A class of solvable models in Condensed Matter Physics
In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expressions for the Green’s fu...
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irk-123456789-1213292017-06-15T03:04:38Z A class of solvable models in Condensed Matter Physics Mancini, F. In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of motion closes and analytical expressions for the Green’s functions are obtained in terms of a finite number of parameters, to be self-consistently determined. Several examples are given. In particular, for these examples it is shown that in the one-dimensional case it is possible to derive by means of algebraic constraints a set of equations which allow us to determine the self-consistent parameters and to obtain a complete exact solution. 2006 Article A class of solvable models in Condensed Matter Physics / F. Mancini // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 393–402. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 71.10.-w, 71.10.Fd, 71.27.+a DOI:10.5488/CMP.9.2.393 http://dspace.nbuv.gov.ua/handle/123456789/121329 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any
dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of
the equations of motion closes and analytical expressions for the Green’s functions are obtained in terms of
a finite number of parameters, to be self-consistently determined. Several examples are given. In particular,
for these examples it is shown that in the one-dimensional case it is possible to derive by means of algebraic
constraints a set of equations which allow us to determine the self-consistent parameters and to obtain a
complete exact solution. |
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Article |
| author |
Mancini, F. |
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Mancini, F. A class of solvable models in Condensed Matter Physics Condensed Matter Physics |
| author_facet |
Mancini, F. |
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Mancini, F. |
| title |
A class of solvable models in Condensed Matter Physics |
| title_short |
A class of solvable models in Condensed Matter Physics |
| title_full |
A class of solvable models in Condensed Matter Physics |
| title_fullStr |
A class of solvable models in Condensed Matter Physics |
| title_full_unstemmed |
A class of solvable models in Condensed Matter Physics |
| title_sort |
class of solvable models in condensed matter physics |
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Інститут фізики конденсованих систем НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121329 |
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A class of solvable models in Condensed Matter Physics / F. Mancini // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 393–402. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT mancinif aclassofsolvablemodelsincondensedmatterphysics AT mancinif classofsolvablemodelsincondensedmatterphysics |
| first_indexed |
2025-07-08T19:39:56Z |
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2025-07-08T19:39:56Z |
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1837108926638718976 |
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Condensed Matter Physics 2006, Vol. 9, No 2(46), pp. 393–402
A class of solvable models in Condensed Matter
Physics
F.Mancini
Dipartimento di Fisica “E.R. Caianiello” Unita’ CNISM di Salerno,
Università degli Studi di Salerno, I–84081 Baronissi (SA), Italy
Received August 3, 2005
In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any
dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of
the equations of motion closes and analytical expressions for the Green’s functions are obtained in terms of
a finite number of parameters, to be self-consistently determined. Several examples are given. In particular,
for these examples it is shown that in the one-dimensional case it is possible to derive by means of algebraic
constraints a set of equations which allow us to determine the self-consistent parameters and to obtain a
complete exact solution.
Key words: strongly correlated electron systems, Hubbard model, Ising model, exact solution
PACS: 71.10.-w, 71.10.Fd, 71.27.+a
One of the most intriguing problems in Condensed Matter Physics is the study of highly inter-
acting fermionic systems [1]. The standard methods based on perturbation theories do not work
well and the attention is paid to the development of alternative approximate formulations. In this
context, it is interesting to cultivate the study of some solvable models, which are themselves of
physical interest and which may furnish some indication on the solution of more complex models.
In a recent paper [2], we have shown that there is a large class of fermionic systems for which
it is possible to find a complete set of eigenoperators and eigenvalues of the Hamiltonian. Then,
the hierarchy of the equations of motion closes and analytical exact expressions for the Green’s
functions (GF) can be obtained.
We consider a system of q species of particles, satisfying the Fermi statistics and localized on
the sites of a Bravais lattice. We suppose that the mass of the particles is very large and/or the
interaction is so strong that the kinetic energy is negligible and the particles are frozen on the lattice
sites. Let ca(i) and c†a(i) be the annihilation and creation operators of the particles of species a in
the Heisenberg picture: i = (i, t), where i stands for the lattice vector Ri. These operators satisfy
the canonical anti-commutation relations
{ca(i, t), c†b(j, t)} = δabδij ,
{ca(i, t), cb(j, t)} = {c†a(i, t), c
†
b(j, t)} = 0.
