7 Rigorous derivation of the mean field Green
functions of the two-band Hubbard model of
superconductivity
Gh Adam and S Adam
Laboratory of Information Technologies, Joint Institute for Nuclear Research,
141980 Dubna, Moscow Region, Russia
Horia Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH),
407 Atomistilor, Magurele–Bucharest, 077125 Romania
E-mail: adamg@jinr.ru, adams@jinr.ru
Abstract. The Green function (GF) equation of motion technique for solving
the effective two-band Hubbard model of high-T
superconductivity in cuprates
[N.M. Plakida et al., Phys. Rev. B, 51, 16599 (1995); JETP, 97, 331 (2003)] rests
on the Hubbard operator (HO) algebra. We show that, if we take into account
the invariance to translations and spin reversal, the HO algebra results in invariance
properties of several specific correlation functions. The use of these properties allows
rigorous derivation and simplification of the expressions of the frequency matrix (FM)
and of the generalized mean field approximation (GMFA) Green functions (GFs) of
the model.
For the normal singlet hopping and anomalous exchange pairing correlation
functions which enter the FM and GMFA-GFs, an approximation procedure based
on the identification and elimination of exponentially small quantities is described. It
secures the reduction of the correlation order to GMFA-GF expressions.
PACS numbers: 74.20.-z, 74.20.Mn, 74.72.-h
Submitted to: J. Phys. A: Math. Gen.
http://arxiv.org/abs/0704.0692v1
Mean field Green functions of Hubbard model of superconductivity 2
1. Introduction
A consistent theoretical model of the high critical temperature superconductivity in
cuprates is to be able to accommodate both the normal and superconducting states
under incorporation of the essential features of these systems (see, e.g., [1] for a review):
strong antiferromagnetic (AFM) superexchange interaction inside the CuO2 planes,
occurrence of two relatively isolated energy bands around the Fermi level, able to develop
dx2−y2 pairing: one stemming from single particle copper dx2−y2 states and the second
one from singlet doubly occupied states generated [2] by crystal field interaction; hopping
conduction for an extremely low density of the free charge carriers.
The p-d model [3], while incorporating all these features, is too cumbersome and
cell-cluster perturbation theory [4, 5] providing a hierarchy of the various interaction
terms was used to derive simpler models from it. Extreme limit cases of this reduction
procedure are various effective one-band t-J models (see, e.g., [6, 7] and references
therein) which, while unveiling the role played by the AFM exchange interaction in the
occurrence of the d-wave pairing, address exclusively the superconducting state.
The reduction of the p-d model to an effective two-band Hubbard model considered
by Plakida et al. [8], corroborated with the use of the equation of motion technique
for thermodynamic Green functions (GF) [9], provided the simplest approach to the
description of both the normal [8, 10] and the superconducting states [11, 12, 13] within
a frame securing rigorous fulfilment of the Pauli exclusion principle for fermionic states.
The Green function technique rests on the Hubbard operator algebra. Its rigorous
implementation onto a system characterized by specific symmetry properties (translation
invariant two-dimensional spin lattice, spin reversal invariance of the observables) results
either in characteristic invariance properties of several correlation functions, or in the
occurrence of some exactly vanishing correlation functions. The use of these results
allows rigorous derivation and simplification of the expressions of the frequency matrix
and of the generalized mean field approximation (GMFA) Green functions of the model.
The obtained expressions contain higher order boson-boson correlation functions
(CFs). For the CFs involving singlets (normal singlet hopping CFs and anomalous
exchange pairing CFs), an approximation procedure which avoids the usual decoupling
schemes and, yet, secures the correlation order reduction to GMFA-GF expressions,
under the identification and elimination of exponentially small quantities, is described.
The organization of the paper is as follows. Sec. 2 summarizes essentials of the
two-band Hubbard model and GMFA-GF equations. Sec. 3 describes the invariance
properties following from the translation invariance of the underlying spin lattice.
Sec. 4 derives invariance properties and constraints following from the invariance of the
macroscopic properties of the system under spin reversal. On the basis of the results of
Sec. 3 and 4, rigorous derivation of the frequency matrix in the (r, ω)-representation is
done in Sec. 5. The derivation of GMFA-GF expressions for the boson-boson correlation
functions involving singlets is discussed in Sec. 6.
Collecting together the results of sections 5 and 6, expressions of the frequency
Mean field Green functions of Hubbard model of superconductivity 3
matrix and of the GMFA Green function matrix are derived in the (q, ω)-representation
in sections 7 and 8 respectively. These results explicitly incorporate both hole-doping
and electron-doping features of the cuprate systems through the singlet hopping and
superconducting pairing terms.
The paper ends with conclusions in section 9.
2. Mean field approximation
The Hamiltonian of the effective two-band singlet-hole Hubbard model [8] is written in
the form
H = E1
Xσσi + E2
X22i +
σ0,0σ
1,i +K22
2σ,σ2
1,i +K21
2σ̄,0σ
1,i + τ
σ0,σ̄2
1,i ) (1)
The summation label i runs over the sites of an infinite two-dimensional (2D) square
array the lattice constants of which, ax = ay, are defined by the underlying single crystal
structure. The spin projection values in the sums over σ are σ = ±1/2, σ̄ = −σ.
The Hubbard operators (HOs) X
i = |iα〉〈iβ| are defined for the four states of the
model at each lattice site i: |0〉 (vacuum), |σ〉 = |↑〉 and |σ̄〉 = |↓〉 (single particle spin
states inside the hole subband), and |2〉 = |↑↓〉 (singlet state in the singlet subband).
The multiplication rule holdsX
i = δβγX
i . The HOs may be fermionic (single
spin state creation/annihilation in a subband) or bosonic (singlet creation/annihilation,
spin or charge densities, particle numbers). For a pair of fermionic HOs, the
anticommutator rule holds {X
i , X
j } = δij(δβγX
i + δηαX
i ) whereas, if one or both
HOs are bosonic, the commutation rule holds [X
i , X
j ] = δij(δβγX
i − δηαX
i ). At
each lattice site i, the constraint of no double occupancy of any quantum state |iα〉 is
rigorously fulfilled due to the completeness relation X00i +X
i = 1.
In (1), E1 = ε̃d−µ denotes the hole subband energy for the renormalized energy ε̃d
of a d-hole and the chemical potential µ. The energy parameter of the singlet subband
is E2 = 2E1 + ∆, where ∆ ≈ ∆pd = εp − εd is an effective Coulomb energy Ueff
corresponding to the difference between the two energy levels of the model.
In the description of the hopping processes, the label 1 points to the hole subband
and 2 to the singlet subband. The hopping energy parameter Kab = 2tpdKab depends on
tpd, the hopping p-d integral, and on energy band dependent form factors, Kab. Inband
(K11,K22) and interband (K21 = K12) processess are present. The Hubbard 1-forms
αβ,γη
1,i =
m (2)
incorporate the overall effects of specific hopping processes (through the labels (αβ, γη)
of the pair of Hubbard operators) involving the lattice site i and its neighbouring sites.
Up to three coordination spheres around the reference site i do contribute [4, 5] to
the sum (2), each being characterized by a small specific value of the overlap coefficients
Mean field Green functions of Hubbard model of superconductivity 4
νij (ν1 for the nearest neighbour (nn), ν2 for the next nearest neighbour (nnn), ν3 for
the third coordination spheres).
The quasi-particle spectrum and superconducting pairing for the Hamiltonian (1)
are obtained [11, 12] from the two-time 4× 4 GF matrix (in Zubarev notation [9])
G̃ijσ(t− t
′) = 〈〈X̂iσ(t) |X̂
′)〉〉 = −iθ(t− t′)〈{X̂iσ(t), X̂
jσ}〉, (3)
where 〈· · ·〉 denotes the statistical average over the Gibbs grand canonical ensemble.
The GF (3) is defined for the four-component Nambu column operator
X̂iσ = (X
⊤ (4)
where the superscript ⊤ denotes the transposition. In (3), X̂
jσ = (X
is the adjoint operator of X̂jσ.
The GF matrix in (r, ω)-representation is related to the expression (3) of the GF
matrix in (r, t)-representation by the non-unitary Fourier transform,
G̃ijσ(t− t
G̃ijσ(ω) e
−iω(t−t′) dω . (5)
The energy spectrum of the translation invariant spin lattice of (1) is solved in
the reciprocal space. The GF matrix in this (q, ω)-representation is related to the GF
matrix in (r, ω)-representation by the non-unitary discrete Fourier transform
G̃ijσ(ω) =
e−iq (rj−ri) G̃σ(q, ω). (6)
For an elemental GF of labels (αβ, γη), we use the notation 〈〈X
i (t)|X
′)〉〉 in
the (r, t)-representation and, similarly, 〈〈X
j 〉〉ω (assuming Hubbard operators at
t = 0), in the (r, ω)-representation. In the (q, ω)-representation, it is convenient to use
the notation Gαβ,γη(q, ω).
We shall consider henceforth the GMFA-GF, G̃0σ(q, ω). Its derivation involves:
(i) Differentiation of the GF (3) with respect to t and use of the equations of motion
for the Heisenberg operators X
i (t).
(ii) Derivation of an algebraic equation for G̃ijσ(ω), Eq. (5).
(iii) Elimination of the contribution of the inelastic processes to the commutator
Ẑiσ = [X̂iσ, H ] entering the equation of motion of G̃ijσ(ω).
(iv) Transformation to (q, ω)-representation of the obtained equation of G̃0ijσ(ω) by
means of the Fourier transform (6).
This finally yields
G̃0σ(q, ω) = χ̃
χ̃ω − Ãσ(q)
χ̃ , (7)
χ̃ = 〈{X̂iσ, X̂
iσ}〉, (8)
Ãσ(q) =
eiq (rj−ri) Ãijσ, rij = rj − ri , (9)
Ãijσ = 〈{[X̂iσ, H ], X̂
jσ}〉 . (10)
The matrix Ãijσ is Hermitian.
Mean field Green functions of Hubbard model of superconductivity 5
3. Translation invariance of the spin lattice
Four consequences follow from the translation invariance of the spin lattice.
