On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1
We revisit the particle-hole symmetry of the one-dimensional (D=1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 XXZ Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temp...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 / M.T. Thomaz, E.V. Corrêa Silva, O. Rojas // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23002:1-6. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1535592019-06-15T01:27:35Z On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 Thomaz, M.T. Corrêa Silva, E.V. Rojas, O. We revisit the particle-hole symmetry of the one-dimensional (D=1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 XXZ Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the XXZ Heisenberg model. The results presented in this paper are valid for any value of the interaction strength parameter V, which describes the attractive/null/repulsive interaction of neighboring fermions. Ми наново переглядаємо симетрiю частинка-дiрка одновимiрної (D = 1) фермiонної безспiнової моделi Габбарда, пов’язуючи цю симетрiю з iнварiантнiстю вiльної енергiї Гельмгольца одновимiрної спiн-1/2 X X Z моделi Гайзенберга, при iнверсiї (перекиданнi) поздовжнього магнiтного поля i при довiльнiй скiнченнiй температурi. В результатi порiвняння двох режимiв ланцюжкової моделi, коли число частинок в одному режимi дорiвнює числу дiрок в iншому, знайдено, що в загальному, їх термодинамiка є подiбною, але не iдентичною: обидвi моделi мають однаковi функцiї питомої теплоємностi та ентропiї, але рiзнi внутрiшню енергiю на вузол, кореляцiйнi функцiї перших сусiдiв i число частинок на вузол. Завдяки цiй симетрiї, рiзниця мiж кореляцiйними функцiями перших сусiдiв є пропорцiйною до z-компоненти намагнiченостi X X Z моделi Гайзенберга. Представленi в цiй статтi результати справедливi для довiльного значення параметра сили взаємодiї V , який описує притягальну/нульову/вiдштовхувальну взаємодiю сусiднiх фермiонiв. 2014 Article On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 / M.T. Thomaz, E.V. Corrêa Silva, O. Rojas // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23002:1-6. — Бібліогр.: 11 назв. — англ. 1607-324X arXiv:1407.2050 DOI:10.5488/CMP.17.23002 PACS: 05.30.Fk, 71.27.+a, 75.10.Pq http://dspace.nbuv.gov.ua/handle/123456789/153559 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We revisit the particle-hole symmetry of the one-dimensional (D=1) fermionic spinless Hubbard model, associating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 XXZ Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon comparing two regimes of that chain model so that the number of particles in one regime equals the number of holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor correlation functions is proportional to the z-component of magnetization of the XXZ Heisenberg model. The results presented in this paper are valid for any value of the interaction strength parameter V, which describes the attractive/null/repulsive interaction of neighboring fermions. |
| format |
Article |
| author |
Thomaz, M.T. Corrêa Silva, E.V. Rojas, O. |
| spellingShingle |
Thomaz, M.T. Corrêa Silva, E.V. Rojas, O. On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 Condensed Matter Physics |
| author_facet |
Thomaz, M.T. Corrêa Silva, E.V. Rojas, O. |
| author_sort |
Thomaz, M.T. |
| title |
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 |
| title_short |
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 |
| title_full |
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 |
| title_fullStr |
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 |
| title_full_unstemmed |
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1 |
| title_sort |
on the particle-hole symmetry of the fermionic spinless hubbard model in d = 1 |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/153559 |
| citation_txt |
On the particle-hole symmetry of the fermionic
