^spU^sp,0+U^chχ^chU^ch,0]_ναμβ_:=V_ναμβ⋆ G^0_βα
    
    	G(k⃗, iω_n) =   [(iω_n + μ)𝕀 - H_0(k⃗) -Σ(k⃗, iω_n)]^-1,

where the non-interacting vertices are zero except for the matrix elements: 
U^sp/ch,0_μμμμ=U_μμ, U^ch,0_μμνν=2U_μν-J_μν and U^sp/ch,0_μνμν=U^sp/ch,0_μννμ=U^sp,0_μμνν=J_μν with μ≠ν. 
No Hartree term is included in Σ since it is already contained 
in the DFT-derived Hamiltonian H_0.

Our multi-orbital extension of TPSC differs from previous formulations <cit.> on the following aspects:
it restricts the self-consistent equations  in the charge channel in Eq. (<ref>) to ensure positivity of the spectral weight and chooses a symmetrized ansatz for U^sp (Eq. (<ref>)).
Furthermore, the set of local spin and charge sum rules (Eq. (<ref>)) 
and bare vertex tensors U^ch,0, U^sp,0 (Eq. (<ref>)) and their dependence on U, J are derived from the Bethe-Salpeter equation for the self-energy Σ within TPSC <cit.>, which is different from the RPA derived expression of U^ch,0, U^sp,0 only in the ijij-element, i≠ j.<cit.>

Our calculations were performed at T=0.015 eV≈174 K since this is the 
lowest accessible temperature before spin fluctuations get too strong and the 
TPSC approximation is not justified anymore. Nevertheless, we checked that the 
results presented below do not change in their trends up to room 
temperature (see Supplemental Material <cit.>).


Results and discussion.-
In Fig. <ref> we show the TPSC spectral function A(k⃗, 
ω) for LiFeAs along Γ-X-M-Γ in the two-iron Brillouin 
zone. To emphasize the changes in the electronic
structure beyond an overall bandwidth renormalization of about a factor of 2,
we also plot the renormalized DFT bandstructure on top. 
We observe that the electronic correlations introduce a downshift
of the hole states around the Fermi level at the Γ-point, while
the electron states at M are slightly shifted up in energy. The inner electron pocket being shifted
by -0.1eV on average while the outer electron pocket is shifted by only -0.01eV.
This leads to an overall shrinking of hole and electron pockets,
corresponding to the “blue/red shift” seen in ARPES measurements<cit.>
compared to the DFT bandstructure.
Apart from orbital dependence the shrinkings are momentum-dependent.
For example, along Γ-X the middle hole pocket
shrinks to approximately 20% of its size compared to the renormalized DFT bandstructure while all other
parts of the Fermi surface shrink to 80-90% of their original size.
The inner hole pocket at Γ, composed of Fe  3d_xz/3d_yz orbital character (see Fig. <ref> (a)),
becomes very diffuse at the Fermi level due to incoherent scattering processes,
leading to a significant reduction of the lifetime of quasi-particle excitations.
This manifests in a broad Fermi surface feature very similar to the one observed in
ARPES<cit.>.
The maximum of the spectral function of the two inner hole pockets at Γ
is shifted basically on top of the Fermi level but retains significant
spectral weight at higher and lower binding energies - the shift being on average 0.18eV for both while for the outer hole pocket it is 0.1eV.
We expect that the inclusion of spin-orbit coupling, which is beyond our current approach,
will split this feature, effectively pushing one hole band below and the other 
above the Fermi level, giving rise to only one central hole Fermi surface pocket,
which would be in very good agreement with previous
ARPES data <cit.> as well as de Haas-van 
Alphen (dHvA) experiments.<cit.>


We can trace back these Fermi surface modifications
to the value of the self-energy at the specific k⃗-points in the Brillouin zone:
The largest contribution to the diagonal elements of 
the self-energy in Eq. (<ref>) stems from V_abab,
which is peaked at  k⃗={ (±π,0) , (0,±π) }. Following 
the argumentation of Ref. Ortenzi2009, this leads to 
a negative (positive) real part of the self-energy in the vicinity of the hole (electron)
pockets and thus to the observed “blue/red shift” and therefore it is
a consequence of non-local 
spin fluctuations.




In Fig. <ref> we show the orbitally-resolved Fermi surface obtained from DFT (within GGA) 
(Fig. <ref> (a)), DFT+TPSC 
(Fig. <ref> (b)) and DFT+“local TPSC” where the 
momentum dependent TPSC self-energy Σ(k⃗,ω) has 
been approximated by its local component 
1/N_k⃗∑_k⃗Σ(k⃗,ω)(Fig. <ref> (c)). 
The DFT Fermi surface reveals 
three well-defined distinct hole pockets centered at Γ  with circular to square shape and two 
electron pockets centered at M. As can already be deduced from the spectral 
function A(k⃗,ω) in Fig. <ref>, the Fermi 
surface experiences appreciable changes due to the TPSC self-energy 
contributions. All pockets are reduced in size,
with the remaining spectral weight of the two center hole pockets
of Fe 3d_xz/3d_yz character at 
Γ becoming incoherent and forming a 
flower-like shape, while the outer hole pocket of 3d_xy character stays coherent as confirmed in ARPES measurements <cit.>.
The electron pockets at M shrink slightly and broaden, since they are mostly composed
of the most incoherent 3d_xz/3d_yz as well.

