a r X i v :c o n d -m a t /0211131v 2 [c o n d -m a t .s u p r -c o n ] 8 N o v 2002
Gossamer supercoductivity and the mean ?eld approximation of a new e?ective
Hubbard model
Yue Yu
Institute of Theoretical Physics,Chinese Academy of Sciences,P.O.Box 2735,Beijing 100080,China
(February 1,2008)We construct a new e?ective two-dimensional Hubbard model by taking the di?erent electron occupancy on site into account.The mean ?eld state of the new Hamiltonian gives rise to the gossamer superconducting state proposed by Laughlin recently [1].PACS numbers:74.20.-z,74.20.Mn,72.-h,71.10.Fd
Since the high temperature superconductor of Cu-O cuprate was discovered,the mechanism of the supercon-ductivity has been attracting much research interesting due to various unusual properties of the high T c supercon-ductor.A well-known model trying to describe the nor-mal state properties is the 2-dimensional Hubbard model or t-J model [2,3].Many analytical and numerical inves-tigations on these two models have been carried out [4].Anderson proposed a resonant valence bond (RVB)state as a mean ?eld state of the models to explain the undoped cuprate as a Mott insulator while the superconductor is thought as a doped Mott insulator [5].Although many progresses have been made along the clues of the Hub-bard model and t-J model,there is no persuasive evidence that the theory of the high T c superconductivity of the cuprates can ?rmly based on these two models and var-ious extended models of them.Especially,the quantum antiferromagnetic state may always be favorable in these two-dimensional models for a small doping.
In a recent work,Laughlin proposed a new scenario for the high T c superconducting cuprate,called the gossamer superconductivity [1].The basic notion is that,instead of the full projection in the RVB state,which forbits the double-occupancy of electrons on a site,a partial projec-tion acting on a BCS-type wave function is introduced.If the partial projection is not far from the full projec-tion,one has a very tiny super?uid density.This is called the gossamer https://www.doczj.com/doc/567606516.html,ughlin’s explicit mi-croscopic wave function for the gossamer superconductor reads
|ΨG =
i
(1?αn i ↑n i ↓)|ΨBCS ,(1)|ΨBCS =
k
(u k +v k c ?k ↑c ?
?k ↓|0 ,
where c ?k σis the Fourier component of the electron cre-ation operator c ?iσon a two-dimensional lattice;n iσ=
c ?
iσc iσis the electron number operator at site i .The pro-jection operator Πα= i (1?αn i ↑n i ↓)for 0≤α≤1is the partial Gutzwiller projection which has an inverse Π?1
α= i (1+βn i ↑n i ↓)if α<1and β=α/(1?α).The BCS superconducting state |ΨBCS is de?ned as usual by
u 2k =
1
E k
),
v 2
k =
1E k
),(2)
u k v k =
?k
ξ2k +?2k
for the elec-tron dispersion ?k ,chemical potential μand the super-conducting gap ?k .It has been pointed out that the state |ΨG may be superconducting even at half ?lling.Furthermore,Laughlin has found that the state |ΨG is the exact ground state of the model Hamiltonian
H G =
k
E k ?b ?k σ?b k σ,
(3)where ?b k σ=Παb k σΠ?1for b k ↑=u k c k ↑+v k c ?k ↓and
b k ↓=u k
c k ↓?v k c ?k ↑annihilate the BCS state.Explic-itly,?b k σreads ?b k ↑=1N j
e i k ·r j
[u k (1+βn j ↓)c j ↑+v k (1?αn j ↑)c ?j ↓],
?b k ↓=
1N j
e i k ·r j
[u k (1+βn j ↑)c j ↓?v k (1?αn j ↓)c ?j ↑].
(4)
Laughlin has shown that for any magnitude of α,the quasiparticle energies remain at E k ,which indicates the psuedogap phenomenon.Right following up Laughlin’s work,Zhang has checked the gossamer superconductivity in a more realistic e?ective Hubbard model [6].It was found that the gossamer superconducting state is simi-lar to the RVB superconducting state,except that the chemical potential is approximately pinned at the mid of the two Hubbard bands away from the half ?lling.
