Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor
We focus on the features of spectra and diagrams of states obtained via exact diagonalization technique for finite ionic conductor chain in periodic boundary conditions. One dimensional ionic conductor is described with the lattice model where ions are treated within the framework of "mixed&quo...
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irk-123456789-1200022017-06-11T03:04:33Z Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor Stasyuk, I.V. Vorobyov, O. Stetsiv, R.Ya. We focus on the features of spectra and diagrams of states obtained via exact diagonalization technique for finite ionic conductor chain in periodic boundary conditions. One dimensional ionic conductor is described with the lattice model where ions are treated within the framework of "mixed" Pauli statistics. The ion transfer and nearest-neighbour interaction between ions are taken into account. The spectral densities and diagrams of states for various temperatures and values of interaction are obtained. The conditions of transition from uniform (Mott insulator) to the modulated (charge density wave state) through the superfluid-like state (similar to the state with the Bose-Einstein condensation observed in hard-core boson models) are analyzed. Робота присвячена вивченню енергетичного спектру та дiаграм станiв, отриманих методом точної дiагоналiзацiї для скiнченного iонного ланцюгового провiдника в перiодичних граничних умовах. Одновимiрний iонний провiдник описується ґратковою моделлю, де iони розглядаються як частинки Паулi, при цьому враховується iонний перенос i двочастинкова взаємодiя мiж найближчими сусiдами. Було розраховано та проаналiзовано спектральнi густини та дiаграми стану такої системи для рiзних температур та величин взаємодiї. Проаналiзовано умови переходу системи з однорiдного (стану т.зв.моттiвського дiелектрика) у модульований стан через стан типу фази з бозе-конденсатом (подiбної до надплинної фази в моделях жорстких бозонiв). 2011 Article Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor / I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23702:1-7. — Бібліогр.: 33 назв. — англ. 1607-324X PACS: 75.10.Pq, 66.30.Dn, 66.10.Ed DOI:10.5488/CMP.14.23702 arXiv:1106.5136 http://dspace.nbuv.gov.ua/handle/123456789/120002 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We focus on the features of spectra and diagrams of states obtained via exact diagonalization technique for finite ionic conductor chain in periodic boundary conditions. One dimensional ionic conductor is described with the lattice model where ions are treated within the framework of "mixed" Pauli statistics. The ion transfer and nearest-neighbour interaction between ions are taken into account. The spectral densities and diagrams of states for various temperatures and values of interaction are obtained. The conditions of transition from uniform (Mott insulator) to the modulated (charge density wave state) through the superfluid-like state (similar to the state with the Bose-Einstein condensation observed in hard-core boson models) are analyzed. |
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Stasyuk, I.V. Vorobyov, O. Stetsiv, R.Ya. |
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Stasyuk, I.V. Vorobyov, O. Stetsiv, R.Ya. Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor Condensed Matter Physics |
| author_facet |
Stasyuk, I.V. Vorobyov, O. Stetsiv, R.Ya. |
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Stasyuk, I.V. |
| title |
Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor |
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Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor |
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Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor |
| title_fullStr |
Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor |
| title_full_unstemmed |
Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor |
| title_sort |
spectral densities and diagrams of states of one-dimensional ionic pauli conductor |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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| citation_txt |
Spectral densities and diagrams of states of one-dimensional ionic Pauli conductor / I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv // Condensed Matter Physics. — 2011. — Т. 14, № 2. — С. 23702:1-7. — Бібліогр.: 33 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT stasyukiv spectraldensitiesanddiagramsofstatesofonedimensionalionicpauliconductor AT vorobyovo spectraldensitiesanddiagramsofstatesofonedimensionalionicpauliconductor AT stetsivrya spectraldensitiesanddiagramsofstatesofonedimensionalionicpauliconductor |
| first_indexed |
2025-07-08T17:03:31Z |
| last_indexed |
2025-07-08T17:03:31Z |
| _version_ |
1837099084008128512 |
| fulltext |
Condensed Matter Physics, 2011, Vol. 14, No 2, 23702: 1–7
DOI: 10.5488/CMP.14.23702
http://www.icmp.lviv.ua/journal
Spectral densities and diagrams of states of
one-dimensional ionic Pauli conductor
I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Sventsitskii Str.,
79011 Lviv, Ukraine
Received January 12, 2011, in final form February 14, 2011
We focus on the features of spectra and diagrams of states obtained via exact diagonalization technique for
finite ionic conductor chain in periodic boundary conditions. One dimensional ionic conductor is described
with the lattice model where ions are treated within the framework of “mixed” Pauli statistics. The ion transfer
and nearest-neighbour interaction between ions are taken into account. The spectral densities and diagrams
of states for various temperatures and values of interaction are obtained. The conditions of transition from
uniform (Mott insulator) to the modulated (charge density wave state) through the superfluid-like state (similar
to the state with the Bose-Einstein condensation observed in hard-core boson models) are analyzed.
