Microscopically derived Ginzburg-Landau theory for magnetic order in the iron pnictides
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec Microscopically derived Ginzburg-Landau theory for magnetic order in the ironpnictides
P. M. R. Brydon, ∗ Jacob Schmiedt, and Carsten Timm † Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden, 01062 Dresden, Germany (Dated: November 15, 2011)We examine the competition of the observed stripe spin density wave (SDW) with other commen-surate and incommensurate SDW phases in a two-band model of the pnictides. Starting from thismicroscopic model, we rigorously derive an expansion of the free energy in terms of the differentorder parameters at the mean-field level. We show that three distinct commensurate SDW statesare possible and study their appearance as a function of the doping and the electronic structure.We show that the stripe phase is generally present, but its extent in the phase diagram dependsstrongly upon the number of hole Fermi pockets that are nested with the electron Fermi pockets.Electron pockets competing for the same portion of a hole pocket play a crucial role. We discussthe relevance of our results for the antiferromagnetism of the pnictides.
PACS numbers: 75.30.Fv, 75.10.Lp, 74.70.Xa
I. INTRODUCTION
The superconductivity of the iron pnictides contin-ues to fascinate the condensed matter community.
Be-cause of their high critical temperatures, particular inter-est has focused upon the so-called 1111 family R FeAsOand the 122 family A Fe As ( R and A are rare-earthand alkaline-earth elements, respectively), which becomesuperconducting by chemical doping or under pressure.The parent compounds are antiferromagnets, withstripe-like magnetic order with respect to the lattice ofFe atoms. Furthermore, the antiferromagnetism is inti-mately linked to an orthorhombic distortion of the crys-tal, as evidenced by the coincidence of the ferromag-netic direction with the shorter crystallographic axis inall 1111 and 122 parent compounds. It has been arguedthat the same condition that favors stable stripe orderalso implies a nematic transition above the N´eel temper-ature T N , where the magnetic fluctuations on each sub-lattice become locked into a stripe configuration, andwhich produces the orthorhombic distortion via the mag-netoelastic coupling. The mechanism for stabilizingthe stripe order is therefore a key problem in pnictidephysics.The microscopic origin of the antiferromagnetism inthe pnictides has been approached in a number of dif-ferent ways. Frustrated local moment models for the Fespins can reproduce the observed magnetic order, but the evidence for the metalicity and relativelyweak correlations of the parent compounds, and the de-velopment of incommensurate (IC) magnetic order upondoping, suggest an itinerant description. Ab initio cal-culations predict, and angle-resolved photoemissionand magneto-oscillation experiments confirm, thatthe Fermi surface of the pnictide parent compounds havequasi-two-dimensional nested electron and hole pockets.Such a system is known to undergo an excitonic insta-bility towards a spin-density-wave (SDW) state, asfor example in chromium.
Many authors have in ad-dition emphasized the importance of the complicated or- bital structure of the Fermi surfaces, but key aspectsof the physics are nevertheless well understood on the ba-sis of simpler orbitally trivial “excitonic” models.
Most itinerant models of the pnictides display at leasttwo nesting instabilities at different wavevectors. Thereis hence competition between the stripe magnetic orderand other SDW phases. Within a minimal two-orbitalmodel, it has been shown that doping away from half-filling or a relatively large ratio of the Hund’s rulecoupling to the Coulomb repulsion can stabilize thestripe state. For excitonic models, on the other hand,Eremin and Chubukov have demonstrated that the el-lipticity of the electron pockets or interactions betweenthe electron bands can stabilize the observed SDW state.The stripe order was nevertheless found to be rather sen-sitive to the number of Fermi pockets involved in theSDW, and its extent in the phase diagram remains un-certain. Competition with a different excitonic instabilityhas also been proposed to stabilize a stripe SDW. In this paper we present a systematic study of themagnetic order in the popular two-band excitonic modelof the pnictides.