(1)
A quite general Hamiltonian for these systems reads as
H =
∑
ia
Va(i)na(i) +
1
2
∑
ij
∑
ab
Vab(i, j)na(i)nb(j)
+
1
3!
∑
ijl
∑
abc
Vabc(i, j, l)na(i)nb(j)nc(l) + · · · , (2)
where Va(i) represents an external field acting on the particle a; na(i) = c†a(i)ca(i) is the particle
density operator of the species a; Vab, Vabc , · · · are the two-body, three-body, . . . potentials. In
general, the potentials will be translationally invariant and will depend only on the differences
between the coordinates of the particles.
c© F.Mancini 393
F.Mancini
The class of systems described by the Hamiltonian (2) is very large and of direct physical
interest. By appropriate choice of the potentials it may describe multiband electronic systems
dominated by charge correlations and by magnetic interactions in the sz-channel, where sz is the
third component of the spin. Furthermore, let n(i) =
∑
a c
†
a(i)ca(i) be the total particle density
and consider the transformation n(i) = q/2 +S(i). Under this transformation the Hamiltonian (2)
is mapped to spin-q/2 Ising-like models.
Now, if the interaction potentials have a finite range, the model Hamiltonian is always solvable.
The proof of this statement is as follows. Firstly, it is immediate to see that the density operator
na(i) does not depend on time
i
∂
∂t
na(i) = [na(i), H ] = 0. (3)
Then, in order to use the equations of motion and GF formalism we must start from the Heisenberg
equation for the fermionic field. It is immediate to see that ca(i) satisfies the equation of motion
i
∂
∂t
ca(i)=Va(i)ca(i) +
∑
jb
Vab(i, j)nb(j)ca(i) +
1
2
∑
jl
∑
bc
Vabc(i, j, l)nb(j)nc(l)ca(i) + · · · (4)
The dynamics has generated other field operators of higher complexity. By taking time deriva-
tives of increasing order, more and more complex operators are generated. These operators might
be named composite operators, as they are all expressed in terms of the original fields ca(i). Due
to the finite range of the interaction, the higher-order field operators generated by the dynam-
ics extend over a finite number of sites. Then, since the particle operator satisfies the algebra
[na(i)]p = na(i) (for p > 1), the number of composite operators is finite and a complete set of
eigenoperators of the Hamiltonian can be found. If n is the number of independent composite
operators, we can construct a n-multiplet operator
ψ(i) =
ψ1(i)
ψ2(i)
...
ψn(i)
(5)
which satisfies the Heisenberg equation
i
∂
∂t
ψ(i) = [ψ(i), H ] = εψ(i), (6)
where the n × n matrix ε will be denominated as the energy matrix. Note that we are using a
vectorial notation: the field ψm(i) is itself a multiplet of rank q. Once the composite operator and
the energy matrix have been determined, an exact solution of the Hamiltonian can be formally
obtained. Let us define the retarded Green’s function
G(i, j) =
〈
R[ψ(i)ψ†(j)]
〉
= θ(ti − tj)
〈
{ψ(i), ψ†(j)}
〉
, (7)
where 〈· · · 〉 denotes the quantum-statistical average over the grand canonical ensemble. By means
of the Heisenberg equation (6), we obtain in momentum space the equation
[ω − ε]G(k, ω) = I(k), (8)
where I(k) is the Fourier transform of the normalization matrix, defined as
I(i, j) =
〈
{ψ(i, t), ψ†(j, t)}
〉
. (9)
The solution of equation (8) is
G(k, ω) =
n
∑
m=1
σ(m)(k)
ω − Em + iδ
, (10)
394
A class of solvable models in CMP
where Em are the eigenvalues of the energy matrix ε. The spectral density matrices σ(m)(k) are
calculated by means of the formula [3]
σ
(m)
αβ (k) = Ωαm
∑
γ
[Ωmγ ]−1Iγβ(k), (11)
where Ω is the matrix whose columns are the eigenvectors of the matrix ε. The correlation function
C(i, j) =
〈
ψ(i)ψ†(j)
〉
can be immediately calculated from (10) and one obtains
C(i, j) =
1
N
∑
k
1
2π
∫
dωeik·(Ri−Rj)−iEm(ti−tj)C(k, ω),
C(k, ω) = π
n
∑
m=1
δ(ω − Em)Tmσ
(m)(k) (12)
with Tm = 1+tanh (βEm/2), β is the inverse temperature. By similar technique we can easily cal-
culate [4] multi-point correlation functions as C(i, j; l1, l2, · · · ls) =
〈
ψ(i)ψ†(j)n(l1)n(l2) · · ·n(ls)
〉
.
