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Theory of coexistence of superconductivity and ferroelectricity Y. Krivolapov,∗ A. Mann, and Joseph L. Birman† Department of Physics, Technion-Israel Institute of Technology, Haifa 32000 Israel (Dated: February 6, 2008; Received textdate;Revised textdate; Accepted textdate;Published textdate) 6 A new investigation of the coexistence and competition of ferroelectricity and superconductivity 0 isreported. InparticularweshowthatthestartingHamiltonianofapreviousstudybyBirmanand 0 Weger(2001)canbeexactlydiagonalized. Theresultdifferssignificantlyfrommean-fieldtheory. A 2 Hamiltonian with a different realization of the coupling between ferroelectricity and superconduc- tivityisproposed. Wereporttheresultsformean-fieldtheoryappliedtothisHamiltonian. Wefind n that theorder parameters are strongly affected by thiscoupling. a J PACSnumbers: 74.20.-z,77.80.-e,64.90.+b,77.90.+k 8 1 Inthepresentpaperwereporttworelatedresultsfrom The Hamiltonian of the ferroelectric sector of the ] n a reexamination of previous work on nearly ferroelectric model is simply a Hamiltonian of a displaced harmonic o superconductors1. Inthat paper acoupling termwasin- oscillator c troducedintotheoriginalHamiltonian(Eq. 27ofRef.1). pr- Aginveisnvaessqtiugeaetzioinngooffththaetcpohuopnloinngstaenrdmnsohocwousptlhinatgittootnhlye HFE =ω0(cid:18)NˆB + 21(cid:19)+γ1(cid:16)Bˆ0†+Bˆ0(cid:17) u electronic pairs, and therefore the Hamiltonian can be s t. diagonalized exactly. However, the double mean-field where Bˆ0†,Bˆ0 have boson commutation relations. a approximation does result in an effective coupling be- Combiningthose two sectorstogether andintroducing m tweenthe two subsystems. (Eq. 34 of Ref. 1). Therefore a gauge invariant coupling between them, Birman and - the results of the analysis of that equation remain valid. WegerobtainthefollowingHamiltonian(Eq.27ofRef.1) d Oursecondresultfollowsfromintroducingadifferentbi- n quadratic coupling term in the Hamiltonian of Eq. 27 of o Ref. 1, which satisfies gauge and inversion symmetries. 1 [c Thistermdoescouplethe twosubsystems. Wewilltreat Hˆ =−2 (εkˆ3k+2∆kˆ2k)+ω0(cid:18)NˆB+ 2(cid:19)+ (1) this new Hamiltonian in mean-field approximation. Xk 1 2 v We start with the model of coexistence of supercon- +γ1 Bˆ0†+Bˆ0 + γ2kˆ22k Bˆ0†+Bˆ0 7 ductivity and ferroelectricity proposed by Birman and (cid:16) (cid:17) Xk (cid:16) (cid:17) 0 Weger1. For convenience we include its main features 4 here. Birman and Weger start with two separate Hamil- If a single k mode is isolated 1 toniansforthesuperconductingandferroelectricsectors. 0 The superconducting sectorisa mean-fieldreducedBCS 1 6 pseudo-spin model (we corrected a misprint in Ref. 1) Hˆ = 2(εˆ3+2∆ˆ2)+ω0 NˆB+ + (2) 0 − (cid:18) 2(cid:19) / 2 at HSC = 2 (εkˆ3k+2∆kˆ2k) +γ1 Bˆ0†+Bˆ0 +γ2ˆ22 Bˆ0†+Bˆ0 m − Xk (cid:16) (cid:17) (cid:16) (cid:17) - Here we notice that the structure of the pseudo-spin d whereεk isthesingle-electronenergyand∆k isthepair- operators in terms of the pair operators implies n ing interaction energy. The pseudo-spin operators ˆpk o obey SU(2) commutation relations and are defined as ˆ2 =ˆ2 =ˆ2 c 1 2 3 : iv ˆ1k =(−i/2)(cid:16)ˆb†k−ˆbk(cid:17) and therefore ˆ21, ˆ22 