The observed shrinking of the hole and electron pockets deviates significantly
from published DFT+DMFT results,
most likely due to the inclusion of non-local correlations in the TPSC approach
which go beyond the DMFT approximation where the self-energy is purely local.
In order to confirm this assumption we separate the local 
from the non-local correlation effects by employing 
a DMFT-like approximation on the TPSC self-energy. We approximate the full
momentum-dependent TPSC self-energy Σ(k⃗, ω) 
by its local component and compare the resulting Fermi surface to
the full result in Fig. <ref> (c). 
The so obtained Fermi surface indeed recovers the result obtained within published DFT+DMFT <cit.>, and, when considering the DFT+DMFT results for the same model as used in this work (see Supplemental Material <cit.>), the agreement is almost perfect 
(compare Fig. <ref> (c) and (d)). DFT+DMFT calculations with a different double counting scheme <cit.> see a more pronounced -although coherent- flower-like shape of spectral weight around Γ but don't account for the “blue/red shift”.
This shows that when taking into account non-local fluctuations,
the local Coulomb interaction gives rise to a significant momentum-dependent self-energy
and can account for the experimentally observed “blue/red shift”.
Interestingly, within the local approximation (local TPSC) the center hole pockets at Γ become again coherent, 
which is also in correspondence with the DMFT result. 
This shows that the quasi-particle scattering rate itself is strongly
momentum and orbital dependent, which has in fact been observed in
recent ARPES experiments<cit.>,
where the inner 3d_xz/3d_yz derived hole Fermi surface have been found to be incoherent
while the outer 3d_xy hole pocket shows Fermi liquid behavior.




Since Fermi-liquid theory predicts a quadratic energy dependence of quasi-particles' 
lifetimes near the Fermi energy, deviations from this energy dependence  
are also a signature for non-Fermi liquid behavior. It is 
therefore compelling to analyze the energy dependencies of the scattering rate within the
TPSC approach. 

For this, we present the quasi-particle lifetime -Z_k⃗Σ”(k⃗,w) with

    Z_k⃗=1/1-∂Σ”(k⃗,iω_n)/∂ w_n|_i ω_n→0^+

in Fig. <ref> at four different k⃗-points in momentum space following 
the  Γ-M path. Along this path the dominating contributions  are (1) d_yz
hole pocket, (2) d_xy hole pocket, (3) d_xy electron pocket and (4) 
d_xz electron pocket (see Fig. <ref>(a) and 
(b)).  The energy dependence of the quasi-particle lifetimes  for the 
d_xz/yz electron and hole pockets (red symbols in Fig. <ref>) 
are in good agreement with the results of Ref. Brouet2016 with 
values between 0.025eV and 0.035eV. The energy 
dependence shows a very shallow linear behavior (fitted slopes are of the order of 10^-3, see Supplemental Material <cit.>) similar to the measurements from Ref. Brouet2016. The  quasi-particle lifetimes of the 
d_xy hole and electron pockets (blue symbols in Fig. <ref>), in 
contrast, show at the considered k-points a quadratic increase in energy as in 
the ARPES measurements of Ref. Brouet2016, suggesting a 
Fermi-liquid-like behavior. Although our data was obtained at
T≈ 174K in contrast to the T=25K in
Ref. Brouet2016, we are confident that our results are still valid at low temperatures, 
since for example in
Ba(Fe_0.92Co_0.08)_2As_2 it has been found that the quasi-particle lifetimes for the hole 
d_xz/yz orbitals showed weak temperature dependence. We also checked how these 
results depend on the k⃗-path and found that small translations along the tip of electron pocket (3) reveal a linear dependence of the quasi-particle lifetime as can already be expected since the quasi-particle 
weight gets incoherent away from the point (3) (see 
Fig. <ref>(b)).


Summary.- In conclusion, we presented a multi-orbital TPSC scheme that 
respects local spin and charge sum rules. This method includes effects
of local and non-local correlations on an equal footing 
within the validity of the local approximation of the 
irreducible 4-point vertex and thus yields momentum- and 
frequency-dependent self-energies. 
We applied this method to the multi-orbital iron-based superconductor LiFeAs
and found that the non-local components of the self-energy are decisive to 
explain its experimentally observed spectral function A(k,ω) and Fermi surface.
Taking into account non-local correlations we observe 
a “blue/red shift" of the electronic structure, where the hole bands 
at the Brillouin zone center are lowered in energy, while the electron bands 
in the corner of the Brillouin zone are slightly shifted upwards, resulting
in an overall reduction of the size of the Fermi surface pockets.
Overall we find very good agreement with ARPES and dHvA experiments,
where the “blue/red shift" was first observed.
We could show that our TPSC approach within a local approximation to the self-energy
recovers the DFT+DMFT result which does not exhibit 
the “blue/red shift", both benchmarking 
 the TPSC result and showing the importance of going beyond the local
picture of DMFT in order to understand the electronic structure of
iron-based superconductors.
Furthermore, we also found a strong momentum and non-quadratic energy dependence
of the electronic scattering rate, in good agreement with recent ARPES measurements.



This work was supported by the German Research Foundation (Deutsche 
Forschungsgemeinschaft) under grant SFB/TR 49. We would like to thank S. 
Bhattacharyya, S. Borisenko, V. Borisov, J. Fink, P.J. Hirschfeld, G. Kotliar and Y. Li for discussions.
R. V. acknowledges the KITP for hospitality
where aspects of this work were discussed.  The research
at KITP was supported in part by NSF Grant No. NSF
PHY-1748958.


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