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A Numerical Renormalization Group Study of the Superconducting and Spin Density Wave Instabilities in a Twoband Model of MFeAsO1 xFx Compounds

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A Numerical Renormalization Group Study of the Superconducting and Spin Density Wave Instabilities in a Twoband Model of MFeAsO1 xFx Compounds
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    a  r   X   i  v  :   0   8   0   5 .   3   3   4   3  v   3   [  c  o  n   d  -  m  a   t .  s  u  p  r  -  c  o  n   ]   2   1   N  o  v   2   0   0   8 A Numerical Renormalization Group Study of the Superconducting and Spin DensityWave Instabilities in a Two-band Model of MFeAsO 1 − x F x  Compounds Fa Wang, 1,2 Hui Zhai, 1,2 Ying Ran, 1,2 Ashvin Vishwanath, 1,2 and Dung-Hai Lee 1,2 1 Department of Physics,University of California at Berkeley, Berkeley, CA 94720, USA 2  Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. (Dated: November 21, 2008)We apply the fermion renormalization group method[1], implemented numerically in Ref.[2], to atwo-band model of FeAs-based materials. At half filling we find the ( π, 0) or (0 ,π ) spin density waveorder and a sub-dominant superconducting pairing tendency. Due to a topological reason, the spindensity wave gap has nodes on the fermi surfaces. Away from half filling we find an unconventional s -wave and a sub-dominant  d x 2 − y 2  pairing instability. The former has  s  symmetry around the holefermi surface but exhibits  s  +  d x 2 − y 2  symmetry around the electron pockets where the 90 degreerotation is broken. The pairing mechanism is inter-pocket pair hopping. Interestingly, the sameinteraction also drives the antiferromagnetism. Recently there is a flurry of interest in the the com-pound MFeAsO 1 − x F x [3]. It has been shown that byvarying the rare earth elements (M)[4], these materialscan be made superconducting with T c  as high as 55K[5].This has stimulated a flurry of interest in these materials.At the present time, preliminary experimental resultshave indicated that these materials exhibit semimetallicantiferromagnetism at stoichiometry (i.e.,  x  = 0). Uponsubstituting O with F the antiferromagnetism diminisheswhile superconductivity appears[6].Structurally MFeAsO 1 − x F x  can be viewed as As-Fe-As tri-layers separated by MO 1 − x F x  spacers. Earlyelectronic structure calculations[7] and angle-integratedphotoemission[10] suggest that the carriers at the fermienergy are essentially Fe in character. Thus, like manyothers, we will focus on a As-Fe-As tri-layer in the fol-lowing discussions. To envision these tri-layers, imaginea square lattice of Fe. The As sit either above or belowthe center of the square plaquettes, and these two typesof plaquette form a checkerboard. Each structural unitcell contains two Fe (dashed rectangle in Fig. (1)(a), andthe basis vectors are  X  = ˆ x  + ˆ y  and  Y  = ˆ x −  ˆ y  where ˆ x and ˆ y  are the basis vectors of the Fe square lattice (seeFig. (1)(a)). In the following the we shall refer to thereciprocal unit cell associated with the basis vector ˆ x,  ˆ y and X , Y  as the “unfolded” and “folded” Brill! ouin zone(BZ) respectively.Aside from the semi-metalicity, there are other impor-tant differences between the antiferromagnetism in thismaterial and that of the cuprates. First, in the unfoldedBZ the magnetic ordering wavector is either (0 ,π ) or( π, 0)[11] instead of the usual ( π,π ). Second, the or-dering moment is quite small ( ∼  0 . 25 µ B )[11] comparedto that of the cuprates. For the superconducting state,penetration depth[12], H c 2 [13], 1 /T  1  of nuclear spins[14], µ SR[15], and point tunneling measurements[16] all in-dicate the presence of line nodes in the superconduct-ing gap. However, more experiments, in particular thosedone on single crystals, will be necessary to check theabove conclusions.On the theoretical side, several electronic structure cal-culations suggest the presence of hole and electron pock-ets at fermi energy. It is suggested that the near nestingof these pockets is responsible for the antiferromagnetismat  x  = 0[7–9]. Estimate of the strength of local