One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice
We study what happens to generalized Wigner crystal, GWC (a regular structure formed by narrow-band electrons on a one-dimensional periodic host lattice), when there is a host lattice random distortion that does not break the host-lattice long-range order. We show that an arbitrarily weak distort...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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Slutskin, A.A. Kovtun, H.A. 2017-06-15T08:56:05Z 2017-06-15T08:56:05Z 2005 One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice / A.A. Slutskin, H.A. Kovtun // Физика низких температур. — 2005. — Т. 31, № 7. — С. 784-795. — Бібліогр.: 19 назв. — англ. 0132-6414 PACS: 73.61.Jc, 75.10.Nr https://nasplib.isofts.kiev.ua/handle/123456789/121675 We study what happens to generalized Wigner crystal, GWC (a regular structure formed by narrow-band electrons on a one-dimensional periodic host lattice), when there is a host lattice random distortion that does not break the host-lattice long-range order. We show that an arbitrarily weak distortion of the kind gives rise to soliton-like GWC defects (discrete solitons, DS) in the ground state, and thereby converts the ordered GWC into a new disordered macroscopic state — lattice Wigner glass (LWG). The ground-state DS concentration is found to be proportional to λ⁴ (λ is the typical host-lattice strain). We show that the low-temperature LWG thermodynamics and kinetics are fully described in DS terms. A new phenomenon of a super-slow logarithmic relaxation in the LWG is revealed. Its time turns out to be tens orders of magnitude greater than the microscopic ones. Analytical dependences of LWG thermodynamic quantities on temperature and λ are obtained for an arbitrary relationship between the relevant Coulomb energies and the electron bandwidth. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкоразмерные и неупорядоченные системы One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice Article published earlier |
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One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
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One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice Slutskin, A.A. Kovtun, H.A. Низкоразмерные и неупорядоченные системы |
| title_short |
One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
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One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
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One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
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One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
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one-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice |
| author |
Slutskin, A.A. Kovtun, H.A. |
| author_facet |
Slutskin, A.A. Kovtun, H.A. |
| topic |
Низкоразмерные и неупорядоченные системы |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
| publishDate |
2005 |
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English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
We study what happens to generalized Wigner crystal, GWC (a regular structure formed by
narrow-band electrons on a one-dimensional periodic host lattice), when there is a host lattice random
distortion that does not break the host-lattice long-range order. We show that an arbitrarily
weak distortion of the kind gives rise to soliton-like GWC defects (discrete solitons, DS) in the
ground state, and thereby converts the ordered GWC into a new disordered macroscopic state —
lattice Wigner glass (LWG). The ground-state DS concentration is found to be proportional to λ⁴
(λ is the typical host-lattice strain). We show that the low-temperature LWG thermodynamics and
kinetics are fully described in DS terms. A new phenomenon of a super-slow logarithmic relaxation
in the LWG is revealed. Its time turns out to be tens orders of magnitude greater than the microscopic
ones. Analytical dependences of LWG thermodynamic quantities on temperature and λ
are obtained for an arbitrary relationship between the relevant Coulomb energies and the electron
bandwidth.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121675 |
| citation_txt |
One-dimensional electron lattice system with a long-range interelectron repulsion on a disordered host lattice / A.A. Slutskin, H.A. Kovtun // Физика низких температур. — 2005. — Т. 31, № 7. — С. 784-795. — Бібліогр.: 19 назв. — англ. |
| work_keys_str_mv |
AT slutskinaa onedimensionalelectronlatticesystemwithalongrangeinterelectronrepulsiononadisorderedhostlattice AT kovtunha onedimensionalelectronlatticesystemwithalongrangeinterelectronrepulsiononadisorderedhostlattice |
| first_indexed |
2025-11-25T20:54:23Z |
| last_indexed |
2025-11-25T20:54:23Z |
| _version_ |
1850542767846981632 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7, p. 784–795
One-dimensional electron lattice system with a long-range
interelectron repulsion on a disordered host lattice
A.A. Slutskin and H.A. Kovtun
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy
of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: slutskin@theor.kharkov.ua
kovtun@theor.kharkov.ua
Received November 29, 2004
We study what happens to generalized Wigner crystal, GWC (a regular structure formed by
narrow-band electrons on a one-dimensional periodic host lattice), when there is a host lattice ran-
dom distortion that does not break the host-lattice long-range order. We show that an arbitrarily
weak distortion of the kind gives rise to soliton-like GWC defects (discrete solitons, DS) in the
ground state, and thereby converts the ordered GWC into a new disordered macroscopic state —
lattice Wigner glass (LWG). The ground-state DS concentration is found to be proportional to �4
(� is the typical host-lattice strain). We show that the low-temperature LWG thermodynamics and
kinetics are fully described in DS terms. A new phenomenon of a super-slow logarithmic relax-
ation in the LWG is revealed. Its time turns out to be tens orders of magnitude greater than the mi-
croscopic ones. Analytical dependences of LWG thermodynamic quantities on temperature and �
are obtained for an arbitrary relationship between the relevant Coulomb energies and the electron
bandwidth.
PACS: 73.61.Jc, 75.10.Nr
1. Introduction
In the last two decades layered and low-dimen-
sional conductors have been a focus of attention. In
such systems the charge carriers are commonly
well-separated from the dopants, so that a mutual
Coulomb repulsion of the charge carriers turns out to
be an essential factor. Among these conductors of es-
pecial interest are lattice (narrow-band) systems
wherein hopping (tunneling) of electrons/holes be-
tween host-lattice sites is suppressed by their mutual
repulsion, and as a consequence of this, the whole
charge carrier ensemble is self-localized. The criterion
for such a Coulomb self-localization (CSL) is the in-
equality
t uc� � ( )a /ree ee0
2� ,
where t is the bandwidth, uc is the typical change of
the Coulomb energy of a charge carrier as it hops be-
tween neighboring host-lattice sites, a0 is the host-
lattice spacing, ree is the mean separation between
charge carriers, and � ee is the mean energy of the
Coulomb repulsion per charge carrier. There exists a
wide class of CSL conductors. Different semicon-
ductor heterostructures [1,2], and organic quasi-one-
dimensional conductors [3] fall into this class. The
latest achievements of the nanotechnology permit cre-
ation of CSL conductors of a new type: arrays of
quantum dots exchanging electrons/holes (see, for
example, [4] and references there), electrons on
helium surface [5], and nets and chains of metal
nano-grains linked by organic molecule wires as
(weak) tunnel junctions [6]. There are also good rea-
sons to suggest that the CSL criterion is fulfilled in
layered metaloxides of a type of high-temperature su-
perconductors, provided the dopant concentration is
not too low.
Impetus to a theoretical research of CSL con-
ductors was given by Hubbard [3], who considered
the ground state of an electron ensemble on a one-di-
mensional (1D) periodic host lattice in the limit
t/uc � 0. Hubbard suggested that in the thermody-
namic limit (the numbers of electrons N and host-lat-
tice sites Ns tend to �) the ground-state electron
structure at a given chemical potential � is a periodic
«generalized Wigner crystal» (GWC) with a rational
© A.A. Slutskin and H.A.Kovtun, 2005
electron density (filling factor) n P/Qe � (P Q, are
arbitrary coprime integers), with spacing Qa0 and P
electrons per unit cell, which does not depend on
the pair potential of the interelectron repulsion v x( )
(x is the interelectron distance), provided v x( ) meets
some simple physically reasonable conditions (Sec. 2).
