DFT+DMFT study on soft moment magnetism and covalent bonding in SrRu_2O_6
Atsushi Hariki, Andreas Hausoel, Giorgio Sangiovanni, Jan Kuneš
aa r X i v : . [ c ond - m a t . s t r- e l ] J un DFT+DMFT study on soft moment magnetism and covalent bonding in SrRu O Atsushi Hariki, Andreas Hausoel, Giorgio Sangiovanni, and Jan Kuneˇs Institute for Solid State Physics, TU Wien, 1040 Vienna, Austria Institute for Theoretical Physics and Astrophysics,University of W¨urzburg, Am Hubland 97074 W¨urzburg, Germany (Dated: September 10, 2018)Dynamical mean-field theory is used to study the three orbital model of the d compound SrRu O both with and without explicitly including the O- p states. Depending on the size of the Hund’scoupling J , at low to intermediate temperatures we find solutions corresponding to Mott or cor-related covalent insulator. The latter can explain the experimentally observed absence of Curiesusceptibility in the paramagnetic phase. At high temperatures a single phase with smoothly vary-ing properties is observed. SrRu O provides an ideal system to study the competition betweenthe local moment physics and covalent bonding since both effects are maximized, while spin-orbitinteraction is found to play a minor role. I. INTRODUCTION
The 4 d and 5 d materials with honeycomb lattice havebeen attracting considerable attention. Na IrO was pro-posed to provide a realization of a topological insula-tor [1] or the Kitaev model [2]. Mazin and collabora-tors pointed out molecular-orbital (MO) features in theelectronic structure of Na IrO [3], which can be alsofound in materials with similar crystal structure, e.g., α -RuCl , Li RuO or SrRu O . The properties of thesematerials result from competition between Hubbard in-teraction, Hund’s exchange, spin-orbit (SO) interaction,band or MO formation, the relative importance of whichdepends strongly on the d -shell filling.In this article, we use dynamical mean-field theory(DMFT) to investigate the d compound SrRu O , anantiferromagnetic insulator with an unusually high N´eeltemperature of 563 K [4] and a soft magnetic moment.A typical insulating magnet is a Mott insulator withrigid local moments that survive well above the tran-sition temperature, giving rise to Curie-Weiss suscepti-bility. Soft moment insulators are rare, nevertheless, anumber of rather different mechanisms exist. Heisenbergmodel build from anti-ferromagnetic dimers was stud-ied by some authors [5, 6] and several realizations werefound [7–9]. Excitonic condensation [10], proposed in d [11, 12] and d [13–15], may also lead to soft momentmagnetism. Yet another scenario is provided by corre-lated covalent insulators [16, 17]. FeSi is a well knownexample of a small gap insulator [18, 19], which devel-ops a local moment response at elevated temperature,although it does not order. The dimerized (bi-layer) Hub-bard model is the simple lattice model exhibiting com-petition between formation of covalent bonds leading toa hybridization gap, on one side, and formation of localmoments and Mott gap, on the other side [20].The density functional (DFT) calculations forSrRu O [21, 22] with a small gap show a tendency toanti-ferromagnetic ordering. The impossibility to stabi-lize a ferromagnetic solution, on the other hand, leadsthe authors of Ref. [22] to discard the Mott insulator local moment picture. This conclusion is supported bythe experimental observation of T -increasing susceptibil-ity above T N [4]. The DFT+DMFT approach was usedin Refs. [22, 23]. We comment on these results in sec-tion IV.The physics of SrRu O involves several competingphenomena: covalent bonding and tendency to formMOs, Hubbard electron-electron repulsion, Hund’s cou-pling and spin-orbit coupling. The authors of Ref. [23]further pointed out the possibility of an orbital selec-tive Mott state. Half-filling of the t g orbitals maxi-mizes Hund’s coupling effect and minimizes the inter-action strength necessary to get the Mott state [24, 25].The effect of spin-orbit coupling is also suppressed bythe half-filling [26]. On the other hand, at half fillingthe honeycomb structure is favorable for the formationof MOs. This is because for n = 4 k + 2, the chemical po-tential of an (non-interacting) n -site cyclic molecule with n electrons falls into the gap.The paper is organized as follows. After introducingthe computational method in Sec. II, we show the calcu-lated results in Sec. III. We discuss the results and per-form a simple model analysis in Sec. IV. Finally, Sec. Vsummarizes our results. II. COMPUTATIONAL METHOD
