Superconducting pairing in resonant inelastic X-ray scattering
SSuperconducting pairing in resonant inelastic X-ray scattering
Yifei Shi , David Benjamin , Eugene Demler and Israel Klich Department of Physics, University of Virginia, Charlottesville, VA 22904, USA Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA
We develop a method to study the effect of the superconducting transition on resonant inelastic X-ray scattering (RIXS) signal in superconductors with an order parameter with an arbitrary symmetrywithin a quasiparticle approach. As an example, we compare the direct RIXS signal below and abovethe superconducting transition for p -wave type order parameters. For a p -wave order parameter witha nodal line, we show that, counterintuitively, the effect of the gap is most noticeable for momentumtransfers in the nodal direction. This phenomenon may be naturally explained as a type of nestingeffect. INTRODUCTION
The description of many-body systems is usually onlypractical in terms of simplified low energy theories. Suchtheories are indispensable and describe a large varietyof measurements such as conductance and magnetic re-sponse. However, measurements based on scatteringtechniques often probe wider energy scales. Indeed, pow-erful probes such as resonant inelastic X-ray scattering(RIXS), are allowing unprecedented access to a widerange of excitations in superconducting and magneticsystems. In particular, the superconducting gap scaleis tiny in comparison with band energies of most mate-rials and often below the experimental energy resolutionscale. It is therefore of interest to ask to which extentdetails of low energy theories, such as the gap functionare observable through RIXS. Recently, it was suggestedthat RIXS can distinguish between different phases ofthe order parameter [1, 2]. This dependence is studiedthrough the dynamical structure factor which is shownto discriminate between singlet and triplet pairing. Thestructure factor itself is related to the RIXS signal onlyin the limit of ultra short core hole life time, for which amore elaborate treatment is needed [3].Here, we set out to examine the effect of supercon-ducting pairing on the RIXS mechanism within a sim-ple mean field BCS picture, which includes the effectof core hole potential and goes beyond the ultra shortcore hole life time approximation which is used to relateRIXS with dynamical structure functions. We derive ageneral formula for the RIXS intensity for an arbitraryquadratic Fermi Hamiltonian, with anomalous pairing ∆,as expressed in Eq. (3) together with (11). This re-sult generalizes the quasi-particle approach of [4], wherethe computation of RIXS spectra was performed usinga model of non-interacting quasiparticles but includingan interaction with a positively-charged core hole via ex-act determinant methods. This formalism allows us tocompute the characteristics of the signal by numericallyevaluating (11). Moreover, the computations can be donefor arbitrary band structures using relatively straightfor-ward numerical means. As a demonstration of the method, throughout thepaper we will concentrate on p wave superconductingstates. p + ip superconductors are of great current in-terest. Such superconductors can support unpaired Ma-jorana fermions at cores of (half quantum) vortices [5, 6],and allow for non-Abelian statistics [7, 8]. Remarkably,we find that the RIXS signal is sensitive to the presenceof a superconducting gap ∆, even down to a scale where∆ is quite small (a few percent) compared to the valueof band parameters. In particular, going through thesuperconducting phase transition ∆ acquires a