Spin texture in a bilayer high-temperature cuprate superconductor
SSpin texture in a bilayer high-temperature cuprate superconductor
Xiancong Lu and D. Sénéchal Department of Physics, Xiamen University, Xiamen 361005, China Département de physique and Institut quantique, Université de Sherbrooke, Sherbrooke, Québec, Canada J1K 2R1
We investigate the possibility of spin texture in the bilayer cuprate superconductor Bi Sr CaCu O + δ using cluster dynamical mean field theory (CDMFT). The one-band Hubbard model with a small interlayerhopping and a Rashba spin-orbit coupling is used to describe the material. The d -wave order parameteris not much affected by the presence of the Rashba coupling, but a small triplet component appears. Wefind a spin texture circulating in the same direction around k = (
0, 0 ) and k = ( π , π ) and stable against thesuperconducting phase. The amplitude of the spin structure, however, is strongly affected by the pseudogapphenomenon, more so than the spectral function itself. While electron-electron interactions are a key ingredi-ent in the study of quantum materials, the presence ofa spin-orbit coupling (SOC) is the source of new emer-gent phenomena, especially in heavy transition metal com-pounds [ ] . The SOC is a key ingredient of the topologicalstates of matter [
2, 3 ] . The interplay or competition be-tween SOC and electron correlations is relevant in systemslike the heavy fermion superlattices [ ] , iridium oxides [ ] ,and optical lattices [ ] , in which exotic phases are expected.Within the Rashba-Hubbard model, a mixed singlet-tripletsuperconducting state [ ] , novel magnetism [
16, 17 ] ,and nontrivial topological properties [
12, 18–20 ] are theo-retically predicted.The spin-orbit coupling also manifests itself in a glob-ally centrosymmetric crystal, which contains subunits inwhich the inversion symmetry is broken locally [ ] , i.e. , a locally non-centrosymmetric crystal. The bilayerSOC system is a typical example, in which the SOC ontwo non-equivalent layers have opposite signs, such as theCeCoIn / YbCoIn hybrid structure [ ] , SrPtAs [
23, 26 ] ,and bilayer transition metal dichalcogenides (TMDs) [ ] . The absence of local inversion symmetry in the bilayersystem can lead to a “hidden” spin polarization [
24, 29 ] ,nontrivial topological states [ ] , and unconventionalsuperconductivity [
23, 28, 33 ] .Recent spin- and angle-resolved photoemission spec-troscopy (SARPES) experiments have shown that, inone of the most studied cuprate superconductorsBi Sr CaCu O + δ (Bi2212), a striking spin texture developsin the Brillouin zone with spin-momentum locking [ ] .The observed spin texture is consistent with the predictionof a bilayer model with opposite Rashba SOC on the twoCuO layers of the unit cell. This has motivated new studiesfocusing on the hidden SOC in high- T c cuprates [ ] .However, correlation effects and a possible competitionbetween the spin texture and d -wave superconductivityhave not been fully considered so far. In this paper, we willaddress these important issues in a fully dynamical studyemploying cluster dynamical mean field theory (CDMFT). Model —
To describe the locally non-centrosymmetricbilayer high- T c cuprate, we use the following tight-binding bilayer Rashba model [ ] : H = H kin + H SOC + H ⊥ + H U . (1)The non-interacting part includes three terms, H kin = (cid:88) l , k , σ (cid:34) ( k ) c † l k σ c l k σ (2) H SOC = (cid:88) l , k , σ , σ (cid:48) λ l g k · σ σσ (cid:48) c † l k σ c l k σ (cid:48) (3) H ⊥ = (cid:88) k , σ t ⊥ ( k ) (cid:148) c †1 k σ c k σ + H.c (cid:151) (4)where c l k σ ( c † l k σ ) is the annihilation (creation) operatorof an electron on the l th layer ( l =