Keeping only interactionterms which lead to an excitonic state, we construct theDyson equation for the single-particle Green’s functionsin an arbitrary commensurate SDW phase treated at themean-field level. By iterating the Dyson equation, weobtain approximate forms for the self-consistency equa-tions for the order parameters valid near T N , which wethen integrate to obtain a Ginzburg-Landau expansionof the free energy. From this we determine the phasediagram for several choices of the non-interacting bandstructure, and show that three different commensurateSDW states are possible. We conclude with a discussionof the relevance to the magnetism of the pnictide parentcompounds. (0,0) ( π ,0) ( π , π ) (0,0) ( k x a , k y a ) -8-4048 e n e r gy ( t c ) c band, ξ h = 0 c band, ξ h = 0.975 f band, ξ e = 1 -1 0 1 k x a / π -101 k y a / π -1 0 1 k x a / π -101 k y a / π Q Q Q Q Q Q ξ e = 1, ξ h = 0 ξ e = 1, ξ h = 0.975 (a)(b) (c) FIG. 1: (color online) (a) Dispersion of the electron and holebands for the cases of a single hole pocket at the Γ point( ξ h = 0) and of hole pockets at both the Γ and M points( ξ h = 0 . Q = ( π/a,
0) and Q = (0 , π/a ). In all panels we set µ = 0. II. MODEL
We model the FeAs planes as a two-dimensional inter-acting two-band system, where one band has electron-likeFermi pockets, while the other has hole-like Fermi pock-ets. Including only interaction terms which directly leadto an excitonic instability, we write the Hamiltonian as H = X k ,σ n ( ǫ c k − µ ) c † k σ c k σ + ( ǫ f k − µ ) f † k σ f k σ o + g V X k , k ′ , q X σ,σ ′ c † k + q ,σ c k σ f † k ′ − q ,σ ′ f k ′ σ ′ + g V X k , k ′ , q n c † k + q , ↑ c † k ′ − q , ↓ f k ′ , ↓ f k , ↑ + H.c. o , (1)where c † k σ ( f † k σ ) creates a spin- σ electron with momen-tum k in the hole-like (electron-like) band. In termsof the single-Fe unit cell, we assume the dispersions ǫ c k = ǫ c + 2 t c (1 − ξ h )[cos( k x a ) + cos( k y a )] + 2 t c ξ h [1 +cos( k x a ) cos( k y a )] and ǫ f k = ǫ f + t f, cos( k x a ) cos( k y a ) − t f, ξ e [cos( k x a ) + cos( k y a )], where a is the Fe-Fe bondlength. In units of t c , we take ǫ c = − . ǫ f = 3 . t f, = 4 .
0, and t f, = 1 .
0. We plot representative bandstructures and Fermi surfaces in Fig. 1. The dimension-less quantities ξ e and ξ h are key control parameters: ξ e controls the ellipticity of the electron pockets, while vary-ing ξ h from 0 to 1 tunes the band structure from a sys- tem with a single hole pocket at the Γ point to a systemwith equally large hole pockets at both the Γ and the Mpoints. For ξ h ≈ Q = ( π/a,
0) and Q = (0 , π/a ) [Fig. 1(b)], while for ξ h ≈ On the otherhand, a Fermi surface with hole pockets at the Γ andM points is realized in the minimal two- orbital model ofthe pnictides and this situation has been extensivelystudied. Furthermore, it is also of relevance tomore sophisticated orbital models where in addition tothe d xz /d yz -derived hole pockets at the Γ point there isusually also a d xy -derived hole pocket at the M point, which may play an important role in generating the SDWorder. Although the orbital content of the Γ and Mhole pockets are very different, mean-field studies sug-gest that the SDW instability is primarily determined bythe nesting properties, hence justifying the orbitally-trivial excitonic model used here.Equation (1) contains a density-density interactionand a term describing correlated transitions betweenthe electron and hole bands, with contact potentials g and g , respectively. At sufficiently low tempera-tures, the system is unstable against an excitonic SDWwith effective coupling g SDW = g + g > For our system the excitonic SDW state has two orderparameters corresponding to electron-hole pairing withrelative wavevector equal to Q and Q , i.e., ∆ = P α,β ∆ ,α,β = (1 /V ) P k P α,β ˆ σσσ α,β h c † k + Q ,α f k β i and ∆ = P α,β ∆ ,α,β = (1 /V ) P k P α,β ˆ σσσ α,β h c † k + Q ,α f k β i ,where ˆ σσσ is the vector of the Pauli matrices. ∆ and ∆ are related to the magnetization of each Fe sub-lattice by m a = ∆ + ∆ , m b = ∆ − ∆ . Whenboth ∆ and ∆ are non-zero, therefore, the magne-tization is the superposition of two SDW states withorthogonal ordering vectors. It has been pointed outthat in the case that ∆ · ∆ = 0 one has to in-troduce additional charge-density-wave (CDW) orderparameters δ c = (1 /V ) P k ,σ h c † k + Q ,σ c k σ i and δ f =(1 /V ) P k ,σ h f † k + Q ,σ f k σ i , where Q = Q + Q . III. FREE ENERGY EXPANSION
