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The low temperature properties of a short-ranged resonating-valence-bond superconductor
Physica C 153-155 (1988) 531-537 North-Holland, Amsterdam
THE LOW TEMPERATUREPROPERTIES OF A SHORT-RANGEDRESONATING-VALENCE-BONDSUPERCONDUCTOR S. KIVELSON and D.S. Rokhsar* Department of Physics, State U n i v e r s i t y of New York at Stony Brook, Stony Brook, NY 11794 *IBM, Thomas J. Watson Research Center, Yorktown Heights, NY 10598 I t is shown that a RVB state is the ground state of a class of f r u s t r a t e d Heisenberg models. An approximate version of t h i s model, the hard core quantum dimer gas, is solved exactly and the nature of the RVB superconducting state is discussed. I.
INTRODUCTION I t is by now f a i r l y clear that in order to understand the properties of the copper oxide superconductors, we must f i r s t understand the properties of the two-dimensional electron-gas with strong electron-electron i n t e r a c t i o n s . In order to do t h i s , i t is useful to study simple models, such as the Hubbard model and others discussed below, so as to determine the possible classes of ground states t h i s system can have. In t h i s note, we summarize the r e s u l t s of an extensive study we have carried out ( I - 5 ) on a class of highly f r u s t r a t e d models which have a novel sort of ground state which we have ident i f i e d as the short-ranged resonating-valencebond-state (SR-RVB). In p a r t i c u l a r , we show that f o r a n o n - h a l f - f i l l e d band, the ground state of these models is superconducting. At present, the experimental s i t u a t i o n in the cuprate superconductors is too murky to allow detailed comparison with experiment; few experimental " f a c t s " , beyond the s t r u c t u r e , have been unambiguously established. Thus, while there are features of the experiments which make us hopeful that the present results are relevant to these materials, we do not consider i t useful at present to make a detailed comparison between theory and experiment. This is e s p e c i a l l y so since a l l our results pertain to the very low temperature properties which are e s p e c i a l l y s e n s i t i v e to effects of sample imperfection. With t h i s in mind, we have focussed on obtaining as complete a solution of our model as possible so as to i d e n t i f y the q u a l i t a t i v e features which can be compared to experiment, and in p a r t i c u l a r those features which lead to high temperature superconductivity. The plan of t h i s paper is as f o l l o w s : In Section I I we describe the i n t u i t i v e picture which underlies the RVB analysis, and in p a r t i c u l a r we discuss the r e l a t i o n between a SR-RVB state, which was o r i g i n a l l y studied in the cont e x t of the spin I / 2 Heisenberg antiferromagnet on a t r i a n g u l a r l a t t i c e by Anderson and Fazekas (6), and the "Princeton" RVB state, which has been studied by Anderson and 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
coworkers (7) in the context of high temperature superconductivity. Section I I I is a s l i g h t diversion in which results (2) on an analogous problem f o r the one-dimensional electron gas are summarized. In Section IV, the properties (3) of a completely f r u s t r a t e d Heisenberg model, o r i g i n a l l y considered by Klein (9), are discussed. In Section IV we(4) elucidate the properties of a s t i l l simpler model, the quantum hard-core dimer gas, which is the low energy part of the model discussed in Section IV. F i n a l l y , in Section VI we summarize our results and make some general observations concerning t h e i r r e l a t i o n to experiment. II.
INTUITIVE PICTURE A valence-bond state is one in which the electron on each s i t e is s i n g l e t paired with the electron on another s i t e as shown in Fig. I ; thus there is a valence bond connecting each occupied s i t e to exactly one other occupied s i t e . The valence bond state is the product state formed of these s i n g l e t s . An RVB is a coherent l i n e a r superposition of valence bond states. Such a state w i l l generally have r e l a t i v e l y low v a r i a t i o n a l energy both because each valence bond state minimizes the part of the energy i n v o l v i n g the pair i n t e r a c t i o n between the two bonded s i t e s , and because i t gains resonance energy from the o f f diagonal matrix elements of the Hamiltonian. In a real sense, which w i l l be made clear below, the valence-bonds can be thought of as preformed, real-space Cooper-pairs. I t is important, from the f i r s t , to d i s t i n guish between two types of RVB state that have been widely discussed: I ) The short-ranged RVB (SR-RVB) in which the only states which have substantial amplitude are states involving short valence-bonds, such as the nearestneighbor valence-bonds shown in f i g . la. 2) The "Princeton" RVB (P-RVB), in which valence-bonds of a l l length play an important r o l e . The P-RVB is the more widely studied(7-8) in the context of high-temperature
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S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
superconductivity. The SR-RVB was f i r s t studied in the context of superconductivity (as d i s t i n c t from i t s magnetic o r i g i n s (6)) by us ( I ) and has been f u r t h e r studied by a v a r i e t y of people. ( I - 5 ) , (10-13) Both RVB states have many things in common: they were both invented by Anderson ( 6 - 7 ) , t h e i r e x c i t a tions ( I ) consist of spin I / 2 , charge zero Fermions (spinons) and spin O, charge e Bosons (holons), and they are both quantum l i q u i d states with no broken t r a n s l a t i o n a l or rotational symmetry. The fundamental difference between them is that in the SR-RVB there is a gap (4,10,12,13) in the s p i n - e x c i t a t i o n spectrum, while the P-RVB is gapless. This difference arises since to make a pair of spinons one must break a valence-bond, which costs a f i n i t e energy in the SR-RVB, but, due to the presence of a r b i t r a r i l y long bonds in the P-RVB, cost v a n i s h i n g l y small energy in t h i s state. There is considerable evidence now (14) that a Neel ordered state, not the SR-RVB, is the ground state of the Heisenberg model on a square lattice. I t is unclear whether the RVB state is the ground state of the Hubbard model away from h a l f - f i l l i n g , but preliminary evidence (15) suggests that i t is not. However, in a l l cases, the SR-RVB has a f a i r l y low v a r i a t i o n a l energy so i t is plausible that the i n t r o d u c t i o n of weak electron-phonon coupling (1,2) or s u f f i c i e n t f r u s t r a t i o n could s t a b i l i z e the SR-RVB. Indeed, in the next section we show that a r b i t r a r i l y weak electron-phonon coupling w i l l do the job in one-dimension, and in Section IV that f r u s t r a t i o n can have t h i s e f f e c t in two dimensions. III.