non-zerovalue and we expect the RIXS spectra to experience asignificant change.Resonant inelastic X-ray scattering is an importanttechnique for the investigation of a large variety of ex-citations in correlated systems. Its main advantage isthe wide range of energy scales to which it is sensitive:from low energy excitations, such as phonons, to chargedexcitations of several eV . Another advantage is thatit is a bulk measurement. The physical mechanism atplay in a RIXS experiment is a second-order photon ab-sorption process, involving a shake-up of the system dueto an abrupt appearance of a core hole potential. Thenon-equilibrium process involved may be rather compli-cated, and thus the interpretation of experimental mea-surements may not be straightforward.In the process, photons with energy ω and momentum q , are scattered, and the outgoing photons have energy ω − ∆ ω , and momentum q + Q (we take (cid:126) = 1 through-out). A complete description of the RIXS intensity wouldrequire the consideration the full interacting dynamicsof the sample, which is too hard to achieve. Below wewill start from the standard approach, using from theKramers-Heisenberg cross section[9]: I ( ω, k , k (cid:48) ) ∝ (cid:88) f | F fg | × δ ( E g − E f + ∆ ω ) , (1)with F fg = (cid:88) l,n e i Q · R n (cid:104) f | d n | l (cid:105)(cid:104) l | d † n | g (cid:105) E g + ω − E l + iΓ . (2)Here, | f (cid:105) , | g (cid:105) are the initial and final state, respectively, of a r X i v : . [ c ond - m a t . s up r- c on ] S e p the electron band, and E f,g are their energies. The opera-tor d n creates a quasiparticle in a conduction band at site R n . The states | l (cid:105) are the set of eigenvectors of the inter-mediate Hamiltonian H n = H + V n , where the remainingcore-hole is interacting with the conduction band througha potential V n . The form of the potential V n may be ar-bitrary. In this paper we used both the local form V n = U c d † n d n , describing an on site interaction with a local corehole, as well as V n = U c d † n d n + U (cid:48) c (cid:80) | R n − R (cid:48) n | =1 d † n (cid:48) d n (cid:48) , toaccount for the effect of the coulomb interaction on theneighboring sites. Here Γ is the inverse of the core-holelifetime, which we take a typical value of order 0 . eV .It is important to note that the Kramers-Heisenbergformula (1) is incomplete in that it doesn’t properly ac-count for the photoelectron-core-hole Coulomb interac-tion (see e.g. [10, 11]). Here, however, we neglect sucheffects as we are only interested in the physics involvingthe band structure. Indeed, these effects (for example,mixing between L and L absorption edges) are morepronounced in lighter elements, while in heavier elementsof interest for high T c superconductivity as well as the p -wave system described here, the L , L separation inenergy is very large (of order 20 eV for Cu and 130 eV for Ru ). AN EXACT DETERMINANT FORMULA USINGMAJORANA FERMIONS
Following [4], we write the intensity as: I ∝ (cid:90) ∞−∞ d s (cid:90) ∞−∞ d t (cid:90) ∞−∞ d τ e i ω ( τ − t ) − i s ∆ ω − Γ( t + τ ) × (cid:88) mn e i Q ( R m − R n ) S mn , (3)with S mn = (cid:104) e i Hτ d n e − i H n τ d † n e i Hs d m e i H m t d † m e − i H ( t + s ) (cid:105) . (4)As long as the various stages in the time evolution aregoverned by quadratic Fermi operators, (4) can be cal-culated by exact diagonalization methods. Considerfermions on a lattice with N = L × L sites, governedby a mean field Hamiltonian: H = (cid:88) i,j h ij d † i d j + ∆ ij d i d j + h.c. (5)To handle arbitrary superconducting pairing ∆ ij , we rep-resent the fermion creation and annihilation operators interms of 2 N Majorana fermions c k defined as: c k = (cid:40) d k + d † k k = 1 , , ...N i( d † k − N − d k − N ) k = N + 1 , N + 2 , ... N , (6)and satisfying the relation { c i , c j } = 2 δ ij . The Hamil-tonian (5) can be re-expressed in terms of the Majorana fermions as H = (cid:88) ij h ij c i c j , (7)with h the antisymmetric matrix: h = 14 (cid:18) i Im( h + 2∆) i Re(2∆ + h ) i Re(2∆ − h ) i Im( h − (cid:19) . (8)Traces involving quadratic Hamiltonians of the form A = a ij c i c j where a ij is an anti-symmetric matrix, canbe calculated by using the counting statistics formulaspresented in, e.g. [12]. As shown in the appendix, thetrace formulaTr( e A ... e A n ) = (cid:112) det(1 + e a ...e a n ) , (9)leads, in the direct RIXS case, to the three distinct con-tributions to S mn , S mn = S mn + S mn + S mn . (10)The contributions are detailed in the Appendix, but wemention that in the absence of a core hole potential S contributes only to the elastic signal, while in the ab-sence of superconducting pairing, S vanishes. We notethat the sign of the square root in equation (9) is deter-mined to be consistent with analyticity of the expressionas function of t, s, τ . The first term is given explicitlyby: S mn = (cid:112) det( F )(Λ n,m + Λ n + N,m + N − iΛ n + N,m − iΛ n,m + N ) × (Γ m,n + Γ m + N,n + N − iΓ m + N,n + iΓ m,n + N ) . (11) Here Λ n,m , Γ n,m are elements of the 2 N × N matricesΛ = e i h s e i h m t e i ( τ − t − s ) h G − (1 − N β ) e − i ( τ − t − s ) h e − i h m t Γ = e i ( τ − t − s ) h N β F − , (12)where N β = e − β h , K = e − i h n τ e i h s e i h m t e i ( τ − t − s ) h , F = 1 − N β + KN β , G = 1 − N β + N β K . Here h m represent the Hamiltonian with core hole at position m (i.e. H m = (cid:80) ij ( h m ) ij c i c j ). We stress that the equations(3,4,11) are valid for any type of mean field pairing andare the main technical result. We now turn to apply thesefor a particular pairing, of p wave form. APPLICATION TO A p + ip SUPERCONDUCTOR
To be concrete, we take a minimal toy model for a p wave superconductor. We use a two-dimensional, spin-less fermionic system, on a square lattice, with super-conducting gap ∆. In the Hamiltonian (5), we chooseband structure parameters sometimes used for Stron-tium Ruthenate, Sr RuO . Following [13], we choose h ii = − µ , h i,i +ˆ x = h i,i +ˆ y = − t , h i,i ± ˆ x ± ˆ y = − t . Toget a p x + ip y superconducting state, we take, to be con-crete, ∆ i,i +ˆ x = ∆, ∆ i,i +ˆ y = i∆, with ( µ, t , t , ∆) =(1 . , . , . , . ε , where ε ∼ . eV [14]. In com-parisons with Sr RuO , the Hamiltonian (5) is associ-ated with the so-called γ band of the d xy orbitals inthe Ruthenate. The signal may also get contributionsfrom additional quasi-1d bands associated with d xz , d yz orbitals, with hopping ∼ ε . We have also carried out ex-plicit calculations for the d xz and d yz bands, however wefocus here on the γ band.To explore the role of the superconducting gap, wecalculated the RIXS intensity across the superconduct-ing phase transition using Eq. (11). As is shown in Fig.1, for Q = 0 . π, , . π, U c = 0, and will be describedbelow.The spectral density flow to higher energy can be sim-ply understood by noting that the main contributions toRIXS intensity occur due to the generation of electron-hole pairs where one of them is close to the Fermi level.In the presence of pairing, states close to the Fermi levelare unavailable - the incoming photon must first over-come the energy gap, and thus the energy difference inthe subsequent electron-hole pair is higher. On the otherhand, as we see below, surprisingly, the increase in inten-sity at a direction Q is not directly related to the pairing∆ Q at that wave vector. THE U C = 0 APPROXIMATION
