Seeing the orbital ordering in Iron-based superconductors with magnetic anisotropy
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Seeing the orbital ordering in Iron-based superconductors with magnetic anisotropy
Yuehua Su and Tao Li Department of Physics, Yantai University, Yantai 264005, P.R.China Department of Physics, Renmin University of China, Beijing 100872, P.R.China (Dated: September 7, 2018)The orbital fluctuation of the conduction electrons in the Iron-based superconductors is found tocontribute significantly to the magnetic response of the system. With the use of a realistic five-bandmodel and group theoretical analysis, we have determined the orbital magnetic susceptibility insuch a multi-orbital system. At n = 6 .
1, the in-plane orbital magnetic susceptibility is predictedto be about 10 µ / eV, which is more than 2 / µ / eV or 4 . × − erg / G mol AS22 ). We find the in-plane orbitalmagnetic response is sensitive to the breaking of the tetragonal symmetry in the orbital space. Inparticular, when the observed band splitting(between the 3 d xz and the 3 d yz -dominated band) isused to estimate the strength of the symmetry breaking perturbation , a 4.5% modulation in thein-plane orbital magnetic susceptibility can be produced, making the latter a useful probe of theorbital ordering in such a multi-orbital system. As a by product, the theory also explains the largeanisotropy between the in-plane and the out-of-plane magnetic response observed universally insusceptibility and NMR measurements. PACS numbers:
An unresolved issue in the study of the Iron-based su-perconductors is the role of their multi-orbital nature. Inmost other superconductors, the orbital degree of free-dom is quenched at low energy in the crystal field envi-ronment. However, both LDA calculation and ARPESmeasurement indicate that in the Iron-based super-conductors all the five Fe 3 d orbital play essential rolein forming the low energy degree of freedom around theFermi surface. Many novel properties of the Iron-basedsuperconductors, especially those in the name of elec-tronic nematicity , have been argued to be related tothe orbital ordering in these systems . Most recently,a two-fold modulation of the magnetic susceptibility inthe Fe-Fe plane is found to develop around a tempera-ture that is significantly higher than the structural phasetransition point . However, it is still a mystery how theobserved electronic nematicity is related to the orbitalordering of the system.Another puzzle about the Iron-based superconduc-tors is the strong anisotropy in their magnetic responseobserved universally in susceptibility and Knight shiftmeasurements . The susceptibility in the Fe-Fe planeis found to be significantly larger than that perpendicularto it. This is very unusual, since the magnetic responseof a transition metal is usually attributed to the spin ofits valence electron and is essentially isotropic. The or-bital magnetic response, on the other hand, is usuallyquenched as a result of the crystal field effect. However,since the crystal field splitting in the Iron-based super-conductors is very small and all the five 3 d orbital areinvolved in the low energy physics , the orbital angu-lar momentum of the conduction electron can contributeto the magnetic response of these systems. Such a con-tribution is intrinsically anisotropic and depends on theelectronic structure of the system, especially on the sym-metry breaking in the orbital space.The purpose of this paper is to evaluate orbital mag- netic response of the Iron-based superconductors froma realistic model and to explore the relation betweenorbital ordering and the electronic nematicity observedin recent torque magnetometry measurement . We findthe orbital magnetic susceptibility in these multi-orbitalsystems is comparable in magnitude with the measuredtotal magnetic susceptibility. More