doi: /PhysRevB

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1 doi: /PhysRvB

2 PHYSICAL REVIEW B VOLUME 53, NUMBER 13 1 APRIL 1996-I Exact rsults on suprconductivity du to intrband coupling Y. Morita* Institut for Solid Stat Physics, Univrsity of Tokyo, Roppongi Minato-ku, Tokyo 106, Japan Y. Hatsugai Dpartmnt of Applid Physics, Univrsity of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan K. Kohmoto Institut for Solid Stat Physics, Univrsity of Tokyo, Roppongi Minato-ku, Tokyo 106, Japan Rcivd 18 July 1995; rvisd manuscript rcivd 8 Dcmbr 1995 W prsnt a family of xactly solvabl modls at arbitrary filling in any dimnsions which xhibit suprconductivity with intrband pairing. By th us of th hiddn SU2 algbra th Hamiltonians wr diagonalizd xplicitly. Th zro-tmpratur phas diagrams and th thrmodynamic proprtis ar discussd. Svral proprtis ar rvald which ar diffrnt from thos of th BCS-typ suprconductor. Suprconductivity is on of th most rmarkabl phnomna in condnsd-mattr physics. Rcntly possibilitis of a suprconductor ar proposd by Kohmoto and Takada. 1 Thy invstigatd th suprconducting instability of insulators by th man-fild tratmnt. A two-band systm which is insulating without intractions bcoms suprconducting by a sufficintly larg intrband attraction. It has many proprtis which ar diffrnt from thos of th BCS-typ suprconductors. 2 Not that th Coopr instability is irrlvant hr, sinc thr is no Frmi surfac. In Rf.3, possibl ralization in organic matrials is discussd, which is an xtnsion of th Littl s ida for th room-tmpratur suprconductor. 5. W hav constructd a family of xactly solvabl modls at arbitrary filling in any dimnsions which includs th modls proposd in Rfs. 1 and 3. W hav obtaind th ground stat and th thrmodynamic quantitis xplicitly. Svral proprtis hav bn rvald. An instability without a Frmi surfac, which was proposd by Kohmoto and Takada, is ralizd in th modls. This instability is quit diffrnt from th Coopr instability. A finit strngth of attraction is ndd to produc th suprconductivity in contrast to th BCS-typ suprconductivity. W also not that thr ar possibilitis that our pictur ralizs in a ralistic and complicatd modl. 4 Lt us considr a two-band modl dscribd by th Hamiltonian HH kin H int, 1 H kin k v kc k v c k v k c kc k c c k c, 2 H int U N k c c v k c k k c v k c c, k whr c k (v) and c k (c) ar th frmion annihilation oprators for th valnc band and th conduction band and (v) (k) and (c) (k) ar th nrgy disprsions of th valnc band and th conduction band, rspctivly. Th momntum vctor k 3 taks valus in th d-dimnsional Brillouin zon. W impos a constraint symmtric condition on th band structur v k c kc, whr C is indpndnt of k. Without loss of gnrality w st C0. W st U positiv and O(N 0 ), whr N is th numbr of th momntum points in th Brillouin zon. Th intraction is an intrband attraction. Th spin dgrs of frdom ar nglctd for simplicity, sinc w do not considr th spin-rlatd quantitis hr. Lt us sktch th procss of th diagonalization. Th diagonalization consists of two stps. At first w show th dcoupling proprty of th Hamiltonian. Nxt w map th systm to an xactly solvabl quantum spin systm. Thn w can construct all th ignvalus and th ignvctors. W rprsnt th stats in th Hilbrt spac diagrammatically s Fig.1. Lt us span th Hilbrt spac by th bas vctors v c pn c p1 v vc q1 c c c qn c 0 S c k1 v c c k3 c v k3 c c km c km 0 D, v c c v k1 c c k2 c k2 whr p 1,,p N v,q 1,,q N cs and k 1,k 2,k 3,,k M D (Th sts S and D will b dfind blow. Hr 0 S is dfind by c (c) k 0 S 0(kS ) and c (v) k 0 S 0(kS ). 