• The definition of the Hubbard 1-form (2) over a translation invariant spin lattice
results in the identity (which secures the hermiticity of the Hamiltonian H):
αβ,γη
1,i = −τ
γη,αβ
1,i . (11)
• The Green function (3) of the model Hamiltonian (1) depends only on the distance
rij = |rj − ri| between the position vectors at the lattice sites i and j [9].
• The one-site statistical averages are independent on the site label i, 〈X
i 〉 = 〈X
(∀ i, j). For this reason, the site label in the one-site averages will be omitted.
• The two-site statistical averages 〈X
j 〉 remain invariant under the interchange
of the site labels i and j,
j 〉 = 〈X
i 〉, i 6= j (12)
4. Spin reversal invariance
The energy spectrum of the system described by the Hamiltonian (1) does not depend
on the specific values σ = ±1/2 of the spin projection. As a consequence, the definition
of the GF (3) either in terms of the σ-Nambu operator (4) or the σ̄-Nambu operator
X̂iσ̄ = (X
⊤ (13)
has to result in mathematically equivalent descriptions of the observables. This means,
however, that the mathematical structures of the frequency matrices Ãijσ, Eq. (10), and
Ãijσ̄ = 〈{[X̂iσ̄, H ], X̂
jσ̄}〉 emerging from the σ̄-Nambu operator (13), have to be related
to each other.
The identification of the existing relationships is constructive: we calculate and
compare the corresponding matrix elements of Ãijσ and Ãijσ̄. The multiplication rules
and the commutation/anticommutation relations satisfied by the Hubbard operators
result in the following general expression of the elemental anticommutators entering
their definitions:
i , H ], X
j } = δijC
λµ,νϕ
i + (1− δij)νijT
λµ,νϕ
ij , (14)
with one-site contributions given by
λµ,νϕ
i = δνµ
δµσ)E1 + δµ2E2
i +K11τ
0ϕ,σ0
1,i −K22τ
2ϕ,σ2
1,i +K21 ·2σ(τ
2ϕ,0σ̄
1,i +τ
0ϕ,2σ̄
1,i )
+δλ2(−E2X
i +K22
σϕ,2σ
1,i +K21
σ̄ϕ,σ0
1,i )−
−δλ0(K11
σϕ,0σ
1,i +K21
σϕ,σ̄2
1,i )
Mean field Green functions of Hubbard model of superconductivity 6
δλσ)E1 + δλ2E2
i +K11τ
ν0,0σ
1,i −K22τ
ν2,2σ
1,i +K21 · 2σ(τ
ν2,σ̄0
1,i +τ
ν0,σ̄2
1,i )
+δµ2(E2X
i +K22
νσ,σ2
1,i +K21
νσ̄,0σ
1,i )−
−δµ0(K11
νσ,σ0
1,i +K21
νσ,2σ̄
1,i )
δϕ0(K11τ
νµ,σ0
1,i + 2σK21τ
νµ,2σ̄
1,i )− δϕ2(K22τ
νµ,σ2
1,i − 2σK21τ
νµ,0σ̄
1,i )
δλ0(K11τ
νµ,0σ
1,i + 2σK21τ
νµ,σ̄2
1,i )− δλ2(K22τ
νµ,2σ
1,i − 2σK21τ
νµ,σ̄0
1,i )
δν0(K11τ
λϕ,0σ
1,i + 2σK21τ
λϕ,σ̄2
1,i )− δν2(K22τ
λϕ,2σ
1,i − 2σK21τ
λϕ,σ̄0
1,i )
δµ0(K11τ
λϕ,σ0
1,i + 2σK21τ
λϕ,2σ̄
1,i )− δµ2(K22τ
λϕ,σ2
1,i − 2σK21τ
λϕ,0σ̄
1,i )
and two-site contributions given by
λµ,νϕ
ij = δνµ
δµσ)(K11X
j −K22X
j )+(−δµ0K11+δµ2K22)
Xλσi X
δλσ)(−K11X
j +K22X
j )+(δλ0K11−δλ2K22)
δν0K11X
j − δν2K22X
+K21 ·2σ
j +δϕ2X
j +δν,−λ(X
δϕ0K11X
j − δϕ2K22X
+K21 ·2σ
j + δν2X
j + δϕ,−µ(X
δλ0K11X
j −δλ2K22X
j +K21 ·2σ(δµ0X
j +δµ2X
δµ0K11X
j −δµ2K22X
j +K21 ·2σ(δλ0X
j +δλ2X
2σ(δλ0δν2X
j −δλ2δν0X
j −δµ0δϕ2X
j +δµ2δϕ0X
The comparison of the results obtained from (14) for the corresponding matrix
elements of Ãijσ and Ãijσ̄ and the use of the translation invariance properties (11)
and (12) result in four distinct kinds of relationships:
• Under the spin reversal σ → σ̄, the following invariance properties hold for the
normal one-site statistical averages:
〈Xσσi 〉 = 〈X
i 〉 (15)
Mean field Green functions of Hubbard model of superconductivity 7
σ2,2σ
1,i 〉 = 〈τ
σ̄2,2σ̄
1,i 〉, 〈τ
0σ̄,σ̄0
1,i 〉 = 〈τ
0σ,σ0
1,i 〉 (16)
σ2,σ̄0
1,i 〉 = 2σ̄〈τ
σ̄2,σ0
1,i 〉 (17)
• The identity 〈C
σ2,0σ
0σ̄,σ̄2
i 〉 = 0 holds, therefrom we get for the one-site anomalous
averages,
〈X02i 〉 = 0 (18)
0σ̄,σ̄2
1,i 〉 = −〈τ
0σ,σ2
1,i 〉 (19)
0σ̄,0σ
1,i 〉 = 〈τ
σ2,σ̄2
1,i 〉 (20)
The first two equations imply that the contributions of the one-site terms 〈X02i 〉
0σ̄,σ̄2
1,i 〉 to the superconducting pairing vanish identically irrespective of the
model details (like, e.g., the relationship between the lattice constants ax and ay).
For a rectangular spin lattice (ax 6= ay), Eq. (20) points to the occurrence of a small
non-vanishing one-site contribution to the superconducting pairing originating
equally in both energy subbands. However, over the square spin lattice (1)
(ax = ay), each term of (20) vanishes for d-wave pairing due to the symmetry
in the reciprocal space [12].
• Under the spin reversal σ → σ̄, the following invariance properties hold for the
two-site statistical averages:
〈Xσσi X
j 〉 = 〈X
j 〉, 〈X
j 〉 = 〈X
j 〉 (21)
〈X22i X
j 〉 = 〈X
j 〉, 〈X
j 〉 = 〈X
j 〉 (22)
〈X02i X
j 〉 = 〈X
j 〉. (23)
• The operator of the number of particles at site i within the singlet subband, Ni, is
the sum of spin σ and σ̄ components,
Ni = niσ + niσ̄, niσ = X
i , niσ̄ = X
i . (24)
Similar relationships hold for the number of particles at site i within the hole
subband, Nhi ,
Nhi = n
iσ + n
iσ̄, n
iσ = X
i , n
iσ̄ = X
i . (25)
Due to the completeness relation,
Ni +N
i = 2, niσ + n
iσ = niσ̄ + n
iσ̄ = 1. (26)
These equalities simply reflect the fact that, at a given lattice site i, there is a single
spin state of predefined spin projection, whereas the total number of spin states
equals two.
Therefore, the operator Ni, Eq. (24), provides unique characterization of the
occupied states within the model [8, 12, 10].
Mean field Green functions of Hubbard model of superconductivity 8
5. Frequency matrix in (r, ω)-representation
A straightforward consequence of the results established in section 4 is the simplest
general expression of the frequency matrix Ãijσ, Eq. (10):
Ãijσ = δij
ĉσ 0̂
0̂ −(ĉσ̄)
+ (1− δij)
D̂ijσ ∆̂ijσ
(∆̂ijσ)
† −(D̂ijσ̄)
. (27)
The one-site 2× 2 matrix ĉσ is Hermitian, its elements do not depend on the particular
lattice site i,
ĉσ =
(E1 +∆)χ2 + a22 2σa21
2σa∗21 E1χ1 + a22
, (28)
and are expressed in terms of the spin reversal invariant quantities
χ2 = 〈niσ〉 = 〈niσ̄〉 (29)
χ1 = 〈n
iσ〉 = 〈n
iσ̄〉 = 1− χ2 (30)
a22 = K11〈τ
0σ̄,σ̄0
1 〉 − K22〈τ
σ2,2σ
1 〉 (31)
a21 = (K11 −K22) · 2σ〈τ
σ2,σ̄0
1 〉+K21(〈τ
0σ̄,σ̄0
1 〉 − 〈τ
σ2,2σ
1 〉). (32)
The normal hopping 2× 2 matrix D̂ijσ is symmetric,
D̂ijσ =
d22ij 2σd
2σd21ij d
Due to the constraints (21)–(22), the charge-spin correlations entering the matrix
elements of (33) get exactly decoupled from each other, such that
d22ij = K22(χ
ij + χ
ij)−K11χ
d11ij = K11[χ
ij + (χ1 − χ2)νij + χ
ij]−K22χ
d21ij = K21[(χ
ij − χ2νij) + χ
ij ]−K21χ
with the three spin reversal invariant weighted boson-boson correlation functions
representing respectively charge-charge (c), spin-spin (S), and singlet-hopping (s-h)
correlations:
χcij = νij〈NiNj〉/4, (34)
χSij = νij〈SiSj〉 (35)
χs−hij = νij〈X
j 〉 (36)
In (35), Si = (S
i , S
i ), with S
i = (X
i )/2 and S
i = X
The anomalous hopping 2× 2 matrix ∆̂ijσ has a very special form namely,
∆̂ijσ =
−K21 · 2σ
(K11 +K22)
(K11 +K22) K21 · 2σ
ij (37)
where the spin reversal invariant weighted boson-boson pairing (pair) correlation
function is given by
ij = νij〈X
i Nj〉 = 2νij〈X
j )〉 = (38)
= − νij〈N
i 〉 = −2νij〈(X
i 〉. (39)
Mean field Green functions of Hubbard model of superconductivity 9
In Eqs. (38) and (39), the derivation of the second expression from the first one makes
use of the spin reversal invariance property (23).