coulombinteraction suggests that this system is on the border be-tween strong and weak correlation[17]. Although LDAtype calculations suggest all five orbitals of Fe contributeto states at the fermi energy[7], this was simplified re-cently where only two out of the five bands are kept whilepreserving the electron and hole pockets[18]. Based ondifferent degree of simplification of the electronic struc-ture, and different approximate treatment of the elec-tronic correlation, a number of groups have studied thesuperconducting pairing instability of these material[19].In addition there are a couple of recent attempts to usethe point group symmetry to narrow down the pairingsymmetries[20]. The wide spread in pairing symmetryconcluded from these studies call for a more unbiasedassessment of the pairing instability.In this paper we study a two band model withHubbard-like and Hund interactions. We study the possi-bility of electronically induced pairing by performing one-loop renomalization group (NRG) calculation[1]. Thisnumerical version of this method (NRG) was applied tothe cuprates by Honerkamp et al[2]. It was shown thatin the framework of one-band Hubbard model, interac-tions which promote  d x 2 − y 2  pairing and ( π,π ) antifer-romagnetic order are generated at low energies. We be-lieve that owing to the weaker correlation, this method ismuch better suited for MFeAsO 1 − x F x . The NRG resultsfor five band model [7] will be the subject of upcomingpublication[21]. Model Hamiltonian -  As Ref. [18] we simplify the elec-tronic structure of the As-Fe-As tri-layter by keeping onlytwo Fe orbitals: 3d xz  and 3d yz . The Arsenic are viewedas merely mediating hopping between these orbitals. Dueto the relative orientation of the Fe and As it is more con-  2venient to use as basis 3d XZ   and 3d YZ   where  x,y  and X,Y   are shown in Fig. (1)(a). Our tight binding modelinclude nearest-neighbor and next-nearest-neighbor hop-pings. From symmetry considerations there are four in-dependent hopping parameters  t 1 ,t ′ 1 ,t 2 ,t ′ 2  as shown inFig. (1)(a). Among them  t ′ 1  is due to the direct overlapof two neighboring Fe orbitals. All the rest three param-eters describe hopping mediated by As. In the followingwe shall label the Fe orbitals 3d XZ   and 3d YZ   as  a  = 1 , 2.The tight-binding Hamiltonian in the unfolded BZ readsˆ H  0  =  k ,s 2  a,b =1 c † a k s K  ab ( k ) c b k s  =  k ,s 2  n =1 ǫ n ( k ) ψ † n k s ψ n k s K  ( k ) =  α ( k ) I   +  b x ( k ) τ  x  +  b z ( k ) τ  z ,  (1)where  c a k s  annihilates a spin  s  electron in orbital  a  andmomentum  k , and  ψ n k s  is the band annihilation opera-tor. The  I,τ  x ,τ  y  in Eq. (1) are the 2  ×  2 identity ma-trix and Pauli matrices respectively. They act on theorbital ( d XZ  ,d YZ  ) space, and  α ( k ) =  µ  + 2 t ′ 1  cos k y  +2cos k x [ t ′ 1  +( t 2  + t ′ 2 )cos k y ],  b x ( k ) = 2 t 1 (cos k x − cos k y ), b z ( k ) =  − 2( t 2  − t ′ 2 )sin k x  sin k y , and  ǫ 1 , 2 ( k ) =  α ( k )  ∓   b x ( k ) 2 +  b z ( k ) 2 . We have checked that when actedupon by the element ( g ) of the point group ( C  4 v ) theband operator  ψ a k s  →  η g ψ ag k s  where  η g  =  ± 1. Afterturning on a proper chemical potential  µ  we get two holepockets around  k  = (0 , 0) and ( π,π ) and two electronpockets at (0 ,π ) and ( π, 0). In the rest of the paper weshall use the following values for the hopping parameters t 1  = 0 . 38 eV,t 2  = 0 . 57 eV,t ′ 2  = 0. The value of   t ′ 1  criti-cally determines the superconducting gap function, andwill be discussed later.Now we consider local interactions including intra-orbital and inter-orbital Coulomb interaction  U  1 , and  U  2 ,Hund’s coupling  J  H   and the pair hopping term. Whensummed together they giveˆ H  int  =  i { U  12  a =1 n i,a, ↑ n i,a, ↓  +  U  2 n i, 1 n i, 2 + J  H  [  s,s ′ c † i 1 s c † i 2 s ′ c i 1 s ′ c i 2 s  + ( c † i 1 ↑ c † i 1 ↓ c i 2 ↓ c i 2 ↑  +  h.c. )] } . Here  i  labels the unit cell,  s,s ′ = ↑ , ↓ , and  n i,a  =  n i,a, ↑  + n i,a, ↓  is the number operator associated with orbital  a .The total Hamiltonian ˆ H   = ˆ H  0  + ˆ H  int  is the startingpoint of our study.We have checked that