0 D is dfind by c (c) k 0 D 0(kD) and c (v) k 0 D 0(kD). Considr a pair which consists of th momntum point k in th valnc band and th momntum point k in th conduction band. W dnot th pair by k, whr k taks valus in th Brillouin zon. Dfin th sts S and D as follows. If k is singl occupid, k blongs to S. And, if k is mpty or doubl occupid, k blongs to D. Not S D and S Dth Brillouin zon. Lt us introduc an oprator P j which is a projction oprator to th Hilbrt spac whr S and D ar fixd to b /96/5313/85615/$ Th Amrican Physical Socity

3 8562 Y. MORITA, Y. HATSUGAI, AND M. KOHMOTO 53 whr H spin is dfind by H spin U N kdj Ŝ k x iŝ k y U N Ŝx iŝ y Ŝ x iŝ y kdj Ŝ k x iŝ k y U N Ŝ2 Ŝ z 2 Ŝ z. 10 Th oprators dfind abov satisfy th rlations S l m k,s ilmn k S n k kk, Ŝ k x 2 Ŝ k y 2 Ŝ k z , FIG. 1. Th classification of th pairs and mapping to a quantum spin. S j and D j. Th indx j dnots how S and D ar fixd. Using th proprtis of P j, rwrit th Hamiltonian as H jp H j jp j jp j HP j. 6 Using th rlation 4, w hav P j HP j P j H kin P j P j H int P j P j H I 1P j P j 1 H II P j, whr 1 is an idntity oprator and H I and H II ar H I ks v v kc k c v k j ks c c kc c k ck, j H II U c N kdj c k v ck kdj c k v c k c. Hr, th kintic trm and th intraction trm dcoupl, th dcoupling proprty of th Hamiltonian. Now w map th systm to an xactly solvabl quantum spin systm s Fig. 1. Hr th SU2 algbra hiddn in spinlss frmions in a two-band systm plays a crucial rol. 6 8 Lt us dfin th spin oprators Ŝ k Ŝ x k iŝ y k, Ŝ k Ŝ x k iŝ y z k and Ŝ k by P j c (v) k c (c) k P j, P j (c) c (v) k c k P j and P j ( 1 (c) 2c (v) k c k c (v) k c (c) k )P j, rspctivly, th total spin oprators Ŝ by kd j Ŝ k (x,y,z) and (Ŝ) 2 by (Ŝ x ) 2 (Ŝ y ) 2 (Ŝ z ) 2. Thn w hav P j 1 H II P j P j 1 H spin P j, whr k and k tak valus in D j. Thus Ŝ x k,ŝ y z k,and Ŝ k (kd j ) ar th componnts of a s 1 2 quantum spin. Now w can idntify k with a sit on which a s 1 2 quantum spin is dfind. In th languag of spin, if th pair k is mpty, th spin on th sit k is up and if th pair k is doubl occupid, th spin on th sit k is down. Not that, sinc k taks valus in D j, all th pairs w now considr ar ithr mpty or doubl occupid. Now diagonaliz H spin which can b idntifid with th Hamiltonian of th quantum spin systm (s 2. 1 Dfin S,S z by an ignstat of (Ŝ) 2 and Ŝ z which satisfis (Ŝ) 2 S,S z S(S1)S,S z and ŜzS,Sz SzS,Sz. Th nrgy is spcifid by S and S z s (10). Thr is, howvr, nontrivial dgnracy which is givn by 2S max!2s1 S max S!S max S1!, whr is th numbr of lmnts in D j and S max is /2. This dgnracy is crucial for th thrmodynamic proprtis. Lt us considr th stat v c pn c p1 v vc q1 c c c qn c 0S,S z, 13 whr p 1,,p N v,q 1,,q N cs j. From th dcoupling proprty (7) and th mapping to th quantum spin systm (9), it can bn sn that this is an ignvctor of H with an ignvalu N v l1 N c v p l m1 c q m U r 2 N 1r N pair N 2 D j N pair 2 1, 14 pair whr N and r(0rn Dj /2,r:intgr) ar dfind by 2S z and /2S,rspctivly. Th total numbr of th frmions is givn by N v N c N pair. Varying th indx j, H is diagonalizd compltly. Now lt us considr th physical proprtis of th systm in th thrmodynamic limit (N ). For simplicity, w considr th half-filld cas, namly, N v N c N pair N. Whn th intraction is absnt, th systm is insulating.

