Effect of non-local correlations on the electronic structure of LiFeAs

TL;DR

This study demonstrates that non-local electronic correlations are crucial for accurately describing the spectral properties and Fermi surface of LiFeAs, highlighting the limitations of local approximations like DMFT.

Contribution

The paper introduces a multi-orbital TPSC approach to explicitly account for non-local correlations in LiFeAs, providing a more accurate theoretical description compared to local methods.

Findings

Non-local correlations significantly influence spectral functions and Fermi surface features.

The TPSC approach captures experimental ARPES and dHvA data better than DMFT.

Non-local correlations are essential for understanding iron-based superconductor properties.

Abstract

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 $A(\vec k,\omega)$, Fermi surface and scattering rates. These findings underline the importance of non-local correlations and benchmark different theoretical approaches for iron-based superconductors.

Figures

Equations

$$\Sigma =\frac{\delta \Phi }{\delta G},\Gamma$$

$$\begin{array}[]{rl}\chi^{0}_{\lambda\mu\nu\xi}(\vec{q},iq_{m})=\left[G_{\nu\lambda}^{0}\star G^{0}_{\mu\xi}\right](\vec{q},iq_{m})\end{array}$$

$$\begin{array}[]{rl}\chi^{\text{sp}}(\vec{q},iq_{m})&=[\mathbb{I}-\chi^{0}(\vec{q},iq_{m})U^{\text{sp}}]^{-1}2\chi^{0}(\vec{q},iq_{m})\\ \chi^{\text{ch}}(\vec{q},iq_{m})&=[\mathbb{I}+\chi^{0}(\vec{q},iq_{m})U^{\text{ch}}]^{-1}2\chi^{0}(\vec{q},iq_{m}),\end{array}$$

$$\begin{array}[]{rl}&\frac{T}{N_{\vec{q}}}\sum\limits_{\vec{q},m}\chi^{\text{sp}}_{\mu\nu\mu\nu}(\vec{q},iq_{m})=\langle n_{\mu\uparrow}\rangle+\langle n_{\nu\uparrow}\rangle-2\langle n_{\mu\uparrow}n_{\nu\downarrow}\rangle,\\ &\frac{T}{N_{\vec{q}}}\sum\limits_{\vec{q},m}\chi^{\text{sp}}_{\mu\mu\nu\nu}(\vec{q},iq_{m})\stackrel{{\scriptstyle\mu\neq\nu}}{{=}}2\langle n_{\mu\uparrow}n_{\nu\uparrow}\rangle-2\langle n_{\mu\uparrow}n_{\nu\downarrow}\rangle,\\ &\frac{T}{N_{\vec{q}}}\sum\limits_{\vec{q},m}\chi^{\text{ch}}_{\mu\mu\nu\nu}(\vec{q},iq_{m})=2\langle(n_{\mu\uparrow}+n_{\mu\downarrow})n_{\nu\uparrow}\rangle-n_{\mu}n_{\nu},\\ &\frac{T}{N_{\vec{q}}}\sum\limits_{\vec{q},m}\chi^{\text{ch}}_{\mu\nu\mu\nu}(\vec{q},iq_{m})\stackrel{{\scriptstyle\mu\neq\nu}}{{=}}\frac{n_{\mu}+n_{\nu}}{2}-\langle(4n_{\mu\uparrow}-2n_{\mu\downarrow})n_{\nu\uparrow}\rangle.\end{array}$$

$$\begin{array}[]{rl}U^{\text{sp}}_{\mu\mu\mu\mu}&=\frac{1}{2}\left(\frac{\langle n_{\mu\uparrow}n_{\mu\downarrow}\rangle}{\langle n_{\mu\uparrow}\rangle\langle n_{\mu\downarrow}\rangle}+\text{particle$\leftrightarrow$hole}\right)U_{\mu\mu}\\ U^{\text{sp}}_{\mu\nu\mu\nu}&=\frac{1}{2}\left[\frac{\langle n_{\mu\uparrow}n_{\nu\downarrow}\rangle}{\langle n_{\mu\uparrow}\rangle\langle n_{\nu\downarrow}\rangle}U_{\mu\nu}+\frac{\langle n_{\mu\uparrow}n_{\nu\uparrow}\rangle}{\langle n_{\mu\uparrow}\rangle\langle n_{\nu\uparrow}\rangle}(U_{\mu\nu}-J_{\mu\nu})\right.\\ &\left.~{}~{}~{}~{}~{}+\text{particle$\leftrightarrow$hole}\right]=U^{\text{sp}}_{\mu\mu\nu\nu}=U^{\text{sp}}_{\mu\nu\nu\mu}.\end{array}$$