specifically, the in-plane orbital magnetic susceptibility is predicted to beabout 10 µ / eV, which accounts for more than 2 / .Furthermore, the in-plane orbital magnetic response isfound to be sensitive to the breaking of the tetragonalsymmetry in the orbital space, making it a useful probeof orbital ordering in these multi-orbital systems. As a byproduct, the observed strong anisotropy between the in-plane and out-of-plane magnetic susceptibility also finda natural explanation from our calculation.The Iron-based superconductors have a very compli-cated band structure. In this study, we adopt the five-band tight-binding model derived from fitting the LDAband structure of the LaFeAsO system. Following thenotations of Ref.20, the band model reads, H kin = X i,j X ν,ν ′ ,σ [ t ν,ν ′ i,j c † i,ν,σ c j,ν ′ ,σ + h.c. ] + X i,ν,σ ε ν n i,ν,σ , (1)where ν, ν ′ = 1 , .., | i = | d Z − R i , | i = | d XZ i , | i = | d Y Z i , | i = | d X − Y i and | i = | d XY i . t ν,ν ′ i,j denotes thehopping integral between the ν -th and ν ′ -th orbital atsite i and site j . Here an unfolded scheme is adoptedas in Ref.20. The X and Y -axis for the Wannier func-tions, which are in the Fe-As bond direction, are rotatedby 45 degree from the x and y -axis of the Fe-Fe squarelattice(see Fig.1). ε ν is the on-site energy of the ν -thorbital. The hopping integral is truncated at the fifthneighbor and the values of the model parameters can befound in Ref.20. XY x y H FIG. 1: The square lattice of the Fe ions (shown as graydots) and the local coordinate system for the atomic orbital.The red and blue dots denote the As ions above and belowthe Fe-Fe plane. φ is the angle between the X -axis and thedirection in which magnetic susceptibility is measured. Inthe tetragonal phase, the point group symmetry around theFe ion is D d , which is broken down to D in the orthogonalphase. The interaction of electron has the following generalform H int = U X i,ν,σ n i,ν,σ n i,ν,σ + ( U ′ − J ) X i,ν ′ = ν,σ n i,ν,σ n i,ν ′ ,σ + U ′ X i,ν ′ = ν,σ n i,ν,σ n i,ν ′ ,σ − J X i,ν ′ = ν,σ c † i,ν,σ c i,ν,σ c † i,ν ′ ,σ c i,ν ′ ,σ − J X i,ν ′ = ν,σ c † i,ν,σ c † i,ν,σ c i,ν ′ ,σ c i,ν ′ ,σ . (2)Here we have included the intra- and inter-orbitalCoulomb repulsion, the Hund’s rule coupling and thepair hopping term and have assumed that U ′ = U − J . n i = P ν,σ n i,ν,σ = P ν,σ c † i,ν,σ c i,ν,σ is the number densityoperator of the electron and σ = − σ .Since the five Fe 3 d orbital | ν i are all real functions,they can not carry current and thus their orbital angularmomentum are quenched in the static limit. However,since the orbital content varies on the Fermi surface,fluctuation in the orbital character and orbital angularmomentum survives in the low energy limit and can con-tribute to the magnetic response of the system. In thefollowing we will calculate such a magnetic response inthe RPA scheme.The orbital magnetic susceptibility is defined throughthe correlation function of the orbital magnetic moment in the following way χ αL ( q , τ ) = −h T τ ˆ L α ( q , τ ) ˆ L α ( − q , i , (3)in which ˆ L α ( q , τ ) denotes the Fourier component of theorbital magnetic moment density in the α direction and α = X, Y, Z . Here we use µ B as the unit of susceptibility.The operator for the orbital magnetic moment on a givensite is defined as ˆ L α = P ν,ν ′ ,σ c † ν,σ l αν,ν ′ c ν ′ ,σ , where l αν,ν ′ is the matrix element of the orbital magnetic moment inthe basis spanned by the five MLWFs.The matrix element l αν,ν ′ can be determined in prin-ciple from a first principle calculation. Here we will besatisfied with the result of a semi-quantitative analysis,for which much simplification can be achieved when sym-metry arguments are adopted. In the following, we willillustrate the steps for l Zν,ν ′ . First, since ˆ L Z is time re-versal odd and the five 3 d orbital are all real, l Zν,ν ′ mustbe