Spin Excitations in the Excitonic Spin-Density-Wave State of the Iron Pnictides

PHYSICAL REVIEW B 80, 174401 ͑2009͒

Spin excitations in the excitonic spin-density-wave state of the iron pnictides

P. M. R. Brydon* and C. Timm† Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany ͑Received 8 September 2009; revised manuscript received 8 October 2009; published 3 November 2009͒ Motivated by the iron pnictides, we examine the spin excitations in an itinerant antiferromagnet where a spin-density wave ͑SDW͒ originates from an excitonic instability of nested electronlike and holelike Fermi pockets. Using the random-phase approximation, we derive the Dyson equation for the transverse susceptibility in the excitonic SDW state. The Dyson equation is solved for two different two-band models, describing an antiferromagnetic insulator and metal, respectively. We determine the collective spin-wave dispersions and also consider the single-particle continua. The results for the excitonic models are compared with each other and also contrasted with the well-known SDW state of the Hubbard model. Despite the qualitatively different SDW states in the two excitonic models, their magnetic response shows many similarities. We conclude with a discussion of the relevance of the excitonic SDW scenario to the iron pnictides.

DOI: 10.1103/PhysRevB.80.174401 PACS number͑s͒: 75.30.Fv, 75.10.Lp

I. INTRODUCTION and holelike Fermi pockets separated by a nesting vector Q The recent discovery of superconductivity in iron pnic- in the presence of interband Coulomb repulsion. Performing tides has sparked a tremendous research effort.1,2 The re- a particle-hole transformation on one of the bands, we obtain an attractive interaction between the particles in one band markably high superconducting transition temperature Tc of some of these compounds,3 their layered quasi-two- and the holes in the other. Within a BCS-type mean-field ͑ ͒ 4 theory, the attractive interaction causes the condensation of dimensional 2D structure, the proximity of superconduc- ͑ ͒ tivity and antiferromagnetism in their phase diagrams,5–8 and interband electron-hole pairs excitons with relative wave 9–13 vector Q, thereby opening a gap in the single-particle exci- the likely unconventional superconducting pairing state 28 are reminiscent of the cuprates.14 It is a tantalizing prospect tation spectrum. Although the interband Coulomb repulsion that the iron pnictides can shed light onto the problem of causes the excitonic instability, additional interband scattering terms are required to stabilize one of several different unconventional high-T superconductivity in general. c density-wave states, such as a SDW or a charge-density For this it is essential to assess the differences between wave ͑CDW͒.29–31 the cuprates and the iron pnictides. For example, the pnic- 15 Several authors have discussed the SDW state of the pnictides have a much more complicated Fermi surface. The tides in terms of an excitonic instability of nested electron antiferromagnetic states in the two families are also qualita- and hole Fermi pockets without regard to the orbital origin of tively different. In the cuprates, superconductivity appears by these bands.31–36 An alternative school of thought empha- doping an insulating antiferromagnetic parent compound. sizes the importance of the complicated mixing of the iron The pnictide parent compounds RFeAsO ͑R is a rare-earth 3d orbitals at the Fermi energy and of the various interorbital ͒ ͑ ͒ 37–40 ion and AFe2As2 A is an alkaline-earth ion are also anti- interactions. These two approaches are not contradictory, ferromagnets but there is compelling evidence that they dis- however, since the excitonic model can be understood as an play a metallic spin-density-wave ͑SDW͒ state: the value of effective low-energy theory for the orbital models.31,34 Fur- the magnetic moment at the Fe sites is small,6,7,16 the com- thermore, even in an orbital model, the SDW state is still pounds display metallic transport properties below the Néel driven by the nesting of electron and hole Fermi pockets. 16–18 temperature TN, and angle resolved photoemission spec- Indeed, at the mean-field level all these models yield quali- troscopy ͑ARPES͒ and quantum oscillation experiments find tatively identical conclusions. A conceptually different pic- 19,20 a reconstructed Fermi surface below TN. ture based on the ordering of localized moments has also The electron-phonon interaction in the pnictides is much been proposed.11,41–43 Although it is hard to reconcile with 21 16,17 too weak to account for the high-Tc values. Instead, the the observed metallic properties and the moderate inter- most likely candidate for the “glue” binding the electrons action strengths,44,45 this picture is consistent with several into Cooper pairs are spin fluctuations,11–13 which are en- neutron-scattering experiments.46,47 At present, it is difficult hanced by the proximity to the SDW state. A proper under- to discriminate between the itinerant ͑excitonic͒ and local- standing of the SDW phase is therefore likely the key to the ized scenarios, as the dynamical spin response of the itiner- physics of the pnictides. Intriguingly, ab initio calculations ant models is unknown. It is therefore desirable to determine suggest that the nesting of electron and hole Fermi pockets is the spin excitations in the excitonic SDW model. responsible for the SDW,15 indicating that, like the supercon- It is the purpose of this paper to examine the transverse ductivity, the antiferromagnetism of these compounds has a spin susceptibility within the excitonic SDW state of a gen- multiband character. The best known material where a SDW eral two-band model. We work within the limits of weak to arises from such a nesting property is chromium22–26 and this moderate correlation strength, using the random-phase ap- mechanism has also been implicated for manganese alloys.27 proximation ͑RPA͒ to construct the Dyson equation for the The SDW in these compounds belongs to a broader class susceptibility. In order to understand the generic features of of density-wave states. Consider a material with electronlike the spin excitations in the excitonic SDW state, we calculate

1 U 4 ⌬ Ј␴͗ † ͘ ͑ ͒ = ͚ ck+Q,␴ck,␴ . 2 2 V k,␴ π /

0 a 0

y Q k -2 The primed sum denotes summation only over the reduced, energy (eV) -4 magnetic Brillouin zone. Diagonalizing the mean-field -1 Hamiltonian, we find two bands in the reduced Brillouin (0,0) (π,0) (π,π) (0,0) -1 0 1 ͱ 2 2 zone with energies EϮ = Ϯ ⑀ +⌬ . In the following, we (a)(k a, k a) (b) kxa/π ,k k x y will assume t=1 eV and U=0.738 eV, which gives a criti- FIG. 1. ͑Color online͒͑a͒ Band structure and ͑b͒ Fermi surface cal temperature for the SDW state of TSDW=138 K and a of the Hubbard model for U=0. In ͑b͒, the nesting vector Q T=0 gap ⌬=21.3 meV. =͑␲/a,␲/a͒ is also shown. The dynamical spin susceptibility is defined by the RPA susceptibilities for the simplest model showing this ␤ 1 ␻ ␶ ␹ ͑ Ј ␻ ͒ ͵ ␶͗ i͑ ␶͒ j͑ Ј ͒͘ i n ͑ ͒ instability. We pay particular attention to the spin waves ij q,q ;i n = d T␶S q, S − q ,0 e , 3 V ͑magnons͒ and damped paramagnons. In the simplest model, 0 however, the SDW state is insulating. We therefore verify the robustness of our results by applying our theory to a system where T␶ is the time-ordering operator and where portions of the Fermi surface remain ungapped in the SDW phase, as in the iron pnictides. We contrast our results ␴i for the excitonic SDW state with those for the SDW phase of 1 s,sЈ i͑ ␶͒ † ͑␶͒ ͑␶͒ ͑ ͒ the single-band Hubbard model, which is commonly used to S q, = ͚ ͚ ck+q,s ck,sЈ . 4 ͱV 2 describe the antiferromagnetic state of the cuprates. k s,sЈ The structure of this paper is as follows. We commence in Sec. II with a brief review of the RPA-level results for the Because of the doubling of the unit cell in the SDW state, the transverse susceptibility in the SDW state of the Hubbard susceptibility in Eq. ͑3͒ is nonzero for q=qЈ and q=qЈ+Q, model.48–50 We then proceed in Sec. III with a general dis- the latter referred to as the umklapp susceptibility.48 Both cussion of the excitonic SDW state in a two-band model and appear in the ladder diagrams for the transverse susceptibil- present the Dyson equation for the transverse susceptibilities. ity, yielding the Dyson equation The RPA susceptibility and spin-wave dispersion is then calculated for the insulating and metallic excitonic SDW mod- ␹ ͑ Ј ␻ ͒ ␦ ␹͑0͒͑ ␻ ͒ els in Secs. III A and III B, respectively. All presented results −+ q,q ;i n = q,qЈ −+ q,q;i n are calculated in the limit of zero temperature. In order to ͑ ͒ + ␦ ␹ 0 ͑q,q + Q;i␻ ͒ properly compare the different models, we choose interaction q+Q,qЈ −+ n strengths such that the zero-temperature SDW gap is the ␹͑0͒͑ ␻ ͒␹ ͑ Ј ␻ ͒ + U −+ q,q;i n −+ q,q ;i n same. We conclude with a comparison with experimental re- ␹͑0͒͑ ␻ ͒␹ ͑ Ј ␻ ͒ sults in Sec. IV and a summary of our work in Sec. V. + U −+ q,q + Q;i n −+ q + Q,q ;i n , ͑5͒ II. HUBBARD MODEL where the superscript ͑0͒ indicates the mean-field suscepti- ␹ ͑ ␻ ͒ ␹ ͑ The Hamiltonian of the Hubbard model reads bilities. Explicit expressions for −+ q,q;i n and −+ q,q ␻ ͒ +Q;i n can be found in Ref. 50. We plot the imaginary part of ␹ ͑q,␻͒=␹ ͑q,q;␻͒ † U † † −+ −+ H = ͚ ⑀ c c ␴ + ͚ c c ↑c c ↓, ͑1͒ ͑ ͒ k k,␴ k, k+q,↑ k, kЈ−q,↓ kЈ, along the line q= qx ,qy =qx in Fig. 2. The calculation of the ␴ V k, k,kЈ,q mean-field susceptibilities in the Dyson equation ͑5͒ was performed over a 10 000ϫ10 000 k-point mesh. In the analytic † ͑ ͒ ͑ ͒ ␻ →␻ ␦ ␦ where ck,␴ ck,␴ creates destroys an electron with momen- continuation i n +i we assume a finite width tum k and spin ␴. We assume a 2D nearest-neighbor tight- =1 meV. Smaller values of ␦ and finer k-point meshes do ⑀ ͑ ͒ binding dispersion k =−2t cos kxa+cos kya , where a is the not produce qualitative or significant quantitative changes in ⑀ lattice constant. We plot the band structure k and the result- our results. ␹ ͑ ␻͒ ing Fermi surface at half filling in Figs. 1͑a͒ and 1͑b͒, re- Im −+ q, in Fig. 2 displays very different behavior for spectively. energies ␻Ͻ2⌬=42.6 meV and ␻Ͼ2⌬. In the former re- At half filling and sufficiently low temperature T, the gion, the dispersion of the collective spin waves is clearly Hubbard model is unstable toward a SDW state with nesting visible as the sharp dark line. The finite width of this line is vector Q=͑␲/a,␲/a͒, which connects opposite sides of the a consequence of the broadening ␦. The dispersion is almost ␲/ Շ Շ ␲/ Fermi surface. We assume a SDW polarized along the z axis flat for 0.1 a qx =qy 0.9 a, where it lies very close to and decouple the interaction term in Eq. ͑1͒ by introducing ␻=2⌬. The distribution of spectral weight for the spin wave the SDW gap is asymmetric, with much greater weight close to q=Q than

174401-2 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒ l gI ) , ( Im og 10 ) χ (meV −+ ω q ω

qax /π =qay /π

FIG. 2. ͑Color online͒ Imaginary part of the transverse susceptibility in the Hubbard model for q=͑qx ,qy =qx͒. The spin-wave dispersion is visible as the dark line running across the ﬁgure at ␻Ͻ2⌬=42 meV. Note the logarithmic color scale.

