Highly anisotropic quasiparticle interference patterns in the spin-density wave state of the iron pnictides
HHighly anisotropic quasiparticle interference patterns in the spin-density wave stateof the iron pnictides
Dheeraj Kumar Singh and Pinaki Majumdar
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India& Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India (Dated: October 17, 2018)We investigate the impurity scattering induced quasiparticle interference in the ( π, ) spin-density wave phaseof the iron pnictides. We use a five orbital tight binding model and our mean field theory in the clean limit cap-tures key features of the Fermi surface observed in angle-resolved photoemission. We use a t-matrix formalismto incorporate the effect of doping induced impurities on this state. The impurities lead to a spatial modulationof the local density of states about the impurity site, with a periodicity of ∼ a Fe − Fe along the antiferromag-netic direction. The associated momentum space quasiparticle interference pattern is anisotropic, with majorpeaks located at ∼ ( ± π/ , , consistent with spectroscopic imaging scanning tunneling microscopy. We tracethe origin of this pattern to an elliptical contour of constant energy around momentum (0,0), with major axisoriented along the (0,1) direction, in the mean field electronic structure. PACS numbers: 74.70.Xa, 75.10.Lp, 75.30.Fv
I. INTRODUCTION
The highly anisotropic electronic properties of the ironpnictides, with broken four-fold rotation symmetry, have beena subject of intense research in recent times. Observed inangle-resolved photoemission spectroscopy (ARPES), neu-tron magnetic resonance (NMR), and transport properties ,such anisotropy is seen both in the low temperature collinearantiferromagnetic (AF) state and the high temperature, un-ordered, ‘nematic’ phase .ARPES reveal a significant energy splitting between the d xz and d yz orbitals below the tetragonal-to-orthorhombictransition - which may precede or coincide with the spindensity wave (SDW) transition. The spin dynamics shows astrong two-fold anisotropy inside the orthorhombic domainsthat are formed below the structural phase transition , op-tical spectra displays a significant in-plane anisotropy uptophotonic energies ∼ eV, and transport measurements showa larger conductivity in the antiferromagnetic direction com-pared to the ferromagnetic direction.Spectroscopic imaging - scanning tunneling microscopy(SI-STM) provides insight into the anisotropic electronicstate. Quasiparticle interference (QPI) probed by SI-STMmeasures the modulation of the local density of states (LDOS)induced by the impurity atoms. QPI patterns in the metallic( π, ) SDW state consist mainly of a quasi-one dimensionalfeature extended along the q x = 0 line with a weaker paral-lel feature at a distance ∼ π/ . Such highly anisotropicfeatures have been attributed to impurity induced states on theanisotropic magnetic background .The QPI probes the response of the ordered state to a stronglocalized perturbation and several attempts have been made toexplain it. A reasonable description of the ARPES and QPIdata imposes constraints on the electronic theory of the ref-erence state. Broadly three frameworks have been used to tomodel the QPI, each with some limitation.