SUMMARYOF THE RESULTS IN ONE DIMENSION The continuum model of the one-dimensional electron gas with a r b i t r a r y short-ranged i n t e r actions has been studied for years. Because the Fermi surface consists of two points, the only possible i n t e r a c t i o n s at the Fermi surface are forward s c a t t e r i n g , called gg, (which f o r an extended-Hubbard-model is g2 L= (U-2V)/2t n) backward s c a t t e r i n g , gl (gl = (U÷2V)/2tn~, and, near the h a l f - f i l l e d -bana, Umklapp scattering g3 ( [- (U+2V)/2to). Zimanyi, Luther and one g3of us (2) have done two new things with t h i s model: f i r s t , we have r e i n t e r preted old r e s u l t s by comparing them to the res u l t s of a v a r i a t i o n a l analysis based on a l i n e a r superposition of nearest-neighbor valence-bond states, and second, we have generalized the model to include both instantaneous repulsions g~. (e.g. Hubbard i n t e r a c t i o n s ) and retarded i f i t e r actions gj (e.g. mediated by the exchange of phonons). A) Old r e s u l t s : I ) There is a t o t a l separation of the spin and charge degrees of freedom which occurs in the continuum l i m i t . The charge e x c i t a t i o n s are
spinless charge e solitons and the spin e x c i t a tions are neutral spin I / 2 s o l i t o n s . This is exactly analogous to spinons and holons. 2) The s i n g l e t superconducting suscept i b i l i t y vanishes as T ÷ 0 unless there is a gap A. > 0 in the s p i n - e x c i t a t i o n spectrum. Conversely, i f As > 0 and i f the system Is doped so that there is no gap in the chargee x c i t a t i o n spectrum, then the s i n g l e t superconducting s u s c e p t i b i l i t y diverges. Thus, while there are states analagous to both the P-RVB and the SR-RVB, only the state analagous to the SR-RVB e x h i b i t s a superconducting tendency in one-dimension. We note that A. > 0 i f and only i f gl < O. 3) Indeed, so long as gc : (g~-2gl) < O, g3 > O, and gl < O, a l l the results can be understood q u a l i t a t i v e l y in terms of nearestneighbor only valence-bond states: for the h a l f - f i l l e d band the ground-state is a bondorder charge-density-wave and the holons and spinons are simply domain walls between the two degenerate senses of the ground-state dimerizat i o n . Note that gc < 0 and g~ > 0 are the expected values of the i n t e r a c t i o n s for a Hubbardl i k e model, but gl < 0 is not reasonable for Coulomb i n t e r a c t i o n s alone. However, i t is a well known property of the renormalization group scaling equations that i f gl is p o s i t i v e i t scales to zero. Thus, i t is not unreasonable to expect that the addition of even weak. retarded i n t e r a c t i o n s w i l l produce a g~TT < O. B) New results with I g j I < < I g i l We have derived scaling equatlons in the presence of weak a t t r a c t i v e retarded and strong repulsive Coulomb i n t e r a c t i o n s . The important r e s u l t is that in the scaling equations for g l , cross terms occur of the form gigk which imply that the rate at which g: scale~ is proportional to the strength of the Jof the Coulomb i n t e r actions. In p a r t i c u l a r , we f i n d that for a small and a t t r a c t i v e bare value of gl = gl ( ° ) , the scaling equations r e s u l t in an extreme magnification of gl so that a f t e r i n t e g r a t i n g out states with energy between E~ and EF, the new value of gl is ,
a~ gl = gl (°)
(E,F/ EF ]
(i)
where A is a number of order I , and g is a suitable average of the Coulomb i n t e r a c t i o n . Since at the same time gl is scaling to zero, the conclusion is that the low energy physics is c o n t r o l l e d by g~ff < O, even i f the bare electron-phonon coupling is very weak. IV.
THE FULLY FRUSTRATEDHEISENBERG MODEL In order to establish that there e x i s t sensible models with a SR-RVB type ground state, J and L Chayes and one of us (3) have
S. Kiuelson and D.S. Rokhsar /Short-ranged resonating-valence-bond superconductor
studied a p a r t i a l l y soluable model of a f r u s t r a t e d spin I / 2 Heisenberg model, the Klein model (9). In general, a spin I / 2 Heisenberg model simply is a model with a spin I / 2 on each s i t e and short-ranged, s p i n - r o t a t i o n a l l y i n v a r i a n t i n t e r a c t i o n between spins on d i f f e r e n t s i t e s . By a f r u s t r a t e d model we mean a model with i n t e r a c t i o n s which f r u s t r a t e any classical ordered spin-state such as the Ferromagnetic or Neel states. While to obtain a soluable model, we must consider a p a r t i c u l a r r a t i o of the various coupling constants, i t is our f e e l i n g that our r e s u l t s pertain to a wide class of models with s u f f i c i e n t f r u s t r a t i o n . Indeed, preliminary numerical r e s u l t s (5) on a Heisenberg model with f i r s t , second, and t h i r d neighbor antiferromagnetic i n t e r a c t i o n s support t h i s contention. The Klein Hamiltonian can most e a s i l y be w r i t t e n in terms of the projection operators P~ onto the highest+spin state of the t o t a l s ~ i n ~ on ~ i t e ~, S~ and i t s nearest neighbors, S~+~
HK = J ~
P~
(2)
where, f o r a square l a t t i c e , each s i t e has 4 nearest neighbors, so P~ is the projection operator onto spin 5/2
and
A This Hamiltonian can also be resolved i n t o a sum of f i r s t , second, and t h i r d neighbor a n t i ferromagnetic i n t e r a c t i o n s plus some short ranged four-spin i n t e r a c t i o n s . I t is manifestly s p i n - r o t a t i o n a l l y - i n v a r i a n t . For the non-halff i l l e d band with s t a t i c holes the K l e i n Hamiltonian can be generalized by omitting the sites occupied by holes from the sum in Eq. (2) and s u i t a b l y modifying the projection operator on neighboring sites to take account of the fact that the highest spin-state is now reduced by I / 2 due to the absence of one neighbor. A few results concerning the ground states of t h i s Hamiltonian f o l l o w almost by inspection. Since H is a sum of p o s i t i v e semi-definite operators, any state which is a n n i h i l a t e d by Hw is a ground state. Such a state must have zer~ energy on each s i t e . The energy on s i t e depends on the spin on s i t e R and i t s nearest neighbors. In any state in which there is a valence-bond connecting any two of these s i t e s ,
533
~R2 is less than i t s maximal value and hence P~ operating on t h i s state is zero. This immediately implies that any nearestneighbor-only valence-bond state is a ground state of HK. What is less obvious, but which nonetheless can be proven (3), is that a l l the nearest-neighbor valence-bond states are l i n e a r l y independent. (They are not orthogonal.) Thus, i f we associate hard-core dimers with nearest-neighbor valence-bonds, then to each dimer covering of the l a t t i c e there corresponds a unique d i s t i n c t groundstate of H.. I t is ~ore d i f f i c u l t to prove that the nearest-neighbor valence-bond states exhaust a l l the ground-states of HK. (Indeed, on a t r i a n g u l a r l a t t i c e they c e r t a i n l y do n o t . ) On a square l a t t i c e a simple argument is s u f f i c i e n t to show that a pure valence-bond state cannot be a ground-state i f i t contains any valence-bonds longer than second neighbor: there are h a l f as many valence-bonds as s i t e s , and each valence-bond can at most cause the energy to be zero on two s i t e s ; a longer bond causes at most one s i t e energy to be zero. I t is even r e l a t i v e l y easy to show f o r free boundary c o n d i t i o n s , that nearest-neighbor valence bond states are a l l the valence-bond ground states. We (3) have proven f o r several l a t t i c e s , i n c l u d i n g the hexagonal l a t t i c e with a r b i t r a r y s t a t i c h o l e s , t h a t the nearestneighbor valence-bond states exhaust the zero energy states, valence bond or otherwise. We are c u r r e n t l y working to extend t h i s proof to the square l a t t i c e . F i n a l l y , we would l i k e to show that there is an energy gap between the ground states of the Klein-Hamiltonian and i t s excited states. This seems i n t u i t i v e l y obvious since only excited state involves a broken bond. We have not, however, proven i t . Some related r e s u l t s which lend f u r t h e r support to the existence of a gap are the proof, due to Klein (9), that in any f i n i t e l i n e a r superposition of nearestneighbor states, the spin-spin c o r r e l a t i o n function is an e x p o n e n t i a l l y f a l l i n g function of distance, and the non-rigorous, but quite convincing demonstration of Komoto and Shapir (I0) that the same c o r r e l a t i o n function is an exponentially f a l l i n g function in the equal amplitude l i n e a r superposition of a l l nearest-neighbor valence-bond states. (see also Ref. 13.) By conventional wisdom, an exponentially f a l l i n g c o r r e l a t i o n function implies a gap A = V~/~, where ~ is the c o r r e l a t i o n length -and Vs is a spinwave velocity. Given the existence of a gap, the ground state of any Hamiltonian which is equal to the Klein Hamiltonian plus a small perturbation can be obtained from degenerate perturbation theory by diagonalizing the perturbation w i t h i n the ground-state manifold of HK. The
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S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
hard core dimer-model considered in the next section could be viewed as a r i s i n g in t h i s fashion. More g e n e r a l l y , our present results establish that there e x i s t s a class of highly f r u s t r a t e d Heisenberg models with a shortranged RVB ground state. THE HARD-CORE QUANTUMDIMER GAS Let us consider quantum hard-core dimers on a square l a t t i c e . The H i l b e r t space is spanned by the orthonermal states {IC > } , where C specifies a dimer c o n f i g u r a t i o n . For s i m p l i c i t y , we study a model dimer Hamiltonian composed only of the most local terms. The Hamiltonian consists of the sum of two pieces: H~. r ' which completely specifies the dy~cs f o r close packed dimers, and Hhole s, which describes additional processes i n v o l v i n g the motions of holes. A) The Close Packed Dimer Gas We f i r s t address the close-packed case: V.