As stated above, many interesting differences in theRIXS signal below and above the SC transition can beobserved already for small U c . In this limit we can com-pute the RIXS more efficiently using perturbation theory.We consider an expansion in terms of U c for F fg in (2).For a simple on-site core hole potential V we write: G = ( H m − E g + i Γ + ω ) − ∼ G (0) − U c G (0) ( d † m d m ) G (0) + ... " ! I n t en s i t y " =0 " =0.05 " =0.1 Q=0.1( π ,0) " ! I n t en s i t y " =0 " =0.05 " =0.1 Q=0.15( π ,0) FIG. 1: Upper panels: RIXS intensity across the tran-sition for a p x + ip y superconductor. (Left panel: Q = 0 . π, Q = 0 . π, U c = 1 ε ,for different values of ∆ (in units of ε ). ∆ increases thespectral weight for higher energy exchanges, shifts thepeak position, and increases the intensity. Notice theshift of the peak position is not linear in ∆. Lower pan-els: spectral shift due to core hole potential as com-pared to U c = 0 for ∆ = 0 . ε .where G (0) = ( H − E g + i Γ+ ω ) − , is the propagator withno core-hole. From here on we take only the lowest ordercontribution, where U c = 0. The theory is then exactlysolvable in terms of the eigenstates of the static prob-lem, and we can calculate the intensity efficiently. Wefirst solve the energy spectrum by switching to momen-tum space and writing the Hamiltonian in the standardBogoliubov-de Gennes form: H = 12 (cid:88) k (cid:2) d † k d k (cid:3) (cid:20) (cid:15) k ∆ k ∆ ∗ k − (cid:15) k (cid:21) (cid:20) d k d †− k (cid:21) (13)where (cid:15) k = − µ − t [cos( k x ) + cos( k y )] − t cos( k x )cos( k y ), and ∆ k = 2i∆[sin( k x ) + i sin( k y )], theHamiltonian is diagonized by a Bogoliubov transforma-tion: d k = u ∗ k b k + v k b †− k d †− k = − v ∗ k b k + u k b †− k (14)the energy of the excitation is now E k = (cid:112) (cid:15) k + | ∆ k | , | u k | + | v k | = 1, and u k v k = ∆ k E k − (cid:15) k , the ground state isannihilated by all b k s, and F fg in Eq. (2) is now givenexplicitly by: F fg = (cid:88) k , k , r e i r · ( k − k + Q ) v k u k E k + ω + i Γ (cid:104) f | b †− k b † k | g (cid:105) (15)From (15) we see that in the quasiparticle picture,FIG. 2: RIXS intensity map on Q , ∆ ω plane, in the(11) and (10) directions (in units of π ). Upper panel isin the normal phase (∆ = 0), lower panel is the super-conducting phase (∆ = 0 . ε ). The calculation is doneusing Eq. (15).FIG. 3: RIXS intensity map for a spinless p x + p y typeof pairing function (∆ k = i k x + sin k y )) with∆ = 0 . ε , in the anti-nodal (1 , , −
1) for Q . In the absence of pairing the two direc-tions should have the same intensity, thus the differencecomes purely from the presence of the superconductinggap. Surprisingly, the effect is more pronounced on thenodal direction where ∆ = 0. the contribution to RIXS intensity comes from pairs ofquasiparticles with momenta k and k + Q , energies E k and − E k + ∆ ω . When there is no pairing term, these arean electron and a hole, and in the presence of a pairingterm, these are the Bogoliubov quasiparticles. Goingto the superconducting phase, the energy spectrumbecomes E k = (cid:112) (cid:15) k + | ∆ k | , when | (cid:15) k | (cid:29) | ∆ k | , we have E k ∼ | (cid:15) k | , which is the case in most of the BrillouinZone as | ∆ | is small compared to other band parameters.Thus the change in the RIXS intensity comes mainlyfrom pairs where at least one quasiparticle is close to theFermi surface, there the energy spectrum and density ofstates change significantly. For a pair of