1, 2) with spin σ = ↑ , ↓ and wave vector k . The dispersion relation on a squarelattice is (cid:34) ( k ) = − t ( cos k x + cos k y ) + t (cid:48) cos k x cos k y − t (cid:48)(cid:48) ( cos 2 k x + cos 2 k y ) − µ , in which the hopping terms upto third nearest neighbor ( t , t (cid:48) , and t (cid:48)(cid:48) ) and the chemicalpotential µ are included. g k = ( − sin k y , sin k x , 0 ) definesthe antisymmetric SOC of Rashba type and σ is the vectorof Pauli matrices. Due to the global inversion symmetry,the SOC in layers 1 ad 2 are opposite in sign: λ = − λ .The interlayer hoping is t ⊥ ( k ) = t z ( cos k x − cos k y ) , whichcauses the bilayer splitting in high- T c cuprate. For Bi2212,the nearest neighbor hopping is t =
360 meV [
39, 40 ] . Inthe remainder of this paper, we set t = t (cid:48) = − t (cid:48)(cid:48) = t z = [ ] . Finally, we set λ = − λ = H U = U (cid:88) r , l n r , l , ↑ n r , l , ↓ (5)in which n r , l , σ is the number of electrons of spin σ at latticesite r of layer l . Only on-site interactions are considered. CDMFT —
In order to reveal the spin texture arisingin model (1), we use cluster dynamical mean-field theory(CDMFT) [ ] with an exact diagonalization solver atzero temperature (or ED-CDMFT). In this approach, the in-finite lattice is tiled into identical units (or supercells ) defin-ing a superlattice. The supercell is made of one or more clusters , each of which coupled to a bath of uncorrelated, a r X i v : . [ c ond - m a t . s t r- e l ] F e b auxiliary orbitals. The parameters describing this bath (en-ergy levels, hybridization, etc.) are then found by imposinga self-consistency condition. ( a )( b ) p y ,2 − ∆ p y ,2 − ∆ ∆ , p x ∆ , p x p y ,1 − ∆ p y ,1 − ∆ ∆ , p x ∆ , p x θ θ θ θ θ θ θ θ ε ε ε ε ε ε ε ε Figure 1. (Color online). The cluster-bath systems used in our im-plementation of ED-CDMFT. Panel (a) : position of the two clustersforming the supercell in 3D. Panel (b) : the cluster (blue) and bath(red) orbitals, with the various bath parameters used in this study.See text for details.
In this work the supercell is made of two superimposed,four-site plaquettes (one per layer), each of which coupledto a bath of eight uncorrelated orbitals. The cluster-bathsystem, or impurity model , is illustrated on Fig. 1 and de-fined by the following Anderson impurity model (AIM): H imp = H c + (cid:88) µ , α θ µ , α (cid:128) c † µ a α + H.c. (cid:138) + (cid:88) αβ ε αβ a † α a β , (6)where H c is the Hamiltonian (1), but restricted to a sin-gle cluster, and c µ and a α destroy electrons on the clus-ter sites and the bath orbitals, respectively. Probing su-perconductivity forces us to use the Nambu formalism, inwhich each degree of freedom is occurring in particle andhole form in a multiplet. Thus, the index µ is a compos-ite index comprising cluster site i , spin and Nambu indices: c i = ( c i ↑ , c i ↓ , c † i ↑ , c † i ↓ ) . This index takes 4 × =
16 valuesin the particular AIM that we use. Likewise, the index α comprises bath orbital index r , spin and Nambu indicesand takes 4 × =
32 values: a r = ( a r ↑ , a r ↓ , a † r ↑ , a † r ↓ ) . θ µα is a complex-valued, 16 ×
32 hybridization matrix betweencluster and bath orbitals, whereas ε αβ is a 32 ×
