Density of states of one-dimensional Pauli ionic conductor
Microscopic one-dimensional noninteracting model for the description of the energy spectrum of the ion subsystem in ionic conductor is considered. The processes of ionic hoppings are described in terms of Pauli operators. Time-dependent correlation functions (b(t)b⁺(0))j in Pauli operators are obt...
Збережено в:
| Дата: | 2007 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2007
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/118383 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Density of states of one-dimensional Pauli ionic conductor / I.V. Stasyuk, I.R. Dulepa // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 259-268. — Бібліогр.: 19 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-118383 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1183832017-05-31T03:07:24Z Density of states of one-dimensional Pauli ionic conductor Stasyuk, I.V. Dulepa, I.R. Microscopic one-dimensional noninteracting model for the description of the energy spectrum of the ion subsystem in ionic conductor is considered. The processes of ionic hoppings are described in terms of Pauli operators. Time-dependent correlation functions (b(t)b⁺(0))j in Pauli operators are obtained using the exact numerical procedure known for time-dependent spin correlation functions. The frequency dependence of autocorrelation function J bb+(ω) is calculated and analysed at the wide range of temperatures. The frequency and temperature dependences of the one-particle density of states are investigated. Розглянуто мiкроскопiчну одновимiрну модель невзаємодiючих частинок для опису енергетичногоспектру iонної пiдсистеми в iонному провiднику. Процеси iонного перескоку описанi у термiнах операторiв Паулi. Використовуючи точний числовий метод, вiдомий для часових спiнових кореляцiйних функцiй, отримано часовi кореляцiйнi функцiї (b(t)b⁺(0))j в операторах Паулi. Обчислено i проаналiзовано частотну залежнiсть автокореляцiйної функцiї J bb+(ω) в широкому iнтервалi температур. Дослiджено частотну i температурну залежностi одночастинкової густини станiв. 2007 Article Density of states of one-dimensional Pauli ionic conductor / I.V. Stasyuk, I.R. Dulepa // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 259-268. — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 75.10.Pq, 66.30.Dn, 66.10.Ed DOI:10.5488/CMP.10.2.259 http://dspace.nbuv.gov.ua/handle/123456789/118383 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 |
Microscopic one-dimensional noninteracting model for the description of the energy spectrum of the ion subsystem
in ionic conductor is considered. The processes of ionic hoppings are described in terms of Pauli operators. Time-dependent correlation functions (b(t)b⁺(0))j in Pauli operators are obtained using the exact numerical procedure known for time-dependent spin correlation functions. The frequency dependence of autocorrelation function J bb+(ω) is calculated and analysed at the wide range of temperatures. The frequency and temperature dependences of the one-particle density of states are investigated. |
| format |
Article |
| author |
Stasyuk, I.V. Dulepa, I.R. |
| spellingShingle |
Stasyuk, I.V. Dulepa, I.R. Density of states of one-dimensional Pauli ionic conductor Condensed Matter Physics |
| author_facet |
Stasyuk, I.V. Dulepa, I.R. |
| author_sort |
Stasyuk, I.V. |
| title |
Density of states of one-dimensional Pauli ionic conductor |
| title_short |
Density of states of one-dimensional Pauli ionic conductor |
| title_full |
Density of states of one-dimensional Pauli ionic conductor |
| title_fullStr |
Density of states of one-dimensional Pauli ionic conductor |
| title_full_unstemmed |
Density of states of one-dimensional Pauli ionic conductor |
| title_sort |
density of states of one-dimensional pauli ionic conductor |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2007 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118383 |
| citation_txt |
Density of states of one-dimensional Pauli ionic conductor / I.V. Stasyuk, I.R. Dulepa // Condensed Matter Physics. — 2007. — Т. 10, № 2(50). — С. 259-268. — Бібліогр.: 19 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyukiv densityofstatesofonedimensionalpauliionicconductor AT dulepair densityofstatesofonedimensionalpauliionicconductor |
| first_indexed |
2025-07-08T13:52:47Z |
| last_indexed |
2025-07-08T13:52:47Z |
| _version_ |
1837087082903764992 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 2(50), pp. 259–268
Density of states of one-dimensional Pauli ionic
conductor
I.V.Stasyuk, I.R.Dulepa
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
Received May 7, 2007
Microscopic one-dimensional noninteracting model for the description of the energy spectrum of the ion sub-
system in ionic conductor is considered. The processes of ionic hoppings are described in terms of Pauli
operators. Time-dependent correlation functions 〈b(t)b+(0)〉j in Pauli operators are obtained using the exact
numerical procedure known for time-dependent spin correlation functions. The frequency dependence of au-
tocorrelation function Jbb+ (ω) is calculated and analysed at the wide range of temperatures. The frequency
and temperature dependences of the one-particle density of states are investigated.
