document: fe - based superconductors ( febs ) represent a non - cuprate class of high-@xmath9 systems with the unconventional superconducting state . the origin of the latter is still debated . in general , 
   febs can be divided into the two subclasses , pnictides and chalcogenides  @xcite , with the square lattice of iron as the basic element , though with orthorhombic distortions in lightly doped materials . 
   iron is surrounded by as or p situated in the tetrahedral positions within the first subclass and by se , te , or s within the second subclass . 
   fermi surface ( fs ) is formed by fe @xmath10-orbitals and excluding the cases of extreme hole and electron dopings it consists of two hole sheets around the @xmath11 point and two electron sheets around the @xmath12 and @xmath13 points in the two - dimensional brillouin zone ( bz ) corresponding to one fe per unit cell ( the so - called 1-fe bz )  @xcite . in the 2-fe bz , 
   electron pockets are centered at the @xmath14 point . 
   nesting between these two groups of sheets leads to the enhanced antiferromagnetic fluctuations with the maximal scattering near the wave vector @xmath2 equal to @xmath12 or @xmath13 in the 1-fe bz or to @xmath15 in the 2-fe bz 
   .    different mechanisms of cooper pairs formation result in the distinct superconducting gap symmetry and structure  @xcite . 
   in particular , the rpa - sf ( random - phase approximation spin fluctuation ) approach gives the extended @xmath16-wave gap that changes sign between hole and electron fs sheets ( @xmath17 state ) as the main instability for the wide range of doping concentrations  @xcite . on the other hand , 
   orbital fluctuations promote the order parameter to have the sign - preserving @xmath6 symmetry  @xcite . 
   thus , probing the gap structure can help in elucidating the underlying mechanism . in this respect , inelastic neutron scattering ( ins ) 
   is a useful tool since the measured dynamical spin susceptibility @xmath18 in the superconducting state carries information about the gap structure . 
   there are been many reports of a well - defined peak in neutron spectra in 1111 , 122 , and 11 systems appearing only for @xmath19 at or around @xmath20  @xcite . 
   the common explanation is that the peak is the spin resonance appearing due to the @xmath7 state . indeed , since @xmath2 connects fermi sheets with different signs of @xmath7 gaps , the resonance condition for the interband susceptibility is fulfilled and the spin resonance peak is formed at a frequency @xmath21 below @xmath22 with @xmath23 being the gap size  @xcite . 
   such simple explanation was indirectly questioned by the angle - resolved photoemission spectroscopy ( arpes ) results and recent measurements of gaps via andreev spectroscopy . 
   latter clearly shows that there are at least two distinct gaps present in 11 , 122 , and 1111 systems  @xcite and even three gaps in lifeas  @xcite . 
   larger gap ( @xmath0 ) is about 9mev and the smaller gap ( @xmath1 ) is about 4mev in baco122 materials . from arpes 
   we know that electron fs sheets and the inner hole sheet are subject to opening the lager gap while the smaller gap is located at the outer hole fs  @xcite . 
   the very existence of the smaller gap rise the question  what would be the spin resonance frequency in the system with two distinct gaps ? 
   naive expectation is that the frequency shifts to the lower gap scale and @xmath24 . 
   then the observed peak in ins in baco122 system at frequency @xmath25mev  @xcite can not be the spin resonance since it is greater than @xmath26mev  @xcite . 
   thus the peak could be coming from the @xmath6 state  @xcite , where it forms at frequencies _ above _ 
   @xmath27 due to the redistribution of the spectral weight upon entering the superconducting state and a special form of scattering in the normal state . here we study this question in details and show that the naive expectation is wrong and that the true minimal energy scale is @xmath28 . 
   latter is consistent with the maximal frequency of the observed peak in ins in baco122 and confirms that it is the true spin resonance evidencing the @xmath7 gap symmetry . 
   the maximal energy scale is @xmath29 . whether the minimal or maximal energy scale will be realized depends on the relation between the exact band structure of a particular material and the wave vector of the spin resonance @xmath2 . 
   to describe spin response in normal and superconducting states of febs , we use random phase approximation ( rpa ) with the local coulomb interactions ( hubbard and hund s exchange ) . in the multiband system , 
   transverse dynamical spin susceptibility @xmath30 is the matrix in orbital ( or band ) indices . 
