The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime
We obtain that when the spinless model is off the half-filling regime (μ ≠ V), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = μ - V. We show numerically that the chemical po...
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| Zitieren: | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime / E.V. Correa Silva, M.T. Thomaz, O. Rojas // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33003: 1–12. — Бібліогр.: 18 назв. — англ. |
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| citation_txt | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime / E.V. Correa Silva, M.T. Thomaz, O. Rojas // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33003: 1–12. — Бібліогр.: 18 назв. — англ. |
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| description | We obtain that when the spinless model is off the half-filling regime (μ ≠ V), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = μ - V. We show numerically that the chemical potential as a function of temperature satisfies a relation similar to one derived from the particle-hole symmetry of the fermionic spinless model. We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order β⁸.
Встановлено, що для безспiнової моделi поза половинним заповненням (µ , V ) вiльну енергiю Гельмгольца можна записати у виглядi двох β-розвинень: одне розвинення походить вiд конфiгурацiї з половинним заповнення, а iнше залежить вiд параметра вiдхилення x = µ − V . Чисельно показано, що
хiмiчний потенцiал як функцiя температури задовольняє спiввiдношення подiбне до того, яке отримується з симетрiї частинка-дiрка фермiонної безспiнової моделi. β-розвинення вiльної енергiї Гельмгольца
одновимiрної фермiонної безспiнової моделi Габбарда продовжено аж до порядку β
⁸
.
|
| first_indexed | 2025-12-07T18:17:04Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 3, 33003: 1–12
DOI: 10.5488/CMP.18.33003
http://www.icmp.lviv.ua/journal
The β-expansion of the D = 1 fermionic spinless
Hubbard model off the half-filling regime
E.V. Corrêa Silva1, M.T. Thomaz2∗, O. Rojas3
1 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
2 Instituto de Física, Universidade Federal Fluminense, Av. Gal. Milton Tavares de Souza s/no, CEP 24210-346,
Niteói-RJ, Brazil
3 Departamento de Física, Universidade Federal de Lavras, Caixa Postal 3037, CEP 37200-000, Lavras-MG, Brazil
Received May 15, 2015
We found that when the spinless model is off the half-filling regime (µ , V ), the Helmholtz free energy (HFE)
can be written as two β-expansions: one expansion comes from the half-filling configuration and another one
that depends on the parameter x = µ−V . We show numerically that the chemical potential as a function of
temperature satisfies a relation similar to the one derived from the particle-hole symmetry of the fermionic
spinless model. We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard
model up to order β8.
Key words: quantum statistical mechanics, strongly correlated electron system, spin chain models
PACS: 05.30.Fk, 71.27.+a, 75.10.Pq
1. Introduction
One-dimensional models are certainly easier to handle than higher-dimensional ones, and for a long
time they have been treated as toy models. In general, these models are a simplified description of a real
physical system. It is often difficult to realize what is missing in those simple models in order to explain
the experimental results.
The development of optical lattices over the last two decades has made possible the physical realiza-
tion of one-dimensional models like the spin-1/2 Ising model [1], thus offering the opportunity for the
experimental verification of the predictions of simplified models like the one-band Hubbardmodel [2, 3],
that partially describes quantum magnetic phenomena.
The simplest one-dimensional fermionic model is the fermionic spinless Hubbard model, the gen-
eralizations of which have been applied to the description of Verwey metal-insulator transitions and
charge-ordering phenomena of Fe3O4, Ti4O7, LiV2O4 and other d -metal compounds [4–6].
In references [7, 8] it is shown that the fermionic spinless Hubbardmodel inD = 1 is mapped onto the
exactly soluble D = 1 spin-1/2 X X Z Heisenberg model in the presence of a longitudinal magnetic field.
The fermionic model has a particle-hole symmetry [8]. In reference [9] we explore the consequences of
that symmetry on the thermodynamic functions of this model in the whole interval of temperature T > 0.
The spin-1/2 X X Z Heisenberg model is an exactly solvable model. Its thermodynamics can be de-
rived from the thermodynamic Bethe ansatz equations [10].
Bühler et al. calculated the β-expansion of the specific heat and the susceptibility, both per site, of
the frustrated and unfrustrated spin-1/2 Heisenberg chain up to order β16 and β24, respectively, in the
absence of an external magnetic field [11] [h = 0 on the r.h.s. of equation (2.2)]. In 2001 Takahashi derived
an integral equation to obtain the HFE of the spin-1/2 X X Z model [12]. The high temperature expansions
∗E-mail: mtt@if.uff.br
© E.V. Corrêa Silva, M.T. Thomaz, O. Rojas, 2015 33003-1
http://dx.doi.org/10.5488/CMP.18.33003
http://www.icmp.lviv.ua/journal
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
of the specific heat and the susceptibility, both per site, of the isotropic spin-1/2 X X X model [13] were
calculated up to order β100 also for h = 0. In the language of the spinless model, the absence of a magnetic
field in the spin-1/2model corresponds to the half-filling case.
In reference [14] we calculated the β-expansion of the Helmholtz free energy (HFE) of the one-dimen-
sional spin-S X X Z Heisenberg model in the presence of a longitudinal magnetic field, S ∈ { 1
2
,1, 3
2
, . . .} up
to order β6. By applying the mapping between the aformentioned one-dimensional fermionic and spin
models, we obtain the expansion of the HFE of the fermionic spinless Hubbard model also up to order
β6. These high temperature expansions are analytic and valid for any set of parameters of the respective
Hamiltonian, thus letting one avoid the numerical solution of a hierarchy of coupled integral for every
set of parameters of the spin-1/2 X X Z model.
