Superconductivity and spin-density-waves in multi-band metals
aa r X i v : . [ c ond - m a t . s up r- c on ] J un Superconductivity and spin-density-waves in multi-band metals
A. B. Vorontsov , M. G. Vavilov and A. V. Chubukov Department of Physics, Montana State University, Bozeman, Montana, 59717, USA Department of Physics, University of Wisconsin, Madison, Wisconsin 53706, USA (Dated: September 3, 2018)We present a detailed description of two-band quasi-2D metals with s -wave superconducting (SC)and antiferromagnetic spin-density wave (SDW) correlations. We present a general approach anduse it to investigate the influence of the difference between the shapes and the areas of the two Fermisurfaces on the phase diagram. In particular, we determine the conditions for the co-existence ofSC and SDW orders at different temperatures and dopings. We argue that a conventional s -waveSC order co-exists with SDW order only at very low T and in a very tiny range of parameters. Anextended s -wave superconductivity, for which SC gap changes sign between the two bands, co-existswith antiferromagnetic SDW over a much wider range of parameters and temperatures, but evenfor this SC order the regions of SDW and SC can still be separated by a first order transition. Weshow that the co-existence range becomes larger if SDW order is incommensurate. We apply ourresults to iron-based pnictide materials, in some of which co-existence of SDW and SC orders hasbeen detected. PACS numbers: 74.25.Dw,74.25.Ha
I. INTRODUCTION
Discovery of new magnetically-active superconduc-tors, iron pnictides, based on FeAs or Fe(Se,S,Te) has further invigorated the on-going discussions aboutco-existence of different ordered electronic states inmetals.
In itinerant electrons systems, the interactionsthat lead to formation of superconducting (SC) and mag-netic spin-density-wave (SDW) orders, “pull” and “push”the same particles, and as a result, influence each other.In particular, two orders may support each other and leadto homogeneous local co-existence of SC and SDW states;or one of them may completely suppress the other order,resulting in a state with spatially separated regions of“pure” SDW or SC orders. The transitions between var-ious states may also be either continuous (second order)or abrupt (first order). The outcome of this interplay de-pends critically on a number of parameters: properties ofthe interactions, such as symmetry of SC pairing, theirrelative strengths, and also on properties the Fermi sur-face (FS), such as its shape or the density of electronicstates.In pnictides this parameter space is vast. First, theseare multi-band materials, with two hole pockets in thecenter, (0 , ± π, , ± π ) points of the unfolded Brillouin zone (BZ)(one Fe atom per unit cell). The shapes of quasi two-dimensional electron pockets are quite distinct in dif-ferent materials, ranging from simple circle-like types inLaOFeP , to cross-like electronic FS in LaOFeAs, to el-lipses in BaFe As and even more complex propeller-like structures in (Ba,K)Fe As (for a descending pointof view on this see Ref. 11). Hole pockets are near-circular, but different hole pockets in the same materialusually have different sizes.Second, multiple FSs also create a number of differ-ent possibilities for electron ordering in the form of SDW, charge density wave (CDW) states, and varioussuperconducting states. The SC states include 1) theconventional s ++ -wave state that has s -wave symmetryin the BZ and gaps of the same sign on electron and holeFSs; 2) the extended s + − -state that looks as s -wave froma symmetry point of view but has opposite signs of thegaps on pockets at (0 ,
0) and ( ± π, and 3) severalSC states with the nodes in the SC gap, of both s -waveand d -wave symmetry. As a result of this complex environment, the inter-play of magnetic and superconducting orders also showssome degree of variations. Most of parent compoundsof iron pnictides are magnetically ordered. Upon dop-ing, magnetism eventually yields to superconductivity,but how this transformation occurs varies significantlybetween different Fe-pnictides. A first-order transi-tion between SC and SDW orders has been reportedfor (La,Sm)O − x F x FeAs.
On the other hand, inelectron-doped Ba(Fe − x Co x ) As recent nuclear mag-netic resonance (NMR), specific heat, susceptibil-ity, Hall coefficient, and neutron scattering experi-ments indicate that SDW and SC phases coexist lo-cally over some doping range. In the same 122 fam-ily, experiments on hole-doped Ba − x K x (FeAs) dis-agree with each other and indicate both co-existence and incompatibility of two orders. Isovalentlydoped 122 material BaFe (As − x P x ) shows the regionof coexistence. The goal of the present work is to understand howthe system evolves from an SDW antiferromagnet to an s ++ /s + − -wave superconductor, and how this evolutiondepends on the shape of the FS, the strengths of the in-teractions, and the structure of the SC order. For thiswe derive and solve a set of coupled non-linear BCS-typeequations for SC and SDW order parameters and com-pare values of the free energy for possible phases.We report several results. First, we find that there ismuch more inclination for co-existence between s + − andSDW orders than between the same-sign s ++ -wave stateand SDW. In the latter case, co-existence is only possi-ble at very low T and in a very tiny range of parameters.Second, the co-existence region generally grows with in-creased strength of SDW coupling relative to supercon-ducting interaction. That the co-existence is only pos-sible when SDW transition comes first has been noticedsome time ago, and our results agree with these find-ings. Third, when SDW order is commensurate, the co-existence is only possible when the following two condi-tions are met simultaneously: hole and electron FSs havedifferent k F (cross-section areas) and different shapes,(e.g., hole pockets are circles and electron pockets areellipses). Even then, SDW and SC orders co-exist onlyin a limited range of parameters and temperatures, seeSections IV and V, Figs. 10 and 15 below. When SDWorder is incommensurate, the difference in k F is a suffi-cient condition, but again, the two orders co-exist in alimited range of parameters/temperatures (Fig. 11).We also analyze in some detail the interplay betweenthe co-existence and the presence of the Fermi surface(i.e. gapless excitations) in the SDW state. The “con-ventional” logic states that superconductivity and mag-netism compete for the Fermi surface and co-exist if SDWorder still leaves a modified Fermi surface on which SCorder can form. We find that the situation is more com-plex and the mere presence of absence of a modified Fermisurface is not the key reason for co-existence. We showthat a more important reason is the effective “attraction”between SDW and SC order parameters, when the devel-opment of one order favors a gradual formation of theother order. Specifically, we show that: • near the point where the transitions from the nor-mal (N) state into SC and SDW states cross, SC candevelop either via the co-existence phase or via a di-rect first order transition between pure SDW and SCstates, In this range, the SDW order parameter issmall and SDW state is definitely a metal, Fig. 15; • at low T , the co-existence phase may develop evenwhen SDW state has no Fermi surface (not countingbands which do not participate in SDW). In this situa-tion there is no Fermi surface for a conventional devel-opment of the SC order, but the system still can lowerthe energy by developing both orders, if there is an“attraction” between them. This is the case for s + − superconductivity and comparable strength of SDWand SC couplings, Figs. 7, 10(a); • the SDW phase at low T can be a metal with ratherlarge Fermi surfaces, yet SC order does not develop.This is the case when SC order is s ++ , Fig. 14.The close connection between the co-existence of thetwo states and the symmetry of the SC state has been dis-cussed earlier in the context of single-band heavy-fermionmaterials. This connection gives a possibility to ob-tain information about the pure states (e.g., about thestructure of the SC gap) from experimental investiga-tions of the SC - SDW interplay, as it has been recently Q = (0,π) e h ξ c ( k )=0 ξ f ( k’ )=0 ∆ c ∆ f k -k k’+Q δ kq E F ξ c ( k ) ξ f ( k+q ) -k’+Q k’ = k+q FIG. 1: (Color online) Left: electronic structure of the twoband model considered in this paper, in the unfolded Brillouinzone. The hole FS is in the center, with SC order parameter∆ c , and the electron FSs are at (0 , π ) and ( π, f . The magnetic order with momentum Q = (0 , π ) hybridize hole and electron FSs separated by Q ,but leaves FSs at ( ± π,
0) intact. Right: by doping or pres-sure one may adjust the size and shape of hole and electronbands, and also SDW order parameter can be incommensu-rate, with momentum Q + q . These effects are described byFS detuning parameter, δ kq = [ ξ f ( k + q ) + ξ c ( k )] / suggested. The structure of the paper is as follows. In the nextsection we define the model and derive generic equationsfor the SDW and SC order parameters and an expres-sion for the free energy. Then we simplify these formulasfor the case of a small splitting between hole and elec-tron FSs and utilize them in Secs. III through V. InSec. III we focus on a pure SDW state, with special at-tention given to the interplay between ellipticity of theFS and the incommensuration of the SDW order. In thenext two sections we discuss possible co-existence of SDWand SC states: in Sec. IV we present numerical resultsobtained in a wide range of temperatures and dopings,and in Sec. V we corroborate this with the analyticalconsideration in the vicinity of the crossing point of SCand SDW transitions, and at T = 0. In Sec. VI we modelthe case when the splitting between the two FSs is notsmall. We present our conclusions in Sec. VII. Some ofthe results reported in this work have been presented inshorter publications. II. MODEL AND ANALYTICAL REASONINGA. General formulation
Since the basic properties of the SC and magnetic SDWinteractions and their interplay should not depend onthe number of bands significantly, we consider a basicmodel of one hole and one electron bands. For pnictidesthis means that we neglect the double degeneracy of holeand electron states at the center and the corners of theBrillouin zone, which does not seem to be essential forsuperconducting or magnetic order.