Hdimer :
Z
{ - J [ l ~ - ~ >
plaquettes
(5)
The f i r s t term is a pure dimer k i n e t i c energy which f l i p s pairs of p a r a l l e l nearest neighbor dimers; the second is a repulsion between such nearest neighbor pairs. The coupling constants J and V are to be regarded as phenomenological parameters. For close packed dimers, the H i l b e r t space breaks up i n t o t o p o l o g i c a l l y d i s t i n c t subspaces labeled by winding numbers ~. and ~ , as described in Ref. lb. Any two s~ates i~ the same topological sector can be connected by a series of pairwise dimerf l i p s (J) and no local rearrangement of bonds can connect states in d i f f e r e n t topological sectors. There e x i s t i n d i v i d u a l configurations which comprise t h e i r own t o p o l o g i c a l l y d i s t i n c t subspaces. These "valence-bond c r y s t a l " (VBC) configurations (Fig. 2) contain no p a r a l l e l nearest neighbor dimers, and are zero energy eigenstates of H,. f o r a r b i t r a r y J and V" ,olmer . . . . For V > J > O, Harm=~ is poslzlve semi-definite, so these VBC's a ~ the only ground states. At V = J, each topological sector possesses a unique zero energy ground state I~>, namely the equal amplitude superposition ZCc~]C > of a l l configurations in that sector. I t is easy to v e r i f y that these states are zero energy eigenstates of These states are the unlque ground Hdimer"states in t h e i r respective • topological sectors: a l l off-diagonal matrix elements of the Hamiltonian Hd. r are nonlme p o s i t i v e , so the ground state must be nodeless, i . e . , a state vector with a l l p o s i t i v e amplitudes. These coherent superpositions are
the precise analogs of the SR-RVB ; the resonance energy is simply the dimer k i n e t i c energy. An important feature of the equal amplitude states discussed above is that any dimer c o r r e l a t i o n function can be computed exactly from the results of Fisher and Stephenson (16) f o r the classical dimer problem. This does not, however, imply that the quantum problem is related to a two dimensional classical problem in the sense that i f we perturb the quantum Hamiltonian s l i g h t l y , the ground state c o r r e l a t i o n functions need not be der i v a b l e from any simple classical s t a t i s t i c a l mechanics problem. For V < J, we can use the equal amplitude states as v a r i a t i o n a l wavefunctions, and obtain an upper bound to the ground state energy of E S ~
(J-V) per s i t e .
Since E is iden-
t i c a l l y zero f o r V > J, the energy has a discontinuous f i r s t d e r i v a t i v e at V = J, implying a f i r s t order t r a n s i t i o n from a VBC to a quantum l i q u i d (RVB) state. When V is large and negative, we expect another t r a n s i t i o n (4) to a d i f f e r e n t c r y s t a l l i n e ground state, the "column phase," which has columns of p a r a l l e l dimers. B. The Doped Dimer Gas Upon doping ( i . e . , removing dimers), additional processes contribute to the Hamiltonian:
H = Hdimer - t
+v h Z
Z [I "-> <- "I + h.c.]
[{--><..(]
(6)
The f i r s t sum, performed over t r i p l e s of sites such that the f i r s t and t h i r d are nearest neighbors of the second, is a "hole k i n e t i c energy" which moves a dimer to an adjacent unoccupied p o s i t i o n . Note that a "black hole" remains on the black s u b l a t t i c e and a red hole remains on the red s u b l a t t i c e . The second sum, performed over nearest-neighbor s i t e s , is a "hole-hole r e p u l s i o n , " which can be rewritten (less transparently) as a local i n t e r a c t i o n between dimers. The ground state of the dimer Hamiltonian can be determined exactly when J = V and Vh = - 2 t , for t and J p o s i t i v e . For these parameters, the equal amplitude superposition z I C > of a l l dimer configurations can be shown to be a nodeless eigenstate of H, and hence the unique ground state. Note that away from close-packing, any configuration can be obtained from any other configuration by repeated a p p l i c a t i o n of the Hamiltonian, so that there is only a s i n g l e , t o p o l o g i c a l l y t r i v i a l sector.
S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor The energy of the equal amplitude superposition is -4t[Nr+N h] where Nr (Nh) is the number of red (black~ holes. Thug the comp r e s s i b i l i t y ~2E/~Ni2 vanishes under the special conditions J = V, Vh = -2t. This is precisely analogous to vanishing c o m p r e s s i b i l i t y of a free Bose gas, and implies phonon e x c i t a tions with a k 2 dispersion. Additional repulsive i n t e r a c t i o n s (as we expect in the oxide superconductors) lead to a Bose condensate with l i n e a r l y dispersing Goldstone modes. I f Vh is less than - 2 t , the system phase separates. Like the free Bose gas, the equal amplitude state possesses off-diagonal long range order, which can be demonstrated by computing the c o r r e l a t i o n function as [ R - R'I÷ ~, where d~~ creates a dimer on the l i n k connecting sites R and R + z. This is (4) bounded from below by I / 4 x2(l - x) 2. C. Excitations in the Single-ModeApproximation F i n a l l y , we have used the FeynmanB i j l (17) single-mode-approximation to calculate the low l y i n g e x c i t a t i o n s . Here we summarize the r e s u l t s : I ) For the n o n - h a l f - f i l l e d band there are density-wave e x c i t a t i o n s analogous to the phonons in s u p e r f l u i d He IV. However,÷these ÷ odes have vanishing energy both f o r k near 0 & = (~,~). I f we think of the system as cons i s t i n g of two i n t e r p e n e t r a t i n g i d e n t i c a l Bose condensates corresponding to the red and black holes, then we can i n t e r p r e t these modes as the in-phase and out-of-phase Goldstone modes associated with the two condensates. The i n phase mode produces a long-wave-length chargedensity o s c i l l a t i o n , and so w i l l be driven to the plasma frequency by the long-range Coulomb i n t e r a c t i o n s . However, the out-of-phase mode should remain gapless even in the presence of Coulomb i n t e r a c t i o n s . At f i r s t s i g h t , i t may appear that t h i s r e s u l t is a special feature of the nearest-neighbor only RVB, where the number of red and black holes are separately conserved. I f we included second (and longer) neighbor valence-bonds (long dimers), then red and black holons can i n t e r c o n v e r t , so one might get a Josephson coupling between the two condensates leading to a gap in the out of phase mode. While we have not yet completely ruled t h i s out, we believe i t does not occur in the presence of s u f f i c i e n t l y low concentrat i o n of longer dimers. The reason f o r t h i s b e l i e f is that i f we associate f i c t i t i o u s holons with the two ends of any long dimer which connects two sites on the same s u b l a t t i c e , then the number of red minus black f i c t i t i o u s plus real holons is s t i l l conserved under any local dynamics. Thus, we expect the gapless, o u t - o f phase mode to be robust. 2) For V = J and Vh = - 2 t , i t follows from the i n f i n i t e compressib~Ity that the two "phononl i k e " modes have dispersion ~ ~ t(6k) 2 f o r
~
535
0k = k, and Ik-QI r e s p e c t i v e l y . For Vh > - 2 t , the modes s t i l l have quadratic dispersion f o r l>>6k>>x where x is the holon concentration but asymptotically for small (Ok), the mode has l i n e a r dispersion,
3) For the close-packed case, there are no low-energy density e x c i t a t i o n s . However, there are e x c i t a t i o n s , which we have c~lled "resonons", with vanishing energy for k every~here along the edge of the B r i l l o u i n zone, k = (k,~) and (~,k). I f we consider the dimers to be f i c t i t i o u s dipoles with the p o s i t i v e l y charged end on the black s u b l a t t i c e , and the negative end on the red, these e x c i t a tions can be thought of as o s c i l l a t i o n s of the local dipole density. Theresonansalso p e r s i s t in the doped system, and may have vanishing energy f o r ~ near ~. VI.