   it can be obtained in the rpa from the bare electron - hole matrix bubble @xmath31 by summing up a series of ladder diagrams : @xmath32^{-1 } \hat\chi_{(0)+-}(\q,\omega ) , \label{eq : chi_s_sol}\end{aligned}\ ] ] where @xmath33 is the momentum , @xmath34 is the frequency , @xmath35 and @xmath36 are interaction and unit matrices in orbital ( or band ) space . 
   exact form of @xmath35 and bare susceptibility @xmath31 depends on the underlying model . 
   later we use two types of tight - binding models for the two - dimensional iron layer . 
   first we study the four - band model of ref .   with the following single - electron hamiltonian @xmath37 where @xmath38 is the annihilation operator of the @xmath10-electron with momentum @xmath39 , spin @xmath40 , and 
   band index @xmath41 , @xmath42 are the on - site single - electron energies , @xmath43 is the electronic dispersion that yields hole pockets centered around the @xmath44 point , and @xmath45 is the dispersion that results in the electron pockets around the @xmath46 point of the 2-febz . 
   parameters are the same as in ref .  . 
   in the superconducting state we assume either the @xmath6 state with @xmath47 or the @xmath7 state with @xmath48 . 
   physical spin susceptibility @xmath49 obtained by calculating matrix elements @xmath50 via equation  ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath51 and with the bare spin susceptibility @xmath52 in the superconducting state ( see ref .   for details ) . 
   we assume here the effective hubbard interaction parameters to be @xmath53 and @xmath54 in order to stay in the paramagnetic phase  @xcite . 
   the model described above is simple enough to gain qualitative description of the spin response of superconductor with unequal gaps . 
   but it lack for the orbital content of the bands that is important for the detailed structure of the susceptibility . 
   that is why we also present results for the tight - binding model from ref . 
   based on the fit to the dft band structure for lafeaso  @xcite . 
   the model includes all five iron @xmath10-orbitals ( @xmath55 , @xmath56 , @xmath57 , @xmath58 , @xmath59 ) enumerated by index @xmath60 and is given by @xmath61 d_{l \k \sigma}^\dagger d_{l ' \k \sigma } ,   \label{eq : h0}\ ] ] where @xmath62 is the annihilation operator of a particle with momentum @xmath39 , spin @xmath40 , and orbital index @xmath60 . 
   later we use numerical values of hopping matrix elements @xmath63 and one - electron energies @xmath64 from ref .  . 
   this model for the undoped and moderately electron doped materials gives fs composed of two hole pockets , @xmath65 and @xmath66 , around the @xmath67 point and two electron pockets , @xmath68 and @xmath69 , centered around @xmath12 and @xmath13 points of the 1-fe bz . similar model for iron pnictides was proposed in ref .  . 
   the general two - particle on - site interaction would be represented by the hamiltonian  @xcite : h_int & = & u _ f , m n_f m n_f m + u _ f , m < l n_f l n_f m + & & + j _ f , m < l _ , 
   d_f l ^d_f m ^d_f l 
   d_f m   + & & + j _ f , m l d_f l ^d_f l ^d_f m d_f m . [ eq : hint ] where @xmath70 , @xmath71 is the number of particles operator at the site @xmath72 , @xmath73 and @xmath74 are the intra- and interorbital hubbard repulsion , @xmath75 is the hund s exchange , and @xmath76 is the so - called pair hopping . 
   we choose the following values for the interaction parameters : @xmath77ev , @xmath78 , and make use of the spin - rotational invariance constraint @xmath79 and @xmath80 . 
   green functions are diagonal in the band basis but not in the orbital basis . 
   let us introduce creation and annihilation operators @xmath81 and @xmath82 of electrons with band index @xmath83 , in terms of which green functions are diagonal , @xmath84 . 
   transformation from the orbital to the band basis is done via the matrix elements @xmath85 , @xmath86 , and for the transverse component of the bare spin susceptibility  @xcite we have [ eq : chipmmu ] & & ^ll,mm_(0)+- ( , ) = -t _ , _ n , , , where @xmath87 and @xmath88 are matsubara frequencies , @xmath89 and @xmath90 are the normal and anomalous ( superconducting ) green s functions , respectively . 
   components of the physical spin susceptibility @xmath91 are calculated using eq . 
   ( [ eq : chi_s_sol ] ) with the interaction matrix @xmath92 from ref .  . 
   since calculation of the cooper pairing instability is not a topic of the present study , here we assume that the superconductivity is coming from some other theory and study either the @xmath6 state with @xmath93 or the @xmath7 state with @xmath94 , where @xmath83 is the band index . 
   here we present results for susceptibilities at the wave vector @xmath95 as functions of frequency @xmath34 obtained via analytical continuation from matsubara frequencies ( @xmath96 with @xmath97 ) 
   .    imaginary part of bare and rpa spin susceptibilities in the four - band model  ( [ eq : h04band ] ) are shown in fig . 