quasiparticles,one close to the Fermi surface, where | (cid:15) k | < ∆ k , with E k ∼ | ∆ k | , and E k ∼ (cid:15) k , we have ∆ ω ∼ | ∆ k | + (cid:15) k .The same pair without the pairing term will contributeto the intensity at ∆ ω ∼ (cid:15) k . In Fig. 2 we show theintensity as a function of Q and ∆ ω , as calculatedfrom the lowest order contribution (15) for the p + ip superconducting state in comparison with its normalstate. The figure shows that for small Q , the intensityis enhanced, which is consistent with having an energygap forcing larger energy transfers for two quasiparticlesnear the Fermi sea.A yet more intriguing situation is that of a super-conducting order like p x + p y which, as opposed to the p x + ip y , exhibits nodal lines. Nodal lines are unusualbut in principle allowed for p-wave systems, both for so-called unitary and non-unitary states [15]. In Fig. 3 theRIXS intensity in the nodal and anti-nodal directions,(1 , −
1) and (1 , k x , k y ), ( k x + q, k y + q ), andenergies (cid:15) , (cid:15) , which contributes to the intensity at∆ ω in the (1 ,
1) direction. Another electron-hole pairwith ( k x , − k y ), ( k x + q, − k y − q ), will have thesame energies, since (cid:15) ( k x , k y ) = (cid:15) ( ± k x , ± k y ), but willcontribute intensity in the (1 , −
1) direction.The effect of the pairing term can be understood bylooking at F fg over the Brillouin zone. In (15), the sum-mation over r gives a delta function and we can write: F fg = (cid:88) k v k u k + Q E k + Q + ω + i Γ (16)where we took | f (cid:105) = b †− k b † k + Q | g (cid:105) . When the system isunpaired, | f (cid:105) describes a particle hole pair whose mo-menta differ by Q and energies differ by ∆ ω .We note that the RIXS intensity is the integral overthe Brillouin zone of the function: F ( k ) = v k u k + Q E k + Q + ω + i Γ δ ( E k + Q + E k − ∆ ω ) . (17) (a) (b)(c) (d) FIG. 4: F k over the Brillouin zone, for a p x + p y pair-ing. The black lines show the original Fermi surface,red regions denote large values of F k from electron likeregions and the blue regions are the associated holelike quasiparticles. (a) and (b): Q = (0 . , . π ,anti-nodal direction, energy transfer ∆ ω = 0 . ε .(a): ∆ = 0, and (b): ∆ = 0 . ε . (c) and (d): Q = (0 . , − . π , nodal direction, energy transfer∆ ω = 0 . ε . (c): ∆ = 0, and (d): ∆ = 0 . ε . Theregion marked with green in (d) is the most affected bythe pairing.To identify the main contributions to the signal in mo-mentum space we now focus on the behavior of F ( k ).In practice, we replace the delta function by: δ ( E ) ∼ e − ( E/E res ) / with E res = 0 . ε , to account for the ex-perimental energy resolution. The result is shown inFig. 4. Because of the symmetry of the Hamiltonian,at ∆ = 0, F k is the same at Q = (0 . , . π and Q = (0 . , − . π , up to 90 ◦ rotation. We can nowsee why for Q = (0 . , . π , in the anti-nodal direc-tion, the effect of pairing is weaker: F k does not changea lot after turning on the pairing, since the significantregions of F k are far from the line k x = k y where thepairing, ∆ k = 2 i (sin( k x ) + sin( k y )) is most significant.However, in the nodal direction, Q = (0 . , − . π , apairing term becomes much more relevant: F k has sig-nificant contributions across the line k x = k y , and inthose regions F k is sensitive to the pairing term (notedby green circles in the plot), resulting in a substantialchange in the RIXS intensity. We thus see that the effectof pairing on intensity is sensitive to the direction of themomentum transfer, and seems to be enhanced in thenodal direction. DISCUSSION AND SUMMARY