In the present article we study the β-expansion of thermodynamic functions of the spinless Hubbard
model off the half-filling regime. We calculate two additional orders in the β-expansion of the HFE of
reference [14] and verify the consequences of those extra terms on the specific heat per site and on the
mean number of spinless fermions per site. We also numerically study the dependence of the chemical
potential on the temperature when the number of particles in the chain is fixed.
In section 2 we present the Hamiltonian of the one-dimensional fermionic spinless Hubbard model
and its mapping onto the D = 1 spin-1/2 X X Z Heisenberg model in the presence of a longitudinal mag-
netic field. We present the relations satisfied by the HFE of the model due to the particle-hole symmetry.
In section 3 we discuss the β-expansion of the specific heat per site and the mean number of spinless
fermions per site off the half-filling regime, and show the parameters of expansions of thermodynamic
functions. In section 4 we use the β-expansion of the mean number of spinless fermions per site to nu-
merically discuss the dependence of the chemical potential on temperaturewhen the number of fermions
in the chain is kept constant. Finally, section 5 has a summary of our results. Appendix A has the β-
expansion of the HFE of the fermionic spinless Hubbard model in D = 1, up to order β8.
2. The fermionic spinless Hubbard model in D = 1 and its exact relations
The fermionic spinless Hubbard model in D = 1 is a very simple anti-commutative model whose
Hamiltonian is [8]:
H(t ,V ,µ)=
N∑
i=1
Hi ,i+1(t ,V ,µ), (2.1a)
in which
Hi ,i+1(t ,V ,µ)≡ t(c
†
i
ci+1 +c
†
i+1
ci )+V ni ni+1 −µni , (2.1b)
the operators ci and c
†
i
, with i ∈ {1,2, . . . , N }, are the destruction and creation fermionic operators, re-
spectively, and N is the number of sites in the periodic chain (HN ,N+1 = HN ,1). Those operators satisfy
anti-commutation relations, {ci ,c
†
j
} = δi j 1li and {ci ,c j } = 0. In this Hamiltonian t is the hopping integral,
V is the strength of the repulsion (V > 0) or attraction (V < 0) between first-neighbour fermions, and µ is
the chemical potential. The operator number of fermions at the i th site of the chain is defined as ni ≡ c
†
i
ci .
It is shown in the literature [7, 8] that the equivalence of the Hamiltonian (2.1a)–(2.1b) and the one
that describes the spin-1/2 X X Z Heisenberg model in D = 1,
HS=1/2(J ,∆,h) =
N∑
i=1
[
J
(
S
x
i S
x
i+1
+S
y
i
S
y
i+1
+∆S
z
i S
z
i+1
)
−hS
z
i
]
, (2.2)
in which S
l =σl /2, l ∈ {x, y, z}, and σl are the Pauli matrices; the parameters of both Hamiltonians satisfy
the relations:
J = 2t , ∆=
V
2t
and h =µ−V . (2.3)
33003-2
The β-expansion of the D = 1 fermionic spinless Hubbard model off the half-filling regime
The Hamiltonians (2.1a)–(2.1b) and (2.2), with their parameters satisfying conditions (2.3), differ by a
constant operator
H(t ,V ,µ) = HS=1/2(J = 2t ,∆=V /2t ,h =µ−V )−N
(
J∆
4
+
h
2
)
1l, (2.4)
in which 1l is the identity operator of the chain.
Let Z (t ,V ,µ;β) and ZS=1/2(J ,∆,h;β) be the partition functions of the fermionic spinless model and
the spin chain model, respectively,
Z (t ,V ,µ;β) = Tr
{
e
−βH(t ,V ,µ)
}
, (2.5a)
ZS=1/2(J ,∆,h;β) = Tr
{
e
−βHS=1/2(J ,∆,h)
}
, (2.5b)
in which β= 1/kT , k is the Boltzmann’s constant and T is the absolute temperature in kelvin.
The functionsW (t ,V ,µ;β) andWS=1/2(J ,∆,h;β) are the HFE’s associated to the Hamiltonians (2.1a)–
(2.1b) and (2.2), respectively, in the thermodynamic limit (N →∞)
W (t ,V ,µ;β) = − lim
N→∞
1
N
1
β
ln
[
Z (t ,V ,µ;β)
]
, (2.6a)
WS=1/2(J ,∆,h;β) = − lim
N→∞
1
N
1
β
ln
[
ZS=1/2(J ,∆,h;β)
]
, (2.6b)
in which N is the number of sites in the chain.
Due to the equality of operators in equation (2.4), we have a relation between the HFE’s (2.6a) and
(2.6b) [8],
W (t ,V ,µ;β) =WS=1/2(J = 2t ,∆=V /2t ,h =µ−V ;β)+
(
V
4
−
µ
2
)
, (2.7)
valid at any non-null temperature T . This relation permits to relate the thermodynamic functions of both
one-dimensional models.
The expression of the functionWS=1/2(J ,∆,h;β) comes from the calculation of the trace of the opera-
tor e−βHS=1/2(J ,∆,h) over all sites in the chain. In the β-expansion of this function, only terms with an even
number of operators S
z
i
at each site give a non-null value to the trace at the i th site, and, therefore, we
obtain that 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 ). (2.8)
Another way to understand the invariance (2.8) ofWS=1/2 is to remember the symmetry of the Hamilto-
nian (2.2) 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 . (2.9)
The identity (2.8) and the condition (2.9) recover the symmetry particle-hole of the fermionic spinless
Hubbardmodel for any values of V and µ. This symmetry is summarized in the relation of the HFE of the
fermionic spinless model at the same potential V and different chemical potentials,
W (t ,V ,µ;β) =W (t ,V ,µ2 =−µ+2V ;β)− (µ−V ). (2.10)
In reference [9] we explore the effect of the relation (2.10) on the thermodynamic functions of the
one-dimensional fermionic spinless model at the same potential V but with chemical potentials µ and
µ2. The results discussed in reference [9] are valid in the whole range of temperatures of T > 0.