Hubbard also postulated a universal algorithm to find
GWC’s. This hypothesis was justified in works [7,8],
wherein Hubbard’s algorithm was put into a compact
form. In recent years Hubbard's results were aug-
mented to the low-temperature thermodynamics of 1D
GWC [9], and the ground state of two-dimensional
(2D) CSL conductors on an arbitrary ideal host lat-
tice [10]. The regular 2D CSL conductor was found to
be characterized by an effective lowering of dimen-
sion, which lies in the fact that even in the case of an
isotropic pair potential of electron–electron repulsion
the structure elements forming the 2D ground-state
structure are electron stripes arranged by Hubbard's
algorithm [10].
The above studies, having revealed a number of un-
usual features of GWC and its 2D modification, have
prompted consideration of the influence of host-lattice
disorder on low-temperature properties of CSL con-
ductors. This is of importance not only because there
are a number of disordered CSL conductors (e.g.,
MOSFETs, some nanostructures), but also in respect
to the known fact (Larkin [11], Imri and Ma [12])
that an arbitrarily weak static random peturbation im-
posed on a continuous Wigner crystal (i.e., a crystal
formed by electrons that are free to move) breaks
(for dimension d 3) the ground-state crystalline
long-range order, producing slowly varying random
distortions that become increasingly smooth as the
perturbation goes to zero. Such a structure was named
«Wigner glass». Unlike the continuous case, the elec-
tron displacements in CSL conductors cannot be less
than the host-lattice spacing a0, so that the Lar-
kin—Imri—Ma argument is inapplicable to clarify
whether lattice electron structures are unstable with
respect to weak random perturbations, and what is the
mechanism of the possible instability. As far as we
know, there is no answer to this question at present.
We are going to fill this gap, considering the low-tem-
perature properties of disordered 1D and 2D CSL con-
ductors in a series of publications. This paper is the
first of the series. Here we study the ground state, ele-
mentary excitations, their lifetimes, and the low-tem-
perature thermodynamics of a 1D CSL conductor (the
temperature T uc�� ) in the limit of strong CSL
(t uc�� ) with the assumption that random host-lat-
tice distortions do not break long-range order of the
arrangement of host-lattice sites. For definiteness, the
charge carriers are further considered to be electrons.
Specifically, we consider the host lattice to be a
chain of sites i Ns� 1,..., with coordinates
X ii i�
�� , (1)
where �� i is a random deviation of the i-th site from
its ideal position X ii
0 � (a0 1� ); � i as a function of i
takes random values on the interval [–1,1]; the disor-
der parameter � is assumed to be � 1 2/ to provide the
periodic correlation in the host lattice. At first glance
such a disorder cannot break the GWC long-range or-
der. However, this seemingly evident statement turns
out to be untrue: we show here that for any finite
� �� 1 the ground-state electron configuration con-
tains soliton-like GWC defects, discrete solitons,
each changing the length of the system (at a given N)
by +1 (the defect of rarefaction) or –1 (the defect
of compression). Being randomly arranged, they do
break the GWC long-range order. This is a novel phe-
nomenon rooted in the discretness of the system. The
key point is that the discrete solitons differ from com-
mon point defects in that a transfer of a discrete
soliton over L unit cells of a GWC leaves behind
a «trace» — a cluster of electrons shifted by one
host-lattice site from their ground-state positions, the
number of the electrons being L. Owing to the host-
lattice disorder the Coulomb energy of the trace is a
sum of L random alternating-sign quantities � �2,
which grows in modulus as L , attaining values � uc
for sufficiently big L. It is this fact that accounts for
discrete solitons coming into existence in the ground
state. Since the random 1D lattice electron structure
just described is bound to be converted into GWC as
� � 0, it can be conceived of, by anology with the
(continual) Wigner glass, as a «lattice Wigner glass»
(LWG). As will be seen from the following, the «gi-
ant» Coulomb energy fluctuations caused by dis-
crete-soliton transfer underlie the distinctive features
of both LWG thermodynamics and kinetics. They
have nothing in common with the energy fluctuations
in the continuous Wigner glass that result from the
above-mentioned smooth random distortions. There-
fore, the macroscopic behavior of LWG is expected to
differ essentially from that of Wigner glass. This is
dramatically manifested in relaxation of high-energy
LWG excitations: our analysis shows that their life-
times are anomalously great, the concentration of the
excitations decreasing with time in a logarithmic
fashion. Broadly speaking, the relaxation is by anni-
hilation of GWC defects of compression and rarefac-
tion. This requires the defects to overcome (by tun-
neling or in a thermal way) a giant fluctuation energy
barrier of many-electron origin. The phenomenon is
somewhat reminiscent of the mechanism of the
nonergodic behavior of spin glass [13,14].
One-dimensional electron lattice system with a long-range interelectron repulsion
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7 785
The layout of the material in our paper is as fol-
lows. In the next Section the Hamiltonian of the sys-
tem under consideration is described. In Sec. 3 we
show that for an arbitrary small � the GWC
long-range order is unstable with respect to formation
of GWC defects at zero temperature, and find the
ground-state dependence of the GWC defects concen-
tration on � and � in the limit t � 0. The LWG
low-temperature thermodynamics (T uc�� ) is consid-
ered in Sec. 4, where the dependence of thermody-
namic quantities on T is found in terms of low-lying
LWG excitations. The anomalously slow relaxation in
LWG is discussed in Sec. 5. In the last Section we dis-
cuss experimental possibilities for studing LWG.
2. Hamiltonian
At a given � the general Hamiltonian H of an elec-
tron system on an 1D host lattice is of the form
H �
�t c ci i
i
( )1 h. c.
� �
� �� �1
2
v X X n n ni
i i
i i i i
i
( ) � . (2)
Here the index i enumerates host-lattice sites, ci
and
ci are operators of creation and annihilation of an
electron at the i-th site, respectively; ni is the opera-
tor of the electron density at the i-th site, which takes
only values 0 and 1, since ne is assumed to be not too
close to 1 2/ ; for this reason the electron spin indexes
are dropped; summation is over all host lattice sites;
the pair potential of an electron–electron repulsion
v x v x( ) ( )�
� 0. The dependence of the site coordi-
nate Xi on i is given by Eq. (1).
To find the ground-state structure in the t uc��
limit under consideration, one can put t � 0, the
Hamiltonian Eq. (2) reducing to
H H� �
� �
��{i m v x x Nm
m m
m( )} ( )
1
2
� , (3)
where the index m N� 1,..., enumerates the electrons,
x Xm i m� ( ) is the coordinate of the m-th electron,
and i(m) is the number of the site at which the m-th
electron is localized. The reduced Hamiltonian
Eq. (3) is a functional of electron configurations
{ ( )} ( ),..., ( )i m i i N� 1 . It differs from that considered
by Hubbard [3] only in that Xi contains the random
addition �� i . To proceed with our studies it is of be-
nefit to formulate briefly, following [3,7,8], the main
properties of the ground state at � � 0.