The basic model is illustrated in Fig. 1. Electronicstructure contains two families of fermions, near one holeand one electron FSs of small and near-equal sizes. Suchtwo-band structure yields the experimentally observedstripe ( π,
0) or (0 , π ) magnetic order which in itinerantscenario appears, at least partly, due to nesting betweenone hole and one electron bands, separated by momen-tum ( π,
0) or (0 , π ). Other hole and electron bands do notparticipate in the SDW order. We assume that SC alsoprimarily resides on the same two FSs, at least close tothe boundary of the SDW phase. The SC order param-eter on the other two bands is not zero, but is smaller.Once doping increases and the system moves away fromSDW boundary, we expect that the magnitudes of theSC order parameter on the two electron bands shouldbecome closer to each other.The basic Hamiltonian includes the free fermion part H , and the fermion-fermion interactions in supercon-ducting and magnetic SDW channels, H = H + H ∆ + H m . (2.1)The free fermion part of the Hamiltonian is H = X k ξ c ( k ) c † k α c k α + X k ′ ξ f ( k ′ ) f † k ′ α f k ′ α , (2.2)where creation/annihilation c † , c -operators correspond tofermions near the central hole pocket (0 , f -operators describe fermions near the electron pocket at Q = (0 , π ) and the fermion dispersions near the pocketsare ξ c ( k ) = µ c − k m c , ξ f ( k ) = k x m fx + k y m fy − µ f (2.3)The momenta k are measured from the center of the BZ,and k ′ are deviations from Q . We assume an inversionsymmetry, ξ c,f ( − k ) = ξ c,f ( k ).The pairing interaction consists of many different pairscattering terms, but the most important one is the pairhopping between the hole and electron pockets, H ∆ = X k , p V cfαββ ′ α ′ ( k , p ) (cid:16) c † k α c †− k β f − p β ′ f p α ′ (2.4)+ f † k α f †− k β c − p β ′ c p α ′ (cid:17) , For definiteness, we consider SC interaction only in thesinglet channel, i.e. V cfαββ ′ α ′ ( k , p ) = V sc k , p ( iσ y ) αβ ( iσ y ) † β ′ α ′ . (2.5)The magnetic interaction between fermions is H m = − X p ′ − p = k ′ − k V sdwαββ ′ α ′ ( p ′ p ; k , k ′ ) × (2.6) × (cid:16) f † p ′ α c p β c † k β ′ f k ′ α ′ + f †− p ′ α c − p β c †− k β ′ f − k ′ α ′ (cid:17) , where we symmetrized the expression with respect to par-ticle hopping between (0 , − (0 , π ) and (0 , − (0 , − π )pockets for later convenience. We take the interactionmatrix in a simple form, V sdwαββ ′ α ′ ( p ′ p ; kk ′ ) = V sdw p ′ p ; kk ′ σ αβ · σ † β ′ α ′ , (2.7)with a constant V sdw p ′ p ; kk ′ = V sdw .The evolution of the interaction couplings with energywas considered in Ref. 13. Here we assume that the in-teractions for low-energy excitations can be representedin terms of fermion couplings to order parameter fieldsin the SC and SDW channels. In the spirit of BCS-typeapproach, we introduce the SC order parameters∆ c ( k ) αβ = ( iσ y ) αβ X p V sc k , p ( iσ y ) † β ′ α ′ h f − p β ′ f p α ′ i , (2.8a)∆ f ( k ) αβ = ( iσ y ) αβ X p V sc k , p ( iσ y ) † β ′ α ′ h c − p β ′ c p α ′ i , (2.8b)and the SDW order parameter directed along ˆ m . Weassume that SDW order parameter has a single orderingmomentum Q = Q + q , in which case it is fully specifiedby ( m q ) αβ = ( m q σ ) αβ = m q ( ˆ m σ ) αβ , where( m q ) αβ = − V sdw X p σ αβ · σ † β ′ α ′ h c † p β ′ f p + q α ′ i = − V sdw X p σ αβ · σ † β ′ α ′ h f †− p − q β ′ c − p α ′ i . (2.8c)Since h c † p α f p + q β i ∼ ( m q σ ) αβ , the corresponding elec-tronic magnetization, m ( R ) = X p σ αβ h h c † p α f p + q β i e i QR + h f † p + q α c p β i e − i QR i is m q cos QR for real m q and is m ′ q cos QR − m ′′ q sin QR for a complex m q = m ′ q + i m ′′ q . In principle, SDW orderparameter may contain several components with differ-ent q , which could give rise to domain-like structures of m ( R ). For recent studies in this direction see Ref. 52.We perform the analysis of the co-existence between SCorder and SDW order with a single q . A more generalform of the SDW order should not qualitatively changethe phase diagram for SC and SDW states, however thisassumption requires further verifications.Using the forms of SC and SDW order parameters, wewrite the free and interaction parts in quadratic forms as H = X k h ξ c ( k ) c † k α c k α + ξ c ( − k ) c †− k α c − k α + ξ f ( k + q ) f † k + q α f k + q α + ξ f ( − k − q ) f †− k − q α f − k − q α i , (2.9) H ∆ = X k h ∆ c ( k ) αβ c † k α c †− k β + ∆ † c ( k ) αβ c − k α c k β + ∆ f ( k + q ) αβ f † k + q α f †− k − q β + ∆ † f ( k + q ) αβ f − k − q α f k + q β i , (2.10) H m = X k h m q ,αβ f † k + q α c k β + m q ,αβ c †− k α f − k − q β + m † q ,αβ c † k α f k + q β + m † q ,αβ f †− k − q α c − k β i . (2.11)The Hamiltonian Eq. (2.1) can be represented in the matrix form H = X k αβ Ψ k α b H k Ψ k β , b H k = ξ c ( k ) ∆ c ( k ) iσ yαβ − ∆ ∗ c ( k ) iσ yαβ − ξ c ( − k ) m ∗ q ( ˆ m σ ) † αβ − m ∗ q ( ˆ m σ T ) † αβ m q ( ˆ m σ ) αβ − m q ( ˆ m σ T ) αβ ξ f ( k + q ) ∆ f ( k + q ) iσ yαβ − ∆ ∗ f ( k + q ) iσ yαβ − ξ f ( − k − q ) , (2.12)with Ψ k α = ( c † k α , c − k α , f † k + q α , f − k − q α ), and Ψ being itsconjugated column. The two diagonal blocks of the ma-trix ˆ H k correspond to a purely SC system with ∆ c and∆ f living on two different bands, and two off-diagonalblocks contain SDW field m q that couples fermions be-tween the two bands.To solve this system of equations for the SC and SDWorder parameters, Eqs. (2.8), we define the imaginary-time Green’s function b G ( k , τ ) αβ = −h T τ Ψ( τ ) k α Ψ(0) k β i ≡ ˆ G cc ˆ G cf ˆ G fc ˆ G ff ! , (2.13)which satisfies the Dyson equation, b G − ( k , ε n ) = iε n − b H k , (2.14)where ε n = πT (2 n + 1) are the Matsubara frequencies.The system of equations is closed by the self-consistencyequations for the SC and SDW order parameters in termsof this Green’s function,∆ c ( k ) = X p V sc k , p T X ε n Tr n ( iσ y ) † ˆ τ + ˆ G ff ( p , ε n ) o (2.15)∆ f ( k ) = X p V sc k , p T X ε n Tr n ( iσ y ) † ˆ τ + ˆ G cc ( p , ε n ) o (2.16) m q = − X p V sdw T X ε n Tr n ( ˆ m σ )ˆ τ ˆ G fc ( p , iε n ) o . (2.17)Henceforth we define Pauli matrices in particle-holespace, ˆ τ , , , ˆ τ ± = (ˆ τ ± i ˆ τ ) /
2, and the following ma- trices in spin- and particle-hole space,ˆ∆ = iσ y ) αβ (∆ iσ y ) † αβ ! , σ = σ σ T ! , (2.18)The expressions above are valid for complex ∆( k ) and m q . Below, to simplify formulas, we assume that ∆’s and m q are real, i.e., consider only “sinusoidal”, cos QR , vari-ations of the SDW order parameter. To lighten the nota-tions, we will also drop the momenta arguments ( k , k + q )in ξ c,f , ∆ c,f and the subscript in m q [still implying thisdependence as it appears in Eq. (2.12)].The equations for components of the Green’s functionare obtained from inversion of Eq. (2.14),ˆ G − cc = ˆ G − c − m ˆ G f , ˆ G fc = ˆ M ˆ G f ˆ G cc , (2.19a)ˆ G − ff = ˆ G − f − m ˆ G c , (2.19b)with definition ˆ G − c − ˆ M − ˆ M ˆ G − f ! ≡ iε n − ξ c ˆ τ − ˆ∆ c − ( m σ )ˆ τ − ( m σ )ˆ τ iε n − ξ f ˆ τ − ˆ∆ f ! . (2.20)To obtain Eq. (2.19) we used the fact that the magneticmatrix ˆ M commutes with purely superconducting parts,[ ˆ M , ˆ G c ] = [ ˆ M , ˆ G f ] = 0, and ˆ M ˆ M = m .The diagonal Green’s functions ˆ G c and ˆ G f are thesame as in a pure superconductor, e.g.,ˆ G f ( ε n ) = ˆ G − f ( − ε n ) D f , D f = ε n + ξ f + ∆ f , (2.21)where for inversion we used the relations { ˆ τ , ˆ∆ } = 0 , [( ˆ m σ )ˆ τ , ˆ∆] = 0 , ˆ∆ = ∆ , (2.22)which are also employed to invert 4 × G cc we haveˆ G cc ( ε n ) = 1ˆ G − c ( ε n ) − m ˆ G − f ( − ε n ) /D f , (2.23)and with the above relations in mind it becomesˆ G cc = ˆ G (1) cc + ˆ G ( τ ) cc + ˆ G (∆) cc , (2.24a)where ˆ G (1) cc = − iε n ( D f + m ) D , (2.24b)ˆ G ( τ ) cc = − ξ c D f − ξ f m D ˆ τ , (2.24c)ˆ G (∆) cc = − ˆ∆ c D f − ˆ∆ f m D , (2.24d)The denominator D = ε n ( D f + m ) + ( ξ c D f − ξ f m ) + (∆ c D f − ∆ f m ) D f = ( ε n + ξ c + ∆ c )( ε n + ξ f + ∆ f )+2 m ( ε n − ξ c ξ f − ∆ c ∆ f ) + m = ( ε n + E )( ε n + E − ) (2.25)gives the energies of new excitations in the system, c.f.Ref. 29. We obtained (explicitly showing k and q here): E ± = ξ kq + δ kq + m + (∆ − kq ) + (∆ + kq ) (2.26) ± q m [(∆ + kq ) + δ kq ] + (∆ − kq ∆ + kq + ξ kq δ kq ) , with ξ kq = ξ f ( k + q ) − ξ c ( k )2 , (2.27) δ kq = ξ f ( k + q ) + ξ c ( k )2 , (2.28) and ∆ − kq = ∆ f ( k + q ) − ∆ c ( k )2 (2.29)∆ + kq = ∆ f ( k + q ) + ∆ c ( k )2 . (2.30)The parameter ξ kq describes the dispersion and parame-ter δ kq describes deviations of the electron and hole FSsfrom perfect nesting, as illustrated in Figs. 1 and 2.For the inter-band part of the Green’s function we ob-tain,ˆ G fc ( ε n ) = ˆ M ˆ G f ˆ G cc == ˆ M ˆ G f ( ε n ) ˆ G − c ( − ε n ) D f − m ˆ G − f ( ε n ) D = ˆ M ˆ G − f ( − ε n ) ˆ G − c ( − ε n ) − m D , (2.31)where for self-consistency Eq. (2.17) we need only thepart proportional purely to ˆ M -matrix, Eq. (2.20),ˆ G ( M ) fc = ( m σ )ˆ τ − ε n + ξ f ξ c + ∆ f ∆ c − m D . (2.32)Expressions for ˆ G ff and ˆ G cf are obtained from Eq. (2.24)and Eq. (2.32) by swapping indices, c ↔ f .We substitute the above expressions for ˆ G (∆) cc ( p ),ˆ G (∆) ff ( p + q ) and ˆ G ( M ) fc ( p ) into the self-consistency equa-tions Eqs. (2.15)-(2.17) and arrive at∆ f ( k ) = T X ε n X p ( − V sc k p ) ∆ c ( p )[ ε n + ξ f ( p + q ) + ∆ f ( p + q )] − ∆ f ( p + q ) m D , (2.33)∆ c ( k ) = T X ε n X p ( − V sc k p + q ) ∆ f ( p + q )[ ε n + ξ c ( p ) + ∆ c ( p )] − ∆ c ( p ) m D , (2.34) m = T X ε n X p V sdw ε n + m − ξ f ( p + q ) ξ c ( p ) − ∆ f ( p + q )∆ c ( p ) D m . (2.35)To calculate the relative stability of different states, one also needs to evaluate the free energy. We follow theLuttinger-Ward and De Dominicis-Martin