32 matrixof one-body terms within the bath, including possible su-perconducting pairing. In principle, the matrix ε αβ couldbe diagonalized (this would change the values of the hy-bridizations θ µα ), but we find it convenient and intuitive to allow pairing operators between bath orbitals.The bath parameters are assumed to be spin indepen-dent, since we are not looking for magnetic ordering. Inorder to probe superconductivity, we include singlet andtriplet pairing operators within the bath. Given two bathorbitals labeled by r and s , the following pairing operatorsare defined:ˆ ∆ rs = a r ↑ a s ↓ − a s ↓ a r ↑ (singlet) (7)ˆ d ( x ) rs = a r ↑ a s ↑ − a r ↓ a s ↓ ˆ d ( y ) rs = i (cid:0) a r ↑ a s ↑ + a r ↓ a s ↓ (cid:1) ˆ d ( z ) rs = (cid:0) a r ↑ a s ↓ + a r ↓ a s ↑ (cid:1) (triplet) (8)In terms of the bath orbital numbering scheme defined onFig. 1b, the pairing terms in H imp are ∆ (cid:0) ˆ ∆ + ˆ ∆ − ˆ ∆ − ˆ ∆ (cid:1) + ∆ (cid:0) ˆ ∆ + ˆ ∆ − ˆ ∆ − ˆ ∆ (cid:1) + ip x (cid:128) ˆ d ( x ) + ˆ d ( x ) (cid:138) + ip y (cid:128) ˆ d ( y ) + ˆ d ( y ) (cid:138) + ip x (cid:128) ˆ d ( x ) + ˆ d ( x ) (cid:138) + ip y (cid:128) ˆ d ( y ) + ˆ d ( y ) (cid:138) + H.c. (9)These terms are included in the one-body matrix ε αβ .The AIM is characterized by 10 variational parameters,all illustrated on Fig. 1b: bath energy levels ε (diago-nal elements of ε αβ ), hybridization amplitudes θ , singletpairing amplitudes ∆ and triplet pairing amplitudes p x , p x , p y , p y . It turns out that, owing to the rather smallvalue of λ in Eq. (3), the triplet bath parameters are toosmall to have an observable effect and can be neglected.This reduces the number of independent bath parametersto six. The two clusters forming the supercell also happento have the same bath parameter values in the convergedsolutions, which is expected from symmetry.For a given set of bath parameters, the AIM (6) may besolved and the electron Green function computed. The lat-ter may be expressed as G c ( ω ) − = ω − t c − Γ c ( ω ) − Σ c ( ω ) (10)where t c is the one-body matrix in the cluster part of theimpurity Hamiltonian H imp , Σ c ( ω ) is the associated self-energy, and Γ c ( ω ) is the bath hybridization matrix: Γ c ( ω ) = θ ω − ε θ † (11)where θ is the 16 ×
32 matrix with components θ µα and ε the 32 ×
32 matrix with components ε αβ . Equation (10)allows us to extract the cluster self-energy Σ c ( ω ) fromcomputed quantities. The fundamental approximation ofCDMFT is to replace the full self-energy of the problem withthe local self-energy Σ . More precisely, when the supercellcontains more than one cluster, as is the case here, the su-percell self-energy is the direct sum of the self-energies of o r d e r p a r a m e t e r d x − y (monolayer) d x − y (bilayer)triplet ( × Figure 2. (Color online) d -wave superconducting order parameteras a function of doping for U =
8, in the single-layer model (opencircles) and the bilayer model with spin-orbit coupling (blacksquares). The triplet component in the bilayer case is roughly 100times smaller (green diamonds). the different clusters: Σ ( ω ) = (cid:76) c Σ c ( ω ) . The lattice Greenfunction is then approximated as G ( ˜k , ω ) = ω − t ( ˜k ) − Σ ( ω ) (12)where ˜k is a wave vector of the reduced Brillouin zone(associated with the superlattice) and t ( ˜k ) is the one-bodyHamiltonian (2)-(4) expressed in that mixed basis of re-duced wave vectors and supercell orbitals. In our system,the matrix G ( ˜k , ω ) has dimension 2 × =
32, because ofthe two clusters forming the supercell.Let us finally summarize the self-consistent procedureused to set the bath parameters, as proposed initiallyin [ ] : (i) trial values of the bath parameters are chosenon the first iteration. (ii) For each iteration, the AIM (6)is solved, i.e., the cluster Green functions G c ( ω ) are com-puted using the Lanczos method, for each cluster. (iii) Thebath parameters are updated, by minimizing the distancefunction: d ( ε , θ ) = (cid:88) c , ω n W ( i ω n ) (cid:2) G c ( i ω n ) − − ¯ G c ( i ω n ) − (cid:3) (13)where ¯ G c ( ω ) is the restriction to cluster c of the projectedGreen function ¯ G , defined as¯ G ( ω ) = (cid:90) d k ( π ) G ( k , ω ) . (14)(iv) We go back to step (ii) and iterate until the bath param-eters or the bath hybridization functions Γ c ( ω ) stop varyingwithin some preset tolerance.Ideally, ¯ G c ( ω ) should coincide with the impurity Greenfunction G c ( ω ) , but the finite number of bath parametersdoes not allow for this correspondence at all frequencies.This is why a distance function d ( ε , θ ) is defined, with em-phasis on low frequencies along the imaginary axis. The weight function W ( i ω n ) is where the method has some ar-bitrariness; in this work W ( i ω n ) is taken to be a constantfor all Matsubara frequencies lower than a cutoff ω c = t ,with a fictitious temperature β − = t / (cid:71) ( k , ω ) , where k now belongsto the original Brillouin zone and (cid:71) is a smaller, 8 × periodization . The simplest pe-riodization scheme is to Fourier transform G directly fromthe supercell to the original Brillouin zone, as follows [ ] : (cid:71) i j ( k , ω ) = N c (cid:88) r , r (cid:48) e − i k · ( r − r (cid:48) ) G r i , r (cid:48) j ( k , ω ) (15)where i , j are composite spin, Nambu and layer indices, andthe difference between k and ˜k is an element K of the recip-rocal superlattice: k = ˜k + K . Note that G ( k , ω ) = G ( ˜k , ω ) since G is by construction a periodic function of the reducedBrillouin zone. Results and discussion —
Figure 2 shows the d -wave or-der parameter, computed from the Green function (12), asa function of hole doping, for U =
8. This is the groundstate average of the following operator:ˆ ∆ x − y = c r , ↑ c r + x , ↓ − c r , ↓ c r + x , ↑ − c r , ↑ c r + y , ↓ + c r , ↓ c r + y , ↑ + H.c.(16)where x and y denote the nearest-neighbor vectors on thesquare lattice. The blue curve is obtained in a single-layermodel, without spin orbit coupling. The black squares areobtained in the current bilayer model, and differ very littlefrom the single layer values, because of the small value ofboth t z and λ . The green diamonds are the average ofthe following triplet operator:ˆ d ( y ) x = c r , ↑ c r + x , ↑ + c r , ↓ c r + x , ↓ + H.c. (17)Note the factor of 100 in the scale. A similar operator de-fined along the y axis with the x component of the triplet d -vector has equal expectation values.The periodized Green function (15) give us access toquantities observable by SARPES, such as the spectral func-tion: A ( k , ω ) = − Im Tr N (cid:128) (cid:71) ( k , ω ) (cid:138) , (18)where Tr N means a trace over spin and layer indices thatexcludes the Nambu sector. One can also extract spin in-formation by projecting the Green function (cid:71) on variousspin directions. We thus define the following spin spectralfunctions: S a ( k , ω ) = − Im Tr N (cid:0) σ a (cid:71) ( k , ω ) (cid:138) ( a = x , y , z ) (19)One can also define the corresponding layer-resolved quan-tities. − π π k y ( a ) µ = n = ( b ) | S ⊥ |− π π k y ( c ) µ = n = ( d ) | S ⊥ |− π π k x − π π k y ( e ) µ = n = − π π k x ( f ) | S ⊥ | Figure 3. (Color online) Spectral function and spin textures for U = n . The left panels(a,c,e) show a color plot of the spectral function at the Fermi levelwithin the Brillouin zone. The non-interacting Fermi surfaces forthe same dopings are shown as a gray dashed line. The right pan-els (b, d, f) show the spin texture: the color plot is the magnitudeof the transverse vector S ⊥ ( k ) = ( S x ( k , 0 ) , S y ( k , 0 )) and the dilutedset of arrows indicates its direction. The color scale is the same forall three densities (blue is lowest, red is highest). Note that thefour-dot structure at n = [ ] ; only the dot touching the non-interactingFermi surface is significant, the other three are “harmonics”. Figure 3 shows the spin texture measured in three of thesuperconducting solutions