The phase of the sigma - pi-pi amplitude in JPsi - omega-pi-pi

a r X i v :c o n d -m a t /9902287v 1 [c o n d -m a t .s t r -e l ] 20 F eb 1999Pairing via Index theoremDung-Hai LeeThis work is motivated by a specific point of view:at short distances and high energies the undoped and underdoped cuprates resemble the π-flux phase of the t-J model.The purpose of this paper is to present a mechanism by which pairing grows out of the doped π-flux phase.According to this mechanism pairing symmetry is determined by a parameter controlling the quantum tunneling of gauge flux quanta.For zero tunneling the symmetry is d x 2−y 2+id xy ,while for large tunneling it is d x 2−y 2.A zero-temperature critical point separates these two limits.PACS numbers:74.25.Jb,79.60.-i,71.27.+aIt is now well appreciated that the unconventional properties of the cuprate superconductors are best man-ifested in the underdoped regime.For example there is a crossover temperature T ∗>T c below which a pseudo spin gap opens up.In addition,recent angle-resolved photoemission spectroscopy shows a common feature be-tween the underdoped and undoped cuprates -a high-binding-energy peak whose dispersion is similar to that of the spinon in the πflux phase [1]of the half-filled t-J Model.[2]Due to this result,there is a spreading feeling that at short distances and high energies the undoped and underdoped cuprates resemble the πflux phase of the antiferromagnet.[3–6]For undoped cuprates this feeling is supported by a re-cent work of Kim and Lee.[6]Starting from the free Dirac spinons of the πflux phase,Kim and Lee show that gauge fluctuation binds spinon and antispinon and causes their condensation into the ordered Neel state at low temper-atures.This work demonstrates that even though the πflux phase physics (i.e.free spinon excitations)is distant from the reality,gauge fluctuations can restore the right low temperature behaviors.In the following we extend this point of view to the underdoped regime.The question we would like to ask is “can gauge fluctuation produce pairing out of the doped π-flux phase”.Interestingly we find a particular mecha-nism which yields either d x 2−y 2+id xy or d x 2−y 2pairing symmetries depending on a parameter controlling the quantum tunneling of gauge flux quanta from +φ0to −φ0.The model:The model we shall consider is the U(1)gauge theory of the doped π-flux phase.The low energy degrees of freedom are:spinons (Dirac-like fermions),holons (non-relavistic bosons),and an U(1)gauge field.The model is defined by:Z = D [¯Ψ±,Ψ±]D [¯b,b ]D [a µ]e −dtd 2x L ,(1)whereL =¯b [(∂t −µb −ia 0)+12|¯bb |2+n =±¯Ψn,α[(∂t −µf −ia 0)+vσ·(p −a )]Ψn,α.(2)In the above ±refers to the two “Dirac points”Q ±≡(π2a );α=↑,↓labels the spin state;Ψ±α= ψ1±αψ2±αare spinon fields;b is the holon field;p ≡∇/i ;and σx ,σy ,σz are Pauli matrices.The fluctuation of a µtends to bind particles of op-posite gauge charges.There are two obvious candidates for the bound states:(a)a spinon-antispinon pair,and (b)a holon-antispinon pair.In the undoped antiferro-magnetic phase,the first scenario is realized.Indeed,the Neel-ordered state can be thought of as a conden-sate of the spinon-antispinon (i.e.magnon)pairs.We view the unusual properties of the underdoped regime as due to the competition and compromise between bind-ing scenarios (a)and (b).A particular example of the compromise is the formation of stripes.