We have confined our discussion here to mean fieldBCS and made no speculation about the suitability ofthe treatment to strongly correlated systems and it’s rel-evance, e. g. to high-Tc superconductors. At this point,it is worth mentioning, among the other probes of super-conducting states, the electronic Raman scattering tech-nique. Electronic Raman scattering essentially measuresthe dynamical structure factor [16]:˜ S ( q , ω ) = 12 π (cid:90) ∞−∞ e − iωt (cid:104) ˜ ρ q (0)˜ ρ q ( t ) (cid:105) , (18)which has a very similar form to the 4-point functionmeasured by RIXS, and describes a similar process. Inthe limit where qξ <<
1, where ξ is the coherent length,there will be a peak around 2∆, which is what we getin the small momentum limit for RIXS using Eq. (17).Thus, RIXS allows for a complementary study to thatof the Raman technique. It is also important to notethat RIXS is especially interesting away from the BCSpicture, where one can see contributions from both bandstructure physics and collective excitations, thus to differ-entiate between such effects it is of particular importanceto have a well developed picture of RIXS in the absenceof collective behavior. In particular it is of great inter-est to see how the present approach may affect resultspertaining to the quasiparticle interpretation of RIXS inthe cuprates. Indeed, although our treatment is within amean-field BCS picture, we remark that the method mayalso be of relevance to the study of cuprates. Most recentstudies of RIXS in the context of cuprates have largelyconsidered cases of insulating phases [17–20]. However,RIXS experiments have been performed over a wide rangeof doping, including systems where itinerant electronsare present, and a description using tools developed forinsulators may be insufficient. For example, in [21], itis shown that contrary to a common interpretation, forBi − Sr IrO .In summary, we developed a general formalism totreat the RIXS intensity for a quadratic Fermi theorywith arbitrary pairing. With the introduction of Majo-rana fermions, all quadratic Hamiltonians can be han-dled within the determinant method. The main formu-las are summarized in the equations (3,4,11) which areready for immediate numerical use. Focusing on p -wavesuperconducting states, we have shown within this ap-proach several intriguing effects on the RIXS signal. Themost important findings are: a non linear shift of theRIXS absorption peak below the superconducting tran-sition, as function of ∆, and, for nodal p-wave pairing, abreaking of symmetry between the nodal and anti-nodaldirections, in which, surprisingly, the effect is more pro-nounced in the nodal direction than the anti-nodal di-rection. We have seen pronounced effects of a gap scaledown to a few percent of the band parameters, unfor-tunately, in actual Sr RuO , the pairing is believed tobe of the order 10 − ε , and the effects discussed here willmost likely be outside experimental resolution in this ma-terial with present techniques. However, the method in-troduced here, allows us to readily study other pairedsystems. Similar effects as described for our toy-modelshould be observable when carrying out RIXS measure-ments below and above a superconducting transition. Acknowledgments