The DFT+DMFT method consists of two steps: (1)construction of an effective model from a converged DFTcalculation and (2) solution of the DMFT self-consistentequation [28]. In this study, we construct two effec-tive models: the d -only model (6 bands) describing Ru t g -like bands near the Fermi energy E F and the dp -model (24 bands) describing Ru t g bands + O 2 p bands.We perform DFT calculation in the experimental P m structure [4, 29], see Fig. 1, using the Wien2K package[30] with the generalized gradient approximation (GGA)[31]. The Wannier functions are constructed using theWIEN2WANNIER [32] and WANNIER90 [33] codes. Forthe d -only case, we construct an effective model includ-ing the spin-orbit interaction (SOI) as well as the one IG. 1. (Color online) The crystal structure of SrRu O viewed from two different directions (a) and (b). The gray, redand green circles represent Ru, O and Sr atoms, respectively.The crystal structure is visualized using VESTA3 [27]FIG. 2. (Color online) Band structure of SrRu O with theenergy windows used in the construction of the d -only modeland the dp model. without SOI, for comparison.The Anderson impurity model (AIM) with the inter-action H int between Ru t g electrons H int = U X γ n γ ↑ n γ ↓ + X γ>γ ′ ,σ [( U − J ) n γσ n γ ′ − σ + ( U − J ) n γσ n γ ′ σ ]+ ν J X γ = γ ′ ,σ ( d † γσ d † γ ′ − σ d γσ d γ ′ − σ − d † γ − σ d † γσ d γ ′ σ d γ ′ − σ ) , (1)is solved using the continuous-time Quantum MonteCarlo method (CT-QMC) in the hybridization expansionformalism [34, 35]. Here, γ and σ represent the t g or-bitals and spin, respectively, with d † γσ ( d γσ ) being thecorresponding creation (annihilation) operators and n γσ the number operators. The calculations for the Slater-Kanamori interaction ( ν = 1) with spin-rotational sym-metry were performed with the w2dynamics [36] code.Most of the presented results were obtained with thedensity-density approximation ( ν = 0) for which weused the computationally efficient segment implementa-tion with recent improvements [37, 38]. FIG. 3. (Color online) Spectral densities of a g and e gπ statesin the PM phase. (a) DFT and, (b) Mott and (c) Covalentinsulating phase in DFT+DMFT calculation at 500K with J = 0 .
16 eV. The energy origin is take as E F . The verticalbars in Fig. (a) represent energy levels of the MOs. In the trigonally distorted RuO octahedra the t g lev-els splits into the e gπ doublet and the a g singlet. The e gπ and a g crystal-field basis is adopted in the calcula-tion without the SOI. In the calculation with the SOI,basis diagonalizing the crystal field + SOI on each Ruatom is adopted.After self-consistency is achieved, physical quantities ofthe system are obtained from the AIM with the renormal-ized bath and the lattice with the converged self-energyΣ( iω n ). Besides the static one-particle observables suchas occupation number or ordered spin moments we calcu-lated the following quantities after the self-consistency isreached. The one-particle spectral densities are obtainedby analytic continuation of the self-energy with maxi-mum entropy method [39, 40]. The reduced (diagonal)density matrix providing the weights of atomic states ismeasured in the QMC simulation as well as the local spinsusceptibility χ loc ( T ) given by χ loc ( T ) = Z /T dτ h S z ( τ ) S z (0) i , (2)where S z ( τ ) is the spin operator at the imaginary time τ , T is the temperature. We also show the screen localmoment m scr ( T ) = p T χ loc ( T ). III. RESULTS
In Fig. 2 we show the DFT band structure with the en-ergy windows of the d -only and the dp models. The corre-sponding spectral density with a hybridization gap at theermi energy E F [21, 22] is shown in Fig. 3a. The widthof the e gπ and a g bands is about 2 . d -only model) are1.92 and 1.08. The MO levels [22] of an atomic hexagonwith nearest-neighbor hopping amplitude of about 400meV are clear in particular in the bonding part of thespectra. In the DMFT calculations, U is fixed to 2.7 eVand 5.3 eV for the d -only and dp model, respectively,the values are consistent with random-phase approxima-tion (cRPA) calculations and previous DMFT studies forRu compounds [41–44]. Hund’s exchange J is treatedas an adjustable parameter. In the dp model we use adouble-counting correction µ dc = ( N orb −
