a r X i v :c o n d -m a t /0303608v 6 [c o n d -m a t .s u p r -c o n ] 5 F e
b 2004
Comment on ’Effect of on-site Coulomb repulsion on superconductivity in the
boson-fermion model’(Phys.Rev.B66,134512(2002))
A.S.Alexandrov
Department of Physics,Loughborough University,Loughborough LE113TU,United Kingdom
The two-dimensional (2D)boson-fermion model (BFM)of high-temperature superconductors,numerically studied by Domanski (Phys.Rev.B 66,134512(2002))is not a superconductor.The critical temperature of the model is zero for any symmetry of the order parameter.The opposite conclusion advocated by Domanski stems from a mean-field approximation (MF A)neglecting the boson lf-energy which is qualitatively erroneous for any-dimensional BFM.flowable
PACS numbers:PACS:71.20.-z,74.20.Mn,74.20.Rp,74.25.Dw
lonely歌词
Recently Domaski [1]and some other authors (e ref-erences in [1])claimed that 2D BFM with hybridid fermions and immobile hard-core bosons is capable to reproduce the pha diagram of cuprates.The model is defined by the Hamiltonian,
H = k ,σ=↑,↓
ξk c †
k ,σc k ,σ+E 0 q
b †q b q +
(1)g N −1/2
q ,k
φk b †q c −k +q /2,↑c k +q /2,↓+H.c. ,
where ξk =−2t (cos k x +cos k y )−µis the 2D energy
spectrum of fermions,E 0≡∆B −2µis the bare bo-son energy with respect to the chemical potential µ,
g is the magnitude of the anisotropic hybridisation inter-action,φk =φ−k is the anisotropy factor,and N is the number of cells.Ref.[1]argued that ’superconductivity is induced in this model from the anisotropic charge ex-change interaction (g φk )between the conduction-band fermions and the immobile hard-core bosons’,and ’the on-site Coulomb repulsion U competes with this pair-ing’reducing the critical temperature T c less than by 25%.The author of Ref.[1]neglected our study of BFM [2],which revealed a devastating effect of the boson lf-energy on T c .Here I show that becau of this effect T c =0K in the model,Eq.(1),even in the abnce of the Coulomb repulsion,U =0,and the mean-field approx-imation of Ref.[1]is meaningless for any-dimensional BFM.
Using MFA Ref.[1]decouples bosons and fermions in Eq.(1)replacing boson operators by c -numbers for q =0.Then T c is numerically calculated using the linearid BCS equation for the fermionic order-parameter ∆k ,
∆k =
˜g 2φk
2ξk ′
,
(2)
where E 0is determined by the atomic density of bosons (n B )as (Eq.(9)in Ref.[1])
tanh
E 0
i Ωn −E 0−Σb (q ,Ωn )
(4)
one must replace incorrect Eq.(3)by
−
k B T
2N
k
φ2k ×
(6)
tanh[ξk −q /2/(2k B T )]+tanh[ξk +q /2/(2k B T )]
2M ∗
+O (q 4)
(7)
for small q with any anisotropy factor compatible with
the point-group symmetry of the cuprates.Here M ∗is the boson mass,calculated analytically in Ref.[2]with the isotropic exchange interaction and parabolic fermion band dispersion (e also Ref.[3]),and =1.The BCS-like equation (2)has a nontrivial solution for ∆k ,if
E 0=−Σb (0,0).奥巴马总统之路
读书报告怎么写(8)
Substituting Eq.(7)and Eq.(8)into the sum-rule,Eq.(5)one obtains a logarithmically divergent integral with re-spect to q ,and
T c =
const
2 Using a’bubble’approximation for the lf-energy Ref.[2]
proved that the Cooper pairing of fermions in BFM is im-
possible without the Bo-Einstein condensation(BEC)
of real bosons.The bubble approximation is actually ex-
act becau of the logarithmic divergence of the Cooperon
diagram,as was also confirmed in Ref.[4].Hence,the
devastating result,Eq.(9)is a direct conquence of the
well-known theorem,which states that BEC is impossible
in2D.
One may erroneously believe that MFA results[1]can
be still applied in three-dimensions,where BEC is pos-
sible.However,increasing dimensionality does not make
MFA a meaningful approximation for the boson-fermion
model.This approximation leads to a naive conclusion
millionsofthat a BCS-like superconducting state occurs below the
critical temperature T c≃µexp(−E0/z c)via fermion
pairs being virtually excited into unoccupied bosonic
states[5,6].Here z c=˜g2N(0)and N(0)is the density
of states(DOS)in the fermionic band near the Fermi
levelµ.However,the Cooper pairing of fermions is im-
possible via virtual unoccupied bosonic states.It occurs
only simultaneously with the Bo-Einstein condensation
of real bosons in the exact theory of3D BFM[2].