␻ ͉␦ ͉ at q=0, reflecting the suppression of long-wavelength spin =cSW q , where cSW is the spin-wave velocity. An expres- 51 D͑ ␻͒ excitations in the SDW state. sion for cSW is obtained by expanding q, about q=Q For ␻Ͼ2⌬, we find a continuum of excitations. It starts and ␻=0.48–50 In agreement with Ref. 49,wefind abruptly at ␻=2⌬, corresponding to the minimum energy for a single-particle excitation across the SDW gap. This mini- −4͑1/U − ⌬2x͒t2␥ mum is the same at all k points lying on the Fermi surface c = ͱ , ͑7͒ SW / shown in Fig. 1͑b͒. By inspection, we see that for every x U value of q, there exist points k and k+q lying on the Fermi surface so that the minimum energy required for any excita- where ␻ ⌬ ␹ ͑ ␻͒ tion is =2 . We also see that Im −+ q, tends to de- ␻ 1 1 crease with increasing . This can be understood in terms of x = ͚Ј , ͑8͒ ͑ ͒ 3 the density of states DOS in the noninteracting model: the V k Ek DOS has a van Hove singularity at the Fermi energy and decreases monotonically as one moves to higher or lower energies. For an occupied state with energy ␻ below the 1 1 o ␥ = ͚Јͭ ͑cos2 k a + cos k a cos k a͒ Fermi energy, the density of unoccupied states with energy V E3 x x y ␻ k k u above the Fermi energy therefore decreases with increasing ␻=␻ −␻ and hence the “density of excitations” con- ⑀2 −2⌬2 u o + k sin2 k aͮ. ͑9͒ tributing to the transverse susceptibility also decreases with E5 x increasing ␻. k Close to q=0, the continuum is bounded from above by The spin-wave velocity is plotted as a function of U in Fig. ␻ the line =vF ·q, where vF is the Fermi velocity along k 4͑a͒ while we compare the low-energy linearized form of the ͑ ͒ ␹ ͑ ␻͒ = kx ,ky =kx . The peak in Im −+ q, at this edge of the spin-wave dispersion to the numerically determined result in continuum is due to single-particle excitations across the Fig. 4͑b͒. As can be seen, the linearized result holds only for Fermi energy in the same branch of the band structure. A small energies ␻Շ0.5⌬. rather weak dispersing feature also appears within the continuum near q=Q, as shown in more detail in Fig. 3. This paramagnon originates from single-particle excitations into the back-folded band. Like the feature at small q, the para-

magnon disperses with the Fermi velocity. The paramagnon ) , ( Im

and spin-wave dispersions curve away from one another in χ what appears to be an avoided crossing. −+ q (meV)

͑ ͒ ␹ ͑ ␻ ͒ ω Solving Eq. 5 for −+ q,q;i n requires the inversion of

ϫ D͑ ␻ ͒ ω a2 2 matrix. The determinant q,i n of this matrix is D͑ ␻ ͒ ͓ ␹͑0͒͑ ␻ ͔͒ q,i n = 1−U −+ q,q;i n ϫ͓ ␹͑0͒͑ ␻ ͔͒ 1−U −+ q + Q,q + Q;i n ͓ ␹͑0͒͑ ␻ ͔͒2 ͑ ͒ − U −+ q,q + Q;i n . 6 qax /π =qay /π ␻ →␻ + Making the analytic continuation i n +i0 , the solution FIG. 3. ͑Color online͒ Imaginary part of the transverse spin D͑ ␻͒ of Re q, =0 yields the spin-wave dispersion. At low susceptibility in the Hubbard model for q=͑qx ,qy =qx͒ close to Q. energies, it has a linear dependence upon ␦q=Q−q, i.e., Note the linear color scale.

174401-3 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒

1.0 2.0 ͑ ͒ CCDW with gCCDW=gcf+g2a−2g2b, and a spin-current- ͑ ͒ 0.9 1.5 density wave SCDW with gSCDW=gcf−g2a. In order to model the iron pnictides, we henceforth assume that the (eV) 0.8

a 1.0

/ SDW state has the largest coupling constant. 0.7 ω/∆

SW 0.5 dispersion In the presence of a SDW polarized along the z axis and c 0.6 low-energy form with nesting vector Q, the effective mean-field Hamiltonian 0.5 0.0 0246 810 0 0.01 0.02 0.03 is written as (a)U (eV) (b) |δqa/π| ␴⌬͑ † ͒ ͑ ͒ HMF = H0 + ͚ ͚ ck,␴fk+Q,␴ + H.c. , 17 ͑ ͒ ␴ FIG. 4. a Spin-wave velocity cSW in the Hubbard model as a k function of U for t=1 eV. ͑b͒ Comparison of the spin-wave dispersion and the low-energy linear form as a function of ␦q=Q−q for where the excitonic gap U=0.738 eV. g ⌬ SDW ␴͗ † ͑͒͘ = ͚ ͚ ck,␴fk+Q,␴ 18 III. EXCITONIC MODEL 2V k ␴ In this section we discuss the excitonic SDW in a general is assumed to be real. The precise relationship between ⌬ two-band model with Fermi-surface nesting. We begin by 29,30 outlining the known results for the mean-field SDW and the magnetization is somewhat complicated. To elu- state.24,25,28–30 We write the Hamiltonian as cidate it, we define the field operator, H = H + H , ͑10͒ 1 0 I ␺ ͑ ͒ ͓␸ ͑ ͒ ␸ ͑ ͒ ͔ ik·r ͑ ͒ ␴ r = ͱ ͚ k,c r ck,␴ + k,f r fk,␴ e , 19 where the noninteracting system is described by V k ͓͑⑀c ␮͒ † ͑⑀f ␮͒ † ͔͑͒ ␸ ͑ ͒ ␣ H0 = ͚ ͚ k − ck␴ck␴ + k − fk␴fk␴ 11 where k,␣ r is a Bloch function for the band . The local k ␴ magnetization M͑r͒ is then † † and c ␴ ͑f ␴͒ creates an electron with spin ␴ and momen- k, k, g␮ ͒ ͑ ␸͒ ͑ ءtum k in the electronlike c band ͑holelike f band͒. The sec- ͑ ͒ B ␸ M r =− ͚ ͚ ͚ r kЈ,b r ͑ ͒ V k,a ond term in Eq. 10 describes the interactions in the model s,sЈ k,kЈ a,b=c,f system. Following Refs. 31 and 34, we take this to consist of ␴ s,sЈ five on-site terms HI =Hcc+Hff+Hcf+HITa+HITb that arise ϫ −i͑k−kЈ͒·rͳ † ʹ ͑ ͒ e a bkЈ,sЈ , 20 naturally in the low-energy effective theory of a multiorbital k,s 2 model. These correspond to intraband Coulomb repulsion, where g is the g-factor and ␮ is the Bohr magneton. Only in g B cc † † ͑ ͒ ␸ ͑r͒ ⌬ Hcc = ͚ c ↑ck,↑c ↓ckЈ,↓, 12 the limit when k,␣ is constant do we find to be simply V k+q, kЈ−q, k,kЈ,q related to the magnetization, ␮ ⌬ 2g B gff † † ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ M r =− cos Q · r ez. 21 Hff = ͚ fk+q,↑fk,↑fkЈ−q,↓fkЈ,↓, 13 V gSDW k,kЈ,q interband Coulomb repulsion, For simplicity, we follow Refs. 13, 23, 27, and 52 in assum- ing constant Bloch functions. g cf † † ͑ ͒ In calculating the susceptibilities, we make use of the Hcf = ͚ ͚ c ␴ck,␴f ␴ fkЈ,␴Ј, 14 V k+q, kЈ−q, Ј single-particle Green’s functions of the mean-field SDW k,kЈ,q ␴,␴Ј state. The two normal ͑diagonal in band indices͒ Green’s and two distinct types of correlated interband transitions, functions are defined by

g ␤ 2a ͑ † † ͒ ͑ ͒ HITa = ͚ c ↑c ↓fkЈ,↓fk,↑ + H.c. , 15 cc † ␻ ␶ k+q, kЈ−q, ͑ ␻ ͒ ͵ ␶͗ ͑␶͒ ͑ ͒͘ i n V G ␴ i =− d T␶c ␴ c ␴ 0 e k,kЈ,q k, n k, k, 0 g i␻ − ⑀f 2b † † ͑ ͒ n k+Q ͑ ͒ HITb = ͚ ͚ c ␴f ␴ ckЈ,␴Јfk,␴. 16 = , 22 V k+q, kЈ−q, Ј ͑i␻ − E ͒͑i␻ − E ͒ k,kЈ,q ␴,␴Ј n +,k+Q n −,k+Q

The interband interaction terms are responsible for a density- ␤ ␻ ␶ wave instability when the electron and hole Fermi pockets ff ͑ ␻ ͒ ͵ ␶͗ ͑␶͒ † ͑ ͒͘ i n Gk,␴ i n =− d T␶fk,␴ fk,␴ 0 e are sufﬁciently close to nesting. A number of different 0 density-wave states are possible:30 a CDW with effective ␻ ⑀c i n − k+Q coupling constant gCDW=gcf−g2a−2g2b, a SDW with cou- = ͑23͒ ͑ ␻ ͒͑ ␻ ͒ pling gSDW=gcf+g2a, a charge-current-density wave i n − E+,k i n − E−,k