method,and consider the functional F = − Sp nb Σ b G + ln[ − ( iε n − ˆ ξ ) + b Σ] o + Φ[ b G ] , (2.36)which, if minimized with respect to b G , gives self-consistency equations, b Σ[ b G ] = 2 δ Φ[ b G ] /δ b G ; and, if min-imized with respect to b Σ, gives the Dyson equation,Eq. (2.14). Here Sp is the trace over two fermion bands,spin, particle-hole matrix structure, and the sum overMatsubara energies and the integral over momenta, and b Σ is the mean field SC and SDW order parameter matrix, b Σ = ˆ∆ c ˆ M ˆ M ˆ∆ f ! . (2.37)The functional Φ[ b G ] producing the self-consistency equa-tions is a quadratic function of b G . Using the self-consistency equations one can explicitly verify that atweak-coupling it can be written as Φ[ b G ] = Sp { b Σ b G } .To deal with the logarithm in Eq. (2.36) one introducesa continuous variable λ instead of ε n , differentiates thelogarithmic term with respect to λ to obtain the Green’sfunction b G ( λ ) = (cid:16) iλ − ˆ ξ − ˆΣ (cid:17) − , and then integratesback to get the difference between a condensed state andthe normal state for fixed external parameters, such astemperature or field, ∆ F (∆ c,f , m ) = − Sp b Σ b G − ∞ Z ε n dλ [ i b G ( λ ) − i b G N ( λ )] (2.38)where b G N is the Green’s function in the normal statewithout either SC or SDW order, and we used the factthat in the normal state b Σ = 0. Substituting into(2.38) the Green’s functions Eqs. (2.24), (2.32), the self-energy Eq. (2.37), and using the self-consistency equa-tions Eqs. (2.33)-(2.35) to eliminate the high-energy cut-offs in order to regularize the ε n -summation and k -integration, one obtains the most general free energyfunctional for given ∆ c,f and m . B. Limit of small Fermi surface splitting
In principle, equations for full Green’s functionsEq. (2.24), (2.32), the self-consistency equationsEqs. (2.33)-(2.35) and the free energy Eq. (2.38), com-pletely describe the system in a very general case. How-ever, to proceed further with the analytics one can reducethe number of summations which is also desirable froma numerical standpoint.The typical approximation is to linearize the dispersionnear the FS and integrate out the momenta in the direc-tion normal to the FS over ξ kq . In the case, when thetwo FSs are reasonably close to each other (when shifted by (0 , π )), and electron and hole dispersions are similar,the values of FS mismatch δ kq are weakly momentumdependent, and can be taken at positions where ξ kq = 0.The consequence of this approximation, which we dis-cuss in some detail in Appendix A, is that δ kq dependsonly on the angle in k -space, but not on ξ kq and henceone can integrate along a particular direction ˆ k over ξ kq ,keeping δ ˆ kq fixed.Within this approximation the DOS for both FSs arethe same, and the magnitudes of ∆ c and ∆ f are equal(the angular dependence of SC gaps is still determinedby that of the SC interactions). There are, indeed, alsohigher order terms, which we neglected in the last linesof Eq. (A2). These terms make hole and electron DOSdifferent from each other, what in turn makes | ∆ c | and | ∆ f | non-equal, but these terms are small in δ k /µ c,f andonly account for sub-leading terms in the free energy, µ c,f are Fermi energies of electron and hole bands, Eq. (2.3).This approximation comes at certain price. When twoFSs are of very different shapes, approximating them assmall deviations from a single line in k -space everywhereis incorrect. This is shown for example in Fig. 2(d), wherethe two FSs are quite different away from the crossingpoints. However in this case one realizes that if at some k -point the two bands are far apart, the effect of theSDW is very small, and we can approximate those FSparts as participating in SC pairing only, with little or nocompetition from the SDW interaction. This can be seenfrom Eq. (2.19) for the Green’s function. For example,for electrons near the FS of the c -band, ξ c → ξ f is largeand ˆ G − cc ≈ ˆ G − c + O ( m /ξ f ), and the corrections due to m can be neglected when we go along c -FS away from theregion where ξ c ≈ ξ f ≈
0. We will return to this issue insection VI, to show that the results are qualitatively thesame whether we consider large or small splitting of theFSs.For small splitting between hole and electron Fermisurfaces, we perform ξ -integration analytically. For thiswe approximate V sc k , p by an isotropic V sc , i.e., take angle-independent SC gap. The sign of V sc can be arbitrary,and we consider separately the two cases:a) V sc >
0: results in the s + − state, with gaps ofopposite signs for electrons and holes,∆ f = − ∆ c = ∆ or ∆ + = 0 , ∆ − = ∆;b) V sc < s ++ state, with the same gaps on two FSs,∆ f = ∆ c = ∆ or ∆ + = ∆ , ∆ − = 0 . In both cases ∆ + ∆ − = 0 and the denominator of theGreen’s function can be written as, D = ( ε n + E )( ε n + E − ) = ( ξ kq +Σ )( ξ kq +Σ − ) , (2.39)whereΣ ± = ε n +∆ + m − δ kq ± r m ∆ s − δ kq ( ε n + ∆ )(2.40)with s = +1 ( s = −
1) corresponding to s ++ ( s + − )state. Closing the integration contours over ξ kq in the self-consistency equations and in the free energy over theupper half-plane and counting poles at + i Σ ± we obtain − sv sc ∆ = πT X | ε n | < Λ * ∆Σ + + Σ − ε n + ∆ + δ kq − s m Σ + Σ − !+ , | v sc | = ln 1 . T c , (2.41)1 v sdw m = πT X | ε n | < Λ * m Σ + + Σ − ε n + m − δ kq − s ∆ Σ + Σ − !+ , v sdw = ln 1 . T s , (2.42) ∆ F (∆ , m )4 N F = ∆ TT c + m TT s − πT X ε n > (cid:28) (Σ + + Σ − ) − ε n − ∆ ε n − m ε n (cid:29) , (2.43)where angle brackets denote remaining momentum av-eraging over directions on the FS. N F is the densityof states at the FS per spin, and v sc = 2 N F V sc and v sdw = N F V sdw are the dimensionless couplings in theSC and SDW channels. Taken alone, v sc leads to a SCstate with critical temperature T c , which is independentof δ ˆ kq as one can check by setting m = 0 in Eq. (2.41),while v sdw leads to an SDW state with transition tem-perature T s which does depend on δ ˆ kq . We define T s asthe SDW transition temperature at perfect nesting, when δ ˆ kq ≡ − s ∆ m in Eq. (2.41) and − s m ∆ inEq. (2.42), is positive for s + − state resulting in effec-tive “attraction” of the two orders, and negative for s ++ state implying that the formation of one order resiststhe appearance of the other. The actual co-existenceof the two orders, however, is a more subtle effect andneeds to be determined from the exact solution of theseequations and the analysis of the free energy. The dif-ference in excitations energies Eq. (2.26) and Eq. (2.40)between s ++ and s + − states is also consistent with previ-ous studies of d - and p - wave superconductivity in heavy-fermion metals, that concluded the SC states with sym-metries P T Q = − s + − , d ), where P is parity[ P ∆( p ) = ∆( − p )] and T Q is the shift by the nestingvector [ T Q ∆( p ) = ∆( p + Q )], are more likely to formco-existence with SDW than those with P T Q = +1 (e.g. s ++ , p ).We will also analyze the quasiparticle density of states(DOS), which is given by the integrals over ξ kq of the di-agonal components of the Green’s function. For examplefor c -fermions g c ( ε n , ˆ k ) = Z dξ kq π G (1) cc (2.44)= − iε n Σ + + Σ − ε n + m + ∆ + δ kq Σ + Σ − ! , which for pure SDW state reduces to g c ( ε n , ˆ k ) = − X ± iε n ± δ ˆ kq q m − ( iε n ± δ ˆ kq ) , (2.45)and actual DOS is obtained by analytic continuation, N ( ǫ, ˆ k ) N F = − Im g ( iε n → ǫ + i + , ˆ k ) . (2.46) III. PURE SDW STATE
In pnictides, parent materials usually have only mag-netic order below a transition temperature T s . Super-conductivity appears at a finite doping, when the SDWtransition is suppressed. Keeping this in mind, we con-sider first a purely SDW state, and analyze how it ismodified when FSs are deformed by addition or removalof electronic carriers, and whether modified FSs are stillpresent in the SDW phase. To remind, we denote by T s the SDW-N transition temperature at perfect nesting,which effectively gives the scale of SDW interaction inthe system. The true instability temperature, which wedenote explicitly by T s ( δ ˆ kq ), is a function of ellipticity,doping, and incommensurability.We begin by presenting explicit formulas for the exci-tation spectrum, the SDW order parameter, and the freeenergy. For ∆ = 0, Σ ± given by (2.40) isΣ ± = (cid:16) ε n ± iδ ˆ kq (cid:17) + m (3.1)and the excitation spectrum consists of four brancheswith energies ± E ± (∆ = 0), where E ± (∆ = 0) = q ξ kq + m ± δ ˆ kq , (3.2)In (3.1) and (3.2) ξ kq = ξ f ( k + q ) − ξ c ( k )2 ,δ ˆ kq = ξ f ( k + q ) + ξ c ( k )2 ≈ ξ f ( k cF + q )2= v F k ( k cF − k fF + q ) . (3.3)We remind that δ ˆ kq describes the mismatch between theshapes of the electron and hole bands and determinestheir nesting properties in ˆ k -direction.Equation (2.42) for the SDW order parameter m sim-plifies to1 v sdw = 2 πT X <ε n < Λ Re 1 q ( ε n + iδ ˆ kq ) + m , (3.4)and the cut-off Λ can be eliminated in favor of T s ,ln TT s = 2 πT X ε n > Re * q ( ε n + iδ ˆ kq ) + m − | ε n | + , (3.5)where the summation over ε n now extends to infinity.Second-order transition temperature T = T s ( δ ˆ kq ) is ob-tained by setting m = 0:ln TT s = 2 πT X ε n > Re * ε n + iδ ˆ kq − ε n + . (3.6)The free energy, Eq. (2.43), becomes ∆ F ( m )4 N F = m TT s − πT X ε n > (cid:18) Re Dq ( ε n + iδ ˆ kq ) + m E − ε n − m ε n (cid:19) = m . T s − πT X <ε n < Λ (cid:16) Re Dq ( ε n + iδ ˆ kq ) + m E − ε n (cid:17) . (3.7)Below we consider several special cases for δ ˆ kq (seeFig. 