To get a workable expression of the frequency matrix, approximations have to be
derived for the boson-boson statistical averages entering the two-site hopping matrix
elements. In the next section we show that the method of reference [12], yielding
the pairing correlation function 〈X02i Nj〉 in terms of GMFA Green functions within
an approach able to identify and rule out exponentially small terms, can be extended to
the singlet hopping correlations 〈X02i X
j 〉 as well.
6. Hopping processes involving singlets
The right approach to the reduction of the order of correlation of the boson-boson
statistical averages 〈X02i X
j 〉 = 〈X
i 〉 goes differently for the hole-doped and
electron-doped cuprates.
• Reduction of the correlation order for hole-doped cuprates
In these systems, the Fermi level (the zero point energy) stays in the singlet
subband. We get the estimates E2 ≃ −∆, E2 − ∆ ≃ −2∆, E2 + ∆ ≃ 0. With
∆ ∼ 3eV , β∆ ∼ 3.5 · 104T−1. Therefore, at T . 300K, the quantities containing the
factor eβE2 ≃ e−β∆ . e−100 < 10−44 are negligible.
We start with the following form of the spectral theorem [9]
〈X02i X
j 〉 =
1 + e−βω
〈〈X02i |X
j 〉〉ω+iε − 〈〈X
j 〉〉ω−iε
, (40)
written for anticommutator retarded (ω + iε), respectively advanced (ω − iε) Green
functions. Their equation of motion in the (r, ω)-representation is
(ω−E2)〈〈X
j 〉〉ω ≃ 2〈X
j 〉+K21
0σ̄,0σ
1,i |X
j 〉〉ω−〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω
where, for the sake of simplicity, the labels ±iε, ε = 0+, describing respectively the
retarded and the advanced Green functions have been omitted. In Eq. (41), the higher
order r.h.s. contributions coming from the inband hopping terms have been dropped off.
Replacing (41) in (40), we get
〈X02i X
j 〉 ≃ K21
1 + e−βω
ω − E2 + iε
0σ̄,0σ
1,i |X
j 〉〉ω+iε−〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε
To evaluate the imaginary part, we use the identity [9]
ω − E2 + iε
ω −E2
− iπδ(ω − E2).
The integrals over the δ-function yield (finite) GF real parts at ω = E2, multiplied by
a thermodynamic factor ∼ e−β∆ ≪ 1. The imaginary part of the hole subband GF
Mean field Green functions of Hubbard model of superconductivity 10
0σ̄,0σ
1,i |X
j 〉〉ω+iε shows a δ-like maximum at ω = E2−∆, where (ω−E2)
−1 ≃ ∆−1 and
the thermodynamic factor reaches a value ∼ e−2∆. The only non-negligible contribution
to the principal part integral comes from the singlet subband GF 〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε the
imaginary part of which shows a δ-like maximum at ω = E2 +∆ ≃ 0. This allows us to
approximate (ω − E2)
−1 ≈ ∆−1 within the integral over the singlet subband GF to get
〈X02i X
j 〉 ≃ (1− δij)
2σ̄〈τ
σ2,σ̄2
1,i X
j 〉 (42)
Replacing this result in Eq. (38) and using (2) we get
ij ≃ (1− δij)
K21νij
4νij ·2σ̄〈X
m6=(i,j)
2σ〈Xσ2i X
m Nj〉
Omitting the three-site terms, we get the two-site approximation of the superconducting
pairing originating in the singlet subband,
ij ≃ (1− δij)
4K21ν
· 2σ̄〈Xσ2i X
j 〉, (44)
which reproduces the well-known two-site exchange term of the t-J model.
For the singlet hopping correlation function, (42) yields the two-site approximation
χs−hij ≃ (1− δij)
2K21ν
· 2σ̄〈Xσ2i X
j 〉 (45)
• Reduction of the correlation order for electron-doped cuprates
The Fermi level (the zero point energy) stays now in the hole subband. We have
the estimates E2 ≃ ∆, E2 +∆ ≃ 2∆, E2 −∆ ≃ 0.
It is convenient now to start with the alternative form of the spectral theorem [9]
i 〉 =
eβω + 1
〈〈X02i |X
j 〉〉ω+iε − 〈〈X
j 〉〉ω−iε
, (46)
with the retarded and advanced GFs following from the same equation (41).
Exponentially small quantities result from the δ-term of (ω − E2 + iε)
−1 and from
the singlet subband GF 〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε. The hole subband GF 〈〈τ
0σ̄,0σ
1,i |X
j 〉〉ω+iε,
yields the non-negligible contribution
i 〉 ≃ (1− δij)
2σ̄〈X
0σ̄,0σ
1,i 〉 (47)
Replacing in (39) and omitting the three-site terms, we get the two-site approximation
of the superconducting pairing originating in the hole subband,
ij ≃ (1− δij)
4K21ν
· 2σ〈X0σ̄i X
j 〉 (48)
Finally, the two-site approximation of the singlet-hopping correlation function is
〈X02i X
j 〉 ≃ (1− δij)
2K21ν
· 2σ̄〈X0σ̄i X
j 〉. (49)
In conclusion, the GMFA superconducting pairing is a second order effect. The
lowest order contribution to it originates in interband hopping correlating annihilation
Mean field Green functions of Hubbard model of superconductivity 11
(or creation) of pairs of spins at neighbouring lattice sites i and j within that energy
subband which crosses the Fermi level.
Similarly, the singlet hopping is a second order effect as well. It mainly proceeds
by interband i ⇄ j single particle jumps from the upper energy subband to the lower
energy subband.
7. Frequency matrix in (q, ω)-representation
The calculation of the matrix elements of Ãσ(q) from Eq. (9) asks for three essentially
different kinds of Fourier transforms, namely,
• The averages of the Hubbard 1-forms entering Eqs. (31) and (32) result in sums of
products of q-space averages and geometrical form factors:
λµ,νϕ
1,i 〉 =
〈XλµXνϕ〉qγα(q) (50)
for label sets {(λµ, νϕ)} ∈ {(0σ̄, σ̄0); (σ2, 2σ); (σ2, σ̄0)}.
The quantity 〈XλµXνϕ〉q denotes the average of the q-space image of the product
of Hubbard operators of labels λµ and νϕ respectively,
〈XλµXνϕ〉q =
1 + e−βω
Gλµ,νϕ(q, ω + iε)−Gλµ,νϕ(q, ω − iε)
Finally, in Eq. (50), γα(q) denote the nn (α = 1), nnn (α = 2), and third
neighbour (α = 3) geometrical form factors, γ1(q) = 2[cos(qxax) + cos(qyay)],
γ2(q) = 4 cos(qxax) cos(qyay), γ3(q) = 2[cos(2qxax) + cos(2qyay)].
• For the two-site weighted singlet hopping (36) and the superconducting pairing (38),
the Fourier transforms result in convolutions of specific averages and geometrical
form factors. The results are as follows:
− Singlet hopping
χs−h(q) =
ν2α ·
Ξkγα(q− k) (52)
where Ξk = 2σ〈X
σ2X σ̄0〉k, while Ξk = 2σ〈X
0σ̄X2σ〉k for hole-doped and electron-
doped cuprates respectively, with averages defined in (51).
− Superconducting pairing
χpair(q) =
ν2α ·
Πkγα(q− k) (53)
where Πk = 2σ̄〈X
σ2X σ̄2〉k, while Πk = 2σ〈X
0σ̄X0σ〉k for hole-doped and electron-
doped cuprates respectively, with averages defined in (51).
• The charge-charge and spin-spin correlation functions (34) and (35) are treated
approximately following [8, 10]:
Mean field Green functions of Hubbard model of superconductivity 12
– The order of the charge-charge correlation function 〈NiNj〉 is lowered using a
Hubbard type I approximation decoupling procedure 〈NiNj〉 ≃ 〈Ni〉〈Nj〉 = 2χ2.
– The spin-spin correlation function 〈SiSj〉 is kept undecoupled, but treated
phenomenologically. Eq. (2) implies the occurrence of up to three non-vanishing
spin-spin correlation functions: nn, χS1 = 〈SiSi±ax/y〉, nnn, χ
2 = 〈SiSi±ax±ay〉, and
χS3 = 〈SiSi±2ax/y〉. These are site independent quantities.
Using the above results, we get from (9) and (27) the mathematical structure of the
frequency matrix Ãσ(q) as follows,
Ãσ(q) =
Êσ(q) Φ̂σ(q)
(Φ̂σ(q))
† −(Êσ̄(q))
. (54)
The normal 2× 2 matrix contributions to Ãσ(q) show the characteristic σ-dependence,
Êσ(q) =
c22 2σc21
2σc∗21 c11
; −(Êσ̄(q))
−c22 2σc
2σc21 −c11
with the σ-independent terms cab carrying normal one-site and two-site matrix elements,
c22 ≡ c22(q) = (E1 +∆)χ2 + a22 + d22(q)
c11 ≡ c11(q) = E1χ1 + a22 + d11(q)
c21 ≡ c21(q) = a21 + d21(q)
dab(q) = Kab
ναγα(q)[χ
α + (−1)
a+bχaχb] +
s−h(q)
The one-site terms are defined by Eqs. (31)–(32) and (50). The exchange energy
parameters are given by
Jab = 4KabK21/∆, {ab} ∈ {22, 11, 21}, (56)
while the singlet hopping contribution χs−h(q) is given by Eq. (52).
The anomalous 2× 2 matrix contributions to Ãσ(q), obtained from (37), show the
characteristic σ-dependence,
Φ̂σ(q) =
−2σξ1b ξ2b
−ξ2b 2σξ1b
; (Φ̂σ(q))
−2σξ1b
∗ −ξ2b
∗ 2σξ1b
with ξ1 = J21, ξ2 = (J11 + J22)/2, whereas b ≡ b(q) is a shorthand notation for the
pairing matrix element (53).
Remark 1 The spin reversal σ → σ̄ symmetry properties of the elemental Green
functions entering the matrix GF (3) are identical to those established for the underlying
frequency matrix Ãσ(q).