$${\Sigma }_{\mu \nu }=$$

$$G(\vec{k},i{\omega }_n)=$$

$$Z_{\vec{k}}=\frac{1}{1-\frac{\partial {\Sigma }^{\prime \prime }(\vec{k},i{\omega }_n)}{\partial w_n}{|}_{i{\omega }_n\to 0^{+}}}$$

Full text

Introduction.- The nature of the electronic structure in iron-based superconductors has been intensively scrutinized since their discovery in 2008Kamihara et al. (2008); Rotter et al. (2008). While ab initio density functional theory (DFT) calculations can provide a qualitative understanding of their bandstructure and Fermi surfaceLebègue (2007); Singh and Du (2008); Mazin et al. (2008), 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 Cao et al. (2008); Si and Abrahams (2008); Qazilbash et al. (2009); Rullier-Albenque et al. (2009); Haule and Kotliar (2009); de’ Medici et al. (2009); Aichhorn et al. (2010); Hansmann et al. (2010); Yin et al. (2011); Ferber et al. (2012a); Werner et al. (2012); Ferber et al. (2012b); Evtushinsky et al. (2016); Bascones et al. (2016); Watson et al. (2017). 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.de’ Medici et al. (2009); Haule and Kotliar (2009); Aichhorn et al. (2010); Hansmann et al. (2010); Yin et al. (2011); Ferber et al. (2012a); Werner et al. (2012); Ferber et al. (2012b); Evtushinsky et al. (2016); Miao et al. (2016); Nourafkan et al. (2016); Watson et al. (2017) 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 $\Gamma$ and electron bands centered at the Brillouin zone edge $M$ (the so-called “blue/red shift”) in a large class of iron-based superconductorsOrtenzi et al. (2009); Borisenko et al. (2010); Lee et al. (2012); Brouet et al. (2016); Kushnirenko et al. (2017), or the recently reportedFink et al. (2019) 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 Borisenko et al. (2010); Wang et al. (2013); Miao et al. (2015); Altenfeld et al. (2018) as well.

Consideration of momentum dependence in the self-energy in real materials’ calculations are scarce but promising, Biermann et al. (2003); Tomczak et al. (2012, 2014); van Roekeghem et al. (2014); Biermann (2014); Li et al. (2016); Sémon et al. (2017) showing, for instance, effects of bandwidth widening and momentum-dependent bandshifts in the systems studied Tomczak et al. (2012); van Roekeghem et al. (2014); Li et al. (2016). 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 Y.M. Vilk and A.-M.S. Tremblay (1997), 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 $\Gamma$ is shifted below the Fermi energy $E_{F}$ as de Haas-van Alphen (dHvA) and ARPES Putzke et al. (2012); Brouet et al. (2016) measurements already suggested. Second, we find a large accumulation of incoherent spectral weight around the $\Gamma$ point as observed in ARPES Borisenko et al. (2010); Lee et al. (2012); Miao et al. (2015); Brouet et al. (2016); Fink et al. (2019). Third, the relative ”blue/red shift” of the bands centered at $\Gamma$ and $M$ respectivelyBorisenko et al. (2010); Lee et al. (2012), is properly described and, fourth, the momentum-averaged TPSC results agree with the results obtained from previous local DFT+DMFT calculations Ferber et al. (2012a); Lee et al. (2012) 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 structureMorozov et al. (2010) within the Generalized Gradient Approximation (GGA)Perdew et al. (1996) using the full-potential linear augmented plane-wave basis from WIEN2K Blaha et al. (2001), we derive an effective low-energy model comprizing the Fe $3d$ orbitals using maximally localized Wannier functions as implemented in Wannier90Mostofi et al. (2014) (see Supplemental Materialsup ). 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)Jiang et al. (2012) on the DFT bandstructure (see Supplemental Materialsup ).