2): • two co-axial circles, q = 0: k cF − k fF = ˆ k ( k cF − k fF ): δ ˆ kq = v F | k cF − k fF | ≡ δ . (3.8)For a fixed δ , circular hole and electron FSs survive inthe SDW phase when m < δ (Fig. 2b), but come closerto each other as m increases and merge at m = δ . Atlarger m all excitations are gapped (Fig. 2a). • FS of different shapes, e.g., one circle and one el-lipse, co-centered: q = 0 with k cF − k fF = ˆ k ( k cF − k fF + ∆ k cos 2 φ ): δ ˆ kq = δ + δ cos 2 φ , δ = v F ∆ k (3.9) a) c) b) q ξ c ( k )=0 ξ f ( k )=0 ξ f ( k+q )=0 d) δ k FIG. 2: (Color online) The appearance of gapless excitationsin the presence of SDW order. The dotted lines indicate FSsfor electrons, ξ f = 0, and holes, ξ c = 0. The dashed curve isan “effective” FS, ξ kq = 0. a) when q = 0 and m is large com-pared to FS mismatch, m > δ ˆ k , all excitations are gapped;b) when m is small, gapless excitations are preserved alongthe two modified FSs at ξ k , = ± ( δ k , − m ) / ( | δ ˆ k , | > m inthe shaded region). Such gapless state, however, only existsat high temperatures, while at low T it is pre-emptied by afirst order transition to the normal state ; c) to prevent thefirst order transition, magnetic order is formed at an incom-mensurate vector Q = Q + q . This improves electron-holenesting on some part of the FS, but allows for gapless excita-tions at the opposite side; d) when the two FSs are of differentshapes, the nested parts become gapped due to SDW order,and on the rest of the FSs the excitations are little affectedby SDW order. The density of states for these cases is shownin Fig. 3. In this case, at small enough m , the FS has a form oftwo hole and two electron pockets. As m gets larger, thepockets shrink and eventually disappear. • two circles of different radii, centers shifted by q : δ ˆ kq = δ ˆ k = δ + v F q = δ + v F q cos( φ − φ ) (3.10)where φ and φ are the directions of v F and ˆ q . In thiscase, when m increases, gapless excitations survive alonga pocket in one region of the k , while excitations with − k become gapped ( Fig. 2c). At large enough m , mod-ified FS disappears and excitations with all momenta k become gapped. This scenario refers to the case whenthe magnetic ordering occurs at a vector, different fromthe nesting vector Q , producing incommensurate SDWstate. It may occur because the electronic system has anoption to choose q = 0 if it minimizes the energy, or be-cause the SDW interaction is peaked at a fixed Q = Q for some reason. Note that Eq. (3.5) for SDW orderis a magnetic analog of Fulde-Ferrell-Larkin-Ovchinnikov(FFLO) state in a paramagnetically limited supercon-ductor. An incommensurate SDW state at finite dop-ings has been studied in application to chromium and itsalloys and, more recently, to pnictide materials. ε / 2π T s N ( ε ) / N F δ δ = 0.13 δ FIG. 3: (Color online) The FS averaged DOS for pure SDWstate and FS mismatch δ ˆ kq = δ + δ cos 2 φ . We use di-mensionless variables denoted by bars, ¯ δ = δ / πT s = 0 . m = m ( T = 0) / πT s = 0 . δ = δ / πT s . For ¯ δ = 0, N ( ǫ ) vanishes below¯ ǫ = ¯ m − ¯ δ and has two sharp BCS peaks at ¯ ǫ = ¯ m ± ¯ δ . Atfinite ¯ δ , each of the two peaks splits into a “band” boundedby two weaker non-analyticities separated by 2¯ δ . The gap inthe DOS behaves as ¯ m − ¯ δ − ¯ δ and closes when ¯ δ + ¯ δ ≥ ¯ m ,metallic states forms. The DOS remains the same if we re-place the ellipticity parameter ¯ δ by the incommensurabilityparameter ¯ q . In general, all three terms are present, and δ ˆ kq = δ + δ cos 2 φ + v F q cos( φ − φ ) . (3.11)In the figures we use dimensionless parameters, that aredenoted by a bar. For isotropic and anisotropic FS dis-tortions, ¯ δ , = δ , πT s , ¯ q = v F q πT s , (3.12)and similarly for other energy variables,¯ m = m πT s , ¯ ǫ = ǫ πT s , ¯∆ = ∆2 πT c . (3.13)We use different notations for prefactors of cos( φ − φ )and cos 2 φ terms to emphasize that they have differentorigin: δ is an “input” parameter defined by the ellipticform of the electron FS due to the electronic band struc-ture, while q is adjustable parameter that minimizes thefree energy of the system. If the minimum of the free en-ergy corresponds to q = 0, SDW order is commensurate,otherwise SDW order is incommensurate.In Fig. 3 we show the DOS N ( ǫ ) for the fixed ¯ δ =0 .
13 and ¯ m = 0 .
28, and different δ . For δ = 0, N ( ǫ )vanishes below ǫ = m − δ and has two BCS-like peaksat ǫ = m ± δ . At finite δ , each of the two peaks spreadsinto a region of width 2 δ bounded by two weaker non-analyticities. The gap in the DOS behaves as m − δ − δ and closes when δ + δ become larger than m . TheDOS and all other results remain the same if we replacethe ellipticity parameter ¯ δ by the incommensurability parameter ¯ q because the angular integral in Eq. (2.45) orEq. (3.6) over momentum directions on the FS coincidesfor cos( φ − φ ) and cos 2 φ terms in δ ˆ k , q , if consideredseparately. The DOS and T s ( δ ˆ kq ) change, however, whenboth δ and q are present simultaneously.Below we discuss the phase diagram for the pure SDWstate to the extend that we will need to analyze potentialco-existence between SDW and SC states, which is thesubject of this paper.It is instructive to consider separately the case whenSDW order is set to remain commensurate for all δ , (i.e., q = 0), and the case when the system can choose q . In our model, the first case is artificial and just setsthe stage to study the actual situation when the value of q is obtained by minimizing the free energy. However, acommensurate magnetic order may be stabilized in theSDW state, if the interaction V sdw is by itself sharplypeaked at the commensurate momentum Q .The results for the case q ≡ T s ( δ , δ ) for several values of δ . All curvesshow that the transition is second-order at high T andfirst-order at small T . The first-order transition lines(dotted lines in Fig. 4(a)) were obtained by solving nu-merically the nonlinear equation for m , substituting theresult into the free energy (3.7) and finding a locationwhere ∆ F ( m ) = 0.To verify that the transition becomes first order at low T , we expanded the free energy in powers of m as ∆ F ( m ) = α m m + Bm + . . . , (3.14)and checked the sign of the B term. The coefficients α m and B are determined from Eq. (3.7), α m = ln TT s − πT X ε n > Re (cid:28) ε n + iδ ˆ k − ε n (cid:29)! ,B = πT X ε n > Re (cid:28) ε n + iδ ˆ k ) (cid:29) , (3.15)where δ ˆ k = δ + δ cos 2 φ . Solid lines in Fig. 4(a) corre-spond to α m = 0. The N-SDW transition is second orderand occurs when α m = 0 if B >
0, but becomes firstorder and occurs before α m becomes negative if B < all fixed δ , for which SDW-N transition is possible, B changes sign along the line α m = 0 and becomes negative at small T . For δ = 0,this occurs at T ∗ s = 0 . T s and ¯ δ ∗ = 0 . δ reduces the transition temper-ature at δ = 0, and at the same time makes the curveflatter allowing for a larger SDW region along δ . Thetransition line becomes completely flat at a critical value δ c = 0 . πT s ) (see below) when T s ( δ , δ c ) = +0.At this point, it spans the interval δ ∈ [0 , δ c ]. The exis-tence of the SDW ordered state at δ = δ c over a finiterange of δ despite that the transition temperature is +00 δ T / T s δ SDW SDWN δ = 0 N0.14 a) b) T s* δ = 00.20.28 FIG. 4: (Color online) The SDW-N transition for commen-surate SDW order. The parameters ¯ δ and ¯ δ describe thedifference between the area of hole and electron pockets andthe ellipticity of the electron pocket, respectively. Here and inall subsequent figures dotted lines mark first-order transitions,solid and dashed lines mark second-order transitions. Panel(a): variation of the transition temperature with δ for fixed δ . The transition is second order at small δ but becomesfirst order at larger δ . At δ = 0, the transition becomes firstorder at T ∗ s ≈ . T s . Panel (b): variation of the transitiontemperature with δ for fixed δ . T s ( δ , δ ) monotonicallydecreases with increasing δ and vanishes at the same value¯ δ ≈ . δ . is a highly non-trivial effect which deserves a separatediscussion. In Fig. 4(b) we show the transition temperature atfixed δ , as a function of the ellipticity parameter δ . Asexpected, T s ( δ ) monotonically decreases with increasingellipticity of the electron band. The SDW order existsup to δ c , at which T s ( δ c ) = +0. The value of δ c isindependent of δ and can be obtained by taking thelimit T → m = 0 and re-writing thisequation as1 v sdw = Re Λ Z dε (cid:28) ε + iδ ˆ k (cid:29) = Re ln 2Λ iδ + p δ − δ . (3.16)The interaction can be eliminated in favor of zero-temperature gap m at δ = δ = 01 v sdw = Λ Z dε p ε + m = ln 2Λ m , (3.17)where from Eq. (3.5) we obtain, at δ = δ = 0: m = 2 πT s e γ E = 0 . × (2 πT s ) (3.18)and γ E ≈ . δ and δ , the value of m at T = 0 remains equal to m as longas δ + δ < m . The combination of Eqs. (3.16) and(3.17) gives δ c = m = 0 . × (2 πT s ), provided that δ T / T s δ = 0SDW0.20 solid: q = 0dashed: q ≠ FIG. 5: (Color online) Same as in Fig. 4, but when the sys-tem is allowed to choose between commensurate and incom-mensurate SDW orders. Solid lines are second-order tran-sition lines into a state with a commensurate SDW order,dashed lines are second order transition lines into an SDWstate with an incommensurate SDW order (the magnetic ana-log of FFLO state). For all δ >
0, incommensuration occursbefore the commensurate transition becomes first order (theonsets of incommensuration and first-order transition coin-cide for δ = 0). Observe that incommensuration develops atprogressively smaller δ as δ increases and T s ( δ ) decreases,but the range of δ over which incommensurate SDW orderexists actually increases with increasing δ . δ < δ c (there exists another solution δ = 2 δ m − m at δ < δ and δ > m /
2, but it corresponds to anunstable state). A similar result has been obtained inthe studies of FFLO transition.