4 53 EXACT RESULTS ON SUPERCONDUCTIVITY DUE TO FIG. 2. Th on-dimnsional two-band modl, whr (c) (k)2tcosk2tg/2 and (v) (k)2tcosk2tg/2. Th zro-tmpratur phas diagram. Lt us first considr th zro-tmpratur phas diagrams. Th ground stat was obtaind by minimizing th nrgy (14). Th comptition btwn th kintic trm and th intraction trm givs a rich phas diagram. W prsnt th phas diagrams of two cass: th on-dimnsional two-band modl, as shown in Fig. 2, and a systm with a constant dnsity of stats, which rsmbls that of th twodimnsional systms Fig. 3. W find thr typs of diffrnt phass as shown in Figs. 2 and 3. All th phass ar sparatd by th first-ordr phas transitions. Th phass ar charactrizd by ( th amplitud of th off-diagonal long-rang ordr for th s-wav suprconductivity 11 which is dfind by 1 dx N 2 dy c x v x v y c y. 15 Hr (c) (x) 1/N k ikx (c) c k and (v) (x) 1/N k ikx c (v) k. Whn taks finit valu, th pairing of lctrons occurs as in th BCS suprconductivity. Th contnts of th thr phass ar as follows: Phas 1: 0.5, which is th uppr bound for. Itis suprconducting. 11,12 Phas 2: It is also suprconducting. Phas 3: 0. Th ground stat is a band insulator as th nonintracting cas. Not that a sufficintly larg attraction is ndd to produc th suprconductivity, which is totally diffrnt from th BCS suprconductivity. Now lt us discuss th Missnr ffct, namly, stimat th suprfluid dnsity N s. N s is dfind by mc/ 2 j/a, whr m dnots th ffctiv mass, j is th currnt dnsity, and A is th vctor potntial. Sinc w hav diagonalizd th Hamiltonian xplicitly, it is straightforward to obtain N s by th us of th Kubo formula. In th insulating phas w obtain N s /N0 and thr is no Missnr ffct. This is th dirct consqunc of th ffctiv mass thorm. 13 In th suprconducting phas w can also obtain N s /N1O(1/U) in th larg U limit, which mans th Missnr ffct. Nxt w considr th thrmodynamic proprtis. For simplicity, w considr a systm with flat bands ( (v) (c).) Whn th two bands dgnrat, namly 0, th thrmodynamic proprtis ar invstigatd by Thoulss. 8 Th grand partition function is Z grand N v,n c 0N v N c N r;intgar 0r /2 whr C and E is dfind by 2N Dj 2r C xp E, 16 pair N 2r FIG. 3. Th modl which has a constant dnsity of stats. Th dnsity of stats and th zrotmpratur phas diagram.