174401-4 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒

while the anomalous ͑band-mixing͒ Green’s functions are ␴ ͑ ͒ ␺†͑ ͒ s,sЈ␺ ͑ ͒ S r = ͚ r sЈ r ␤ s 2 ␻ ␶ s,sЈ fc ͑ ␻ ͒ ͵ ␶͗ ͑␶͒ † ͑ ͒͘ i n Gk,␴ i n =− d T␶fk,␴ ck+Q,␴ 0 e 0 1 † ␴ −iq·r = ͚ ͚ ͚ ak+q,s s,sЈbk,sЈe ␴⌬ 2V a,b=c,f k,q Ј = , ͑24͒ s,s ͑ ␻ ͒͑ ␻ ͒ i n − E+,k i n − E−,k 1 ͑ ͒ −iq·r ͑ ͒ = ͱ ͚ ͚ Sa,b q e , 27 ␤ V a,b=c,f q ␻ ␶ cf ͑ ␻ ͒ ͵ ␶͗ ͑␶͒ † ͑ ͒͘ i n fc ͑ ␻ ͒ G ␴ i =− d T␶c ␴ f ␴ 0 e = G ␴ i . ͑ ͒ k, n k, k+Q, k+Q, n where Sa,b q is a generalized spin operator. The dynamical 0 spin susceptibility is then deﬁned by ͑25͒ ␤ 1 j ␻ ␶ ␹ ͑ Ј ␻ ͒ ͵ ͗ i ͑ ␶͒ ͑ Ј ͒͘ i n q,q ;i = ͚ ͚ T␶S q, S − q ,0 e The functions EϮ are the dispersion relations for the recon- ij n a,b aЈ,bЈ ,k V a,b Ј Ј 0 structed bands, a ,b ␹abaЈbЈ͑ Ј ␻ ͒ ͑ ͒ = ͚ ͚ ij q,q ;i n . 28 1 c f ͱ c f 2 2 a,b Ј Ј EϮ = ͓⑀ + ⑀ Ϯ ͑⑀ − ⑀ ͒ +4⌬ ͔. ͑26͒ a ,b ,k 2 k+Q k k+Q k ␹abaЈbЈ͑ ␻ ͒ The generalized susceptibilities q,qЈ;i n are calcu- ⌬ Ϸ⑀c ij For energies much larger than we have E+,k k+Q and lated using the RPA. We are only concerned with the trans- Ϸ⑀f E−,k k. verse susceptibility, which is obtained by summing the lad- The total spin operator is written as der diagrams. This yields the Dyson equation

Ј Ј ͑ ͒ ¯¯͑ ͒ ¯͑ ͒ ¯ ͑ ͒ ␹aba b ␦ ͑␦ ␦ ␹abba 0 ␦ ¯␦ ␹abba 0 ͒ ␦ ͑␦ ␦ ␹abba 0 ␦ ¯␦ ␹abba 0 ͒ −+,00 = q,qЈ aЈ,b bЈ,a −+,00 + aЈ,b bЈ,¯a −+,00 + q+Q,qЈ aЈ,b bЈ,¯a −+,0Q + aЈ,b bЈ,a −+,0Q ͑␹abcc͑0͒␹ccaЈbЈ ␹abcc͑0͒␹ccaЈbЈ͒ ͑␹abff͑0͒␹ffaЈbЈ ␹abff͑0͒␹ffaЈbЈ͒ + gcc −+,00 −+,00 + −+,0Q −+,Q0 + gff −+,00 −+,00 + −+,0Q −+,Q0 ͑␹abcf͑0͒␹fcaЈbЈ ␹abcf͑0͒␹fcaЈbЈ ␹abfc͑0͒␹cfaЈbЈ ␹abfc͑0͒␹cfaЈbЈ͒ + gcf −+,00 −+,00 + −+,0Q −+,Q0 + −+,00 −+,00 + −+,0Q −+,Q0 ͑␹abcf͑0͒␹cfaЈbЈ ␹abcf͑0͒␹cfaЈbЈ ␹abfc͑0͒␹fcaЈbЈ ␹abfc͑0͒␹fcaЈbЈ͒ + g2a −+,00 −+,00 + −+,0Q −+,Q0 + −+,00 −+,00 + −+,0Q −+,Q0 ͑␹abcc͑0͒␹ffaЈbЈ ␹abcc͑0͒␹ffaЈbЈ ␹abff͑0͒␹ccaЈbЈ ␹abff͑0͒␹ccaЈbЈ͒ ͑ ͒ + g2b −+,00 −+,00 + −+,0Q −+,Q0 + −+,00 −+,00 + −+,0Q −+,Q0 , 29

where we have adopted the short-hand notation ¯ ͑ ͒ 1 1 ¯ ␹abb¯a 0 =− ͚ ͚ Gbb ͑i␯ ͒G¯aa ͑i␯ − i␻ ͒. −+,00 ␤ k,↑ n k+q,↓ n n V k i␯ ␹abaЈbЈ͑0͒ ␹abaЈbЈ͑0͒͑ ␻ ͒ ͑ ͒ n −+,mn = −+ q + m,q + n;i n , 30 The next two terms are the umklapp susceptibilities which, as in the Hubbard model, are the product of a normal and an ␹abaЈbЈ ␹abaЈbЈ͑ Ј ␻ ͒ ͑ ͒ −+,mn = −+ q + m,q + n;i n . 31 anomalous Green’s function,

abaЈbЈ͑0͒ 1 1 Note that ␹ does not depend on qЈ. We have also ␹abba¯͑0͒ bb ͑ ␯ ͒ ¯aa ͑ ␯ ␻ ͒ −+,mn −+,0Q =− ͚ ͚ Gk,↑ i n Gk+q,↓ i n − i n , ͑ ͒ ͑ ͒ V ␤ ␯ introduced the notation ¯a=c f when a= f c . The first line k i n of Eq. ͑29͒ gives the mean-field susceptibilities, obtained by using Wick’s theorem to contract the correlation function in ¯ 1 1 ¯ ͑ ͒ ␹abba͑0͒ bb ͑ ␯ ͒ aa ͑ ␯ ␻ ͒ Eq. 28 into products of two mean-field Green’s functions. −+,0Q =− ͚ ͚ Gk,↑ i n Gk+q,↓ i n − i n . V ␤ ␯ The first two terms on the right-hand side of Eq. ͑29͒ are the k i n correlators resulting from the product of two normal Green’s ͑ ͒ functions, The remaining lines of Eq. 29 give the ladder sums for the various interactions: on the second line we have the intraband Coulomb interactions, on the third line the interband 1 1 ␹abba͑0͒ bb ͑ ␯ ͒ aa ͑ ␯ ␻ ͒ −+,00 =− ͚ ͚ Gk,↑ i n Gk+q,↓ i n − i n Coulomb interaction, and on the last two lines the two types V ␤ ␯ k i n of correlated transitions. In Figs. 5͑a͒ and 5͑b͒, we show a ␹cffc diagrammatic representation of the Dyson equation for −+,00 ␹cccc and from the product of two anomalous Green’s functions, and −+,00, respectively. Note that the Dyson equation is also

174401-5 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒

cc ff ff f ff ff ff ff f χcffc ccfc fffc cffc gcc χ gff χ gcf χ ccc00 = c + cccQ0 + Q0 cc+ 00 c c c f f c c cc c f fc f fcfc ff cffc f f fcfc f gcf χ g2a χ g2a χ + c 00 cc+ 00cc+ 00 c ff f c c f f c c f fffccfc fffc f g2bχ g 2b χ + cccQ0 + Q0 c (a) f c c f cc ff ff c c c c ccccc c χcccc cccc ffcc cfcc gcc χ gff χ gcf χ c 00 c = c c + c 00 cc+ 00 c + c Q0 c c c f f c c cc c f fc c ccc cc fccc cfcc fccc gcf χ g2a χχg2a + c Q0 c + ccQ0 + ccQ0 ff f c c f f c c f cccc cccc ffcc g2bχ g 2b χ + c 00c + c 00 c (b) f c c f

͑ ͒ ͑ ͒ ␹cffc ͑ ͒ ␹cccc FIG. 5. Diagrammatic representation of the Dyson Eq. 29 for a −+,00 and b −+,00. The curved lines are the mean-ﬁeld Green’s functions in the SDW state. When the label b=c, f follows the label a=c, f in the direction of the arrow, the corresponding Green’s function is Gab.

abba͑0͒ ␤ valid in the normal state in which case the ␹ are the 1 + ␻ ␶ −+,00 ␹inter͑ ␻ ͒ ͵ ␶͗ − ͑ ␶͒ ͑ ͒͘ i n q,i = ͚ d T␶S q, S − q,0 e only nonzero mean-ﬁeld susceptibilities. −+ n a,¯a b,¯b V a,b=c,f 0 From the structure of the Dyson equation, we see that ␹abaЈbЈ Ј ෈͕ ͖ = ␹cffc + ␹cfcf + ␹fcfc + ␹fccf . ͑34͒ −+,mn is only nonzero for q=q and m, n 0,Q .We −+,00 −+,00 −+,00 −+,00 observe that Eq. ͑29͒ may then be written as four indepen- dent sets of coupled equations for We note that the Dyson equation for the interband susceptibilities has been previously obtained in Refs. 23 and 52 for ͕␹cccc ␹ffcc ␹fccc ␹cfcc ͖ ͑ ͒ the case where only the interband Coulomb interaction Eq. −+,00, −+,00, −+,Q0, −+,Q0 , 32a ͑14͒ is nonzero. From Eq. ͑28͒ we see that the total trans- ͕␹ffff ␹ccff ␹cfff ␹fcff ͖ ͑ ͒ verse susceptibility is the sum of the intraband and interband −+,00, −+,00, −+,Q0, −+,Q0 , 32b contributions,

͕␹fccf ␹cfcf ␹cccf ␹ffcf ͖ ͑ ͒ ␹ ͑ ␻ ͒ ␹intra͑ ␻ ͒ ␹inter͑ ␻ ͒ ͑ ͒ −+,00, −+,00, −+,Q0, −+,Q0 , 32c −+ q,i n = −+ q,i n + −+ q,i n . 35