The TPSC method considers the Luttinger-Ward functional $\Phi[G]$Martin and Schwinger (1959); Baym (1962), which is a functional of the interacting Green’s function $G$ and yields the self-energy $\Sigma$ and two-particle irreducible four-point vertex $\Gamma$ as functional derivatives

[TABLE]

Within the TPSC approach one approximates the vertex $\Gamma$ to be static and momentum independentY.M. Vilk and A.-M.S. Tremblay (1997) (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,Arya et al. (2015); Aizawa et al. (2015); Zantout et al. (2018); Mertz et al. (2018); Aizawa and Kuroki (2018); Nishiguchi et al. (2018) nearest-neighbor interaction,Davoudi and Tremblay (2006) and multi-orbitalMiyahara et al. (2013) generalizations of the Hubbard model and has provided valuable insights on the pseudogap physics in the cuprates Tremblay (2012) and unconventional superconductivity. Kyung et al. (2003); Ogura and Kuroki (2015); Zantout et al. (2018)

In the multi-orbital generalization of the TPSC method similar to the original formulationMiyahara et al. (2013) we first introduce the non-interacting susceptibility $\chi^{0}$ given by

[TABLE]

where $G^{0}$ denotes the non-interacting Green’s function in orbital-space, $\star$ denotes a convolution over frequency and momentum and $q_{m}=2m\pi T$ the $m$-th bosonic Matsubara frequency. The interacting susceptibility $\chi$ is decomposed into the spin and charge channel ( $\chi^{\text{sp}}$ and $\chi^{\text{ch}}$ respectively) and reads

[TABLE]

where the inversion of a 4-index tensor is given as the matrix inverse after combining the first and last two indices of $\lambda\mu\nu\xi$ into a superindex $(\lambda\mu)(\nu\xi)$.

We only consider the $U^{\text{ch/sp}}_{\alpha\alpha\beta\beta}$ and $U^{\text{ch/sp}}_{\alpha\beta\alpha\beta}=U^{\text{ch/sp}}_{\alpha\beta\beta\alpha}$ matrix elements of the renormalized irreducible vertices in the spin $U^{\text{sp}}$ and the charge channel $U^{\text{ch}}$ 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

[TABLE]

In order to solve this underdetermined set of equations we employ an ansatz for the spin vertex $U^{\text{sp}}$ that is motivated by the Kanamori-Brueckner screening Y.M. Vilk and A.-M.S. Tremblay (1997); Miyahara et al. (2013) and introduce an additional particle-hole symmetrization to keep all equations within TPSC particle-hole symmetric:

[TABLE]

The local spin vertex $U^{\text{sp}}$ can be obtained by iterating the equations above. For the charge channel we optimize $U^{\text{ch}}$ in order to fulfill the corresponding charge sum rules, restricting to positive values of $U^{\text{ch}}$ 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 sup ) the difference between left-hand side and right-hand side of the charge sume rules (Eq. (4)).

After the determination of the spin and charge vertices the self-energy $\Sigma$ and interacting Green’s function $G$ are then given as

[TABLE]

where the non-interacting vertices are zero except for the matrix elements: $U^{\text{sp/ch,0}}_{\mu\mu\mu\mu}=U_{\mu\mu}$, $U^{\text{ch,0}}_{\mu\mu\nu\nu}=2U_{\mu\nu}-J_{\mu\nu}$ and $U^{\text{sp/ch,0}}_{\mu\nu\mu\nu}=U^{\text{sp/ch,0}}_{\mu\nu\nu\mu}=U^{\text{sp,0}}_{\mu\mu\nu\nu}=J_{\mu\nu}$ with $\mu\neq\nu$. No Hartree term is included in $\Sigma$ since it is already contained in the DFT-derived Hamiltonian $H_{0}$.

Our multi-orbital extension of TPSC differs from previous formulations Miyahara et al. (2013) on the following aspects: it restricts the self-consistent equations in the charge channel in Eq. (3) to ensure positivity of the spectral weight and chooses a symmetrized ansatz for $U^{sp}$ (Eq. (5)). Furthermore, the set of local spin and charge sum rules (Eq. (4)) and bare vertex tensors $U^{ch,0},U^{sp,0}$ (Eq. (6)) and their dependence on $U,J$ are derived from the Bethe-Salpeter equation for the self-energy $\Sigma$ within TPSC Y.M. Vilk and A.-M.S. Tremblay (1997), which is different from the RPA derived expression of $U^{ch,0},U^{sp,0}$ only in the $ijij$-element, $i\neq j$.Miyahara et al. (2013)

Our calculations were performed at T=0.015 eV$\approx$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 sup ).