1) ¯ U ¯ n , where N orb is the number of interacting orbitals on a Ru site(6 in our case), ¯ U is the averaged Coulomb interaction,and ¯ n is the average self-consistent occupation per Ru d orbitals [45, 46]. A. Paramagnetic phase diagram
First, we present results for the system constrainedto the PM phase by symmetrization over spin in eachDMFT iteration. Fig. 4b shows the phase diagram inthe temperature T vs Hund’s value J plane for the d -onlymodel with the density-density interaction. Below about T = 1000 K, we obtained two distinct phases: the Mottinsulator (MI) and the covalent insulator (CI) [16, 17].The MI phase, realized for large J , and the CI phase,realized for small J , are separated by a coexistence regionwhere two stable solutions are found. The MI phase ischaracterized by presence of fluctuating local moments.The MI charge gap is opened due to a low-energy peak inthe self-energy. The local moments are absent in the CIphase. The CI charge (pseudo) gap originates from thehybridization. We quantify these features in the followingsection. The charge gap in MI phase of about 0.5 eV,Fig. 3b, is rather large compared to the experimentalgap of about 36 meV [41]. The pseudo gap is obtainedin the CI phase, Fig. 3c, which appears to match theexperiment better. We note here that the gap in the CIphase is almost independent of J . The coexistence regionis likely a non-generic feature. Its appearance is limitedto the d -only model with the density-density interactionwhile it is absent in the dp model with density-densityinteraction as well as in the d -only model with Slater-Kanamori interaction in the studied parameters.With Slater-Kanamori interaction, Fig. 4a, the hys-teretic transition between CI and MI is replaced by con-tinuous crossover. Except for the disappearance on theMI phase from most of the coexistence region the density-density and Slater-Kanamori interactions lead to similarresults. As expected for a d system [26], the inclusionof SOI has minor effect consisting in small shift (about0.025 eV) of the CI/MI crossover to larger J . J (eV) T ( K ) T ( K ) J (eV) (a)(b)
High T phase Mott insulator
Covalent insulator Coexistence regionCovalent insulator
Mott insulator High T phase
FIG. 4. (Color online) Phase diagram in the Hund’s J and temperature plane of the d -only model with (a) Slater-Kanamori interaction and (b) density-density interaction.The squares mark the points where actual calculations areperformed. Color plot is used for better visibility of differentphases. B. Charge and spin dynamics in the paramagneticphase
The existence of the coexistence regime allows us tocompare the characteristics of CI and MI states for thesame parameters. The atomic state weights in the MIand CI phases, characteristic for these phases in general,are shown in Fig. 5a,b. These are the diagonal elementsof the site-reduced density matrix, which measure the rel-ative time spent by the system in a given atomic state.The MI phase is dominated by the high-spin S = 3 / d sector. The CI phase exhibits largercharge fluctuations – more weight in the d and d sec-tors – and much more even population of the d states.These differences reflect the formation of atomic high-spin states in MI and MOs in CI. Unsurprisingly, the hightemperature weights are somewhere between the two.Next, we show the local susceptibility χ loc ( T ) obtainedwith the density-density interaction. First, χ loc ( T ) of the d -only model is shown in Fig. 6a. In the MI phase, the χ loc ( T ) shows the Curie 1 /T behavior with a local mo-ment m scr close to the atomic value of 3 µ B . At high " % $ & ’( " )* +** ,* -* .* /**0-*0,*0+*0)*0-*0,*0+*0)*0-*0,*0+*0)
12) 12+ 12, 12- 12.3*04563*0/-63*05-6 (a)(b)(c)
FIG. 5. (Color online) The weights of the atomic eigenstatesfor (a) Mott insulating phase and (b) covalent insulatingphase at 500K, and (c) high temperature (1000K). J = 0 . N = 3 block isshown in the parentheses. The two states with the highestweights correspond to S z = ± / (b)(d)(a)(c) ( d -only) ( d -only)( dp ) ( dp ) FIG. 6. (Color online) Local susceptibility χ loc ( T ) (left) andscreened magnetic moment m scr (right) for the d -only model(top) and the dp model (bottom). In Figs. (a) and (b), theresults for the Mott and the covalent solutions are shown to-gether in the coexistence region. (a)(b) FIG. 7. (Color online) Spin correlation function χ ( τ ) in (a)Mott insulating phase and (b) covalent insulating phase as afunction of τ /β , where β = 1 /k B T . The d -only model with J = 0 .