Key words: Pauli statistics, correlation functions, particle density of states, ionic conductor
PACS: 75.10.Pq, 66.30.Dn, 66.10.Ed
1. Introduction
An ion transfer in solids has become an active area of research nowadays. Different theoretical
models have been developed to perform the necessary investigations in this direction. One of the
most important aspects in this field is the problem of determining the energy spectrum, e. g. the
particle density of states, and the ionic conductivity. The latter is connected with the process of
site-to-site ionic hopping and strongly depends on temperature. It is important to mention here
that the activation energy of conductivity is determined not only by the height of potential barrier
which the ion must overcome changing its position but is also effected by the nearest-neighbor
configuration. An interaction energy should be taken into account in theoretical investigations of
the ionic hopping in solids.
Among several basic models of ionic hopping conduction, the lattice gas model [1–5] is the
mostly used. This model has been introduced by G. D. Mahan three decades ago to calculate the
ionic conductivity in superionic conductors. The conductivity value obtained was in agreement with
the found experimental data [1]. At present the lattice gas model is applied, besides ionic conductors
(see, for example, [6]), to numerous physical processes such as intercalation [7], adsorption [8] and
others.
Perturbation expansion in powers of hopping parameter [2,5] was used to calculate the correla-
tion functions for the lattice gas model. Mostly, only the first order perturbation corrections were
taken into account, i.e. the term which should describe the ionic hopping is neglected in the model
Hamiltonian at statistical averaging. The second order perturbation expansion was taken into ac-
count in [5]. However, the nearest-neighbor interaction between ions was not considered. Taking
into account the known exact solutions for Ising model, an explicit expression for the time depen-
dent correlation function was obtained within the framework of lattice gas model [1] using current
density operators to determine the ionic conductivity. However, the approach was applicable only
in a narrow interval of the ion concentrations. Expansion in terms of the hopping parameter was
also used in [9] in describing the conductivity of superprotonic conductor; the effect of phonon
assisted hopping on the activation energy of protonic conductivity was taken into account. Sum-
marizing, one can conclude that the problem of developing the model approaches that explicitely
take into account the ionic hopping in the zero-order Hamiltonian still remains unresolved.
c© I.V.Stasyuk, I.R.Dulepa 259
I.V.Stasyuk, I.R.Dulepa
Statistics of particles, which are described by the lattice gas model, as well as its consequences
turn out to be another interesting question. In a series of papers devoted to ionic conductivity,
starting with [1], the particles were described by Pauli creation and annihilation operators. This
corresponds to an ordinary restriction on the ion occupation number at a certain lattice site
and reflects the Bose origin of particles (the approach that corresponds to the picture of hard-
core bosons [10]). Besides, the spinless fermion lattice model, where particles obey the Fermi
statistics, is also used in describing the objects of the ionic conductor type [11]. The comparison
of the mentioned approaches has not been made so far. In particular, it touches upon the possible
difference in the energy spectrum structure and thermodynamics of the model.
In the present paper we consider the one-dimensional model of the ion hopping between near-
est sites. For simplicity the nearest-neighbor interactions between ions are neglected. Our aim is
to calculate the correlation functions and analyse the frequency dependences of the one-particle
densities of states in the case when ions on the lattice are described using the Pauli statistics. The
Pauli correlation functions are obtained using the exact numerical method based on the pseudospin
representation and the fermionization procedure1.
In section 2 we consider the relation between pseudospin and Pauli operators; the model is
discussed and the Jordan-Wigner transformation is applied. The time-dependent autocorrelation
functions built on Pauli operators are obtained in section 3. This makes it possible to obtain
frequency dependent densities of states at a given site, temperature and local energy of particles.
The frequency and the temperature dependence of the density of states are analysed in section 4.
The comparison is made with the results that can be obtained within the fermion lattice model.