purely imaginary. Second, since ˆ L Z is odd under theaction of the three generators of the D d point grouparound each Fe ion, namely R x ( π ), σ X and σ Y , whilethe five 3 d orbital transform as R x ( π ) : | i| i| i| i| i → | i−| i−| i−| i| i σ X : | i| i| i| i| i → | i−| i| i| i−| i and σ Y : | i| i| i| i| i → | i| i−| i| i−| i , the only none-zero matrix elements are l Z , = − l Z , and l Z , = − l Z , . Thus ˆ L Z can be generally written asˆ L Z = i X σ [ γ c † ,σ c ,σ + γ c † ,σ c ,σ ] + h.c., in which γ and γ are two real numbers. Following thesame line of reasoning one find thatˆ L X = i X σ [ γ c † ,σ c ,σ + γ c † ,σ c ,σ + γ c † ,σ c ,σ ] + h.c. ˆ L Y = − i X σ [ γ c † ,σ c ,σ + γ c † ,σ c ,σ − γ c † ,σ c ,σ ] + h.c., with the three real coefficients γ , , left undetermined.To have an estimate of the values of the five coefficients γ ,.., , we approximate the five MLWFs | ν i , ν = 1 , .., d orbital in the atomic limit. Theseatomic orbital are related to the spherical harmonics of l = 2 in the following ways(apart from the radial part ofthe wave function which is not used in determining thematrix element of ˆ L α ) | i = | , i| i = 1 √ | , − i − | , i ) | i = i √ | , − i + | , i ) | i = 1 √ | , − i + | , i ) | i = i √ | , − i − | , i ) , where | , m i ∝ Y m are the spherical harmonics of l = 2.Since h , m ′ | ˆ L Z | , m i = mδ m,m ′ and h , m | ˆ L + | , m ′ i = p − m ′ ( m ′ + 1) δ m,m ′ +1 (here ˆ L + = ˆ L X + i ˆ L Y ), we have γ = − γ = γ = − γ = − γ = √ . We will use these values in the following calculation.The bare orbital magnetic susceptibility is readily ob-tained as χ ,αL ( T ) = lim q → N X k ,m,m ′ f ( ξ k + q ,m ′ ) − f ( ξ k ,m ) ξ k ,m − ξ k + q ,m ′ (cid:12)(cid:12) L α k ,m,m ′ (cid:12)(cid:12) . (4)Here ξ k ,m = ǫ k ,m − µ is the band energy of the m -th band( m = 1 , ...,
5) and µ is the chemical potential. L α k ,m,m ′ = P ν,ν ′ l αν,ν ′ u ∗ k ,ν,m u k ,ν ′ ,m ′ and u k ,ν,m is the m -th eigenvector of the band Hamiltonian at momentum k .As a comparison, the Pauli spin susceptibility is given by χ ,αS ( T ) = lim q → N X k ,m (cid:20) f ( ξ k + q ,m ) − f ( ξ k ,m ) ξ k ,m − ξ k + q ,m (cid:21) . Unlike the orbital magnetic susceptibility, the Pauli spinsusceptibility has contribution only from intra-band pro-cess. Thus at low temperature the spin susceptibilityis solely determined by the electronic state around theFermi surface, while the orbital magnetic susceptibilitydepends on electronic states both on and far away fromthe Fermi energy. As a result, both the temperature andthe doping dependence of the orbital magnetic responseshould be much weaker than that of the spin magneticresponse.Now we consider the RPA correction of the orbitalmagnetic susceptibility. The orbital magnetic excita-tion of the system has the general form of ˆ O νν ′ = i P σ ( c † ν,σ c ν ′ ,σ − c † ν ′ ,σ c ν,σ ). Without losing generality, we assume ν ′ > ν . There are in total 10 such excitationsand all of them are time reversal odd and spin singlet.The correlation function between these excitations canbe defined in the following way χ νν ′ ,υυ ′ O ( q , τ ) = −h T τ ˆ O νν ′ ( q , τ ) ˆ O υυ ′ ( − q , i , and the corresponding bare susceptibility in the staticlimit χ ,νν ′ ,υυ ′ O ( T ) is given by an expression similar toEq.(4), except that the matrix element | L α k ,m,m ′ | shouldbe replaced by O νν ′ ,υυ ′ k ,m,m ′ = ( u ∗ k ,ν ′ ,m ′ u k ,ν,m − u ∗ k ,ν,m ′ u k ,ν ′ ,m ) × ( u ∗ k ,υ,m u k ,υ ′ ,m ′ − u ∗ k ,υ ′ ,m u k ,υ,m ′ ) . The RPA correction of χ νν ′ ,υυ ′ O is contributed by theinter-orbital Coulomb repulsion, the Hund’s rule couplingand the pair hopping term. The RPA kernel is extremelysimple and is given by V νν ′ ,υυ ′ = ( U ′ − J )4 δ νν ′ ,υυ ′ (see Sup-plementary material A). The RPA corrected susceptibil-ity can be written formally as χ O = χ O − V χ O , in which χ O , χ O and V are all to be