(i) In an effective band approach LDOS modulation isstrongest along the ferromagnetic direction while experimen-tally it is in the AF direction. Corresponding contours of con- stant energy (CCE) consist mainly of a circular pocket around Γ , smaller pockets located inside, and the electron pocketaround Y. (ii) A five-orbital model used to study QPI ei-ther produces patterns without a clear modulation or showsmodulation at an energy ω ∼ − meV, much larger thanin the experiments. In one of the studies, details of the recon-structed FSs are not provided, while in another one large FSsconsists of parallel running structures extending from Γ to thezone boundary near X . (iii) First principles calculations indicate QPI peaks at (0, ± π /4) and therefore the correctwavelength of modulation, ∼ a Fe − Fe , but again along the ferromagnetic direction. In this case, FSs consist of crescentlike structure around Γ with the broader part facing Y. So, ei-ther the wavelength, or the orientation, or the energy of theQPI modulations remain inconsistent with experiments.In this paper, we report on the QPI in the ( π, )-SDW stateof an electron-doped iron pnictide. We use mean field the-ory on a five-orbital tight-binding model to describe the or-dered state and a t-matrix calculation to quantify (single) im-purity effects. We find the following: (i) Our mean fieldbands have several features consistent with the ARPES mea-surements, e.g , a large elliptical pocket around Γ and adja-cent four smaller pockets. (ii) The QPI is highly anisotropic,consisting of quasi-one dimensional peak structures runningnearly along q x = ± π/ . (iii) The real-space features consistof LDOS modulation with periodicity ∼ a Fe − Fe along theantiferromagnetic direction as observed in the STM measure-ments. The period of modulation along the AF direction isrobust against change in the quasiparticle energy though thestrongest modulation can shift to other direction. II. MODEL AND METHOD
In order to study QPI in the SDW state, we consider a five-orbital tight-binding Hamiltonian defined in the Fe-As planes, a r X i v : . [ c ond - m a t . s t r- e l ] A p r he kinetic part of which is given by H = (cid:88) k (cid:88) µ,ν (cid:88) σ ε µν k d † k µσ d k νσ (1)in the plane-wave basis. Here, the operator d † k µσ ( d k µσ ) cre-ates (destroys) an electron with spin σ and momentum k inthe µ -th orbital. Matrix elements ε µν k , which include both thehopping matrix elements and the momentum independent on-site orbital energies, are taken from Ref.[28]. The set of d -orbitals, to which orbitals µ and ν belong, consists of d xz , d yz , d xy , d x − y , and d z − r .The interaction part includes standard onsite Coulomb in-teractions H int = U (cid:88) i ,µ n i µ ↑ n i µ ↓ + ( U (cid:48) − J (cid:88) i ,µ<ν n i µ n i ν − J (cid:88) i ,µ<ν S i µ . S i ν + J (cid:88) i ,µ<ν,σ d † i µσ d † i µ ¯ σ d i ν ¯ σ d i νσ . (2) U and U (cid:48) are the intra-orbital and the inter-orbital Coulombinteraction, respectively. J is the Hund’s coupling, with thecondition U (cid:48) = U − J imposed for a rotation-invariant inter-action.The mean-field Hamiltonian for the ( π, -SDW state in thetwo-sublattice basis is given by H mf = (cid:88) k σ Ψ † k σ (ˆ ζ k σ + ˆ M k σ )Ψ k σ . (3) ζ ll (cid:48) k σ are the matrix elements due to the kinetic part while M ll (cid:48) k σ = − sσ ∆ ll (cid:48) δ ll (cid:48) + J − U n ll (cid:48) δ ll (cid:48) . l , l (cid:48) ∈ s ⊗ µ with s and µ belonging to the sublattice and orbital bases, respectively.Off-diagonal elements of ∆ ll (cid:48) and n ll (cid:48) are small for the param-eters considered here, and hence neglected. s and σ in frontof ∆ ll (cid:48) δ ll (cid:48) take value 1 (-1) for A (B) sublattice and ↑ -spin( ↓ -spin), respectively. The electron field operator is defined as Ψ † k ↑ = ( d † A k ↑ , d † A k ↑ , ..., d † B k ↑ , d † B k ↑ , ... ) , where subscriptindices 1, 2, 3, 4, and 5 stand for orbitals d z − r , d xz , d yz , d x − y , and d xy , respectively. The exchange fields are givenas ll = U m l + J (cid:80) l (cid:54) = l (cid:48) m l (cid:48) . Orbital charge density andmagnetization are determined in a self-consistent manner bydiagonalizing the Hamiltonian.The change caused in the Green’s function because of a sin-gle impurity with δ -potential is given by δ ˆ G ( k , k (cid:48) , ω ) = ˆ G ( k , ω ) ˆ T ( ω ) ˆ G ( k (cid:48) , ω ) (4)using t -matrix approximation. ˆ G ( k , ω ) = (ˆ I − ˆ H (cid:48) mf ) − isthe Green’s function in the SDW state with ˆ H (cid:48) σmf = (cid:18) ˆ ε k sgn¯ σ ˆ∆sgn¯ σ ˆ∆ ˆ ε k + Q (cid:19) . (5) ˆ I is a 10 ×
10 identity matrix and Q = ( π, . ˆ H (cid:48) mf is ob-tained from (ˆ ζ k + ˆ M k ) using a unitary transformation Next, T ( ω ) = (ˆ − ˆ V ˆ G ( ω )) − ˆ V , (6) with ˆ G ( ω ) = 1 N (cid:88) k ˆ G ( k , ω ) (7)and ˆ V = V imp (cid:18) ˆ ˆ ˆ ˆ (cid:19) . (8)Here, ˆ is a 5 × δρ ( q , ω ) in theDOS due to the impurity scattering is given by δρ ( q , ω ) = i π (cid:88) k g ( k , q , ω ) (9)with g ( k , q , ω ) = Tr δ ˆ G ( k , k (cid:48) , ω ) − Tr δ ˆ G ∗ ( k (cid:48) , k , ω ) , (10)where k − k (cid:48) = q . The real-space QPI can be obtained as δρ ( r i , ω ) = 1 N (cid:88) k δρ ( q , ω ) e i k · r i . (11)In the following, intraorbital Coulomb interaction parame-ter ( U ) is taken as . eV with J = 0 . U to keep the totalmagnetic moment per site less than unity. Band filling n e isfixed at 6.03 (3 % electron doping). Self-consistently obtainedorbital magnetizations are m r − x = 0 . , m xz = 0 . , -0.4-0.2 0 0.2 0.4 E ( e V ) k (a) (0,0) ( (cid:47) /2,0) ( (cid:47) /2, (cid:47) ) (0,0) (0, (cid:47) ) ( (cid:47) /2, (cid:47) ) FIG. 1. (a) Electronic dispersion along the high-symmetry directions,(b) Reconstructed Fermi surfaces consisting of several pockets nearand around Γ as well as around ( π , 0), and (c) orbital-resolved den-sity of states in the ( π, -SDW state. a) -65meVq q q P P (b) -55meV (c) -45meV(d) -35meV (e) -25meVq q P (f) highlow-15meVq q P q FIG. 2. Constant energy maps of the spectral functions A ( k , ω ) inthe unfolded Brillouin zone from − meV (top left) to − meV(bottom right) in step of meV. The arrows represent scatteringwavevectors in the SDW state. Note that q , q , and q are notshown, which are the intrapocket scattering vectors for the tiny CCEsP , P , and for the subpockets in CCE P , respectively. m yz = 0 . , m xy = 0 . , and m x − y = 0 . . Or-bital charge densities are n r − x = 1 . , n xz = 1 . , n yz = 1 . , n xy = 1 . , and n x − y = 1 . . Strengthof the impurity potential V imp is set to be meV. Varyingstrength will change the intensity while the basic structure ofthe QPI is expected to remain the same. A mesh size of 300 ×
300 in the momentum space is used for all the calculations.