33003-3
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
In reference [14] we use the method of reference [15] to calculate the β-expansion of the spin-S
X X Z Heisenberg model in D = 1, in the presence of a longitudinal magnetic field up to order β6, with
S ∈ { 1
2
,1, 3
2
, . . . } For a summary of the results of reference [15] we suggest to the reader reference [16].
Relation (2.7) permits to derive the HFE of the chain of spinless fermions from the β-expansion presented
in reference [14] up to order β6.
In this article we introduce a new set of rules for algebraic calculation using the method of reference
[15] that enables us to calculate the β-expansion of the HFE of the fermionic spinless Hubbard model in
D = 1 up to order β8.
In equation (A.1) we present the β-expansion of the HFE of the one-dimensional fermionic spinless
Hubbard model up to order β8. This result is calculated using the method of reference [15] for arbi-
trary values of the parameters in the Hamiltonian (2.1a)–(2.1b). The coefficient of the βn term, with
n ∈ {−1,0,1, . . . ,8}, in expansion (A.1) is exact. The polynomial form of the HFE expansion in β and in
the parameters of the Hamiltonian (2.1b) can be easily handled by any computer algebra system. Ther-
modynamic functions of the model can be derived from the appropriate derivatives of the HFE.
We explicitly verified that expansion (A.1) satisfies the relation (2.10), which is valid separately for
each coefficient of the βl terms of this HFE, with l ∈ {1,2, . . . ,8}.
3. Discussion on the β-expansion of the HFE of the model
The β-expansion (A.1) of the HFE of the fermionic spinless Hubbard model in D = 1 permits the
derivation of the β-expansion of various thermodynamic functions. In this article we discuss only two
thermodynamic functions: the specific heat per site C (t ,V ,µ;β) = −β2∂2[βW ]/∂β2, and the mean num-
ber of spinless fermions per site ρ(t ,V ,µ;β) =−∂W /∂µ. (From this point on, it will be ommitted that those
functions are calculated per site.) The expansion (A.1) is two orders higher in β than the β-expansion of
the HFE of the one-dimensional spin-1/2 X X Z Heisenberg model, in the presence of a longitudinal mag-
netic field, presented in reference [14]. In what follows we make a simple comparison, the β-expansions
of the specific heat and themean number of particles, derived from the expansion of the HFE in reference
[14] and equation (A.1), are compared to their respective exact expressions of two simple limiting cases,
and the interval of β in which there is a good agreement between them is determined.
In order to verify the range of convergence of each expansion, we compare them to the respective
thermodynamic function of two limiting cases of the Hamiltonians (2.1a)–(2.1b) and (2.2): the free spin-
less fermion model [17] and the spin-1/2 Ising model in the presence of a longitudinal magnetic field [18].
We do not need any extra computational effort to exactly calculate these two limiting cases for arbitrary
values of the parameters in their respective Hamiltonians.
Let C7 and C9 be the specific heat and the β-expansion up to order β
7 and β9, respectively, derived
from the HFE of reference [14] and equation (A.1). We have compared the expansions C7 and C9 to
the specific heat of the free spinless fermion model [14] and the spin-1/2 Ising model [17, 18], both in
D = 1. In order to measure the difference between each specific heat of the exactly soluble models and
its expansions C7 and C9, we define the percentage difference,
δDCk ≡ 100%×
(
CM −Ck
CM
)
, k ∈ {7, 9}, (3.1)
withM ∈ {Ising, Free}. Let CIsing and CFree be the specific heat of the spin-1/2 Ising model and that of the
free spinless fermion model, respectively.
Table 1 compares the expansions C7 and C9 to the exact specific heat of the free spinless fermion
model, showing the percentage differences of the expansions of this thermodynamic function to the exact
result for t = 1, V = 0 and µ = 0. Table 2 compares the exact specific heat of the spin-1/2 Ising model, in
the presence of a longitudinal magnetic field, in D = 1, mapped onto the fermionic spinless Hubbard
model to the expansions C7 and C9 of this model, for t = 0, V = 0.5 and µ= 0.8. From data in tables 1 and
2 we conclude that the addition of two more orders in β in the previous expansion of the specific heat
increases the interval in β where this expansion is a good approximation of the exact expression of the
specific heat. Certainly, this improvement depends on the values of the set (t ,V ,µ).
33003-4
The β-expansion of the D = 1 fermionic spinless Hubbard model off the half-filling regime
Table 1. Comparison of the percentage difference (3.1) of the expansions C7 and C9 of the specific heat
of the free spinless fermion model for t = 1, V = 0 and µ= 0.
|t |β 0.5 0.82
δD C7(%) – 0.35 – 7.49
δD C9(%) 0.04 2.38
Table 2. Comparison of the percentage difference (3.1) of the expansions C7 and C9 of the specific heat
corresponding to the mapping onto the spin-1/2 Ising model in the presence of a longitudinal magnetic
field for t = 0, V = 0.5 and µ= 0.8.
|t |β 1.6 1.91
δD C7(%) – 2.10 – 6.70
δD C9(%) 0.54 2.32
Let ρ6(t ,V ,µ;β) and ρ8(t ,V ,µ;β) be the β-expansions up to order β6 and β8, respectively, of the av-
erage number of spinless fermions derived from the HFE of reference [14] and the equation (A.1). The
effect on the convergence β-intervals due to the terms β7 and β8 in ρ8(t ,V ,µ;β) can be determined by
comparison of the expansions ρ6(t ,V ,µ;β) and ρ8(t ,V ,µ;β) to the exact expression of this termodynamic
function on the mapping of the fermionic spinless Hubbard model onto on the spin-1/2 Ising model, in
the presence of a longitudinal magnetic field. In analogy to (3.1), the percentage difference regarding the
functions ρ6(t ,V ,µ;β), ρ8(t ,V ,µ;β) and ρIsing can be defined as
δDρk ≡ 100%×
(
ρIsing−ρk
ρIsing
)
, k = 7 or 9. (3.2)
Here, ρIsing is the mean value of the number of spinless fermions derived from the exactly soluble spin-
1/2 Ising model.
