88 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

spins on nearby sites form into “resonating” singlet pairs to retain some exchange energy and have a su ciently liquid–like character that the “holes” can propagate through them coherently, and, finally, superconduct at low temperatures. This is the main idea of a paper by Anderson et al. published early in 1987 on the resonating–valence–bond (RVB) theory of high–Tc superconductivity [150]. The antiferromagnetic order of the parent compound does not invalidate the essence of the argument; it is important that in the doped case mobile holes frustrate the tendency for the spins to order and stabilize the singlet liquid phase. One important consequence of the RVB theory is the so–called spin–charge separation. Note that this scenario points to a strong deviation from standard metallic behavior because the usual Fermi liquid is hard to reconcile with quasiparticles propagating with both spin and charge. On the other hand, in our theory the system remains a Fermi liquid for T → 0; deviations from the Fermi liquid in the normal state above Tc (similar to the marginal–Fermi liquid (MFL) mentioned earlier) are the result of an anomalous behavior of the quasiparticle scattering rate τ −1 calculated self-consistently. This will be discussed in connection with Fig. 3.22.

2.5.3 Projected Trial Wave Functions and the RVB Picture

In [150] Anderson et al. proposed a trial wave function as a description of the RVB state mentioned above:

where PG = i(1 − ni↑ni↓) is the Gutzwiller projection operator. This operator has the e ect of suppressing all amplitudes in |ψ0 with double occupation of the sites i, thereby enforcing the constraint of the t–J model exactly. The unprojected wave function contains variational parameters and its choice is guided by mean–field theory. We discuss here mainly the projected wave function because the underlying concepts are quite simple. The projection operator is a relatively complicated object to treat analytically, but the properties of the trial wave function may be handled using Monte Carlo techniques.

Historically, the notation of a linear superposition of spin singlet pairs, called an RVB, was introduced by Anderson as a possible ground state for the S = 1/2 antiferromagnetic Heisenberg model on a triangular lattice [151]. This type of lattice is of special interest because an Ising–like ordering of the spins is frustrated. An important concept associated with the RVB picture is the notion of spinons and holons, and spin–charge separation: it was postulated that the spin excitations in an RVB state are S = 1/2 fermions which Anderson called spinons. Note that this is in contrast to the N´eel state, in which excitations are S = 1 magnons or S = 0 singlet excitations. On the other hand, the concept of spinons is related to one–dimensional spin chains, where spinons act as domain walls and are well understood. In two dimensions, the concept is not well established, but if the singlet bonds are “liquid”,

two S = 1/2 spins formed by breaking a single bond can drift, with the liquid of singlet bonds filling in the space between them. Thus they behave as free particles and are called spinons.

What happens in the half–filled case, in which the problem reduces to the Heisenberg model? It was soon found that the d–wave BCS state is a good candidate for a trial wave function ([152] and references therein) by using the Variational Monte Carlo method for Gutzwiller states [153]:

Hd−wave = (−tij − µ) f † fiσ + c.c. + ∆ij f †↑f †↓ − f †↓f †↑ + c.c. ,

iσ i j i j

ij σ

(2.216) where tij = t for nearest neighbors, and ∆ij = ∆0 for j = i + xˆ and −∆0 for j = i + yˆ. The corresponding spectrum consists of the well-known BCS result

Ek = (εk − µ)2 + ∆2k , (2.217)

where εk = −2t(cos kx + cos ky ) and ∆k = ∆0(cos kx − cos ky ). At half filling (µ = 0), |ψ0 is the usual BCS wave function

Since then, many mean–field wave functions have been discovered which yield an identical energy and dispersion. This can be explained as being to a certain local SU (2) symmetry [154]. Among these wave functions is the important staggered–flux phase, in which the hopping parameter tij is complex, tij = t0 exp(i(−1)ix +iy Φ0), and the phase is arranged in such a way that it describes free fermion hopping with a flux ±4Φ0 [43]. One can show that if tan Φ0 = ∆0/t0, the eigenvalues are identical to those in (2.217). Note that the case Φ0 = π/4, the so–called π flux phase, is special in such a way that it does not break the lattice translational symmetry.

What are the properties of the projected wave function? First, the superfluid density vanishes linearly with the doping x. This is expected, since the projection operator is designed to yield an insulator at half–filling. Second, the momentum distribution has a jump near the noninteracting Fermi surface. This is interpreted as the quasiparticle weight z according to Fermi liquid theory. It vanishes smoothly as x → 0. Third, using a sum rule and assuming Fermi liquid behavior for the nodal quasiparticles, one can estimate the corresponding Fermi velocity, which is found to be in reasonable agreement with experiment. Recently Lee and coworkers analyzed the question of whether there are signs of the orbital currents and the SU (2) symmetry mentioned above in the projected d–wave superconductor. Since this state does not break time-reversal and translational symmetry, there is no static current, of course. However, Lee and coworkers found fluctuations of the orbital current that are entirely a consequence of the projection [155]. Physically speaking, this result is similar to a hole moving around a Cu–O plaquette

90 2 Theory of Cooper Pairing Due to Exchange of Spin Fluctuations

that experiences a Berry phase owing to the noncollinearity of the spin quantization axes of the (instantaneous) spin configurations. Thus the flux Φ0 of the staggered–flux phase has its origin in the coupling between the kinetic energy of a hole and the corresponding spin chirality.