Equations (10) and (12) are an exact solution of the model Hamiltonian (2). However, the
knowledge of the GF is not fully achieved yet. The algebra of the field ψ(i) is not canonical: as a
consequence, the normalization matrix I(k) in the equation of motion (8) contains some unknown
static correlation functions. Generally, these correlators are expectation values of operators not
belonging to the chosen basis and should be self-consistently calculated. According to the scheme of
calculations proposed by the composite operator method [3,5], one way of calculating the unknown
correlators is by specifying the representation where the GF are realized. The knowledge of the
Hamiltonian and of the operatorial algebra is not sufficient to completely determine the GF. The GF
refer to a specific representation (i.e., to a specific choice of the Hilbert space) and this information
must be supplied to the equations of motion that alone are not sufficient to completely determine
the GF. The procedure is as follows. From the algebra it is possible to derive several relations
among the operators. We shall call algebra constraints (AC) all possible relations among the
operators dictated by the algebra. This set of relations valid at microscopic level must be also
satisfied at macroscopic level, when expectation values are considered. These considerations lead
to some self-consistent equations which will be used to fix the unknown correlators appearing in
the normalization matrix. An immediate set of rules is given by the equation
〈
ψ(i)ψ†(i)
〉
=
1
N
∑
k
1
2π
∫
dωC(k, ω), (13)
where the l.h.s. is fixed by the AC and by the boundary conditions compatible with the phase
under investigation, while in the r.h.s. the correlation function C(k, ω) is computed by means of
the expression (12). The use of (13) will lead to a set of exact relations among the correlation
functions. Unfortunately, for the class of systems considered, the number of equations is always
less than the number of correlation functions and extra equations are needed. To this end we have
recently found [2,4] a set of AC capable of giving other relations among the correlation functions.
Let us suppose that there exist some operators, O, which project out of the Hamiltonian a reduced
part
OH = OH0 . (14)
When H0 and HI = H −H0 commute, an important rule which descends from the AC (14) is that
the quantum statistical averages over the complete Hamiltonian H must coincide with the average
over the reduced Hamiltonian H0
Tr{Oe−βH} = Tr{Oe−βH0}. (15)
For a variety of one-dimensional models, we have found that the use of the condition (15) allows
us to close the set of equations for the correlation functions and to obtain an exact solution of the
395
F.Mancini
model. For dimensions higher than one these conditions are not sufficient and more equations are
necessary.
We now apply the above procedure to some specific models, obtained by special choice of the
interaction potentials. We shall consider a d-dimensional hypercubic Bravais lattice.
First example
We consider a q-state model described by the Hamiltonian
H = −µ
∑
i
n(i) + dV
∑
i
n(i)nα(i), (16)
where µ is the chemical potential, n(i) =
∑
a na(i) is the total particle density. Hereafter, for a
generic operator Φ(i) we use the notation Φα(i) =
∑
j αijΦ(j, t), where αij is the projector on the
first-nearest neighbor sites. This model might describe a system of q-electron bands, interacting
through an intersite Coulomb potential of strength V . Alternatively, by means of the transformation
n(i) = q/2 + S(i) the Hamiltonian (16) takes the form
H = −h
∑
i
S(i) − J
∑
i
S(i)Sα(i) + E0 , (17)
where J = −dV , h = µ−qdV , E0 = q/2(−µ+q/2·dV )N ;N is the number of sites. Hamiltonian (17)
is just the spin-q/2 Ising model with nearest neighbor interactions in the presence of an external
magnetic field h. The following recursion rule can be established [4] for the operator [nα(i)]p
[nα(i)]p =
2qd
∑
m=1
A(p)
m [nα(i)]m, (18)
where the coefficients A
(p)
m are rational numbers, satisfying the relation
∑2qd
m=1A
(p)
m = 1, that can
be easily determined by the algebra and by the structure of the lattice [4]. Owing to this rule, the
set of eigenoperators of the Hamiltonian is given by
ψ(i) =
c(i)
c(i)nα(i)
...