2. Model
The ion transfer from the given site to nearest-neighbor sites of a lattice in one dimension is
described by the Hamiltonian
H =
N
∑
i=1
(ε0 − µ)b+
i bi + t0
N
∑
i=1
(b+
i bi+1 + b+
i+1bi), (1)
where ε0 is the ion energy at the site, µ is chemical potential. We consider the chain with N sites,
t0 is the transfer parameter.
The ions on the sites are described by the Pauli creation and annihilation b+
i and bi operators
which are neither fermion nor boson operators, but correspond exactly to S = 1/2 spin operators.
The ion presence or absence at the i -th site is equivalent to the mathematical problem with spin
up or down. The commutation relations for the Pauli operators are as follows:
{b+
i , bi} = 1, [b+
i , b+
j ] = [bi, bj ] = [b+
i , bj ] = 0, (bi)
2 = (b+
i )2 = 0.
We perform the transformation from these operators to spinless Fermi operators on the same lattice
by means of the Jordan-Wigner transformation:
bi = S−
i = ai(−1)
�
j6i−1
nj , b+
i = S+
i = a+
i (−1)
�
j6i−1
nj
and [b+
i , bi] = 2Sz
i , Sz
i = b+
i bi − 1/2 = ni − 1/2. a+
i , ai are Fermi operators. The sign factor can
be written as follows:
(−1)
�
j6i−1
nj =
i−1
∏
j=1
(1 − 2a+
j aj) =
i−1
∏
j=1
(−2Sz
j )
and
b+
i bi+1 = a+
i
i−1
∏
j=1
(1 − 2a+
j aj)
i
∏
k=1
(1 − 2a+
k ak)ai+1 = a+
i ai+1, b+
i+1bi = a+
i+1ai .
1First the “fermionization” was done in [12] for transformation between the spin variables and noninteracting
lattice fermions. Jordan-Wigner transformation for spin systems in one and two dimensions was considered in [13].
It yields an explicit solution for spin correlation functions [14–18].
260
DOS of one-dimensional Pauli ionic conductor
Thus, we have free spinless fermions, whose Hamiltonian reads
H =
N
∑
i=1
(ε0 − µ)a+
i ai + t0
N
∑
i=1
(a+
i ai+1 + a+
i+1ai). (2)
Let us consider an infinite chain (N → ∞) and make Fourier transformation of operators in (2)
ai =
1√
N
∑
q
eiqRiαq, a+
i =
1√
N
∑
q
e−iqRiα+
q ;
then H has the diagonal form
H =
∑
q
λ(q)α+
q αq, λ(q) = ε0 − µ + 2t0 cos (qa). (3)
Here a is the lattice constant , q = (2πn)/(Na), n = 1, . . . , N (throughout this paper a = 1).
It is easy to consider the thermodynamics of such a system in this representation. The grand
canonical potential is
Ω = − 1
β
ln Sp e−βH = − 1
β
ln
∏
q
∑
nq=0,1
e−βλ(q)nq = − 1
β
∑
q
ln
(
1 + e−βλ(q)
)
.
Then the average occupation number
〈n〉 =
1
π
∫ π
0
〈nq〉dq, 〈nq〉 =
1
2
− 1
2
tanh
βλ(q)
2
(4)
and 〈Sz〉 = 〈n〉 − 1/2.
3. The time-dependent correlation functions on Pauli operators
Let us consider the correlation functions built on Pauli operators. In order to calculate them we
can use the results known for the time-dependent spin correlation functions. The exact numerical
method of calculating such functions is described in [14–18]. In our case the parameter ε = ε0 − µ
which can be introduced here plays the role of external field in spin representation. We investigate
the ion chains with sufficiently large number of sites N = 400, 600, 1000 to analyse the limit of
large N .
Let us write the Hamiltonian (2) as follows:
H =
∑
i,j
a+
i Aijaj , (5)
where Aij = (ε0 − µ)δij + t0(δj,i+1 + δj,i−1). The Hamiltonian (5) can be reduced to the diagonal
form by a linear transformation
a+
i =
∑
k
gikη+
k , ai =
∑
k
gikηk, H =
∑
i,j,k,k′
gikgjk′Aijη
+
k ηk′ , (6)
where η+
k , ηk are also Fermi operators.