spinless Hubbard model in D = 1 / M.T. Thomaz, E.V. Corrêa Silva, O. Rojas // Condensed Matter Physics. — 2014. — Т. 17, № 2. — С. 23002:1-6. — Бібліогр.: 11 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT thomazmt ontheparticleholesymmetryofthefermionicspinlesshubbardmodelind1 AT correasilvaev ontheparticleholesymmetryofthefermionicspinlesshubbardmodelind1 AT rojaso ontheparticleholesymmetryofthefermionicspinlesshubbardmodelind1 |
| first_indexed |
2025-07-14T04:39:19Z |
| last_indexed |
2025-07-14T04:39:19Z |
| _version_ |
1837595844285562880 |
| fulltext |
Condensed Matter Physics, 2014, Vol. 17, No 2, 23002: 1–6
DOI: 10.5488/CMP.17.23002
http://www.icmp.lviv.ua/journal
On the particle-hole symmetry of the fermionic
spinless Hubbard model in D = 1
M.T. Thomaz1, E.V. Corrêa Silva2, O. Rojas3
1 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/no, CEP 24210-346,
Niterói-RJ, Brazil
2 Departamento de Matemática, Física e Computação, Faculdade de Tecnologia, Universidade do Estado do Rio
de Janeiro, Rodovia Presidente Dutra km 298 s/no, Pólo Industrial, CEP 27537-000, Resende-RJ, Brazil
3 Departamento de Ciências Exatas, Universidade Federal de Lavras, Caixa Postal 3037, CEP 37200-000,
Lavras-MG, Brazil
Received January 10, 2014, in final form February 25, 2014
We revisit the particle-hole symmetry of the one-dimensional (D = 1) fermionic spinless Hubbard model, as-
sociating that symmetry to the invariance of the Helmholtz free energy of the one-dimensional spin-1/2 X X Z
Heisenberg model, under reversal of the longitudinal magnetic field and at any finite temperature. Upon com-
paring two regimes of that chain model so that the number of particles in one regime equals the number of
holes in the other, one finds that, in general, their thermodynamics is similar, but not identical: both models
share the specific heat and entropy functions, but not the internal energy per site, the first-neighbor correlation
functions, and the number of particles per site. Due to that symmetry, the difference between the first-neighbor
correlation functions is proportional to the z-component of magnetization of the X X Z Heisenberg model. The
results presented in this paper are valid for any value of the interaction strength parameter V , which describes
the attractive/null/repulsive interaction of neighboring fermions.
Key words: quantum statistical mechanics, strongly correlated electron system, spin chain models
PACS: 05.30.Fk, 71.27.+a, 75.10.Pq
The one-band Hubbard model [1, 2] partially describes quantum magnetic phenomena; the complex-
ity of real materials, however, imposes severe limitations on the direct comparison of experimental and
theoretical results. It is not always clear which missing terms should be included in the fermionic Hamil-
tonian to account for the diversity of phenomena in a strongly correlated electron system.
The development of optical lattices over the last two decades has made the experimental simulation
of chainmodels possible. The three-dimensional Hubbardmodel at low temperatures has been simulated
by a fermionic quantum gas trapped in an optical lattice [3, 4]. A review of the simulation of the Fermi-
Hubbard model with fermionic atoms in optical lattices can be found in [5]. The simulation of a one-
dimensional spin-1/2 Ising model by a degenerate Bose gas of rubidium atoms confined in an optical lat-
tice can be found in [6]. The simplest one-dimensional fermionic model is the fermionic spinless Hubbard
model, the generalizations of which have been applied to the description of Verwey metal-insulator tran-
sitions and charge-ordering phenomena of the Fe3O4, Ti4O7, LiV2O4 and other d -metal compounds [7, 8].
In this paper we revisit the consequences of the particle-hole symmetry on the thermodynamics of
the one-dimensional fermionic spinless Hubbard model in the whole range of temperatures, by mapping
it into the exactly solvableD = 1 spin-1/2 X X Z Heisenberg model. Appendix A shows the β-expansion of
the Helmholtz free energy (HFE) of this model, up to the order β6
[9].