͕␹cffc ,␹fcfc ,␹ccfc ,␹fffc ͖. ͑32d͒ Since the interband and intraband susceptibilities involve −+,00 −+,00 −+,Q0 −+,Q0 qualitatively different types of excitations, considering these ␹abaЈbЈ separately offers greater physical insight into the magnetic Note that this includes −+,mn for mn=QQ and mn=0Q by symmetry; all other possible transverse susceptibilities van- response than the total susceptibility. ish. The first two sets contain the contributions to the intra- In the following sections we discuss the transverse sus- band susceptibility, which involve spin-flip transitions within ceptibility for two different models of the band structure. For the c and f bands, simplicity, we restrict ourselves to the case gcc=gff=g2b=0, as these interactions do not drive the SDW instability. We ␤ emphasize, however, that the preceeding results are valid for 1 ␻ ␶ ␹intra͑ ␻ ͒ ͵ ␶͗ − ͑ ␶͒ + ͑ ͒͘ i n −+ q,i n = ͚ d T␶Sa,a q, Sb,b − q,0 e any choice of couplings in both the normal and SDW states. V a,b=c,f 0 Except where stated otherwise, we furthermore set g2a=0, as ␹cccc ␹ccff ␹ffcc ␹ffff ͑ ͒ at reasonable coupling strengths we find very little change in = −+,00 + −+,00 + −+,00 + −+,00. 33 the transverse susceptibility upon varying gcf and g2a while The last two sets contain the contributions to the interband keeping gSDW=gcf+g2a fixed. Unless explicitly mentioned, susceptibility, which involve spin-flip transitions between the we have used a 10 000ϫ10 000 k-point mesh and a width c and f bands, ␦=1 meV to calculate the mean-field susceptibilities.

174401-6 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒

8 1 electron band at k=0.77Q. For the energies considered here, f band to excellent approximation the interband susceptibility de- 4 c band Q pends only on ͉␦q͉=͉Q−q͉. π /

0 a 0 ͑ ͒

y For q near Q, Fig. 7 a shows a spin-wave dispersion k which appears to intersect the continuum and continue as a

energy (eV) -4 paramagnon. Figure 9 reveals, however, that the situation is -8 -1 more complicated: the spin-wave dispersion does not inter- (0,0) (π,0) (π,π) (0,0) -1 0 1 (a)(k a, k a) (b) k a/π sect the continuum but instead flattens out as it approaches x y x ␻=2⌬ and disappears at qϷ0.985Q. As in the Hubbard model, the paramagnon and spin-wave dispersions appear to FIG. 6. ͑Color online͒͑a͒ Band structure and ͑b͒ Fermi surface of the noninteracting insulating excitonic model. In ͑b͒, the nesting avoid one another. The paramagnon nevertheless seems to vector Q=͑␲/a,␲/a͒ is also shown. connect to significant weight lying just inside the continuum region at the intersection point with the spin-wave dispersion. A. Insulating SDW state We now turn our attention to the intraband contribution to We first examine an excitonic model with perfect nesting the transverse susceptibility in Fig. 7͑b͒. Apart from a for- ⑀c ⑀f bidden region close to q=0, this appears almost like a mirror between the electron and hole bands, i.e., k =− k+Q for all k. Although hardly realistic, at the mean-field level it exactly image of the interband susceptibility, albeit much reduced in maps onto the BCS model after particle-hole weight. In particular, we note a V-shaped dispersing feature 28 ␹intra͑ ␻͒ transformation. It is therefore useful for obtaining physi- at q=0.46Q, the tendency of Im −+ q, to increase with cally transparent results and is frequently encountered in the increasing ␻, and a dispersing feature at the edge of the q literature.28,29,31,34,35 We assume the 2D band structure Ϸ0 forbidden region, which resembles the paramagnon close to q=Q. The presence of the V-shaped feature is particularly ⑀c =2t͑cos k a + cos k a͒ + ⑀ , ͑36a͒ interesting, as the discussion above indicates that it is due to k x y 0 interband excitations. Thus we find that interband excitations give a significant contribution to the intraband susceptibility. ⑀f ͑ ͒ ⑀ ͑ ͒ k =2t cos kxa + cos kya − 0, 36b This is confirmed by examining the Dyson equation for ␹cccc ͑ ͒ −+,00, cf. Fig. 5 b : for gcc=gff=g2b=g2a=0, as assumed ⑀ here, the intraband susceptibilities do not appear on the right- where we set t=1 eV and 0 =3 eV. The band structure and Fermi surface at half filling are shown in Figs. 6͑a͒ and 6͑b͒, hand side of the equation so that the RPA enhancement of ␹cccc ␹cfcc respectively. Below we will take g =1.8 eV, for which −+,00 stems only from the umklapp susceptibilities −+,Q0 SDW ␹fccc the mean-field equations yield a SDW with nesting vector and −+,Q0. The coupling to these terms in the Dyson equa- ͑␲/ ␲/ ͒ tions is through the anomalous Green’s functions Gcf and Q= a, a , critical temperature TSDW=138 K, and T fc =0 gap ⌬=21.3 meV. The system is insulating at T=0, with G , which reflect the mixing of the states in the electronlike the SDW gap completely removing the Fermi surface. and holelike bands separated by the nesting vector Q in the We plot the imaginary parts of the interband, intraband, SDW phase. Consequently, the intraband susceptibility is ͑ ͒ Q and total transverse susceptibilities for q= qx ,qy =qx in similar to the interband susceptibility but shifted by . Figs. 7͑a͒–7͑c͒, respectively. We consider first the interband The total transverse susceptibility in Fig. 7͑c͒ clearly contribution. For q sufficiently close to Q, we find a con- shows the partial symmetry of the response about q=Q/2 ␹ ͑ ␻͒ tinuum of single-particle excitations. In contrast to the results but also the asymmetric distribution of weight. Im −+ q, for the Hubbard model ͑Fig. 2͒, the magnitude of the trans- for ͉q͉Ͻ͉Q͉/2 is roughly one order of magnitude smaller verse susceptibility in this region tends to increase with in- than at qЈ=Q−q. creasing ␻. This can again be explained in terms of the DOS of the noninteracting model, which now increases as the en- Spin-wave velocity ergy is raised ͑lowered͒ away from the Fermi energy up The calculation of the spin-wave velocity proceeds as for ͑down͒ to a van Hove singularity at 3 eV ͑−3 eV͒ in the the Hubbard model. For g =g =g =0, solving the Dyson electronlike ͑holelike͒ band. The density of excitations con- cc ff 2b equations for the interband susceptibilities in Eqs. ͑32c͒ and tributing to the susceptibility therefore also increases with ␻. ͑32d͒ again involves the inversion of a 2ϫ2 matrix, which As the SDW state is insulating, with a minimum energy of has the determinant 2⌬ required to excite a quasiparticle across the gap, the continuum is sharply bounded at ␻=2⌬=42.6 meV. The con- D͑ ␻͒ ͓ ␹cffc͑0͒ ␹cfcf͑0͔͓͒ ␹fccf͑0͒ q, = 1−gcf −+,00 − g2a −+,00 1−gcf −+,00 tinuum is also bounded by a dispersing V-shaped feature ␹fcfc͑0͔͒ ͓ ␹fcfc͑0͒ ␹fccf͑0͔͒ with minimum at q=0.54Q, which is not seen for the Hub- − g2a −+,00 − gcf −+,00 + g2a −+,00 ͑ ͒ ͑ ͒ bard model. The absence of any weight at small q is antici- ϫ ͓g ␹cfcf 0 + g ␹cffc 0 ͔. ͑37͒ pated from the band structure in Fig. 6, which shows that the cf −+,00 2a −+,00 minimum wave vector for an interband transition with en- Expanding this determinant about ␻=0 and ␦q=Q−q=0, ␻ ͉␦ ͉ ergy ␻Ͻ400 meV is qϷ0.5Q. The V-shaped feature is we obtain the low-energy linear form =cSW q of the spin- plotted in detail in Fig. 8͑a͒. As shown in Fig. 8͑b͒,itisdue wave dispersion. For the band structure considered here, the to the weak nesting of the hole band at k=0.23Q with the spin-wave velocity is given by

174401-7 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒ o m(,) , ( Im log 10 χ i −+ nter (meV) ω q ω

π π (a) qax / =qay / o m(,) , ( Im log 10 ) χ intra −+ meV ( ω q ω

π π (b) qax / =qay / o m(,) , ( Im log 10 χ (meV) −+ ω q ω

π π (c) qax / =qay /

FIG. 7. ͑Color online͒ Imaginary part of ͑a͒ the interband, ͑b͒ the intraband, and ͑c͒ the total transverse susceptibility in the insulating excitonic model for q=͑qx ,qy =qx͒. The spin-wave dispersion is visible as the dark line at ␻Ͻ2⌬ close to Q in ͑a͒ and ͑c͒. Note the logarithmic color scale.

2 2 2 ⑀c 2t a ͓a ͑g − g ͒ − g ͔ 1 k+Q c = ͱ 3 0 2a cf 2a , ͑38͒ a = ͚ ͭ t cos k a SW ͑ 2 ͒͑ 2 2 ͒ 3 2V 3 x a1 +2a0a2 g2a − gcf −2a2g2a k 4Ek 2⌬4 + ⌬2͑⑀c ͒2 − ͑⑀c ͒4 where k+Q k+Q 2 ͮ ͑ ͒ + 5 sin kxa , 41 Ek 1 ⌬2 1 ⑀c k+Q ͑ ͒ a0 = ͚ 3 , a1 = ͚ 3 , 39 and 4V k Ek 4V k Ek ͱ͑⑀c ͒2 ⌬2 ͑ ͒ Ek = k+Q + . 42 1 1 ͑ ͒ We plot c as a function of g for different values of g a2 = ͚ 3 , 40 SW SDW cf 8V k Ek in Fig. 10͑a͒. The behavior of the spin-wave velocity for

174401-8 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒

1.28

m(,) Im 1.275 ) χ ) i −+ nter 1.27 1.5 meV eV ( ( q 1.4

(eV) g =0 a

ω 1.265

ω cf / a / 1.3 gcf = 0.25gSDW SW SW c c 1.26 g = 0.5g 1.2 cf SDW g = 0.75g 1.255 0 102030 cf SDW g = g g (eV) cf SDW (a) qax /π =qay /π SDW 1.25 3 1 1.5 2 2.5 3 (a) gSDW = gcf + g2a (eV) 2 Q 2.0 1