and ˆ23 are invariant to any rotation X r ˆ2k =(1/2) ˆb†k+ˆbk of the form exphnˆ·−→j i. a (cid:16) (cid:17) We use this property to diagonalize the Hamiltonian ˆ3k =( 1/2)(nˆk+nˆ−k 1) analytically by applying the following unitary transfor- − − mation where ˆb andˆb† are pair operators defined by k k eβ2(Bˆ0†2−Bˆ02)eα(Bˆ0†−Bˆ0)+iθˆ1He−iθˆ1−α(Bˆ0†−Bˆ0)e−β2(Bˆ0†2−Bˆ02) † ˆb†k =aˆ†k↑aˆ†−k↓, ˆbk =(cid:16)ˆb†k(cid:17) , nˆk ≡aˆ†k↑aˆk↑ =HD+ˆ2(2εsinθ−4∆cosθ)+ + Bˆ0†+Bˆ0 ω0αe−2β +γ1e−β +4αγ2ˆ22e−2β + oapnderaâkt↑orasnfdoraˆ†kw↑aavreevtehceteolrec~kt,rospninan(n↑i)h,ileattcio.nandcreation +(cid:16)(cid:16)Bˆ0†2+Bˆ02(cid:17)(cid:17)(cid:0)(cid:16)−ω20 sh2β+γ2ˆ22e−2β(cid:17) (cid:1) 2 HereHD is aHamiltoniandiagonalinthe j3,NB basis: with | i HD =2ˆ3(2∆sinθ−εcosθ)+ ξ =[ Γ1/ω0 Γ2(1 msinθ)]/ω0 − − − +ω0(cid:18)NˆBch2β+sh2β+ 21 +αe−2β(cid:19)+ ∆′ =∆ 1−γ2P2 Γ1 =γ1(cid:0) 2γ2P∆2(cid:1) Γ2 =4γ2 +2αγ1e−β +γ2ˆ22e−2β 2NˆB+1+4α2 − (cid:16) (cid:17) Here both order parameters are affected by the SC-FE and if we want the non-diagonal parts to vanish we re- coupling strength (γ2). quire To sum up this part, we have obtained an exact an- alytical solution of the ferroelectricity and superconduc- 2εsinθ =4∆cosθ tivitycoexistencemodel,usedinRef.1. We showedhere ω0αe−2β +γ1e−β +4αγ2ˆ22e−2β =0 thatcontraryto themean-fieldresults,anexactsolution γ2ˆ22e−2β = ω20 sh2β opfartahmeemteordiesladffeemctoendsbtryattehsetchoautpltinhge,fbeurrtotehleectsruipceorcrodner- ducting order parameter is not. If one were to initially Theseyieldthefollowingsetofcoupledequationsforthe model the SC-FE system by Eq. 34 of Ref. 1, the results parameters reported in Ref. 1 would remain valid. tanθ = 2∆, α= γ1eβ , e4β = ω0+γ2. pliRngetuterrnmingintoathqeuaonrtiguimnalmqeucehsatnioicnalofHtahmeipltroonpiearn,cowue- ε −ω0+γ2 ω0 needtorespectgaugeandinversionsymmetry. Sincethe Inserting them back into the transformed Hamiltonian superconductinggapparameterisacomplexquantitythe 2 gives added term should correspond to the bilinear ∆ . We | | now take this as ˆ+ˆ−. The coupling term for the ferro- HD =−2ˆ3√εε22−+44∆∆22 − γ2γ+12ω0+ ewleilcltcroicrrpeosplaornizdattioonP,2wahnidchcarenspbeecttsakiennvearssionBˆ0†sy+mBmˆ0etr2y. 1 (cid:16) (cid:17) + ω0(ω0+γ2)(cid:18)NˆB+ 2(cid:19) Thus the bilinear coupling becomes ˆ+ˆ− Bˆ0†+Bˆ0 2. p (cid:16) (cid:17) and this is the decoupled Hamiltonianof a squeezed and We note that ˆ+ˆ− = ˆ21+ˆ22+j3 = 21 +ˆ3 (Using ˆ−ˆ+ shifted harmonic oscillator and a spin of one-half in a would have changed ˆ32into −ˆ3). The coupling becomes magnetic field. The energy spectrum is given by γ2 ˆ3+ 21 Bˆ0†+Bˆ0 . The Hamiltonian is now (cid:0) (cid:1)(cid:16) (cid:17) ε2 4∆2 γ2 H (m,n)= 2m − 1 + D − √ε2+4∆2 − γ2+ω0 Hˆ = 2(εˆ3+2∆ˆ2)+ω0 NˆB+ 1 + (4) 1 − (cid:18) 2(cid:19) + ω0(ω0+γ2)(cid:18)n+ 2(cid:19) 1 2 We can now use tphe exact eigenstates of the Hamilto- +γ1(cid:16)Bˆ0†+Bˆ0(cid:17)+γ2(cid:18)ˆ3+ 2(cid:19)(cid:16)Bˆ0†+Bˆ0(cid:17) nian to compute the order parameters of the system: TheinteractionterminthisHamiltoniandiffersfromthe ηFE ≡DBˆ0†+Bˆ0E=2αe−β =−ω0γ+1γ2 (3) cisornroetspeoxnadcitnlygtseorlmvab(Eleq,.t2h7e)reinforReefw.e1.mTahkiesaHammeialtno-nfiiealnd 1 ∆ approximationin the form ηSC ˆ2 = sinθ = ≡h i −2 −√ε2+4∆2 2 2 While the order parameter of the ferroelectric sector is ˆ3(cid:16)Bˆ0†+Bˆ0(cid:17) ≃(cid:28)(cid:16)Bˆ0†+Bˆ0(cid:17) (cid:29)ˆ3+ (5) affected by the SC-FE coupling strength, the order pa- 2 2 rameter of the superconducting sector is not. This oc- + ˆ3 Bˆ0†+Bˆ0 Bˆ0†+Bˆ0 ˆ3 . 