understood as 10 × V is a diagonal matrix in the spaceof ˆ O ν,ν ′ , χ O is not). The orbital magnetic susceptibil-ity can be obtained from the combinations of the matrixelement of χ O . For example, χ ZL = χ , O + 4( χ , O + χ , O ) . The orbital magnetic susceptibility in other direction canbe obtained in a similar way.The observation of the two-fold modulation in the in-plane magnetic susceptibility indicates that the tetrago-nal symmetry of the system is broken down to orthog-onal. This can happen either through orbital ordering,or through nematicity in spin correlation . Here weassume it happens through orbital ordering, since the or-bital magnetic response is much more sensitive to it thanto spin nematicity. The form of the symmetry break-ing perturbation in the orthogonal phase can be largelydetermined by group theoretical arguments. Among thefive 3 d orbital, the 3 d Z − R , 3 d XY and 3 d X − Y or-bital each form a one dimensional representation of the D d point group. The 3 d XZ and 3 d Y Z orbital form atwo-dimensional representation which becomes reduciblewhen the symmetry is lowered to orthogonal. We thusfocus on symmetry breaking terms in the space spannedby the 3 d XZ and 3 d Y Z orbital. A group theoretical anal-ysis then shows that up to nearest neighboring hoppingterms, the only allowable symmetry breaking perturba-tion in the orthogonal phase takes the form (see Supple-mentary information B)∆ H = η X i,σ ( c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ )+ η X i,δ,σ d δ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ )+ η X i,δ,σ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ ) , (5)in which δ = ± x, ± y is the vector between nearest neigh-boring Fe sites. d δ is the d-wave form factor and d ± x = 1,d ± y = −
1. Here, η is the strength of the on-site symme-try breaking perturbation. η and η are the strengthsof the d-wave intra-orbital and s-wave inter-orbital hop-ping terms between nearest neighboring Fe sites. FromARPES measurement , it is found that the splitting be-tween the 3 d xz and the 3 d yz -dominated band is zero atthe Γ point and maximizes at the X and Y point. Amongthe three perturbations in Eq.(5), only the d-wave intra-orbital hopping term is consistent with such a momen-tum dependence. For example, both the η or η -typeperturbation would result in an nonzero band splittingat the Γ point, which is not observed. Furthermore, the η -type perturbation has no effect at the X and Y point,where the observed band splitting reaches its maximum.We thus set η = η = 0. This leaves us η as the onlyundetermined parameter.We are now at the position to present the numericalresults. Our calculation is done at a fixed band fillingof n = 6 .
1. The chemical potential is determined bysolving the mean field particle number equation at eachtemperature. We have set U = 1 . J = 0 . η from theobserved band splitting, we note that the band widthof the Iron-based superconductors is significantly smallerthan the prediction of band structure calculation. Wethus fit the relative rather than the absolute magnitude ofthe band splitting. According to ARPES measurement,the maximal band splitting between the 3 d yz and 3 d xz -dominated band is about one half of the dispersion of the3 d yz -dominated band between the Γ and X point . To fitsuch a splitting, we set η = 30 meV. The calculated banddispersion along the Γ − X and Γ − Y direction is shownin Fig.2, which looks very similar to the experimentalresult . The temperature dependence of η is modeledby the mean field form of η ( T ) = η (0) p − ( T /T c ) ,in which T c is to be understood as the mean field criticaltemperature of orbital ordering. We set T c = 150K inour calculation .In the tetragonal phase, the orbital magnetic sus-ceptibility is found to be isotropic in the Fe-Fe planeand is almost temperature and doping independent for6 . ≤ n ≤ . n = 6 .