Mean field Green functions of Hubbard model of superconductivity 13
8. GMFA Green function
From Eqs. (15) and (18) it follows that the matrix χ̃, Eq. (8), is diagonal and spin
reversal invariant, with two nonvanishing matrix elements,
χ̂ 0̂
0̂ χ̂
, χ̂ =
, 0̂ =
, (58)
where χ2 and χ1 are given by Eqs. (29) and (30) respectively.
Replacing in (7) the expressions (58) of the matrix χ̃ and (54) of the frequency
matrix Ãσ(q), we get a structure of the GMFA-GFmatrix obeying the general symmetry
properties established in [11],
G̃0σ(q, ω) =
Ĝ0σ(q, ω) F̂
σ (q, ω)
(F̂ 0σ (q, ω))
† −(Ĝ0σ̄(q,−ω))
, (59)
where the argument ω carries, in fact, the complex value ω + iε, ε = 0+. (Hence the
elemental GFs containing the argument ω point to retarded GFs, while those containing
the argument −ω point to advanced GFs.)
The normal 2× 2 matrix Ĝ0σ(q, ω) shows the characteristic σ-dependence,
Ĝ0σ(q, ω) =
g22(q, ω) 2σg21(q, ω)
2σg∗21(q, ω) g11(q, ω)
D(q, ω)
with the σ-independent components gab(q, ω) found from
gab(q, ω) = Aabω
3 +Babω
2 + Cabω +Dab, {ab} ∈ {22, 11, 21}.
Here the coefficients Aab are given respectively by
A22 = χ2, A11 = χ1, A21 = 0,
while Bab, Cab, Dab are q-dependent coefficients:
B22(q) = c22, B11(q) = c11, B21(q) = c21
C22(q) = − [χ2(c
11 + ξ
1 |b|
2) + χ1(|c21|
2 + ξ22 |b|
2)]/χ21
C11(q) = − [χ1(c
22 + ξ
1 |b|
2) + χ2(|c21|
2 + ξ22 |b|
2)]/χ22
C21(q) = [c21(χ2c11 + χ1c22)− ξ1ξ2|b|
2]/(χ1χ2)
D22(q) = −[c11(c22c11 − |c21|
2) + (c22ξ
1 + c11ξ
2 + 2ℜ(c21)ξ1ξ2)|b|
2]/χ21
D11(q) = −[c22(c22c11 − |c21|
2) + (c11ξ
1 + c22ξ
2 + 2ℜ(c21)ξ1ξ2)|b|
2]/χ22
D21(q) = {c21(c22c11 − |c21|
2)− [c∗21ξ
1 + c21ξ
2 + (c22 + c11)ξ1ξ2]|b|
2}/(χ1χ2)
The anomalous 2× 2 matrix F̂ 0σ (q, ω) shows the characteristic σ-dependence,
F̂ 0σ (q, ω) =
2σf22(q, ω) f21(q, ω)
−f21(q,−ω) 2σf11(q, ω)
D(q, ω)
Mean field Green functions of Hubbard model of superconductivity 14
with the elemental GFs fab(q, ω) given by
faa(q, ω) = (Paaω
2 +Raa)b, {aa} ∈ {22, 11},
f21(q, ω) = (P21ω
2 +Q21ω +R21)b.
Here, P22 = −ξ1, P11 = ξ1, and P21 = −ξ2 are q-independent, while
R22(q) = [(c
11 + c
21)ξ1 + 2c11c21ξ2 + ξ1(ξ
1 − ξ
2)|b|
2]/χ21
R11(q) = −[(c
22 + c
)ξ1 + 2c22c
21ξ2 + ξ1(ξ
1 − ξ
2)|b|
2]/χ22
R21(q) = [(c11c
21 + c22c21)ξ1 + (c22c11 + |c21|
2)ξ2 − ξ2(ξ
1 − ξ
2)|b|
2]/(χ1χ2)
Q21(q) = [(χ2c21 − χ1c
21)ξ1 + (χ2c11 − χ1c22)ξ2]/(χ1χ2).
The denominator D(q, ω) occurring in Eqs. (60) and (61), which is proportional to the
determinant of the matrix χ̃ω − Ãσ(q) in (7), shows the following monic bi-quadratic
dependence in ω:
D(q, ω) = (ω2 − uω + v)(ω2 + uω + v), (62)
where v = v(q) and u = u(q) are found respectively from
(c22c11−|c21|
2)− (ξ21 − ξ
2)|b|
[(c22+c11) + 2ℜ(c21)]
2ξ21 −
−4(c22+c11)ℜ(c21)ξ1(ξ1 − ξ2)− 4|c21|
2(ξ21−ξ
/(χ21χ
2) (63)
u2 − 2v =
(c211 + ξ
1 |b|
(c222 + ξ
1 |b|
(|c21|
2 + ξ22 |b|
2). (64)
A necessary consistency condition to be satisfied by the parameters of the model at any
vector q inside the Brillouin zone is v2(q) ≥ 0.
Remark 2 The zeros of the determinant of the GMFA-GF,
D(q, ω) = 0 (65)
provide the GMFA energy spectrum of the system.
At every wave vector q inside the Brillouin zone, this yields for the superconducting
state the energy eigenvalue set
{Ω1(q), Ω2(q), −Ω2(q), −Ω1(q)},
Ω1,2(q) = (u/2)±
(u/2)2 − v. (66)
In the normal state (b = 0), Eqs. (63) and (64) reduce respectively to
v0 = (c22/χ2)(c11/χ1)− |c21|
2/(χ1χ2)
u0 = (c22/χ2) + (c11/χ1)
such that the energy spectrum is given by the roots of the second order equation
ω2 − u0ω + v0 = 0 solved in [8].
Finally, if we assume a pure Hubbard model (i.e., energy band independent hopping
parameters, K11 = K22 = K21 ≡ t, [10]), then a significant simplification of the
equations derived in the last two sections is obtained. The normal 2× 2 matrix Êσ(q)
Mean field Green functions of Hubbard model of superconductivity 15
becomes symmetric and so is the normal GMFA-GF Ĝ0σ(q, ω). Moreover, there is a
single exchange energy parameter in (57), ξ1 = ξ2 ≡ J = 4t
2/∆, which simplifies the
anomalous 2×2 frequency matrix to Φ̂σ(q) =
−1 2σ
Jb, such that the quantities
u and v in the expression (62) of the GF determinant reduce to
v2 = [(c22c11 − c
2 + (c22 + c11 + 2c21)
2J2|b|2]/(χ21χ
2) (67)
u2 − 2v = [χ22c
11 + χ
22 + 2χ1χ2c
21 + J
2|b|2]/(χ21χ
2). (68)
A non-negative value v ≥ 0 always follows from Eq. (67), however, the reality of the
solutions (66) needs investigation of the domain of variation of the adjustable parameters
of the model.
9. Conclusions
The two-band Hubbard model of the high Tc superconductivity in cuprates [8, 12] uses
Hubbard operator algebra on a physical system characterized by specific invariance
symmetries with respect to translations and spin reversal.
In the present paper we have shown that the system symmetries result either in
invariance properties or exact vanishing of several characteristic statistical averages. The
vanishing of the one-site anomalous matrix elements is shown to be a property which
is embedded in the Hubbard operator algebra. Another worth mentioning consequence
following from the spin reversal invariance properties of the two-site statistical averages
is the exact decoupling from each other of the charge and spin correlations entering
the matrix elements of the frequency matrix. The use of these results allowed rigorous
derivation and simplification of the expression of the frequency matrix of the generalized
mean field approximation (GMFA) Green function (GF) matrix of the model.
For the higher order boson-boson averages 〈X02i X
j 〉 and 〈X
i Nj〉, which enter
respectively the normal singlet hopping and anomalous exchange pairing contributions
to the frequency matrix, an approximation procedure resulting in GMFA-GF expressions
was described. The procedure avoids the current decoupling schemes [14, 15]. Its
principle, first formulated in [12], consists in the identification and elimination of
exponentially small contributions to the spectral theorem representations of these
statistical averages.
A point worth noting is that the proper identification of exponentially small
quantities asks for the use of different starting expressions of the spectral theorem for
the hole-doped and electron-doped cuprates.
The results of the reduction procedure may be summarized as follows:
• The singlet hopping is a second order effect which may be described as interband
i ⇄ j single particle jumps from the upper to the lower energy subband.
• The GMFA superconducting pairing is a second order effect, the lowest order
contribution to which originates in interband hopping correlating the annihilation
Mean field Green functions of Hubbard model of superconductivity 16
(creation) of spin pairs at neighbouring lattice sites i and j within that energy
subband which crosses the Fermi level.
The derivation of the most general and simplest possible expressions of the
frequency matrix and of the GMFA-GF matrix in the (q, ω)-representation enables
reliable numerical investigation of the consequences coming from the adjustable
parameters of the model (the degree of hole/electron doping, the energy gap ∆, the
hopping parameters).
Another open question of the GF approach to the solution of the present model is
the use of the Hubbard operator algebra to get rigorous derivation and simplification of
the Dyson equation of the complete Green function. As shown previously in [12], the
self-energy corrections induce a spin fluctuation d-wave pairing originating in kinematic
interaction in the second order.
These investigations are underway and results will be reported in a forthcoming
paper.
Acknowledgments
The authors would like to express their gratitude to Prof. N.M. Plakida for useful
advice and critical reading of the manuscript. Partial financial support was secured by
the Romanian Authority for Scientific Research (Project 11404/31.10.2005 - SIMFAP).