The spinless fermionic Hubbard model in D = 1 is a very simple anti-commutative model the Hamil-
tonian of which is [10]:
H(t ,V ,µ) = t
N∑
i=1
(
c†
i ci+1 +c†
i+1ci
)
+V
N∑
i=1
ni ni+1 −µ
N∑
i=1
ni , (1)
©M.T. Thomaz, E.V. Corrêa Silva, O. Rojas, 2014 23002-1
http://dx.doi.org/10.5488/CMP.17.23002
http://www.icmp.lviv.ua/journal
M.T. Thomaz, E.V. Corrêa Silva, O. Rojas
in which (ci ,c†
i ), with i ∈ {1,2, . . . , N }, are fermionic annihilation and creation operators, respectively, and
N is the number of sites in the chain. These operators satisfy anticommutation relations, {ci ,c†
j } = δi j 1li
and {ci ,c j } = 0, in which t is the hopping integral, V is the strength of the repulsion (V > 0) or attrac-
tion (V < 0) between first-neighbour fermions, µ is the chemical potential and ni = c†
i ci is the operator
number of fermions at the i th
site of the chain.
Sznajd and Becker [10] have shown that the Hamiltonian (1) has a particle-hole symmetry; conse-
quently, the HFE of this model,W (t ,V ,µ;β), satisfies the relation
W (t ,V ,µ;β) =W (t ,V ,−µ+2V ;β)− (µ−V ), (2)
in which β= 1
kT , k is the Boltzmann’s constant and T is the absolute temperature in Kelvin. The relation
(2) is valid for any values (positive, null or negative) of V and µ. Equation (2) provides the condition for
having the same number of particles and holes at the same potentialV but at distinct chemical potentials,
〈ni 〉(t ,V ,µ;T ) = 1−〈ni 〉(t ,V ,−µ+2V ;T ), (3)
in which 〈ni 〉 is the average number of fermionic particles at each site of the chain at temperature T .
Haldane [11] showed the equivalence of the model (1) and the spin-1/2 X X Z Heisenberg model in
D = 1. More recently, Sznajd and Becker [10] also used the inverse Wigner-Jordan transformation to
show that the Hamiltonian (1) is mapped onto the Hamiltonian of the one-dimensional spin-1/2 X X Z
Heisenberg model with a longitudinal magnetic field,
HS=1/2(J ,∆,h) =
N∑
i=1
[
J
(
Sx
i Sx
i+1 +Sy
i Sy
i+1 +∆Sz
i Sz
i+1
)−hSz
i
]
, (4)
in which Sl = σl
2 , l ∈ {x, y, z}, and σl
are the Pauli matrices. The norm of the spin operator~S is ||~S|| =
p
3
2 .
The Hamiltonians (1) and (4) are related by
H(t ,V ,µ) = HS=1/2(J = 2t ,∆=V /(2t ),h =µ−V )−N
(
J∆
4
+ h
2
)
1l (5)
and 1l is the identity operator of the chain. This relation shows a constant shift between the energy spec-
trum of these two models.
LetWS=1/2(J ,∆,h;β) be the HFE associated to the Hamiltonian (4) of the D = 1 spin-1/2 X X Z Heisen-
berg model. A direct consequence of (5) is that
W (t ,V ,µ;β) =WS=1/2(J = 2t ,∆=V /(2t ),h =µ−V ;β)+
(
V
4
− µ
2
)
. (6)
At finite temperature (T , 0), the HFE of the one-dimensional S = 1/2 X X Z Heisenberg model is an
even function of the longitudinal magnetic field h,
WS=1/2(J ,∆,−h;T ) =WS=1/2(J ,∆,h;T ). (7)
Such invariance ofWS=1/2 comes from the symmetry of the Hamiltonian (4) upon reversal of the external
magnetic field, h →−h, and of the spin operators,~Si →−~Si , in which i ∈ {1,2, · · ·N }.
Consider, for a given magnetic field h and a fixed value (positive, null or negative) of V , the chemical
potential µ so that h = µ−V . For a reversed magnetic field, the corresponding chemical potential µ2 for
which −h =µ2 −V is
µ2 =−µ+2V. (8)
The identity (7) and the condition (8) recover the result (2) satisfied by the HFE of the spinless Hubbard
model for any values ofV andµ. Notice that in the half-filling condition (µ=V ), we haveµ2 =µ, and there
is no visible consequence of the symmetry (7).