the bare ˆ H   has no superconduct-ing instability for realistic interaction parameters. Thuswe decide to perform a numerical renormalization group(NRG) calculation where high energy electronic excita-tions are recursively integrated out. The hope is thatthe effective interaction generated at low energy wouldshow the sign of superconducting instability. in the fol-lowing we present the result of such a calculation. Tech-nical details of the NRG can be found in Ref. [2]. In t ′ 1 1 − t 1 t 2 t ′ 2 x y X  Y   (a) (b) FIG. 1: (a) The in-plane projection of   d XZ  and  d YZ  orbitalsand the four independent hopping parameters. (b)  b  as afunction of   k . The red curves are the fermi surfaces. brief, we divide the first BZ into N patches, and at eachrenormalization iteration we sum over the five one-loopFeynman diagrams labeled (c1)-(c3) in Fig. (2). The es-sential complication in the present study is the presenceof two different bands and four disjoint pieces of fermisurface. It turns out that it is easier to work with thefolded BZ. This is because folding doubles the numberof bands so that each band has only one fermi surface(Fig. (2)(a)) and the BZ patching scheme of Ref.[2] canbe directly applied. Because of the existence of four band!s, we need to keep track of the band indices as well asthe momenta in the interaction vertex. Thus we need tocompute 256 ×  N  3 different interaction vertices at eachrenormalization step in contrast to  N  3 in the single bandcase[2]. Although our RG calculation is performed withthe folded BZ we shall present our results using the un-folded BZ for simplicity. The antiferromagnetism and superconducting pairing tendency at half filling.  At stoichiometry (x=0) the twobands derived from the  d XZ   and  d YZ   orbitals accom-modate 2 electrons per unit cell. For  t ′ 1  = 0 the elec-tron and hole pockets are perfectly nested by (0 ,π ) , ( π, 0)in the unfolded BZ. In a theory like ours, the antiferro-magnetism is due to the the above nesting. For  t ′ 1   = 0the nesting is imperfect. Hence one might expect a fullspin density wave (SDW) gap for  t ′ 1  = 0 and a partiallygapped fermi surface for  t ′ 1   = 0. However, as we willshow in the following, even for the former case there arenodes in the SDW gap, and the reason is topological.First, we say a few words about how the results areobtained. In the calculation we compute the renor-malized interaction vertex function  V   ( k 1 ,a ; k 2 ,b ; k 3 ,c ; d )for two incoming electrons with opposite spins. Here a,b,c,d  = 1 ,.., 4 are the band indices, and  k 1 ,.., k 3 are momenta on the Fermi surfaces. From this ver-tex function we can extract the renormalized interac-tion in the Cooper or SDW channels as follows. Forthe singlet/triplet Copper channel  V   SC s,t  ( k ,a ; p ,b ) = V   ( k ,a ; − k ,a ; p ,b ; b )  ±  V   ( − k ,a ; k ,a ; p ,b ; b ); for theSDW channel  V   SDW  ( k ,a,d ; p ,c,b ) =  − 2 V   ( k ,a ; p  +  3 - ππ - π π k  Y  k   X  01234 5 67891011121314151617181901234 5678910111213141516171819012345 67891011121314 151617181901234567891011121314 1516171819 (a) Folded BZ FS 0FS 1FS 2FS 30 π 0  π k   y k   x FS 0FS 1FS 2FS 3 (b) 1/4 Unfolded BZ Q =( π ,0) (c1)(c2)(c3) FIG. 2: (a) The division of the first BZ into N (=20) patches.The color curves denote the fermi surface( t ′ 1  =  − 0 . 05). (b)The strongest inter-pocket pair tunneling(for  t ′ 1  = 0). As in-dicated by the red and blue arrows, a  ± q  pair on one pocketfermi surface are scattered to a  ± k  pair on another pocket.Interestingly, the same scattering also drives the antiferro-magnetism. This can be seen by noting that the momen-tum transfer between the incoming spin up(red) and outgoingspin down(blue) electrons is exactly the nesting wavevector( π, 0) or (0 ,π ) as indicated by the dashed line. (c1)-(c3) Thesummed Feynman diagrams. Here the arrowed solid denoteGreen’s function and the dashed lines represent the renormal-ized interactions. Q ,b ; p ,c ; d ), where  Q  is the ordering wavevecctor, bandindices  b  =  c  ±  1 mod 4 and  d  =  a  ±  1 mod 4. Since k  and  p  only takes  N   discrete values, we can treat  V   SC s,t and  V   SDW  as matrices. The few lowest eigenvalues as afunction of the RG evolution (panel (a)) and their finalassociated eigenvectors (panel (b)) of these matrices areplotted in Fig. (3).In panel (a) of Fig. (3) we show the N=20 RG evolu-tion of the scattering amplitudes in the SDW and twodifferent types of superconducting pairing channels as afunction of   a ln(Λ 0 / Λ). Here  a  =  − 1 / ln(0 . 