   [ fig : imchi4band ] . 
   first , we discuss result for equal gaps on electron ( @xmath98 , @xmath99 ) and hole ( @xmath100 , @xmath101 ) fss , @xmath102 . 
   since @xmath103 describes particle - hole excitations and in the superconducting state all excitations are gapped below approximately @xmath104 ( at @xmath105 ) , then @xmath106 becomes finite only after that frequency . for the @xmath6 state 
   , there is a gradual increase of the spin response for @xmath107 . for the @xmath7 state 
   , @xmath2 connects fss with different signs of gaps , @xmath108 , and within rpa  ( [ eq : chi_s_sol ] ) this results in the spin resonance peak  divergence of @xmath109 at a frequency @xmath110 , see fig . 
   [ fig : imchi4band ] , bottom panel . 
   ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . 
   two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . [ 
   fig : imchi4band],title="fig : " ]   ( top ) and @xmath109 with @xmath111 in the 2-fe bz for the four - band model in the normal , @xmath6 and @xmath7 superconducting states . 
   two cases of superconducting states are shown : equal @xmath23 s with @xmath102 , and unequal gaps with @xmath112 and @xmath113 . 
   [ fig : imchi4band],title="fig : " ]    now let s consider the case of unequal gaps with a small gap scale on outer hole fs , @xmath113 , and a larger gap scale on all other fss , @xmath112 . as seen from fig . 
   [ fig : imchi4band ] , top panel , for the @xmath7 state the discontinuous jump and , thus , @xmath114 , moved to lower frequencies . 
   this new energy scale clearly tracked down in the @xmath6 state as the starting point of the susceptibility gradual increase . 
   it is equal to @xmath115 , where @xmath0 and @xmath1 being the larger and smaller gap scales . 
   consequently , the spin resonance peak in @xmath7 moved to lower frequencies , @xmath116 , see fig . 
   [ fig : imchi4band ] , bottom panel . 
   additional feature is the hump around the @xmath117 energy scale . 
   note that the susceptibility in the @xmath6 state havent changed much compared to the equal gaps case .     in the superconducting state , @xmath118 , as a function of momentum @xmath39 along the @xmath119 direction , i.e. @xmath120 . 
   scattering wave vector @xmath2 entering the spin susceptibility is also shown . 
   [ fig:5orbdelta ] ]     with @xmath121 in the 1-fe bz for the five - orbital model in the normal , @xmath6 and @xmath7 superconducting states . 
   two cases of superconducting states are shown : equal gaps with @xmath122 , and unequal gaps with @xmath123 and @xmath124 , where @xmath125 . 
   latter case is shown in the inset , where gaps at the fs are plotted together with the wave vector @xmath2 . 
   [ fig:5orbimchi ] ]    to demonstrate where the new energy scale is coming from we turn our attention to the five - orbital model  ( [ eq : h0 ] ) . 
   its energy spectrum near the fermi level in the superconducting state , @xmath118 , is shown in fig . 
   [ fig:5orbdelta ] . 
   we consider here the case of unequal gaps with the smaller gap @xmath124 on the outer hole fs and larger gaps @xmath123 on inner hole and electron fss . to be consistent with the experimental data , we choose @xmath126 , 
   see the inset in fig . 
   [ fig:5orbimchi ] . 
   naturally , the two energy scales , @xmath127 and @xmath128 , appear in the energy spectrum @xmath129 and they are connected with hole @xmath66 and electron @xmath130 bands , respectively . on the other hand , the susceptibility @xmath131 contains scattering _ between _ hole and electron bands with the wave vector @xmath2 . 
   the energy gap that have to be overcome to excite electron - hole pair is the indirect gap with the scale @xmath4 . 
   that is why spin excitations in the @xmath6 state start with the frequency proportional to the indirect gap @xmath132 , see fig . 
   [ fig:5orbimchi ] . the same is true for the discontinuous jump in @xmath133 for the @xmath7 state  it shifts to frequency @xmath134 . 
   this , together with the corresponding @xmath135 singularity in @xmath136 , produce the spin resonance peak in rpa at frequency @xmath8 . 
   such shift of resonance peak to lower frequencies compared to the equal gaps situation is seen in fig . 