0 0.54Q 1.5 energy (eV) -1

-2 1.0 ω/Δ

-3 0 0.25 0.5 0.75 1 dispersion (excitonic model) 0.5 (b) kxa/π, ky = kx low-energy form dispersion (Hubbard model) FIG. 8. ͑Color online͒͑a͒ Imaginary part of the interband trans- 0.0 verse susceptibility in the insulating excitonic model for q 0 0.01 0.02 =͑qx ,qy =qx͒ close to 0.54Q. Note the linear color scale. ͑b͒ Band structure along the Brillouin-zone diagonal, showing the nesting (b) |δqa/π| ͑ ͒ responsible for the dispersing feature in a . ͑ ͒͑ ͒ FIG. 10. Color online a Spin-wave velocity cSW as a function of g =g +g in the insulating excitonic model. Inset: c for a ӷ Ն SDW cf 2a SW gSDW t is included as an inset. For gSDW gcf we always larger range of g . ͑b͒ Spin-wave dispersion in the excitonic 2 Ͼ Ͼ SDW ﬁnd cSW 0; for sufﬁciently large gcf gSDW, however, we model and its low-energy linear form for gSDW=1.8 eV as a func- 2 Ͻ have cSW 0 which indicates that the system becomes un- tion of ␦q=q−Q. Shown for comparison is the spin-wave disper- stable toward a different ground state. This is not surprising, sion in the Hubbard model from Fig. 4͑b͒ for the same T=0 gap. Ͼ Ͼ Ͼ as for gcf gSDW 0 g2a the effective coupling gSDW is smaller than that for the CDW or SCDW. In the opposite Fig. 10͑b͒ we plot the spin-wave dispersion in the excitonic Ͼ Ͼ Ͼ ␦q case g2a gSDW 0 gcf the coupling constants for the SDW model as a function of . Compared to a Hubbard model and CCDW states are equal and always greater than those for with identical T=0 gap, the spin-wave dispersion has both a the CDW and SCDW, and so the SDW remains stable. In higher low-energy velocity and remains approximately linear up to higher energies ͓see Fig. 4͑b͔͒. ͑ ͒ Although Eq. 38 is a rather complicated function of gcf Ͻ and g2a, for gSDW 3 eV the spin-wave velocity in the excitonic model shows remarkably little dependence upon the

m(,) Im interaction constants, in contrast to the Hubbard model re- )

χ ͑ ͒ sults in Fig. 4 a . Instead, the value of cSW is ﬁxed by the i −+ nter ⌬Ӷ ͑

meV band structure: for an excitonic gap t the weak-coupling (

q ͒ Ϸ /ͱ limit we have to excellent approximation c ˜vF 2

ω SW ω where ˜vF is the average Fermi velocity. This is anticipated by the results of Refs. 22 and 23 for chromium, where it was ͱ / ͑ ͒ found that cSW= vevh 3, where ve͑h͒ is the electron hole Fermi velocity and the factor of 1/ͱ3 arises because a three- qa/π =qa/π dimensional Fermi surface is considered. It is also consistent x y ␹ ͑ ␻͒ with our observation that −+ q, is rather insensitive to FIG. 9. ͑Color online͒ Imaginary part of the interband transverse the choice of gcf and g2a for small gSDW. susceptibility in the insulating excitonic model for q=͑qx ,qy =qx͒ The behavior of cSW in the strong-coupling regime of the close to Q. The spin-wave dispersion is visible as the thick black Hubbard and excitonic models is also qualitatively different. ӷ ͑ ͒ ͱ line in the lower left-hand corner. Note the linear color scale. In the former, the U t limit of Eq. 7 gives cSW= 2J

174401-9 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒ where J=4t2 /U is the exchange integral of the corresponding 4 1 effective Heisenberg model.48–50 In the excitonic model, however, the inset of Fig. 10͑a͒ reveals that c has only 0 Q’

SW π f band / ӷ a 0 weak dependence upon the interaction strength for gSDW t. y

c band k A strong-coupling expansion of Eq. ͑38͒ gives the limiting -4 Q ͱ energy (eV) value cSW= 2ta. The interpretation of the strong-coupling -8 -1 limit in the excitonic model is not straightforward: as (0,0) (π,0) (π,π) (0,0) -1 0 1 g →ϱ, simultaneous occupation of the c and f states on k a/π SDW (a) (k a, k a) (b) x the same site is forbidden but double occupation of the c and x y f states is allowed. Since we work at half filling, one might FIG. 11. ͑Color online͒͑a͒ Band structure and ͑b͒ Fermi surface expect a checkerboard orbital ordering with filled c states on of the noninteracting metallic excitonic model. In ͑b͒, the nesting one sublattice and filled f states on the other, which is in- vectors Q=͑␲/a,0͒ and QЈ=͑0,␲/a͒ are also shown. compatible with SDW order. However, it has been shown in a spinless two-band model that such a state is unstable to- identical area. The band structure and Fermi surface are il- ward an excitonic insulator or a phase with either the c or f lustrated in Figs. 11͑a͒ and 11͑b͒, respectively. Note that the ⑀ 53 states fully occupied for 0 0. How this result would hole pocket is nearly but not quite perfectly nested with both change in the presence of spin is not clear. In any case, the electron pockets. We impose a single-Q SDW with ordering ӷ ͑␲/ ͒ gSDW t limit seems somewhat unphysical without also con- vector Q= a,0 . For gSDW=1.873 eV we find a mean- sidering gcc and gff to be large and so we do not further field state with critical temperature TSDW=132 K and T=0 discuss the strong-coupling regime here. gap ⌬=21.3 meV. In the T=0 SDW state both the hole The evaluation of the intraband susceptibilities proceeds pocket at the zone center and the electron pocket at ͑␲/a,0͒ similarly but here the denominator is D͑q+Q,␻͒. This are completely gapped while the electron pocket at ͑0,␲/a͒ yields an identical spin-wave dispersion but shifted to q=0. remains intact. As in the Hubbard model, however, the spin wave has van- The imaginary parts of the interband, intraband, and total ͑ ͒ ishing weight close to the zone center and is barely visible as transverse susceptibilities for q= qx ,0 are shown in Figs. it exits the continuum in the lower right-hand corner of Fig. 12͑a͒–12͑c͒, respectively. Our results are very similar to 7͑b͒. those for the insulating SDW model in Fig. 7. The slightly higher magnitude of the transverse susceptibility is due to the B. Metallic SDW state greater density of states in the electronlike band. The simi- larity is not surprising, as the relevant excitations in both It is more generally the case that the nesting condition models have identical origin, i.e., excitations between states ⑀c Ϸ−⑀f is only approximately satisfied. Furthermore, k k+Q close to two Fermi pockets which are gapped by an excitonic there may be portions of the Fermi surface that do not par- SDW instability. The states close to the ungapped Fermi ticipate in the excitonic instability, as is the case in pocket do not contribute to the interband susceptibility for chromium.24–26 The pnictides also have a complicated Fermi the plotted range of ͑q,␻͒. Although these states do contrib- surface involving several bands. Although the numerous ute to the intraband susceptibility for small values of q, they models for the band structure differ in their are only negligibly mixed with states in the holelike band details,12,13,15,36–40 there is general agreement that in the “un- and thus are not RPA enhanced by the interband interactions. folded” Brillouin zone corresponding to the 2D iron sublat- In contrast to the insulating SDW state studied in Sec. tice the nesting of hole pockets at k=͑0,0͒ with electron III A, here the interband susceptibility does not just depend pockets at ͑␲/a,0͒ or ͑0,␲/a͒ is primarily responsible for on ͉␦q͉: although it is identical for q=͑q ,0͒ and the SDW. In the physical, tetragonal Brillouin zone, both x ͑␲/a,␲/a−q ͒ by tetragonal symmetry, and quantitatively ͑␲/a,0͒ and ͑0,␲/a͒ are folded back onto the M point, lead- x very similar along q=͑␲/a−q /ͱ2,Ϯq /ͱ2͒, away from ing to two electron pockets around that point.36,40 The wave x x these high-symmetry lines in q space we find that the con- vectors in the present paper refer to the unfolded zone. Ap- tinuum can extend to significantly lower energies. This parently only one of the electron pockets undergoes the ex- is shown in Fig. 13, where we plot Im ␹inter͑q,␻͒ for q citonic instability, yielding a SDW with ordering vector Q −+ =͑␲/a−˜q cos ␪,˜q sin ␪͒ with ␪=␲/8. Although the re- =͑␲/a,0͒, say. The other electron pocket at QЈ=͑0,␲/a͒ sponse for ␻Ͼ100 meV is very similar to that in Fig. 12͑a͒, remains ungapped. We can capture the basic features of this we see that the lower edge of the continuum is not constant scenario within a two-band model by including one hole at ␻=2⌬ but instead shows higher and lower thresholds pocket around ͑0,0͒ and treating the two electron pockets as which coincide only at special values of q. belonging to the same band. We thus assume the band struc- The origin of this threshold behavior is the imperfect nest- ture ing of the Fermi pockets. Consider Fig. 14͑a͒, which shows ⑀c =2t cos k a cos k a + ⑀ , ͑43a͒ the superimposed hole and back-folded electron Fermi pock- k x y c ͑ ͒ ͑ ͒ ets: along k= kx ,0 or 0,ky , the width of the hole Fermi ⑀f ͑ ͒ ⑀ ͑ ͒ pocket is greater than that of the electron one while the re- k =2t cos kxa + cos kya + f . 43b ͑ ͒ ͑ ͒ verse is true for k= kx ,ky =kx or kx ,ky =−kx . In the former ⑀ ⑀ We take t=1 eV, c =1.5 eV, f =−3.5 eV, and fix the dop- case, the intersection of the noninteracting electronlike and ing at n=1.916, which gives electron and hole pockets of holelike dispersions therefore occurs above the Fermi energy

174401-10 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒ o m(,) , ( Im log 10 χ inter −+ (meV) ω q ω

π, (a) qax /=0 qy o m(,) , ( Im log 10 χ intra −+ (meV) ω q ω

π, (b) qax /=0 qy o m(,) , ( Im log 10 ) χ meV −+ ( q ω ω

π, (c) qax /=0 qy

FIG. 12. ͑Color online͒ Imaginary part of the ͑a͒ interband, ͑b͒ intraband, and ͑c͒ total transverse susceptibility in the metallic excitonic model for q=͑qx ,0͒. The spin-wave dispersion is visible as the dark line at ␻Ͻ2⌬ close to Q in ͑a͒ and ͑c͒. Note the logarithmic color scale.