We define the single-particle Green’s functions of theexcitonic SDW system by G a,b Q ,σ,σ ′ ( k , iω n ) = − Z β dτ h T τ a k + Q ,σ ( τ ) b † k ,σ ′ (0) i e iω n τ , (2)where a, b = c, f . Treating the SDW and CDW orders atthe mean-field level, we write the Dyson equation for theGreen’s functions as G a,b Q n ,σ,σ ′ ( k , iω n ) = δ a,b δ σ,σ ′ δ Q n , G a (0) ( k , iω n ) + g δ a G a (0) ( k + Q n , iω n ) G a,b Q n + Q ,σ,σ ′ ( k , iω n ) − g SDW X m =1 , X α ∆ m,α,σ G a (0) ( k + Q n , iω n ) G a,b Q n + Q m ,α,σ ′ ( k , iω n ) , (3)where c = f and f = c , and the Green’s functions of thenon-interacting system are G a (0) ( k , iω n ) = ( iω n − ǫ a k + µ ) − . The order parameters can be expressed in termsof the Green’s functions as ∆ m,σ,σ ′ = 1 V X k β X n G f,c Q m ,σ ′ ,σ ( k , iω n ) e iω n + , (4a) δ ν = c,f = 1 V X k ,σ β X n G ν,ν Q ,σ,σ ( k , iω n ) e iω n + . (4b)where β = 1 /k B T . In general, it is not possible to analyt-ically solve Eq. (3) for the Green’s functions. By iteratingthe Dyson equation, however, we are able to expand theGreen’s functions in terms of the order parameters. In-serting this expansion into the self-consistency (“gap”)equations (4), and truncating it above a given order, wehence obtain an approximate form of the self-consistencyequations valid close to T N (assuming a second-ordertransition, as is the case here). Since the self-consistency equations are obtained from the stationary points of thefree energy with respect to the order parameters, we canconstruct a Ginzburg-Landau expansion for the free en-ergy F by integrating them, F = F + α (cid:0) | ∆ | + | ∆ | (cid:1) + β (cid:0) | ∆ | + | ∆ | (cid:1) + β | ∆ | | ∆ | + β | ∆ · ∆ | +( γ c δ c + γ f δ f ) ∆ · ∆ + α cf δ c δ f + α c δ c + α f δ f , (5)where F is independent of the order parameters. Wekeep only second-order terms involving the CDW orderparameters, since the system is far away from a pureCDW instability and a CDW only emerges as a secondaryorder parameter. We also neglect gradient terms sincewe are only interested in homogeneous states. The co-efficients in Eq. (5) are written in terms of the non-interacting Green’s functions as follows: α = 2 g SDW " g SDW V X k β X n G c (0) ( k , iω n ) G f (0) ( k + Q , iω n ) , (6a) β = g V X k β X n h G c (0) ( k , iω n ) G f (0) ( k + Q , iω n ) i , (6b) β = 2 g V X k β X n (cid:26) G c (0) ( k , iω n ) G c (0) ( k + Q , iω n ) h G f (0) ( k + Q , iω n ) i + G f (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) h G c (0) ( k , iω n ) i − G c (0) ( k , iω n ) G c (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) o , (6c) β = 4 g V X k β X n G c (0) ( k , iω n ) G c (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) , (6d) γ c = 4 g g V X k β X n G c (0) ( k , iω n ) G f (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) , (6e) γ f = 4 g g V X k β X n G c (0) ( k , iω n ) G c (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) , (6f) α cf = g , (6g) α c = − g V X k β X n G f (0) ( k + Q , iω n ) G f (0) ( k + Q , iω n ) , (6h) α f = − g V X k β X n G c (0) ( k , iω n ) G c (0) ( k + Q , iω n ) . (6i)The CDW order parameters can be integrated out, re-sulting in the renormalization of β → e β = β + α c γ f + α f γ c − α cf γ c γ f α cf − α c α f . (7)The Ginzburg-Landau expansion of the free energy,Eq. (5), and the expressions for the coefficients in termsof a specific microscopic model, Eq. (6), are the first ma-jor results of our paper. IV. PHASE DIAGRAM