1, the bare or-bital magnetic susceptibility in the Fe-Fe plane is foundto be about 7.3 µ / eV, which is enhanced to 10 µ / eV E k ( e V ) k ( /a) X/Y d xy d xz d yz d -r E F -X -Y FIG. 2: Overlay of the band dispersion along the Γ − X andΓ − Y direction in the orthogonal phase. The orbital characteris indicated by the color of the lines and the dispersion inthe tetragonal phase is plotted in thin lines for reference. Inthe calculation we have set η = 30 meV. The dashed lineindicates the Fermi level at n = 6 . after RPA correction. This is already comparable to theobserved total in-plane magnetic susceptibility at 200Kin 122 systems, which is about 4 . × − erg / G mol AS (or 14 µ / eV) . As a comparison, the bare Pauli spinsusceptibility is only about 2 µ / eV.When a symmetry breaking perturbation of the η -type is turned on, a two-fold modulation shows up in thein-plane orbital magnetic susceptibility. The angular de-pendence of the in-plane susceptibility at T = T c / φ denotes the angle between the X -axis and the direction in which the magnetic suscep-tibility is measured. The relative strength of the mod-ulation is about 2.6% before RPA correction and is en-hanced to 4.5% after RPA correction. The principle axesof the modulation are along the direction of the nearestFe-Fe bond, which is just what we should expect from ourmodel construction. The temperature dependence of thesusceptibility in the principle axes are shown in Fig.3b.These predictions are in good agreement with the resultof the recent torque magnetometry measurement . Thusthe magnetic anisotropy provide a realistic probe of theorbital ordering in the Iron-based superconductors.A robust prediction of our theory is the stronganisotropy between the in-plane and the out-of-plane or-bital magnetic susceptibility. At n = 6 .
1, the bare or-bital magnetic susceptibility in Z direction is found tobe about 3.8 µ / eV, which is enhanced to 4.5 µ /eV af-ter RPA correction. This is only about the half of thevalue of the in-plane orbital magnetic susceptibility. Wefind the ratio between the in-plane and out-of-plane or-bital magnetic susceptibility is also almost temperatureand doping independent for 6 . ≤ n ≤ . . This behavior can be eas-ily understood if we decompose the measured magneticsusceptibility into an isotropic component that is linearlytemperature dependent and a temperature independentcomponent that is anisotropic, or, χ α ( T ) = χ αL + χ S ( T ) . (6)It is then quite natural to associate the anisotropic com-ponent χ αL with the orbital magnetic response, which isessentially temperature independent. The isotropic com-ponent χ S ( T ) should then be attributed to the spin mag-netic response, whose linear temperature dependence isstill an unresolved issue in the field. RPA Bare () ( B / e V ) ( ) (a) x y ( T ) ( B / e V ) T (K) (b)
FIG. 3: (a)The in-plane modulation of the orbital magneticsusceptibility before and after RPA correction. (b) The tem-perature dependence of the RPA-corrected orbital magneticsusceptibility along the two principle axes of the orthogonalphase.