III. RESULTS
Fig.1(a) and (b) show the electronic dispersion and theFermi surface (FS) in the SDW state. The FS consists ofan ellipse-like hole pocket around Γ , with major axis inthe (0, 1) direction, and tiny electron pockets situated at ≈ ( ± π/ , and (0 , ± π/ but outstretched along (0 , and (1 , directions, respectively. Interestingly, similar pocketsalthough larger in size have been reported by the ARPESexperiments . In addition, there are electron pocketsaround ( , ± π ). Some of the above characteristics of FSs leadto significant anisotropy in the QPI. Fig.1(c) shows that the d xy , d yz , and d zx orbitals dominate at the Fermi level.In order to understand the QPIs, it will be useful to lookat the CCEs of the spectral functions, which are shown inFig.2(a)-(f) as a function of energy with step of meV upto − meV starting from − meV. Near − meV, CCEs con-sists of an ellipse-like pocket P around (0, 0) and two tinypockets P along k y = 0 mapped onto each other by 180 ◦ rotation owing to the C symmetry. Thus, there are four setsof scattering vectors - intrapocket scattering vectors q due toP , interpocket scattering vectors q connecting the pocketsof P , interpocket scattering vectors q connecting P and P and intrapocket scattering vectors q (not shown in Fig.3(a))due to P . Corresponding QPI pattern is expected to havea two-dimensional nature because of a near cancellation of two opposite tendencies in which scattering vectors q triesto create a one-dimensional pattern along the (0 , directionwhile q and q do the same along the (1 , direction. On thecontrary, pattern consists of two parallel peak structures run-ning along q x = const and passing through q x ≈ ( ± π/ , )(Fig.3(a)).In addition, there are small elliptical pockets located near q x ≈ ( ± π/ , ). Here, it is important to note that only thosescattering vectors are important, which connect parts of theCCEs having same dominating orbitals because only intraor-bital scattering has been incorporated owing to the symmetryconsideration .An important change in the QPI patterns occurs upon in-creasing the energy as shown in Fig.3(d). This happens pri-marily because of the appearance of a new set of CCEs inthe form of tiny pockets P , which emerge out of the ellipti-cal pocket P . Since P is in the proximity of band extrema,pattern generated corresponding to the scattering vectors con-necting these pockets should dominate the overall QPI patternbecause of a larger phase space available for the scatteringprocesses. Therefore, the balance maintained by the two op-posite tendencies described above is perturbed now. This re-sults into a highly anisotropic QPIs (Fig.3(d)) dominated bywavevectors q , q and q (Fig.3(c)). q and q are the set ofinterpocket and intrapocket scattering vectors for P pockets,respectively, whereas q is another set of interpocket scatter-ing vectors connecting P and P .On increasing energy further, an additional set of CCEs ap-pear near ( , ± π ) as seen Fig.2(f), which may also containsvery small subpockets (Fig.1(b)). As these are in the vicin-ity of local band extrema and parallel to the k y = 0 , two-dimensional characteristics is imparted to the QPIs as noticedin Fig.3(f). Dominating QPI wavevectors are due to the intersubpocket scattering vectors q and q as well as due to intrasubpocket scattering vectors q not shown (Fig.1(b)).Several aspects of the QPI obtained here comparewell with those of SI-STM measurements carried out for (a) -65meV q q q (b) -55meV -1.6 1.1 (c) -45meV -1.7 1.5 (d) -35meVq (e) -25meVq q (f) -15meVq q FIG. 3. QPI maps in the unfolded Brillouin zone for different ener-gies ω upto − meV (top left) to − meV (bottom right) in steps of meV. The arrows denote the QPI wavevectors in the SDW state.