We point out that the quantity−µ+2V , which appears on the r.h.s. of (8), also appears as an argument
of W (the HFE of D = 1 spinless fermionic Hubbard model) in the r.h.s. of (2), which in its turn comes
23002-2
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1
from the particle-hole symmetry of the Hamiltonian (1). On the other hand, (7) comes from the fact that
the HFE of the D = 1 spin-1/2 X X Z Heisenberg model is insensitive to a reversal of the longitudinal
magnetic field.
Equation (3) can be interpreted as follows: the number of particles in the chain under a potential V
and a chemical potential µ equals the number of holes in the chain under the same potential V and
a chemical potential µ2 given by (8). Those configurations correspond to distinct distributions of the
fermionic particles in the chain, and certainly have some different thermodynamic functions at tempera-
ture T . In what follows, we explore the consequences of the equality (7) in the thermodynamic functions
of those two configurations.
The specific heat C and the entropy S, both per site, are related to the HFE of the model (1) by
C (t ,V ,µ;β) =−β2 ∂2
∂β2
[
β W (t ,V ,µ;β)
]
and
S
k =β2 ∂
∂βW (t ,V ,µ;β), respectively. Due to equation (6) we ob-
tain
C (t ,V ,µ;T ) =C (t ,V ,−µ+2V ;T ) (9a)
and
S(t ,V ,µ;T ) = S(t ,V ,−µ+2V ;T ). (9b)
Both (9a) and (9b) are valid in the whole range of temperatures T > 0. This can be verified at each order
of the β-expansion of the thermodynamic functions derived from the expansion (16) of the HFE of the
model, shown in appendix A.
However, not all thermodynamic functions of the model (1) are identical for the chemical potentials µ
and µ2, at the same potentialV . The internal energy per site ε(t ,V ,µ;β) = ∂
∂β [βW (t ,V ,µ;β)] distinguishes
the distributions of the fermionic particles in the chain:
ε(t ,V ,µ2;β) =−V +µ+ε(t ,V ,µ;β). (10)
Notice that the difference of internal energies per site does not depend on the temperature. This equality
is valid for any temperature T > 0 and it is verified at each order of the expansion in β for this thermo-
dynamic function, obtained from (16).
In the spin-1/2Heisenberg model (4), the parallel and anti-parallel configurations of spin with respect
to the external magnetic field can be distinguished, for instance, by the average value of the z-component
of the spin operator Sz
i at a site and the correlation function of odd powers of such operators. In terms of
fermionic operators, we have Sz
i = ni − 1
2 1li , in which 1li is the identity operator at the i -th site.
For the spinless fermionic Hubbard model, the first-neighbor correlation function Gi ,i+1(t ,V ,µ;T ) ≡
〈ni ni+1〉 also relates configurations in which the number of particles in one equals the number of holes
in the other, for two values of the chemical potential, µ and µ2.
The two-point correlation function Gi ,i+1 is related to the HFE by
Gi ,i+1(t ,V ,µ;T ) = ∂W (t ,V ,µ;T )
∂V
. (11)
From relation (6), the symmetry condition (7) and the definition of the z-component of the magneti-
zation of the D = 1 spin-1/2 X X Z Heisenberg model,
M S=1/2
z (J ,∆,h;T ) = −∂WS=1/2(J ,∆,h;T )
∂h
= 〈Sz
i 〉(J ,∆,h;T ), (12)
in which i ∈ {1,2, · · · , N } and 〈Sz
i 〉(J ,∆,h;T ) is the mean value of the z-component of the spin-1/2 operator
at the i th
site of the chain and at temperature T , we obtain
Gi ,i+1(t ,V ,µ2;T )−Gi ,i+1(t ,V ,µ;T ) =−2M S=1/2
z (J ,∆,h;T ), (13)
where on its r.h.s. we have the z-component of the magnetization M S=1/2
z in the presence of a longitu-
dinal magnetic field. (Notice that Mz is a one-point function of the model, whereas Gi ,i+1 is a two-point
23002-3
M.T. Thomaz, E.V. Corrêa Silva, O. Rojas
function.) Equation (13) is valid for each order of the β-expansion of the functionGi ,i+1(t ,V ,µ;β), derived
from the expansion (16).