obtained, for three different val-ues of density n (underdoped, optimally doped and over-doped), on the first layer. The left panel shows a color plotof the spectral function (18) at the Fermi level (the non-interacting Fermi surface is indicated by a dashed line). Theright panel shows the magnitude of the projection on the x - y plane of the spin spectral function (19), on which wesuperimposed a (diluted) vector plot indicating the direc-tion of the spin in the x - y plane, as given by the vector ( A ( k , 0 )) max ( | S ⊥ ( k ) | ) Figure 4. (Color online) Maximum value of the spectral function A ( k , 0 ) and of the spin texture | S ⊥ ( k ) | over the Brillouin zone, forlayer 1, as a function of doping, normalized to the same quantitiesat the last value of doping computed ( x = x → S ⊥ ( k ) = ( S x ( k , 0 ) , S y ( k , 0 )) . On this plot the arrows onlyprovide a direction, not a magnitude. The color scale is thesame for all three values of electron density in the figure.The structure of the spin texture is similar for all three casesillustrated: it is made of a clockwise rotating pattern around k = (
0, 0 ) and around k = ( π , π ) , on the first layer. Thecorresponding quantities on the second layer are obtainedsimply by reversing the arrows.As expected, the amplitude of the spin texture is maxi-mal in the vicinity of the non-interacting Fermi surface, withthe important proviso that the pseudogap phenomenon sup-presses the spectral weight (and the amplitude of the spintexture) away from the diagonals as we go in the under-doped regime. Exactly on the Fermi surface, at least nearthe diagonals, the spin texture vanishes as it must changedirection, leading to split maxima along the diagonals onthe right panels of Fig. 3. Figure 4 shows the drop of thespectral weight (indeed its maximum value over the Bril-louin zone at the Fermi level) as doping x decreases (bluecircles). This suppression is even more pronounced for themaximum value of | S ⊥ | (red squares).Overall, the known physics of the two-dimensional, one-band Hubbard model is not affected by the presence of thespin-orbit coupling between the two layers or by the inter-layer coupling. The d -wave order parameter (Fig. 2) is notaffected in any visible way and the triplet component of theorder parameter is two orders of magnitude smaller thanthe singlet component. The spin-orbit coupling and theassociated spin texture do not interfere with the pseudo-gap physics illustrated by the concentration of the spectralweight along the diagonal as one gets closer to half-filling.The spin-orbit coupling is too small to make the systemtopological [ ] , and we checked that the Chern number,computed by adding the real part of the zero-frequency self-energy to the non-interacting Hamiltonian [
48, 49 ] , is in-deed zero in all the cases studied. The amplitude of the spintexture, however, is more suppressed in the underdoped re-gion than the spectral function is (roughly twice as much).Discussions with Yuehua Su and A.-M. Tremblay aregratefully acknowledged. Computing resources were pro-vided by Compute Canada and Calcul Québec. X.L. is sup-ported by the National Natural Science Foundation of China(Grant No. 11974293) and the Fundamental ResearchFunds for Central Universities (Grant No. 20720180015). [ ] W. Witczak-Krempa, G. Chen, Y. B. Kim, and L. Balents, An-nual Review of Condensed Matter Physics , 57 (2014). [ ] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045 (2010). [ ] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057 (2011). [ ] M. Shimozawa, S. K. Goh, T. Shibauchi, and Y. Matsuda,Reports on Progress in Physics , 074503 (2016). [ ] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Annual Review