͓Fig. 14͑c͔͒ while in the latter it occurs below the Fermi on ␦q. For example, the states near the Fermi surface ͓ ͑ ͔͒ ͑ ͒ ͑ ͒ ͑ ͒ energy Fig. 14 d . In the reconstructed bands of the exci- in Fig. 14 a at k= 0,ky and kx ,ky =kx are connected tonic model, the SDW gap is always centered at the point of by ␦q=͓˜q cos͑␲/8͒,−˜q sin͑␲/8͔͒ with ˜q=0.17␲/a; from intersection of the original bands, as can be seen from Eq. Figs. 14͑c͒ and 14͑d͒ we see that the minimum energy for ͑26͒ and in Figs. 14͑c͒ and 14͑d͒. In general, the difference single-particle excitations with this wave vector is 1.2⌬ between the Fermi energy and the bottom of the recon- =25.6 meV, which marks the lowest edge of the continuum ⌬ structed electronlike band, +,k, and the difference between in Fig. 13. The upper threshold originates from the maximum the Fermi energy and the top of the reconstructed holelike energy connecting the top of the holelike band and the bot- ⌬ ⌬ band, −,k, will be unequal. The minimum energy for an tom of the electronlike band, which for this q is 2.8 interband excitation with wave vector Q+␦q is therefore =59.6 meV. For the remainder of this paper we shall restrict ͑⌬ ⌬ ͒ ␦ mink Ϯ,k + ϯ,k+␦q . For q along the high-symmetry direc- ourselves to high-symmetry directions. tions mentioned above, the tetragonal symmetry of the Fermi Since only one-electron pocket is gapped, the directions ⌬ ͑ ͒ ͑ ͒ pockets ensures that this minimum energy is 2 . Away from qx ,0 and 0,qy are not equivalent. The imaginary part of ͑ ͒ these directions, however, the energy difference depends the interband transverse susceptibility along q= 0,qy is

174401-11 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒ o m(,) , ( Im log 10 χ i −+ nter (meV) ω q ω

||/Q − q a π

FIG. 13. ͑Color online͒ Imaginary part of the interband transverse susceptibility in the metallic excitonic model for q=͑␲/a −˜q cos ␪,˜q sin ␪͒ for ␪=␲/8. The spin-wave dispersion is visible as the dark line at ␻Ͻ2⌬ close to Q. Note the logarithmic color scale. shown in Fig. 15. This is the direction toward the second ungapped electron pocket and the gapped hole pocket is nesting vector QЈ=͑0,␲/a͒, which was not selected by the smaller than 2⌬, thus giving a lower threshold for the con- SDW instability. For q sufficiently close to QЈ, we thus find tinuum near QЈ. the response generated by transitions between states near the Another significant difference concerns the spin-wave dis- ͑gapped͒ hole Fermi pocket and states near the ungapped persion. The spin-wave dispersion near Q is visible in the electron Fermi pocket. At ␻տ2⌬, this is very similar to the lower left-hand corner of Fig. 12͑a͒, and it intersects the interband susceptibility near Q ͓Fig. 12͑a͔͒, reflecting the continuum and appears to continue as a paramagnon. From small changes to the band structure at high energies upon Fig. 15, we see that there is also a gapless Goldstone mode at q=QЈ. This mode is gapless since it rotates the single-Q opening of the SDW gap. The differences are more striking Ј at lower energies. In particular, comparing Fig. 15 to Fig. SDW into a superposition of Q and Q SDWs, which is degenerate with the single-Q SDW in our tetragonal model. 12͑a͒, we see that the continuum extends to lower energies Although there appears to be a spin-wave branch around QЈ, close to QЈ than close to Q. The minimum energy required it is not as distinct as in Fig. 12͑a͒ due to the lower threshold for a single-particle excitation between the states near the of the continuum. We therefore plot the interband transverse susceptibilities for a finer q resolution near Q and QЈ in Figs. 1.2 16͑a͒ and 16͑b͒, respectively. As expected from the discus- 0.2 ) 1.0 -1 0.8 sion above, the former is qualitatively identical to Fig. 9. The π / c latter, in contrast, shows several different features: the spin- a 0.6 y ε

k 0.1 k+Q wave dispersion does not curve away from the edge of the f 0.4

DOS (eV continuum but rather intersects it with little change in veloc- εk 0.2 0 0.0 ity and the spin-wave and paramagnon features approach 0 0.1 0.2 -3 -2 -1 0 1 2 3 much closer to one another than for qϷQ. Although it is not k a/π ω/Δ (a) x (b) clear from Fig. 16͑b͒, the spin-wave and paramagnon disper- 2 sions do not intersect, and the spin waves become damped at ␻Ϸ1.7⌬. To obtain the spin-wave dispersion, we must again solve 1 Re D͑q,␻͒=0 with D͑q,␻͒ given by Eq. ͑37͒. We have not Δ+,k been able to obtain analytical expressions for the spin-wave

Δ velocity, however, as the Fermi distribution functions appear- )/ Δ+,k 0 ing in the mean-ﬁeld susceptibilities cannot be expanded as a f Taylor series in ␦q due to the ungapped electron Fermi

ε−µ Δ

( −,k ε k Ϸ Ϸ Ј c pocket. Plotting the dispersions at q Q and q Q in Fig. ε Δ k+Q −,k 17, we see that the velocity at Q is roughly 25% higher than -1 E ⌬ -,k at QЈ. Despite the variation in Ϯ,k, there is no anisotropy of E+,k the low-energy spin-wave velocity. The difference between (c) (d) the results for ␪=0 and ␪=␲/8 at higher energies is due to -2 0.22 0.23 0.16 0.17 the lower edge of the continuum in the latter case. Whereas the spin waves close to Q have a very similar dispersion kxa/π, ky =0 kxa/π=kya/π compared to the insulating model ͓Fig. 10͑b͔͒, the dispersion FIG. 14. ͑Color online͒͑a͒ Electron and hole Fermi pockets close to QЈ has two noticeable kinks at ␻=0.65⌬ and ␻ superimposed upon one another. ͑b͒ Density of states close to the =1.25⌬. As shown in the inset, these kinks coincide with Fermi energy. ͑c͒ Comparison of the band structure near the inter- abrupt changes in Im D͑q,␻͒:ImD͑q,␻͒ becomes ﬁnite at ⑀c ⑀f ␻ ⌬ ␻ ⌬ section of k+Q and k in the normal and SDW phases for k =0.65 and starts to sharply increase at =1.25 . The =͑kx ,0͒. ͑d͒ Sameasin͑c͒ but for k=͑kx ,ky =kx͒. ﬁrst feature corresponds to the onset of Landau damping as

174401-12 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒ o m(,) , ( Im log 10 χ i −+ nter (meV) ω q ω

qay /=0π, qx

FIG. 15. ͑Color online͒ Imaginary part of the interband transverse susceptibility in the metallic excitonic model for q=͑0,qy͒. The spin-wave dispersion is visible as the dark line at ␻Ͻ2⌬ close to QЈ ͑qy =0͒. Note the logarithmic color scale. the spin-wave branch enters the continuum. The second fea- tively. In the former case, we see the steplike start of the ture is a result of the DOS, as discussed in the following continuum at ␻=2⌬. The peak at this energy is due both ͑ paragraph. to the remnant of the spin-wave branch at least for qx Examining the lower edge of the continuum in both pan- =0.98␲/a͒ and to the enhancement of the DOS at the edge ␹ ͑ ␻͒ ␻ els of Fig. 16, we see that whereas the continuum disappears of the SDW gap. The finite value of Im −+ q, for sharply at ␻=2⌬ near Q, it appears to vanish more smoothly Ͻ2⌬ is an artifact of the finite width ␦. The susceptibility near QЈ. In the latter case there are two distinct thresholds, near QЈ is qualitatively different: the lower threshold of the ␲/ continuum is at ␻ =0.65⌬ and immediately above this the which are particularly visible around qy =0.99 a. To exam- 1 ␹ ͑ ␻͒ ␻ susceptibility increases continuously as ͱ␻−␻ .At␻ ine this more closely, we plot Im −+ q, as a function of 1 2 for fixed q near Q and QЈ in Figs. 18͑a͒ and 18͑b͒, respec- =1.3⌬, the susceptibilility abruptly starts to increase more steeply. The locations of these two thresholds correspond to the kinks in the spin-wave dispersion. The rapid increase in ␹ ͑ ␻͒ ␻ Im −+ q, above 2 accounts for the strong increase in the damping ͑see inset of Fig. 17͒.

m(,) Im As for the interband susceptibility near Q, the origin of )

χ ⌬ the lower threshold is the variation in Ϯ,k. The difference is i −+ nter meV ( 2.0 q ω

ω Q-δq, θ=0 Q-δq, θ=π/8 1.5 Q’-δq, θ=0

(a) qax /=0π, qy 0.012

1.0 )} ω/Δ ω , 0.008 q ( D 0.004

m(,) Im 0.5 -Im{ ) χ 0.000 i −+ nter 0 0.5 1 1.5 2 meV

( ω/Δ q 0.0 ω

ω 0 0.02 |δqa/π|

FIG. 17. ͑Color online͒ Spin-wave dispersion in the metallic excitonic model close to Q for ␦q=͑−˜q cos ␪,˜q sin ␪͒ with ␪=0 (b) qay /=0π, qx ͑solid red line͒ and ␪=␲/8 ͑thin dotted red line͒. The dispersion for ␪=␲/4 is indistinguishable from the ␪=0 case. We also plot the FIG. 16. ͑Color online͒ Imaginary part of the interband trans- spin-wave dispersion close to QЈ ͑dashed black line͒. Inset: imagi- verse susceptibility in the metallic excitonic model for ͑a͒ q nary part of D͑q,␻͒ for each dispersion. We have used a 20 000 =͑qx ,0͒ close to Q and ͑b͒ q=͑0,qy͒ close to QЈ. In both panels the ϫ20 000 k-point mesh and a width ␦=0.1 meV to calculate the spin-wave dispersion is visible as the thick black line in the bottom mean-field susceptibilities. Note that the finite value of Im D͑q,␻͒ left-hand corner. Note that in ͑b͒ that the continuum region starts at for the spin waves close to Q for ␻Ͻ2⌬ is an artifact of the finite ␻Ϸ0.6⌬. In both panels we use a linear color scale. width ␦.