120 135 150 165 180 120 135 150 165 180 (a) ω = -65meV
120 135 150 165 180 120 135 150 165 180 (b) ω = -55meVR a
120 135 150 165 180 120 135 150 165 180 -0.02 0 0.02 (c) ω = -45meV
120 135 150 165 180 120 135 150 165 180 (d) ω = -35meV
120 135 150 165 180 120 135 150 165 180 (e) ω = -25meV
120 135 150 165 180 120 135 150 165 180 -0.02 0 0.02 (f) ω = -15meV FIG. 4. Real-space QPIs for the set of parameters as in Fig.3. LDOS modulation along the antiferromagnetic direction ( x -axis) with thewavelength R x ≈ a Fe − Fe . Although the period of modulation along the antiferromagnetic direction remains almost unchanged, the strongmodulation direction is sensitive to the quasiparticle energy. Ca(Fe − x Co x ) As . For ω = − meV, a central peak struc-ture runs along q x = 0 and consists of three main peakswhich themselves are made of multiple peaks either coincid-ing or placed very closely: one at (0, 0), and other two placedequidistant from it. Additionally, there are parallel runningsatellite peak structures situated at ≈ ( ± π/ , 0), which arepart of an elliptical QPI patterns. The quasi-one dimensionalnature of the pattern is found in a wide energy window. Thesefeatures are in agreement with the STM measurements. Wealso note that they very sensitive to energy as evident fromFig.3(e), when they become very weak for ω = − meVon increasing the quasiparticle energy further. Features ofthe CCEs especially the existence of pockets along k x = 0 , (c) q y / π q x / π ω = 0meVq -1-0.5 0 0.5 1-0.5 0 0.5 k y / π k x / π (a) ω = -40meV xzyzxyq q y / π q x / π (b) ω = -40meV q FIG. 5. (a) CCEs for ω = -40meV with dominant orbital character inthe SDW state and (b) corresponding momentum-space QPI patterns.(c) QPI patterns observed in 3 % electron-doped CaFe As for ω =0 meV by the STM. which play a crucial role in imparting the quasi-one dimen-sionality to the main peak structure, have also been noticedin the ARPES measurements. We have also examined therole of an orbital splitting term in the Hamiltonian, whichis found to bring in only minor deviations because of a rel-atively large sized P and also due to the suppression of pock-ets P . Thus, it is the significant band reconstruction in theSDW state which is responsible for the experimentally ob-served anisotropy in the QPI.Fig.4 shows corresponding real-space QPI pattern. Thewavelength of LDOS modulation along x (antiferromagneticdirection) is R x ∼ a Fe − Fe for all energy values consideredhere though modulation may be weak or strong depending onthe energy. A strong modulation along x is seen for energies ω = − meV and − meV, with parallel running peak struc-tures along q x = const and passing through q x = 2 π/R x ≈ ( ± π/ , ) in qualitative agreement with the impurity inducedelectronic structure observed by the SI-STM experiments.However, it becomes stronger along x ≈ y upon increasingthe energy and corresponds to a strong modulation of DOSin the momentum space along a direction tilted away from q x = const . IV. DISCUSSIONA. Physical mechanism
Our study highlights the role of redistributed orbital weightalong the CCEs. In the unordered state, the hole pock-ts around Γ have predominantly d xz and d yz character dis-tributed in way to respect the four-fold rotational symmetry.On the contrary, the CCE pocket around Γ in the SDW state isdominated by the d xz orbital. For ω = − meV, the scatter-ing vectors connecting the regions near the vertices along theminor axis of elliptical pocket leads to the most intense regionin the QPI. This happens primarily because of two reasons.First, a larger phase space is available when compared withthe case of scattering vectors connecting the vertices along themajor axis. Secondly, only intraorbital scattering processesare taken into account. On increasing energy, a