As a consequence of the symmetry in equation (7), the magnetizationM S=1/2
z is an odd function of the
magnetic field h,
M S=1/2
z (J ,∆,−h;T ) =−M S=1/2
z (J ,∆,h;T ). (14)
By writing equation (14) in terms of fermionic operators, Sz
i = ni − 1li
2 , one obtains
〈ni 〉(t ,V ,µ;T ) = 1−〈ni 〉(t ,V ,−µ+2V ;T ), (15)
thus, recovering equation (3).
In summary, we have verified that the particle-hole symmetry of the one-dimensional spinless
fermionic Hubbard model (1) is associated to the invariance of the HFE of the D = 1 spin-1/2 X X Z
Heisenberg model (4) with respect to a reversal of the longitudinal external magnetic field.
The thermodynamics of the chain off the half-filling condition (µ , V ) with chemical potentials µ
and µ2, under the same potential V , are not identical; rather, some thermodynamic functions permit
their distinction. Although the number of fermionic particles in the chain differ for µ and µ2, we obtain
unexpected results, expressed in (9a) and (9b), where both configurations exhibit the same specific heat
and entropy per site at any finite temperature T and at any value of V . Distinction arises from other
thermodynamic functions of the chain, though: the values of the internal energy per site of these two
distributions of particles in the chain differ by a constant that is independent of the temperature; and
the difference of their first-neighbour correlation functions is a one-point function proportional to the
z-component of magnetization per site of the spin-1/2model (4).
The equality of the number of particles in the chain for the chemical potential µ and the number
of holes in the chain for the chemical potential µ2 is a consequence of the odd parity of magnetization
M S=1/2
z (J ,∆,h;β) under reversal of the magnetic field h →−h, for any temperature.
The results presented here are valid for any value of V (negative, null or positive) and any value of
temperature T > 0, verified at each order of the β-expansion of the respective thermodynamic function.
These results are also valid at very low temperatures and could be checked in an optical lattice simulation
of the one-dimensional fermionic spinless Hubbard model.
Acknowledgements
E.V. Corrêa Silva thanks CNPq (Fellowship CNPq, Brazil, Proc. No. 303876/2010-7) for partial financial
support. O. Rojas thanks CNPq and FAPEMIG for partial financial support.
A. The HFE of the one-dimensional spinless fermionic Hubbard model
up to order β6
In reference [9] we calculated the β-expansion of the HFE of the normalized one-dimensional spin-S
of the X X Z Heisenberg model with single-ion anisotropy term in the presence of a longitudinal magnetic
field up to the order β6
, in terms of the rescaled spin operator~s =~S/
p
S(S +1).