Below we consider the pair potential v x( ) � 0 to be
an everywhere convex function of x that goes to zero
with an increase of x faster than x
1; its dependence
on x is otherwise arbitrary. In such a case the
ground-state space structure of the 1D electron gas
on the ideal host lattice X ii
0 � is given by the
above-mentioned universal Hubbard's algorithm that
is expressed by the simple formula
i m i m n m/nH e e( ) ( , ) [ ]� �
� , (4)
where [...] is the integer part of a number, and � is
an arbitrary phase that reflects the arbitrariness of
the choice of the first electron site coordinate i( )1 . As
was mentioned in the Introduction, only rational
n P/Qe � survive at zero temperature and a given �,
the function i m P/QH ( , ) determining the GWC.
The ground-state dependence of ne on � is a devil’s
staircase: for each pair P Q, there is an interval
� � �
( , ) ( , )P Q P Q within the limits of which
ne ( )� remains equal to P/Q. The left and right end-
points of the devil’s-staircase interval is a decrease
and an increase in the Coulomb energy of the system
as N changes by
1 and
1, respectively. The length of
the interval �� � ��
depends only on Q:
�� � QuQ,
u k v Qk v Qk v QkQ
k
�
�
�
�
1
1 2 1[ ( ) ( ) ( )].
(5)
As is seen from this expression, uQ � uc, and hence,
�� vanishes rapidly together with uc asQ tends to �.
3. The ground state of LWG
3.1. GWC instability
In this Section we deal with the Hamiltonian of
Eq. (3). First we consider devil’s-staircase intervals
with n /Qe � 1 . Generalization to an arbitrary ne �
� P/Q is performed without difficulty (Sec. 5). In the
case of P � 1 the Hubbard’s algorithm Eq. (4) gives
the simple periodic dependence i m /Q iH m( , )1 � ��
�
Qm � with the � �
0 1,...,Q . (We will call this
the «�-configuration».) If the ground-state electron
arrangement i mg( ) was the same as in the GWC (i.e.,
i m ig m( ) � � ) despite the random small displacements
�� i (� �� 1), a weak disorder of the form Eq. (1)
would affect only the energy of the �-configuration
EGWC. The latter does not depend on � to an accuracy
of fluctuation corrections � N:
E Nº NGWC �
O( ),
º º º�
0 �, º v kQ
k
� � �� � � ��
�
�
�2 2
1
( ) .
(6)
786 Fizika Nizkikh Temperatur, 2005, v. 31, No. 7
A.A. Slutskin and H.A. Kovtun
Here º is the mean energy per electron, º0 is that for
the ideal host lattice, º� is the addition to º0 pro-
duced by a weak host-lattice disorder, � �... is the
average on the � distribution, and �� �v x d v/dx( ) 2 2.
Let us show that the periodic (in site coordinates)
�-configuration is unstable under conditions of a weak
disorder. To reveal this, let us find out what happens
to the energy Eq. (3) if some �-configuration cluster
formed by electrons with numbers m m1 2,..., is shifted
as a whole by �1: i im m
� �� �1, m m m1 2 .
The energy of the cluster E m m( , , )1 2 � is the sum
E m m º E m mb i( , , ) ( , , )1 2 1 2� ��
(7)
whose first term is the cluster boundary energy, i.e.,
the energy of the interaction of the cluster with its
surroundings. The second term is the internal cluster
energy:
E m m m m º E m mi ( , , ) ( ) ~( , , )1 2 2 1 1 2� ��
; (8)
~( , , )E m m1 2 � is its fluctuating part. Expanding energy
Eq. (3) in powers of �, we obtain
~( , ; ) ( )E m m º º
m m
m
m1 2
1
2
� ��
�
� , (9)
º v Q m m i im
m
m m
� � ��
� �� ��
�
� �
�
�
��
2
2
2
( ( )) [ ( ) ( )]
O ( )�3 .
The shift of the cluster, on the one hand, creates
two defects at the ends of the cluster, which are pairs
of electrons separated by distances Q
1 (rarefaction
defect, «
dimer») and Q
1 (compression defect,
«
dimer»). Using expression (5), one can obtain that
in the case of �dimers well-separated dimer formation
increases the cluster boundary energy by
� � � �º /Q º m º mb �
� ~ ( , ) ~ ( , ), (10)
where ��/Q is the total energy of dimer formation at
� � 0; ~ ( , )º m� � is a small (� �uc) random addition
produced by the host-lattice disorder. On the other
hand, the shift changes ~( , ; )E m m1 2 � by
� �� � � �
� � � �, , ( , ) ( ) ,� �
�
�� �
�� ��m m º º
m m
m
m m1 2
1
2
1.
(11)
This energy change, being a sum of random quanti-
ties º ºm m
� ��
with the zero average, fluctuates in
m m1 2, , taking both positive and negative values. Its
modulus | |,�� �� � �2
2 1
1 2u m mc
/( )
therewith grows
as m m2 1
increases. Therefore, despite the smallness
of � there are many sufficiently long clusters for which
�� � �, �
�ºb 0, and hence, their shifts decrease the
energy of the system. What it means is that the GWC
is unstable relative to dimer formation for an arbi-
trarily small �.
3.2. The space structure of the LWG ground state
As follows from the general theory of GWCs
[3,7,8] (see also [9]), in the case under consideration
(P � 1) the �dimers are the GWC defects with the
least energies of formation at a given pressure. This
fact together with the above reasoning suggests that
for any finite � �� 1 the ground-state configuration
i mg( ) consists of long periodic electron clusters, seg-
ments of �-configurations, alternating with �dimers.
One should thus seek i mg( ) among functions i m( )
of the form
i m Qm m m m( ) ( )�
�� � �
1 2 ,
m m1 2
1 1� ��
, � � �� �
��
1 ,
(12)
where index � enumerates periodic clusters of the
configuration i m( ), �� �
1 or 1 depending on whe-
ther ( )�
1 -th and �-th clusters are joined by – or
dimer, respectively. The coordinates m g1
� , m g2
� of the
end electrons of ground-state clusters are obviously
random, and, hence, there is no long-range inter-
electron correlation in the ground state. This is just
the LWG mentioned in the Introduction, the discrete
solitons being �dimers. The LWG correlation radius
Lg � 1/ n ng g( )
(in units of r Qee � ), where
n /Ng g
� �� N is the ground-state concentration of �di-
mers, N �
g is their number in the ground state.
The next step is to find ng
� . To this end one should
consider the general expression for the energy E of the
configuration (12). It is appropriate to compare E
with EGWC (Eq. (6)), mapping the configuration on
a set of �-configurations. In so doing �-th cluster is
considered as the cluster of �� -configuration
( [ ]� � �� � ��
Q /Q is the reduced phase less thanQ)
with the end electrons m m /Q12 12, , [ ]� � ���
. The map-
ping gives
E E E E�
GWC dim
~, (13)
where Edim is the total energy of dimer formation,
and ~E is the fluctuating part of the energy.
To the zero approximation in �
E n u n u Ndim �
( ) (14)
(n /N� �� N is the �dimer concentration for a given
configuration (12), N � is the number of the
�dimers), and
One-dimensional electron lattice system with a long-range interelectron repulsion
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7 787
u
u
zQ
� � �( ) ( )�
2
1 0�
z /�
�
�
�
�
�
!
2
2
( ) ,� � � �
� �
�
(15)
are the �dimer formation energies at a given �; the
variable z takes values on the interval [–1,1] as � is
varied from �
to �
; uQ has been defined in Eq. (5).