Key words: Pauli statistics, spectral density, diagrams of state, ionic conductor
PACS: 75.10.Pq, 66.30.Dn, 66.10.Ed
1. Introduction
Ionic conductors are a wide class of physical and biological objects ranging from ice to DNA
membranes. One of the most interesting subclasses of these are superionic conductors that exhibit
high temperature phase with high conductivity that arises due to the motion of ions [1] or pro-
tons [2]. Theoretical description of systems with ionic conductivity is most frequently based on
the lattice models. Some of them treat ions as Fermi-particles focusing on different aspects of the
ionic subsystem like long-range interactions [3–5] or interaction with phonons [6, 7]. Some recent
attempts have also been made towards short-range interactions between particles [8–12].
However, a more correct consideration of ions should be based on the mixed statistics of
Pauli [13] since these particles are bosons by nature but they also obey the Fermi rule. Due
to the special commutation rules, the utilization of Pauli operators generates additional mathe-
matical complexities. On the other hand, this approach might be very effective. For instance, it
has been shown that the lattice model of Pauli particles is capable of describing the appearance
of superfluid-like state (that corresponds to superionic phase) in the system even in the absence
of interaction between particles [14–16]. On the other hand, the lattice model of Pauli particles is
similar to the hardcore Bose-Hubbard model widely used for the description of ionic conductivity
phenomena as well as for the modelling of energy spectrum of absorbed ions on a crystal surface
and intercalation in crystals [17]. Bose-Hubbard model also exhibits the transition from Mott in-
sulator state to superfluid-like state [18–24]. Some of the authors also observe the possibility of
formation of intermediate “supersolid” phase that may appear on the phase diagrams alongside the
transition from dielectric (CDW) to superfluid phase.
In this work we focus on the diagrams of state for one-dimensional ionic conductor described
by the system of Pauli particles. Our lattice model includes ion transfer as well as the interaction
between nearest-neighbouring ions. We calculate the single-particle spectral densities of the finite
system in periodic boundary conditions and obtain the diagrams of state analyzing the features
of this spectra. The conditions of transition from Mott insulator (MI)-like state to the modulated
charge density wave (CDW) state through the superfluid(SF)-like state (similar to the state with
the Bose-Einstein (BE) condensation observed in hard-core boson models) are discussed.
c© I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv, 2011 23702-1
http://dx.doi.org/10.5488/CMP.14.23702
http://www.icmp.lviv.ua/journal
I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv
2. The model for ionic conductor
Let us consider the chain of heavy immobile ionic groups (large circles in figure 1) and light ions
that move along this chain occupying positions denoted by small circles in figure 1. The subsystem
of light ions is described with the following Hamiltonian
Ĥ = t
∑
i
(c+i ci+1 + c+i+1ci) + V
∑
i
nini+1 − µ
∑
i
ni . (2.1)
This model takes into account the nearest-neighbour ion transfer (with hopping parameter t) and
interaction between ions that occupy nearest-neighbouring positions (with corresponding parame-
ter V ). If this Hamiltonian is considered within the framework of Fermi statistics, the corresponding
model is known as spinless-fermion model. This model is widely used in the theory of strongly cor-
related electron systems [25] as well as for the description of ionic conductors [26]. A more complex
two-sublattice case of this model can be applied to a proton conductor [27]. More correct consid-
eration of ions should be based on “mixed” Pauli statistics and this approach is used onwards. In
this case, the model (2.1) is equivalent to the extended hard-core boson model, i.e., boson Hubbard
model with repulsive interaction between nearest neighbours and infinite on-site repulsion [28]. The
latter is often applied to the investigation of the problems of BE-condensation and superfluidity.
i i+1 i+2i-1
Figure 1. The model for one-dimensional ionic conductor. Large circles denote heavy ionic groups
while the small ones denote light movable ions.
3. Exact diagonalization technique
We calculate the spectral densities of one-dimensional ionic Pauli conductor using exact diag-
onalization technique. For the chain of N sites, we introduce the many-particle states
| n1,an1,b . . . nN,anN,b〉. (3.1)
The Hamiltonian matrix on the basis of these states is the matrix of the order 2N × 2N and is
constructed as follows:
Hmn =
N∑
i=1
[
t
(
H(1)
mn +H(2)
mn
)
+ Ṽ H(3)
mn − µH(4)
mn
]
, (3.2)
where
H(1)
mn = 〈n1 . . . |c
+
i ci+1|n
′
1 . . .〉 = δ(ni−n′
i−1)δ(ni+1−n′
i+1+1)
×
∏
l 6=i;i+1
δ(nl−n′
l),
H(2)
mn = 〈n1 . . . |c
+
i+1ci|n
′
1 . . .〉 = δ(ni−n′
i+1)δ(ni+1−n′
i+1−1)
×
∏
l 6=i;i+1
δ(nl−n′
l),
H(3)
mn = 〈n1 . . . |nini+1|n
′
1 . . .〉 = δ(ni−1)δ(n′
i−1)δ(ni+1−1)
×δ(n′
i+1−1)
∏
l 6=i;i+1
δ(nl−n′
l),
H(4)
mn = 〈n1 . . . |ni|n
′
1 . . .〉 = δ(ni−1)δ(n′
i−1)
∏
l 6=i
δ(nl−n′
l).