The form of T s ( δ ) near δ c can be obtained analyti-cally by re-writing the condition α m = 0 in (3.15) asln TT s + 2 πT X ε n > (cid:28) δ cos φε n ( ε n + δ cos φ ) (cid:29) = 0 , (3.19)integrating explicitly over φ , re-expressing 1 / p ε n + δ as (2 /π ) R ∞ dx/ ( x + ε n + δ ), and performing the sum-mation over ε n before the integration over x . Carryingout this procedure, we obtain T s ( δ ) ≈ δ c | ln (1 − δ /δ c ) | (3.20)We see that T s very rapidly increases at deviations from δ c . For δ = 0 . δ c (¯ δ = 0 . T s ( δ ) ≈ . T s , in good agreement with Fig. 4(a).Also, one can easily show that at T = 0 fermionicexcitations in the SDW state are all gapped when m >δ + δ . When δ + δ > m , the SDW state possess Fermisurfaces and gapless fermionic excitations.We next consider the case when the system is freeto choose between commensurate and incommensurateSDW orders and may develop incommensurate order tolower the free energy. In Fig. 5 we show the transitiontemperature T s ( δ ) for fixed δ . We found that, for all δ , first order transition is overshadowed by a transition1into an incommensurate SDW state. For δ = 0, incom-mensuration develops exactly where B changes sign, andthe transition into incommensurate SDW state remainssecond order for all δ . For δ >
0, incommensurationdevelops before B changes sign, and the transition intoincommensurate SDW state remains second order oversome range of δ but eventually becomes first order atlarge δ and low T . The full phase diagram also containsa transition line (not shown in Fig. 5) separating alreadydeveloped commensurate and incommensurate SDW or-ders.To analyze the interplay between the appearance ofincommensurate SDW order and the sign change of B ,we again expand the free energy in powers of m butnow allow incommensuration parameter δ to be non-zero, i.e., replace in the coefficients in Eq. (3.14), δ ˆ k = δ + δ cos(2 φ ) with δ ˆ k , q = δ ˆ k + q cos( φ − φ ). In general,for small q , α m ( δ ˆ k , q ) = α ( δ ˆ k )+ α ( δ ˆ k ) q + α ( δ ˆ k ) q + O ( q ) , (3.21)with α ( δ ˆ k ) given by (3.15). When α and B are posi-tive, the N-SDW transition is second order, and is intoa commensurate SDW state when α > α changes sign andbecomes negative. If B changes sign while α is still pos-itive, the SDW-N transition becomes first order beforeincommensuration develops.To understand the phase diagram, it is sufficient toconsider small δ . Expanding all coefficients in powers of δ we obtain α ( δ ˆ k ) = α , + α , δ + O ( δ ) , (3.22a) α ( δ ˆ k ) = α , + α , cos 2 φ δ + O ( δ ) , (3.22b) α ( δ ˆ k ) = α , + O ( δ ) ,B = α , + O ( δ ) , (3.22c)where α , = ln TT s + 2 πT X ε n > δ ε n ( ε n + δ ) ! , (3.23a) α , = α , = πT X ε n > ε n ε n − δ ( ε n + δ ) , (3.23b) α , = 32 2 πT X ε n > ε n ( δ − ε n ) δ ( ε n + δ ) , (3.23c) α = −
316 2 πT X ε n> ε n ε n − δ ε n + 5 δ ( ε n + δ ) . (3.23d)We see from Eqs. (3.22) that for δ = 0, B and α ( δ ˆ k )change sign simultaneously, at the point where α , = α , = 0. However, when δ = 0, α ( δ ˆ k ) changes sign be-fore B becomes negative because α ( δ ˆ k ) contains a termlinear in δ , whose prefactor can be made negative by ad-justing φ . This explains why in Fig. 5 incommensuration begins while B is still positive. Also, we verified that nearthe onset points for incommensuration, α ( δ ˆ k ) >
0, i.e.,in this range the transition into incommensurate SDW issecond order. At larger δ , the incommensurate transi-tion eventually becomes first order. IV. SDW+SC STATE, NUMERICAL ANALYSIS
In the next two sections we look at potential co-existence of SDW and the s + − or s ++ states, when thesystem is doped and the SDW state is suppressed. Thesuperconducting T c is doping independent, so at somedoping SDW and SC transition temperatures cross. Nearthis point, the two orders either support or suppress eachother and either co-exist or are separated by a first-ordertransition.In this section we present numerical results in the ex-tended range of temperatures and dopings, in the nextsection we corroborate them with analytical considera-tion in the vicinity of the crossing point, when both orderparameters are small, and at T = 0. A. Coexistence with s ± state We look first at the s + − state. In this case the sys-tem of coupled self-consistency equations for ∆ and m is obtained from Eqs. (2.41)-(2.43) by taking Σ ± =( E n ± iδ ˆ kq ) + m and E n = p ε n + ∆ ,ln TT c = 2 πT X ε n > Re * ( E n + iδ ˆ kq ) /E n q ( E n + iδ ˆ kq ) + m − | ε n | + , (4.1a) δ T / T c δ SDW SC + − N SDW N δ = 0 δ = 0 a) b) SC + − FIG. 6: (Color online) The phase diagram of SDW and SC s + − states when only a commensurate SDW order is allowed( q = 0). We set T s /T c = 3 and varied either the relativeradius of circular hole and electron pockets (a) or the form ofone of the pockets (b). The pure SC s + − and SDW states areseparated by first order transition, and there is no co-existenceregion. T / T c δ δ (a )(a )(a ) (b ) δ = 0.1 δ = 0.2 δ = 0.1 δ = 0.05 δ = 0.25 δ = 0.17 SDW SC + − N SC+SDW (b )(b ) FIG. 7: (Color online) Appearance of co-existence when both δ and δ are finite. We set T s /T c = 2 and q = 0. Panels(a1)-(a3) – phase diagrams in variables T, δ at fixed δ , pan-els (b1)-(b3) – phase diagrams in variables T, δ at fixed δ .Panels (a1), (b1) – there appears a region near T = 0, whereSDW and SC s + − orders co-exist. Panels (a2),(b2) – the co-existence region broadens and reaches T = T c . Panels (a3),(b3) – the transition at low T becomes first order betweenpure SDW and SC states, but narrow co-existence region isstill present near T c . A complimented zero-temperature phasediagram is presented in Fig.10. ln TT s = 2 πT X ε n > Re * q ( E n + iδ ˆ kq ) + m − | ε n | + . (4.1b)We remind that T c is the transition temperature for thepure SC state, and T s is the transition temperature forthe pure SDW state at δ ˆ kq = 0.These equations are solved numerically to find all pos-sible states (∆ , m ) and their energies evaluated usingEq. (2.43). The main results for this part are presentedin Figs. 6-12.
1. Commensurate SDW state
Figure 6 shows the results for the case when SDW or-der is set to be commensurate (i.e., q = 0) and the FSsare either co-axial circles (panel a), or of different shapeswith equal k F (panel b). In the first case, δ = 0 and δ = 0, in the second case δ = 0 and δ = 0. We seethat in both cases pure SDW and SC states are separatedby a first-order transition. We verified that in both casesfermionic excitations in the SDW state are fully gapped δ T / T c δ -0.200.20 0.5 1 1.5 2 T / T c T s / T c = 5 SC + − SC + SDW N ∆ F m ∆ T / T c = 0.2 m ∆ δ =0.17 ∆ F SDW δ =0.2 (a) (b)(c) FIG. 8: (Color online) (a) Same as in Fig. 7(b) but for T s /T c = 5. (b,c) SDW and SC gaps, in units ¯ m and¯∆ = ∆ / πT c , and the free energy as functions of ¯ δ alongthe line T /T c = 0 . δ = 0 .
17 (c). at T = 0 and thus there are no Fermi surfaces. From thisperspective, the results presented in Figure 6 are consis-tent with the idea that co-existence requires the presenceof the Fermi surfaces in the SDW state. However, we willsee next that the situation in the cases when both δ and δ are non-zero is more complex.This is demonstrated in Fig. 7 which shows the phasediagram for T s /T c = 2 as a function of δ for a set offixed δ (panels (a1)-(a3)), and as a function of δ fora set of fixed δ (panels (b1)-(b3)). For all cases, pureSDW state is fully gapped at T = 0, so naively one shouldnot expect a co-existence state. However, as is evidentfrom the figure, the phase diagram does involve the co-existence phase, which can be either at low T (including T = 0), or near T = T c , depending on the parameters. Inparticular, as δ in panels (a) or δ in panels (b) increase,the co-existence state first appears at low T , while athigher T the pure SDW and SC states are still separatedby first-order transition (panels (a1) and (b1)). Thenthe co-existence region grows, and extends up to T = T c (panels (a2) and (b2)). At even larger δ or δ , SDWand SC states are separated by the first-order transitionat low T , but the co-existence phase still survives near T c .In Fig. 8 we show the phase diagram for ¯ δ = 0 . T s /T c = 5 together with the plots of SDW and SCorder parameters and the free energy. We see the samebehavior as in Fig. 7 (a2) – there is a co-existence phasefor all T up to T c . In Fig. 9 we show the changes in thequasiparticle DOS at low T = 0 . T c as the system evolvesfrom the SDW state to the SC state via the co-existenceregion.Finally, in Fig. 10, we show the zero-temperature phasediagram in variables δ and δ for T s /T c = 2 and T s /T c =5, together with the locus of points where T s ( δ , δ ) = T c .The phase diagram was obtained by numerically solving3 ε / 2π T s DO S DO S m +δ ± δ δ = 0.2 δ δ ∆ m −δ ± δ ∆ max SDWSDW+SC
FIG. 9: (Color online) FS averaged DOS as a function of en-ergy at
T /T c = 0 . δ (i.e., different dop-ings). We set ¯ δ = 0 . T s /T c = 5. The characteristic val-ues of the SDW order parameter for this range of parametersis ¯ m ≈ .
28 (zero- T limit). Upper panel: DOS for small ¯ δ ,when the system remains in the pure SDW state. This figureis similar to Fig. 3. The DOS vanishes below ǫ = m − δ − δ and has ln − non-analytic behavior at ǫ = m ± δ + δ , andsudden drops at ǫ = m ± δ − δ . Lower panel: DOS for larger δ , when SDW and SC orders co-exist. Sharp peaks at small ǫ are due to opening of the superconducting gap ∆. OnceSDW order disappears at ¯ δ ≈ .
21, the DOS acquires BCSform with the maximal gap ∆ max / πT c ≈ . Eqs. (4.1) and evaluating the free energy at
T /T c = 0 . T = 0 first moves tothe left, shrinks, and disappears at ¯ δ ≈ .
24. Similarly,in panels (b), the co-existence range shrinks to a pointat ¯ δ ≈ .
16, and at larger δ the transition betweenSDW and SC phases at T = 0 becomes first order. Inthe next section we present the results of complimentaryanalytical studies of the phase diagram at T = 0 andnear T c . These results are in full agreement with thenumerical analysis in this section.Observe that for T s /T c = 5 the left boundary ofthe co-existence region is located very close to the line δ + δ = m (dashed line in Fig. 10, ¯ δ + ¯ δ = 0 . T s /T c , the co-existence regionat T = 0 virtually coincides with the region where SDWstate has a Fermi surface. However, for smaller T s /T c = 2(Fig. 7; left panel in Fig. 10), co-existence clearly occursalready in the parameter range where SDW excitationsare all gapped. The co-existence for T s /T c = 2 is there-fore not the result of the “competition for the Fermi sur-face”, but rather the consequence of the fact that thesystem can gain in energy by reducing the SDW order δ δ SDW
SDW + SC + − SC + − SC + − T s / T c = 2 T s / T c = 5 SDWSC + − SC + − S D W + S C + − FIG. 10: (Color online) The zero temperature phase diagram:SDW, SC states and their co-existence region for various δ and δ and two different T s /T c = 2 (left) and T s /T c = 5(right). We only allow the system to develop a commensurateSDW order ( q = 0). The dashed line denotes first appearanceof gapless excitations in the pure SDW state [ m = δ + δ ,Eq. (2.45)]. The co-existence region at T = 0 (shaded area)extends down to δ = 0, but is not present at small δ . Thewidth of the co-existence region increases with the relativestrength of SDW interaction, as determined by ratio T s /T c .The squares mark the location of the crossing point between T s ( δ , δ ) and T c . Open squares indicate that the SDW-SCtransition near T c is first order, while filled squares signal thepresence of the co-existence phase near T c . Note that theregions where co-existence phase is present at T = 0 and near T c are not identical. parameter (still keeping all fermionic excitations gapped)and creating a non-zero SC order parameter. The gainof energy in this situation can best be interpreted as theconsequence of the attraction between the two orders.