Results and discussion.- In Fig. 1 we show the TPSC spectral function $A(\vec{k},\omega)$ for LiFeAs along $\Gamma-X-M-\Gamma$ 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 $\Gamma$-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 measurementsBorisenko et al. (2010); Brouet et al. (2016) compared to the DFT bandstructure. Apart from orbital dependence the shrinkings are momentum-dependent. For example, along $\Gamma-X$ 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 $\Gamma$, composed of Fe $3d_{xz}/3d_{yz}$ orbital character (see Fig. 2 (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 ARPESBorisenko et al. (2010); Lee et al. (2012); Miao et al. (2015); Brouet et al. (2016); Fink et al. (2019). The maximum of the spectral function of the two inner hole pockets at $\Gamma$ 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 Brouet et al. (2016); S. Kushnirenko et al. (2018) as well as de Haas-van Alphen (dHvA) experiments.Putzke et al. (2012)

We can trace back these Fermi surface modifications to the value of the self-energy at the specific $\vec{k}$-points in the Brillouin zone: The largest contribution to the diagonal elements of the self-energy in Eq. (6) stems from $V_{abab}$, which is peaked at $\vec{k}=\{(\pm\pi,0),(0,\pm\pi)\}$. Following the argumentation of Ref. Ortenzi et al., 2009, 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. 2 we show the orbitally-resolved Fermi surface obtained from DFT (within GGA) (Fig. 2 (a)), DFT+TPSC (Fig. 2 (b)) and DFT+“local TPSC” where the momentum dependent TPSC self-energy $\Sigma(\vec{k},\omega)$ has been approximated by its local component $\frac{1}{N_{\vec{k}}}\sum_{\vec{k}}\Sigma(\vec{k},\omega)$(Fig. 2 (c)). The DFT Fermi surface reveals three well-defined distinct hole pockets centered at $\Gamma$ with circular to square shape and two electron pockets centered at $M$. As can already be deduced from the spectral function $A(\vec{k},\omega)$ in Fig. 1, 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 $\Gamma$ becoming incoherent and forming a flower-like shape, while the outer hole pocket of $3d_{xy}$ character stays coherent as confirmed in ARPES measurements Borisenko et al. (2010); Brouet et al. (2016). 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 $\Sigma(\vec{k},\omega)$ by its local component and compare the resulting Fermi surface to the full result in Fig. 2 (c). The so obtained Fermi surface indeed recovers the result obtained within published DFT+DMFT Ferber et al. (2012a); Lee et al. (2012), and, when considering the DFT+DMFT results for the same model as used in this work (see Supplemental Material sup ), the agreement is almost perfect (compare Fig. 2 (c) and (d)). DFT+DMFT calculations with a different double counting scheme Nourafkan et al. (2016) see a more pronounced -although coherent- flower-like shape of spectral weight around $\Gamma$ 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 $\Gamma$ 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 experimentsBrouet et al. (2016); Fink et al. (2019), 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 $-Z_{\vec{k}}\Sigma^{\prime\prime}(\vec{k},w)$ with

[TABLE]

in Fig. 3 at four different $\vec{k}$-points in momentum space following the $\Gamma-M$ 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. 2(a) and (b)). The energy dependence of the quasi-particle lifetimes for the $d_{xz/yz}$ electron and hole pockets (red symbols in Fig. 3) are in good agreement with the results of Ref. Brouet et al., 2016 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 sup ) similar to the measurements from Ref. Brouet et al., 2016. The quasi-particle lifetimes of the $d_{xy}$ hole and electron pockets (blue symbols in Fig. 3), in contrast, show at the considered $k$-points a quadratic increase in energy as in the ARPES measurements of Ref. Brouet et al., 2016, suggesting a Fermi-liquid-like behavior. Although our data was obtained at $T\approx 174\text{K}$ in contrast to the $T=25\text{K}$ in Ref. Brouet et al., 2016, we are confident that our results are still valid at low temperatures, since for example in Ba(Fe0.92Co0.08)2As2 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 $\vec{k}$-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. 2(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 $A(k,\omega)$ 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.