The free energy in Eq. (5) admits three possible com-mensurate SDW states which we name following Ref. 29: • A magnetic stripe (MS) state where only one of theexcitonic order parameters is nonzero, e.g., ∆ = 0, ∆ = 0. This corresponds to the ordering in thepnictides. This state minimizes the free energy if2 β < min { β + e β , β } . • An orthomagnetic (OM) state where | ∆ | = | ∆ | and ∆ ⊥ ∆ . This corresponds to a “flux” typeordering of the magnetic moments. This state min-imizes the free energy if β < min { β , β + e β } . • A spin and charge order (SCO) state where | ∆ | = | ∆ | and ∆ k ± ∆ . In this state only one sublat-tice of the Fe plane has non-zero moments, whichorder in a checkerboard pattern. When g = 0,the spin order induces a CDW with ordering vec-tor Q . This state minimizes the free energy if β + e β < min { β , β } .From close examination of Eqs. (6b) and (6c) we observethat if ǫ f k = ǫ f k + Q or ǫ c k = ǫ c k + Q for all k we have 2 β = β , and hence the MS and OM states are degenerate.These conditions are satisfied for the electron and holebands at ξ e = 0 and ξ h = 1, respectively. In particular,if ξ h = 1 the degeneracy of the MS and OM states islifted by arbitrarily small ellipticity of the electron Fermipockets, as pointed out in Ref. 43.The free energy in Eq. (5) allows us to determine thephase diagram of the model close to T N . In Fig. 2 wepresent phase diagrams showing the ordered state re-alized at a temperature T = T − N infinitesimally below T N as a function of ξ e and of the doping relative tohalf-filling, δn , for various values of ξ h . In construct-ing the phase diagrams, we adjust g SDW such that foreach value of ξ e the maximum critical temperature ofthe commensurate SDW as a function of the doping δn is k B T opt N = 0 . t c , which gives a reasonable ratioof k B T opt N to the bandwidth. The variation of the op-timal doping level δn opt , where T N is maximal, with ξ e is shown by black dotted lines. The boundaries be-tween the different commensurate SDW phases are deter-mined by the conditions on the β , β and e β mentioned ξ e δ n ICIC δ n opt SCO MS ξ h = 0 OMMS/OM (a) ξ e -0.1-0.0500.050.1 δ n δ n opt ICMS MSOMSCO SCOMS/OM ξ h = 0.95 (b) PM ξ e -0.0500.05 δ n δ n opt IC ICPM OMMS MS SCO ξ h = 0.975SCOMS/OM (c) FIG. 2: (color online) Magnetic order at T = T − N as a functionof ξ e and δn for (a) ξ h = 0, (b) ξ h = 0 .
95, and (c) ξ h = 0 . ξ e = 0 thethick solid line indicates degenerate MS and OM solutions.For ξ e = 0, solid lines indicate phase boundaries, while thedotted line indicates the optimal doping δn opt . δ n T N / T op t N MSOMIC ξ e = 1, ξ h = 0 (a) -0.15 -0.1 -0.05 0 0.05 0.1 δ n T N / T op t N MSOMSCOIC (b) ξ e = 1, ξ h = 0.95 -0.1 -0.05 0 0.05 0.1 δ n T N / T op t N MSOMSCOIC (c) ξ e = 1, ξ h = 0.975 FIG. 3: (color online) Critical temperature T N of the SDW states as a function of doping δn for ξ e = 1 and (a) ξ h = 0,(b) ξ h = 0 .
95, and (c) ξ h = 0 . k B T opt N = 0 . t c . above, where the coefficients in Eq. (6) were evaluatedfor g = g and using a 1000 × k -point mesh. Inall phase diagrams we find regions where IC SDW or-der occurs. Since the IC SDW ordering vector is likelyclose to the commensurate SDW vector, the bound-ary between the two phases is determined by solving1 + ( g SDW /V ) P k (1 /β ) P n G c (0) ( k , iω n ) G f (0) ( k + Q + δ q , iω n ) = 0 for the critical temperature of the Q + δ q SDW state, where δ q = (0 . π/a, , . π/a ). Whenthe critical temperature of the Q + δ q SDW exceeds thatof the commensurate SDW, an IC SDW is assumed to berealized. We similarly find the critical doping for whichthere is no IC SDW order, and the system remains para-magnetic (PM) down to zero temperature. Note that wedisregard states with T N < . T opt N .We first consider the phase diagram for ξ h = 0[Fig. 2(a)], which corresponds to a Fermi surface witha single hole pocket and two electron pockets at optimaldoping as shown in Fig. 1(b). A commensurate SDWstate is realized here for δn ≈ δn opt ± .