c(i)[nα(i)]2qd
, [ψ(i), H ] = εψ(i), (19)
where the energy matrix ε can be calculated by means of the equation of motion and the recursion
rule (18). The eigenvalues Em of ε are given by
Em = −µ+ (m− 1)V, (m = 1, 2, · · ·2qd+ 1). (20)
The retarded Green’s functions and the correlation functions can be exactly calculated by applying
the scheme of calculations illustrated above. Then, by using the AC (15) it is possible to derive
the self-consistent equations
κ(k−1) − λ(k−1) =
1
2
2qd+1
∑
m=1
Tmσ
(m)
1,k , (k = 1, . . . , 2qd+ 1), (21)
where Tm = 1 + tanh (βEm/2) and σ(m) are the spectral density matrices, that can be calculated
by means of (11) and are expressed in terms of the quantities κ(p). Equations (21) give a set of
exact relations, valid for any value of q and d, among the correlation functions
κ(p) = 〈[nα(i)]p〉 , (p = 0, 1, · · ·2qd),
λ(p) = 〈n(i)[nα(i)]p〉 . (22)
396
A class of solvable models in CMP
Unfortunately, the number of equations is not sufficient and we need other conditions. We have
studied the cases of q = 1 ,2, 3 in [4,6,7], respectively. In all these cases for one-dimensional systems,
it is possible to find by means of the algebraic constraints (15) the necessary equations to close
the set. These extra conditions are obtained by exploiting the following algebraic conditions. Let
us divide the Hamiltonian as
H = H0 + 2V n(i)nα(i), (23)
where, due to translational invariance, i is a generic site of the infinite chain. Then, we have
q = 1, c†(i)e−βH = c†(i)e−βH0 , (24)
q = 2, ξ†(i)e−βH = ξ†(i)e−βH0 ,
η†(i)e−βH = η†(i)
{
1 +
4
∑
m=1
fm[nα(i)]m
}
e−βH0 , (25)
q = 3, ξ†(i)e−βH = ξ†(i)e−βH0 ,
η†(i)e−βH = η†(i)
{
1 +
6
∑
m=1
fm[nα(i)]m
}
e−βH0 ,
ζ†(i)e−βH = ζ†(i)
{
1 +
6
∑
m=1
(2fm + gm)[nα(i)]m
}
e−βH0 . (26)
For q = 2 the definitions are:
ξa(i) = [1 − n(i)]ca(i), ηa(i) = n(i)ca(i). (27)
The operators ξa(i) and ηa(i) induce the transitions 0 ⇔ 1, 1 ⇔ 2, respectively. For q = 3 the
definitions are:
ξa(i) = [1 − n(i) +D(i)]ca(i), ηa(i) = [n(i) − 2D(i)]ca(i), ζa(i) = D(i)ca(i), (28)
where D(i) is the double occupancy operator, defined as
D(i) = n1(i)n2(i) + n1(i)n3(i) + n2(i)n3(i). (29)
The projection operators ξa, ηa and ζa induce the transitions among states with different particle
numbers: 0 ⇔ 1, 1 ⇔ 2, 2 ⇔ 3, respectively. The quantities fm and gm are the known functions
of βV .