small d yz rich region (Fig.5(a)) appears along the elliptical CCE as anew band crosses the quasiparticle energy. Because of thenew band’s extrema and associated large spectral weight, QPIpattern due to the scattering vectors connecting d yz rich re-gions prevails over the others (Fig.5(b)). Consequently, themost intense region moves towards the vertices along the ma-jor axis (Fig.3(b)-(d)) a features present also in the STM re-sults (Fig.5(c)).In this work, the focus was on the LDOS modulation. An-other important issue is the modulation in the local magnetiza-tion induced by in the vicinity of the impurity. This has beenaddressed in a recent work within a self-consistent approachfor single non-magnetic impurity . The study found that theimpurity induces magnetic nanostructures with checkerboard-type order inside, extended along the antiferromagnetic direc-tion with a significant LDOS modulation at the ends. Our re-sult on the LDOS modulation in real-space is also consistentwith this study. B. Comparison with earlier work
Anisotropy in the QPI patterns of the SDW state is not un-expected because of the breaking of four-fold rotational sym-metry. However, the details of the patterns depend on the elec-tronic structure. A plausible description of the QPI patternsrestrict the modeling of the electronic structure. The failure ofalmost all the earlier work in reproducing the nearly parallelrunning satellite peak structures along q x ≈ ± π /4 highlightsthe limitation of the electronic structure used. In our work,these structures result from the elliptical CCE around (0 , with a major role played by the scattering vectors lying nearlyparallel to the minor axis of length ∼ π /4. Thus, we believethat the Fermi pocket around Γ , the existence of which hasalso been suggested by the ARPES measurements, is likely to be elliptical in shape. C. Unresolved issues
For 3 % doping on the parent state the a Fe − Fe × a Fe − Fe nanostructures would contain more than one impurity atomon the average. Therefore, the interference between scatter-ing events from multiple impurities could be important for themeasured QPI patterns. The present t-matrix approach unfor-tunately does not access these effects. Recently, a frameworkto study QPI in the presence of interacting multiple impuritieshas been discussed . However, in many instances, single im-purity treatment has yielded QPI patterns which successfullydescribe the qualitative features of STM measurements. TheLDOS modulation obtained in this work with the periodicity ∼ a Fe − Fe is another such example.LDOS modulation with the experimentally observed peri-odicity is reproduced successfully in our results along the an-tiferromagnetic direction, and is robust against any change inthe quasiparticle energy. However, the strongly modulated di-rection exhibits sensitivity to the quasiparticle energy, whichis due to the fast change in the CCEs. In the experiments,however, the strongly modulated direction is robustly alongthe antiferromagnetic direction despite the change in energy.This may indicate that CCEs change comparatively slowly inthe real systems as a function of energy. V. CONCLUSIONS
We have investigated the quasiparticle interference pat-tern in the ( π, )-SDW state using a five-orbital tight-bindingmodel of electron-doped iron pnictides. With a realistic recon-structed bandstructure, which includes an ellipse-like constantenergy contour around (0, 0) and additional nearby smallerpockets, we find highly anisotropic QPI patterns. Because thescattering vectors oriented along the minor axis of the ellip-tical CCE (of length π/ ) connects d xz rich segments, QPIpeak structures are obtained at ≈ ( ± π/ , ), running parallelto the q y axis. The corresponding real-space pattern consistsof LDOS modulation along the antiferromagnetic directionwith periodicity ∼ a Fe − Fe . Both the features are in agree-ment with STM results for the doped iron pnictides.We acknowledge use of the HPC Clusters at HRI. M. Yi, D. Lu, J.