97), Λ 0  is theinitial and Λ is the running energy cutoff. The bare inter-action parameters used are  U  1  = 4 . 0 , U  2  = 2 . 5 , J  H   = 0 . 7eV. Clearly as Λ 0 / Λ increases the interaction that drivesSDW (black dots) grows in magnitude the fastest. Theform factor  f  SDW  ( k ) associated with the SDW order(ˆ∆ SDW,a,b  =   k f  SDW,a,b ( k ) ψ † b k +( π,π ) ↑ ψ a k ↓ ,  b  =  a  ±  1mod 4) is shown in panel (b1) of this figure. Interest-ingly despite the perfect nesting of the fermi surfaces (seeFig.2(b)) there are two nodes !To understand the srcin of these nodes we observethat the top of the hole pockets (situated at  k  = (0 , 0)and ( π,π )) is doubly degenerate. According to Berry [22]the band eigenfunctions must exhibit non-trivial phaseas  k  moves around these degenerate  k  points. Omit-ting the identity term, the  K  ( k ) in Eq. (1),  K  ( k ) = b z ( k ) τ  z  + b x ( k ) τ  x , is that of a spin 1/2 in a  k -dependentmagnetic field. A plot of   b  as a function of   k  for t ′ 1  =  t ′ 2  = 0 is given in Fig. (1)(b). Clearly, as  k  cir-cles around (0 , 0) or ( π,π ) the direction of   b  winds twicearound the unit circle. This “double-winding” explainsthe fact that the degeneracy of the band dispersion islifted “quadratically” as  k  deviates from (0 , 0) or ( π,π ).On the contrary, the bottom of electron pockets around(0 ,π ) and ( π, 0) are non-degenerate, and  b  exhibits nowinding around them. Now let us consider switching ona SDW order parameter to nest, say, the fermi surfacesaround (0 , 0) and ( π, 0). Let  q  be the momentum aroundthe (0 , 0)-fermi surface, and  | ψ ( q )   and  | ψ ( q  + ( π, 0))  be, respectively, the band eigenstates associated withthe two fermi surfaces. The following matrix element∆( q ) =   ψ ( q ) | M  ( q ) | ψ ( q  + ( π, 0))   determines the SDWgap. Here  M  ( q ) is a 2  ×  2 matrix acting in the or-bital space (here we have assumed that after choosinga spin quantization axis, the ordered moments lies, say,in the  ±  x-direction). From Fig. (1)(b) one can see that H  ( q ) =  H  ( − q ) and  H  (( π, 0)+ q ) =  H  (( π, 0) − q ). How-ever, due to the double-winding behavior of   b  around(0 , 0) there is a non-trivial Berry phase after  k  made ahalf circle around the srcin, i.e.,  | ψ ( − q )   =  −| ψ ( q )  .On the other hand the no-winding of   b  around ( π, 0) im-plies  | ψ ( − q + ( π, 0))   =  | ψ ( q + ( π, 0))  . Consequently if  M  ( q ) is inversion symmetric, i.e.,  M  ( q ) =  M  ( − q ), wehave ∆( q ) =  − ∆( − q ) .  Under the assumption that themagnetically ordered phase preserves time reversal plusa spatial translation (hence ∆( q ) is real), this implies thegap function must change sign twice as  q  moves aroundthe fermi surface. Hence there must be two diametri-cally opposite nodes. Explicit mean-field calculation[23]using the bare  H   shows that nodes are situated at theintersection between the ordering wave vector and the(0 , 0)-fermi surface. This agrees with the form factor of Fig. (3)(b1).Next, we come to superconducting pairing. As shownin Fig. (3)(a) even for half-filling there are growing in-teraction that drives superconductivity! However, theseinteraction are sub-dominant compared with the inter-action that promotes antiferromagnetism. Our resultshows that the two most favorable pairing symmetryare an unconventional  s  (u- s ) and  d x 2 − y 2  like. The u- s -wave pairing has  s  symmetry around the hole fermisurface but exhibits  s  +  d x 2 − y 2  symmetry around theelectron fermi surface (where the 90 degree rotation sym-metry is broken)[8, 20]. It turns out that dependingon the value of   t ′ 1  it is possible for the gap functionto have nodes on the electron fermi surface. We shallreturn to this point shortly. The form factors  f  SC,a ( k )(∆ SC,a  =  f  SC,a ( k )[ ψ a ↑ ( k ) ψ a ↓ ( − k ) − ↑↔↓ ]) of these pair-ing symmetry are shown in Fig. (3)(b2,b3).  