ofCondensed Matter Physics , 195 (2016). [ ] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A.Duine, Nature Materials , 871 (2015). [ ] P. A. Frigeri, D. F. Agterberg, A. Koga, and M. Sigrist, Phys.Rev. Lett. , 097001 (2004). [ ] T. Yokoyama, S. Onari, and Y. Tanaka, Phys. Rev. B ,172511 (2007). [ ] Y. Yanase and M. Sigrist, Journal of the Physical Society ofJapan , 124711 (2008). [ ] Y. Tada, N. Kawakami, and S. Fujimoto, Journal of the Phys-ical Society of Japan , 054707 (2008). [ ] A. Greco and A. P. Schnyder, Phys. Rev. Lett. , 177002(2018). [ ] X. Lu and D. Sénéchal, Phys. Rev. B , 245118 (2018). [ ] R. Ghadimi, M. Kargarian, and S. A. Jafari, Phys. Rev. B ,115122 (2019). [ ] K. Nogaki and Y. Yanase, Phys. Rev. B , 165114 (2020). [ ] S. Wolf and S. Rachel, Phys. Rev. B , 174512 (2020). [ ] X. Zhang, W. Wu, G. Li, L. Wen, Q. Sun, and A.-C. Ji, NewJournal of Physics , 073036 (2015). [ ] A. Greco, M. Bejas, and A. P. Schnyder, Phys. Rev. B ,174420 (2020). [ ] A. Farrell and T. Pereg-Barnea, Phys. Rev. B , 035112(2014). [ ] M. Laubach, J. Reuther, R. Thomale, and S. Rachel, Phys.Rev. B , 165136 (2014). [ ] E. Marcelino, Phys. Rev. B , 195112 (2017). [ ] M. H. Fischer, F. Loder, and M. Sigrist, Phys. Rev. B ,184533 (2011). [ ] D. Maruyama, M. Sigrist, and Y. Yanase, Journal of the Phys-ical Society of Japan , 034702 (2012). [ ] M. Sigrist, D. F. Agterberg, M. H. Fischer, J. Goryo, F. Loder,S.-H. Rhim, D. Maruyama, Y. Yanase, T. Yoshida, and S. J.Youn, Journal of the Physical Society of Japan , 061014(2014). [ ] X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger, Nature Physics , 387 (2014). [ ] S. K. Goh, Y. Mizukami, H. Shishido, D. Watanabe, S. Ya-sumoto, M. Shimozawa, M. Yamashita, T. Terashima,Y. Yanase, T. Shibauchi, A. I. Buzdin, and Y. Matsuda, Phys.Rev. Lett. , 157006 (2012). [ ] Y. Nishikubo, K. Kudo, and M. Nohara, Journal of the Phys-ical Society of Japan , 055002 (2011). [ ] A. M. Jones, H. Yu, J. S. Ross, P. Klement, N. J. Ghimire,J. Yan, D. G. Mandrus, W. Yao, and X. Xu, Nature Physics , 130 (2014). [ ] C.-X. Liu, Phys. Rev. Lett. , 087001 (2017). [ ] J. M. Riley, F. Mazzola, M. Dendzik, M. Michiardi,T. Takayama, L. Bawden, C. Granerød, M. Leandersson,T. Balasubramanian, M. Hoesch, T. K. Kim, H. Takagi,W. Meevasana, P. Hofmann, M. S. Bahramy, J. W. Wells, andP. D. C. King, Nature Physics , 835 (2014). [ ] S. Nakosai, Y. Tanaka, and N. Nagaosa, Phys. Rev. Lett. ,147003 (2012). [ ] T. Das and A. V. Balatsky, Nature Communications , 1972(2013). [ ] X.-Y. Dong, J.-F. Wang, R.-X. Zhang, W.-H. Duan, B.-F. Zhu,J. O. Sofo, and C.-X. Liu, Nature Communications , 8517(2015). [ ] J. Ishizuka and Y. Yanase, Phys. Rev. B , 224510 (2018). [ ] K. Gotlieb, C.-Y. Lin, M. Serbyn, W. Zhang, C. L. Smallwood,C. Jozwiak, H. Eisaki, Z. Hussain, A. Vishwanath, and A. Lan-zara, Science , 1271 (2018). [ ] Z. M. Raines, A. A. Allocca, and V. M. Galitski, Phys. Rev. B , 224512 (2019). [ ] T. Hitomi and Y. Yanase, Journal of the Physical Society ofJapan , 054712 (2019). [ ] W. A. Atkinson, Phys. Rev. B , 024513 (2020). [ ] N. Harrison, B. J. Ramshaw, and A. Shekhter, Scientific Re-ports , 10914 (2015). [ ] R. S. Markiewicz, S. Sahrakorpi, M. Lindroos, H. Lin, andA. Bansil, Phys. Rev. B , 054519 (2005). [ ] I. K. Drozdov, I. Pletikosi´c, C. K. Kim, K. Fujita, G. D. Gu,J. C. S. Davis, P. D. Johnson, I. Božovi´c, and T. Valla, NatureCommunications , 5210 (2018). [ ] A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B ,R9283 (2000). [ ] G. Kotliar, S. Y. Savrasov, G. Pálsson, and G. Biroli, Phys. Rev.Lett. , 186401 (2001). [ ] A. Liebsch, H. Ishida, and J. Merino, Phys. Rev. B , 165123(2008). [ ] D. Sénéchal, in
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