Table 3 has been generatedwith the percentage difference of the expansions ρ6(t ,V ,µ;β), ρ8(t ,V ,µ;β)
to the ρIsing , for t = 0, V = 0.5 and µ = 0.8. Data shows that for the function ρ(t ,V ,µ;β) the presence of
two orders in its β-expansion does not really increase the region where the expansion is a good approxi-
mation of the exact result, although ρ8(t ,V ,µ;β) is closer to the correct result.
In general, the β-expansions of the thermodynamic functions associated to a givenmodel get worse as
the parameters of the Hamiltonian increase. Let us choose two sets of values of parameters in the Hamil-
tonian (2.1a)–(2.1b) that map onto the spin-1/2 Ising model in the presence of a longitudinal magnetic
field:
(t = 0,V = 0.5,µ= 0.7) ≡ (1), (3.3a)
(t = 0,V = 0.5,µ= 0) ≡ (2). (3.3b)
Table 3. Comparison of the percentage difference (3.2) of the expansions ρ6 and ρ8 of the of the mean
number of spinless fermions per site corresponding to the mapping of the fermionic spinless model onto
the spin-1/2 Ising model in the presence of a longitudinal magnetic field for t = 0, V = 0.5 and µ= 0.8.
β 2.5 2.7 3
δD ρ6(%) – 0.46 – 0.78 –1.62
δD ρ8(%) 0.35 0.67 1.64
33003-5
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
In what follows we use the notations:
C
(1)
9
≡ C9(t = 0,V = 0.5,µ= 0.7;β), (3.4a)
C
(2)
9
≡ C9(t = 0,V = 0.5,µ= 0;β), (3.4b)
ρ(1)
9
≡ ρ9(t = 0,V = 0.5,µ= 0.7;β), (3.4c)
ρ(2)
9
≡ ρ9(t = 0,V = 0.5,µ= 0;β). (3.4d)
Figure 1 show the percentage differences of C9 and ρ9, given by (3.1) and (3.2), respectively, to their
respective exact expressions for the set of values (3.3a) and (3.3b). Figure 1 show that for both thermo-
dynamic functions the percentage differences increase more rapidly for µ = 0 than for µ = 0.7. How to
explain that a higher value of µ yields a larger interval in βwhere the expansions of the thermodynamic
functions are better approximations of the exact functions?
Figure 1. (a): percentage differences of C
(1)
9
(solid line) and C
(2)
9
(dashed line) to the specific heat of the
spin-1/2 Ising model. (b): percentage differences of ρ(1)
9
(solid line) and ρ(2)
9
(dashed line) to the mean
value of spinless fermions also derived from the spin-1/2 Ising model.
In order to understand the convergence behavior of the expansions of the functions C (t ,V ,µ;β) and
ρ(t ,V ,µ;β), we define the parameter
x ≡µ−V = h. (3.5)
as a measure of how much the chain is off the half-filling regime (i.e., µ=V ). Rewriting the relation (2.7)
between the HFE’s of the one-dimensional fermionic spinless Hubbard model and the spin-1/2 X X Z
Heisenberg model in D = 1 in the presence of a longitudinal magnetic field in terms of the parameter x,
we obtain
W (t ,V ,µ=V + x;β) =−
(
V
4
+
x
2
)
+WS=1/2
(
J = 2t ,∆=V /2t ,h = x;β
)
. (3.6)
The functionWS=1/2(J ,∆,h;β) has a Taylor expansion in βwhose coefficient of the βn term is a prod-
uct of powers of the parameters in Hamiltonian (2.2), J n1∆
n2 hn3 , with n1 +n2 +n3 = n+1. This thermo-
dynamic function can be written as an expansion in any of the parameters: J ,∆,h and β. The expansion
of WS=1/2 around h = 0 = x corresponds to an expansion of W (t ,V ,µ = V + x;β) about the half-filling
configuration, µ=V .
Expanding the HFEW (t ,V ,µ=V + x;β) about x = 0 yields
W (t ,V ,µ=V + x;β) ≡W (t ,V ,µ=V ;β)+W̃ (t ,V , x;β), (3.7a)
33003-6
The β-expansion of the D = 1 fermionic spinless Hubbard model off the half-filling regime
in which
W̃ (t ,V , x = 0;β) = 0. (3.7b)
The symmetry relation (2.8) and the definition (3.7a) permit to conclude that
W̃ (t ,V , x;β) =−
x
2
+ω(t ,V , x2
;β). (3.8)
From the form the HFE in equation (3.7a) is written, one can affirm that the thermodynamic quantities
of the chain off the half-filling regime can be expressed as a contribution of the half-filling configuration
plus an amount due to how off the system is from the half-filling regime (that depends, naturally, on the
parameter x).