It is a pleasure to thank AmitKeren, David Ellis and Marton Kanasz-Nagy for discus-sions. The work of IK and YS was supported by the NSFCAREER grant DMR-0956053. DB and ED acknowl-edge support from Harvard-MIT CUA, NSF Grant No.DMR-1308435, AFOSR Quantum Simulation MURI, theARO-MURI on Atomtronics, ARO MURI Quism pro-gram. [1] P. Marra, S. Sykora, K. Wohlfeld, and J. van den Brink,Physical review letters , 117005 (2013).[2] P. Marra, J. van den Brink, and S. Sykora, ScientificReports , 25386 EP (2016), URL http://dx.doi.org/10.1038/srep25386 .[3] C. Jia, K. Wohlfeld, Y. Wang, B. Moritz, and T. P. De-vereaux, Physical Review X , 021020 (2016).[4] D. Benjamin, I. Klich, and E. Demler, Physical ReviewLetters , 247002 (2014).[5] J. Alicea, Reports on Progress in Physics , 076501(2012).[6] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics , 3045 (2010).[7] D. Ivanov, Physical Review Letters , 268 (2001).[8] J. Alicea, Physical Review B , 125318 (2010).[9] L. J. Ament, M. van Veenendaal, T. P. Devereaux, J. P.Hill, and J. van den Brink, Reviews of Modern Physics , 705 (2011).[10] J. Zaanen, G. Sawatzky, J. Fink, W. Speier, and J. Fug-gle, Physical Review B , 4905 (1985).[11] P. Kr¨uger and C. R. Natoli, Physical Review B ,245120 (2004).[12] I. Klich, Journal of Statistical Mechanics: Theory andExperiment , P11006 (2014).[13] S. Lederer, W. Huang, E. Taylor, S. Raghu, andC. Kallin, Physical Review B , 134521 (2014).[14] H. Kontani, T. Tanaka, D. Hirashima, K. Yamada, andJ. Inoue, Physical review letters , 096601 (2008).[15] A. P. Mackenzie and Y. Maeno, Reviews of Modern Physics , 657 (2003).[16] M. Klein and S. Dierker, Physical Review B , 4976(1984).[17] J. van den Brink, EPL (Europhysics Letters) , 47003(2007).[18] C.-C. Chen, B. Moritz, F. Vernay, J. N. Hancock,S. Johnston, C. J. Jia, G. Chabot-Couture, M. Greven,I. Elfimov, G. A. Sawatzky, et al., Physical review letters , 177401 (2010).[19] F. Vernay, B. Moritz, I. Elfimov, J. Geck, D. Hawthorn,T. Devereaux, and G. Sawatzky, Physical Review B ,104519 (2008).[20] C. J. Jia, E. A. Nowadnick, K. Wohlfeld, Y. F. Kung,C. C. Chen, S. Johnston, T. Tohyama, B. Moritz, andT. P. Devereaux, Nature communications (2014).[21] M. Guarise, B. Dalla Piazza, H. Berger, E. Giannini,T. Schmitt, H. Rønnow, G. Sawatzky, J. van den Brink,D. Altenfeld, I. Eremin, et al., Nature Communications (2014).[22] D. Benjamin, I. Klich, and E. Demler, Physical ReviewB , 035151 (2015).[23] M. Kan´asz-Nagy, Y. Shi, I. Klich, and E. A. Demler,arXiv preprint arXiv:1508.06639 (2015).[24] J.-i. Igarashi and T. Nagao, Physical Review B ,064402 (2014). APPENDIX: CALCULATING S mn WITHPAIRING
Here we give further details regarding the derivationof (11). Explicitly, S xy = (cid:104) e iHτ d y e − iH y τ d † y e iHs d x e iH x t d † x e − iH ( t + s ) (cid:105) = tr[ e iHτ d y e − iH y τ d † y ...... e iHs d x e iH x t d † x e − iH ( t + s ) − βH ] / tr[ e − βH ] . (19)Here, the core-holes act at sites x and y .We first focus on the numerator. When replacing allthe fermions with Majorana operators, we get a combi-nation of terms such as: Num == Σ qmnp tr[ e iHτ c q e − iH y τ c m e iHs c n e iH x t c p e − iH ( t + s ) − βH ]= Σ qmnp tr[ c q e X c m e X c n e X c p e X ] . (20)Defining ξx = x + N , then the nonzero elements of Σ areΣ yyxx = Σ ξy,ξy,ξx,ξx = Σ ξy,ξy,x,x = Σ y,y,ξx,ξx = Σ y,ξy,x,ξx = Σ ξy,y,ξx,ξx = − Σ y,ξy,ξx,x = − Σ ξy,y,x,ξx = − Σ y,y,x,ξx = Σ y,ξy,x,x = Σ ξy,ξy,ξx,x = Σ ξy,y,ξx,ξx = i Σ y,y,ξx,x = Σ ξy,y,x,x = Σ ξy,ξy,x,ξx = Σ y,ξy,ξx,ξx = − i . Using the relation: c m e A i,j c i c j = e A i,j c i c j c m (cid:48) ( e A ) m,m (cid:48) (same indices are summed over), we can move all theMajorana fermions to the right, yielding:Num = Σ qmnp ( e X ) p,p (cid:48) ( e X e X ) n,n (cid:48) ( e X e X e X ) m,m (cid:48) × tr[ e Z ij c i c j c m (cid:48) c n (cid:48) c p (cid:48) c q ] (21)where e Z ij c i c j = e X e X e X e X . Now the task is tocalculate traces of the form: T mnpq = tr[ e Z ij c i c j c m c n c p c q ]= tr[ e Z ij c i c j ( δ mn + c m c n − c