Second example
Let us consider two species of particles, say a and b, and consider the Hamiltonian
H = −µ
∑
i
n(i) + U
∑
i
D(i) + dV
∑
i
n(i)nα(i), (30)
where n(i) = na(i)+nb(i) andD(i) = na(i)nb(i) are the total particle density and double occupancy
operators, respectively. This Hamiltonian is just the extended Hubbard model in the ionic limit,
where U and V are the on-site and inter-site Coulomb interactions, respectively. The two species
of particles, a and b, are in this case the electrons with spin up and down, respectively. By means
of the transformation n(i) = 1 + S(i) , (30) can be cast in the form
H = −h
∑
i
S(i) + ∆
∑
i
S2(i) − J
∑
i
S(i)Sα(i) + E0 , (31)
397
F.Mancini
where J = −dV , h = µ− 2dV − U/2, ∆ = U/2, E0 = (−µ+ dV )N . Hamiltonian (31) is just the
Ising spin-1 model with nearest-neighbor interactions in the presence of a crystal field ∆ and an
external magnetic field h [8–11]. We now define the composite operators
ψ(ξ)(i) =
ξ(i)
ξ(i)nα(i)
...
ξ(i)[nα(i)]4d
, ψ(η)(i) =
η(i)
η(i)nα(i)
...
η(i)[nα(i)]4d
, (32)
where ξ(i) and η(i) are the Hubbard operators, defined in (27). By means of (18), these fields are
eigenoperators of the Hamiltonian (30)
i
∂
∂t
ψ(ξ)(i) = [ψ(ξ)(i), H ] = ε(ξ)ψ(ξ)(i), i
∂
∂t
ψ(η)(i) = [ψ(η)(i), H ] = ε(η)ψ(η)(i), (33)
where ε(ξ) and ε(η) are the energy matrices, of rank (4d+ 1) × (4d+ 1) , which can be calculated
by means of the equations of motion and the recursion rule (18). The eigenvalues of the energy
matrices are given by
E(ξ)
m = −µ+ (m− 1)V, E(η)
m = −µ+ U + (m− 1)V, (m = 1, 2, · · · 4qd+ 1). (34)
The retarded and the correlation functions can be exactly calculated by applying the scheme of
calculations illustrated above. Then, by using the AC (13) it is possible to derive the self-consistent
equations
κ(k−1) −
1
2
λ(k−1) =
1
2
4d+1
∑
m=1
[
T (ξ)
m σ
(ξ,m)
1,k + T (η)
m σ
(η,m)
1,k
]
, (k = 1, · · · 4d+ 1), (35)
where T
(a)
m = 1 + tanh
(
βE
(a)
m /2
)
with a = ξ, η. σ(a,m) are the spectral density matrices, that can
be calculated by means of (11) and are expressed in terms of the quantities κ(p) and λ(p). Equations
(35) give a set of exact relations, valid for any dimension, among the correlation functions (22).
Unfortunately, the number of equations is not sufficient and we need other conditions. Our study
[6] has shown that for the one-dimensional case, it is possible to find by means of the algebraic
constraints (15) the necessary equations to close the set (35). These extra conditions are obtained
by exploiting the following algebraic conditions
ξ†(i)e−βH = ξ†(i)e−βH0 ,
D(i)e−βH = D(i)
{
1 +
4
∑
m=1
(2fm + gm)[nα(i)]m
}
e−βH0 ,
(36)
where H0 = H − 2V n(i)nα(i).
Third example
The model (30) can be generalized by considering 3- and 4-body potentials. In particular, by
considering one-dimensional systems, let us take
H = −µ
∑
i
n(i) − γ
∑
i
D(i) + V
∑
i
n(i)nα(i)
+ U
∑
i
D(i)Dα(i) − U
∑
i
D(i)nα(i). (37)
This Hamiltonian can be mapped to the following model
H = −h
∑
i
S(i) + ∆
∑
i
S2(i) − J
∑
i
S(i)Sα(i) −K
∑
i
S2(i)S2α(i) + E0, (38)
398
A class of solvable models in CMP
where J = U/4 − V , K = −U/4, h = µ + γ/2 + U/2 − 2V , ∆ = −U/2 − γ/2, E0 = (−µ + V )N .