16 is employed in the calculation. temperatures, m scr is somewhat reduced due to admix-ture of other atomic states, see Fig. 6b. In the CI phase,on the other hand, the χ loc ( T ) is linearly increasing with T until it reaches the maximum around 800K and startsto decrease. The peak position in χ loc ( T ) is almost in-dependent of J , but correlates with the size of the (non-interacting) hybridization gap, as discussed in Sec. IV.The m scr in the CI phase does not have much of a phys-ical meaning.The behavior of χ loc is determined by the local spin-spin correlation χ loc ( τ ) = h S z ( τ ) S z (0) i , shown in Fig. 7for J = 0 .
16. The χ loc ( τ ) in the MI phase shows a typical τ -constant behavior. In the CI phase, the χ loc ( τ ) rapidlydecays from relatively large instantaneous values of h S z i ,reflecting the sizeable weights of the high-spin states inFig. 5b. This shows that h S z i itself is not enough to drawconclusions about the nature of magnetic response.The χ loc ( T ) of the dp model is shown in Fig. 6c. Withrescaled parameters it exhibits similar behavior as in the d -only model. Especially, the χ loc ( T ) for J = 0 .
40 eVgives a peak around 800 K similar to the one of the d -only model at J = 0 .
16 eV. In the dp model, the MIphase appears for J > .
45 eV. The low-
T m scr is onlyabout 2.3 µ B (Fig. 6d), reflecting the different Wannierbasis and explicit presence of the O- p states, shown inFig. 8. IG. 8. (Color online) The a g wannier function in the d -onlymodel (top) and dp model (bottom). The gray, red and greencircles represent Ru, O and Sr atoms, respectively. The wan-nier function in the d -only model has a considerable weighton neighboring O sites. C. Anti-ferromagnetic phase
When the paramagnetic constraint is lifted the sys-tem picks the G-type AF order at lower temperatures.Figs. 9a,c show the ordered (staggered) moment per atomin the AF phase. Both the d -only and dp calculationsoverestimate the N´eel temperature T N substantially. TheSlater-Kanamori interaction reduces T N by about 200 K,which still leaves a substantial overestimation. This is,however, not unexpected. The lack of non-local corre-lations [47] in our approach, particularly in a materialwith layered (quasi-2D) structure, can be blamed.The ordered moment also appears to be substantiallylarger than the experimentally reported value of 1.3 µ B .The two quantities are, however, not directly compara-ble. Unlike uniform magnetization which is unique, thestaggered moment corresponds to and depends on over-lapping Wannier orbitals [24]. In particular, the moredelocalized orbitals of the d -only model may be quitebad for comparison with experimental moments. There-fore we calculate the ordered moments in non-overlappingatomic spheres. The average spin magnetic moment inthe muffin-tin sphere ( R MT = 1 .