CONCLUSIONS: SUMMARYAND RELATION TO EXPERIMENT We have argued on plausible physical grounds that systems with s u f f i c i e n t f r u s t r a t i o n , or with strong enough coupling to high frequency phonons, and e s p e c i a l l y when the e l e c t r o n i c band is nearly, but not quite h a l f f i l l e d , may have a SR-RVB ground state. To support these h e u r i s t i c arguments, we have shown with reasonable c e r t a i n t y that there e x i s t models which, while not r e a l i s t i c models of cuprate superconductors, are not a l l that f a r from such models and which have SR-RVB ground states. By studying these simple models, we have obtained a more complete characterization of the low energy properties of the SR-RVB state. Since i t is clear that the undoped parent compounds of the cuprate superconductors do not have SR-RVB ground states, we w i l l focus in t h i s section on summarizing the r e s u l t s which pertain to the nature of the SR-RVB superconductor (See also Ref. 18). The superconducting order parameter of a SR-RVB state involves the in-phase motion of the red and black holons, and so leads to f l u x quantization in u n i t s of hc/2e; in t h i s respect i t is i n d i s t i n g u i s h a b l e from a conventional superconductor. There is the i n t r i g u i n g p o s s i b i l i t y , which we are c u r r e n t l y t r y i n g to v e r i f y , that there are other charge neutral components of the order parameter associated with the out-of-phase motion of the red and black holons. I f t h i s can be established, i t could lead to macroscopically observable differences between the cuprate superconductors and conventional superconductors. We have analyzed the elementary e x c i t a t i o n s r e l a t i v e to the SR-RVB ground state. I ) At the highest energies we have analyzed (18), there are charged q u a s i - p a r t i c l e e x c i t a t i o n s ; these are p o s i t i v e l y and negatively charged s o l i t o n p a i r s , or in other words holon
536
S. Kiuelson and D.S. Rokhsar / Short-ranged resonating-ualence-bond superconductor
a n t i - h o l o n pairs. As with any theory based on the large U Hubbard model, the energy to create such e x c i t a t i o n s is AE = U -8t + ~ ( t 2 / U ) . Photo-production of holon a n t i - h o l o n - p a i r s may account f o r the peak in the optical absorption observed at ~ ~ 2 eV. 2) The lowest energy q u a s i - p a r t i c l e e x c i t a t i o n s consist of spinonanti-spinon pairs. I t is in t h i s sense that the SR-RVB can be thought of as a BCS-like condensate of Cooper pairs of spinons; when you break one apart you obtain a pair of spinons. That the spinons are neutral is the central feature of the model that is responsible f o r the p o s s i b i l i t y of high temperature superc o n d u c t i v i t y and the concomitant shortnsuperconducting c o r r e l a t i o n length ~o ~ 15 A; because the spinons are neutral there is no Coulomb cost to forming such small Cooper pairs. And because the holons are not paired, the charge remains as spread-out as possible in the superconducting state! Moreover, the gap in the spinon-spectrum is analogous to the superconducting gap in a conventional superconductor. As a r e s u l t , the optical absorption should not be of the Bardeen-Mattis form; there is zero o s c i l l a t o r strength f o r d i r e c t photo-generation of spinon-pairs. The spinon gap should be d i r e c t l y observable in experiments which do not have dipole selection rules such as NMR I / T I , t u n n e l i n g , and Raman scattering. The spifions should be observable (19) in optical absorption only through i n d i r e c t , Holsteinl i k e processes. 3) The remaining e x c i t a t i o n s we have characterized are c o l l e c t i v e Bosonic e x c i t a t i o n s . As in a conventional superconductor, there is a long-wave-length charge density o s c i l l a t i o n which is driven to the plasma frequency by the long-range Coulomb forces. The e f f e c t i v e mass of the holons can be deduced from the putative dopant concentrat i o n and the magnitude of the London penetrat i o n depth to be m* ~ 5-10 mc. In a d d i t i o n , there is the out-of-phase motion of the red and black holons, which we have shown is gapless in the nearest-neighbor-bands-only RVB state, and which we have argued p l a u s i b l e , but nonr i g o r o u s l y , remains gapless in any SR-RVB state. This mode does not couple strongly to any convenient probe although i t may lead to NQR r e l a x a t i o n at low temperatures, and must cert a i n l y couple to sound. I t couples weakly to neutrons and proably could in p r i n c i p l e be seen in i n e l a s t i c x-ray or electron scattering. The observation of t h i s mode would be dramatic evidence of a SR-RVB state. F i n a l l y , there are the resonons which are gapless in the i n s u l a t ing state, and which should remain low energy e x c i t a t i o n s , perhaps even gapless at ~ = (~,~) in the superconducting state. Of course, a l l these additional states should produce an anomalous c o n t r i b u t i o n to the low temperature s p e c i f i c heat. While such anomalies have been widely observed in the cuprate superconductors,
i t is not yet clear whether they are i n t r i n s i c . F i n a l l y , we have said nothing here about the t r a n s i t i o n to the superconducting state; in f a c t we have l i t t l e to say. We speculate that above Tr , there is no gap in the spino~ spectrum, ann the system may be well described in terms of a P-RVB state. I f t h i s is so, then at T~, the spinons pair and form a BCS-IIke condensate, whlch opens a gap in the spinon spectrum and permits the holons to Bose condense into a s u p e r f l u i d state. However, considerable f u r t h e r work is necessary to test whether t h i s scenario is v a l i d f o r any model system, much less f o r any real superconductor. ACKNOWLEDGEMENTS SK was funded in part by NSF grant # DMR-83-18051. REFERENCES (1) a) S.A. Kivelson, D.S. Rokhsar, and J.P. Sethna, Phys. Rev. B35, 8865 (1987); b) Euro. Phys. Lett. (submitted). (2) G.T. Zimanyi, S.A. Kivelson, and A. Luther, Phys. Rev. L e t t . (submitted). (3) J. Chayes, L. Chayes, and S.A. Kivelson, (work in progress). (4) S.A. Kivelson and D.S. Rokhsar, Phys. Rev. L e t t . (submitted). (5) S. Sondhi, F. Figuerido, E. Ramos, A. Karlhede, S.A. Kivelson, and M. Rocek, (work in progress). (6) P.W. Anderson, Mat. Res. Bul. 8, 153 (1973). P. Fazekas and P.W. Anderson, P h i l . Mag. 3(], 432 (1974). (7) P.W. Anderson, Science 235, 1196 (1987), G. Baskaran, Z. Zhou, and P.W. Anderson, Solid State Commun. 63, 973 (1987). (9) D.J. K l e i n , J. Phys. A 15, 661 (1982). (I0) M. Komoto and Y. Shapir--Cpreprint). ( I I ) B. Sutherland, preprints. (12) V. Kalmeyer and R.B. Laughlin, Phys. Rev. L e t t . 5__9_9,2095 (1987). (13) D.H. Lee and S.A. Kivelson, work in progress. (14) S. Chakravarty, D. Nelson, and B . I . Halperin, Phys. Rev. Lett. 60, I057 (1988). (15) J.E. Hirsch, p r e p r i n t . (16) M.E. Fisher and J. Stephenson, Phys. Rev. 132, 1411 (1963). (17) R.P. Feynman, S t a t i s t i c a l Mechanics: A Set of Lectures, Addison-Wesley (1972). (18) S.A. Kivelson, Phys. Rev. B36, 7237 (1987). (19) C. K a l l i n , W-K Wu, J. B e r l i ~ k i , S.A. Kivelson, P. Allen (work in progress).
