i v L_ I E M C E IC/88/230
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
THE EFFECT OF RVB FLUX CONFIGURATION ON THE SUPERCONDUCTING STATE
G. Baskaran
INTERNATIONAL E. Tosatti ATOMIC ENERGY and AGENCY YuLu
UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 198 6 MIRAM ARE-TRIESTE
IC/88/230 ABSTRACT
International Atomic Energy Agency and The RVB state is a singlet quantum liquid which can support periodic or fluid-like quantum circulation around the elementary plaquette in two dimensions In the 3 +id phase United Nations Educational Scientific and Cultural Organization there is a half flux quanta of circulation in every elementary plaquette. We show that the INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS holons in the presence of this lattice of circulation have a two-valley structure, as opposed to "s" or "d" states, which have a single valley. Using an effective holon Hamittonian, we study the mean-field properties of the two-valley holon condensate, and compare its energy with the single valley case. A transition from the " G. Baskaran ** International Centre for Theoretical Physics, Trieste, Italy, E. Tosatti International Centre for Theoretical Physics, Trieste, Italy and International School for Advanced Studies, Trieste, Italy and Yu Lu "* International Centre for Theoretical Physics, Trieste, Italy. MIRAMARE - TRIESTE August 1988 * To be published in the Proceedings of the Adriatico Research Conference, "Towards the Theoretical Understanding of the High Temperature Superconductivity", edited by S. Lundqvist, E. Tosatti, M. Tosi and Yu Lu, WSPC, Singapore. ** Permanent address: Institute of Mathematical Sciences, Madras 600113, India. ** Permanent address: Institute of Theoretical Physics Academia Sinica, Beijing, People's Republic of China. i. The deep and fasunaliiig siau.ii.f.' ul !lsc Resonating Valence Bond (RVB) state — the electron operators cig, ctj remain invariant and hence the Hamiltonian. Since we are recently revived by Anderson [ 1 >2j as a candidate model for high temperature superconductors interested in the limit VIt » 1, the double occupancy occurs only as a virtual fluctuation and can •-- is attracting increasing attention. In the RVB model, the elementary excitations differ be eliminated to leading order in UU by a canonical transformation giving the effective completely from those of a regular Fermi liquid. They are neutral, spin 1/2 "xcitati ..s Hamiltonian for less than half-filled band (spinons), and charged, spinless excitations (holons) [3,4]. The RVB state itself couid be seen as a condensate of holons, strongly interacting with spinons. Although qualitative •^^ io jo i ' ' ' £-4 i iu,t us bristly suuiitiadzc ihe titet'-vt holon theory [ 10J. flic problem we want to study is the Hubbard model writien as s. +h.c. J Integration over the fields Ai °(r ) enforces the local particle number conservation (6). A'Sicit, a - ± 1, e. ,dt Lifid ilt} are btisoii and feirmon cptrators, respectively, with ihe We combine the holon hopping and exchange terms as follows to get 1 COIlStl ill [| [ ^C t t X"" t t X" t ' t -/ > {e.es J +h.c.)-J > s s s s =-J 7 (s s +— e e )* *-t • J ja io •*-( io jo j 2_ 2 t ' + > ' V t t Zt (9) '•'•:-•••• • : • .1 •••''; •.»"••:!.•» nv ••', tl'-. 'ii'-.:,ut:inj of itit -.I !\v Dublin faiiJii1^ !i;»ltn!i£ HI-, ti ; a ..!i ifneractiitn icrni can be linearized using the Hubbard-Stratonovich transformation to gel Based on the above, Marston [11] has introduced a Chern-Simons type of term in the action in terms of % for the 2+1 dimensional problem which restricts the dynamically generated * c- fluxes of the % fle'd to integer or half-integer value. We take a different approach and replace the X field by Ay V / *» Ky" ^'y^'1 (lO) Now we also have to perform the functional integral with respect to the complex collective field where p.. is the radial fluctuation of the gauge field and cr is an Ising variable with values ± 1. In this approximation sgn (XijXjtXiJCij) = ± 1 (17) (11) and the Marston identity is approximately satisfied. In other words we replace the (/(I) group Thus the field A can be thought as a dynamically generated U{\) vector