2. Commensurate vs. incommensurate SDW state
One of the results of our consideration so far is that,if we keep an SDW order commensurate, a finite regionof SDW + SC phase appears only when both δ and δ are non-zero. If we allow the system to choose theordering momentum of the SDW state, the co-existenceregion widens and appears even if we set δ = 0. Weillustrate this in Fig. 11, where we plot the phase diagramat δ = 0 for two different values of T s /T c . In agreementwith Fig. 5, at T < T ∗ s , the system chooses an SDWstate with a non-zero q . We see that, in this situation,there appears a region where SC state co-exist with anincommensurate SDW state. The co-existence regionwidens up when the ratio T s /T c increases, and for largeenough T s /T c extends down to T = 0. In Fig. 12 we set δ to be non-zero (¯ δ = 0 .
2) and allowed the system tochoose q which minimizes the free energy. The resultsare quite similar to the case when q = 0. We see that the4 δ T / T c δ T c a) T s / T c = 3 N SDWqSDW SC+SDWq SC + − N SDWq
SDWT s* SC + SDWq SC + − b) T s / T c = 5 FIG. 11: (Color online) The phase diagram for δ = 0, whenthe system can choose the value of q . Incommensurate SDWorder appears below T ∗ s = 0 . T s and leaves some parts of theFS ungapped, allowing for co-existing SC order. (a) T s /T c =3. The SDW+SC phase appears only in a small region near T c . At low T the system still undergoes a first order transitionbetween commensurate SDW and SC states. (b) For larger T s /T c = 5 (weaker SC interaction) the co-existence regionwidens and extends down to T = 0. The q = 0 SDW statehas the lowest energy at T = 0 for ¯ δ . . δ T / T c q cos φ + 0.2 cos2 φ SC + − N SC + SDW
SDW
FIG. 12: (color online) Same as in Fig. 8(a) but now we allowthe system to choose the value of q . The phase diagram fromFig. 8 is shown by dashed lines. A finite q emerges belowa particular T and moves the co-existence region to larger δ , together with the SDW-N transition. This broadens theco-existence region, and slightly changes the shape of T c ( δ )inside the magnetic dome. SDW and SC orders do co-exists in the parameter rangewhich extends from the crossing point down to T = 0.The width of the co-existence region widens a bit whenwe allow the system to choose q , but qualitatively, thebehavior in Figs. 8 and 12 is the same. Note, in our two-band model, the ellipticity of of the electron FS breaksthe rotational symmetry and favors the direction of q along the ellipse’s major axis, see Eq. (3.22b).To summarize, SDW and SC + − phases do co-exist in arange of finite dopings, but the width of the co-existenceregion depends on the amount of ellipticity of the electronband and the ratio of T s /T c . At larger T s /T c the widthof the co-existence region increases for fixed δ , and there is optimal δ at which the width is the largest. The factthat the system can lower the energy by making SDWorder incommensurate also acts in favor of co-existence,but qualitatively the picture remains the same as in thecase when q is set to be zero. B. Minimal co-existence with s ++ state We next look at the SC state with gaps of the samesigns on two FSs. Such states seem unlikely for pnictides,because they require a negative sign of the interband pairhopping term. Still, it would be interesting to investi-gate consequences of attractive SC interaction betweenelectron and hole bands.The expressions for Σ ± in this case is slightly morecomplicated and less illuminating than those for s + − state, although quite similar, and so are the self-consistency equations, which we do not write here, butwhich are obtained from Eqs. (2.41)-(2.43) in a way com-pletely analogous to Eqs. (4.1). We first present the re-sults for δ = 0, Fig. 13. We found that co-existenceregion does not appear even if we allow SDW order tobecome incommensurate. There are commensurate andincommensurate SDW phases on the phase diagram, andSC ++ phase, but the transition between SC and SDWphases remains first order. In other words, the appear-ance of gapless excitations in the SDW phase due to in-commensuration at large δ does not seem to favor amixed superconducting and magnetic state, in sharp con-trast to the case of s + − SC, where incommensurationinduces co-existence, see Fig. 11(b).For a non-zero δ , there might appear a tiny region ofco-existence at low temperatures. We illustrate this inFig. 14, where in panel (a) we plot the phase diagram for¯ δ = 0 . q = 0. (When the system is allowed tochoose q , the results change minimally, in a way similar δ T / T c SDWq-N IISDW-SDWq ISDW-SC I SC-N II T s / T c = 5 SDW SC ++ SDWq transition order
FIG. 13: (Color online) The phase diagram for a conventional s ++ SC order parameter, at δ = 0, and varying δ . We allowthe system to choose q . The SDW+SC state does not appear,even when SDW order becomes incommensurate. δ T / T c δ δ T s / T c = 5 SC ++ SC + SDW N SDW δ =0.2 (a) S D W + S C SC ++ SDWSC ++ (b) T → FIG. 14: (Color online) (a) Same as in Fig. 13 but for fixed¯ δ = 0 . q = 0. For most of the phase diagram, thebehavior is the same as for δ = 0, but there appears a verytiny range of SC + SDW phase at the lowest T . The transi-tion to purely SC state is first order at all T . (b) The zero-temperature phase diagram. SDW and SC s ++ states areseparated by first order transition virtually everywhere ex-cept a small region at finite δ and δ , where SDW+SC stateemerges. to Fig. 12). In panel (b) of this figure we show where theregion of SDW+SC ++ exists for different δ and δ . Wesee that the range of co-existence is very narrow, and wealso found that the difference in free energies between apure SDW state and SDW+SC state is very small due tosmall value of the SC order parameter.Observe also that the co-existence region in Fig. 14 isto the left of the line δ + δ = m at which a Fermi sur-face appears in the SDW state (a dashed line in Fig. 14b).In other words, s ++ superconductivity does not emergeeven when there is a Fermi surface in the SDW state.This shows once again that the presence or absence of theFermi surface in the SDW state is not the primary reasonfor the presence or absence of the SDW+SC phase. Thetrue reason is energetic – the SDW+SC state can eitherlower or increase the energy compared to pure state de-pending on whether SDW and SC orders attract or repeleach other. The absence of the co-existence phase even inthe range where SDW state has a Fermi surface is a clearindication that there is the “repulsion” between SDWand SC orders, if the SC order is s ++ , Eqs. (2.41)-(2.42).The same conclusion was recently reached by Fernandes et al . V. SDW + SC, ANALYTICAL RESULTS
We corroborate the numerical analysis in the preced-ing Section with the analytical analysis. We first presentthe results of Ginzburg-Landau (GL) description near thepoint where second-order SDW-N and SC-N transitionsmeet, then consider the phase diagram at T = 0, andfinally combine the two sets of results and compare ana-lytical phase diagram with Fig. 7. A. Ginzburg-Landau analysis
We begin with the GL analysis near the point where T s ( δ , δ ) = T c . Near this point, both the SDW and SCorder parameters are small and we can expand the freeenergy, Eq. (2.43), to the fourth order in m and ∆ andcompare different phases. For simplicity, in this sectionwe assume that the SDW order is commensurate. Anextension to a finite q complicates the formulas but doesnot change the outcome.The expansion of the free energy, Eq. (2.43) in powersof m and ∆ yields F = α ∆ ∆ + α m m + A ∆ + Bm + 2 C ∆ m . (5.1)where F = ∆ F ( m, ∆) / (4 N F ). Coefficients α ∆ , α m , A ,and B in Eq. (5.1) are identical for both s + − and s ++ SC states: α ∆ = ln TT c , (5.2) α m = ln TT s + 2 πT X ε n > * δ k ε n ( ε n + δ k ) +! , (5.3)and A = πT X ε n > ε n , (5.4) B = πT X ε n > * ε n ε n − δ k ( ε n + δ k ) + . (5.5)The difference between s + − and s ++ SC orders appearsonly in the coefficient C . For s + − state we have C (+ − ) = πT X ε n > * ε n − δ k ε n ( ε n + δ k ) + , (5.6)while for s ++ C (++) = πT X ε n > * ε n + δ k ε n ( ε n + δ k ) + . (5.7)Note, that, although both C -coefficients are positive,this does not preclude co-existence in Eq. (5.1), and wefind below that the sign of parameter χ = AB − C ismore important for co-existence. We will demonstratethat since C (++) > C (+ − ) , χ is positive for a broaderrange of parameters in s + − state than that in s ++ state.In fact, χ remains always negative in s ++ state. Below wewill use the notion that χ > m = 0 and ∂ F /∂ ∆ = 0,has the free energy and SC order parameter F ∆ = − α A , ∆ = − α ∆ A ; (5.8)62) a pure SDW state, defined by ∆ = 0 and ∂ F /∂m = 0,has the free energy and SDW order parameter F m = − α m B , m = − α m B . (5.9)In addition, the free energy may also have either asaddle point or a global minimum when both ∆ = 0 and m = 0. To see this, we write the free energy Eq. (5.1) inequivalent form, F = α m (cid:18) m + CB ∆ (cid:19) + B (cid:18) m + CB ∆ (cid:19) + (cid:18) α ∆ − CB α m (cid:19) ∆ + (cid:18) A − C B (cid:19) ∆ , which is now a sum of two independent parts for ∆ and M ≡ m + ( C/B )∆ . For an extremum state, given by ∂ ∆ F = ∂ m F = 0, the stationary values of order param-eters,∆ = − α ∆ B − α m C AB − C ) , M = m + CB ∆ = − α m B , (5.10)determine the free energy, F m &∆ = − BM − AB − C B ∆ . (5.11)When both coefficients in Eq. (5.11) are positive, B > , χ = AB − C > , (5.12)the mixed state, Eq. (5.10), corresponds to the minimumof the free energy, which is smaller than the minima forpure SC or SDW states: F m &∆ = F m − B ( α ∆ B − α m C ) AB − C = F ∆ − A ( α m A − α ∆ C ) AB − C . (5.13)Consequently, in the phase diagram, the pure SDWand SC states are separated by a SDW+SC phase, andthe transitions into this intermediate state are second-order. However, if B > χ <
0, the mixed phase,Eq. (5.10), corresponds to the saddle point of the freeenergy and is not thermodynamically stable phase. Inthis case, pure SDW and SC phases are separated by afirst-order transition line. When
B <
0, one needs to ex-pand further in m to determine the phase diagram. Wewill not discuss the case B < δ ˆ k = δ + δ cos 2 φ which we considered in the previous Sections. We remindthat δ = 0 corresponds to co-circular FSs with differentchemical potentials, while δ = 0 corresponds to FS ge-ometry in which k cF = k fF , but the electron pocket iselliptical.At perfect nesting δ = δ = 0, and the system devel-ops an SDW order at T s > T c . Deviations from perfect nesting lead to two effects. First, as we already said,the magnitude of α m is reduced because SDW instabilityis suppressed when nesting becomes non-perfect. Super-conducting α ∆ is not affected by δ ˆ k , and eventually winsover SDW. Second, coefficients B and C evolve with δ ˆ k and, as a result, the sign of χ = AB − C depends onvalues of δ and δ .The GL expansion is applicable only in the vicinityof points at which the temperatures of the SDW-N andSC-N transitions coincide T s ( δ ˆ k ) = T c . This conditiontogether with Eq. (3.6) establish the relation between δ and δ at which one needs to compute the parameters B and C . s + − superconductivity To get an insight on how χ evolves with δ ˆ kq , wefirst assume that T s /T c is only slightly larger than one( T s /T c = 1 + δt ), in which case T s ( δ ˆ k ) = T c at small δ and δ , and we can expand A , B , and C in powers of δ and δ . Specifically, we have from Eq. (3.6) δt = 7 ζ (3)4 π T s (cid:18) δ + 12 δ (cid:19) = 0 . m (cid:18) δ + 12 δ (cid:19) (5.14)where ζ (3) is a Riemann Zeta function. Collecting termsup to the fourth order in the expansion, we obtain χ = 132 π T c (cid:16) s h δ k i − s h δ k i (cid:17) , (5.15)where s = 5 X n ≥ n + 1) X n ≥ n + 1) ,s = 9 X n ≥ n + 1) . (5.16)The sums are expressed in terms of the Riemann-Zetafunction ζ (3) , ζ (5), and ζ (7) and give s ≈ .