n c m )( δ pq + c p c q − c q c p )]= tr[ e Z ij c i c j ( MN + M δ pq + N δ nm + δ mn δ pq )] , (22)where M = M ij c i c j . M = | m (cid:105)(cid:104) n | − | n (cid:105)(cid:104) m | , N = | p (cid:105)(cid:104) q | −| q (cid:105)(cid:104) p | . Now that M and N are anti-symmetric matricesand we can write M = ∂∂α e α M | α =0 , and use the traceformula (9) to calculate T . First we find:tr( e Z ij c i c j ddα e αM ij c i c j | α =0 ) = ∂∂α tr( e Z ij c i c j e αM ij c i c j ) | α =0 = 12 (cid:113) det(1 + e Z e αM )tr[ 4 e Z M e Z e αM ] | α =0 = 2 (cid:113) det(1 + e Z ) { (1 + e − Z ) − nm − (1 + e − Z ) − mn } (23)Next, we define B = e − Z and D = det(1+ e Z ). Then, ∂∂β ∂∂α tr( e Z e αM e βN )= √D{ BM )tr( BN ) − BN BM ) + 8tr(
BM N ) } (24)The last step we take α = 0, and β = 0. Plugging theresult from the above two equations into Eq. (22), wefind: T mnpq = √D{ ( B nm − B mn + δ mn )( B qp − B pq + δ pq )+2 B qm ( δ np − B np ) + 2 B pn ( δ mq − B mq ) − B pm ( δ nq − B nq ) − B qn ( δ mp − B mp ) } (25)Noticing that since Z is anti-symmetric, B nm + B mn = δ mn , we get: T mnpq = 4 √D ( B nm B qp + B qm B pn − B pm B qn ) (26)We plug (26) back into Eq.(21): S xy = (27)Σ qmnp ( e X ) p,p (cid:48) ( e X e X ) n,n (cid:48) ( e X e X e X ) m,m (cid:48) T m (cid:48) n (cid:48) p (cid:48) q . We see that S xy is comprised of 3 terms correspondingto the terms on the right hand side of Eq. (26). We firstfocus on the first term: S = (28)4Σ qmnp √D B nm B qp ( e X ) p,p (cid:48) ( e X e X ) n,n (cid:48) ( e X e X e X ) m,m (cid:48) . It will be convenient to denote K ≡ e − ih n τ e ihs e ih m t e i ( τ − t − s ) h , and N β ≡ e βh .With this notation we have e Z = K N β − N β , B = (1 + − N β N β K − ) − . And we find: S = Σ qmnp ( e X e X e X B T ( e X e X ) T ) mn ( e X B T ) pq , (29) where T is the matrix transpose. In order to get conve-nient expressions in the low temperature limit ( β → ∞ ),we have to calculate e X B ( e X ) T . Using that for anti-symmetric matrix h , e − h = ( e h ) T , we write: e − βh B T ( e − βh ) T = N β − N β − N β N β K − − N β N β K − − N β N β = K − N β + (1 − N β ) K − (1 − N β )= 11 − N β + N β K (1 − N β ) , and e − βh B T = N β − N β
11 + K N β − N β = N β − N β + KN β . Using the above results and summing over m, n, p, q , wehave: S = (cid:112) det( F )(Λ y,x + Λ ξy,ξx − i Λ ξy,x + i Λ y,ξx ) × (Γ y,x + Γ ξy,ξx + i Γ ξy,x − i Γ y,ξx ) (30)whereΛ = e ihs e ih x t e i ( τ − t − s ) h G − (1 − N β ) e − i ( τ − t − s ) h e − ih x t Γ = e i ( τ − t − s ) h N β F − , (31)and F = 1 − N β + KN β , G = 1 − N β + N β K . Similarly,the second term is written as: S = Σ qmnp ( e X e X B T e − X ) pn ( e X e X e X B T ) mq = (Λ (2) y,y + Λ (2) ξy,ξy + i Λ (2) ξy,y − i Λ (2) y,ξy ) × (Γ (2) x,x + Γ (2) ξx,ξx + i Γ (2) ξx,x − i Γ (2) x,ξx ) , (32)where Λ (2) = e − ihs Λ e ih x t and Γ (2) = e ihs e ih x t Γ. For thethird term S , S = Σ qmnp ( e X e X B T ) nq ( e X e X e X B T e − X ) mp = (Λ (3) x,y − Λ (3) ξx,ξy − i Λ (3) ξx,y − i Λ (3) x,ξy ) × (Γ (3) x,y − Γ (3) ξy,ξx + i Γ (3) ξx,y + i Γ (3) x,ξy ) (33)where Λ (3) = Λ e ih x t and Γ (3) = e ih x t Γ.The terms S and S have a special behavior when ei-ther the core-hole potential or the superconducting pair-ing vanishes as follows:(I) S does not contribute to the inelastic signal whenthe core-hole potential U c is 0: In that case, K = I , and S only depends on t and τ , so S only contributes to theelastic scattering.(II) S vanishes when there is no pairing, in that casethe matrices Λ (3) and Γ (3) have the special property thatΛ (3) ( x, y ) = Λ (3) ( ξx, ξy ), Λ (3) ( x, ξy ) = − Λ (3) ( ξx, y ), sothat S3