S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
I
>/_X ,
~
(~
•. . . " - - - - p, CO) Figure la)
Figure 2.
Typical nearest-neighbor only valence bond state, b) Typical short-range valence-bond state. c) Typical valence-bond state which enters the P-RVB,
Valence bond crystal state.
537
THE LOW TEMPERATUREPROPERTIES OF A SHORT-RANGEDRESONATING-VALENCE-BONDSUPERCONDUCTOR S. KIVELSON and D.S. Rokhsar* Department of Physics, State U n i v e r s i t y of New York at Stony Brook, Stony Brook, NY 11794 *IBM, Thomas J. Watson Research Center, Yorktown Heights, NY 10598 I t is shown that a RVB state is the ground state of a class of f r u s t r a t e d Heisenberg models. An approximate version of t h i s model, the hard core quantum dimer gas, is solved exactly and the nature of the RVB superconducting state is discussed. I.
INTRODUCTION I t is by now f a i r l y clear that in order to understand the properties of the copper oxide superconductors, we must f i r s t understand the properties of the two-dimensional electron-gas with strong electron-electron i n t e r a c t i o n s . In order to do t h i s , i t is useful to study simple models, such as the Hubbard model and others discussed below, so as to determine the possible classes of ground states t h i s system can have. In t h i s note, we summarize the r e s u l t s of an extensive study we have carried out ( I - 5 ) on a class of highly f r u s t r a t e d models which have a novel sort of ground state which we have ident i f i e d as the short-ranged resonating-valencebond-state (SR-RVB). In p a r t i c u l a r , we show that f o r a n o n - h a l f - f i l l e d band, the ground state of these models is superconducting. At present, the experimental s i t u a t i o n in the cuprate superconductors is too murky to allow detailed comparison with experiment; few experimental " f a c t s " , beyond the s t r u c t u r e , have been unambiguously established. Thus, while there are features of the experiments which make us hopeful that the present results are relevant to these materials, we do not consider i t useful at present to make a detailed comparison between theory and experiment. This is e s p e c i a l l y so since a l l our results pertain to the very low temperature properties which are e s p e c i a l l y s e n s i t i v e to effects of sample imperfection. With t h i s in mind, we have focussed on obtaining as complete a solution of our model as possible so as to i d e n t i f y the q u a l i t a t i v e features which can be compared to experiment, and in p a r t i c u l a r those features which lead to high temperature superconductivity. The plan of t h i s paper is as f o l l o w s : In Section I I we describe the i n t u i t i v e picture which underlies the RVB analysis, and in p a r t i c u l a r we discuss the r e l a t i o n between a SR-RVB state, which was o r i g i n a l l y studied in the cont e x t of the spin I / 2 Heisenberg antiferromagnet on a t r i a n g u l a r l a t t i c e by Anderson and Fazekas (6), and the "Princeton" RVB state, which has been studied by Anderson and 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
coworkers (7) in the context of high temperature superconductivity. Section I I I is a s l i g h t diversion in which results (2) on an analogous problem f o r the one-dimensional electron gas are summarized. In Section IV, the properties (3) of a completely f r u s t r a t e d Heisenberg model, o r i g i n a l l y considered by Klein (9), are discussed. In Section IV we(4) elucidate the properties of a s t i l l simpler model, the quantum hard-core dimer gas, which is the low energy part of the model discussed in Section IV. F i n a l l y , in Section VI we summarize our results and make some general observations concerning t h e i r r e l a t i o n to experiment. II.
INTUITIVE PICTURE A valence-bond state is one in which the electron on each s i t e is s i n g l e t paired with the electron on another s i t e as shown in Fig. I ; thus there is a valence bond connecting each occupied s i t e to exactly one other occupied s i t e . The valence bond state is the product state formed of these s i n g l e t s . An RVB is a coherent l i n e a r superposition of valence bond states. Such a state w i l l generally have r e l a t i v e l y low v a r i a t i o n a l energy both because each valence bond state minimizes the part of the energy i n v o l v i n g the pair i n t e r a c t i o n between the two bonded s i t e s , and because i t gains resonance energy from the o f f diagonal matrix elements of the Hamiltonian. In a real sense, which w i l l be made clear below, the valence-bonds can be thought of as preformed, real-space Cooper-pairs. I t is important, from the f i r s t , to d i s t i n guish between two types of RVB state that have been widely discussed: I ) The short-ranged RVB (SR-RVB) in which the only states which have substantial amplitude are states involving short valence-bonds, such as the nearestneighbor valence-bonds shown in f i g . la. 2) The "Princeton" RVB (P-RVB), in which valence-bonds of a l l length play an important r o l e . The P-RVB is the more widely studied(7-8) in the context of high-temperature
532
S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
superconductivity. The SR-RVB was f i r s t studied in the context of superconductivity (as d i s t i n c t from i t s magnetic o r i g i n s (6)) by us ( I ) and has been f u r t h e r studied by a v a r i e t y of people. ( I - 5 ) , (10-13) Both RVB states have many things in common: they were both invented by Anderson ( 6 - 7 ) , t h e i r e x c i t a tions ( I ) consist of spin I / 2 , charge zero Fermions (spinons) and spin O, charge e Bosons (holons), and they are both quantum l i q u i d states with no broken t r a n s l a t i o n a l or rotational symmetry. The fundamental difference between them is that in the SR-RVB there is a gap (4,10,12,13) in the s p i n - e x c i t a t i o n spectrum, while the P-RVB is gapless. This difference arises since to make a pair of spinons one must break a valence-bond, which costs a f i n i t e energy in the SR-RVB, but, due to the presence of a r b i t r a r i l y long bonds in the P-RVB, cost v a n i s h i n g l y small energy in t h i s state. There is considerable evidence now (14) that a Neel ordered state, not the SR-RVB, is the ground state of the Heisenberg model on a square lattice. I t is unclear whether the RVB state is the ground state of the Hubbard model away from h a l f - f i l l i n g , but preliminary evidence (15) suggests that i t is not. However, in a l l cases, the SR-RVB has a f a i r l y low v a r i a t i o n a l energy so i t is plausible that the i n t r o d u c t i o n of weak electron-phonon coupling (1,2) or s u f f i c i e n t f r u s t r a t i o n could s t a b i l i z e the SR-RVB. Indeed, in the next section we show that a r b i t r a r i l y weak electron-phonon coupling w i l l do the job in one-dimension, and in Section IV that f r u s t r a t i o n can have t h i s e f f e c t in two dimensions. III.