References
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[2] F.C. Zhang F C and T.M. Rice T M 1988 Phys. Rev. B 37 3759
[3] Emery V J 1987 Phys. Rev. Lett. 58 2794; Varma C M Schmitt-Rink S, and Abrahams E 1987
Solid State Commun. 62 681
[4] Feiner L J, Jefferson J H and Raimondi R 1996 Phys. Rev. B 53 8751
[5] Yushankhai V Yu, Oudovenko V S and Hayn R 1997 Phys. Rev. B 55 15562
[6] Plakida N M and Oudovenko V S 1999 Phys. Rev. B 59, 11949
[7] Plakida N M 2001 JETP Lett. 74 36
[8] Plakida N M, Hayn R, and Richard J -L 1995 Phys. Rev. B 51 16599
[9] Zubarev D N 1960 Sov. Phys. Usp. 3 320
[10] Plakida N M and Oudovenko V S 2007 JETP 104 230
[11] Plakida N M 1997 Physica C 282–287 1737
[12] Plakida N M, Anton L, Adam S, and Adam Gh 2003 ZhETF 124, 367; English transl.: 2003 JETP
97 331
[13] Plakida N M 2006 Fiz. Nizkikh Temp. 32 483
[14] Roth L M 1969 Phys. Rev. 184 451
[15] Beenen J and Edwards D M 1995 Phys. Rev. B 52 13636; Avella A, Mancini F, Villani D and
Matsumoto H 1997 Physica C 282–287 1757; Di Matteo T, Mancini F, Matsumoto H and
Oudovenko V S 1997 Physica B 230–232 915; Stanescu T D, Martin I and Phillips Ph, 2000
Phys. Rev. B 62 4300
	Introduction
	Mean field approximation
	Translation invariance of the spin lattice
	Spin reversal invariance
	Frequency matrix in (r, )-representation
	Hopping processes involving singlets
	Frequency matrix in (q, )-representation
	GMFA Green function
	Conclusions

  The Green function (GF) equation of motion technique for solving the
effective two-band Hubbard model of high-T_c superconductivity in cuprates
[N.M. Plakida et al., Phys. Rev. B, v. 51, 16599 (1995); JETP, v. 97, 331
(2003)] rests on the Hubbard operator (HO) algebra. We show that, if we take
into account the invariance to translations and spin reversal, the HO algebra
results in invariance properties of several specific correlation functions. The
use of these properties allows rigorous derivation and simplification of the
expressions of the frequency matrix (FM) and of the generalized mean field
approximation (GMFA) Green functions (GFs) of the model.
  For the normal singlet hopping and anomalous exchange pairing correlation
functions which enter the FM and GMFA-GFs, an approximation procedure based on
the identification and elimination of exponentially small quantities is
described. It secures the reduction of the correlation order to GMFA-GF
expressions.

Introduction
A consistent theoretical model of the high critical temperature superconductivity in
cuprates is to be able to accommodate both the normal and superconducting states
under incorporation of the essential features of these systems (see, e.g., [1] for a review):
strong antiferromagnetic (AFM) superexchange interaction inside the CuO2 planes,
occurrence of two relatively isolated energy bands around the Fermi level, able to develop
dx2−y2 pairing: one stemming from single particle copper dx2−y2 states and the second
one from singlet doubly occupied states generated [2] by crystal field interaction; hopping
conduction for an extremely low density of the free charge carriers.
The p-d model [3], while incorporating all these features, is too cumbersome and
cell-cluster perturbation theory [4, 5] providing a hierarchy of the various interaction
terms was used to derive simpler models from it. Extreme limit cases of this reduction
procedure are various effective one-band t-J models (see, e.g., [6, 7] and references
therein) which, while unveiling the role played by the AFM exchange interaction in the
occurrence of the d-wave pairing, address exclusively the superconducting state.
The reduction of the p-d model to an effective two-band Hubbard model considered
by Plakida et al. [8], corroborated with the use of the equation of motion technique
for thermodynamic Green functions (GF) [9], provided the simplest approach to the
description of both the normal [8, 10] and the superconducting states [11, 12, 13] within
a frame securing rigorous fulfilment of the Pauli exclusion principle for fermionic states.
The Green function technique rests on the Hubbard operator algebra. Its rigorous
implementation onto a system characterized by specific symmetry properties (translation
invariant two-dimensional spin lattice, spin reversal invariance of the observables) results
either in characteristic invariance properties of several correlation functions, or in the
occurrence of some exactly vanishing correlation functions. The use of these results
allows rigorous derivation and simplification of the expressions of the frequency matrix
and of the generalized mean field approximation (GMFA) Green functions of the model.
The obtained expressions contain higher order boson-boson correlation functions
(CFs). For the CFs involving singlets (normal singlet hopping CFs and anomalous
exchange pairing CFs), an approximation procedure which avoids the usual decoupling
schemes and, yet, secures the correlation order reduction to GMFA-GF expressions,
under the identification and elimination of exponentially small quantities, is described.
The organization of the paper is as follows. Sec. 2 summarizes essentials of the
two-band Hubbard model and GMFA-GF equations. Sec. 3 describes the invariance
properties following from the translation invariance of the underlying spin lattice.
Sec. 4 derives invariance properties and constraints following from the invariance of the
macroscopic properties of the system under spin reversal. On the basis of the results of
Sec. 3 and 4, rigorous derivation of the frequency matrix in the (r, ω)-representation is
done in Sec. 5. The derivation of GMFA-GF expressions for the boson-boson correlation
functions involving singlets is discussed in Sec. 6.
Collecting together the results of sections 5 and 6, expressions of the frequency
Mean field Green functions of Hubbard model of superconductivity 3
matrix and of the GMFA Green function matrix are derived in the (q, ω)-representation
in sections 7 and 8 respectively. These results explicitly incorporate both hole-doping
and electron-doping features of the cuprate systems through the singlet hopping and
superconducting pairing terms.
The paper ends with conclusions in section 9.
2. Mean field approximation
The Hamiltonian of the effective two-band singlet-hole Hubbard model [8] is written in
the form
H = E1
Xσσi + E2
X22i +
σ0,0σ
1,i +K22
2σ,σ2
1,i +K21
2σ̄,0σ
1,i + τ
σ0,σ̄2
1,i ) (1)
The summation label i runs over the sites of an infinite two-dimensional (2D) square
array the lattice constants of which, ax = ay, are defined by the underlying single crystal
structure. The spin projection values in the sums over σ are σ = ±1/2, σ̄ = −σ.
The Hubbard operators (HOs) X
i = |iα〉〈iβ| are defined for the four states of the
model at each lattice site i: |0〉 (vacuum), |σ〉 = |↑〉 and |σ̄〉 = |↓〉 (single particle spin
states inside the hole subband), and |2〉 = |↑↓〉 (singlet state in the singlet subband).
The multiplication rule holdsX
i = δβγX
i . The HOs may be fermionic (single
spin state creation/annihilation in a subband) or bosonic (singlet creation/annihilation,
spin or charge densities, particle numbers). For a pair of fermionic HOs, the
anticommutator rule holds {X
i , X
j } = δij(δβγX
i + δηαX
i ) whereas, if one or both
HOs are bosonic, the commutation rule holds [X
i , X
j ] = δij(δβγX
i − δηαX
i ). At
each lattice site i, the constraint of no double occupancy of any quantum state |iα〉 is
rigorously fulfilled due to the completeness relation X00i +X
i = 1.
In (1), E1 = ε̃d−µ denotes the hole subband energy for the renormalized energy ε̃d
of a d-hole and the chemical potential µ. The energy parameter of the singlet subband
is E2 = 2E1 + ∆, where ∆ ≈ ∆pd = εp − εd is an effective Coulomb energy Ueff
corresponding to the difference between the two energy levels of the model.
In the description of the hopping processes, the label 1 points to the hole subband
and 2 to the singlet subband. The hopping energy parameter Kab = 2tpdKab depends on
tpd, the hopping p-d integral, and on energy band dependent form factors, Kab. Inband
(K11,K22) and interband (K21 = K12) processess are present. The Hubbard 1-forms
αβ,γη
1,i =
m (2)
incorporate the overall effects of specific hopping processes (through the labels (αβ, γη)
of the pair of Hubbard operators) involving the lattice site i and its neighbouring sites.
Up to three coordination spheres around the reference site i do contribute [4, 5] to
the sum (2), each being characterized by a small specific value of the overlap coefficients
Mean field Green functions of Hubbard model of superconductivity 4
νij (ν1 for the nearest neighbour (nn), ν2 for the next nearest neighbour (nnn), ν3 for
the third coordination spheres).
The quasi-particle spectrum and superconducting pairing for the Hamiltonian (1)
are obtained [11, 12] from the two-time 4× 4 GF matrix (in Zubarev notation [9])
G̃ijσ(t− t
′) = 〈〈X̂iσ(t) |X̂
′)〉〉 = −iθ(t− t′)〈{X̂iσ(t), X̂
jσ}〉, (3)
where 〈· · ·〉 denotes the statistical average over the Gibbs grand canonical ensemble.
The GF (3) is defined for the four-component Nambu column operator
X̂iσ = (X
⊤ (4)
where the superscript ⊤ denotes the transposition. In (3), X̂
jσ = (X
is the adjoint operator of X̂jσ.
The GF matrix in (r, ω)-representation is related to the expression (3) of the GF
matrix in (r, t)-representation by the non-unitary Fourier transform,
G̃ijσ(t− t
G̃ijσ(ω) e
−iω(t−t′) dω . (5)
The energy spectrum of the translation invariant spin lattice of (1) is solved in
the reciprocal space. The GF matrix in this (q, ω)-representation is related to the GF
matrix in (r, ω)-representation by the non-unitary discrete Fourier transform
G̃ijσ(ω) =
e−iq (rj−ri) G̃σ(q, ω). (6)
For an elemental GF of labels (αβ, γη), we use the notation 〈〈X
i (t)|X
′)〉〉 in
the (r, t)-representation and, similarly, 〈〈X
j 〉〉ω (assuming Hubbard operators at
t = 0), in the (r, ω)-representation. In the (q, ω)-representation, it is convenient to use
the notation Gαβ,γη(q, ω).
We shall consider henceforth the GMFA-GF, G̃0σ(q, ω). Its derivation involves:
(i) Differentiation of the GF (3) with respect to t and use of the equations of motion
for the Heisenberg operators X
i (t).
(ii) Derivation of an algebraic equation for G̃ijσ(ω), Eq. (5).
(iii) Elimination of the contribution of the inelastic processes to the commutator
Ẑiσ = [X̂iσ, H ] entering the equation of motion of G̃ijσ(ω).
(iv) Transformation to (q, ω)-representation of the obtained equation of G̃0ijσ(ω) by
means of the Fourier transform (6).
This finally yields
G̃0σ(q, ω) = χ̃
χ̃ω − Ãσ(q)
χ̃ , (7)
χ̃ = 〈{X̂iσ, X̂
iσ}〉, (8)
Ãσ(q) =
eiq (rj−ri) Ãijσ, rij = rj − ri , (9)
Ãijσ = 〈{[X̂iσ, H ], X̂
jσ}〉 . (10)
The matrix Ãijσ is Hermitian.