174401-13 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒

10 experiments, only transverse excitations contribute to the q a = 0.98π x neutron-scattering cross section, allowing us to write it as q a = 0.975π q =(q ,0) x x 2 8 q a = 0.97π d ␴ x ϰ ͉F͑q͉͒2͓n ͑␻͒ +1͔Im ␹ ͑q,␻͒, ͑44͒ )} B −+ qxa = 0.965π d⍀dE ω

, q a = 0.96π 6 x ͑ ͒ ͑␻͒ q q a = 0.955π where F q is a form factor and nB is the Bose-Einstein

( x distribution function. A direct, quantitative comparison be- −+

χ 4 tween theory and experiment would require a more realistic model for the low-energy band structure than the one we are using. We nevertheless make several general remarks relat- Im{ 2 ing what we have learnt about the spin excitations in the (a) excitonic SDW state to the experimental results. 0 We ﬁrst review the experimental situation. Despite considerable variation in the Néel temperature within the qya = 0.98π AFe As ͑A=Ca, Sr, or Ba͒ family, the static magnetic prop- qya = 0.975π q = (0,q ) 2 2 y erties of these compounds are rather similar. In particular, qya = 0.97π )} antiferromagnetism only occurs in the presence of an ortho- qya = 0.965π ω 4 , rhombic distortion, which ﬁxes the ordering vector Q. Ex- qya = 0.96π q

( periments on the low-energy spin dynamics are also in broad qya = 0.955π

−+ agreement: there is a strongly dispersing spin wave close to 46,47,54–57 47,54–57 χ Q, the spin-wave velocity is anisotropic, and 2 the spin-wave dispersion has a gap of energy 6–10 46,47,54–57 Im{ meV. At present, however, there is considerable disagreement over the high-energy excitations. For CaFe2As2,it (b) was reported55 that the spin wave is strongly damped at en- 0 ergies above 100 meV, suggesting the presence of a particle- 01234 hole continuum. On the other hand, although Zhao et al.47 ω/Δ found similar spin-wave velocities, they did not observe any significant jump in the damping of the spin wave below 200 ͑ ͒ FIG. 18. Color online Imaginary part of the transverse suscep- meV, which would indicate the intersection of the spin-wave tibility as a function of ␻ at various values of q near ͑a͒ Q and ͑b͒ dispersion with the continuum. The results for BaFe2As2 QЈ in the metallic excitonic model. We have calculated the mean- 46 ϫ show greater inconsistency, with reports of strong spin ex- field susceptibilities using a 30 000 30 000 k-point mesh and a citations possibly up to 170 meV in stark disagreement with width ␦=0.2 meV. claims of spin-wave damping by continuum excitations at energies as low as 24 meV.57 The results of Refs. 57 and 55 are most consistent with that here the threshold originates from the minimum energy itinerant antiferromagnetism, as the existence of a particle- required for a single-particle excitation between the states hole continuum is a key feature of this scenario. Interpreting near the gapped hole pocket and the states near the ungapped the latter experiment55 in terms of the excitonic model, we ␻ ͑⌬ ⌬ ͒ ͑ ͒ ⌬Ϸ electron pocket, 1 =mink +,k , −,k . From Figs. 14 c and deduce a SDW gap of 50 meV. This is nearly twice the ͑ ͒ ␻ Ϸ ⌬ ⌬Ϸ 14 d we deduce 1 0.6 , closely matching the lower estimate 30 meV of the T=0 gap based on ARPES for ͑ ͒ ␹ ͑ ␻͒ 20 threshold in Fig. 18 b . The strong increase in Im −+ q, SrFe2As2. Although a SDW gap of only 12 meV for ␻ Ϯ␻ BaFe As , which we could infer from Ref. 57, seems low, we above 2 is due to the peaks in the DOS located at 2 on 2 2 either side of the Fermi energy, shown in Fig. 14͑b͒ because have seen above that spin-wave damping sets in at energies ⌬ of this DOS enhancement, the density of excitations between much smaller than 2 , depending upon the details of the states close to the gapped and ungapped Fermi pockets is reconstructed band structure. In order to fit the results of Ref. increased above ␻ . 47 into the excitonic picture, however, we require a SDW 2 gap of at least 100 meV implying a rather high value of the ⌬/ տ ratio kBTSDW 7. These results instead support a local- moment picture.11,41–43 The absence of the continuum is nev- IV. EXPERIMENTAL SITUATION ertheless surprising since ARPES shows clear evidence for quasiparticle bands at low energies, which suggests a possible resolution:20,58 the imperfect nesting of the elliptical This work ultimately aims to shed light upon the nature of electron pockets with the circular hole pocket is expected to the antiferromagnetism in the iron pnictides, in particular, the yield incompletely gapped Fermi surfaces in the SDW state, extent to which it is itinerant or localized in character. There which implies that continuum excitations are present down are several published results of inelastic neutron scattering to zero energy. As such, the spin waves would be damped at examining the spin excitations in the antiferromagnetic state all energies and the jump in the damping characteristic of the 47,54,55 56 46,57 of CaFe2As2, SrFe2As2, and BaFe2As2. In these entry into the continuum is absent. Such an explanation is of

174401-14 SPIN EXCITATIONS IN THE EXCITONIC SPIN-… PHYSICAL REVIEW B 80, 174401 ͑2009͒ course at odds with Ref. 55, indicating the need for further though the spin excitations in more sophisticated models will work to clarify the experimental situation. differ in their details from those presented here, we never- The reported 40% anisotropy of the spin-wave velocity theless think that our results will remain qualitatively correct within the ab plane47,55 is quite remarkable. Although this and will thus be valuable in interpreting future experiments. effect is absent from our results due to the tetragonal symmetry of the Fermi pockets, it nevertheless seems rather too V. SUMMARY large to be accounted for by the expected elliptical shape of the electronlike Fermi pockets in the pnictides.47 Experimen- We have presented an analysis of the zero-temperature tal results also do not show a second spin-wave branch at QЈ, transverse spin excitations in the excitonic SDW state of as found here for the metallic SDW model. Both observa- two-band 2D models with nested electronlike and holelike tions are likely due to the orthorhombic distortion in the Fermi pockets. Using the RPA, we have derived the Dyson SDW phase, which lifts the degeneracy of the ͑␲,0͒ and equation for the spin susceptibility and have shown that the ͑0,␲͒ SDW,36 and do not imply a failure of the excitonic total spin susceptibility can be divided into contributions scenario. from interband and intraband excitations. We have solved the We finally remark upon the gap in the spin-wave disper- Dyson equation in the special case when only the interac- sion in the pnictides. Due to the absence of magnetic aniso- tions responsible for the SDW are nonzero. While the inter- tropy is our model, we always find Goldstone modes in the band excitations are then directly enhanced by the interac- SDW phase. As demonstrated in Fishman and Liu’s study of tions, the intraband excitations are still indirectly enhanced manganese alloys,27 a gap is possible in an excitonic SDW due to the mixing of the electronlike and holelike states in state in the presence of magnetoelastic coupling. The mag- the SDW phase. The susceptibility exhibits collective spin- netoelastic coupling in the pnictides is indeed strong, as evi- wave branches close to the SDW ordering vector Q and also, denced by the role of the orthorhombic distortion in fixing with much smaller weight, close to q=0, as well as a con- the polarization and the ordering vector of the SDW,6,7,57 tinuum of single-particle excitations at energies above a suggesting that it might be responsible for the spin-wave threshold on the order of the SDW gap. gap. Depending upon the noninteracting band structure, the In summary, the neutron-scattering data for the antiferro- opening of the excitonic gap can result in qualitatively dif- magnetic state in the pnictides are currently unable to decide ferent SDW states. This has been illustrated by considering upon the origin and character of the magnetism. We have two models, one which becomes insulating in the SDW state shown that the excitonic SDW scenario gives spin-wave ex- and another which remains metallic due to the presence of an citations in qualitative agreement with experiments. An ob- ungapped portion of the Fermi surface. For comparison, we vious direction of future work is therefore to examine the have also performed the corresponding calculations for a 2D spin excitations based on more realistic band structures. Con- Hubbard model with the same mean-field SDW gap. Differ- sidering the imperfect nesting of the electron and hole pock- ences in the spin excitations between the insulating and me- ets in the pnictides, it will be particularly interesting to ad- tallic models occur only at low energies and mainly close to dress the possibility of incommensurate SDW order.34 The the nesting vector QЈ between the ͑gapped͒ hole pocket and effects of the interactions not directly contributing to the the ungapped electron pocket, which is essentially unaffected SDW instability should be included. Comparison of our re- by the SDW formation. We have also discussed data from sults with those obtained within a model explicitly account- neutron-scattering experiments in light of our results. We ing for the orbital character of the bands is also important. conclude that the available data do not yet allow us to dis- Furthermore, the orthorhombic distortion and a magnetoelas- tinguish between an excitonic SDW and a local-moment sce- tic coupling should be implemented for greater realism. Al- nario for the antiferromagnetic order in the pnictides.

*[email protected] A. Green, G. F. Chen, G. Li, Z. Li, J. L. Luo, N. L. Wang, and †[email protected] P. Dai, Nature Mater. 7, 953 ͑2008͒; J. Zhao, Q. Huang, C. de la 1 Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am. Cruz, J. W. Lynn, M. D. Lumsden, Z. A. Ren, J. Yang, X. Shen, Chem. Soc. 130, 3296 ͑2008͒. X. Dong, Z. Zhao, and P. Dai, Phys. Rev. B 78, 132504 ͑2008͒. 2 M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101, 7 Q. Huang, Y. Qiu, W. Bao, M. A. Green, J. W. Lynn, Y. C. 107006 ͑2008͒. Gasparovic, T. Wu, G. Wu, and X. H. Chen, Phys. Rev. Lett. 3 Z.-A. Ren, W. Lu, J. Yang, W. Yi, X.-L. Shen, Z.-C. Li, G.-C. 101, 257003 ͑2008͒; A. Jesche, N. Caroca-Canales, H. Rosner, Che, X.-L. Dong, L.-L. Sun, F. Zhou, and Z.-X. Zhao, Chin. H. Borrmann, A. Ormeci, D. Kasinathan, H. H. Klauss, H. Luet- Phys. Lett. 25, 2215 ͑2008͒. kens, R. Khasanov, A. Amato, A. Hoser, K. Kaneko, C. Krell- 4 J. W. Lynn and P. Dai, Physica C 469, 469 ͑2009͒. ner, and C. Geibel, Phys. Rev. B 78, 180504͑R͒͑2008͒. 5 C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. Ratcliff II, J. L. 8 H. Luetkens, H.-H. Klauss, M. Kraken, F. J. Litterst, T. Dell- Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, and P. mann, R. Klingeler, C. Hess, R. Khasanov, A. Amato, C. Baines, Dai, Nature ͑London͒ 453, 899 ͑2008͒. M. Kosmala, O. J. Schumann, M. Braden, J. Hamann-Borrero, 6 J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn, Y. Chen, M. N. Leps, A. Kondrat, G. Behr, J. Werner, and B. Büchner, Na-