   [ fig:5orbimchi ] , where the spin response @xmath109 for the cases of equal and distinct gaps is shown . 
   the changes in the band structure and/or doping level can result in the change of the indirect gap . in particular , since for the hole doping hole fss become larger the wave vector @xmath2 may connect states on the electron fs and on the _ inner _ hole fs . 
   gaps on both these fss are determined by @xmath0 and thus the indirect gap will be equal to @xmath5 . 
   this sets up a maximal energy scale for the spin resonance , i.e. @xmath29 .    thus we conclude that depending on the relation between the wave vector @xmath2 and the exact fs geometry , the indirect gap in most febs can be either @xmath4 or @xmath5 . 
   the peak in the dynamical spin susceptibility at the wave vector @xmath2 will be the true spin resonance if it appears below the indirect gap scale , @xmath8 . 
   now we can compare energy scales extracted from arpes , andreev spectroscopy , and inelastic neutron scattering . 
   latter gives peak frequency @xmath137mev in bafe@xmath138co@xmath139as@xmath140 with @xmath141k  @xcite . 
   for the same system , gap sizes extracted from arpes are @xmath142mev and @xmath143mev  @xcite , and for a similar system with @xmath144k , @xmath145mev and @xmath146mev  @xcite . 
   gap sizes extracted from andreev spectroscopy are @xmath147mev and @xmath148mev in bafe@xmath149co@xmath150as@xmath140 with @xmath151k  @xcite . 
   evidently , @xmath152 and we can safely state that the peak in ins is the spin resonance . for the hole doped systems , peak frequency in ins is about @xmath153mev in ba@xmath154k@xmath155fe@xmath140as@xmath140 with @xmath156k  @xcite . 
   there is a slight discrepancy between gap sizes extracted from arpes and andreev spectra . 
   former gives @xmath157mev and @xmath158mev in the same material with @xmath159k  @xcite , thus @xmath152 . 
   gap sizes from andereev spectroscopy are @xmath160mev and @xmath161mev in ba@xmath162k@xmath163fe@xmath140as@xmath140 with lower @xmath164k  @xcite . in this case , @xmath165 but @xmath166 and we still can assume that the peak in ins is the spin resonance . 
   however , in such a case definitive conclusion can be given only by the calculation of spin response for the particular experimental band structure . 
   for more extensive review of available experimental data on @xmath167 and gap scales , see the supplemental material  @xcite .    on the separate note , we would like to mention that the appearance of a hump structure in the superconducting state at frequencies larger than the main peak frequency ( the so - called double resonance feature ) may be related to the @xmath128 energy scale , see fig . 
   [ fig : imchi4band ] . 
   such hump structure was observed in nafe@xmath168co@xmath169as  @xcite and fete@xmath170se@xmath170  @xcite . somehow similar structure was found in polarized inelastic neutron studies of bafe@xmath171ni@xmath172as@xmath140  @xcite and ba(fe@xmath173co@xmath174)@xmath140as@xmath140  @xcite , but its origin may be related to the spin - orbit coupling  @xcite rather than the simple @xmath128 energy scale . 
   another explanation of the double resonance feature is related to the pre - existing magnon mode , i.e. the dispersive low - energy peak in underdoped materials is associated with the spin excitations of the magnetic order with the intensity enhanced below @xmath9 due to the suppression of the damping  @xcite . 
   we analysed the spin response of febs with two different superconducting gap scales , @xmath175 . 
   spin resonance appears in the @xmath7 state below the indirect gap scale @xmath3 that is determined by the sum of gaps on two different fermi surface sheets connected by the scattering wave vector @xmath2 . in the @xmath6 state , 
   spin excitations are absent below @xmath3 unless additional scattering mechanisms are assumed  @xcite . for the fermi surface geometry characteristic to the most of febs materials , 
   the indirect gap is either @xmath4 or @xmath5 . 
   this gives the simple criterion to determine whether the experimentally observed peak in inelastic neutron scattering is the true spin resonance  if the peak frequency @xmath21 is less than the indirect gap @xmath3 , then it is the spin resonance and , consequently , the superconducting state has the @xmath7 gap structure . 
   comparison of energy scales extracted from ins , andreev spectroscopy , arpes and other techniques allowing to determine superconducting gaps , for most materials gives confidence that the observed feature in ins is the spin resonance peak . 
   however , sometimes it is not always clear experimentally which gaps are connected by the wave vector @xmath2 . even without knowing this exactly 
   , one can draw some conclusions . 