Mean field Green functions of Hubbard model of superconductivity 5
3. Translation invariance of the spin lattice
Four consequences follow from the translation invariance of the spin lattice.
• The definition of the Hubbard 1-form (2) over a translation invariant spin lattice
results in the identity (which secures the hermiticity of the Hamiltonian H):
αβ,γη
1,i = −τ
γη,αβ
1,i . (11)
• The Green function (3) of the model Hamiltonian (1) depends only on the distance
rij = |rj − ri| between the position vectors at the lattice sites i and j [9].
• The one-site statistical averages are independent on the site label i, 〈X
i 〉 = 〈X
(∀ i, j). For this reason, the site label in the one-site averages will be omitted.
• The two-site statistical averages 〈X
j 〉 remain invariant under the interchange
of the site labels i and j,
j 〉 = 〈X
i 〉, i 6= j (12)
4. Spin reversal invariance
The energy spectrum of the system described by the Hamiltonian (1) does not depend
on the specific values σ = ±1/2 of the spin projection. As a consequence, the definition
of the GF (3) either in terms of the σ-Nambu operator (4) or the σ̄-Nambu operator
X̂iσ̄ = (X
⊤ (13)
has to result in mathematically equivalent descriptions of the observables. This means,
however, that the mathematical structures of the frequency matrices Ãijσ, Eq. (10), and
Ãijσ̄ = 〈{[X̂iσ̄, H ], X̂
jσ̄}〉 emerging from the σ̄-Nambu operator (13), have to be related
to each other.
The identification of the existing relationships is constructive: we calculate and
compare the corresponding matrix elements of Ãijσ and Ãijσ̄. The multiplication rules
and the commutation/anticommutation relations satisfied by the Hubbard operators
result in the following general expression of the elemental anticommutators entering
their definitions:
i , H ], X
j } = δijC
λµ,νϕ
i + (1− δij)νijT
λµ,νϕ
ij , (14)
with one-site contributions given by
λµ,νϕ
i = δνµ
δµσ)E1 + δµ2E2
i +K11τ
0ϕ,σ0
1,i −K22τ
2ϕ,σ2
1,i +K21 ·2σ(τ
2ϕ,0σ̄
1,i +τ
0ϕ,2σ̄
1,i )
+δλ2(−E2X
i +K22
σϕ,2σ
1,i +K21
σ̄ϕ,σ0
1,i )−
−δλ0(K11
σϕ,0σ
1,i +K21
σϕ,σ̄2
1,i )
Mean field Green functions of Hubbard model of superconductivity 6
δλσ)E1 + δλ2E2
i +K11τ
ν0,0σ
1,i −K22τ
ν2,2σ
1,i +K21 · 2σ(τ
ν2,σ̄0
1,i +τ
ν0,σ̄2
1,i )
+δµ2(E2X
i +K22
νσ,σ2
1,i +K21
νσ̄,0σ
1,i )−
−δµ0(K11
νσ,σ0
1,i +K21
νσ,2σ̄
1,i )
δϕ0(K11τ
νµ,σ0
1,i + 2σK21τ
νµ,2σ̄
1,i )− δϕ2(K22τ
νµ,σ2
1,i − 2σK21τ
νµ,0σ̄
1,i )
δλ0(K11τ
νµ,0σ
1,i + 2σK21τ
νµ,σ̄2
1,i )− δλ2(K22τ
νµ,2σ
1,i − 2σK21τ
νµ,σ̄0
1,i )
δν0(K11τ
λϕ,0σ
1,i + 2σK21τ
λϕ,σ̄2
1,i )− δν2(K22τ
λϕ,2σ
1,i − 2σK21τ
λϕ,σ̄0
1,i )
δµ0(K11τ
λϕ,σ0
1,i + 2σK21τ
λϕ,2σ̄
1,i )− δµ2(K22τ
λϕ,σ2
1,i − 2σK21τ
λϕ,0σ̄
1,i )
and two-site contributions given by
λµ,νϕ
ij = δνµ
δµσ)(K11X
j −K22X
j )+(−δµ0K11+δµ2K22)
Xλσi X
δλσ)(−K11X
j +K22X
j )+(δλ0K11−δλ2K22)
δν0K11X
j − δν2K22X
+K21 ·2σ
j +δϕ2X
j +δν,−λ(X
δϕ0K11X
j − δϕ2K22X
+K21 ·2σ
j + δν2X
j + δϕ,−µ(X
δλ0K11X
j −δλ2K22X
j +K21 ·2σ(δµ0X
j +δµ2X
δµ0K11X
j −δµ2K22X
j +K21 ·2σ(δλ0X
j +δλ2X
2σ(δλ0δν2X
j −δλ2δν0X
j −δµ0δϕ2X
j +δµ2δϕ0X
The comparison of the results obtained from (14) for the corresponding matrix
elements of Ãijσ and Ãijσ̄ and the use of the translation invariance properties (11)
and (12) result in four distinct kinds of relationships:
• Under the spin reversal σ → σ̄, the following invariance properties hold for the
normal one-site statistical averages:
〈Xσσi 〉 = 〈X
i 〉 (15)
Mean field Green functions of Hubbard model of superconductivity 7
σ2,2σ
1,i 〉 = 〈τ
σ̄2,2σ̄
1,i 〉, 〈τ
0σ̄,σ̄0
1,i 〉 = 〈τ
0σ,σ0
1,i 〉 (16)
σ2,σ̄0
1,i 〉 = 2σ̄〈τ
σ̄2,σ0
1,i 〉 (17)
• The identity 〈C
σ2,0σ
0σ̄,σ̄2
i 〉 = 0 holds, therefrom we get for the one-site anomalous
averages,
〈X02i 〉 = 0 (18)
0σ̄,σ̄2
1,i 〉 = −〈τ
0σ,σ2
1,i 〉 (19)
0σ̄,0σ
1,i 〉 = 〈τ
σ2,σ̄2
1,i 〉 (20)
The first two equations imply that the contributions of the one-site terms 〈X02i 〉
0σ̄,σ̄2
1,i 〉 to the superconducting pairing vanish identically irrespective of the
model details (like, e.g., the relationship between the lattice constants ax and ay).
For a rectangular spin lattice (ax 6= ay), Eq. (20) points to the occurrence of a small
non-vanishing one-site contribution to the superconducting pairing originating
equally in both energy subbands. However, over the square spin lattice (1)
(ax = ay), each term of (20) vanishes for d-wave pairing due to the symmetry
in the reciprocal space [12].
• Under the spin reversal σ → σ̄, the following invariance properties hold for the
two-site statistical averages:
〈Xσσi X
j 〉 = 〈X
j 〉, 〈X
j 〉 = 〈X
j 〉 (21)
〈X22i X
j 〉 = 〈X
j 〉, 〈X
j 〉 = 〈X
j 〉 (22)
〈X02i X
j 〉 = 〈X
j 〉. (23)
• The operator of the number of particles at site i within the singlet subband, Ni, is
the sum of spin σ and σ̄ components,
Ni = niσ + niσ̄, niσ = X
i , niσ̄ = X
i . (24)
Similar relationships hold for the number of particles at site i within the hole
subband, Nhi ,
Nhi = n
iσ + n
iσ̄, n
iσ = X
i , n
iσ̄ = X
i . (25)
Due to the completeness relation,
Ni +N
i = 2, niσ + n
iσ = niσ̄ + n
iσ̄ = 1. (26)
These equalities simply reflect the fact that, at a given lattice site i, there is a single
spin state of predefined spin projection, whereas the total number of spin states
equals two.
Therefore, the operator Ni, Eq. (24), provides unique characterization of the
occupied states within the model [8, 12, 10].
Mean field Green functions of Hubbard model of superconductivity 8
5. Frequency matrix in (r, ω)-representation
A straightforward consequence of the results established in section 4 is the simplest
general expression of the frequency matrix Ãijσ, Eq. (10):
Ãijσ = δij
ĉσ 0̂
0̂ −(ĉσ̄)
+ (1− δij)
D̂ijσ ∆̂ijσ
(∆̂ijσ)
† −(D̂ijσ̄)
. (27)
The one-site 2× 2 matrix ĉσ is Hermitian, its elements do not depend on the particular
lattice site i,
ĉσ =
(E1 +∆)χ2 + a22 2σa21
2σa∗21 E1χ1 + a22
, (28)
and are expressed in terms of the spin reversal invariant quantities
χ2 = 〈niσ〉 = 〈niσ̄〉 (29)
χ1 = 〈n
iσ〉 = 〈n
iσ̄〉 = 1− χ2 (30)
a22 = K11〈τ
0σ̄,σ̄0
1 〉 − K22〈τ
σ2,2σ
1 〉 (31)
a21 = (K11 −K22) · 2σ〈τ
σ2,σ̄0
1 〉+K21(〈τ
0σ̄,σ̄0
1 〉 − 〈τ
σ2,2σ
1 〉). (32)
The normal hopping 2× 2 matrix D̂ijσ is symmetric,
D̂ijσ =
d22ij 2σd
2σd21ij d
Due to the constraints (21)–(22), the charge-spin correlations entering the matrix
elements of (33) get exactly decoupled from each other, such that
d22ij = K22(χ
ij + χ
ij)−K11χ
d11ij = K11[χ
ij + (χ1 − χ2)νij + χ
ij]−K22χ
d21ij = K21[(χ
ij − χ2νij) + χ
ij ]−K21χ
with the three spin reversal invariant weighted boson-boson correlation functions
representing respectively charge-charge (c), spin-spin (S), and singlet-hopping (s-h)
correlations:
χcij = νij〈NiNj〉/4, (34)
χSij = νij〈SiSj〉 (35)
χs−hij = νij〈X
j 〉 (36)
In (35), Si = (S
i , S
i ), with S
i = (X
i )/2 and S
i = X
The anomalous hopping 2× 2 matrix ∆̂ijσ has a very special form namely,
∆̂ijσ =
−K21 · 2σ
(K11 +K22)
(K11 +K22) K21 · 2σ
ij (37)
where the spin reversal invariant weighted boson-boson pairing (pair) correlation
function is given by
ij = νij〈X
i Nj〉 = 2νij〈X
j )〉 = (38)
= − νij〈N
i 〉 = −2νij〈(X
i 〉. (39)
Mean field Green functions of Hubbard model of superconductivity 9
In Eqs. (38) and (39), the derivation of the second expression from the first one makes
use of the spin reversal invariance property (23).