Then a traditional eigenvalue problem takes place
∑
i
gsiAil = Esgsl, (7)
which is solved using the standard numerical procedure. The Hamiltonian in diagonal form in this
case is H =
∑N
k=1 Ekη+
k ηk. Our task is to calculate the time dependent Pauli pair correlation
functions. The relation
〈b(t)b+(0)〉i = 〈S−(t)S+(0)〉i (8)
261
I.V.Stasyuk, I.R.Dulepa
between Pauli and spin correlation functions is used. In this case
〈b(t)b+(0)〉i = 2[〈Sx(t)Sx(0)〉i + i〈Sx(t)Sy(0)〉i] (9)
and 〈Sx(t)Sx(0)〉i = 〈Sy(t)Sy(0)〉i, 〈Sx(t)Sy(0)〉i = −〈Sy(t)Sx(0)〉i.
Therefore, the calculation of the pair correlation functions on the Pauli operators is reduced
to the calculation of the spin correlation functions of appropriate components. However, the com-
puting of the spin correlation function maps into the calculation of the many-particle correlation
function on Fermi operators.
Let us write down the fermionization of the Sx
i , Sy
i operators:
Sx
i =
1
2
(
b+
i + bi
)
=
1
2
i−1
∏
j=1
(−2Sz
j )(a+
i + ai),
Sy
i =
1
2i
(b+
i − bi) =
1
2i
i−1
∏
j=1
(−2Sz
j )(a+
i − ai),
Sz
i =
1
2
[b+
i , bi] = −1
2
(b+
i + bi)(b
+
i − bi) = −1
2
(a+
i + ai)(a
+
i − ai).
It is known that by substituting the expression for Sz
i into Sx
i and Sy
i we can obtain a relation
between spin correlation functions and the ones on the Fermi operators:
〈Sx(t)Sx(0)〉i =
1
4
〈
i−1
∏
j=1
ϕ+
j (t)ϕ−
j (t)ϕ+
i (t)
i−1
∏
j=1
ϕ+
j ϕ−
j ϕ+
i
〉
, (10)
〈Sx(t)Sy(0)〉i =
i
4
〈
i−1
∏
j=1
ϕ+
j (t)ϕ−
j (t)ϕ+
i (t)
i−1
∏
j=1
ϕ+
j ϕ−
j ϕ−
i
〉
, (11)
where ϕ±
j = a+
j ± aj . Wick’s theorem can be applied to expand 〈Sx(t)Sx(0)〉i and 〈Sx(t)Sy(0)〉i
into the sum of the products of elementary average values:
〈
ϕ+
i (t)ϕ±
j
〉
=
N
∑
k=1
gikgjk
[
eiEkt
1 + eβEk
± e−iEkt
1 + e−βEk
]
,
〈
ϕ+
i (t)ϕ+
j
〉
= −
〈
ϕ−
i (t)ϕ−
j
〉
,
〈
ϕ+
i (t)ϕ−
j
〉
= −
〈
ϕ−
i (t)ϕ+
j
〉
.
The expansions (9) and (10) can be more compactly expressed as a pfaffian [14–16].
The analytical results for these one-site spin correlation functions 〈Sx(t)Sx(0)〉i, 〈Sx(t)Sy(0)〉i
were obtained by Capel and Perk [19] at the infinite temperature β = 0 in the limiting case N → ∞
〈Sx(t)Sx(0)〉 =
1
4
e−t20t2 cos ((ε0 − µ)t) , 〈Sx(t)Sy(0)〉 =
1
4
e−t20t2 sin ((ε0 − µ)t) .
Substituting these relations into (8) we have
〈b(t)b+(0)〉 =
1
2
eεte−t20t2 . (12)
For finite values of N, we obtain the correlation functions using a numerical method for calcula-
tion of pfaffians [15]. The transfer parameter t0 is set equal to unity in this paper, t0 = 1; all energy
quantities are given in relation to t0. Here we do not investigate the dependence of correlation on
a site number. We consider only the one-site correlation at a separate site (j=50, taken as an
example) and the boundary effects are neglected.
The time dependent correlation function 〈b(t)b+(0)〉j at jth site for the ion chain with N = 600
and with the energy values ε = ε0−µ = 0.0001, ±0.5 at β = 0.001 is presented in figure 1. The real
262
DOS of one-dimensional Pauli ionic conductor
Figure 1. Time dependent correlation function 〈b(t)b+〉j at β = 0.001.
parts of Pauli correlation function are the same at ε = ±0.5 and the imaginary parts of 〈b(t)b+〉j
change the sign. Im 〈b(t)b+(0)〉j is equal practically to zero at ε = 0.0001 and the correlation
function is given by its real part.