The decomposition (3.7a) and the definitions of the specific heat and the mean number of fermions
permit us to write those functions in terms of the parameter x,
C (t ,V ,µ=V + x;β) =C (t ,V ,µ=V ;β)+∆C (t ,V , x;β), (3.9a)
in which
∆C (t ,V , x;β) ≡−β2 ∂2[βW̃ (t ,V , x;β)]
∂β2
, (3.9b)
and
ρ(t ,V ,µ=V + x;β) =
1
2
+∆ρ(t ,V , x;β) (3.10a)
with
∆ρ(t ,V , x;β) ≡−
∂
∂x
[
W̃ (t ,V , x;β)+
x
2
]
. (3.10b)
Returning to the set of values (3.3a) and (3.3b) for the parameters of Hamiltonian (2.1a)–(2.1b) we
notice that the values of x for those sets are, respectively,
x(1)
= 0.2 and x(2)
=−0.5. (3.11)
Notice that the absolute value of x(1) is smaller than the absolute value of x(2), and this explains why the
good approximations of those two functions are obtained in intervals of β that are larger for the set (3.3a)
than those for the set (3.3b). This result is clearly shown in figure 1.
In order to verify that x is one of the possible parameters of an expansion of the function ρ(T,V ,µ=
V + x;β) rather than the chemical potential µ, we calculate the percentage weight of the β8 term in its
β-expansion. Let us denote the β-expansion of ∆ρ by
∆ρ8(t ,V , x;β) ≡
8∑
l=1
al (t ,V , x) βl
. (3.12)
The percentage weight δW a8 of the β
8 term in the expansion of ∆ρ is
δW a8(t ,V , x;β) ≡ 100%
a8(t ,V , x)β8
∆ρ(t ,V , x;β)
. (3.13)
Let βmax be the maximum value of the variable β for which |δW a8(t ,V , x;β)| <∼ 4%, and for which we
expect that the expansion should be still a good approximation to the exact function ∆ρ(t ,V , x;β). Table 4
shows the values of |t |βmax and the corresponding value of δW a8 for different values of x/|t | for t = 1
and V /|t | = 0.5. The second column in this table shows the two distinct values of µ for which the same
value of |t |βmax is obtained. In particular, for x/|t | = ±0.5 we have the chemical potentials µ = 0 and
µ= 1 yielding the same value of |t |βmax.
33003-7
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
Table 4. The values of |t |βmax calculated from the percentage weight δa8 (3.13) for t = 1 and V /|t | = 0.5.
x
|t |
µ
|t |
|t |βmax δW a8(%)
± 0.1 0.4 1.12 – 4.13
0.6
±0.5 0 0.98 + 4.03
1
±1 – 0.5 0.77 + 4.03
1.5
In order to discuss the value ofβmax for which the specific heat can bewell described by its expansion,
we define the coefficients of the β-expansions of C (t ,V ,µ=V + x;β) and of the function ∆C (t ,V , x;β),
C9(t ,V ,µ=V + x;β) ≡
9∑
l=2
cl (t ,V , x)βl (3.14a)
and
∆C9(t ,V , x;β) ≡
9∑
l=2
gl (t ,V , x)βl
. (3.14b)
We also define the percentage weight δW c9 of the term of order β
9 in the expansion C9 as
δW c9(t ,V , x;β) ≡ 100%
c9(t ,V , x)β9
C9(t ,V ,V + x;β)
, (3.15)
in order to determine the value of βmax for the specific heat.
Table 5 shows the values of |t |βmax for the specific heat where δW c9
<
∼ 4%. The calculations have been
done with t = 1 and V /|t | = 0.5. Again we obtain that for x = ±0.5, the β- interval, where the expansion
C9 is a good approximation of the exact expression of this thermodynamic function, is the same for µ= 0
and µ= 1.
The function ∆C in equation (3.9b) measures the difference between the specific heat in the half-
filling regime and that function at the chemical potential µ=V +x. It also has a β-expansion that depends
on x. In order to verify the value of βmax for the function ∆C9, we define the percentage weight of the
term of order β9 in this function,
δW g9(t ,V , x;β) ≡ 100%
g9(t ,V , x)β9
∆C9(t ,V , x;β)
. (3.16)
Table 5. The percentage weight δW c9 of the term of order β
9 in the expansion C9(t ,V ,V + x;β). The
values of |t |βmax are calculated for t = 1 and V /|t | = 0.5.
x
|t |
µ
|t |
|t |βmax δW c9(%)
± 0.1 0.4 0.68 – 4.10
0.6
±0.5 0 0.69 – 4.06
1
±1 – 0.5 0.65 + 4.05
1.5
33003-8
The β-expansion of the D = 1 fermionic spinless Hubbard model off the half-filling regime
Table 6. The percentage weight δW g9(%) of the term of order β9 in the expansion of ∆C9(t ,V ,V +x;β).
The values of |t |βmax are calculated for t = 1 and V /|t | = 0.5.
x
|t |
µ
|t | |t |βmax δW g9(%)
± 0.1 0.4 0.56 – 4.03
0.6
±0.5 0 0.56 + 4.05
1
±1 – 0.5 0.47 + 4.00
1.5
Table 6 shows the values of |t |βmax for the function ∆C9(t ,V , x;β) with |t | = 1 and V /|t | = 0.5. We verify
that the values of |t |βmax for the functions C9(t ,V ,V + x;β) and ∆C9(t ,V , x;β) can be different.
4. The temperature dependence of the chemical potential
The chemical potential µ is one of the parameters in the Hamiltonian (2.1a)–(2.1b). For a given fixed
value of µ, the relation ρ(t ,V ,µ;β) =−∂W (t ,V ,µ;β)/∂µ permits the determination, from the expansion
(A.1), how the mean number of spinless fermions varies with the temperature.