025 for all ξ e . At ξ e = 0 the condition ǫ f k = ǫ f k + Q is satisfied, and hencethe OM and MS states are degenerate. These states havethe lowest free energy at underdoping and near optimaldoping, but at overdoping the SCO is realized. Uponswitching on a finite ξ e , the degeneracy of the MS andOM states is lifted, and the MS state is found to have thelower free energy near optimal doping and at overdoping,while the OM state is stable at underdoping. The SCOstate is rapidly suppressed by a finite ξ e .The phase diagrams for the case of two hole and twoelectron Fermi pockets [see Fig. 1(c)] is shown in Figs.2(b) and (c) for ξ h = 0 .
95 and ξ h = 0 . g SDW needed to produce the SDW state is roughly athird smaller than for the single-hole-pocket scenario.For small ξ e , a commensurate SDW is nevertheless real-ized over a much greater doping range than in the single-hole-pocket case. The OM phase is stable near optimaldoping, with the MS phase found at moderate doping,and the SCO found at stronger doping. At larger val- ues of ξ e , however, we find a strong tendency towards ICorder in the ξ h = 0 .
95 case, with commensurate ordercompletely absent for ξ e > .
75. In contrast, the com-mensurate SDW in the ξ h = 0 .
975 case is present for all ξ e and is always realized about optimal doping.In Fig. 3 we plot the critical temperature of the SDWstates as a function of doping δn for constant- ξ e cutsthrough the three phase diagrams in Fig. 2. For the ξ h = 0 case [Fig. 3(a)] we note that there are substantialIC SDW “shoulders” to the commensurate SDW dome,which extend up to T ≈ . T opt N . Although IC SDWstates are also found at strong underdoping or overdop-ing in the ξ h = 0 .
975 case [Fig. 3(c)], they are realizedover a smaller doping range relative to the commensu-rate states and do not extend to such high temperaturescompared to the single-hole-pocket scenario. As shownin Fig. 3(b), however, slightly reducing ξ h leads to ICstates appearing at optimal doping. V. DISCUSSION
To summarize our main results, we have shown thatin a two-band model of the pnictides there are three dis-tinct commensurate SDW states: The MS, OM, and SCOphases. In a model with a single hole pocket and two elec-tron pockets, the MS state dominates the phase diagram,but the OM and SCO phases are possible away from op-timal doping. For a model with two hole pockets, in con-trast, the OM phase is stable at optimal doping, althoughthe MS phase remains at under- and overdoping. Sinceonly the MS state is observed experimentally, we henceconclude that the model with a single hole pocket givesa more reasonable description of the physics. We never-theless note that such a model displays a rather strongtendency towards IC SDW order, which is not observedexperimentally. We consider the results for the single-hole-pocket casein more detail. In agreement with Ref. 43 we foundthat MS order was realized near optimal doping for ar-bitrarily small ellipticity of the electron Fermi pockets. (a) (b) (c)optimal dopingunderdoping overdoping
FIG. 4: (color online) Schematic diagram of the nesting of thetwo electron pockets (blue dotted and red dashed lines) witha single hole pocket (black solid line). We show the situationfor (a) underdoping, (b) optimal doping, and (c) overdoping.The small shaded circles indicate the region of best nesting ofthe electron pockets with the hole pocket.
Away from optimal doping, however, states consisting ofa superposition of commensurate SDWs with orthogo-nal ordering vectors Q , Q are realized. This can beunderstood via the following argument. At strong un-derdoping, we expect that the best nesting between thehole pocket and the elliptical electron pockets occurs forthe states near the major axis of the electron pockets,as shown in Fig. 4(a). Similarly, for strong overdopingthe best nesting occurs for the states near the minor axisof the electron pockets [Fig. 4(c)]. In both cases, theSDW gaps for the two nesting vectors involve states farapart on the hole Fermi surface, and so there should belittle competition between them. Near optimal doping,however, the nested electron Fermi pockets compete forthe same states on the hole Fermi surface [Fig. 4(b)], andhence it is more favorable for only a single electron pocketto participate in the SDW. We note that the variation ofthe nesting “hotspots” with doping is expected to havesignificant consequences for the a - b resistivity anomalyin the pnictides. The addition of a second hole pocket at the M pointstrongly affects the physics. The doping range of com-mensurate SDW states is significantly expanded, and theOM state is realized near optimal doping with a MS phaseappearing upon doping. This is consistent with previousinvestigations of the two-orbital model.