In our calculation, we have used a five-band model de- rived from the band structure of the LaFeAsO system.However, the best known susceptibility data on singlecrystalline sample are all taken from the 122 system. Itis thus better to perform the calculation with a material-specific band structure for the 122 systems. While this isan interesting possibility and should be pursued in thefuture, we note that the basic structure of the bandsin both the 1111 and the 122 systems are quite simi-lar. Since the orbital magnetic response is contributedby the whole band rather than the electronic state nearthe Fermi level only, we expect the 1111 and 122 systemto exhibit similar orbital magnetic response. Anotherway to improve our calculation is to use the matrix el-ement of ˆ L α calculated from first principle code, ratherthan approximating them with those in the basis spannedby the atomic orbital. However, since the form the ma-trix element is largely determined by symmetry, we donot expect such more advanced calculation to change theconclusion of this paper in a qualitative way. Indeed, wefind that our results are not sensitive to the small varia-tion of the parameters γ ,., .In summary, we have shown that the orbital angularmomentum of the conduction electrons in the Iron-basedsuperconductors contributes significantly to the magneticresponse of the system. In particular, the theory predictsthat the orbital magnetic susceptibility accounts for morethan 2 / M. Yi, D. Lu, J.H. Chu, J. Analytis, A. Sorini, A. Kemper,B. Moritz, S.K. Mo, R.G. Moore, M. Hashimoto, W.S.Lee, Z. Hussain, T. Devereaux, I.R. Fisher, and Z.X. Shen,Proc. Natl. Acad. Sci. , 6878 (2011). Y. Zhang, F. Chen, C. He, B. Zhou, B. P. Xie, C. Fang,W. F. Tsai, X. H. Chen, H. Hayashi, J. Jiang, H. Iwasawa,K. Shimada, H. Namatame, M. Taniguchi, J. P. Hu, D. L.Feng, Phys. Rev. B , 054510 (2011). T. Shimojima, K. Ishizaka, Y. Ishida, N. Katayama, K.Ohgushi, T. Kiss, M. Okawa, T. Togashi, X.-Y. Wang, C.-T. Chen, S. Watanabe, R. Kadota, T. Oguchi, A. Chainaniand S. Shin, Phys. Rev. Lett. , 057002 (2010). T.M. Chuang, M.P. Allan, J. Lee, Y. Xi, N. Ni, S. Bud’ko, G.S. Boebinger, P.C. Canfield, and J.C. Davis, Science , 181 (2010). J.H. Chu, J.G. Analytis, D. Press, K. De Greve, T.D.Ladd, Y. Yamamoto, I.R. Fisher, Phys. Rev. B , 214502(2010). J.H. Chu, J.G. Analytis, K. De Greve, P.L. McMahon, Z.Islam, Y. Yamamoto, and I.R. Fisher, Science , 824(2010). J.H. Chu, H.H. Kuo, J.G. Analytis and I.R. Fisher, Science , 710 (2012). S. Kasahara, H. J. Shi, K. Hashimoto, S. Tonegawa, Y.Mizukami, T. Shibauchi, K. Sugimoto, T. Fukuda, T.Terashima, A.H. Nevidomskyy and Y. Matsuda Nature , 382 (2012). C.C. Lee, W.G. Yin and W. Ku, Phys. Rev. Lett. ,267001 (2009). W. Lv, J. Wu, and P. Phillips, Phys. Rev. B , 224506(2009). C.C. Chen, J. Maciejko, A.P. Sorini, B. Moritz, R. Singhand T. P. Devereaux, Phys. Rev. B , 100504 (2010). W. Lv, F. Kruger, and P. Phillips, Phys. Rev. B , 045125(2010). A. H. Nevidomskyy, arxiv.org:1104.1747 (2011). G. Wu, H. Chen, T. Wu, Y.L. Xie, Y.J. Yan, R.H. Liu,X.F. Wang, J.J. Ying and X.H. Chen, J. Phys.: Cond.Matter , 422201 (2008). X.F. Wang, T. Wu, G. Wu, H. Chen, Y.L. Xie, J.J. Ying,Y.J. Yan, R.H. Liu, and X.H. Chen, Phys. Rev. Lett. ,117005 (2009). J.Q. Yan, A. Kreyssig, S. Nandi, N. Ni, S.L. Bud’ko,A. Kracher, R.J. McQueeney, R.W. McCallum, T.A. Lo-grasso, A.I. Goldman, and P.C. Canfield, Phys. Rev. B ,024516 (2008). Z. Li, D.L. Sun, C.T. Lin, Y.H. Su, J.P. Hu, G.Q. Zheng,Phys. Rev. B , 140506 (2011). C. Xu, M. Muller, and S. Sachdev, Phys. Rev. B , 020501(2008). R. M. Fernandes, A. V. Chubukov, J. Knolle, I. Eremin,and J. Schmalian, Phys. Rev. B , 024534 (2012). K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H.Kontani, and H. Aoki, Phys. Rev. Lett. , 087004 (2008). Here R x ( π ) and R y ( π ) are the π rotations along the x and y axis, σ X and σ Y are the mirror planes normal to the X and Y axis. R x ( π ), σ X and σ Y form a set of generatorsof the symmetry group D d of the tetragonal phase and R x ( π ) and R y ( π ) form a set of generators of the symmetrygroup D of the orthogonal phase. R. Klingeler, N. Leps, I. Hellmann, A. Popa, U. Stock-ert, C. Hess, V. Kataev, H.J. Grafe, F. Hammerath, G.Lang, S. Wurmehl, G. Behr, L. Harnagea, S. Singh, andB. B¨uchner, Phys. Rev. B , 024506 (2010). G.M. Zhang, Y.H. Su, Z.Y. Weng, D.H. Lee, and T. Xiang,EuroPhys. Lett. S.P. Kou, T. Li, and Z.Y. Weng, EuroPhys. Lett. M.M. Korshunov, I. Eremin, D.V. Efremov, D.L. Maslov,and A.V. Chubukov,Phys. Rev. Lett. 102, 236403 (2009).