96 a.u.) at site l is givenby m l = X α,β,nγ,n ′ γ ′ σ αβ h d † nγα d n ′ γ ′ β i ρ lnγ,n ′ γ ′ ρ lnγ,n ′ γ ′ = h w n ′ γ ′ | P l MT | w nγ i , (3)where w nγ is the Wannier function (WF) on lattice site n carrying the orbital flavor γ and P l MT is the projec-tion operator on the muffin-tin sphere on site l . Selectedvalues of ρ lnγ,n ′ γ ′ are shown in Table I. The dominant (a) on WF ( d -only) (b) within MT ( d -only)(d) within MT ( dp )(c) on WF ( dp ) FIG. 9. (Color online) Temperature dependence of the mag-netic moment µ for the d -only model (top) and the dp model(bottom), respectively. The µ computed on wannier func-tions (WF) and within muffin-tin (MT) sphere are compared.The density-density interaction is employed in the calcula-tion. The inset in (a) shows a comparison of the (full) Slater-Kanamori and density-density calculations with J = 0 . contribution to the reduction of m l from its value perWannier orbital comes from the l = n = n ′ terms, i.e.the leakage of WFs from the muffin-tin spheres. Thenearest-neighbor terms with l = n = n ′ cause correctionof about 0.5%. The contribution from the overlap terms l = n = n ′ is negligibly small in the dp model and is pre-cisely zero in the d -only model. These results show thatDFT+DMFT yields ordered moments of the same size asthe LDA [21, 22] and that the size of the ordered momentcannot be used as a criterion to distinguish between theMott and covalent insulator scenarios. IV. DISCUSSION AND MODEL ANALYSIS
Previously, SrRu O has been studied with several ex-perimental and theoretical methods. The basic experi-mental features of SrRu O are insulating behavior, AForder and the lack of local moment response above T N .The DFT calculations [21, 22] found a small hybridiza-tion gap and tendency to the AF ordering. Moreover, the a) ρ nnγ,nγ ′ ❍❍❍❍❍ γ γ ’ a g e (1) gπ e (2) gπ a g e (1) gπ e (2) gπ ρ lnγ,nγ ′ ( n ∈ n.n of l in ab plane) ❍❍❍❍❍ γ γ ’ a g e (1) gπ e (2) gπ a g e (1) gπ -0.006 0.005 0.000 e (2) gπ ρ l onthe muffin-tin sphere l for the d -only model (see Eq (3)). fact that a ferromagnetic DFT solution cannot be stabi-lized [21, 22] indicates that the material is not a localmoment magnet. Streltsov et al. [22] performed a basicDFT+DMFT calculation and obtained a local momentMott insulator – consistent with our results for large J .Okamoto et al. [23] used DFT+DMFT and the dp model.Varying the double-counting correction they arrived atan orbitally selective Mott state, which according to theauthors provides the best description of SrRu O . Ourresults lead to the correlated covalent insulator scenario,which is qualitatively different. Given the available data,we do not think it is possible to decide which of the twoscenarios is more realistic. Since we did not find the se-lective Mott state in our calculations we conclude thatthe main difference between our model and Ref. 23 is theform of the interaction. While the present interactiondoes not distinguish between the t g orbitals, the authorsof Ref. 23 use about 5% stronger repulsion between the a g electrons than between the e gπ ones.The correlated insulator picture provides a natural ex-planation for the lack of local moment behavior. Theoverall shape of χ loc ( T ) in the 3-orbital is the same asin MO one-orbital model [16] and follows from the com-petition between formation of local moments on one sideand bonding/anti-bonding orbitals on the other. Themulti-orbital nature of the atoms and Hund’s coupling J favor the local moment formation. Nevertheless, nu-merical data show that the position of the χ loc ( T ) max-imum is rather insensitive to J . Model DMFT calcula-tions in Fig. 10 show that the maximum shifts when thehybridization gap is changed.While our results show that DMFT is able to capturethe competition between formation of hybridization gapand local moments, the question remains how accuratesuch a description is in systems consisting of molecular (c) (d) FIG. 10. (Color online) The spectral densities of (a) model Awith a tiny gap at E F and (b) model B with simple semiel-lipses. (c) χ loc ( T ) in model A and B. Model A gives a similar χ loc ( T ) of CI in realistic model (Fig. 6a,c), while model Bshows a Curie 1 /T behavior. (d) hybridization gap depen-dence: χ loc ( T ) in model A with a large hybridization gap(0.1eV). The peak is shifted about 400 K, as indicated by anarrow. Here, U = 2 . J = 0 .