Writing Eqs. (14),(15), we allow for the fact that a
small variation ��ne of ne about n /Qe � 1 gives rise
to Q ne� �dimers (this follows immediately from Eq.
(4)), i.e.,
N N Q�
0 ( )N N ; (16)
N N /Qs0 � is the ground-state number of electrons
at � � 0.
The energy ~E, unlike the two previous terms in
Eq. (13), depends upon the dimer coordinates m1
� es-
sentially:
~ ~( , ; )E E m m� �
�
� � ��1 2 . (17)
As follows from Eq. (9), the quantities ~( , ; )E m m1 2
� � ��
are sums of l m m�
� ��
2 1 � ( )n n /
1 2 random
terms with zero average and moduli � �2uc. Since
~( , ; )E m m1 2
� � �� fluctuate randomly, it is easy to con-
clude that in the limit n� �� 1 the cluster endpoints
m m1 2
� �, can be chosen in such a way that terms of the
sum (17) are all negative, and hence, at a given n�
the minimum ~Emin of ~E with respect to dimer ar-
rangement is bound to be � 0.
Random sequences º º ºN1 2
� � �, ,..., (� �
0 1,...,Q ),
whose terms are involved in expression (9), are all sta-
tistically equivalent. Therefore, ~Emin does not depend
on the sequence { } , ,...,� � �� � 1 2 being a function of
the dimer density n n
and system parameters
only. Taking this into account together with the fact
that | ~( , ; )|E m m1 2
� � �� � � �
2 1 2u lc
/ , one can write ~Emin
as
~ ( )E u n n N/
min �
��
2 1 2 , (18)
where energy factor u� � uc is uniquely determined
byQ, v x( ), and the moments of the � distribution (see
below).
The ground-state concentrations of �dimers, ng
� ,
are those which minimize the sum ~E Emin dim
. Using
Eqs. (14), (15), and (18), we find that for any � � 0
n z
u
u z
z
z
g
Q
�
� �
� �
"
#
$
%
$
( ) ( )
,
,
� �
2
2
4
21
1 0
0 0 1
;
n z
z
u
u z
z
g
Q
�
� �
� �
"
#
$
%
$
( )
,
( )
,
0 1 0
1
0 1
2
2
4
2
� � .
(19)
It remains to find the explicit form of u�. This re-
quires rather sophisticated calculations. Here we only
outline them. Since u� does not depend on { }�� , it is
convenient to choose the sequence { }�� with alternat-
ing �� signs, considering the minimum of ~E u
N in-
stead that of ~E; here u is a given arbitrary positive pa-
rameter, and N N N� �
is the variable that should
be found as the result of the minimization. This value,
N N� , can be calculated in terms of a random walk of
the sum � 01 1, ( , )m (see expression (11)) as a function
of m. So doing, one should take into account that the
~E u
N minimum is realized if dimers of a given sign
are all positioned at the points m� (� � 1 2, , ...,N )
such that � �01 011 1, ,( , ) ( , )m m
� first attains u value
at some points m m mr� �� � (a random walk toward
the right) and m m ml� �� � (a random walk toward
the left) without taking zero value on intervals
[ , ]m mr
� � and [ , ]m ml
� � . The total number N of points
m� is found on the basis of the probability theory
methods [15] that are used in consideration of the
so-called first passage and renewal problems. The fac-
tor u� is obtained in view of the fact that the minimum
of ~E u
N is also min ( )N N N
u u/
�
1 2 . The net re-
sult of the calculation is the following:
u D/
/
� &� ( )2 2 1 2
; (20)
D � uc
2 is the diffusion coefficient for the random
walk of �� � �, �/
2, which is given by the formula
D v v� � �
� �
� �
� �1
2 4 2 2
2
4 2 23( ) ( );� � � �
v v Qk
k
k
1
0
� ��
�
�
�
�
� ( ) , v v Qk
k
k
2
0
2� ��
�
�
�
�
�( ( )) .
(21)
Existence of dimers in the ground state not only
renders a long-range order impossible, but also ac-
counts for the difference between the ground-state
electron density of LWG ne ( , )� � and that of GWC
ne ( , )� 0 . According to Eq. (16) we have ne ( , )� �
� � �n n /Qe
g( , )� 0 . As is seen from Eq. (19), dimers
of one sign are fully replaced by those of the opposite
one as � passes through the point �, both ng
( )� 0
and ng
( )� 0 being equal to n u /ug q� ( )� �2 4. This
results in a jump of the electron density by 2 1Q ng
at
� �� , i.e., in a first-order transition in �.
Another remarkable point is that ng
� grows signifi-
cantly as � approaches the endpoints �� . Close to
them a mutual repulsion of dimers (which is negligi-
788 Fizika Nizkikh Temperatur, 2005, v. 31, No. 7
A.A. Slutskin and H.A. Kovtun
bly small for | |� �
� � ��) should be taken into ac-
count. That problem is beyond the scope of this paper.
From the above expressions for u� and uQ it follows
that the factor u /uQ�
2 2 appearing in Eq. (19) is �1 both
for Q �1 and Q �� 1, i.e., ng
� � �4 for | |1 � z � 1.
According to Eq. (19) the LWG correlation radius
Lg � 1 1 4/n /g
� � � remains �� 1 even if � is not too
small. This suggests that the description of LWG in
terms of dimers works up to � � 1 2/ .
4. Low-lying elementary excitations and
the low-temperature thermodynamics of LWG
4.1. The low-lying excitation spectrum
The low-lying LWG excitations are those whose
energies are much less than the typical increase (uc) in
the energy of the system as an interior electron of a
regular claster is shifted by one site or more. They are
produced by ground-state dimer displacements, and
hence, they are of a many-electron nature. To find
the low-lying excitation spectrum let us consider
the state | s� created by displacement of the �dimer
adjacent, for definiteness, to the left end of the �-th
cluster from its ground-state position, say, to the right
by s � 1 2, , ... steps. This occurs as s electrons
m m sg g1 1 1� �,...,
are shifted to the left (
dimer) or
to the right (
dimer) neighboring sites all together.
The Coulomb energy of this electron «trace» is the sum
u s s s� � �� �� � ��
( ) ( ) ~ ( ), (22)
where
�� � �
� �
� �
�
��
( ) ( , ),s m m sg g� 1 1 1 , (23)
�� �, � is defined by Eq. (11);
~ ( ) ~ ( , ) ~ ( , )� � ��
�
�
�
�
�
� ��
s º m s º mg g1 1 , (24)
~ ( , )º m� � is the quantity appearing in Eq. (10); u( )0 �
� 0. From this point on, we does not show a depend-
ence on � or the dimer sign, provided it does not
cause misunderstanding. The explicit form of ~( )� s
is of no importance for further consideration; what
matters is only the fact that ~( )� s takes random
values within some interval [ , ]
� �� �/ /2 2 whose
width � � � �uc. The quantity �( )s can be considered
as the «coordinate» of a «particle» that executes a
random walk with the typical step �� � � �
2uc �� � as
the «time» s increases. With an increase in s energy
�( )s grows as �2u sc , so that it becomes more than
� � for s � s� ��
2 (s Lg� �� ). Since �( )s randomly
undergoes «returns» into the interval [ , ]
� �� �/ /2 2 ,
there are inevitably such s at which u s t( ) � . There-
fore, the low-lying excitations should be considered
with regard to the electron hopping, despite the in-
equality t uc�� .