-H. Chu, J. G. Analytis, A. P. Sorini, A. F. Kemper,B. Moritz, S.-K. Mo, R. G. Moore, M. Hashimoto, W.-S. Lee, Z.Hussain, T. P. Devereaux, I. R. Fisher, and Z.- X. Shen, Proc. Natl.Acad. Sci. , 6878 (2011). T. Shimojima, K. Ishizaka, Y. Ishida, N. Katayama, K. Ohgushi, T.Kiss, M. Okawa, T. Togashi, X.-Y. Wang, C.- T. Chen, S. Watan-abe, R. Kadota, T. Oguchi, A. Chainani, and S. Shin, Phys. Rev.Lett. 104, 057002 (2010). M. Fu, D. A. Torchetti, T. Imai, F. L. Ning, J.-Q. Yan, and A. S. Sefat, Phys. Rev. Lett., , 247001 (2012). J-H. Chu, J.-H. Chu, J. G. Analytis, K. De Greve, P. L. McMahon,Z. Islam, Y. Yamamoto, and I. R. Fisher, Science , 824 (2010). M. A. Tanatar, E. C. Blomberg, A. Kreyssig, M. G. Kim, N. Ni, A.Thaler, S. L. Bud’ko, P. C. Canfield, A. I. Goldman, I. I. Mazin,and R. Prozorov, Phys. Rev. B , 184508 (2010). E. C. Blomberg, M. A. Tanatar, R. M. Fernandes, I. I. Mazin, B.Shen, H.-H. Wen, M. D. Johannes, J. Schmalian, and R. Prozorov,Nat. Commun. , 1914 (2013). S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt,A. Thaler, N. Ni, S. L. Budko, P. C. Canfield, J. Schmalian, R.J. McQueeney, and A. I. Goldman, Phys. Rev. Lett., , 057006(2010). M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, andR. P¨ottgen, Phys. Rev. B, , 020503R (2008). M. Nakajima, T. Liang, S. Ishida, Y. Tomioka, K. Kihou, C. H.Lee, A. Iyo, H. Eisaki, T. Kakeshita, T. Ito, and S. Uchida, Proc.Natl. Acad. Sci. U.S.A. , 12238 (2011). L. Capriotti, D. J. Scalapino, and R. D. Sedgewick, Phys. Rev. B , 014508 (2003). S. Sykora and P. Coleman, Phys. Rev. B , 054501 (2011). A. Kreisel, Peayush Choubey, T. Berlijn, W. Ku, B. M. Andersen,and P. J. Hirschfeld, Phys. Rev. Lett. , 217002 (2015). Y.-Y. Zhang, C. Fang, X. Zhou, K. Seo, W.-F. Tsai, B. A.Bernevig, and J. Hu, Phys. Rev. B , 094528 (2009). Y. Yamakawa and H. Kontani Phys. Rev. B 92, 045124 (2015). P. J. Hirschfeld, D. Altenfeld, I. Eremin, and I. I. Mazin Phys. Rev.B , 184513 (2015). T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L. Budko, G.S. Boebinger, P. C. Canfield, and J. C. Davis, Science , 181(2010). E. P. Rosenthal, E. F. Andrade, C. J. Arguello, R. M. Fernandes, L.Y. Xing, X. C. Wang, C. Q. Jin, A. J. Millis and A. N. Pasupathy,Nature Physics , 225 (2014). X. Zhou, C. Ye, P. Cai, X. Wang, X. Chen, and Y. Wang, Phys. Rev. Lett. , 087001 (2011). M. P. Allan, T-M. Chuang, F. Massee, Yang Xie, Ni Ni, S. L.Budko, G. S. Boebinger, Q. Wang, D. S. Dessau, P. C. Canfield,M. S. Golden, and J. C. Davis, Nature Physics , 220 (2013). M. N. Gastiasoro, P. J. Hirschfeld, and B. M. Andersen Phys. Rev.B , 100502 (2014). J. Knolle, I. Eremin, A. Akbari, and R. Moessner, Phys. Rev. Lett. , 257001 (2010). S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalapino, NewJ. Phys. , 025016 (2009). N. Plonka, A. F. Kemper, S. Graser, A. P. Kampf, and T. P. Dev-ereaux, Phys. Rev. B , 174518 (2013). H.-Y. Zhang and J.-X. Li, Phys. Rev. B , 075153 (2016). M. Yi, D. H. Lu, J. G. Analytis, J.-H. Chu, S.-K. Mo, R.-H. He,M. Hashimoto, R. G. Moore, I. I. Mazin, D. J. Singh, Z. Hussain,I. R. Fisher, and Z.-X. Shen, Phys. Rev. B , 174510 (2009). Q. Wang, Z. Sun, E. Rotenberg, F. Ronning, E. D. Bauer, H. Lin,R. S. Markiewicz, M. Lindroos, B. Barbiellini, A. Bansil, and D.S. Dessau, arXiv:1009.0271 I. I. Mazin, S. A. J. Kimber, and D. N. Argyriou, Phys. Rev. B ,054501 (2011). H. Ikeda, R. Arita, and J. Kunes, Phys. Rev. B , 054502 (2010). S. Ghosh and A. Singh, New J. Phys. , 063009 (2015). D. K. Singh and T. Takimoto, J. Phys. Soc. Jpn. , 044703(2016). A. K. Mitchell, P. G. Derry, and D. E. Logan, Phys. Rev. B91