261 and s ≈ . δ ˆ k = δ + δ cos 2 φ and averaging overmomentum direction φ on the FSs, we obtain χ ≈ π T c (cid:0) − . δ + 6 . δ δ − . δ (cid:1) . (5.17)We see that for δ = δ = 0, χ = 0, i.e., for a perfectnesting the system cannot distinguish between first ordertransition and SDW+SC phase. This result, first noticedin Ref. 29, implies that the phase diagram is quite sen-sitive to the interplay between δ and δ . We see from(5.17) that in the two limits when either δ = 0 or δ = 0, χ <
0, i.e., the transition is first order. This agrees withthe numerical analysis in the previous Section. We em-phasize that in both limits, a small SDW order, which weconsider here, still preserves low-energy fermionic states7 T c / T s δ δ T c / T s χ>0χ<0 χ<0 χ <0, B<0 FIG. 15: (Color online) The three-dimensional plot of theSDW-SC crossing surface, T s ( δ , δ ) = T c . At each point onthe surface we show the sign of B and χ = AB − C . Inthe region where χ >
0, the transition between SDW andSC states occurs via the co-existence region. For χ <
0, pureSDW and SC states are separated by the first-order transition.When
B <
0, the SDW-N conversion is of the first order andthe present GL analysis is invalid. near the modified FSs. Fermions near these FSs do havea possibility to pair into s + − state. However, SDW+SCstate turns out to be energetically unfavorable. We par-ticularly emphasize that the ellipticity of electron disper-sion is not sufficient for the appearance of the SDW+SCphase near T c ∼ T s .When both δ = 0 and δ = 0, there is a broad range0 . < δ δ < . , (5.18)where χ > T s ( δ , δ ) = T c , and the boundaries in Eq. (5.18) setthe critical values of δ and δ as functions of T s /T c .Combining Eqs. (5.18) and (5.14), we obtain that co-existence occurs for0 . m √ δt < δ < . m √ δt (5.19)To verify that this result holds at larger values of δ and δ , we computed χ without expanding in δ ˆ kq . Weplot the resulting phase diagram in Fig. 15. The result isqualitatively the same as Eq. (5.17): for δ = 0 or δ = 0, χ < δ and δ are non-zero,there exists a region where χ > -0.200.20.4 ( π T c ) B ( δ , δ ) B (δ,0) B (δ,δ) B (0,δ) T c / T s -8-40 ( π T c ) χ ( δ , δ ) χ(δ,0)χ(δ,δ)χ(0,δ) FIG. 16: (Color online) The case of s ++ SC. Panel (a): thebehavior of χ ( δ , δ ) = AB − C for different T s /T c for threecases: χ ( δ, χ (0 , δ ), and χ ( δ, δ ). For each case, δ is chosento satisfy the condition T s ( δ , δ ) = T c for a given T s /T c . Wesee that in all three cases, χ < T s /T c is. Panel B – the coefficient B ( δ , δ ) along the line T s ( δ , δ ) = T c . GL analysis is only valid when B > s ++ superconductivity We performed the same calculations for a conventional,sign-preserving s -wave superconductivity. The key dif-ference with the s + − case is that now χ = AB − C isnon-zero already when δ = δ = 0. Substituting A and B from (5.4) and (5.5) and C from (5.7) we obtain χ ( δ ˆ kq = 0) = − ζ (3)128 π T c < δ and δ , χ remainsnegative and the transition between SDW and SC statesis first-order. This result was first obtained by Fernandes et al. in Ref. 29. These authors also argued, based ontheir numerical analysis of the free energy, that thereis no SDW+SC phase for s ++ gap even when δ = δ are not small. We analyzed the sign of χ for larger δ and δ using our analytical formulas and confirmed theirresult. In Fig. 16 we show the behavior of χ ( δ , δ ) at thetransition point T s ( δ , δ ) = T c for three representativecases: χ ( δ , χ (0 , δ ), and χ ( δ , δ = δ ). In all cases,when B > χ ( δ , δ )remains negative.We caution, however, that the absence of co-existencebetween s ++ SC and SDW states within GL model doesnot imply that the two states are always separated byfirst-order transition. GL analysis is only valid near T s ( δ , δ ) = T c , when both orders are weak. The situ-ation at lower T has to be analyzed without expandingin m and ∆. And, indeed, we did find a small co-existenceregion T = 0, see Fig. 14.8 B. Zero-temperature limit
We consider only the case of s ± SC and the limit whenrelevant δ and δ are small, i.e., when T s /T c = 1 + δt and δt ≪
1. We compare energies for pure SDW and SCstate and for the co-existence state and find the regionwhere the co-existence state is energetically favorable.For this, we first verified that, at small δ and δ , thevalues of SDW and SC order parameters at T = 0 remainthe same as at δ = δ = 0, i.e., m = m = 0 . × (2 πT s ) and ∆ = ∆ = m ( T c /T s ). These values onlychange at large enough δ and δ , e.g., m changes when δ + δ > m .The free energies of pure SDW and SC states for m, ∆ > δ + δ can be straightforwardly evaluated at T = 0 by replacing the frequency sums in (2.43) by inte-grals. We obtain F ( m ) = − m δ δ m mm (5.21) F (∆) = − ∆ (cid:18) ∆∆ (cid:19) (5.22)These free energies have minima at m = m and ∆ = ∆ ,respectively. At the minima, F ( m ) = − m δ δ F (∆ ) = − ∆ − (cid:18) T c T s (cid:19) m . (5.23)Observe that F (∆ ) < F ( m ) when T c = T s . This isthe consequence of the fact that SDW magnetism is de-stroyed by doping and ellipticity, while superconductivityis unaffected.Comparing F ( m ) and F (∆ ), we find that the firstorder transition between pure SDW and SC states occursat m δt = δ + δ / . (5.24)If there is no intermediate co-existence phase, the SDWstate is stable for δ ≤ p δtm − δ /
2, while SC state isstable for larger values of δ .We next determine when the intermediate state ap-pears at T = 0. For this we expand the free energy nearthe SDW and SC states in powers of ∆ and m , respec-tively. We then obtain, near the SDW state, F ( m, ∆) = F ( m ) + a ∆ ∆ + b ∆ ∆ , (5.25)and near the SC state F ( m, ∆) = F (∆ ) + a m m + b m m . (5.26)We verified that b ∆ and b m are positive, while a m and a ∆ can be of either sign. The key issue is what are thesigns of a m and a ∆ at the point where F ( m ) = F (∆ ). We found that, to leading order in δt , a m = a ∆ = a atthis point, and a is given by a = δt (cid:0) − z + 7 z (cid:1) , z = δ m δt (5.27)Note that to obtain a we had to expand to order δt . Byvirtue of Eq. (5.24), δ = m √ δt √ − z , i.e. we have toconsider z ≤ a is positive, both pure states are stable, andthere is a first-order transition between them. When a <
0, the pure SDW and SC states are already unstable atthe point where F ( m ) = F (∆ ), what implies that whenwe vary δ at a fixed δ , there is a range of δ around δ = m √ δt √ − z in which the co-existence state hasa lower energy than the pure states. From (3.1) we seethat a > z < /
7, while a < / < z < δ , this implies that the transition at T = 0 isfirst order between pure states when δ < . m √ δt ,while at larger δ , pure SDW and SC phases are separatedalong δ line by the region of the co-existence phase. Thewidth of the co-existence phase initially increases as δ increases, but then begins to shrink and vanishes when δ approaches δ = 1 . m √ δt ( z approaches one frombelow). At this point, the co-existence region shrinks to apoint δ = 0. At larger δ , the SC state has lower energythan the SDW state for all values of δ If we keep δ fixed but vary δ , the co-existence rangeappears at δ = +0 ( z = 1) and exists up to δ =0 . m √ δt ( z = 1 / δ ( z < / C. The phase diagram
We now combine the results of GL analysis near thecrossing point and at T = 0 into the phase diagrams.For definiteness, we set δt = T s /T c − T and δ for different fixed δ . The results of this subsection hasto be compared with the phase diagrams presented inpanels (a1)-(a3) in Fig. 7, see also Fig. 10.From the analysis in the preceding two subsections,we found five critical values of δ : two are obtained fromthe GL analysis of the range of the co-existence phase,and are given by (5.19), two are critical values at whichthe co-existence phase first appears and then disappearsat T = 0, and the last one is the maximum value of δ at which T s ( δ = 0 , δ ) = T c . From (5.14) this valueis δ = 1 . m √ δt . Arranging these five values fromthe smallest to the largest, we obtain the following set ofphase diagrams at small δt :(a) For δ < . m √ δt , there is no intermediatephase, and pure SDW and SC transitions are sepa-rated by a line of a fist-order transition. The line istilted towards smaller δ at smaller T : it originatesat δ = m √ δt (1 . − . δ / ( m δt )) / at T = T c and ends up at δ = m √ δt (1 − . δ / ( m δt )) / .9 Q e h N sdw FIG. 17: (color online) A partially-gapped SDW state, whereonly a fraction N sdw of the electron and hole FSs is nested.On the remaining parts the dispersions ξ f and ξ c are verydifferent. The SDW state appearing below T s gaps excitationsonly in the shaded/boxed areas (see Fig. 2(d)), while on therest of the FSs the dispersions are close to the original ξ f and ξ c . The SC state below below T c does not compete with SDWin non-nested regions, but competes with SDW state in thenested (boxed) regions. (b) For 0 . m √ δt < δ < . m √ δt < , the inter-mediate phase appears near T = 0 and extends tosome T < T c . At larger T , the transition remainsfirst order. This behavior is consistent with thepanel (a1) in Fig. 7(c) For 0 . m √ δt < δ < . m √ δt , the interme-diate phase occupies the whole region T < T c . Thisbehavior is consistent with the panel (a2) in Fig. 7(d) For 1 . m √ δt < δ < . m √ δt , SC statewins at T = 0 for all δ . There is phase transi-tion at finite T . The transition is first order be-tween pure SDW and SC state at smaller T , butthe co-existence phase still survives near T c . Thisbehavior is consistent with the panel (a3) in Fig. 7(e) For 1 . m √ δt < δ < . m √ δt , the co-existence phase near T c disappears, and the transi-tion becomes first-order along the whole line sepa-rating SDW and SC states. (f) δ > . m √ δt , T s ( δ , δ ) becomes smaller than T c for all δ , and the system only develops a SCorder.This behavior is also totally consistent with Fig. 10: alldifferent phase diagrams are reproduced if we take hori-zontal cuts at different δ . We see therefore that numeri-cal and analytical analysis is in full agreement with eachother.The only result of numerical studies not reproduced insmall δt analytical expansion is the existence of a range of δ where the transition between the SDW phase and theco-existence phase is second order, while the transitionbetween the SC phase and the co-existence phase is firstorder, see Fig. 10. To reproduce this effect in analyticaltreatment, we would have to expand to the next orderin δt . Note in this regard that it is evident from Fig.10 that the width of the range where one transition isfirst order and another is second order shrinks as T s /T c decreases. VI. PARTIAL SDW STATE