This Hamiltonian is just the one-dimensional Blume-Emery-Griffiths model [12]. We now define
the composite operators
ψ(ξ)(i) =
ξ(i)
ξ(i)nα(i)
ξ(i)[nα(i)]2
ξ(i)[nα(i)]3
ξ(i)[nα(i)]4
ξ(i)Dα(i)
ξ(i)[Dα(i)]2
, ψ(η)(i) =
η(i)
η(i)nα(i)
η(i)[nα(i)]2
η(i)[nα(i)]3
η(i)[nα(i)]4
η(i)Dα(i)
η(i)[Dα(i)]2
, (39)
where ξ(i) and η(i) are the Hubbard operators, [cf. (27)]. These fields are eigenoperators of the
Hamiltonian (37). The corresponding eigenvalues are
E(ξ)
m =
−µ
−µ+ V
−µ+ 2V
−µ− U + 2V
−µ− 1
2U + 2V
−µ− U + 4V
−µ− 1
2U + 3V
, E(η)
m =
−µ− γ
−µ− γ − 1
2U + V
−µ− γ − U + 2V
−µ− γ + 2V
−µ− γ − 1
2U + 2V
−µ− γ − U + 4V
−µ− γ − U + 3V
. (40)
The retarded and the correlation functions can be exactly calculated by applying the scheme of
calculations illustrated above. Then, by using the AC (13) it is possible to derive [13] the self-
consistent equations
λ(0) − δ(1) =
1
2
7
∑
m=1
T (η)
m σ
(η,m)
1,1 ,
κ(k−1) −
1
2
λ(k−1) =
1
2
7
∑
m=1
[
T (ξ)
m σ
(ξ,m)
1,k + T (η)
m σ
(η,m)
1,k
]
, (k = 1, · · · 5),
δ(k−5) −
1
2
θ(k−5) =
1
2
7
∑
m=1
[
T (ξ)
m σ
(ξ,m)
1,k + T (η)
m σ
(η,m)
1,k
]
, (k = 6, 7). (41)
where the definitions are the same as in example 2, and the new correlation functions are defined as
δ(p) = 〈[Dα(i)]p〉
θ(p) = 〈n(i)[Dα(i)]p〉 , (p = 1, 2). (42)
σ(a,m) are the spectral density matrices and are expressed in terms of the quantities κ(p), λ(p), δ(p),
θ(p). Equations (41) give a set of exact relations among the correlation functions (22) and (42).
By exploiting the algebraic condition ξ†(i)e−βH = ξ†(i)e−βH0 it is possible to derive [13] the
self-consistent equations
C
(ξ)
1,3 = C
(ξ)
1,1
[
1
2
X1 +X2 +
1
2
X2
1
]
,
C
(ξ)
1,4 = C
(ξ)
1,1
[
1
4
X1 +
3
2
X2 +
3
2
X1X2 +
3
4
X2
1
]
,
C
(ξ)
1,5 = C
(ξ)
1,1
[
1
8
X1 +
7
4
X2 +
9
2
X1X2 +
7
8
X2
1 +
3
2
X2
2
]
,
C
(ξ)
1,7 = C
(ξ)
1,1
[
1
2
X2 +
1
2
X2
2
]
, (43)
399
F.Mancini
where
X1 =
C
(ξ)
1,2
C
(ξ)
1,1
, X2 =
C
(ξ)
1,6
C
(ξ)
1,1
, (44)
C
(ξ)
1,k is an equal time correlation function C
(ξ)
1,k =
〈
ψ
(ξ)
1 (i)ψ
(ξ)†
1 (i)
〉
, expressed in terms of the
parameters κ(p), λ(p), δ(p), θ(p) by means of the relation
C
(ξ)
1,k =
1
2
7
∑
m=1
T (ξ)
m σ
(ξ,m)
1,k . (45)
Fourth example
Let us consider the case of two particles, characterized by spin σ =↑, ↓ (+,−), and the choice
the potentials as
Vσ(i) = −µ− σh, Vσ,σ′ (i, j) = 2V d(2δσ,σ′ − 1)αi,j , (46)
h is the intensity of the external magnetic field. The Hamiltonian is
H = −µ
∑
i
n(i) + dV
∑
i
n3(i)n
α
3 (i) − h
∑
i
n3(i), (47)
where n3(i) = n↑(i) − n↓(i) is the third component of the spin density operator. Let us restrict
the analysis to one-dimensional systems. The following recursion rule can be established for the
operator [nα
3 (i)]p
[nα
3 (i)]p =
4
∑
m=1
A(p)
m [nα
3 (i)]m, (48)
where the coefficients A
(p)
m are rational numbers, satisfying the relation
∑4
m=1A
(p)
m = 1, that can
be easily determined by the algebra and the structure of the lattice. Owing to this rule, the set of
eigenoperators of the Hamiltonian (47) is given by
ψ(i) =
(
ψ↑(i)
ψ↓(i)
)
, ψσ(i) =
cσ(i)