potential. This field element exp(i'0..) by a Z2 group element a. (=± 1). represents the singlet fluctuations in the system. If we formally integrate over the the spinon and holon fields, we get an effective action Physically the field X, represents the quantum fluctuation of the singlet bonds, whereas which is a function of the fields %. (r) and ^^(T) only. Minimization of this functional gives us the product XiX-^CiJCti represents the quantum circulation associated with this singlet the mean field or the saddle point equations. fluctuation. In terms of theses variables the extended s (or d ) and s +id mean field solutions Marston [11] has made an important observation that correspond to 4 2 Sgn (CT..ff..fft,O\. y= t \*oj and =-1 <19> where respectively. In the next section we discuss the holon condensation in the s +id and s phases .;.• and their energies in the mean field approximation. We will call the above the Marston identity. If we were to replace the r^ operator by the II. Holon Condensation corresponding conjugate variable (as an approximation) The Hamiltonian of the system is suggesting that + the rest of the spinon part where B = 0 or K . We notice that the spinons and the holons interact with the same gauge field. Hence, their kinetic This means that the quantum fluctuations in the gauge field tend to concentrate around the energies have the same form, but their band widths have a ratio of Jit. In our mean field analysis flux value of integer or half-integer quanta. In other words the function we choose a particular mean field for;j, concentrate on holons alone and do not attempt to study P (z) s Tr exp(-/) H) The remarkable feature of spectrum (24) is to have two minimal valleys, centered around the inequivalent degenerate /t-points A= (0,1/4) 2jtfa, and B= (0, -1/4 )2>tfa. Ai The inter-siie holon-holon repulsion U, as discussed above, is generated by the coupling between both places, the holon kinetic energy attains its minimal value -r 2 Tz, and this leads to the holon and spinon kinetic energies. In principle, a hard-core repulsion preventing unphysical necessity of a two-valley holon condensation, which is outlined below. holon double occupancy should also be present, but that is omitted here, as it has negUgible effect Omitting the chemical potential term, the interaction part of the Hamiltonian (21) is for S=nIN « 1, and we shall return to this point later on. (25) As discussed above, the holon Hamiltonian can describe different RVB states , +f .*' with depending upon the choice of the gauge field %ji • or equivalently upon the choice of A. The V {k,k-.q,q- ) =V -t; )] outcome depends solely on the elementary square lattice plaquette product (14). In particular, 0 = 0 corresponds both to the RVB s state (1,1,1,1 ), and d state (1,-1,1,-1). They (26) have the same holon energy, though they may differ in their background 5=0 energy, which is Restricting k, k', q and q', to the close neighborhood of valleys A and B , it is easy probably lower for the d state [7] . On the other hand, 6 = x corresponds to the to check that out of 16 terms generated by (25), only four survive, namely AAAA, BBBB, "s+id", or chiral, or flux, phase which is realized, e.g., by the (complex) choice (1,1,1,1) AABB. and BBAA. Four other terms, ABAB. ABBA, BAAB, and BABA, are in principle exp(/» /4 ), or by the (real) choice (1,1,1, -1). It is well known, and it will also be apparent nonzero, but negligibly smalt in comparison .The remainigeigh t terms vanish identically, due to below, that the hoton kinetic energy of the s +id state is not as good as that of the d state. ^-conservation. Thus the s +id holon Hamiltonian becomes 1 S;jice the two states arc related by SU(2) symmetry [13] and therefore degenerate at 5=0, this has led to the (reasonable) belief that s +id state is altogether disfavored for any finite hole density 8 . However, we here show that the joint effect of the holon hopping -/ Xi ^ of the (27) holon-holon intersite repulsion U can reverse this conclusion for S and U large enough. Our approximate treatment of Hamiltonian (21) for the s+id case proceeds as follows. We first transform the one-particle operators