How should the chemical potential vary for a given temperature T , keeping the chain in thermal
equilibrium at this temperature, so that the chain keeps its number of fermions per site? The relation
between ρ(t ,V ,µ;β) and W (t ,V ,µ;β) permits to rewrite the expansion ρ8(t ,V ,µ;β) as a polynomial in
the chemical potential µ of order µ7, written as
ρ8 = n0(t ,V ;β)µ0
+n1(t ,V ;β)µ1
+·· ·+n7(t ,V ;β)µ7
. (4.1)
The coefficients nl (t ,V ;β), with l ∈ {0,1, . . . ,7}, are known and— differently from the coefficients of the
β-terms in the expansion (A.1)— they have corrections from higher orders in β.
In order to derive the dependence of the function µ on the variables ρ8, t ,V and β, one must obtain
the roots of a 7th degree polynomial in µ. Figure 2 show our numerical results for the dependence of µ
on the temperature T for t = 0 and t = 1, for fixed values of V /|t | and ρ8.
Figure 2. The chemical potential µ(t ,V ,ρ8 ;T ) as function of the temperature T . (a): for t = 1 and V /|t | =
0.8. (b): for t = 0 and V /|t | = 1.
33003-9
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
Figure 3. (a): comparison of the specific heat curves CIsing (solid line), C8 (dotted line) and C24 (dashed
line). (b): comparison of the average number of fermions ρIsing (solid line), ρ7 (dotted line) and ρ23
(dashed line). In both panels, t = 0, V = 0.5 and µ= 0.9.
By comparing the curves in each graph of figure 2, we obtain the relation
µ(t ,V ,ρ = 0.5+δ;T ) = 2V −µ(t ,V ,ρ = 0.5−δ;β), (4.2)
with δ ∈ [−0.5,0.5]. This is similar to equation (2.9), derived from the hole-particle symmetry of the one-
dimensional fermionic spinless Hubbard model.
5. Conclusions
The one-dimensional fermionic spinless fermionic Hubbard model is the simplest fermionic model,
and it has the particle-hole symmetry. This model can be mapped onto the spin-1/2 X X Z Heisenberg
model in the presence of a longitudinal magnetic field in D = 1. Some years ago we derived the β-
expansion of the HFE of the latter up to order β6 [14]. In this article we have extended the β-expansion of
the HFE of both models up to order β8. Each β term in the expansion satisfies the condition (2.10) derived
from the particle-hole symmetry of the one-dimensional fermionic model.
We have used the expansion (A.1) of the HFE of the fermionic spinless model (2.1a)–(2.1b) to study
how the interval of convergence (in β) of the specific heat per site [C (t ,V ,µ;β)] and of the mean number
of spinless fermions per site [ρ(t ,V ,µ;β)] is modified by the presence of two more orders in β in their
respective expansions.
An interesting result that we obtain for the β expansions of the thermodynamic functions comes
from the relation (2.7) between the HFE of the fermionic spinless model and the spin-1/2 model. When
the chain is off the half-filling regime (µ , V ), the relation (2.7) permits to write the thermodynamic
functions of the chain in this regime as two β-expansions: the expansion of the function in the half-filling
(µ= V ) plus another expansion that depends on the set of parameters (t ,V , x = µ−V ;β). The parameter
x is a measure of how off the chain is from the half-filling regime. This fact explains why the expansions
C9(t ,V ,µ;β) and ρ8(t ,V ,µ;β) with µ= 0 have shorter β intervals of convergence than those for |µ| > 0.
We have numerically obtained the dependence of the chemical potential µ on the temperature T
when the mean value of fermions per site ρ is kept fixed. We have verified that the relation (4.2), satisfied
by µ(T ) for ρ = 0.5±δ with δ ∈ [−0.5,+0.5], is similar to equation (2.9) derived from the particle-hole
symmetry of the fermionic spinless Hubbard model in D = 1.
Finally, we point out that the presentβ-expansion of the HFE of the one-dimensional spinless Hubbard
model is valid for any set of parameters of its Hamiltonian, including the casesV > 0 (repulsion) andV < 0
(attraction).
33003-10
The β-expansion of the D = 1 fermionic spinless Hubbard model off the half-filling regime