I. SUPPLEMENTARY MATERIALSA. The form of the RPA kernel for orbitalmagnetic excitations
The ten orbital magnetic excitation of the form ˆ O νν ′ = i P σ ( c † ν,σ c ν ′ ,σ − c † ν ′ ,σ c ν,σ ) are all time reversal odd andspin rotational invariant. In the absence of time rever-sal symmetry breaking they form a subspace within thespace of all orbital excitations. It is thus sufficient torestrict our consideration in this subspace.The RPA correction to the orbital magnetic responseis contributed by the inter-orbital Coulomb term, theHund’s rule coupling term and the pair hopping term.For example, the inter-orbital Coulomb term has the fol- lowing mean field decoupling( ν ′ > ν ), U ′ n i,ν,σ n ı ,ν ′ ,σ ∼ − U ′ h c † i,ν,σ c i,ν ′ ,σ i c † i,ν ′ ,σ c i,ν,σ − U ′ h c † i,ν ′ ,σ c i,ν,σ i c † i,ν,σ c i,ν ′ ,σ + U ′ h c † i,ν,σ c i,ν ′ ,σ ih c † i,ν ′ ,σ c i,ν,σ i . When expressed in terms of ˆ O ν,ν ′ , we have U ′ X σ n i,ν,σ n ı ,ν ′ ,σ ∼ − U ′ h ˆ O ν,ν ′ i ˆ O ν,ν ′ + U ′ h ˆ O ν,ν ′ ih ˆ O ν,ν ′ i . Thus the RPA kernel is diagonal in the subspace of ˆ O ν,ν ′ .Following the same steps, it can be shown that the RPAcorrection contributed by the last two terms in Eq.(2)cancels with each other. B. The form of the symmetry breakingperturbation in the orthogonal phase
The form of the symmetry breaking perturbation in theorthogonal phase can be determined from the followinggroup theoretical arguments. We first consider the formof the on-site symmetry breaking term. The point grouparound each Fe ion in the orthogonal phase is D and hasfour one dimensional irreducible representations. Amongthe five MLWFs, | Z − R i and | XY i both belong tothe identity representation, | X − Y i belongs to the B representation, the linear combinations | XZ i + | Y Z i and | XZ i − | Y Z i belong to the B and B representation.Thus symmetry allowed on-site Fermion bilinear termshave the general form of H = X i,σ ( β c † i, ,σ c i, ,σ + β c † i, ,σ c i, ,σ + β c † i, ,σ c i, ,σ )+ β X i,σ ( c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ )+ β X i,σ ( c † i, ,σ + c † i, ,σ )( c i, ,σ + c i, ,σ )+ β X i,σ ( c † i, ,σ − c † i, ,σ )( c i, ,σ − c i, ,σ ) (7)In the tetragonal phase, the local symmetry aroundeach Fe ion is promoted to D d , which has four one di-mensional representations and a two dimensional repre-sentation. Among the five MLWFs, | Z − R i belongs tothe identity representation, | XY i and | X − Y i belongto the B and B representation, the linear combinations | XZ i + | Y Z i and | XZ i − | Y Z i form the two componentsof the two dimensional representation. For this reason,the bilinear form c † i, ,σ c i, ,σ , c † i, ,σ c i, ,σ , c † i, ,σ c i, ,σ , and c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ all belong to the identity repre-sentation of D d . When these symmetric perturbationsare removed from Eq.