At t � 0 the dimer, owing to the smallness of t/uc,
can be considered as a quasiparticle whose Hamil-
tonian H dim acts in the subspace of states | s�
(s-space). As follows from the general expression (2),
it is of the form
H dim �
��
��
��� �t s s s s u s s s
s s
(| | | | ) ( )| |1 1 , (25)
where the first term is the kinetic-energy operator,
u s( ) plays the role of an effective potential; the sum-
mation is over the whole s-space. The state vectors of
the low-lying excitations and their energies are eigen-
vectors | '� of H dim and its eigenvalues �exc, respec-
tively. Due to the randomness of the «potential» u s( )
the dimer is Anderson localized in the s-space, so that
the eigenvectors
| | ( )|' '� � � � ��l l
s
s s(
can be classified in the coordinates l � �0 1, ,... of
points (localization centers) about which the station-
ary wave functions ( l s( ) are centered. Since
| ( ) ( )|u s u s
1 � � �, the Anderson parameter gov-
erning the | 'l � structure is ) �� � /t � �u /tc . It can
be both less than and more than 1 despite the t/uc
smallness. If ) �� 1, the kinetic energy operator is a
weak perturbartion. In such a case
( � )l sls /( ) ( )�
O 1 , � )exc( ) ( ) ( )l u l /�
O 1 ,
and the localization radius r is thus equal to 1. For
) �� 1 the 1D Anderson localization under a weak ran-
dom potential takes place. In this limit
r � )
2� ( )t/uc
2 2�
. (26)
For the following consideration of the thermody-
namics it is necessary to know the typical separation
�min between the least of �exc and the ground-state
energy.
If ) �� 1, it is determined, as is seen from
Eq. (22)–(24), by the number of «returns» R of the
random sum �( )s to the above-mentioned domain
| ( )|� �s � � . Since �( )s values fall into this domain
randomly, we have �min� � �/R. By definition
R d s
/
/
s
Lg
�
�
* �� �
�
�
�
�
2
2
0
P( , ),
where P( , )s d� � is the probability that �( )s will fall
into the interval [ , ]� � �
d after s steps; for s �� 1 it
satisfies the diffusion equation
+ + � + +P D P/ s /2 2� (27)
One-dimensional electron lattice system with a long-range interelectron repulsion
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7 789
with the diffusion coefficient D � �2D � ( )�� 2 (D is
given by expression (21)). Solving Eq. (27) subject
to the boundary condition P( , )s /
�� � 2 0 (this is
necessary since �exc( ) ( )l u l, is positive) and substi-
tuting the solution into the expression for R, we find
�min � � ��3uc �� . (28)
It should be added that the main contribution to R is
given by s � s�. Therefore, the localization center of
the eigenvector with the minimal excitation energy
�min is generally � s� distant from the ground-state
position of the dimer.
For ) �� 1 the localization radius according to
Eq. (26) is �� s�, the eigenenergies �exc( )l randomly
filling a band of width � t as l varies within a domain
of a length � s�. Therefore, in this limit
�min � t/s� � �2t. (29)
4.2. The LWG thermodynamics
According to the above consideration, in the limit-
ing case T uc�� LWG is in essence an ensemble of in-
dependent Anderson localized dimers. Their number
dependence on T can be ignored with accuracy to ex-
ponentially small (in T/uc) corrections. The thermo-
dynamic potential - -� ( , )� T of such a system is ex-
pressed in terms of a distribution of excitation
energies ��
exc( )l over the dimer ensemble:
- -�
�
�
�
�
!
�
�
�
�
�
!
!
.
�
�
* �g
g
k
kT
T
N ln exp1
1
�
. W d d( , ,...) ...� � � �1 2 1 2 , (30)
where - g is the ground-state value of -;
W( , ,...)� �1 2 is the probability density that the least
excitation energy of the system is �1, the next one is
�2, etc.; the symbol prime means that the integration
is performed over the region � �1 2� � ... from �k � 0
( , , ...k � 1 2 ) to the upper bound of the dimer energy
spectrum (it is � uc). The upper bound of the sum-
mation over k � Lg . We do not show it as it is imma-
terial to an accuracy of exponentially small term
�
exp ( )u /Tc .
As was proved by Molchanov [16], there is no mu-
tual repulsion of neighboring energy levels in 1D An-
derson localized systems, unlike Wigner–Dyson sta-
tistical ensembles [17]. This holds for the energy
spectrum considered, since L rg �� . In such a case the
eigenenergies are arranged quite randomly, similar to
particle coordinates in an 1D ideal gas, andW is com-
pletely determined by the density of states (per
dimer), g( )� , with a given energy �:
W w
k
k k( , ,...) ( , )� � � �1 2
1
1�
�
/ , (31)
where the two-level correlation function w( , )� ��
obeys the Poisson law
w g g d( , ) ( ) exp ( )� � � � �
�
�
� �
�� ��
�
�
�
��
�
!
!!
�
* . (32)
The states density g( )� increases monotonically
with a growth in � from
g( )0 � 1/�min � 1 2/ u tc� �( )
(33)
to values � 1 4/ � uc. The most important dimen-
sionless parameter governing - temperature depend-
ence is thus Tg( )0 . In the limit Tg( )0 1�� the
thermodynamical potential can be expanded in pow-
ers of Tg( )0 . To find the n-th term of this series, all
exponents exp ( )
�k/T with k n� appearing in ex-
pression (30) should be discarded. So doing, we find
that to the first nonvanishing approximation - is of
the form
- -�
g
gg T
&2
2
12
0N ( ) . (34)
Consequently, both the entropy S and the heat capac-
ity C per site tend to zero as T with T � 0. In view of
Eqs. (19), (33), and (34) we obtain
S C n g Tg, ( )�
&2
6
0 � � �2 1( )u t Tc
,
i.e., the proportionality coefficient is linear or qua-
dratic in � depending on ) �� 1 or ) �� 1.
In the opposit limit Tg( )0 1�� , the excitation spec-
trum can be considered as a continuous one. This gives
- -�
�
�
�
��
�
!
!!
�
*g
gT /T g dN ln exp ( ) ( )
0
� � � . (35)
We have expanded the integration to infinity as the
integrand is reduced rapidly for � �� T. Expression
(35) shows that in the case Tg( )0 1�� C depends
slightly on T, being � ng .
5. Super-slow relaxation in LWG
Below we consider low-temperature (T uc�� ) re-
laxation of strongly nonequilibrium LWG states
formed by excitations with energies � uc. The simplest
excitation of this type arises if an interior electron of
some regular ground-state cluster is shifted by one site
from its ground-state position. Such an excitation can
be considered as a pair of bound «odd» (as compared
to the ground state)
dimer and
dimer. The electron
790 Fizika Nizkikh Temperatur, 2005, v. 31, No. 7
A.A. Slutskin and H.A. Kovtun
shifted reverts to its ground-state site emitting pho-
nons with energies � uc. Obviously, this takes a mi-
croscopic time of the order of reciprocal Debye fre-
quency. The situation is drastically changed if some
external perturbation, for example, light excitation
disrupts the pair, separating the «odd» �dimers by a
sufficiently large distance (within the limits of the
cluster). A dimer cannot disappear by itself, since this
would cause, by virtue of dimer's topological nature, a
simultaneous shift of a macroscopic number of elec-
trons. Therefore, the «odd» dimers can only disappear
by their mutual annihilation. We will show below
that under slight-disorder conditions this process
takes an anomalously long time*.