[11]In this case holon-antispinon and spinon-antispinon exist in differ-ent spatial regions.Another example is the formation of pairs of physical holes.In this case two holons and two antispinons (hence (a)and (b))occur in the same composite.This pairing mechanism is the subject of the rest of the paper.The index theoremLet a (x )be a static gauge field.The eigenvalue prob-lemσ·(p −a )Ψ=E Ψ(3)has N φ−1normalizable zero-energy solutions,where N φ= d 2x (∇×a (x ))/φ0.[7]Pairing by index theoremLet us imagine starting from the half-filled π-flux phase,and allow ∇×a to nucleate N +1flux quanta.By index theorem,such flux quanta produce 4N zero-energy1spinon levels.(4N ,because of two Dirac points and two spin directions.)At half-filling 2N of these levels are oc-cupied.As the result the flux quanta lift 2N spinons to the Fermi energy so that they can be removed by dop-ing.We note that in the above argument there is a free choice,namely,the polarity of the flux quanta.Now let us come to finite doping (x >0).In order to produce the right number of zero-energy spinons we have to allow a density,x/2,of flux quanta in ∇×a .In order for these spinons to be true zero-energy these flux quanta must have the same polarity.In the above discussion the flux quanta are implicitly assumed to be static.In the following we shall relieve that assumption.We shall treat ∇×a as a fluctuating flux density satisfying<∇×a (r ,t )>=B =x2B2φ0and a ext,0=0.By integratingout the spinon excitations above the mean-field vacuum,we obtain the following effective action for δa µ:S eff =dtd 2x {12κ2b 2−iφ0∇×δa .(8)Physically Eq.(8)implies that two antispinons are bound to a quantum of ∇×a .Next we look at the holons.By putting Eq.(7)to-gether with the remaining boson terms in Eq.(2)we obtain:S b =dtd 2x {¯b [(∂t −iδa 0−µb )−12κ1|e |2+12πǫµνλδa µ∂νδa λ}.(9)Eq.(9)describes semions in an external magnetic field ∇×a ext corresponding to filling factor ν=2.For semions this is the right filling factor for the formation of quantum Hall liquid.Thus at low energies2δρholon =1gaugeflux.Under such condition the minimum energy isobtained when the polarity of theflux quanta order ferro-magnetically.The reversal of a singleflux quantum costsan energy equals to that of breaking two d x2−y2+id xypairs.Such energy cost is reminiscent to the spin-flipenergy of an Ising ferromagnet.In this analogy instanton acts as a transversefield onthe Ising spins.Thus the model describing the dynamicsof gaugeflux is a transverse-field-Ising-like model:H=−K <ij>σz iσz j−h iσx i.(12)In Eq.(12)K prefers T-breaking while h tends to restoreT.The parameter that determines time-reversal symme-try breaking is R=K[1]I.Affleck and J.B.Marston,Phys.Rev.B37,3774,(1988).[2]F.Ronning et al,preprint.[3]T.C.Hsu,Phys.Rev.B41,11379(1990).[4]ughlin,Phys.Rev.Lett.79,1726(1997).[5]X-G Wen and P.A.Lee,Phys.Rev.Lett.76,503(1996);P.A.Lee,N.Nagaosa,T-K Ng and X-G Wen,Phys.Rev.576003(1998).[6]D.H.Kim and P.A.Lee,cond-mat/9800130.[7]Y.Aharonov and A.Casher,Phys.Rev.A19,2461(1979).[8]ughlin,Science,242,525(1988);ughlin,Physica C234,280(1994).[9]D.S.Rokhsar,Phys.Rev.Lett.70,493(1993).[10]P.B.Wiegmann,cond-mat/9504094.[11]For a review,see e.g.,S.A.Kivelson and V.J.Emery,cond-mat/9809082.[12]S.Doniach and M.Inui,Phys.Rev.B41,6668(1990);V.J.Emery and S.A.Kivelson,Nature374,434(1995).[13]So far we focused on the dopedπflux phase.Similar ar-guments can also be applied to the staggeredflux phaseof Wen and Lee[5].For the staggeredflux phase thereare two spices of boson instead of one.These two kindsof boson carry opposite gauge charge,and have equaldensity(x/2)in the mean-field vacuum.In the spirit dis-cussed above each quantum of gaugeflux converts a“b2boson”into a b1type,[5]and binds two b1bosons withtwo antispinons.[14]ughlin,cond-mat/9709195.3。

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