to it-space )- +T )) \i,b (22) where sublattices a and b are shown for the real gauge (1,1,1,-1) on Fig. 1. There, a double bond means hopping energy -t, and a single bond, hopping energy /. There are two holon branches, or bands, corresponding to two orthogonal choices of uk and vt in (22). For a purposes of holon condensation, we consider only the lower branch, obtained by vt =1, sin * (23) Fig. 1 The enlarged unit cell and (he reduced Brillouin zone with two minimal vaileys ck = where k -sums are restricted to small neighborhoods of A and B, and £ t' = 2 Jl - ek . We (34) should stress that in (27) and what follows the wave vector k is measured relative to valley A orB. yielding By straightforward generalization of Bogoiiubov's standard method, we introduce a and (35) b condensates '•» (28) with and retain terms up to bilinear, yielding (36) For Jt—K) we have the following limiting forms >(*a>*>) (29) (37) For the asymmetric condensate na = n, nb= 0, we have instead where (38) V (Jt)=V_(i) = 2E/ e + with JJ^ = COSifcJ+ COSjt . (30) 2- 165 (39) . With the identity Again for Jfc-»0, we have (31) lil, £k = 2 tk . (40) and the gauge transformation Since the dispersion (40) is stiffer than (37), it will lead to a larger zero-point energy, anu it, -ie> a (32) therefore the symmetrical condensate is favored. We now wish to compare our s+id state energy with that of the single valley condensate we can readily diagonalize the bilinear form (29). On symmetry grounds, we expect that both of the d (or s ) RVB state. The latter is obtained in a totally parallel fashion , to be no = nb=n 12, and na = n, nb = 0 (or vice versa) should be total energy extrema, one a ^^^ t 1 t A T \ minimum, the other a maximum. By diagonalizing H in both cases, we will find where the k minimum lies. For n = n = n 12, we introduce a b with a fl + 2 k = < * V / V * • p.=(at-b.)jj2, (33) followed by a standard Bogoliubov transformation In the limit k-*0 we have ordered pattern of half flux quanta. One can easily envisage other mean field solutions where the fll*l. (43) half flux quanta are arranged periodically with larger periodicity. Introduction of holes may Comparison of the s+id energy (35) with the d energy (41) (assuming the s state to have induce interaction among the fluxes leading to a rich floating or incommensurate type behavior of already higher energy at 8 = 0), can now be done. It is important to note the shift AE = It (2 - the ftuxoids. The density of fluxoids is not a conserved quantity. This idea is similar to that of $2) of the relative energy scales, and the different (/-dependent dispersions (37) against (43). W i Laughlin [15], Mele[16], Anderson[17] and Wiegmann[18]. Our eventual aim is to study the melting and the change in the nature of the flux lattice as more and more holes are introduced. reach the following conclusions: However, we must defer this to future work. In this paper we have studied the holon condensation in the mean field approximation and 1. The d state will prevail for very small 5, because it gains a kinetic energy AE per in the dilute limit. We treated the nearest neighbor repulsion alone and did not consider the holon over the s+id state; on-site hard-core repulsion. But in the spirit of Fermi's pseudopotential the on-site hard core repulsion is inessential to the first approximation. This is because the large nearest neighbor 2. The interaction-dominated zero-point energy is clearly lower for the s+id state (two repulsion U strongly screens the central site where another hole has negligible chance to reach. excitation branches, both lower than the single branch of the d state). This must lead to The inter-site h-h repulsion U replaces in the present approach what Anderson and a transition from d to s+id for either increasing doping 6, or for increasing Hubbard coworkers[19] have termed a "hard core repulsion in the momentum space". In our picture, the U\ effect of this repulsion would correspond to a depletion away from valley minima caused by scattering by the pseudopotential, possibly le, ding to a larger depletion of the zero momentum component of the distribution function. 