Acknowledgements
O. Rojas thanks CNPq and FAPEMIG for partial financial support.
A. The HFE of the one-dimensional fermionic spinless Hubbard model
up to order β8
We have applied the method of reference [15] to calculate the β-expansion of the HFE associated to
the Hamiltonian (2.1a)–(2.1b) and to the Hamiltonian (2.2) [see equation (2.7)].We have also implemented
a new set of rules that permit the algebraic computation of the HFE of the one-dimensional fermionic
spinless Hubbard model up to order β8,
W (t ,V ,µ;β) = −
ln (2)
β
+
1
4
V −
1
2
µ+
(
−
5
32
V 2
+1/4V µ
1
8
µ2
−
1
4
t 2
)
β
+
(
1
16
V 3
−
1
8
V 2µ+
1
16
V µ2
−
1
16
V t 2
)
β2
+
(
−
31
3072
V 4
+
1
96
V 3µ+
1
64
V 2µ2
+
7
96
V 2t 2
−
1
48
V µ3
−1/8V µ t 2
+
1
192
µ4
+
1
16
µ2t 2
+
1
32
t 4
)
β3
+
(
−
1
128
V 5
+
7
192
V 4µ−
23
384
V 3µ2
−
7
256
V 3t 2
+
1
24
V 2µ3
+
1
16
V 2µ t 2
−
1
96
V µ4
−
1
32
V µ2t 2
+
1
32
V t 4
)
β4
+
(
287
36864
V 6
−
239
7680
V 5µ+
139
3072
V 4µ2
−
21
2560
V 4t 2
−
31
1152
V 3µ3
+
7
192
V 3µ t 2
+
5
1536
V 2µ4
−
23
384
V 2µ2t 2
−
47
1536
V 2t 4
+
1
480
V µ5
+
1
24
V µ3t 2
+
1
16
V µ t 4
−
1
2880
µ6
−
1
96
µ4t 2
−
1
32
µ2t 4
−
1
144
t 6
)
β5
+
(
−
29
10240
V 7
+
389
46080
V 6µ−
119
30720
V 5µ2
+
1603
92160
V 5t 2
−
7
576
V 4µ3
−
53
768
V 4µ t 2
+
41
2304
V 3µ4
+
157
1536
V 3µ2t 2
+
83
23040
V 3t 4
−
17
1920
V 2µ5
−
13
192
V 2µ3t 2
−
1
64
V 2µ t 4
+
17
11520
V µ6
+
13
768
V µ4t 2
+
1
128
V µ2t 4
−
11
768
V t 6
)
β6
+
(
−
108527
165150720
V 8
+
17
645120
µ8
+
17
9216
t 8
+
1709
46080
V 5µ t 2
−
1439
30720
V 4µ2t 2
+
19
1152
V 3µ3t 2
−
83
1536
V 3µ t 4
+
49
4608
V 2µ4t 2
+
73
1024
V 2µ2t 4
−
17
1920
V µ5t 2
−
17
384
V µ3t 4
−
17
576
V µ t 6
+
9241
1290240
V 7µ−
8939
368640
V 6µ2
−
12877
1290240
V 6t 2
+
3539
92160
V 5µ3
−
2311
73728
V 4µ4
+
191
12288
V 4t 4
+
287
23040
V 3µ5
−
73
46080
V 2µ6
+
31
2880
V 2t 6
−
17
80640
V µ7
+
17
11520
µ6t 2
+
17
1536
µ4t 4
+
17
1152
µ2t 6
)
β7
+
(
817
92160
V 6µ t 2
−
841
20480
V 5µ2t 2
+
85
1152
V 4µ3t 2
+
473
7680
V 4µ t 4
−
299
4608
V 3µ4t 2
−
1393
15360
V 3µ2t 4
+
107
3840
V 2µ5t 2
+
23
384
V 2µ3t 4
−
5
1152
V 2µ t 6
−
107
23040
V µ6t 2
−
23
1536
V µ4t 4
+
5
2304
V µ2t 6
+
4619
3096576
V 9
−
24649
2580480
V 8µ+
125497
5160960
V 7µ2
+
367
4128768
V 7t 2
−
4259
138240
V 6µ3
+
3449
184320
V 5µ4
−
9991
645120
V 5t 4
−
53
23040
V 4µ5
−
443
138240
V 3µ6
+
65
13824
V 3t 6
+
31
20160
V 2µ7
−
31
161280
V µ8
+
29
4608
V t 8
)
β8
+O(β9
). (A.1)
33003-11
E.V. Corrêa Silva, M.T. Thomaz, O. Rojas
References
1. Simon J., Bakr W.S., Ma R., Tai M.E., Preiss P.M., Greiner M., Nature, 2011, 472, 307; doi:10.1038/nature09994.
2. Hubbard J., Proc. R. Soc. Lond. A, 1963, 276, 238; doi:10.1098/rspa.1963.0204.
3. Hubbard J., Proc. R. Soc. Lond. A, 1964, 277, 237; doi:10.1098/rspa.1964.0019.
4. Verwey E.J.W., Haaymann P.W., Physica, 1941, 8, 979; doi:10.1016/S0031-8914(41)80005-6.
5. Kobayashi K., Susaki T., Fujimori A., Tonogai T., Takagi H., Europhys. Lett., 2002, 59, 868;
doi:10.1209/epl/i2002-00123-2
6. Fulde P., Penc K., Shannon N., Ann. Phys. (Berlin), 2002, 11 892;
doi:10.1002/1521-3889(200212)11:12<892::AID-ANDP892>3.0.CO;2-J.
7. Haldane F.D.M., Phys. Rev. Lett., 1980, 45, 1358; doi:10.1103/PhysRevLett.45.1358.
8. Sznajd J., Becker K., J. Phys.: Condens. Matter, 2005, 17, 7359; doi:10.1088/0953-8984/17/46/020.
9. Thomaz M.T., Corrêa Silva E.V., Rojas O., Condens. Matter Phys., 2014, 17, 23002; doi:10.5488/CMP.17.23002.
10. Takahashi M., Thermodynamics of One-Dimensional Solvable Models, Cambridge Univ. Press, Cambridge, 1999.
11. Bühher A., Elstner N., Uhrig G.S., Eur. Phys. J. B, 2000, 16, 475; doi:10.1007/s100510070206.
12. Takahasi M., In: Physics and Combinatorics, Kirillov A.K., Liskova N. (Eds.), World Scientific, Singapore, 2001,
p. 299.
13. Shiroishi M., Takahashi M., Phys. Rev. Lett., 2002, 89, 117201; doi:10.1103/PhysRevLett.89.117201.
14. Rojas O., de Souza S.M., Corrêa Silva E.V., Thomaz M.T., Eur. Phys. J. B, 2005, 47, 165;
doi:10.1140/epjb/e2005-00310-5.