5.1. The Hamiltonian and eigenvectors of a separate
pair of «odd» dimers
To the main approximation in t/uc the state space
of the pair is a set of state vectors
| � ( ) � ( )|s � �
�
D s D s g1 0 ,
where |0g � is the ground-state eigenvector, s s
,
are arbitrary integers meeting the condition m g1
�
s s m g, 2
� (here the reduced coordinates of the
regular-clauster end electrons defined in Sec. 3 are
used; we denote them m12, from this point on); � ( )D s
and � ( ) � ( )D s D s
� 1 are the operators that shift all
electrons with numbers 1 s one site leftwards and
rightwards, respectively. The integers s� play the role
of the coordinates of the «odd» �dimer. The Hamil-
tonian of the pair H pair can be written as
H pair � ���
��
� � �
� �t u /Q| | ( )| |s s s s s
s s s
�� ; (36)
the first term is the operator of the kinetic energy of
the pair; the symbol � � �s s means summation over all
nearest neighbors of s; the «potential» energy of the
pair
u u s u s( ) ( ) ( )s �
� � , � �
sgn ( )s s ,
u s m s º s� �
� � �� �
�
� � ��( ) ( , ) ~ ( ),� 1 ,
(37)
where �� �, � is the sum of random quantities defined
by Eq. (11); ~ ( ) ~ ( , )º s º s� �� � (see Eq. (10)) oscillates
randomly with the amplitude �uc. The third term in
Eq. (36) is the energy of pair formation at � � 0.
The structure of Hamiltonian (36) is akin to that of
the one-dimer Hamiltonian (25). It immediately fol-
lows that the
dimer and
dimer of a pair are both An-
derson localized in the s-representation, the localiza-
tion radius r being � 1 2
( )t/ uc� . Classifying the
pair's eigenvectors |0� by coordinates l l
, of the
�dimer localization centers, we can write |0� as
| | ( )|0 0� � � � ��l l
s
s s( . (38)
Here l �
{ , }l l , the summation is over all s, ( l s( ) is
the pair's wave function, which is localized in the
s-space region | |l s
� r. In the general case that the
pair's length l l lp �
| | �� r one can replace � in
Eq. (37) with sgn ( )l l
. This allows ( l s( ) to be
factorized:
( � �l s( ) ( ) ( )�
l ls s , (39)
the �dimer wave function � l s
� �( ) satisfying the
Schrödinger equation
t s s u s sl l l( ( ) ( )) ( ) ( )� � ��
� � �� � � � �
�1 1
� � � ��
E l sl( ) ( )� . (40)
The eigenenergy Ep ( )l of the Hamiltonian (36) is re-
lated to the eigenvalues E l� �( ) by the formula
E E l E l /Qp ( ) ( ) ( )l �
�
�� 0. (41)
If ) �� 1 (r � 1), we have E l E l
,( ) ( )
,
� � �, ( , )1 l l or � � �, ( , )
1 l l depending on whe-
ther l l
� or l l
� . For ) � 1 (r t/ uc� ( )� 2) the
sum E l E l
( ) ( ) represents � � �, ( , )� �1 l l� be-
havior in outline. Therefore, Ep ( )l as a function of lp
undergoes, together with �� �, �1, random alter-
nating-sign fluctuations whose amplitude grows as
�2 1 2u lc p
/ with increasing lp . The minimal energy Ep
of the energy spectrum of the pair is generally � uc,
the pair's length in the state |0 l � with E Ep p( )l �
being � Lg .
5.2. Transition frequencies
At T � 0 the annihilation of the pair is by phonon
emission. First one should consider the frequency �( )l
of the direct quantum transition | |0 0l � � �g for an ar-
bitrary |0 l �. According to Fermi’s golden rule, we
have
�( ) | ( )|l l� A 2, (42)
where
A He g( ) | |l l� � �0 0- ph (43)
is the amplitude of the transition | |0 0l � � �g , He- ph
is the Hamiltonian of the electron-phonon interac-
One-dimensional electron lattice system with a long-range interelectron repulsion
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7 791
* A pair of separated «odd» dimers can also be thought of as an excitation arising as some subcluster of a regular cluster is
shifted as a whole by one site. As will be clear from the following, the lifetime of an excitation produced by a shift of the
subcluster by more than one site exceeds the lifetime of a pair of dimers.
tion, and the line over | |A 2 symbolizes averaging over
all finite phonon states with energy Ep ( )l .
Since the electron-phonon interaction is of a local
character, only ( l s( ) with | |s s
� 1or s s
� con-
tribute to expression (43). To estimate �( )l it is suffi-
cient to consider ( l ( , )s s , using the factorized ex-
pression (39). Since the typical lp is Lg , it is necessary
to clarify the asymptotic behavior of � l s
�
( ) for
| |l s Lg� �
� . So doing, one should take into consider-
ation that the difference u s u l� � � �
� �( ) ( ), being a
random sum of | |l s� �
terms, undergoes fluctuations
with the typical amplitude �u u l sc�
� ��2 1 2| | / . It is
these effective-potential fluctuations rather than the
dimer Anderson localization that govern the decreas-
ing of � l s
�
( ) when �u exceeds � � �* �
u tc . The lat-
ter is the typical spread of eigenenergies of a reduced
Schrödinger equation which would come from Eq. (40)
with the randomly oscillating bounded function
~ ( )º s� � (see Eqs. (10) and (37)) in place of u s� �
�( ).
This takes place for | | ( )* *l s s / uc� �
�� � � �� 2 2
( *s Lg� �� ). In such a case � l s
�
( ) is nonzero owing to
dimer tunneling through the fluctuation barrier.
Basing on the Schrödinger equation (40), we find that
in the above forbidden region
� l /
b l s
s B
t
l s�
� �
�
�
�
�
�
�
!
!
� �
( )
| |
| |
1 2
, (44)
where the factor
B b
s
r t
�
�
�
�
�
�
!
!
exp ln
* *
1
� ��
,
results from falling of the dimer wave function (over
a distance � s�
*) due to the Anderson localization; b b, 1
are some constants � 1. Here we omit a pre-exponen-
tial factor � 1. Substitution of expression (40) in
Eqs. (39) and (42) gives
� � 2( ) exp ( )l �
0
4l B lp p , 2
�
� 4
2 1 2
b
l
t
p
/
ln . (45)
The pre-exponential factor � 30 � D is expressed in
terms of electron–phonon interaction characteristics;
3D is the Debye frequency. The expression holds for
l sp � �
*. The exponential decline of �( )l with increas-
ing lp is in essence of a many-electron origin. It be-
comes clear if one recalls that in the state |0 l � there
are � ��lp 1 electrons shifted about their ground-
state positions. The annihilation of a pair's dimers oc-
curs if all electrons shifted revert to the ground-state
sites simultaneously. Naturally, this results in small
�( )l values.