cσ(i)nα
3 (i)
cσ(i)[nα
3 (i)]2
cσ(i)[nα
3 (i)]3
cσ(i)[nα
3 (i)]4
, (49)
i
∂
∂t
ψσ(i) = ε(σ)ψσ(i). (50)
The energy matrix ε(σ) is a 5 × 5 matrix with the following expression
ε(σ) =
−µ− σh 2σV 0 0 0
0 −µ− σh 2σV 0 0
0 0 −µ− σh 2σV 0
0 0 0 −µ− σh 2σV
0 − 1
2σV 0 5
2σV −µ− σh
. (51)
The eigenvalues of the matrix ε(σ) are
E(σ)
m =
−µ− σh
−µ− σh− 2V
−µ− σh− V
−µ− σh+ V
−µ− σh+ 2V
. (52)
400
A class of solvable models in CMP
The retarded Green’s functions and the correlation functions can be exactly calculated by applying
the scheme of calculations illustrated above. Then, by using the AC (13) it is possible to derive
the self-consistent equations
2κ(k−1) − λ(k−1) =
1
2
5
∑
m=1
[T (↑)
m σ
(↑,m)
1,k + T (↓)
m σ
(↓,m)
1,k ],
θ(k−1) =
1
2
5
∑
m=1
[T (↑)
m σ
(↑,m)
1,k − T (↓)
m σ
(↓,m)
1,k ], (k = 1, · · · 5), (53)
where T
(σ)
m = 1 + tanh
(
βE
(σ)
m /2
)
and σ(σ,m) are the spectral density matrices, that can be calcu-
lated by means of (11)and are expressed in terms of the quantities κ(p). Equations (53) give a set
of exact relations among the correlation functions
κ(p) = 〈[nα
3 (i)]p〉 ,
λ(p) = 〈n(i)[nα
3 (i)]p〉 ,
θ(p) = 〈n3(i)[n
α
3 (i)]p〉 , (p = 0, 1, . . . , 4). (54)
For this model we can also use the AC (15) in order to derive extra equations capable of closing
the set (53) for the correlation functions. Details will be presented elsewhere.
Summarizing, we have shown that a large class of systems of localized particles, satisfying Fermi
statistics and subject to finite-range interactions, is always solvable, in the sense that a complete
finite set of eigenoperators and eigenvalues of the Hamiltonian can be found. This knowledge allows
us to derive analytical expressions for the Green’s functions and for the correlation functions and
a set of exact relations among the correlation functions can be derived. As an illustration we have
considered several examples. In all the studied models we have shown that for the case of one
dimension it is possible to use algebraic constraints which permit to close the set of self-consistent
equations and to obtain exact solutions. For higher dimensions more self-consistent equations are
needed. This problem is now under investigation.
References
1. Fulde P. Electron Correlations in Molecules and Solids. Springer-Verlag, Berlin, 1995.
2. Mancini F., Europhys. Lett., 2005, 70, 485.
3. Mancini F., Avella A., Eur. Phys. J. B, 2003, 36, 37.
4. Mancini F., Eur. Phys. J. B, 2005, 45, 497.
5. Mancini F., Avella A., Adv. Phys., 2004, 53, 537.
6. Mancini F., Eur. Phys. J. B, 2005, 47, 527.
7. Avella A., Mancini F., Eur. Phys. J. B, 2006, 50, 527.
8. Blume M., Phys. Rev., 1966, 141, 517.
9. Capel H., Physica, 1966, 32, 966.
10. Capel H., Physica, 1967, 33, 295.
11. Capel H., Physica, 1967, 37, 423.
12. Blume M., Emery V., Griffiths R., Phys. Rev. A, 1971, 4, 1071.
13. Mancini F., Mancini F.P., in preparation, 2005.
401
F.Mancini
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402
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