In the present work we have applied equation (6) to equation (B) of reference [9], with ||~S|| =
p
3
2 , to
derive the β-expansion, up to the order β6
, of the HFE of the one-dimensional fermionic spinless Hubbard
model. We have obtained
W (t ,V ,µ;β) = − ln2
β
− 1
2
µ+ 1
4
V
+
(
−1
4
t 2 + 1
4
V µ− 5
32
V 2 − 1
8
µ2
)
β
+
(
1
16
V µ2 − 1
16
t 2V + 1
16
V 3 − 1
8
V 2µ
)
β2
23002-4
On the particle-hole symmetry of the fermionic spinless Hubbard model in D = 1
+
(
− 1
48
V µ3 + 1
16
t 2µ2 + 7
96
t 2V 2 + 1
64
V 2µ2 + 1
96
V 3µ
− 31
3072
V 4 + 1
32
t 4 − 1
8
t 2V µ+ 1
192
µ4
)
β3
+
(
− 7
256
t 2V 3 − 23
384
V 3µ2 − 1
128
V 5 − 1
32
t 2V µ2 + 1
32
t 4V
+ 1
24
V 2µ3 + 1
16
t 2V 2µ+ 7
192
V 4µ− 1
96
V µ4
)
β4
+
(
− 47
1536
t 4V 2 − 1
96
t 2µ4 − 1
32
t 4µ2 − 239
7680
V 5µ
− 1
144
t 6 + 287
36864
V 6 − 31
1152
V 3µ3 − 1
2880
µ6
− 21
2560
t 2V 4 + 1
480
V µ5 + 7
192
t 2V 3µ+ 5
1536
V 2µ4
− 23
384
t 2V 2µ2 + 1
24
V t 2µ3 + 139
3072
V 4µ2 + 1
16
V t 4µ
)
β5
+
(
13
768
V t 2µ4 − 1
64
t 4V 2µ+ 157
1536
t 2V 3µ2 − 29
10240
V 7
+ 1
128
V t 4µ2 − 13
192
t 2V 2µ3 − 7
576
V 4µ3 − 53
768
t 2V 4µ
+ 17
11520
V µ6 + 83
23040
t 4V 3 + 1603
92160
t 2V 5 − 17
1920
V 2µ5
− 119
30720
V 5µ2 + 389
46080
V 6µ− 11
768
V t 6
+ 41
2304
V 3µ4
)
β6 +O(β7). (16)
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M.T. Thomaz, E.V. Corrêa Silva, O. Rojas
До симетрiї частинка-дiрка фермiонної безспiнової моделi
Габбарда в D = 1
M.T. Томас1, Е.В. Корреа Сiльва2, О. Рохас3
1 Iнститут фiзики, Федеральний унiверситет Флумiненсе, Нiтерой-РЖ, Бразилiя
2 Факультет математики, фiзики та iнформатики, технологiчний факультет, Державний унiверситет
Рiо-де-Жанейро, Ресенде-РЖ, Бразилiя
3 Факультет точних наук, Федеральний унiверситет м. Лаврас, Лаврас-МЖ, Бразилiя
Ми наново переглядаємо симетрiю частинка-дiрка одновимiрної (D = 1) фермiонної безспiнової моделi
Габбарда, пов’язуючи цю симетрiю з iнварiантнiстю вiльної енергiї Гельмгольца одновимiрної спiн-1/2
X X Z моделi Гайзенберга, при iнверсiї (перекиданнi) поздовжнього магнiтного поля i при довiльнiй скiн-
ченнiй температурi. В результатi порiвняння двох режимiв ланцюжкової моделi, коли число частинок
в одному режимi дорiвнює числу дiрок в iншому, знайдено, що в загальному, їх термодинамiка є подi-
бною, але не iдентичною: обидвi моделi мають однаковi функцiї питомої теплоємностi та ентропiї, але
рiзнi внутрiшню енергiю на вузол, кореляцiйнi функцiї перших сусiдiв i число частинок на вузол. Завдя-
ки цiй симетрiї, рiзниця мiж кореляцiйними функцiями перших сусiдiв є пропорцiйною до z-компоненти
намагнiченостi X X Z моделi Гайзенберга. Представленi в цiй статтi результати справедливi для довiльно-
го значення параметра сили взаємодiї V , який описує притягальну/нульову/вiдштовхувальну взаємодiю
сусiднiх фермiонiв.
Ключовi слова: квантово-статистична механiка, сильноскорельована електронна система, моделi
спiнових ланцюжкiв
23002-6
The HFE of the one-dimensional spinless fermionic Hubbard model up to order 6
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