The time dependence of the one-site Pauli correlation function becomes more complicated at
finite temperatures. The case β = 1 is presented in figure 2; Im〈b(t)b+(0)〉j is nonzero for ε = 0.
The oscillation appears at the increase of ε ; the main minimum value of the imaginary part of the
Pauli correlation function decreases with the energy ε0 − µ increase (figure 2b).
Figure 2. Time dependent correlation function 〈b(t)b+〉j at β = 1. (a): Re〈b(t)b+(0)〉j ; (b):
Im〈b(t)b+(0)〉j .
4. Particle density of states
The commutation relations for Pauli operators are intermediate between Fermi and Bose stati-
stics. It is interesting to calculate the particle density of states (DOS) and compare with the
mentioned pure cases. So, here we consider such a particle DOS as ρc
j (ρa
j ) calculated similarly to
the case of bosons (fermions).
It is well known that the particle DOS can be obtained from the imaginary part of the two-time
Zubarev Green’s function:
ρij(ω) = −2Im 〈〈bi | b+
j 〉〉ω+iδ . (13)
At first we consider the commutator Green’s function
Gij(t, t
′
)c = 〈〈bi(t) | b+
j (t′)〉〉 = −iΘ(t − t′)〈[bi(t), b
+
j (t′)]〉. (14)
263
I.V.Stasyuk, I.R.Dulepa
The Fourier transforms of the Green’s function in the spectral representation:
〈〈bi | b+
j 〉〉ω+iδ =
1
2π
∫ +∞
−∞
Jbb+(ω′)
1 − e−βω′
ω − ω′ + iδ
dω′ (15)
is connected with the one-site autocorrelation function
Jbb+(ω) =
∫ +∞
−∞
eiω(t−t′)〈b(t)b+(t′)〉jd(t − t′).
The latter can be expressed in terms of spin operators
Jbb+(ω) =
∫ +∞
−∞
eiωt〈S−(t)S+(0)〉jdt. (16)
Taking into account that
〈S−(t)S+(0)〉j = 〈S−(0)S+(−t)〉j = 〈S−(0)S+(t)〉∗j = 〈S−(−t)S+(0)〉j = 〈S−(t)S+(0)〉∗j ,
we have in Pauli operators
Jbb+(ω) =
∫ ∞
0
[
eiωt〈b(t)b+(0)〉j + e−iωt〈b(−t)b+(0)〉j
]
dt = 2Re
∫ ∞
0
eiωt
〈
b(t)b+(0)
〉
j
dt. (17)
This expression is used for the numerical calculation of the particle autocorrelation function based
on the procedure presented in the previous section.
In the analytical (N → ∞) case (11) the autocorrelation function calculated using the expression
(11) is as follows:
Jbb+(ω) =
√
π
2t0
exp
( (ω − ε)2
4t20
)
. (18)
Figure 3. Frequency dependence of Jbb+(ω) at β = 0.001.
The frequency dependence of the Jbb+(ω) at β = 0.001; ε = 0.0001, ±0.5 is presented in figure 3.
Here we compare the results of numerical calculations with the analytical ones (17) obtained at
β = 0. As it is shown in figure 3 on the example of the ε = 0 case, numerical curves and analytical
data are identical. The autocorrelation function Jbb+(ω) possesses a gaussian shape with maximum
value at ω = ε; its positions shift to positive or negative values depending on the sign of the ε
parameter.
264
DOS of one-dimensional Pauli ionic conductor
Substitution of (14) into (12) gives the following expression for DOS
ρc
j(ω) =
(
1 − e−βω
)
Jbb+(ω). (19)
So, the calculation of the frequency dependent DOS ρc
j is reduced to the calculation of the
autocorrelation function (18) with a factor 1 − e−βω.
Figure 4. Frequency dependent DOS ρc
j , ρa
j at β = 0.001.