23702-2
Spectral densities and diagrams of states of 1D ionic Pauli conductor
This matrix is diagonalized numerically
U−1HU = H̃ =
∑
p
λpX̃
pp, (3.3)
where λp are eigenvalues of the Hamiltonian, X̃pp are Hubbard-operators. The same transformation
is applied to the creation and annihilation operators
U−1ciU =
∑
pq
ApqX̃
pq , U−1c+i U =
∑
pq
A∗
rsX̃
rs (3.4)
which are required to construct one-particle Green’s function ≪ ci,a|c
+
i,a ≫ that contains infor-
mation about one-particle energy spectrum of the system. For Pauli creation and annihilation
operators, this Green’s function can be constructed in two ways, i.e., commutator Green’s function
≪ ci(t)|c
+
i (t
′) ≫= −iΘ(t− t′)〈[ci(t), c
+
i (t
′)]〉 (3.5)
and anticommutator Green’s function
≪ ci(t)|c
+
i (t
′) ≫= −iΘ(t− t′)〈{ci(t), c
+
i (t
′)}〉. (3.6)
Imaginary part of these Green’s functions are one-particle spectral densities (also referred to as
densities of states or DOS)
ρ(ω) = −
1
πN
N∑
i=1
Im ≪ ci,a|c
+
i,a ≫
= −
1
πN
N∑
i=1
Im
[
1
Z
∑
pq
ApqA
∗
pq
e−βλp − ηe−βλq
ω − (λq − λp)
]
, (3.7)
where Z =
∑
p e
−βλp . Spectral densities in (3.7), obtained from commutator η = 1 (3.5) and
anticommutator η = −1 (3.6) Green’s functions, respectively, exhibit a discrete structure, i.e.,
consist of several δ-peaks due to the finite size of a cluster. Therefore, we apply the periodic
boundary conditions to the cluster and introduce a small parameter ∆ to broaden the δ-peaks
according to Lorentz distribution
δ(ω) →
1
π
∆
ω2 +∆2
. (3.8)
4. Results and discussion
We perform calculations of one-particle spectral densities of one-dimensional ionic Pauli con-
ductor (2.1) for the chain of ten sites (N = 10) in periodic boundary conditions. To test the results
we compare them with the exact solution obtained by means of fermionization procedure [14] in
the absence of nearest neighbour interaction (V = 0) for different values of chemical potential (fig-
ure 2). The redistribution of statistical weight with the change of chemical potential level can be
observed, and a good level of agreement is achieved. The analysis of spectral densities is a way to
distinguish the different states of an ionic subsystem and the corresponding conditions. In the case
of half-filling, as we turn on the interaction starting with spectral density whose shape corresponds
to SF state, we observe the development of the gap in the spectra (figure 3). A similar effect was
found for ionic and proton conductors described by the similar models within the framework of
Fermi statistics. It was connected with the splitting of spectra due to charge ordering with doubling
of the lattice period [11, 12]. The detailed analysis of spectral densities of Fermi and Pauli models
of ionic conductor can be found in [29]. So, here we have a transition from SF to CDW state. This
23702-3
I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv
-2 0 2 4
-0,1
0,0
0,1
0,2
Figure 2. Commutator spectral densities of non-interacting (V = 0) ionic Pauli conductor for
different values of chemical potential (right figure, ε = −µ) compared to exact results obtained
in [14] via fermionization procedure (left figure, ε = ε0 − µ). t = 1, T = 0.2,∆ = 0.4. Spectral
density on the left figure is scaled to 2π, while on the right figure it is scaled to unity.
-6 -4 -2 0 2 4 6
-0,1
0,0
0,1
V=0
V=1
V=4
Figure 3. Commutator spectral density of ionic Pauli conductor for different values of nearest-
neighbour interaction V at half filling (µ = 0). t = 1, T = 0.2, ∆ = 0.4.
transition was described analytically for T = 0 in [30, 31]. We have investigated the case of T 6= 0
when this transition manifests itself as a crossover.
The existence of a gap in the spectra which separates the bottom of the energy band from the
chemical potential level is also the sign of the presence of homogenous MI state. The vanishing
of the gap and the appearance of a negative branch points to the transition to SF-like state
(see, for example, [32]). According to these criteria we analyze the spectral densities at different
temperatures and values of interaction and build the corresponding diagrams of state (figure 4).