How can we realize mathematically the RVB picture? In the slave boson method, one can decompose the electron operator into a neutral spin–1/2 fermion operator and a charge-e spinless boson operator (c†iσ = fiσ† bi), and enforce the non-double-occupancy constraint by a Lagrange multiplier. In other words, a “slave” boson operator that keeps track of a moving hole has been introduced. The t–J Hamiltonian then becomes

Here λi is the Lagrange multiplier. Note that a crucial simplification is that the constraint enforces on average λi → λ = const i. At this point the situation seems to be more complicated, but now a mean-field analysis can be applied. Fukuyama and coworkers [156] and Kotliar and Liu [157] have obtained a phase diagram consisting of a superconducting phase with a d– wave order parameter ( b = 0, ∆i,i+x = −∆i,i+y ), a “spin-gap” phase ( b = 0, ∆ij = 0), the so–called uniform RVB (uRVB) phase ( b = 0, ∆ij = 0), and a Fermi liquid–like phase ( b = 0, ∆ij = 0). In the superconducting phase, the bosons are condensed and the fermions are paired. The spin–gap phase is a metallic phase, since the bosons have not yet condensed, but the fermions are paired, yielding a gap for magnetic excitations. Remarkably, this approach captures some rough qualitative features of the hole-doped cuprates (see Fig. 1.4). Note that the key feature of the mean–field theory is that the spin carrying fermions (spinons) and the charge–carrying bosons (holons) are decoupled. In other words, one has a full spin–charge separation of the kind mentioned above.

2.5.4 Current Research and Discussion

In order to compare the above approach with our theory, let us point out some problems of the RVB picture. (a) The mean–field theory does not capture all energy scales accurately. In particular, the Bose–Einstein temperature at which holons acquire macroscopic coherence, i.e. b = 0, comes out too high. Thus the superconducting transition temperature Tc is too high. (b) There are fictitious phase transitions between certain mean field phases which should be just crossovers; for example, between the spin gap phase and the uniform RVB phase. (c) The mean–field theory can hardly explain why the normal state is a poor metal. (d) The mean–field theory loses most of the antiferromagnetic

correlations. These problems are related to current research in which the key is to include fluctuations around the RVB mean fields.

The gauge fields reflect the fact that the t–J model is invariant under the local transformation fi → eiΘi fi, bi → eiΘi bi, since ciσ = b†i fiσ is obviously a gauge singlet. With the inclusion of gauge fields, the RVB approach takes the following schematic form in the continuum limit:

Here LF and LB are mean field spinon and holon Lagrangians, a0 refers to λi in (2.219), and a denotes the spatial part of the local gauge transformation. Note that the main di erence between this Lagrangian and the corresponding QED Lagrangians is that the “kinetic term” for the gauge fields, Fµν2 , is absent. Thus the apparent coupling is infinitely strong. This enforces the no– double–occupancy constraint and the constraint that the boson current is canceled by fermion backflow [158].

The gauge field acquires dynamics from the fermions and bosons. Integrating out the matter fields, one finds that the gauge propagator in the Coulomb gauge is given by

where ΠF,B and ΠF,B00 are transverse and longitudinal polarization functions of fermions and bosons, respectively. Therefore, the dynamics of the gauge field depend on the mean–field ground states and excitations of spinons and holons, and, vice versa, the gauge field a ects the dynamics of the matter fields. Moreover, because of the gauge field, the fermions and bosons are no longer decoupled, yielding only a “quasi” spin–charge separation. Within this picture, one can argue that the magnetic properties of cuprates are related to spinons interacting with a gauge field, while the transport properties should be calculated mainly in terms of holons interacting with a gauge field.

One important problem is related to the type of the dominant fluctuations. Lee and coworkers find that the staggered–flux fluctuations may yield new collective modes [159], a prediction that has to be tested experimentally. Another question is related to the degree and control of the “quasi”–spin–charge separation mentioned above. In this connection, Herbut et al. have argued that there should exist no gapless spinons, for example [160]. Finally, as long as gauge fluctuations are treated as Gaussian, the Io e–Larkin law holds which predicts that the superfluid density ns behaves as ns ax − bx2T . The quadratic term, however, which arises from the Gaussian fluctuations