Ref.[6]indicated that the gossamer superconducting state may possibly be a good variational state of the e?ec-tive Hubbard model.Nevertheless,it is still in question to make an explicit relation between the Hamiltonian (3)and that of the e?ective Hubbard model used in ref.[6].In this paper,we try to provide a new e?ective Hub-bard model which is the extension of the Hubbard model and t-J model.Our extending method is other than all the previous extended models of those two models.We consider the case that the lattice sites are allowed to be
1
double-occupied by electrons with opposite spins.Be-cause of the on-site Coulomb repulsion between electrons,the electrons hopping between sites becomes dependent on the occupancy of sites.There are three possible hop-ping processes that may have di?erent hopping probabil-ities (see,Fig.1).On the other hand,when electrons are paired,the coupling between pairs may also be de-pendent on the the occupancy of the sites (Fig.2)while the hopping may be dependent on the paring gap.What is new by employing such a new e?ective Hubbard model is that the Laughlin gossamer superconducting state may be viewed as the mean ?eld state of this https://www.doczj.com/doc/567606516.html,ly,there is a gossamer superconducting phase in the system described by this new e?ective Hubbard Hamiltonian.To extend the Hubbard model,we ?rst write down the Hamiltonian of the e?ective Hubbard model
H eh =? i =j ;σ
(t ij c ?iσc jσ
+h.c )+U
i
n i ↑n i ↓?μ
i,σ
n iσ+
i =j
J ij (S i ·S j ?
1
2c ?στσσ′
c σ
′is the local spin operator.
One can
rewrite this Hamiltonian as
H eh =? i =j ;σ
(t ij c ?iσ
c jσ+h.c )?1
N
k ?k e i k ·(r i ?r j )
.Notice that the chemical po-tential and the gap-dependence,we can estimate δ(a )t ij by
t (1)
ij =
αN
k
[a 2ξk +b 2(E k ?ξk )]e i k ·(r i ?r j ),
where the coe?cients a l and b l ,in principle,depend on
U and J ij but we are unable to calculate them at present.We leave them to be determined later.The α-dependence in eq.(8)is because in the limit U →0,α→0while the hopping probability becomes independent of the site-occupancy.
The coupling between the pairing operators ?D
ij and ?D ?ij
may also dependent on the occupancy of the sites i and j (See Fig.2):
?J ij =J ij +J (1)ij
(n i +n j )+J (2)ij n i n j ,(9)
where
J (1)
ij =A 1αJ ij ,J (2)
ij =A 2α2J ij ,
(10)
with the undetermined coe?cients A l .The extended ef-fective Hubbard model we are considering is given by
H e?=? i =j ;σ
(?t ij,σc ?iσc jσ+h.c )?
12
i =j
???ij ?D ij
?1
2α2
,b 2=?
α2+β2
2N
k
[(β?α)E k +(β+α)ξk ],
μG =
1
where
?t G
ij,σ
=
k E k
2
(v2k?u2k)?(αv2k+βu2k)n iˉσ
+
1
N
u k v k e i k·(r i?r j)
×(1+(α+β)n iˉσ+αβn iˉσn jσ). Substituting the de?nition(4)of?b kσinto the gossamer superconducting Hamiltonian(3),one can directly check that the gossamer superconducting Hamiltonian(3)is exactly the same as the mean?eld Hamiltonian(15),i.e.,
H G MF=H G.(17) Thus,we?nd that the gossamer superconducting state is indeed a?xed point of the system described by the e?ec-tive Hubbard model(11).If the parameters of the hop-ping,exchange,chemical potential and interaction are not far from their?xed point values,the system may exhibit a gossamer superconductity.
In conclusions,we have constructed a new e?ective Hubbard model in which the di?erent hopping proba-bilities and the pairing couplings are introduced due to the di?erent site-occupancy at sites.We showed that there is a superconducting phase in such a system since the mean?eld Hamiltonian of this system in a proper choice of the parameters is just the gossamer supercon-ducting Hamiltonian.Beside the superconducting phase, this new e?ective Hubbard model is anticipated to have a fruitful phase structure.In ref.[6],it was shown that there is a critical interaction strong U c that separates the gossamer superconducting phase from the Mott insultor phase.In this new e?ective Hubbard model,we see that even the superconducting gap vanishes or the super?uid density is completely suppressed(α=1),the dispersion of electron as well as the interaction between electrons for the latter are dependent on the occupancy of the site that electrons lying on.We expect these unusual behaviors of the electrons may cause unusual normal state properties and relate to the anomalous features in the cuprates.It is possible that the phase diagram of this model may have a better overlap to the cuprate superconductors.The further works are in progress.
The author is grateful to useful discussions with Shao-jin Qin,Tao Xiang and Lu Yu.He would like to thank Shiping Feng for him to draw the author’s attention to gossamer superconductivity.This work was partially supported by the NSF of China.
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