The foregoing argument can be easily extended to
find the frequencies �( , )l l� of quantum transitions
| |0 0l l� � �� , which occur with emission of phonons
with energies E Ep p( ) ( )l l
� . The expression for
�( , )l l� is of the form
� � 2( , ) exp ( )l l� �
� � �0
4d B d , 2
�
�
�� 4
2 1 2
b
d
t
/
ln ,
(46)
where d l l� � ��
�| |. It is implied that only one differ-
ence, d
or d
, is nonzero. Otherwise, the transition
frequencies are exponentially small as compared with
those given by Eq. (46).
5.3. Relaxation of a separate pair of «odd» dimers
To find the time 4p of the annihilation of a pair,
one should keep track of how a pair's density matrix
� ( , ) ,5 5 4 �� �l l l varies with time 4. The quantities 5 4( , )l
that are the probabilities of finding the pair in the
states |0 l � at an instant of time 4 satisfy a Pauli-like
kinetic equation
+
+
� � �
� ��5 4
4
� 5 � � 5 4
( , )
( , ) ( ) ( ( ) ( )) ( , )
l
l l l l l l
l
,
(47)
where
� �� �
�
� ��( ) ( , )l l l
l
. (48)
Here symbols � and � mean that the summation
extends only to l� for which E Ep p( ) ( )l l�
� 0 and
� 0, respectively. If l d rp, � � , the transition frequen-
cies � �( ), ( , )l l l� � �0; for l d rp, � �� they are deter-
mined by Eqs. (45) and (46).
The annihilation of a pair is governed by two equa-
tions. One is for the probability of existence of the
pair at an instant 4,
P( ) ( , )4 5 4� �
l
l
(the summation is over all l). Another is for the en-
ergy-average value
6 4 5 4( ) ~ ( ) ( , )� �Ep
l
l l ,
where ~ ( )Ep l is the eigenenergy measured from the
minimal energy Ep . Both equations follow immedi-
ately from Eq. (47):
dP/d4 � 5 4�
�
l
l l( ) ( , ); (49)
d
d
E Ep p
6 4
4
� 5 4
( )
( ( ) ( )) ( , ) ( , )
,
�
� �
�
�
�
l l
l l l l l
��� 5 4
l
l l l( ) ~ ( ) ( , )Ep 0. (50)
792 Fizika Nizkikh Temperatur, 2005, v. 31, No. 7
A.A. Slutskin and H.A. Kovtun
These equations show that the zero-temperature life-
time of the pair 4p is conditioned by which of two
processes is the faster: a fall in 6 4( ) caused by
| |0 0l l� � �� transitions, or decreasing of 6 4( ) to-
gether with P( )4 as a result of direct transitions
| |0 0l � � �g . In the first case the annihilation occurs
in two stages: in the first stage P( )4 remains close to
unity, while 6 4( ) approaches a vicinity of zero; in the
next stage P( )4 vanishes, 5 4( , )l being nonzero only at
such l for which ~ ( )Ep l , 0. For this scenario 4p does
not depend on the initial state 5 �( , )l l l0 �
0
. In the
second case a pronounced dependence of 4p on the
initial l l� 0 should be expected.
In Appendix it is shown that
4 � �p /�
�1 min ( ( ) ( ))
l
l l (51)
(the minimum in l is implied), i.e., 4p does not de-
pend on l0, and hence, the first of the above-men-
tioned scenarios takes place. In such a case ~ ( )Ep l
value at the minimum point l l� min of the function
� ��
( ) ( )l l is bound to be among a few eigenenergies
closest to zero. This can be justified on the following
grounds. As follows from item 5.1, ~ ( )Ep l as a func-
tion of l executes a random walk with a diffusion co-
efficient of the order of D appearing in Eq. (27). Sim-
ple estimates based on this fact show that the number
Nl of eigenenergies between a given ~ ( )Ep l and zero
is � �~ ( ) ~ ( )E L / E / up g p cl lD �4 . Since, on the one
hand, the mean separation of points l� at which
E Ep p( ) ( )l l� � is � L /Ng l , and on the other hand,
�( , )l l� falls exponentially as d� increases, we see (in
view of the definition (48)) that the less is Nl the
less is � ��
( ) ( )l l , i.e., at the minimum point l min
we really have N
l min � 1. Hence | |l l
min min � Lg .
Taking into account this fact together with Eqs. (45)
and (46), we arrive thus to conclusion that
| ln | ln ( )� 4 �0
4
p cu /t�
. (52)
This 4p value is anomalously great: it is more than
3D
1 by tens of orders of magnitude even if � is not too
small (� � 1 3/ –1 4/ ).
At finite temperatures activation transitions caused
by phonon absorbtion can provide annihilation of
dimers by bringing them at distances � r, whatever the
initial lp may be. If the pair's length lp �� s �
* , drawing
dimers into the proximity requires them to overcome a
fluctuation energy barrier � �2 1 2u lc p
/ . Within an ex-
ponential factor the activation transition frequency is
� �act �
exp ( )2 1 2u l /Tc p
/ (53)
In the temperature region T �� � �
* the pre-exponen-
tial factor omitted is determined by the Mott's vari-
able-range-hopping [18] over distances � s�
*. For the
typical l Lp g� we obtain for the activation time of
the dimer annihilation 4 �act act� 1/
4act � exp ( )u /Tc .
The time 4act becomes less than the tunnel relax-
ation time Eq. (52) when � � ( )T/uc
/1 4. At helium
temperatures and in the most realistic case uc � 102–
103 K this gives � � 1 5/ . However, 4act remains an-
omalously great.
5.4. Relaxation of an ensemble of odd-dimer pairs
Experimentally, it is possible to create (e.g., by op-
tical excitation) a set of �dimer pairs whose density is
�� �ng , but is �� ne . It should be elucidated how the
density of pairs n( )4 or, quite the same, the concentra-
tion of alternating «odd»
and
dimers reduces with
time 4. The decrease in n( )4 is expected to be so slow
(this is verified below) that the ensemble of pairs has
time to come to the partial equilibrium corresponding
to current n( )4 . (In such a case the energy of the state
at an instant 4 is approximately equal to the energy
Eq. (18) at n n� � ( )4 .) This allows a «reduction of
the description», so that we may restrict ourselves to
consideration of the differential equation
dn/d n n4 3�
( ) , (54)
wherein the transition frequency 3( )n comes from
expression (45) (the dimer tunneling prevails) or
from Eq. (53) (the activation prevails) by substitu-
tion of L /n( ) ( )4 4� 1 for lp . Correspondingly, there
are two different asymptotic dependences of n on 4. If
the activation can be neglected, we have
n
b
�
ln ln
ln
� 4
� 4
0
0
. (55)
In the case that the activation is paramount,
Eq. (54) gives
n u /Tc�
( ) ln� � 42 2 2
0 . (56)
Expressions (55) and (56) have been written to
the main logarithmic approximation. They hold for
L s( ) *4 ��� , L Lg( )4 � . The logarithmic decreasing of
n( )4 with time is a salient feature of LWG.
The super-slow relaxation in LWG and its logarith-
mic time-dependence recall to some extent those in
spin glass [13], which features a nonergodic behavior
and an infinite spectrum of relaxation times caused by
a macroscopic multi-valley degeneracy of the ground
state. As is known, a spin-glass valley is a state whose
energy is close to the ground-state one, but its space
structure differs from that of the ground state macro-
scopically. Transition from the valley to the ground
state takes place only if a macroscopically great num-
ber of spin flips occur simultaneously. In the LWG the
One-dimensional electron lattice system with a long-range interelectron repulsion
Fizika Nizkikh Temperatur, 2005, v. 31, No. 7 793
analogues of the spin-glass valleys are the above-men-
tioned electron subclusters containing a great (but by
no means macroscopic) numbers of electrons. This
gives grounds to consider the «spin-glass» LWG fea-
tures as a quasi-nonergodic behavior.