Most signif-icantly, from our calculation the pairing mechanism can be unambiguously determined - the pairing are all driven by the inter-pocket pair tunneling  [24]. An example of thestrongest such process is shown in Fig. (2)(b). As indi-  4cated by the red and blue arrows, a  ± q  pair on one fermisurface are scattered to a  ± k  pair on another. Interest-ingly, the same scattering also drives the antiferromag-netism. This can be seen by noting that the momentumtransfer between the incoming spin up and outgoing spindown electrons is exactly the nesting wavevector ( π, 0) or(0 ,π ) as indicated by the dashed line. Thus the same in-teraction also drives antiferromagnetism! One might askwould’t pairing and SDW interaction require oppositesign? No, for the inter band pairing interaction eithersign will do. This is because the pairing order param-eters can choose opposite signs on the two bands thusbenefit from the positive interaction. At half-filling when -25-20-15-10-5 0 0 40 80 120 160 200   r  e  n  o  r  m  a   l   i  z  e   d   i  n   t  e  r  a  c   t   i  o  n a log( Λ 0  /  Λ ) (a) sd x 2 -y 2 SDW 0 5 10 15-0.2-0.1 0 0.1 0.2 0.3 SDW(b1) f   SDW,0,1 f SDW,2,1 f SDW,0,3 f SDW,2,3 -0.2-0.1 0 0.1 0.2 0.3 0 5 10 15 s(b2)   f SC,0 f SC,1 f SC,2 f SC,3  0 5 10 15-0.2-0.1 0 0.1 0.2 0.3 d x 2 -y 2 (b3)   f SC,0 f SC,1 f SC,2 f SC,3 FIG. 3: (a)The N=20 RG evolution of the scattering ampli-tudes in the SDW and two different types of superconductingpairing channels as a function of   a ln(Λ 0 / Λ). The parameter t ′ 1  is set to zero. (b1) The SDW form factor is plotted as themomentum moves around the (0 , 0) or ( π,π ) fermi surface.The vertical dash lines are high symmetry directions. Thecurves are obtained by interpolating the data points. Herethe flat (zero) form factor are associated with the fermi sur-faces that are not nested by the ordering wavevector ( π, 0) or(0 ,π ). (b2,b3) The form factor of the two most prominent su-perconducting pairing. The most favorable pairing symmetryis u- s -wave and the next one is  d x 2 − y 2 -wave. the nesting is sufficiently good (e.g., when  t ′ 1  is absent),antiferromagnetism overwhelms the superconducting in-stability. We propose the reason weak superconductivityobserved in the stoichiometric compound LaFePO is be-cause As ↔ P replacement damages nesting hence allowsuperconductivity to prevail in the competition with an-tiferromagnetism. The superconducting pairing away from half filling   Inpanel (a) and (b) of Fig. (4) we show the N=20 RG evolu-tion of the two most favorable singlet and the best tripletsuperconducting scattering amplitudes for  x  = 0 . 13.The  t ′ 1  parameter we used here is 0 . 12 eV, and the bareinteraction parameters are  U  1  = 4 . 0 ,U  2  = 2 . 5 ,J  H   = 0 . 7eV. Due to the removal of nesting, the antiferromagneticscattering (not shown) is no longer dominant. Likehalf-filling, the most favorable pairing symmetry is the -4-3-2-1 0 0 40 80 120 160   r  e  n  o  r  m  a   l   i  z  e   d   i  n   t  e  r  a  c   t   i  o  n a log( Λ 0  /  Λ ) (a)t 1 ’=0.12, x=0.13,electron doped sd x 2 -y 2 triplet 0 5 10 15-0.2-0.1 0 0.1 0.2 0.3 s(b) f   SC,0 f SC,1 f SC,2 f SC,3 FIG. 4: (a) The N=20 RG evolution of the scattering am-plitude associated with the two most favorable singlet chan-nel (u- s  and  d x 2 − y 2 ) and the top triplet channel. (b) Theform factors of u- s . Note that it changes sign on the electronpocket. singlet u- s . The pairing mechanism is the inter-pocketpair hopping shown in Fig. (2)(b). At  t ′ 1  = 0 . 12 eV theform factor changes sign as shown in Fig. (4)(b). As aresult  the superconducting gap has nodes on the electron pocket  . For all parameters we have studied, the tripletpairing channel is never favored. In Fig. (4)(a) we showthe RG evolution of the best triplet pairing amplitude,and it never becomes competitive with the u- s  channel.We emphasize that while the u- s  gap function alwaysshow a full gap on the hole pockets, it can be gappedor gapless on the electron pocket depending on thevalue of   t ′ 1 . 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