SUMMARYOF THE RESULTS IN ONE DIMENSION The continuum model of the one-dimensional electron gas with a r b i t r a r y short-ranged i n t e r actions has been studied for years. Because the Fermi surface consists of two points, the only possible i n t e r a c t i o n s at the Fermi surface are forward s c a t t e r i n g , called gg, (which f o r an extended-Hubbard-model is g2 L= (U-2V)/2t n) backward s c a t t e r i n g , gl (gl = (U÷2V)/2tn~, and, near the h a l f - f i l l e d -bana, Umklapp scattering g3 ( [- (U+2V)/2to). Zimanyi, Luther and one g3of us (2) have done two new things with t h i s model: f i r s t , we have r e i n t e r preted old r e s u l t s by comparing them to the res u l t s of a v a r i a t i o n a l analysis based on a l i n e a r superposition of nearest-neighbor valence-bond states, and second, we have generalized the model to include both instantaneous repulsions g~. (e.g. Hubbard i n t e r a c t i o n s ) and retarded i f i t e r actions gj (e.g. mediated by the exchange of phonons). A) Old r e s u l t s : I ) There is a t o t a l separation of the spin and charge degrees of freedom which occurs in the continuum l i m i t . The charge e x c i t a t i o n s are
spinless charge e solitons and the spin e x c i t a tions are neutral spin I / 2 s o l i t o n s . This is exactly analogous to spinons and holons. 2) The s i n g l e t superconducting suscept i b i l i t y vanishes as T ÷ 0 unless there is a gap A. > 0 in the s p i n - e x c i t a t i o n spectrum. Conversely, i f As > 0 and i f the system Is doped so that there is no gap in the chargee x c i t a t i o n spectrum, then the s i n g l e t superconducting s u s c e p t i b i l i t y diverges. Thus, while there are states analagous to both the P-RVB and the SR-RVB, only the state analagous to the SR-RVB e x h i b i t s a superconducting tendency in one-dimension. We note that A. > 0 i f and only i f gl < O. 3) Indeed, so long as gc : (g~-2gl) < O, g3 > O, and gl < O, a l l the results can be understood q u a l i t a t i v e l y in terms of nearestneighbor only valence-bond states: for the h a l f - f i l l e d band the ground-state is a bondorder charge-density-wave and the holons and spinons are simply domain walls between the two degenerate senses of the ground-state dimerizat i o n . Note that gc < 0 and g~ > 0 are the expected values of the i n t e r a c t i o n s for a Hubbardl i k e model, but gl < 0 is not reasonable for Coulomb i n t e r a c t i o n s alone. However, i t is a well known property of the renormalization group scaling equations that i f gl is p o s i t i v e i t scales to zero. Thus, i t is not unreasonable to expect that the addition of even weak. retarded i n t e r a c t i o n s w i l l produce a g~TT < O. B) New results with I g j I < < I g i l We have derived scaling equatlons in the presence of weak a t t r a c t i v e retarded and strong repulsive Coulomb i n t e r a c t i o n s . The important r e s u l t is that in the scaling equations for g l , cross terms occur of the form gigk which imply that the rate at which g: scale~ is proportional to the strength of the Jof the Coulomb i n t e r actions. In p a r t i c u l a r , we f i n d that for a small and a t t r a c t i v e bare value of gl = gl ( ° ) , the scaling equations r e s u l t in an extreme magnification of gl so that a f t e r i n t e g r a t i n g out states with energy between E~ and EF, the new value of gl is ,
a~ gl = gl (°)
(E,F/ EF ]
(i)
where A is a number of order I , and g is a suitable average of the Coulomb i n t e r a c t i o n . Since at the same time gl is scaling to zero, the conclusion is that the low energy physics is c o n t r o l l e d by g~ff < O, even i f the bare electron-phonon coupling is very weak. IV.
THE FULLY FRUSTRATEDHEISENBERG MODEL In order to establish that there e x i s t sensible models with a SR-RVB type ground state, J and L Chayes and one of us (3) have
S. Kiuelson and D.S. Rokhsar /Short-ranged resonating-valence-bond superconductor
studied a p a r t i a l l y soluable model of a f r u s t r a t e d spin I / 2 Heisenberg model, the Klein model (9). In general, a spin I / 2 Heisenberg model simply is a model with a spin I / 2 on each s i t e and short-ranged, s p i n - r o t a t i o n a l l y i n v a r i a n t i n t e r a c t i o n between spins on d i f f e r e n t s i t e s . By a f r u s t r a t e d model we mean a model with i n t e r a c t i o n s which f r u s t r a t e any classical ordered spin-state such as the Ferromagnetic or Neel states. While to obtain a soluable model, we must consider a p a r t i c u l a r r a t i o of the various coupling constants, i t is our f e e l i n g that our r e s u l t s pertain to a wide class of models with s u f f i c i e n t f r u s t r a t i o n . Indeed, preliminary numerical r e s u l t s (5) on a Heisenberg model with f i r s t , second, and t h i r d neighbor antiferromagnetic i n t e r a c t i o n s support t h i s contention. The Klein Hamiltonian can most e a s i l y be w r i t t e n in terms of the projection operators P~ onto the highest+spin state of the t o t a l s ~ i n ~ on ~ i t e ~, S~ and i t s nearest neighbors, S~+~
HK = J ~
P~
(2)
where, f o r a square l a t t i c e , each s i t e has 4 nearest neighbors, so P~ is the projection operator onto spin 5/2
and
A This Hamiltonian can also be resolved i n t o a sum of f i r s t , second, and t h i r d neighbor a n t i ferromagnetic i n t e r a c t i o n s plus some short ranged four-spin i n t e r a c t i o n s . I t is manifestly s p i n - r o t a t i o n a l l y - i n v a r i a n t . For the non-halff i l l e d band with s t a t i c holes the K l e i n Hamiltonian can be generalized by omitting the sites occupied by holes from the sum in Eq. (2) and s u i t a b l y modifying the projection operator on neighboring sites to take account of the fact that the highest spin-state is now reduced by I / 2 due to the absence of one neighbor. A few results concerning the ground states of t h i s Hamiltonian f o l l o w almost by inspection. Since H is a sum of p o s i t i v e semi-definite operators, any state which is a n n i h i l a t e d by Hw is a ground state. Such a state must have zer~ energy on each s i t e . The energy on s i t e depends on the spin on s i t e R and i t s nearest neighbors. In any state in which there is a valence-bond connecting any two of these s i t e s ,
533
~R2 is less than i t s maximal value and hence P~ operating on t h i s state is zero. This immediately implies that any nearestneighbor-only valence-bond state is a ground state of HK. What is less obvious, but which nonetheless can be proven (3), is that a l l the nearest-neighbor valence-bond states are l i n e a r l y independent. (They are not orthogonal.) Thus, i f we associate hard-core dimers with nearest-neighbor valence-bonds, then to each dimer covering of the l a t t i c e there corresponds a unique d i s t i n c t groundstate of H.. I t is ~ore d i f f i c u l t to prove that the nearest-neighbor valence-bond states exhaust a l l the ground-states of HK. (Indeed, on a t r i a n g u l a r l a t t i c e they c e r t a i n l y do n o t . ) On a square l a t t i c e a simple argument is s u f f i c i e n t to show that a pure valence-bond state cannot be a ground-state i f i t contains any valence-bonds longer than second neighbor: there are h a l f as many valence-bonds as s i t e s , and each valence-bond can at most cause the energy to be zero on two s i t e s ; a longer bond causes at most one s i t e energy to be zero. I t is even r e l a t i v e l y easy to show f o r free boundary c o n d i t i o n s , that nearest-neighbor valence bond states are a l l the valence-bond ground states. We (3) have proven f o r several l a t t i c e s , i n c l u d i n g the hexagonal l a t t i c e with a r b i t r a r y s t a t i c h o l e s , t h a t the nearestneighbor valence-bond states exhaust the zero energy states, valence bond or otherwise. We are c u r r e n t l y working to extend t h i s proof to the square l a t t i c e . F i n a l l y , we would l i k e to show that there is an energy gap between the ground states of the Klein-Hamiltonian and i t s excited states. This seems i n t u i t i v e l y obvious since only excited state involves a broken bond. We have not, however, proven i t . Some related r e s u l t s which lend f u r t h e r support to the existence of a gap are the proof, due to Klein (9), that in any f i n i t e l i n e a r superposition of nearestneighbor states, the spin-spin c o r r e l a t i o n function is an e x p o n e n t i a l l y f a l l i n g function of distance, and the non-rigorous, but quite convincing demonstration of Komoto and Shapir (I0) that the same c o r r e l a t i o n function is an exponentially f a l l i n g function in the equal amplitude l i n e a r superposition of a l l nearest-neighbor valence-bond states. (see also Ref. 13.) By conventional wisdom, an exponentially f a l l i n g c o r r e l a t i o n function implies a gap A = V~/~, where ~ is the c o r r e l a t i o n length -and Vs is a spinwave velocity. Given the existence of a gap, the ground state of any Hamiltonian which is equal to the Klein Hamiltonian plus a small perturbation can be obtained from degenerate perturbation theory by diagonalizing the perturbation w i t h i n the ground-state manifold of HK. The
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S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
hard core dimer-model considered in the next section could be viewed as a r i s i n g in t h i s fashion. More g e n e r a l l y , our present results establish that there e x i s t s a class of highly f r u s t r a t e d Heisenberg models with a shortranged RVB ground state. THE HARD-CORE QUANTUMDIMER GAS Let us consider quantum hard-core dimers on a square l a t t i c e . The H i l b e r t space is spanned by the orthonermal states {IC > } , where C specifies a dimer c o n f i g u r a t i o n . For s i m p l i c i t y , we study a model dimer Hamiltonian composed only of the most local terms. The Hamiltonian consists of the sum of two pieces: H~. r ' which completely specifies the dy~cs f o r close packed dimers, and Hhole s, which describes additional processes i n v o l v i n g the motions of holes. A) The Close Packed Dimer Gas We f i r s t address the close-packed case: V.