14 eV is commonly employedin the DMFT calculation.FIG. 11. (Color online) The model of coupled hexagons fol-lowing Mazin et al. [3]. Each of the t g orbitals belongs toone of the three hexagons that meet at the Ru atom. Thehexagons are formed by the bonds of maximal hopping, inter-hexagon hopping is neglected. The hexagons are coupled byinter-orbital Coulomb interaction on each atom. building blocks such as dimers or hexagons. If the inter-layer hopping in bi-layer models is smaller or compara-ble to the intra-layer one [16, 17], the DMFT descriptionshould not be worse than it is for a single-layer system.However, when the inter-layer hopping substantially ex-ceeds the intra-layer one, DMFT with a ’dimer site’ is amore natural description. The ’atom’ and ’dimer’ DMFTmay lead to rather different results [48]. The discussedDMFT studies of SrRu O assume that the non-localorrelations within the Ru-hexagons can be neglected.An opposite extreme would be a model in Fig. 11 of iso-lated hexagons (neglecting the inter-hexagon hopping)carrying six electrons each, coupled by Coulomb interac-tion at the vertices. Comparing the magnetic propertiesof such model with the present DMFT may provide fur-ther insight in the physics of SrRu O and materials withquasi-molecular structure in general. V. CONCLUSION
Using DFT+DMFT approach we have studied 3-orbital d -only and dp models of SrRu O . Dependingon the value of the Hund’s exchange J we find at tem-peratures below approximately 800-1000 K either Mottinsulator with local moments (large J ) or a covalent in-sulator (small J ). Above 1000 K a single regime withproperties smoothly varying with J is observed. Com-paring to the experimental observations we conclude thatthe covalent insulator regime with T -increasing local sus- ceptibility is realized in SrRu O . We point out thatan alternative scenario based on orbitally selective Mottphysics was proposed in Ref. 23. We find strong ten-dency to anti-ferromagnetic order for all studied valuesof the Hund’s exchange J with T N substantially exceed-ing the experimental value, which we attribute to layeredcrystal structure and the mean-field nature of our theory.The role of possibly strong non-local correlation reflectingthe molecular orbital nature of the compound remains anopen question. ACKNOWLEDGMENTS
The authors thank V. Pokorn´y, A. Sotnikov and J.Fernandez Afonso for fruitful discussions. This workhas received funding from the European Research Coun-cil (ERC) under the European Union’s Horizon 2020research and innovation programme (grant agreementNo.646807-EXMAG). [1] A. Shitade, H. Katsura, J. Kuneˇs, X.-L. Qi, S.-C. Zhang,and N. Nagaosa, Phys. Rev. Lett. , 256403 (2009).[2] J. Chaloupka, G. Jackeli, and G. Khaliullin,Phys. Rev. Lett. , 027204 (2010).[3] I. I. Mazin, H. O. Jeschke, K. Foyevtsova, R. Valent´ı, andD. I. Khomskii, Phys. Rev. Lett. , 197201 (2012).[4] C. I. Hiley, D. O. Scanlon, A. A. Sokol, S. M. Woodley,A. M. Ganose, S. Sangiao, J. M. De Teresa, P. Manuel,D. D. Khalyavin, M. Walker, M. R. Lees, and R. I.Walton, Phys. Rev. B , 104413 (2015).[5] S. Sachdev and R. N. Bhatt,Phys. Rev. B , 9323 (1990).[6] T. Sommer, M. Vojta, and K. W. Becker,Eur. Phys. J. B , 329 (2001).[7] M. Matsumoto, B. Normand, T. M. Rice, and M. Sigrist,Phys. Rev. Lett. , 077203 (2002).[8] M. Kofu, J.-H. Kim, S. Ji, S.-H. Lee, H. Ueda,Y. Qiu, H.-J. Kang, M. A. Green, and Y. Ueda,Phys. Rev. Lett. , 037206 (2009).[9] H.-T. Wang, B. Xu, and Y. Wang,J. Phys. Condens. Matter , 4719 (2006).[10] J. Kuneˇs, J. Phys. Condens. Matter , 333201 (2015).[11] G. Khaliullin, Phys. Rev. Lett. , 197201 (2013).