The origin of the simultaneous condensation of the
fermionic and bosonicfields in3D BFM lies in the soften-
ing of the boson mode at T=T c caud by its hybridiza-
tion with fermions.Indeed,Eq.(8)does not depend on
the dimensionality,so that the analytical continuation
of Eq.(4)to real frequenciesωyields the partial boson
DOS asρ(ω)=(1−2n B)δ(ω)at T=T c for q=0in
any-dimensional BFM for any coupling with fermions.
Taking into account the boson damping and dispersion
shows that the boson spectrum significantly changes for
英汉字典all momenta.Continuing the lf-energy,Eq.(6)to real
frequencies yields the he imaginary part
of the lf-energy)as[2]
γ(q,ω)=πz c
cosh(−qξ+ω/(4k B T c)) ,(10)
whereξ=v F/(4k B T c)is a coherence length.The damp-ing is significant when qξ<<1.In this regionγ(q,
ω)=ωπz c/(8k B T c)is comparable or even larger than the bo-son energyω.Hence bosons look like overdamped dif-fusive modes,rather than quasiparticles in the long-wave limit[2,3],contrary to the erroneous conclusion of Ref.[7],that there is’the ont of coherent free-particle-like motion of the bosons’in this limit.Only outside the long-wave region,the damping becomes small.Indeed, using Eq.(10)one obtainsγ(q,ω)=ωπz c/(2qv F)<<ω, so that bosons at q>>1/ξare well defined quasiparti-cles with a logarithmic dispersion,ω(q)=z c ln(qξ)[2]. As a result the boson dispersion is distributed over the whole energy interval from zero up to E0,but not a delta-function at E0even in the weak-coupling limit.
The main mathematical problem with MFA stems from the density sum rule,Eq.(5)which determines the chem-ical potential of the system and conquently the bare boson energy E0(T)as a function of temperature.In the framework of MFA one takes the bare boson en-ergy in Eq.(2)as a temperature independent parameter, E0=z c ln(µ/T c),or determines it from the conrva-tion of the total number of particles,Eq.(5)neglecting the boson lf-energyΣb(q,Ω)[1,5,6]).Then Eq.(2) looks like the conventional mean-field BCS equation,or the Ginzburg-Landau equation(near the transition)with a negative coefficientα∝T−T c at T<T c in the linear term with respect to∆(T).Hence,one concludes that the pha transition is almost the conventional BCS-like transition,at least at E0≫T c[5,6],and,using t
he Gor’kov expansion in powers of∆,finds afinite upper criticalfield H c2(T)[8].Thefindings are mathemati-cally and physically wrong.Indeed,the term of the sum in Eq.(5)withΩn=0is given by the integral
T d q E0+Σb(q,0).(11)
The integral converges,if and only if E0 −Σb(0,0).In fact,E0+Σb(0,0)is strictly zero in the Bo-condend state,becauµb=−[E0+Σb(0,0)]corresponds to the boson chemical potential relative to the lower edge of the boson energy spectrum.More generally,µb=0corre-sponds to the appearance of the Goldstone-Bogoliubov mode due to a broken symmetry below T c.This exact result makes the BSC equation(2)simply an identity[2] withα≡0at any temperature below T c.On the other hand,MFA violates the density sum-rule,predicting the wrong negativeα(T)below T c.
Sinceα(T)=0,the Levanyuk-Ginzburg parameter[9] is infinite,Gi=∞.It means that the pha transi-tion is never a BCS-like cond-order pha transition even at large E0and small g.In fact,the transition is driven by the Bo-Einstein condensation of real bosons with q=0,which occur due to the complete softening of their spectrum at T c in3D BFM.Remarkably,the conventional upper criticalfield,determined as thefield, where a non-trivial solution of the linearid Gor’kov equation occ
mbsurs,is zero in BFM,H c2(T)=0,becau α(T)=0below T c.It is not afinite H c2(T)found in Ref.[8]using MFA.Even at temperatures well below T c the condend state is fundamentally different from the BCS-like MFA ground state,becau of the pairing of bosons[10].The pair-boson condensate significantly modifies the thermodynamic properties of the condend BFM compared with the MFA predictions.
This qualitative failure of MFA might be rather un-expected,if one believes that bosons in Eq.(1)play the same role as phonons in the BCS superconductor.This is not the ca for two reasons.Thefirst one is the den-sity sum-rule,Eq.(5),for bosons which is not applied to phonons.The cond being that the boson lf-energy is given by the divergent(at T=0)Cooperon diagram, while the lf-energy of phonons isfinite at small cou-pling.
I have to conclude that the numerical work by Doman-ski[1]does not make any n in any dimension.There is nothing in2D BFM to compete with becau the model
3
revi
is not a superconductor even without the Coulomb re-pulsion.The MFA results[1,8]do not make any n in three dimensions either,becau the divergent lf-energy has been neglected in calculatin
g T c and H c2(T).The common wisdom that at weak coupling the boson-fermion model is adequately described by the BCS the-ory,is negated by our results.
[1]T.Doma´n ski,Phys.Rev.B66,134512(2002).
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