Hdimer :
Z
{ - J [ l ~ - ~ >
plaquettes
(5)
The f i r s t term is a pure dimer k i n e t i c energy which f l i p s pairs of p a r a l l e l nearest neighbor dimers; the second is a repulsion between such nearest neighbor pairs. The coupling constants J and V are to be regarded as phenomenological parameters. For close packed dimers, the H i l b e r t space breaks up i n t o t o p o l o g i c a l l y d i s t i n c t subspaces labeled by winding numbers ~. and ~ , as described in Ref. lb. Any two s~ates i~ the same topological sector can be connected by a series of pairwise dimerf l i p s (J) and no local rearrangement of bonds can connect states in d i f f e r e n t topological sectors. There e x i s t i n d i v i d u a l configurations which comprise t h e i r own t o p o l o g i c a l l y d i s t i n c t subspaces. These "valence-bond c r y s t a l " (VBC) configurations (Fig. 2) contain no p a r a l l e l nearest neighbor dimers, and are zero energy eigenstates of H,. f o r a r b i t r a r y J and V" ,olmer . . . . For V > J > O, Harm=~ is poslzlve semi-definite, so these VBC's a ~ the only ground states. At V = J, each topological sector possesses a unique zero energy ground state I~>, namely the equal amplitude superposition ZCc~]C > of a l l configurations in that sector. I t is easy to v e r i f y that these states are zero energy eigenstates of These states are the unlque ground Hdimer"states in t h e i r respective • topological sectors: a l l off-diagonal matrix elements of the Hamiltonian Hd. r are nonlme p o s i t i v e , so the ground state must be nodeless, i . e . , a state vector with a l l p o s i t i v e amplitudes. These coherent superpositions are
the precise analogs of the SR-RVB ; the resonance energy is simply the dimer k i n e t i c energy. An important feature of the equal amplitude states discussed above is that any dimer c o r r e l a t i o n function can be computed exactly from the results of Fisher and Stephenson (16) f o r the classical dimer problem. This does not, however, imply that the quantum problem is related to a two dimensional classical problem in the sense that i f we perturb the quantum Hamiltonian s l i g h t l y , the ground state c o r r e l a t i o n functions need not be der i v a b l e from any simple classical s t a t i s t i c a l mechanics problem. For V < J, we can use the equal amplitude states as v a r i a t i o n a l wavefunctions, and obtain an upper bound to the ground state energy of E S ~
(J-V) per s i t e .
Since E is iden-
t i c a l l y zero f o r V > J, the energy has a discontinuous f i r s t d e r i v a t i v e at V = J, implying a f i r s t order t r a n s i t i o n from a VBC to a quantum l i q u i d (RVB) state. When V is large and negative, we expect another t r a n s i t i o n (4) to a d i f f e r e n t c r y s t a l l i n e ground state, the "column phase," which has columns of p a r a l l e l dimers. B. The Doped Dimer Gas Upon doping ( i . e . , removing dimers), additional processes contribute to the Hamiltonian:
H = Hdimer - t
+v h Z
Z [I "-> <- "I + h.c.]
[{--><..(]
(6)
The f i r s t sum, performed over t r i p l e s of sites such that the f i r s t and t h i r d are nearest neighbors of the second, is a "hole k i n e t i c energy" which moves a dimer to an adjacent unoccupied p o s i t i o n . Note that a "black hole" remains on the black s u b l a t t i c e and a red hole remains on the red s u b l a t t i c e . The second sum, performed over nearest-neighbor s i t e s , is a "hole-hole r e p u l s i o n , " which can be rewritten (less transparently) as a local i n t e r a c t i o n between dimers. The ground state of the dimer Hamiltonian can be determined exactly when J = V and Vh = - 2 t , for t and J p o s i t i v e . For these parameters, the equal amplitude superposition z I C > of a l l dimer configurations can be shown to be a nodeless eigenstate of H, and hence the unique ground state. Note that away from close-packing, any configuration can be obtained from any other configuration by repeated a p p l i c a t i o n of the Hamiltonian, so that there is only a s i n g l e , t o p o l o g i c a l l y t r i v i a l sector.
S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor The energy of the equal amplitude superposition is -4t[Nr+N h] where Nr (Nh) is the number of red (black~ holes. Thug the comp r e s s i b i l i t y ~2E/~Ni2 vanishes under the special conditions J = V, Vh = -2t. This is precisely analogous to vanishing c o m p r e s s i b i l i t y of a free Bose gas, and implies phonon e x c i t a tions with a k 2 dispersion. Additional repulsive i n t e r a c t i o n s (as we expect in the oxide superconductors) lead to a Bose condensate with l i n e a r l y dispersing Goldstone modes. I f Vh is less than - 2 t , the system phase separates. Like the free Bose gas, the equal amplitude state possesses off-diagonal long range order, which can be demonstrated by computing the c o r r e l a t i o n function
~
535
0k = k, and Ik-QI r e s p e c t i v e l y . For Vh > - 2 t , the modes s t i l l have quadratic dispersion f o r l>>6k>>x where x is the holon concentration but asymptotically for small (Ok), the mode has l i n e a r dispersion,
3) For the close-packed case, there are no low-energy density e x c i t a t i o n s . However, there are e x c i t a t i o n s , which we have c~lled "resonons", with vanishing energy for k every~here along the edge of the B r i l l o u i n zone, k = (k,~) and (~,k). I f we consider the dimers to be f i c t i t i o u s dipoles with the p o s i t i v e l y charged end on the black s u b l a t t i c e , and the negative end on the red, these e x c i t a tions can be thought of as o s c i l l a t i o n s of the local dipole density. Theresonansalso p e r s i s t in the doped system, and may have vanishing energy f o r ~ near ~. VI.