   for example , if one of the gaps is @xmath0 , then there are three cases possible : ( 1 ) @xmath176 and the peak at @xmath21 is the spin resonance , ( 2 ) @xmath177 and the peak is definitely not a spin resonance , and ( 3 ) @xmath29 and the peak is most likely the spin resonance but the definitive conclusion can be drawn only from the calculation of the dynamical spin susceptibility for the particular experimental band structure .    we would like to thank h. kontani , s.a . 
   kuzmichev , t.e . 
   kuzmicheva , v.m . 
   pudalov , and i.s . 
   sandalov for useful discussions . 
   mmk is grateful to b. keimer and max - planck - institut fr festkrperforschung for the hospitality during his visit . 
   we acknowledge partial support by rfbr ( grant 16 - 02 - 00098 ) , and government support of the leading scientific schools of the russian federation ( nsh-7559.2016.2 ) . 
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   _ review of experimental data on the peak in inelastic neutron scattering ( ins ) and gaps extracted from various experimental techniques . _    in the article 
   , we have analysed the spin response of febs with two different superconducting gap scales , @xmath175 . 
   spin resonance appears in the @xmath7 state below the indirect gap scale @xmath3 that is determined by the sum of gaps on two different fermi surface sheets connected by the scattering wave vector @xmath2 . for the fermi surface geometry characteristic to the most of febs materials , 
   the indirect gap is either @xmath4 or @xmath5 . 
   this gives the simple criterion to determine whether the experimentally observed peak in inelastic neutron scattering is the true spin resonance  if the peak frequency @xmath21 is less than the indirect gap @xmath3 , then it is the spin resonance and , consequently , the superconducting state has the @xmath7 gap structure .    sometimes it is not always clear experimentally which gaps are connected by the wave vector @xmath2 . even without knowing this exactly , one can draw some conclusions . 
   for example , if one of the gaps is @xmath0 , then there are three cases possible : ( 1 ) @xmath176 and the peak at @xmath21 is the spin resonance , ( 2 ) @xmath177 and the peak is definitely not a spin resonance , and ( 3 ) @xmath29 and the peak is most likely the spin resonance but the definitive conclusion can be drawn only from the calculation of the dynamical spin susceptibility for the particular experimental band structure .    here 
   we combine data on the peak frequency @xmath21 and maximal and minimal gap sizes @xmath0 and @xmath1 available in the literature . 
   results are presented in table  [ tab ] . 
   unfortunately , for many materials either the ins data or gaps estimations are absent . 
   this gives a whole set of tasks for future experiments . 
   here are some conclusions , which we can make :    1 .   in electron - doped bafe@xmath178co@xmath179as@xmath140 system , nafe@xmath178co@xmath179as system , and fese , @xmath116 and , 
   thus the peak in ins is the true spin resonance evidencing sign - changing gap . 
   some hole doped ba@xmath178k@xmath179fe@xmath140as@xmath140 materials satisfy @xmath176 condition , and some satisfy @xmath180 condition . 
   latter comes especially from newer tunneling  @xcite and andreev reflection  @xcite data reveling smaller gap values . 
   the fact that @xmath180 is still consistent with the sign - changing gap , but as we mentioned before , the calculation of the spin response for the particular experimental band structure is required to make a final conclusion . 
   3 .   the only case when @xmath181 is fete@xmath170se@xmath170 . according to our analysis , there should be no sign - changing gap structure . 
   but before concluding this since this is the single case only , gap data coming from @xmath83sr  @xcite should be double checked by independent techniques . 
   interesting to note , that arpes in all cases gives gaps values larger than extracted from other techniques . 
   natural question arise  whether the arpes overestimates or all other methods underestimates superconducting gaps ? 
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summary: we study the spin resonance in superconducting state of iron - based materials within multiband models with two unequal gaps , @xmath0 and @xmath1 , on different fermi surface pockets . 
   we show that due to the indirect nature of the gap entering the spin susceptibility at the nesting wave vector @xmath2 the total gap @xmath3 in the bare susceptibility is determined by the sum of gaps on two different fermi surface sheets connected by @xmath2 . for the fermi surface geometry characteristic to the most of iron pnictides and chalcogenides , the indirect gap is either @xmath4 or @xmath5 . in the @xmath6 state , 
   spin excitations below @xmath3 are absent unless additional scattering mechanisms are assumed . 
   the spin resonance appears in the @xmath7 superconducting state at frequency @xmath8 . 
   comparison with available inelastic neutron scattering data confirms that what is seen is the true spin resonance and not a peak inherent to the @xmath6 state .