174401-15 P. M. R. BRYDON AND C. TIMM PHYSICAL REVIEW B 80, 174401 ͑2009͒

ture Mater. 8, 305 ͑2009͒. 34 V. Cvetkovic and Z. Tesanovic, EPL 85, 37002 ͑2009͒; Phys. 9 G. Mu, X.-Y. Zhu, L. Fang, L. Shan, C. Ren, and H.-H. Wen, Rev. B 80, 024512 ͑2009͒. Chin. Phys. Lett. 25, 2221 ͑2008͒. 35 A. B. Vorontsov, M. G. Vavilov, and A. V. Chubukov, Phys. Rev. 10 L. Shan, Y. Wang, X. Zhu, G. Mu, L. Fang, C. Ren, and H.-H. B 79, 060508͑R͒͑2009͒. Wen, EPL 83, 57004 ͑2008͒. 36 P. M. R. Brydon and C. Timm, Phys. Rev. B 79, 180504͑R͒ 11 Q. Si and E. Abrahams, Phys. Rev. Lett. 101, 076401 ͑2008͒. ͑2009͒. 12 K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, 37 S. Raghu, X.-L. Qi, C.-X. Liu, D. J. Scalapino, and S.-C. Zhang, and H. Aoki, Phys. Rev. Lett. 101, 087004 ͑2008͒. Phys. Rev. B 77, 220503͑R͒͑2008͒. 13 M. M. Korshunov and I. Eremin, Phys. Rev. B 78, 140509͑R͒ 38 J. Lorenzana, G. Seibold, C. Ortix, and M. Grilli, Phys. Rev. ͑2008͒. Lett. 101, 186402 ͑2008͒. 14 P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78,17 39 Y. Ran, F. Wang, H. Zhai, A. Vishwanath, and D.-H. Lee, Phys. ͑2006͒. Rev. B 79, 014505 ͑2009͒. 15 D. J. Singh and M.-H. Du, Phys. Rev. Lett. 100, 237003 ͑2008͒; 40 R. Yu, K. T. Trinh, A. Moreo, M. Daghofer, J. A. Riera, S. Haas, I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, ibid. and E. Dagotto, Phys. Rev. B 79, 104510 ͑2009͒. 101, 057003 ͑2008͒. 41 T. Yildirim, Phys. Rev. Lett. 101, 057010 ͑2008͒. 16 M. A. McGuire, A. D. Christianson, A. S. Sefat, B. C. Sales, M. 42 G. S. Uhrig, M. Holt, J. Oitmaa, O. P. Sushkov, and R. R. P. D. Lumsden, R. Jin, E. A. Payzant, D. Mandrus, Y. Luan, V. Singh, Phys. Rev. B 79, 092416 ͑2009͒. Keppens, V. Varadarajan, J. W. Brill, R. P. Hermann, M. T. 43 F. Krüger, S. Kumar, J. Zaanen, and J. van den Brink, Phys. Rev. Sougrati, F. Grandjean, and G. J. Long, Phys. Rev. B 78, B 79, 054504 ͑2009͒. 094517 ͑2008͒; M. A. McGuire, R. P. Hermann, A. S. Sefat, B. 44 T. Kroll, S. Bonhommeau, T. Kachel, H. A. Dürr, J. Werner, G. C. Sales, R. Jin, D. Mandrus, F. Grandjean, and G. J. Long, New Behr, A. Koitzsch, R. Hübel, S. Leger, R. Schönfelder, A. K. J. Phys. 11, 025011 ͑2009͒. Ariffin, R. Manzke, F. M. F. de Groot, J. Fink, H. Eschrig, B. 17 R. H. Liu, G. Wu, T. Wu, D. F. Fang, H. Chen, S. Y. Li, K. Liu, Büchner, and M. Knupfer, Phys. Rev. B 78, 220502͑R͒͑2008͒. Y. L. Xie, X. F. Wang, R. L. Yang, L. Ding, C. He, D. L. Feng, 45 W. L. Yang, A. P. Sorini, C.-C. Chen, B. Moritz, W.-S. Lee, F. and X. H. Chen, Phys. Rev. Lett. 101, 087001 ͑2008͒. Vernay, P. Olalde-Velasco, J. D. Denlinger, B. Delley, J.-H. Chu, 18 J. K. Dong, L. Ding, H. Wang, X. F. Wang, T. Wu, G. Wu, X. H. J. G. Analytis, I. R. Fisher, Z. A. Ren, J. Yang, W. Lu, Z. X. Chen, and S. Y. Li, New J. Phys. 10, 123031 ͑2008͒. Zhao, J. van den Brink, Z. Hussain, Z.-X. Shen, and T. P. De- 19 S. E. Sebastian, J. Gillett, N. Harrison, P. H. C. Lau, C. H. vereaux, Phys. Rev. B 80, 014508 ͑2009͒. Mielke, and G. G. Lonzarich, J. Phys.: Condens. Matter 20, 46 R. A. Ewings, T. G. Perring, R. I. Bewley, T. Guidi, M. J. 422203 ͑2008͒; J. G. Analytis, R. D. McDonald, J.-H. Chu, S. C. Pitcher, D. R. Parker, S. J. Clarke, and A. T. Boothroyd, Phys. Riggs, A. F. Bangura, C. Kucharczyk, M. Johannes, and I. R. Rev. B 78, 220501͑R͒͑2008͒. Fisher, Phys. Rev. B 80, 064507 ͑2009͒. 47 J. Zhao, D. T. Adroja, D.-X. Yao, R. Bewley, S. Li, X. F. Wang, 20 D. Hsieh, Y. Xia, L. Wray, D. Qian, K. Gomes, A. Yazdani, G. F. G. Wu, X. H. Chen, J. Hu, and P. Dai, Nat. Phys. 5, 555 ͑2009͒. Chen, J. L. Luo, N. L. Wang, and M. Z. Hasan, arXiv:0812.2289 48 J. R. Schrieffer, X. G. Wen, and S. C. Zhang, Phys. Rev. B 39, ͑unpublished͒. 11663 ͑1989͒. 21 L. Boeri, O. V. Dolgov, and A. A. Golubov, Phys. Rev. Lett. 49 A. Singh and Z. Tešanović, Phys. Rev. B 41, 614 ͑1990͒. 101, 026403 ͑2008͒; Physica C 469, 628 ͑2009͒. 50 A. V. Chubukov and D. M. Frenkel, Phys. Rev. B 46, 11884 22 P. A. Fedders and P. C. Martin, Phys. Rev. 143, 245 ͑1966͒. ͑1992͒. 23 S. H. Liu, Phys. Rev. B 2, 2664 ͑1970͒. 51 P. W. Anderson, Phys. Rev. 86, 694 ͑1952͒. 24 T. M. Rice, Phys. Rev. B 2, 3619 ͑1970͒. 52 H. Hasegawa, J. Low Temp. Phys. 31, 475 ͑1978͒. 25 N. I. Kulikov and V. V. Tugushev, Sov. Phys. Usp. 27, 954 53 C. D. Batista, Phys. Rev. Lett. 89, 166403 ͑2002͒. ͑1984͒. 54 R. J. McQueeney, S. O. Diallo, V. P. Antropov, G. D. Samolyuk, 26 E. Fawcett, Rev. Mod. Phys. 60, 209 ͑1988͒; E. Fawcett, H. L. C. Broholm, N. Ni, S. Nandi, M. Yethiraj, J. L. Zarestky, J. J. Alberts, V. Y. Galkin, D. R. Noakes, and J. V. Yakhmi, ibid. 66, Pulikkotil, A. Kreyssig, M. D. Lumsden, B. N. Harmon, P. C. 25 ͑1994͒. Canfield, and A. I. Goldman, Phys. Rev. Lett. 101, 227205 27 R. S. Fishman and S. H. Liu, Phys. Rev. B 58, R5912 ͑1998͒; ͑2008͒. Phys. Rev. B 59, 8672 ͑1999͒; 59, 8681 ͑1999͒. 55 S. O. Diallo, V. P. Antropov, T. G. Perring, C. Broholm, J. J. 28 L. V. Keldysh and Y. V. Kopaev, Sov. Phys. Solid State 6, 2219 Pulikkotil, N. Ni, S. L. Bud’ko, P. C. Canfield, A. Kreyssig, A. I. ͑1965͒; J. des Cloizeaux, J. Phys. Chem. Solids 26, 259 ͑1965͒; Goldman, and R. J. McQueeney, Phys. Rev. Lett. 102, 187206 D. Jérome, T. M. Rice, and W. Kohn, Phys. Rev. 158, 462 ͑2009͒. ͑1967͒. 56 J. Zhao, D.-X. Yao, S. Li, T. Hong, Y. Chen, S. Chang, W. 29 B. A. Volkov, Y. V. Kopaev, and A. I. Rusinov, Sov. Phys. JETP Ratcliff II, J. W. Lynn, H. A. Mook, G. F. Chen, J. L. Luo, N. L. 41, 952 ͑1976͒. Wang, E. W. Carlson, J. Hu, and P. Dai, Phys. Rev. Lett. 101, 30 D. W. Buker, Phys. Rev. B 24, 5713 ͑1981͒. 167203 ͑2008͒. 31 A. V. Chubukov, D. V. Efremov, and I. Eremin, Phys. Rev. B 78, 57 K. Matan, R. Morinaga, K. Iida, and T. J. Sato, Phys. Rev. B 79, 134512 ͑2008͒. 054526 ͑2009͒. 32 Q. Han, Y. Chen, and Z. D. Wang, EPL 82, 37007 ͑2008͒. 58 D. H. Lu, M. Yi, S.-K. Mo, J. G. Analytis, J.-H. Chu, A. S. 33 T. Mizokawa, T. Sudayama, and Y. Wakisaka, J. Phys. Soc. Jpn. Erickson, D. J. Singh, Z. Hussain, T. H. Geballe, I. R. Fisher, Suppl. C 77, 158 ͑2008͒. and Z.-X. Shen, Physica C 469, 452 ͑2009͒.

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