To get a workable expression of the frequency matrix, approximations have to be
derived for the boson-boson statistical averages entering the two-site hopping matrix
elements. In the next section we show that the method of reference [12], yielding
the pairing correlation function 〈X02i Nj〉 in terms of GMFA Green functions within
an approach able to identify and rule out exponentially small terms, can be extended to
the singlet hopping correlations 〈X02i X
j 〉 as well.
6. Hopping processes involving singlets
The right approach to the reduction of the order of correlation of the boson-boson
statistical averages 〈X02i X
j 〉 = 〈X
i 〉 goes differently for the hole-doped and
electron-doped cuprates.
• Reduction of the correlation order for hole-doped cuprates
In these systems, the Fermi level (the zero point energy) stays in the singlet
subband. We get the estimates E2 ≃ −∆, E2 − ∆ ≃ −2∆, E2 + ∆ ≃ 0. With
∆ ∼ 3eV , β∆ ∼ 3.5 · 104T−1. Therefore, at T . 300K, the quantities containing the
factor eβE2 ≃ e−β∆ . e−100 < 10−44 are negligible.
We start with the following form of the spectral theorem [9]
〈X02i X
j 〉 =
1 + e−βω
〈〈X02i |X
j 〉〉ω+iε − 〈〈X
j 〉〉ω−iε
, (40)
written for anticommutator retarded (ω + iε), respectively advanced (ω − iε) Green
functions. Their equation of motion in the (r, ω)-representation is
(ω−E2)〈〈X
j 〉〉ω ≃ 2〈X
j 〉+K21
0σ̄,0σ
1,i |X
j 〉〉ω−〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω
where, for the sake of simplicity, the labels ±iε, ε = 0+, describing respectively the
retarded and the advanced Green functions have been omitted. In Eq. (41), the higher
order r.h.s. contributions coming from the inband hopping terms have been dropped off.
Replacing (41) in (40), we get
〈X02i X
j 〉 ≃ K21
1 + e−βω
ω − E2 + iε
0σ̄,0σ
1,i |X
j 〉〉ω+iε−〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε
To evaluate the imaginary part, we use the identity [9]
ω − E2 + iε
ω −E2
− iπδ(ω − E2).
The integrals over the δ-function yield (finite) GF real parts at ω = E2, multiplied by
a thermodynamic factor ∼ e−β∆ ≪ 1. The imaginary part of the hole subband GF
Mean field Green functions of Hubbard model of superconductivity 10
0σ̄,0σ
1,i |X
j 〉〉ω+iε shows a δ-like maximum at ω = E2−∆, where (ω−E2)
−1 ≃ ∆−1 and
the thermodynamic factor reaches a value ∼ e−2∆. The only non-negligible contribution
to the principal part integral comes from the singlet subband GF 〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε the
imaginary part of which shows a δ-like maximum at ω = E2 +∆ ≃ 0. This allows us to
approximate (ω − E2)
−1 ≈ ∆−1 within the integral over the singlet subband GF to get
〈X02i X
j 〉 ≃ (1− δij)
2σ̄〈τ
σ2,σ̄2
1,i X
j 〉 (42)
Replacing this result in Eq. (38) and using (2) we get
ij ≃ (1− δij)
K21νij
4νij ·2σ̄〈X
m6=(i,j)
2σ〈Xσ2i X
m Nj〉
Omitting the three-site terms, we get the two-site approximation of the superconducting
pairing originating in the singlet subband,
ij ≃ (1− δij)
4K21ν
· 2σ̄〈Xσ2i X
j 〉, (44)
which reproduces the well-known two-site exchange term of the t-J model.
For the singlet hopping correlation function, (42) yields the two-site approximation
χs−hij ≃ (1− δij)
2K21ν
· 2σ̄〈Xσ2i X
j 〉 (45)
• Reduction of the correlation order for electron-doped cuprates
The Fermi level (the zero point energy) stays now in the hole subband. We have
the estimates E2 ≃ ∆, E2 +∆ ≃ 2∆, E2 −∆ ≃ 0.
It is convenient now to start with the alternative form of the spectral theorem [9]
i 〉 =
eβω + 1
〈〈X02i |X
j 〉〉ω+iε − 〈〈X
j 〉〉ω−iε
, (46)
with the retarded and advanced GFs following from the same equation (41).
Exponentially small quantities result from the δ-term of (ω − E2 + iε)
−1 and from
the singlet subband GF 〈〈τ
σ2,σ̄2
1,i |X
j 〉〉ω+iε. The hole subband GF 〈〈τ
0σ̄,0σ
1,i |X
j 〉〉ω+iε,
yields the non-negligible contribution
i 〉 ≃ (1− δij)
2σ̄〈X
0σ̄,0σ
1,i 〉 (47)
Replacing in (39) and omitting the three-site terms, we get the two-site approximation
of the superconducting pairing originating in the hole subband,
ij ≃ (1− δij)
4K21ν
· 2σ〈X0σ̄i X
j 〉 (48)
Finally, the two-site approximation of the singlet-hopping correlation function is
〈X02i X
j 〉 ≃ (1− δij)
2K21ν
· 2σ̄〈X0σ̄i X
j 〉. (49)
In conclusion, the GMFA superconducting pairing is a second order effect. The
lowest order contribution to it originates in interband hopping correlating annihilation
Mean field Green functions of Hubbard model of superconductivity 11
(or creation) of pairs of spins at neighbouring lattice sites i and j within that energy
subband which crosses the Fermi level.
Similarly, the singlet hopping is a second order effect as well. It mainly proceeds
by interband i ⇄ j single particle jumps from the upper energy subband to the lower
energy subband.
7. Frequency matrix in (q, ω)-representation
The calculation of the matrix elements of Ãσ(q) from Eq. (9) asks for three essentially
different kinds of Fourier transforms, namely,
• The averages of the Hubbard 1-forms entering Eqs. (31) and (32) result in sums of
products of q-space averages and geometrical form factors:
λµ,νϕ
1,i 〉 =
〈XλµXνϕ〉qγα(q) (50)
for label sets {(λµ, νϕ)} ∈ {(0σ̄, σ̄0); (σ2, 2σ); (σ2, σ̄0)}.
The quantity 〈XλµXνϕ〉q denotes the average of the q-space image of the product
of Hubbard operators of labels λµ and νϕ respectively,
〈XλµXνϕ〉q =
1 + e−βω
Gλµ,νϕ(q, ω + iε)−Gλµ,νϕ(q, ω − iε)
Finally, in Eq. (50), γα(q) denote the nn (α = 1), nnn (α = 2), and third
neighbour (α = 3) geometrical form factors, γ1(q) = 2[cos(qxax) + cos(qyay)],
γ2(q) = 4 cos(qxax) cos(qyay), γ3(q) = 2[cos(2qxax) + cos(2qyay)].
• For the two-site weighted singlet hopping (36) and the superconducting pairing (38),
the Fourier transforms result in convolutions of specific averages and geometrical
form factors. The results are as follows:
− Singlet hopping
χs−h(q) =
ν2α ·
Ξkγα(q− k) (52)
where Ξk = 2σ〈X
σ2X σ̄0〉k, while Ξk = 2σ〈X
0σ̄X2σ〉k for hole-doped and electron-
doped cuprates respectively, with averages defined in (51).
− Superconducting pairing
χpair(q) =
ν2α ·
Πkγα(q− k) (53)
where Πk = 2σ̄〈X
σ2X σ̄2〉k, while Πk = 2σ〈X
0σ̄X0σ〉k for hole-doped and electron-
doped cuprates respectively, with averages defined in (51).
• The charge-charge and spin-spin correlation functions (34) and (35) are treated
approximately following [8, 10]:
Mean field Green functions of Hubbard model of superconductivity 12
– The order of the charge-charge correlation function 〈NiNj〉 is lowered using a
Hubbard type I approximation decoupling procedure 〈NiNj〉 ≃ 〈Ni〉〈Nj〉 = 2χ2.
– The spin-spin correlation function 〈SiSj〉 is kept undecoupled, but treated
phenomenologically. Eq. (2) implies the occurrence of up to three non-vanishing
spin-spin correlation functions: nn, χS1 = 〈SiSi±ax/y〉, nnn, χ
2 = 〈SiSi±ax±ay〉, and
χS3 = 〈SiSi±2ax/y〉. These are site independent quantities.
Using the above results, we get from (9) and (27) the mathematical structure of the
frequency matrix Ãσ(q) as follows,
Ãσ(q) =
Êσ(q) Φ̂σ(q)
(Φ̂σ(q))
† −(Êσ̄(q))
. (54)
The normal 2× 2 matrix contributions to Ãσ(q) show the characteristic σ-dependence,
Êσ(q) =
c22 2σc21
2σc∗21 c11
; −(Êσ̄(q))
−c22 2σc
2σc21 −c11
with the σ-independent terms cab carrying normal one-site and two-site matrix elements,
c22 ≡ c22(q) = (E1 +∆)χ2 + a22 + d22(q)
c11 ≡ c11(q) = E1χ1 + a22 + d11(q)
c21 ≡ c21(q) = a21 + d21(q)
dab(q) = Kab
ναγα(q)[χ
α + (−1)
a+bχaχb] +
s−h(q)
The one-site terms are defined by Eqs. (31)–(32) and (50). The exchange energy
parameters are given by
Jab = 4KabK21/∆, {ab} ∈ {22, 11, 21}, (56)
while the singlet hopping contribution χs−h(q) is given by Eq. (52).
The anomalous 2× 2 matrix contributions to Ãσ(q), obtained from (37), show the
characteristic σ-dependence,
Φ̂σ(q) =
−2σξ1b ξ2b
−ξ2b 2σξ1b
; (Φ̂σ(q))
−2σξ1b
∗ −ξ2b
∗ 2σξ1b
with ξ1 = J21, ξ2 = (J11 + J22)/2, whereas b ≡ b(q) is a shorthand notation for the
pairing matrix element (53).