2 h i(cid:16) (cid:17) −(cid:28)(cid:16) (cid:17) (cid:29)h i curs since in the coupling term γ2ˆ22 Bˆ0†+Bˆ0 , ˆ22 is a (cid:16) (cid:17) Inserting Eq. 5 into Eq. 4 results in a solvable bilinear c-number and not an operator. Hamiltonian Using a double mean-field approximation for the Hamiltonian (Eq. 34 of Ref. 1) and variational coherent 1 sfotalltoewiwnagvreelfautnioctnisonfo,rBthiremoarndearnpdarWamegeetre1rsobtained the HˆMF =ε′ˆ3−4∆ˆ2+ω0(cid:18)NˆB+ 2(cid:19)+γ1(cid:16)Bˆ0†+Bˆ0(cid:17)+ (6) m(∆′/ε+ξΓ2/ε) 2 2 ηFE =2ξ ηSC = 1+(∆′/ε+ξΓ2/ε)2 +γ2′ (cid:16)Bˆ0†+Bˆ0(cid:17) −γ2(cid:28)(cid:16)Bˆ0†+Bˆ0(cid:17) (cid:29)hˆ3i q 3 This set of equations can be reduced to a polynomial 0.5 10 |η | equation, which we solved numerically. Using these nu- SC merical results we investigate the behavior of the order 0.4 |ηFE| 8 parameters 0.3 6 η|SC η|FE ηFE Bˆ0†+Bˆ0 =2αe−β (8) | 0.2 4 | ≡D E 1 ηSC ˆ2 = sinθ ≡h i −2 0.1 2 0 0 Fig. 1showsthe dependence ofthe twoorderparame- −−11 00 11 22 33 44 55 66 77 γ2 ters on the coupling coefficient between the ferroelectric andsuperconductingsubsystems. Thereisasmoothevo- FIG.1: Theabsolutevaluesofthesuperconductingorderpa- lution of each of the order parameters with the coupling rameter and the ferroelectric order parameter as a function coefficient γ2. Both order parameters are non-vanishing ofthecouplingconstant γ2.Theleft sideoftheordinatecor- at γ2 = 0. We note that γ2 is bounded by two critical responds to small negative values of γ2. Here ∆ = ε = γ1 = values, beyond which there are no real solutions for the ω0 =1. set of self-consistent equations (7). At the positive criti- calvaluethesuperconductinggapparameterη reaches SC its maximum, and the polarizationorder parameter η FE Here reachesitsminimum. Fornegativevaluesofγ2thesuper- conducting order parameter vanishes very rapidly, while ε′ =−2ε+γ2(cid:28)(cid:16)Bˆ0†+Bˆ0(cid:17)2(cid:29) tγh2eafpeprrrooealcehcetsriictsorndeegratpiavreacmriettiecralshvaalrupely. diverges, when 1 In conclusion we note that our new model for SC-FE γ2′ =γ2(cid:18)hˆ3i+ 2(cid:19) coexistence/competitiondoesagreewiththespiritofthe Matthiasconjecturethateachofthesecooperativeeffects This mean-field Hamiltonian, Eq. 6, differs from Eq. 34 tends to exclude or suppress the other2. in Ref. 1 andcanbe diagonalizedusing rotation,squeez- ing and displacement transformations; Wˆ =eα(Bˆ0†−Bˆ0)eβ2(Bˆ0†2−Bˆ02)eiθˆ1 These yield the following set of (self-consistent) coupled Acknowledgments equations for the parameters α,β and θ tanθ = −4∆, α= −γ1eβ , e4β = ω0+γ2′. JLB thanks the Institute of Theoretical Physics of ε′ ω0+γ2′ ω0 Technion for its support and hospitality. Also support (7) from PSC-CUNY is acknowledged. ∗ Electronic address: [email protected] 2 B. Matthias, in Ferroelectricity, edited by E. Weller (Else- † Permanent address: Department of Physics, City College, vier, NewYork,1976) CUNY,New York,New York10031 1 J.L.BirmanandM.Weger,Phys.Rev.B64,174503(2001).