3. A tendency towards a d —» s+id transition may also be anticipated as a function of temperature, in the appropriate doping range, the softness of the s+id state excitations In connection with the existing theoretical approaches, it is interesting to note that our two-valley s+id condensate might provide a more fundamental justification for the RVB implying larger entropy. two-sublattice structure ("red" and "black" sites) postulateded by Kivelson et al.[3,20]. 1'lta.iy the present study is preliminary and incomplete. Important issues remain to be Recent analogies with fully relativistic field theory problems [ 21, 22] can also be pursued investigated, such as a) possible coexistence of a boson pair order parameter with tf - to implv two distinct holon species, somewhat similar to the two helicity states for the weak single-boson condensate, also in connection with 2e flux quantization; b) possible current \j/ty0( l±;s)^v in the unified electroweak theory [23]. mechanisms leading to phasing of the two condensates A and B, and/or to a gap in the excitation The -1 signature of the plaquette is the crucial ingredient in the s+id holon problem. It spectrum, including additional Coulomb interaction. seems likely that the counterpart of this " Berry phase " in the spinon spectrum should have to do with the famous topological term in the effective spin-wave Lagrangian [21,22,24,25]. III. Discussion Finally, two experimental facts recentlyemerged , namely (i) the presence of a remarkable plateau (around- 60 K ) of Tc versus oxygen deficiency in the 123compounds [26]; (ii) r a 2 Now we return to the discussion of the physical meaning of the gauge field X-t- F° a nonzero T specific heat term, most notably in the new Bi-based superconductors[27]; may square lattice Heisenberg antiferromagnet (AFM) with s =1/2 there is overwhelming evidence be interpreted as indirect evidence pointing to the appearance of the s+id state. Moreover, a that there is an AFM long range order [14]. However, introduction of few pereents of holes two-valley condensate might atso lead to brund new phenomena such as internal Joseph son should enhance greatly the singlet fluctuation and may destroy the long range order converting effect, intervalley phase modulation modes, etc. the system into an RVB liquid. It has been thought that the RVB liquid is a structureless singlet fluid. Our analysis, Acknowledgments however, suggests that the RVB fluid still has a structure which is related to the periodic or possibly random disposition of the circulation <£, J; p^ ; >- +id phase contains an One of the authors (G.B.) would like to thank Professor Abdus SaJam, the Inter- national Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. 1 0 1 1 kht-hil'.i.i hS i. t V,- i .. In .r:. Sciiv.ce 235, ' !96 (1J87) .: i.hi. ii.iii: ..'. i\i\it, J , , W. yu*Ji.> tun, /jn niij ii^cmit.- ivl'i/'. LO !.«; ^jt>iiiin.u by ;N ^i r.i.. ^ri.inj;, C. lii'iiii.'1'. M. Rn.:. and ti Stiibj. Supeitonducuvliy Jt-icaco iind '.(•.i.-.tiiv.*lug>, I, N>J. i i iVaiii; Ci.KoUiarA J Liu, Phys. Rev. B, lo be published. M i.lJ ttegcr siui A F: Yciimg, ^hys. Rev. B57, 5978 U9M8), S ' ii)!!j;. i'. iJi^-.:. >i. .: Clidfldia, ajid P.W. Anderson, Phys Rev, b. tu U> pubhsiitd. : ! •' .; I 1.,,.:i.ii.i [J(.i,ol ;.vi^;M»iur.173, Orai'..)-;iL:. n ; v,-J;;;i. hiiu; 0 ! J. 1^88 T . ^e!e Nov-"1 .;/!'.i^"?*i'ii'VI. Grafvavallj.j, Swrtj.-r, Jur>e ! / ? '/••• , .i.itf •/•;' ii i. ;.! i i. -iiiilfii , •'. \ • i: ^'ici-z^arii: Ni^.Vi Syr.ipoiiiiit. 73. OijftavaHr.il, L\ r;, W.. V-.-.-j A••••*•. ;-.I.I;II i ii.-.:. i,eil./» S, M.~i ~. : J1. , 2.? !'ti V>i ii.^'iui!.1! !'*,>.: J'..1-'. iiM. 6I>, K2H!yaK); I iJiyai: ,i!.». vm. K. V A. P !,->•)», t'i;.. , • .,i. 12/^., 112 (!988). l\ <• j.iil liiCii, j ••! .Hi- .-. I1."! -i.lltiCiiUUIl. i!.d 10 U. pjtiil;. . ': 24. Z, Zou, l^nu^ciOii pnvpr.ii! !988, 25 K Wu !. Yu, and CM Zhu, Mod. Phys. Lett, B, lo lie JJUWished, 1988. ' "ry«i?i: Stampato in proprio nella tipografia del Centro Internazionale di Fisica Teoricafermion with single-particle energy variables sia, where, however, some self-consistency was included, is that of Fukuyama[12]. Thus our working holon Hamiltonian is (24)