Remark 1 The spin reversal σ → σ̄ symmetry properties of the elemental Green
functions entering the matrix GF (3) are identical to those established for the underlying
frequency matrix Ãσ(q).
Mean field Green functions of Hubbard model of superconductivity 13
8. GMFA Green function
From Eqs. (15) and (18) it follows that the matrix χ̃, Eq. (8), is diagonal and spin
reversal invariant, with two nonvanishing matrix elements,
χ̂ 0̂
0̂ χ̂
, χ̂ =
, 0̂ =
, (58)
where χ2 and χ1 are given by Eqs. (29) and (30) respectively.
Replacing in (7) the expressions (58) of the matrix χ̃ and (54) of the frequency
matrix Ãσ(q), we get a structure of the GMFA-GFmatrix obeying the general symmetry
properties established in [11],
G̃0σ(q, ω) =
Ĝ0σ(q, ω) F̂
σ (q, ω)
(F̂ 0σ (q, ω))
† −(Ĝ0σ̄(q,−ω))
, (59)
where the argument ω carries, in fact, the complex value ω + iε, ε = 0+. (Hence the
elemental GFs containing the argument ω point to retarded GFs, while those containing
the argument −ω point to advanced GFs.)
The normal 2× 2 matrix Ĝ0σ(q, ω) shows the characteristic σ-dependence,
Ĝ0σ(q, ω) =
g22(q, ω) 2σg21(q, ω)
2σg∗21(q, ω) g11(q, ω)
D(q, ω)
with the σ-independent components gab(q, ω) found from
gab(q, ω) = Aabω
3 +Babω
2 + Cabω +Dab, {ab} ∈ {22, 11, 21}.
Here the coefficients Aab are given respectively by
A22 = χ2, A11 = χ1, A21 = 0,
while Bab, Cab, Dab are q-dependent coefficients:
B22(q) = c22, B11(q) = c11, B21(q) = c21
C22(q) = − [χ2(c
11 + ξ
1 |b|
2) + χ1(|c21|
2 + ξ22 |b|
2)]/χ21
C11(q) = − [χ1(c
22 + ξ
1 |b|
2) + χ2(|c21|
2 + ξ22 |b|
2)]/χ22
C21(q) = [c21(χ2c11 + χ1c22)− ξ1ξ2|b|
2]/(χ1χ2)
D22(q) = −[c11(c22c11 − |c21|
2) + (c22ξ
1 + c11ξ
2 + 2ℜ(c21)ξ1ξ2)|b|
2]/χ21
D11(q) = −[c22(c22c11 − |c21|
2) + (c11ξ
1 + c22ξ
2 + 2ℜ(c21)ξ1ξ2)|b|
2]/χ22
D21(q) = {c21(c22c11 − |c21|
2)− [c∗21ξ
1 + c21ξ
2 + (c22 + c11)ξ1ξ2]|b|
2}/(χ1χ2)
The anomalous 2× 2 matrix F̂ 0σ (q, ω) shows the characteristic σ-dependence,
F̂ 0σ (q, ω) =
2σf22(q, ω) f21(q, ω)
−f21(q,−ω) 2σf11(q, ω)
D(q, ω)
Mean field Green functions of Hubbard model of superconductivity 14
with the elemental GFs fab(q, ω) given by
faa(q, ω) = (Paaω
2 +Raa)b, {aa} ∈ {22, 11},
f21(q, ω) = (P21ω
2 +Q21ω +R21)b.
Here, P22 = −ξ1, P11 = ξ1, and P21 = −ξ2 are q-independent, while
R22(q) = [(c
11 + c
21)ξ1 + 2c11c21ξ2 + ξ1(ξ
1 − ξ
2)|b|
2]/χ21
R11(q) = −[(c
22 + c
)ξ1 + 2c22c
21ξ2 + ξ1(ξ
1 − ξ
2)|b|
2]/χ22
R21(q) = [(c11c
21 + c22c21)ξ1 + (c22c11 + |c21|
2)ξ2 − ξ2(ξ
1 − ξ
2)|b|
2]/(χ1χ2)
Q21(q) = [(χ2c21 − χ1c
21)ξ1 + (χ2c11 − χ1c22)ξ2]/(χ1χ2).
The denominator D(q, ω) occurring in Eqs. (60) and (61), which is proportional to the
determinant of the matrix χ̃ω − Ãσ(q) in (7), shows the following monic bi-quadratic
dependence in ω:
D(q, ω) = (ω2 − uω + v)(ω2 + uω + v), (62)
where v = v(q) and u = u(q) are found respectively from
(c22c11−|c21|
2)− (ξ21 − ξ
2)|b|
[(c22+c11) + 2ℜ(c21)]
2ξ21 −
−4(c22+c11)ℜ(c21)ξ1(ξ1 − ξ2)− 4|c21|
2(ξ21−ξ
/(χ21χ
2) (63)
u2 − 2v =
(c211 + ξ
1 |b|
(c222 + ξ
1 |b|
(|c21|
2 + ξ22 |b|
2). (64)
A necessary consistency condition to be satisfied by the parameters of the model at any
vector q inside the Brillouin zone is v2(q) ≥ 0.
Remark 2 The zeros of the determinant of the GMFA-GF,
D(q, ω) = 0 (65)
provide the GMFA energy spectrum of the system.
At every wave vector q inside the Brillouin zone, this yields for the superconducting
state the energy eigenvalue set
{Ω1(q), Ω2(q), −Ω2(q), −Ω1(q)},
Ω1,2(q) = (u/2)±
(u/2)2 − v. (66)
In the normal state (b = 0), Eqs. (63) and (64) reduce respectively to
v0 = (c22/χ2)(c11/χ1)− |c21|
2/(χ1χ2)
u0 = (c22/χ2) + (c11/χ1)
such that the energy spectrum is given by the roots of the second order equation
ω2 − u0ω + v0 = 0 solved in [8].
Finally, if we assume a pure Hubbard model (i.e., energy band independent hopping
parameters, K11 = K22 = K21 ≡ t, [10]), then a significant simplification of the
equations derived in the last two sections is obtained. The normal 2× 2 matrix Êσ(q)
Mean field Green functions of Hubbard model of superconductivity 15
becomes symmetric and so is the normal GMFA-GF Ĝ0σ(q, ω). Moreover, there is a
single exchange energy parameter in (57), ξ1 = ξ2 ≡ J = 4t
2/∆, which simplifies the
anomalous 2×2 frequency matrix to Φ̂σ(q) =
−1 2σ
Jb, such that the quantities
u and v in the expression (62) of the GF determinant reduce to
v2 = [(c22c11 − c
2 + (c22 + c11 + 2c21)
2J2|b|2]/(χ21χ
2) (67)
u2 − 2v = [χ22c
11 + χ
22 + 2χ1χ2c
21 + J
2|b|2]/(χ21χ
2). (68)
A non-negative value v ≥ 0 always follows from Eq. (67), however, the reality of the
solutions (66) needs investigation of the domain of variation of the adjustable parameters
of the model.
9. Conclusions
The two-band Hubbard model of the high Tc superconductivity in cuprates [8, 12] uses
Hubbard operator algebra on a physical system characterized by specific invariance
symmetries with respect to translations and spin reversal.
In the present paper we have shown that the system symmetries result either in
invariance properties or exact vanishing of several characteristic statistical averages. The
vanishing of the one-site anomalous matrix elements is shown to be a property which
is embedded in the Hubbard operator algebra. Another worth mentioning consequence
following from the spin reversal invariance properties of the two-site statistical averages
is the exact decoupling from each other of the charge and spin correlations entering
the matrix elements of the frequency matrix. The use of these results allowed rigorous
derivation and simplification of the expression of the frequency matrix of the generalized
mean field approximation (GMFA) Green function (GF) matrix of the model.
For the higher order boson-boson averages 〈X02i X
j 〉 and 〈X
i Nj〉, which enter
respectively the normal singlet hopping and anomalous exchange pairing contributions
to the frequency matrix, an approximation procedure resulting in GMFA-GF expressions
was described. The procedure avoids the current decoupling schemes [14, 15]. Its
principle, first formulated in [12], consists in the identification and elimination of
exponentially small contributions to the spectral theorem representations of these
statistical averages.
A point worth noting is that the proper identification of exponentially small
quantities asks for the use of different starting expressions of the spectral theorem for
the hole-doped and electron-doped cuprates.
The results of the reduction procedure may be summarized as follows:
• The singlet hopping is a second order effect which may be described as interband
i ⇄ j single particle jumps from the upper to the lower energy subband.
• The GMFA superconducting pairing is a second order effect, the lowest order
contribution to which originates in interband hopping correlating the annihilation
Mean field Green functions of Hubbard model of superconductivity 16
(creation) of spin pairs at neighbouring lattice sites i and j within that energy
subband which crosses the Fermi level.
The derivation of the most general and simplest possible expressions of the
frequency matrix and of the GMFA-GF matrix in the (q, ω)-representation enables
reliable numerical investigation of the consequences coming from the adjustable
parameters of the model (the degree of hole/electron doping, the energy gap ∆, the
hopping parameters).
Another open question of the GF approach to the solution of the present model is
the use of the Hubbard operator algebra to get rigorous derivation and simplification of
the Dyson equation of the complete Green function. As shown previously in [12], the
self-energy corrections induce a spin fluctuation d-wave pairing originating in kinematic
interaction in the second order.
These investigations are underway and results will be reported in a forthcoming
paper.
Acknowledgments
The authors would like to express their gratitude to Prof. N.M. Plakida for useful
advice and critical reading of the manuscript. Partial financial support was secured by
the Romanian Authority for Scientific Research (Project 11404/31.10.2005 - SIMFAP).
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Phys. Rev. B 62 4300
	Introduction
	Mean field approximation
	Translation invariance of the spin lattice
	Spin reversal invariance
	Frequency matrix in (r, )-representation
	Hopping processes involving singlets
	Frequency matrix in (q, )-representation
	GMFA Green function
	Conclusions