[12] A. Jain, M. Krautloher, G. H. Ryu, D. L. Chen, D. L.Abernathy, J. T. Park, A. Ivanov, J. Chaloupka, G. Khal-iullin, B. Keimer, and B. J. Kim, arXiv:1703.04926 .[13] J. Kuneˇs and P. Augustinsk´y,Phys. Rev. B , 235112 (2014).[14] A. Sotnikov and J. Kuneˇs, Sci. Rep. , 30510 (2016).[15] T. Yamaguchi, K. Sugimoto, and Y. Ohta,J. Phys. Soc. Jpn. , 043701 (2017).[16] J. Kuneˇs and V. I. Anisimov,Phys. Rev. B , 033109 (2008).[17] M. Sentef, J. Kuneˇs, P. Werner, and A. P. Kampf,Phys. Rev. B , 155116 (2009). [18] Z. Schlesinger, Z. Fisk, H.-T. Zhang, M. B. Maple, J. Di-Tusa, and G. Aeppli, Phys. Rev. Lett. , 1748 (1993).[19] K. Ishizaka, T. Kiss, T. Shimojima, T. Yokoya,T. Togashi, S. Watanabe, C. Q. Zhang, C. T.Chen, Y. Onose, Y. Tokura, and S. Shin,Phys. Rev. B , 233202 (2005).[20] C. Fu and S. Doniach, Phys. Rev. B , 17439 (1995).[21] D. J. Singh, Phys. Rev. B , 214420 (2015).[22] S. Streltsov, I. I. Mazin, and K. Foyevtsova,Phys. Rev. B , 134408 (2015).[23] S. Okamoto, M. Ochi, A. Arita, J. Yan, and N. Trivedi,arXiv:1703.04926 .[24] J. Mravlje, M. Aichhorn, and A. Georges,Phys. Rev. Lett. , 197202 (2012).[25] L. de’ Medici, J. Mravlje, and A. Georges,Phys. Rev. Lett. , 256401 (2011).[26] K.-H. Ahn, K. Pajskr, K.-W. Lee, and J. Kuneˇs,Phys. Rev. B , 064416 (2017).[27] K. Momma and F. Izumi,J. Appl. Crystallogr. , 1272 (2011).[28] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozen-berg, Rev. Mod. Phys. , 13 (1996).[29] C. I. Hiley, M. R. Lees, J. M. Fisher, D. Thompsett,S. Agrestini, R. I. Smith, and R. I. Walton,Angewandte Chemie International Edition , 4423 (2014).[30] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, andJ. Luitz, WIEN2k, An Augmented Plane Wave + Lo-cal Orbitals Program for Calculating Crystal Properties(Karlheinz Schwarz, Techn. Universitat Wien, Austria,2001), ISBN 3-9501031-1-2 .[31] J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett. , 3865 (1996).[32] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S.Lee, I. Souza, D. Vanderbilt, and N. Marzari,Comput. Phys. Commun. , 2309 (2014).33] J. Kuneˇs, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, andK. Held, Comput. Phys. Commun. , 1888 (2010).[34] P. Werner, A. Comanac, L. de’ Medici, M. Troyer, andA. J. Millis, Phys. Rev. Lett. , 076405 (2006).[35] E. Gull, A. J. Millis, A. I. Lichtenstein,A. N. Rubtsov, M. Troyer, and P. Werner,Rev. Mod. Phys. , 349 (2011).[36] N. Parragh, A. Toschi, K. Held, and G. Sangiovanni,Phys. Rev. B , 155158 (2012).[37] L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann,and O. Parcollet, Phys. Rev. B , 075145 (2011).[38] H. Hafermann, K. R. Patton, and P. Werner,Phys. Rev. B , 205106 (2012).[39] X. Wang, E. Gull, L. de’ Medici, M. Capone, and A. J.Millis, Phys. Rev. B , 045101 (2009).[40] M. Jarrell and J. Gubernatis,Phys. Rep. , 133 (1996). [41] W. Tian, C. Svoboda, M. Ochi, M. Matsuda,H. B. Cao, J.-G. Cheng, B. C. Sales, D. G.Mandrus, R. Arita, N. Trivedi, and J.-Q. Yan,Phys. Rev. B , 100404 (2015).[42] L. Vaugier, H. Jiang, and S. Biermann,Phys. Rev. B , 165105 (2012).[43] J. Mravlje, M. Aichhorn, T. Miyake,K. Haule, G. Kotliar, and A. Georges,Phys. Rev. Lett. , 096401 (2011).[44] G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini,Phys. Rev. Lett. , 106402 (2016).[45] V. Kˇr´apek, P. Nov´ak, J. Kuneˇs, D. Novoselov,D. M. Korotin, and V. I. Anisimov,Phys. Rev. B , 195104 (2012).[46] J. Kuneˇs, V. I. Anisimov, A. V. Lukoyanov, and D. Voll-hardt, Phys. Rev. B , 165115 (2007).[47] G. Rohringer, A. Toschi, A. Katanin, and K. Held,Phys. Rev. Lett. , 256402 (2011).[48] S. Biermann, A. Poteryaev, A. I. Lichtenstein, andA. Georges, Phys. Rev. Lett.94