CONCLUSIONS: SUMMARYAND RELATION TO EXPERIMENT We have argued on plausible physical grounds that systems with s u f f i c i e n t f r u s t r a t i o n , or with strong enough coupling to high frequency phonons, and e s p e c i a l l y when the e l e c t r o n i c band is nearly, but not quite h a l f f i l l e d , may have a SR-RVB ground state. To support these h e u r i s t i c arguments, we have shown with reasonable c e r t a i n t y that there e x i s t models which, while not r e a l i s t i c models of cuprate superconductors, are not a l l that f a r from such models and which have SR-RVB ground states. By studying these simple models, we have obtained a more complete characterization of the low energy properties of the SR-RVB state. Since i t is clear that the undoped parent compounds of the cuprate superconductors do not have SR-RVB ground states, we w i l l focus in t h i s section on summarizing the r e s u l t s which pertain to the nature of the SR-RVB superconductor (See also Ref. 18). The superconducting order parameter of a SR-RVB state involves the in-phase motion of the red and black holons, and so leads to f l u x quantization in u n i t s of hc/2e; in t h i s respect i t is i n d i s t i n g u i s h a b l e from a conventional superconductor. There is the i n t r i g u i n g p o s s i b i l i t y , which we are c u r r e n t l y t r y i n g to v e r i f y , that there are other charge neutral components of the order parameter associated with the out-of-phase motion of the red and black holons. I f t h i s can be established, i t could lead to macroscopically observable differences between the cuprate superconductors and conventional superconductors. We have analyzed the elementary e x c i t a t i o n s r e l a t i v e to the SR-RVB ground state. I ) At the highest energies we have analyzed (18), there are charged q u a s i - p a r t i c l e e x c i t a t i o n s ; these are p o s i t i v e l y and negatively charged s o l i t o n p a i r s , or in other words holon
536
S. Kiuelson and D.S. Rokhsar / Short-ranged resonating-ualence-bond superconductor
a n t i - h o l o n pairs. As with any theory based on the large U Hubbard model, the energy to create such e x c i t a t i o n s is AE = U -8t + ~ ( t 2 / U ) . Photo-production of holon a n t i - h o l o n - p a i r s may account f o r the peak in the optical absorption observed at ~ ~ 2 eV. 2) The lowest energy q u a s i - p a r t i c l e e x c i t a t i o n s consist of spinonanti-spinon pairs. I t is in t h i s sense that the SR-RVB can be thought of as a BCS-like condensate of Cooper pairs of spinons; when you break one apart you obtain a pair of spinons. That the spinons are neutral is the central feature of the model that is responsible f o r the p o s s i b i l i t y of high temperature superc o n d u c t i v i t y and the concomitant shortnsuperconducting c o r r e l a t i o n length ~o ~ 15 A; because the spinons are neutral there is no Coulomb cost to forming such small Cooper pairs. And because the holons are not paired, the charge remains as spread-out as possible in the superconducting state! Moreover, the gap in the spinon-spectrum is analogous to the superconducting gap in a conventional superconductor. As a r e s u l t , the optical absorption should not be of the Bardeen-Mattis form; there is zero o s c i l l a t o r strength f o r d i r e c t photo-generation of spinon-pairs. The spinon gap should be d i r e c t l y observable in experiments which do not have dipole selection rules such as NMR I / T I , t u n n e l i n g , and Raman scattering. The spifions should be observable (19) in optical absorption only through i n d i r e c t , Holsteinl i k e processes. 3) The remaining e x c i t a t i o n s we have characterized are c o l l e c t i v e Bosonic e x c i t a t i o n s . As in a conventional superconductor, there is a long-wave-length charge density o s c i l l a t i o n which is driven to the plasma frequency by the long-range Coulomb forces. The e f f e c t i v e mass of the holons can be deduced from the putative dopant concentrat i o n and the magnitude of the London penetrat i o n depth to be m* ~ 5-10 mc. In a d d i t i o n , there is the out-of-phase motion of the red and black holons, which we have shown is gapless in the nearest-neighbor-bands-only RVB state, and which we have argued p l a u s i b l e , but nonr i g o r o u s l y , remains gapless in any SR-RVB state. This mode does not couple strongly to any convenient probe although i t may lead to NQR r e l a x a t i o n at low temperatures, and must cert a i n l y couple to sound. I t couples weakly to neutrons and proably could in p r i n c i p l e be seen in i n e l a s t i c x-ray or electron scattering. The observation of t h i s mode would be dramatic evidence of a SR-RVB state. F i n a l l y , there are the resonons which are gapless in the i n s u l a t ing state, and which should remain low energy e x c i t a t i o n s , perhaps even gapless at ~ = (~,~) in the superconducting state. Of course, a l l these additional states should produce an anomalous c o n t r i b u t i o n to the low temperature s p e c i f i c heat. While such anomalies have been widely observed in the cuprate superconductors,
i t is not yet clear whether they are i n t r i n s i c . F i n a l l y , we have said nothing here about the t r a n s i t i o n to the superconducting state; in f a c t we have l i t t l e to say. We speculate that above Tr , there is no gap in the spino~ spectrum, ann the system may be well described in terms of a P-RVB state. I f t h i s is so, then at T~, the spinons pair and form a BCS-IIke condensate, whlch opens a gap in the spinon spectrum and permits the holons to Bose condense into a s u p e r f l u i d state. However, considerable f u r t h e r work is necessary to test whether t h i s scenario is v a l i d f o r any model system, much less f o r any real superconductor. ACKNOWLEDGEMENTS SK was funded in part by NSF grant # DMR-83-18051. REFERENCES (1) a) S.A. Kivelson, D.S. Rokhsar, and J.P. Sethna, Phys. Rev. B35, 8865 (1987); b) Euro. Phys. Lett. (submitted). (2) G.T. Zimanyi, S.A. Kivelson, and A. Luther, Phys. Rev. L e t t . (submitted). (3) J. Chayes, L. Chayes, and S.A. Kivelson, (work in progress). (4) S.A. Kivelson and D.S. Rokhsar, Phys. Rev. L e t t . (submitted). (5) S. Sondhi, F. Figuerido, E. Ramos, A. Karlhede, S.A. Kivelson, and M. Rocek, (work in progress). (6) P.W. Anderson, Mat. Res. Bul. 8, 153 (1973). P. Fazekas and P.W. Anderson, P h i l . Mag. 3(], 432 (1974). (7) P.W. Anderson, Science 235, 1196 (1987), G. Baskaran, Z. Zhou, and P.W. Anderson, Solid State Commun. 63, 973 (1987). (9) D.J. K l e i n , J. Phys. A 15, 661 (1982). (I0) M. Komoto and Y. Shapir--Cpreprint). ( I I ) B. Sutherland, preprints. (12) V. Kalmeyer and R.B. Laughlin, Phys. Rev. L e t t . 5__9_9,2095 (1987). (13) D.H. Lee and S.A. Kivelson, work in progress. (14) S. Chakravarty, D. Nelson, and B . I . Halperin, Phys. Rev. Lett. 60, I057 (1988). (15) J.E. Hirsch, p r e p r i n t . (16) M.E. Fisher and J. Stephenson, Phys. Rev. 132, 1411 (1963). (17) R.P. Feynman, S t a t i s t i c a l Mechanics: A Set of Lectures, Addison-Wesley (1972). (18) S.A. Kivelson, Phys. Rev. B36, 7237 (1987). (19) C. K a l l i n , W-K Wu, J. B e r l i ~ k i , S.A. Kivelson, P. Allen (work in progress).
S. Kivelson and D.S. Rokhsar / Short-ranged resonating-valence-bond superconductor
I
>/_X ,
~
(~
•. . . " - - - - p, CO) Figure la)
Figure 2.
Typical nearest-neighbor only valence bond state, b) Typical short-range valence-bond state. c) Typical valence-bond state which enters the P-RVB,
Valence bond crystal state.
537