The system is in MI homogenous state at high temperatures and far away from half-filling (at
large δ). As the temperature decreases or one comes closer to half-filled case, the system undergoes
transition to SF-like state which corresponds to the appearance of the negative branch without any
gap on the spectral density. At further temperature decrease and closer to half filling, we observe a
transition to the state with the gap on the spectral density that corresponds to the CDW-ordering
(the negative branch still exists). As the interaction strength V increases, such a region becomes
broader while the region of SF-like state becomes smaller. On the other hand, with the decrease
of V , the CDW state diminishes and disappears at V ≈ t. It should be mentioned that the sequence
23702-4
Spectral densities and diagrams of states of 1D ionic Pauli conductor
of states that the system is going through at the increase of mean occupancy δ, corresponds to the
phase diagram obtained in [30, 31, 33]. We have also performed a detailed analysis of the transition
to SF-like state at different temperatures and values of interaction (figure 5). It is interesting that
at weak interactions (V < 1), the increase of interaction strength facilitates the formation of
SF-like state while further increase of V suppresses this transition. The shift of the curves that
separate MI- and SF-states towards smaller values of δ with the increase of interaction strength
that we have obtained is also observed on the phase diagrams obtained by other authors [30, 31].
0.00 0.05 0.10 0.15 0.20
0.0
0.5
1.0
1.5
2.0
MI
SF
T
V=3
CDW
0.00 0.05 0.10 0.15 0.20
0.0
0.5
1.0
1.5
2.0
V=4
T
MI
SF
CDW
-4 -2 0 2 4
-0,10
-0,05
0,00
0,05
0,10
T=0.5, V=4, =-1
SF
-6 -4 -2 0 2 4 6
0,00
0,05
0,10
0,15
0,20
0,25
MI
T=0.5, V=4, =-2.5
Figure 4. Diagrams of state for different interactions and the spectral densities for negative δ
that correspond to CDW, SF and MI states. t = 1,∆ = 0.25. δ = 〈n〉−1/2 denotes the deviation
from half-filling.
23702-5
I.V. Stasyuk, O. Vorobyov, R.Ya. Stetsiv
0,0
0,5
1,0
1,5
2,0
2,5
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35
T
V =4 3 2 0 1
0 1 2 3 4
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
V
T
T
0.2
0.5
1.0
1.5
2.0
2.2
Figure 5. The curves that separate SF- and MI- states at different temperatures and values of
interaction. SF-state is on the left hand side of the curves and below them. t = 1, ∆ = 0.25.
δ = 〈n〉 − 1/2 denotes the deviation from half-filling.
5. Conclusions
We have performed the analysis of diagrams of states of one-dimensional Pauli ionic conductor
using an exact diagonalization technique. We have shown that the system undergoes transition
from Mott insulator to superfluid-like state and then to CDW-sate. At weak interaction, the latter
transition may vanish.
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33. Hen I., Iskin M., Rigol M., Phys. Rev. B, 2010, 81, 064503; doi:10.1103/PhysRevB.81.064503.
Спектральнi густини та дiаграми стану одновимiрного
iонного провiдника Паулi
I.В. Стасюк, О. Воробйов, Р.Я. Стецiв
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Робота присвячена вивченню енергетичного спектру та дiаграм станiв, отриманих методом точ-
ної дiагоналiзацiї для скiнченного iонного ланцюгового провiдника в перiодичних граничних умо-
вах. Одновимiрний iонний провiдник описується ґратковою моделлю, де iони розглядаються як
частинки Паулi, при цьому враховується iонний перенос i двочастинкова взаємодiя мiж найближ-
чими сусiдами. Було розраховано та проаналiзовано спектральнi густини та дiаграми стану такої
системи для рiзних температур та величин взаємодiї. Проаналiзовано умови переходу системи з
однорiдного (стану т. зв. моттiвського дiелектрика) у модульований стан через стан типу фази з
бозе-конденсатом (подiбної до надплинної фази в моделях жорстких бозонiв).
Ключовi слова: статистика Паулi, густина станiв, iонний провiдник
23702-7
http://dx.doi.org/10.1103/PhysRevLett.88.167208
http://dx.doi.org/10.1088/0953-8984/4/38/010
http://dx.doi.org/10.1002/pssb.2221010131
http://dx.doi.org/10.1103/PhysRevB.50.362
http://dx.doi.org/10.1063/1.3284416
http://dx.doi.org/10.1103/PhysRevB.61.12474
http://dx.doi.org/10.1103/PhysRevB.80.134508
http://dx.doi.org/10.1103/PhysRevB.77.235120
http://dx.doi.org/10.1103/PhysRevB.81.064503
Introduction
The model for ionic conductor
Exact diagonalization technique
Results and discussion
Conclusions
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