(7), we get the symmetric breakingperturbation in the orthogonal phase, which now takesthe form of∆ H = λ X i,σ ( c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ )+ λ X i,σ ( c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ ) , in which λ = β − β , λ = β .The above argument can be easily generalized to deter-mined the form the symmetry breaking perturbation onvarious bonds. In particular, we find there are in total 13independent symmetry breaking perturbations on near-est neighboring Fe-Fe bonds. The form of these termsare ∆ H = ∆ H s + ∆ H p + ∆ H d , in which∆ H s = κ X i,δ,σ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ )+ X i,δ,σ ( κ c † i, ,σ c i + δ, ,σ + κ c † i, ,σ c i + δ, ,σ )∆ H p = κ X i,δ,σ (p δ c † i, ,σ c i + δ, ,σ + p ′ δ c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ (p δ c † i, ,σ c i + δ, ,σ + p ′ δ c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ (p ′ δ c † i, ,σ c i + δ, ,σ + p δ c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ (p ′ δ c † i, ,σ c i + δ, ,σ + p δ c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ (p δ c † i, ,σ c i + δ, ,σ − p ′ δ c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ (p δ c † i, ,σ c i + δ, ,σ − p ′ δ c † i, ,σ c i + δ, ,σ ) , and ∆ H d = κ X i,δ,σ d δ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ )+ κ X i,δ,σ d δ c † i, ,σ c i + δ, ,σ + κ X i,δ,σ d δ c † i, ,σ c i + δ, ,σ + κ X i,δ,σ d δ c † i, ,σ c i + δ, ,σ . Here p δ , p ′ δ are p-wave form factors, d δ is the d-waveform factor. The value of these form factors are illus-trated in Fig.4 If we restrict our consideration to the subspacespanned by the d XZ and d Y Z orbital, then up to nearest p p’ d +1+1-1 -1 +1+1 -1 -1 +1+1 -1-1
FIG. 4: An illustration of the p-wave and d-wave form factordefined in the main text. neighboring hopping term, the only allowable symmetrybreaking perturbation has the following form∆ H = η X i,σ ( c † i, ,σ c i, ,σ + c † i, ,σ c i, ,σ )+ η X i,δ,σ d δ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ )+ η X i,δ,σ ( c † i, ,σ c i + δ, ,σ + c † i, ,σ c i + δ, ,σ ) , in which η = λ = β − β , η = κ , η = κ . C. The temperature and doping dependence of theanisotropy ratio
Unlike the spin magnetic response, the orbital mag-netic response is contributed by both intra-band andinter-band process. As a result, the orbital magnetic re-sponse is much less sensitive to the variation of temper-ature and doping concentration of the system. In Fig.5,we present the temperature and doping dependence ofthe RPA-corrected in-plane orbital magnetic susceptibil-ity and the ratio between the in-plane and the out-of-plane orbital magnetic susceptibility.From the figure it is clear that both quantities haveonly small temperature and doping dependence. Morespecifically, the relative change of the in-plane orbitalmagnetic susceptibility for 6 . ≤ n ≤ .
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T (K) n (a)
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T (K) n (b) FIG. 5: The temperature and doping dependence of the RPA-corrected in-plane orbital magnetic susceptibility(a) and theratio between the in-plane and out-of-plane orbital magneticsusceptibility(b) for 6 ..
T (K) n (b) FIG. 5: The temperature and doping dependence of the RPA-corrected in-plane orbital magnetic susceptibility(a) and theratio between the in-plane and out-of-plane orbital magneticsusceptibility(b) for 6 .. ≤ n ≤ ..