The form of function ρc
j(ω) in the case β = 0.001 is shown in figure 4 at ε = 0.0001, ±0.5. The
function changes its sign at ω = 0 and reaches maximum and minimum values at positive and
negative frequencies , respectively. At ε > 0 the spectral weight is larger at ω > 0 and increases at
the increase of ε. For comparison in the same figure we show a plot of the “fermi-like” density of
states
ρa
j (ω) = (1 + e−βω)Jbb+(ω) (20)
as a function of frequency. The function ρa
j (ω) is connected with the anticommutator Green’s
function
Ga
ij(ω + iε) = −iθ(t − t′)
〈
{bi(t), b
+
j (t′)}
〉
(21)
by the relation similar to the relation (13). Its frequency behaviour is different from the ρc
j(ω)
function coinciding up to a factor of 2 with the Jbb+ plot in the β = 0 limit.
In figure 5 the frequency dependence of the autocorrelation function Jbb+(ω) and density of
states ρc
j(ω) is given in the case of finite temperatures (β = 1 and β = 5) for different values of ε.
At the temperature decrease (that corresponds to the increase of the β parameter) the additional
peaks appear on the frequency dependence of DOS (in the region of positive/negative frequencies
at ε > 0/ε < 0). Their intensities grow at larger values of |ε|. This situation is also illustrated
in figure 6 where a series of plots of the Jbb+(ω) function at ε = 0.5 at different temperatures
is presented. At temperature lowering, two distinct peaks are developed. The similar two-peak
structure also takes place at β � 1 for “fermi” DOS ρa
j . As an example, the plots of this function
at ε = −2 are shown in figure 7 at different values of β.
The reason for such a behaviour of DOS at low temperatures is as follows. When T → 0 and
ε � 1 (we consider here, for definition, the case ε > 0), the b+
i and bi operators describe the
creation and annihilation of the magnon-like excitations (at ε > 0 the ground state of the system
is realized at Sz
i = −1/2); their commutation relations become boson-like at low concentrations
of “magnons”. In this case the Hamiltonian (1) can be diagonalized at N → ∞ in terms of b+
i , bi
operators by means of Fourier transformation with the eigenvalue spectrum, which coincides with
formula (3). The density of states ρc
j(ω) is given now by the known expression
ρc
j(ω) =
1
N
∑
q
δ (ω − λ(q)) =
1
π
1
√
4t20 − (ω − ε)2
(22)
265
I.V.Stasyuk, I.R.Dulepa
ε
ε
ε
ε
ε
ε
ω
ρ
ω
ε
ε
ε
ε
ε
ε
ε
ε
ε
ε
ρ
ω
ε
ε
ε
ε
ε
ε
ε
ρ
ω
Figure 5. Jbb+(ω), ρc
j(ω) at β = 1 (a, b), ρc
j(ω) at β = 5 (c, d).
ω
ε= −0.5
β=0
2: β=1
3: β=3
4: β=5
5: β=7
6: β=10
7: β=15
Figure 6. Autocorrelation functions Jbb+(ω) at ε = 0.5 for indicated temperatures.
Figure 7. Frequency dependences ρa
j (ω) in the case ε = −2 at different temperatures (a) and
ρc
j(ω)|β→∞ at different values of ε (b).
266
DOS of one-dimensional Pauli ionic conductor
at |ω − ε| 6 2t0. The function (21) diverges at the edges of the “magnon” band (ω = ε ± 2t0,
or ω − ε = ±1 in dimensionless units). This result indicates the limiting shape of the two-peak
structure of the ρc
j(ω) function when temperature goes to zero in the case of large values of ε.
Similarly we can interpret the form of the ρa
j function at low temperatures. However, this
function differs, on the whole, by its appearance from the ρc
j one. Moreover, if the b+
i , bi operators
were considered as fermi-operators from the very beginning (which corresponds to the frequently
used fermion lattice model), the eigenvalue spectrum would be the same as (3) and the fermion
density of states ρa
j (ω) would be given by formula analogous to (21). Contrary to the case of Pauli
operators, this should be the case in the whole temperature region.
5. Conclusions
The simple noninteracting model of ionic hopping in the one-dimensional chain is analysed in
the given paper based on the lattice gas model. The particles are described using the Pauli statistics.