5 8564 Y. MORITA, Y. HATSUGAI, AND M. KOHMOTO 53 FIG. 4. Th tmpratur dpndnc of th ordr paramtr and th hat capacity whn 0.3 and U2. and N!! 2r1 C!N v!n c! r! r1! EN v N c N pair N v N c U N r 2 1r N pair N 2 D j N pair 2 1. In th thrmodynamic limit (N ) w us th saddl-point mthod. Th chmical potntial is st 0 and th systm is half-filling. A dirct calculation lads to analytic forms of th thrmodynamic quantitis. For xampl, (T) is givn by y 2 1 T0.5 y 2 ay1, 17 whr y is th largst root of ln x 1 2UT 1 (x1)(x1)(x 2 ax1) 1 and a /T /T. As shown in Fig. 4, th scond-ordr phas transition occurs at a finit tmpratur. Th critical tmpratur T c is proportional to U whn U. Th ntropy S(T) pr unit cll is givn by ST/Nlny 2 ay1t 1 /T /T y y 2 ay1 0.5UT 1 y2yay2 1 y 2 ay Th hat capacity T(S/T) V pr unit cll is shown in Fig. 4. In th suprconducting phas it bhavs as Axp(B/T) at a sufficintly low tmpratur, whr A is a constant and BU/22 is th xcitation gap. In th hightmpratur phas it is a dcrasing function of T, sinc th band widths ar finit. W find that (T0)/T c and C/C n ar not univrsal in contrast to th BCS-typ suprconductivity, whr C is th jump of th hat capacity at TT c and C n is th hat capacity at TT c 0. A mor dtaild study of th thrmodynamic proprtis will b prsntd lswhr. Th half-filld cas considrd hr sms to b most prospctiv to b ralizd. Th crucial point is th origin of th attractiv intraction. On of th possibl candidats is th xciton mchanism proposd in Rfs. 3 and 5. Thr th attraction is nvisagd as arising from a polarizabl mdium sandwichd btwn th two chains, whr th ffctiv intraction btwn lctrons in diffrnt chains bcoms statically attractiv. This is bcaus lctrons shar positiv charg inducd in th mdium. Thy hav confirmd that thr ar cass in which this attractiv intraction is strongr than th dirct Coulomb rpulsion btwn lctrons in diffrnt chains. In Rf. 9, anothr xampl of attraction was proposd in th two-band rpulsiv Hubbard modl. Th lctrons in on band xprinc attractiv intraction mdiatd by an accompanying Mott-insulator band. Thn, if w considr th filling othr than half-filling, th xcitonlctron intraction which lads to th attraction is rducd considrably by scrning. Thus th half-filling cas is bst for our purpos. Without th scrning, a strong attraction is rathr asily achivd. 10 In summary, a rcnt proposal by Kohmoto and Takada of th pairing stat btwn a conduction lctron and a valnc lctron was invstigatd through a family of xactly solvabl modls. W obtaind all th ignvalus and th ignvctors xplicitly. Th zro-tmpratur phas diagrams wr obtaind. Th suprconducting instability without a Frmi surfac which was proposd by Kohmoto and Takada wr confirmd. It was also provd that a sufficintly larg attraction btwn stats in th two bands is ndd to produc suprconductivity. Th thrmodynamic proprtis wr also dicussd. Th proprtis ar quit diffrnt from thos of th BCS-typ suprconductor. Th modls w considr may b ralizd in spcially synthsizd doubl-chain organic matrials. Although w hav prsntd th rsults for th cass whr fully analytical tratmnts ar possibl, th rsults for th mor gnral cass ar not diffrnt from th prsnt cass in ssntial ways. Thy will b prsntd lswhr.

6 53 EXACT RESULTS ON SUPERCONDUCTIVITY DUE TO * Elctronic addrss: morita@kodama.issp.u-tokyo.ac.jp Elctronic addrss: hatsugai@tansi.cc.u-tokyo.ac.jp 1 M. Kohmoto and Y. Takada, J. Phys. Soc. Jpn. 59, J. Bardn, L. N. Coopr, and J. R. Schriffr, Phys. Rv. 108, Y. Takada and M. Kohmoto, Phys. Rv. B 41, K. Kuroki, H. Aoki, and Y. Takada, J. Phys. Soc. Jpn. 61, W. A. Littl, Phys. Rv. 134, A P. W. Andrson, Phys. Rv. 112, Y. Wada, F. Takano, and N. Fukuda, Prog. Thor. Phys. 19, D. J. Thoulss, Phys. Rv. 117, H. Aoki and K. Kuroki, Phys. Rv. B 42, D. Davis, H. Gutfrund, and W. A. Littl, Phys. Rv. B 13, C. N. Yang, Rv. Mod. Phys. 34, G. Swll, J. Stat. Phys. 61, S, for xampl, N. W. Ashcroft and N. D. Mrmin, Solid Stat Physics Saundrs Collg, Philadlphia, PA, 1976, p. 765.