In previous Sections we considered the situation whenthe splitting between hole and electron FSs is small. Wenow consider how the phase diagram is modified if insome k -regions hole and electron FSs are quite apart fromeach other (after we shift the hole FS by Q ). Such re-gions are far from nesting and we make a simple assump-tion that they are not affected by SDW. We then splitthe FS into nested parts where commensurate SDW stateexists and a SC order can exist as well, and non-nestedparts, where only SC order is possible. We present thisschematically in Fig. 17. The nested parts lie in some in-tervals of angles φ with total circumference ∆ φ , and haveweight N sdw < N total = 1 ( ∆ φ/ π = N sdw /N total ).The free energy and the self-consistency equations thencan be written as sums of the two contributions. The firstsum is over the FS part that has only SC order parameter,and in the second sum we integrate over part of the FSwith both orders. ∆ F (∆ , m )4 N F = ∆ TT c + N sdw m TT s − πT X ε n > (1 − N sdw ) (cid:20)p ε n + ∆ − | ε n | − ∆ | ε n | (cid:21) (6.1) − πT X ε n > Z ∆ φ dφ π Re (cid:20) (Σ + + Σ − ) − | ε n | − ∆ | ε n | − m | ε n | (cid:21) , ∆ ln TT c = 2 πT X ε n > (1 − N sdw ) " ∆ p ε n + ∆ − ∆ | ε n | + 2 πT X ε n > Z ∆ φ dφ π Re (cid:20) (cid:18) ∂ Σ + ∂ ∆ + ∂ Σ − ∂ ∆ (cid:19) − ∆ | ε n | (cid:21) , (6.2) N sdw m ln TT s = 2 πT X ε n > Z ∆ φ dφ π Re (cid:20) (cid:18) ∂ Σ + ∂m + ∂ Σ − ∂m (cid:19) − m | ε n | (cid:21) . (6.3)0 δ T / T c δ -0.100.10.20.30 0.5 1 1.5 2T / T c -0.100.10.20.3 T s / T c = 2 δ =0.13N sdw = 0.8-0.4( δ /0.3) SC + − SDW ∆ FSDW+SC ∆ m ∆ m T / T c = 0.2 b)c)a) ∆ F FIG. 18: (color online) (a) The phase diagram for s + − su-perconductivity, and SDW order parameter existing only inboxed regions of the FS in Fig. 17, with the relative width N sdw . We manually set the doping dependence of N sdw tobe N sdw = 0 . − . δ / .
3) and neglected the effect of thisvariation of N sdw on T s . Observe that SC and SDW ordersco-exist in a wide range of δ and T . (b,c) the order parame-ters, ∆ and m , and the free energy, F , as functions of δ at aconstant temperature, T /T c = 0 . T at a constant ¯ δ = 0 .
13 (c). As a function of δ , SDW orderparameter starts decreasing when SC order appears, and thenjumps to zero and the system becomes a pure SC. The self-consistency equations (6.2) and (6.3) areobtained by minimization of the functional ∆ F , ∂ ( ∆ F ) /∂ ∆ = 0 and ∂F/∂m = 0, and these expressionsreduce to previous formulas (2.41)-(2.43) for N sdw = 1.We find that the results are very similar to what wefound within the approximation of a small FS splitting.The typical picture is shown in Fig. 18.The only differences from Fig. 8 in this case are the co-existence of SC and SDW states already at zero doping δ = 0, and weak first order transition to purely SC state.We also analyzed s ++ SC order and again found a muchweaker tendency for co-existence, similar to Fig. 14.
VII. CONCLUSIONS
To conclude, we presented a general theoretical de-scription of the interplay between itinerant SDW andSC orders in two-band metals. Within the mean-fieldapproach we derived coupled self-consistency equationsfor the order parameters and the expression for the freeenergy, which is necessary to determine the stability ofdifferent phases.We considered the FS geometry with one hole and oneelectron bands of different shapes (a simplified FS geome-try for Fe-pnictides) and investigated the phase diagramsand the stability of the SDW+SC states for: (a) differ-ent gap structures of the SC state, Figs. 8, 10 vs. 14; (b) variations in the relative strength of SDW and SC inter-actions, Figs. 10, 11; (c) ellipticity of electron pockets,Figs. 7, 8, 10; and (d) incommensuration of SDW order,Figs. 11, 12. We considered the case when the transi-tion temperature to pure SDW state, T s , is higher thanthe critical temperature T c of a pure SC state. In theopposite case, T s < T c , the SC state develops first andsuppresses SDW state.We found that the SC s ± state with extended s -wavesymmetry has much stronger affinity with the SDW statethan the traditional s ++ state. A co-existence region of s ++ SC state with SDW is tiny and the co-existence isanyway very weak in terms of energy gain compared tothe pure SDW state. The transition from the pure SDWstate to the pure SC state is always first order, Fig. 14.For s ± gap, there is a stronger inclination towards co-existence with SDW state due to effective “attraction”between the two orders. We found that, depending onthe interplay between different effects (e.g., ellipticityand doping), the transition between SDW and SC or-ders is either first order or continuous, via the interme-diate SDW+SC phase, in which both order parametersare non-zero, Figs. 6-8. We note that the co-existence region gets larger withincreased strength of the SDW interaction relative to itsSC counterpart, described by the ratio T s /T c . Thus gen-erally we should see better co-existence between SDWand SC states, if T s is increasingly larger than T c , Fig. 10.Our results are in a disagreement with a common be-lief that, because SDW and SC states compete for theFermi surface, the SDW+SC state should emerge when apure SDW state next to the boundary of the co-existenceregion still has a modified Fermi surface at T = 0, andshould not emerge when fermionic excitations in the pureSDW phase are fully gapped at zero temperature. Wefound that the key reason for the existence of the mixedSDW+SC state is the “effective attraction” between theSDW and SC orders, while the presence or absence of theFermi surface in the SDW state at T = 0 matters less.Specifically, we found cases when SDW and SC ordersdo co-exist even when fermionic excitations in the pureSDW phase are fully gapped at T = 0, Fig. 10(a), andwe also found, for s ++ pairing, that there might be noco-existence down to T = 0 even when the pure SDWphase has a Fermi surface, Fig. 14.The phase diagrams for s + − gap are quite consistentwith the experimental findings in pnictides. For example,first order transition in Fig. 6 looks very similar to phasediagram of 1111 materials (La,Sm)OFeAs, where FSs aremore cylindrical. The co-existence region in Fig. 8 corre-lates well with doped 122 materials based on BaFe As ,where hole and electron FSs are less nested. And Fig. 10shows that one can get both SDW+SC phase and first-order transitions for the same SC state and the samefamily of materials. Our key result is that the way thedoping is introduced into the sample will determine thenature of the FS changes, and the path it will take in the( δ , δ )-plane: whether through a first order transition or1through a co-existence region. In other words, we arguethat there is strong correlation between how exactly FSsevolve upon doping and whether or not SC and SDWstates co-exist.The final remark. In the literature, there existsa notion of “homogeneous” and “inhomogeneous” co-existence of SC and SDW orders. The latter is ametastable state when the two orders exist in different spatial parts of the material. What we emphasize is thatthe other kind, “homogeneous” co-existence of SC andSDW orders in real space, is in fact “inhomogeneous”in momentum space: the SC and SDW orders dominateexcitation gaps on different parts of the FS. A. Acknowledgment
We acknowledge with thanks useful discussions withD. Agterberg, V. Cvetkovic, R. Fernandes, I. Eremin,I. Mazin, J. Schmalian, O. Sushkov, and Z. Tesanovic.A.V.C. acknowledges the support from nsf-dmr 0906953.
Appendix A: Electron and Hole dispersion for smallFS splitting
In this Appendix, we discuss in detail the approxima-tion we used for the dispersions of holes and electronsfor the case when the splitting between hole and electronFSs is small.SDW and SC orders mix c -fermions with momenta k and f -fermions with momenta k + q . The generic expres-sions for the two dispersions are ξ c ( k ) = µ c − ( k ) m c , ξ f ( k + q ) = ( k + q ) x m fx + ( k + q ) y m fy − µ f (A1)When the two FSs are circles of non-equal size, m fx = m fy = m f , we have ( µ c , m c ) ≈ ( µ f , m f ) ≈ ( µ, m ), but m c = m f and µ c = µ f . The approximation we used inthe text implies that ξ c ( k ) = µ c − ( k + q ) m c = µ c + µ f − k (cid:18) m c + 1 m f (cid:19) − kq (cid:18) m c + 1 m f (cid:19) + µ c − µ f − k (cid:18) m c − m f (cid:19) + kq (cid:18) m c + 1 m f (cid:19) ≈ µ c + µ f − k (cid:18) m c + 1 m f (cid:19) − kq (cid:18) m c + 1 m f (cid:19) + µ c − µ f k F m ( m c − m f ) + k F q mξ f ( k + q ) = ( k + q ) m f − µ f = k (cid:18) m c + 1 m f (cid:19) + kq (cid:18) m c + 1 m f (cid:19) − µ c + µ f µ c − µ f − k (cid:18) m c − m f (cid:19) + kq (cid:18) m f − m c (cid:19) ≈ k (cid:18) m c + 1 m f (cid:19) + kq (cid:18) m c + 1 m f (cid:19) − µ c + µ f µ c − µ f k F m ( m c − m f ) + k F q m (A2)Introducing ξ kq and δ ˆ kq defined in (2.30), we obtain ξ kq = k (cid:18) m c + 1 m f (cid:19) + kq (cid:18) m c + 1 m f (cid:19) − µ c + µ f δ ˆ kq = µ c − µ f k F m ( m c − m f ) + k F q m = 12 v F (cid:16) k cF − k fF − q (cid:17) (A3)We emphasize that, within this approximation, ξ kq and δ ˆ kq are two independent variables, one depends on thedeviation along the FS in the transverse direction, an- other depends on the angle along the FS. 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