The above consideration is extended to the case of
an arbitrary rational n P/Qe � with P � 1 (see Secs. 1
and 2) almost without modifications. The only differ-
ence is that in the general case the discrete solitons
(GWC defects of compression and rarefaction) arising
in the system with a change of the electron number are
not simple �dimers, but more complicated structures
[10]. However, this is immaterial since only the for-
mation energy of the GWC defect �
�( )� � /Q
( ( )� �� �� P/Q ) is relevant.
The long-range order in GWC is also broken if the
system is exposed to an external weak random poten-
tial. The above line of argument can be extended
to this situation without change. Bearing this in mind,
it is easy to find that the ground-state dimer concen-
tration depends on the potential amplitude quadrati-
cally.
6. Conclusion
Our results show that the low-temperature thermo-
dynamics and kinetics of LWG are fully described in
terms of GWC point defects of a many-electron origin
(discrete solitons). It is the soliton nature of these de-
fects (for definiteness, we will consider them to be
dimers) that imparts to the LWG its distinctive fea-
tures differing qualitatively from those of the known
Wigner glass: the spontaneous dimer formation in the
ground state caused by an arbitrarily weak disorder of
the host lattice, first-order phase transitions in � at
zero temperature, and the almost nonergodic macro-
scopic behavior. These facts raise questions inviting
further investigation. Below we outline some of them.
First of all, the transport and high-frequency prop-
erties of LWG should be clarified, taking into account
that each �dimer carries the fractional charge
e e/Q* � � (e is the free electron charge). Conduction
in LWG is by dimer transfer over distances
� � �L /ng
g1 ( )� . This requires dimers to overcome a
fluctuation barrier of the order of the energy of dimer
formation u� ( )� (see Eq. (15)), so that the LWG
static conductivity �LWG is expected to be propor-
tional to
exp ( )ln
( )
�
�
��
�
!!
�L
u
tg �
�
or to exp
( )
�
�
��
�
!!
�u
T
�
,
depending on which of the two mechanisms prevails:
dimer tunneling through the barrier or the activation
transitions. Since Lg( )� and u� fall when � ap-
proaches the endpoints �� of the devil's-staircase
intervals (see Eqs. (15) and (19)), �LWG is bound to
show pronounced splashes in vicinities of �� . These
giant oscillations of �LWG as a function of � are not
only of interest by themselves, but also can be an
effective experimental instrument for investigating
the super-slow (logarithmic) relaxation of a strongly
nonequilibrium LWG state. At a fixed � a proper way
to observe the relaxation is keeping track of photo-
absorbtion of low-intensity light*.
Since the fluctuation energy barriers responsible
for LWG resistance cover long intervals of the order
of L /g � 1 4� , a noticeable deviation from Ohm’s law
is expected to arise even for low strengths E of the ap-
plied electric field. Namely, this takes place if e Lg
*E
becomes comparable with the typical barrier height
u� . The electric intensities that meet this condition,
E � ��4 3u /e, rapidly decline as � decreases. They are
especially low in vicinities of the endpoints of the
devil's-staircase intervals. It is remarkable that this
nonlinear effect can be pronounced even though the
LWG conductance is low.
Results obtained in Sec. 3 (see also the note at the
end of Sec. 5) show that the length of space correla-
tion in LWG L Qg is in inverse proportion to the
square of the typical energy of a random static pertur-
bation, i.e., it behaves in the same way as the correla-
tion length in continuous Wigner glass [12], wherein
the inverse quadratic dependence takes place for any
dimension d 3. This raises a very interesting ques-
tion whether such an accordance between discrete
(lattice) electron systems and continuous ones re-
mains for d � 1, and moreover, whether discrete elec-
tron systems with d � 1 are unstable with respect to an
arbitrarily weak random external perturbation. Our
preliminary studies leads us to the conclusion that,
owing to the phenomenon of the effective lowering of
dimension mentioned in Sec. 1, the 2D modification of
GWC is stable in this sense: at a given � a random dis-
tortion of the host lattice breaks the long-range order
of this system only if the disorder parameter exceeds
some critical value, which vanishes at the endpoints of
the devil’s-staircase intervals. This suggests that mac-
roscopic behavior of a 2D electron lattice system on a
weakly disordered host lattice has nothing in common
with that of 2D Wigner glass.
We are going to carry out detailed studies of the
above problems in the immediate future.
The authors are grateful to V.D. Natsik and
L.A. Pastur for fruitful discussions.
794 Fizika Nizkikh Temperatur, 2005, v. 31, No. 7
A.A. Slutskin and H.A. Kovtun
* Matrix elements of photoinduced one-step dimer transitions are not small.
Appendix
To find the general expression for the relaxation
time 4p of a pair of dimers, it is convenient to enumer-
ate eigenenergies E( )l in increasing order: Ei �
� �
E Ei i( )l 1, i � 1 2, ,...; energy E1 is the least of
E( )l . In these terms the kinetic equation (47) takes
the form
d /dρ ρ( ) � ( )4 4 � 4� , (A.1)
where ρ � ( , ,...)5 51 2 , 5 4 5 4i i( ) ( , )� l ,
� ( ) ( )� 4 � 5 4ρ �
1
� ik
k i
k , (A.2)
and � ik is a triangular matrix:
�
� �
� �ik
i k ik
i i i
i
i
k i k i
k i
�
� � �
�
"
�
� �
�
( , ), ; , ;
( ) , .
l l
l
0
1
#
$
%
$
(A.3)
The general solution of Eq. (A.1) can be expressed
in terms of eigenvectors f j and �f j (j � 1 2, , ...) of the
operator �� and the conjugate one ��� ( � �� �ik ki ), re-
spectively. They satisfy the equations
� , �� �f f f fj j j j j j� � � � �7 7 , (A.4)
where
7 j jj j ij
i
j
� �
�
�
�� � �( )l
1
0 (A.5)
are the eigenvalues of the operators �� and ���.
Taking into account that � � � �� �f fj j jj| � (� �a b|
means the scalar product � i i ia b here and further on),
from Eqs. (A.1) and (A.4) we obtain
ρ ρ( ) | exp ( )4 4� � � ��
j
j j jf f0 7 , (A.6)
where ρ ρ0 0� ( ).
Since 7 j are all negative, this expression tends
asymptotically to
ρ ρ( ) | exp ( )* * *4 4� � � �f f
j j j0 7 (A.7)
as 4 � �. Here j* is the number of the eigenvalue
with the least modulus. (As follows from the reason-
ing after formula (51), j*� 1. Hence, 4p j
� 1/| |*7 .
This coincides with expression (51).
As to the prefactor � � �f
j*
|ρ0 , it is easy to show (in
view of the triangular form of the matrix � ik) that it is
� 1 for all ρ0 of the form 5 �0 0i ii� , except i j0 � *. In
the latter (very special) case the relaxation time is less
than 1/| |*7
j
, but its logarithm is of the same order of
magnitude.
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