The correlation functions constructed on the Pauli operators b+
i , bi(0) of creation and annihilation
of particles are investigated. The one-particle densities of states are calculated. An approach for
correlation function 〈b(t)b+(0)〉j is based on fermionization procedure which permits to formulate
the exact calculation procedure. Numerical results for 〈b(t)b+(0)〉j at β = 0 are in good agreement
with analytical estimates at N → ∞. The frequency dependences of autocorrelation function
Jbb+(ω) as well as frequency dependences of densities of states are obtained for a fixed lattice site,
temperature and particle energy. Special attention is paid to the changes in density of states at
the decrease of temperature. The transformation of the one-particle spectrum (from gaussian-like
density of states at high enough temperatures to the two-peak structure at low temperatures and
sufficiently large energy difference ε = ε0 − µ) is analysed. The correlation between the two-peak
DOS and density of states ρc
j at T → 0, |ε| → ∞, that corresponds to the boson gas of the magnon-
like excitations, is established. A comparison with the DOS in the fermion picture shows significant
differences in spectrum and its spectral features.
Acknowledgements
The authors are greatly indebted to Taras Krokhmalskii for help, collaboration and numerical
assistance.
267
I.V.Stasyuk, I.R.Dulepa
References
1. Mahan G. D., Phys. Rev. B, 1976, 14, No. 2, 780.
2. Yonashiro K., Tomoyose T., Sakai E., J. Phys. Soc. Jpn., 1983, 52, No. 11, 3837.
3. Tomoyose T., J. Phys. Soc. Jpn., 1990, 60, No. 4, 1263.
4. Tomoyose T., J. Phys. Soc. Jpn., 1997, 66, No. 8, 2383.
5. Tanaka T., Majed A. Sawatarie, Barry J. H., Shatma N. L. and Munera C. H., Phys. Rev. B, 1986,
34, 3773.
6. I. V. Stasyuk, O. L. Ivankiv, N. I. Pavlenko, J. Phys. Stud., 1997, 1, 418.
7. Yu. I. Dublenych, Phys. Rev. B., 2005, 71, 012411.
8. Rikvold P. A., Zhang J., Sung Y. E., Wieckowski A. Preprint arXiv:cond-mat/9506025, 1995.
9. I. V. Stasyuk, N. I. Pavlenko, J. Chem. Phys., 2001, 114, No. 10, 4607.
10. R. Micnas and S. Robaszkiewicz,Phys. Rev. B., 1992, 45, No. 17, 9902.
11. I. V. Stasyuk, O. Vorobyov, B. Hilczer, Solid State Ionics, 2001, 145, 211.
12. Lieb E., Schultz T., Mattis D., Ann. Phys., 1961, 16, 407.
13. Derzhko O., J. Phys. Stud., 2001, 5, No. 1, 49.
14. Derzhko O., Krokhmalskii T., Phys. Stat. Sol. B, 1998, 208, 221.
15. Derzhko O., Krokhmalskii T., Phys. Stat. Sol. B, 2000, 217, 927.
16. Derzhko O., Krokhmalskii T., J. Phys. A, 2000, 33, 3063.
17. Stolze J., Vogel M., Phys. Rev. B, 2000, 61, 4026.
18. Derzhko O., Krokhmalskii T. and Stolze J., J. Phys. A, 2002, 35, 3573.
19. Capel H. W., Perk J. H. H., Physica A, 1977, 87, 211.
Густина станiв одновимiрного iонного провiдника Паулi
I.В.Стасюк, I.Р.Дулепа
Iнститут фiзики конденсованих систем НАН України, 79011 Львiв, вул. I.Свєнцiцького, 1
Отримано 7 травня 2007 р.
Розглянуто мiкроскопiчну одновимiрну модель невзаємодiючих частинок для опису енергетичного
спектру iонної пiдсистеми в iонному провiднику. Процеси iонного перескоку описанi у термiнах опе-
раторiв Паулi. Використовуючи точний числовий метод, вiдомий для часових спiнових кореляцiйних
функцiй, отримано часовi кореляцiйнi функцiї 〈b(t)b+(0)〉j в операторах Паулi. Обчислено i проана-
лiзовано частотну залежнiсть автокореляцiйної функцiї Jbb+ (ω) в широкому iнтервалi температур.
Дослiджено частотну i температурну залежностi одночастинкової густини станiв.
Ключовi слова: статистика Паулi, кореляцiйнi функцiї, частинкова густина станiв, iонний провiдник
PACS: 75.10.Pq, 66.30.Dn, 66.10.Ed
268
|