15. Rojas O., de Souza S.M., Thomaz M.T., J. Math. Phys., 2002, 43, 1390; doi:10.1063/1.1432484.
16. Moura-Melo W.A., Rojas O., Corrêa Silva E.V., de Souza S.M., Thomaz M.T., Physica A, 2003, 322, 393;
doi:10.1016/S0378-4371(02)01749-1.
17. Rojas O., de Souza S.M., Corrêa Silva E.V., Thomaz M.T., Braz. J. Phys., 2001, 31, 577 (Please notice a misprint in
the HFE of this reference; the constant ∆ in equation (25) should read ∆/2).
18. Baxter R.J., Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982, Chapter 2.
β-розвинення D = 1 фермiонної безспiнової моделi
Габбарда поза половинним заповненням
Е.В. Кореа Сiльва1, М.Т. Томаз2 , O. Рохас3
1 Технологiчний факультет, Державний унiверситет м. Рiо-де-Жанейро, Резендi, Бразилiя
2 Iнститут фiзики, Федеральний унiверситет Флумiненсе, Нiтерой, Бразилiя
3 Вiддiл фiзики, Федеральний унiверситет Лаврас, Лаврас, Бразилiя
Встановлено, що для безспiнової моделi поза половинним заповненням (µ , V ) вiльну енергiю Гельм-
гольца можна записати у виглядi двох β-розвинень: одне розвинення походить вiд конфiгурацiї з по-
ловинним заповнення, а iнше залежить вiд параметра вiдхилення x = µ−V . Чисельно показано, що
хiмiчний потенцiал як функцiя температури задовольняє спiввiдношення подiбне до того, яке отримує-
ться з симетрiї частинка-дiрка фермiонної безспiнової моделi. β-розвинення вiльної енергiї Гельмгольца
одновимiрної фермiонної безспiнової моделi Габбарда продовжено аж до порядку β8.
Ключовi слова: квантова статистична механiка, сильно скорельвана електронна система, моделi
спiнових ланцюжкiв
33003-12
http://dx.doi.org/10.1038/nature09994
http://dx.doi.org/10.1098/rspa.1963.0204
http://dx.doi.org/10.1098/rspa.1964.0019
http://dx.doi.org/10.1016/S0031-8914(41)80005-6
http://dx.doi.org/10.1209/epl/i2002-00123-2
http://dx.doi.org/10.1002/1521-3889(200212)11:12%3C892::AID-ANDP892%3E3.0.CO;2-J
http://dx.doi.org/10.1103/PhysRevLett.45.1358
http://dx.doi.org/10.1088/0953-8984/17/46/020
http://dx.doi.org/10.5488/CMP.17.23002
http://dx.doi.org/10.1007/s100510070206
http://dx.doi.org/10.1103/PhysRevLett.89.117201
http://dx.doi.org/10.1140/epjb/e2005-00310-5
http://dx.doi.org/10.1063/1.1432484
http://dx.doi.org/10.1016/S0378-4371(02)01749-1
Introduction
The fermionic spinless Hubbard model in D=1 and its exact relations
Discussion on the -expansion of the HFE of the model
The temperature dependence of the chemical potential
Conclusions
The HFE of the one-dimensional fermionic spinless Hubbard model up to order 8
|
| id | nasplib_isofts_kiev_ua-123456789-154206 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T18:17:04Z |
| publishDate | 2015 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Correa Silva, E.V. Thomaz, M.T. Rojas, O. 2019-06-15T10:00:51Z 2019-06-15T10:00:51Z 2015 The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime / E.V. Correa Silva, M.T. Thomaz, O. Rojas // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33003: 1–12. — Бібліогр.: 18 назв. — англ. 1607-324X PACS: 05.30.Fk, 71.27.+a, 75.10.Pq DOI:10.5488/CMP.18.33003 arXiv:1510.06531 https://nasplib.isofts.kiev.ua/handle/123456789/154206 We obtain that when the spinless model is off the half-filling regime (μ ≠ V), the Helmholtz free energy (HFE) can be written as two β-expansions: one expansion comes from the half-filling configuration and another one that depends on the parameter x = μ - V. We show numerically that the chemical potential as a function of temperature satisfies a relation similar to one derived from the particle-hole symmetry of the fermionic spinless model. We extend the β-expansion of the HFE of the one-dimensional fermionic spinless Hubbard model up to order β⁸. Встановлено, що для безспiнової моделi поза половинним заповненням (µ , V ) вiльну енергiю Гельмгольца можна записати у виглядi двох β-розвинень: одне розвинення походить вiд конфiгурацiї з половинним заповнення, а iнше залежить вiд параметра вiдхилення x = µ − V . Чисельно показано, що
 хiмiчний потенцiал як функцiя температури задовольняє спiввiдношення подiбне до того, яке отримується з симетрiї частинка-дiрка фермiонної безспiнової моделi. β-розвинення вiльної енергiї Гельмгольца
 одновимiрної фермiонної безспiнової моделi Габбарда продовжено аж до порядку β
 ⁸
 . O. Rojas thanks CNPq and FAPEMIG for partial financial support. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime β-розвинення D = 1 фермiонної безспiнової моделi Габбарда поза половинним заповненням Article published earlier |
| spellingShingle | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime Correa Silva, E.V. Thomaz, M.T. Rojas, O. |
| title | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime |
| title_alt | β-розвинення D = 1 фермiонної безспiнової моделi Габбарда поза половинним заповненням |
| title_full | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime |
| title_fullStr | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime |
| title_full_unstemmed | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime |
| title_short | The β-expansion of the D=1 fermionic spinless Hubbard model off the half-filling regime |
| title_sort | β-expansion of the d=1 fermionic spinless hubbard model off the half-filling regime |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154206 |
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