In contrast,our results are inconsistent with the degenerate OM andMS phases found in Ref. 43. This is due to the fact that inthe model of Ref. 43 the Γ hole pocket is mapped exactlyonto the M pocket by translation of Q , i.e., the degener-acy condition ǫ c k = ǫ c k + Q is always satisfied. Despite theapparent differences, the phase diagram can be under-stood in a similar way to the single-hole-pocket case. Atoptimal doping the nesting of the electron pockets is notoptimal for either hole pocket, but instead correspondsto overdoping for the smaller pocket and underdopingfor the larger pocket, thus allowing the OM phase. In-deed, this perfectly describes the nesting properties ofthe two-orbital model at half-filling. Upon doping thesystem, the nesting with one of the hole pockets is opti-mized, while the nesting is the other hole pocket becomes extremely poor. As such, only a single hole pocket par-ticipates in the nesting, and so the MS state is stable.The relative size of the two hole pockets is thus crucial:If the two pockets are too dissimilar in size, there will bea tendency for the T N ( δn ) curves in Fig. 3 to split intotwo separate domes with commensurate order near theirmaxima. The strong tendency towards an IC SDW stateat optimal doping suggests that the ξ h = 0 .
95 case is closeto this limit. For somewhat lower ξ h , i.e., when the holepocket at the M point is much smaller than the pocket atthe Γ point, the pocket at the M point only shows goodnesting with the electron pockets at very large hole dop-ing. At realistic doping levels, the smaller hole pocketis essentially irrelevant for the SDW formation and thesingle-hole-pocket model is applicable.Finally, we consider the implications of our results forthe hypothesized nematic state in the pnictides. Indeed,a major motivation for our study is the connection of thisphase with the MS SDW state. This can be seen viathe following naive argument: After integrating out theCDW, we write the free energy Eq. (5) in terms of thesublattice magnetizations m a and m b , F = F + 12 α (cid:0) | m a | + | m b | (cid:1) + 116 (2 β + β ) (cid:0) | m a | + | m b | (cid:1) + 14 (2 β − β ) ( m a · m b ) + 116 e β (cid:0) | m a | − | m b | (cid:1) , (8)and identify the Ising nematic degree of freedom as ϕ = m a · m b . In the mean-field theory presented inthis paper, ϕ is only non-zero in the MS phase. The in-clusion of sufficiently strong magnetic fluctuations, how-ever, allows the nematic order parameter to be non-zero above T N , so long as the coefficient of ϕ in Eq. (8)is negative, i.e 2 β < β . This is the same condition thatensures that the MS phase has lower free energy than theOM phase, and hence implies that a nematic transitionoccurs at T ≥ T N when the SDW state shows stripe or-der. It is therefore intriguing that we find that 2 β − β changes sign as a function of doping in all cases. This sug-gests a strong doping dependence of the nematic phase:In particular, for a scenario with a single hole pocket weexpect that the nematic phase at underdoping will beweaker and occur closer to T N compared to overdoping,or may even be absent. Since this is apparently not ob-served experimentally, our results might imply that themagnetoelastic coupling plays a more direct role in thestructural transition. VI. SUMMARY
In this paper we have presented a weak-coupling studyof the magnetic order in a two-band model of the ironpnictides. Using the Dyson equation for the Green’s func-tion of an arbitrary commensurate SDW state treated atthe mean-field level, we have obtained an expansion of thefree energy valid close to the critical temperature. Wehave shown that this allows three commensurate SDWstates: The experimentally relevant stripe MS phase, theflux OM phase, and the SCO phase for which only onesublattice orders. The competition of these phases withone another and with IC SDW states has been studied asa function of the doping and the variation of key featuresof the non-interacting electronic structure.In particular, we have examined systems containingtwo elliptical electron pockets and either a single holepocket at the Γ point or hole pockets at both the Γ andthe M points. In the former case we find that the MSstate is stable at optimal doping, while a superposition oftwo orthogonal SDW states is stable away from optimaldoping. In the latter scenario, however, the OM phaseis realized near optimal doping, while the MS state isstable at moderate doping and the SCO state appears at higher doping levels. The doping dependence of the asso-ciated phase diagrams can be understood in terms of thechanging nesting properties of the Fermi surface. Ourresults indicate that the single-hole-pocket model is inbetter agreement with experiments, presumably becauseany hole pocket at the M point in the real materials issmall and poorly nested with the electron pockets. Thesingle-hole-pocket picture also suggest that the proposednematic state in the pnictides should be highly asymmet-ric with respect to electron vs. hole doping.
Acknowledgments
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