Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities
We present an exact theory that describes how magnetic impurities change the behavior of the thermal
 conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic impurities
 and a large concentration of impurities with similar values of the co...
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| Опубліковано в: : | Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
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| Цитувати: | Thermal conductivity of a quantum spin-1/2
 antiferromagnetic chain with magnetic impurities / A.A. Zvyagin // Физика низких температур. — 2008. — Т. 34, № 3. — С. 273–277. — Бібліогр.: 15 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860239096599805952 |
|---|---|
| author | Zvyagin, A.A. |
| author_facet | Zvyagin, A.A. |
| citation_txt | Thermal conductivity of a quantum spin-1/2
 antiferromagnetic chain with magnetic impurities / A.A. Zvyagin // Физика низких температур. — 2008. — Т. 34, № 3. — С. 273–277. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We present an exact theory that describes how magnetic impurities change the behavior of the thermal
conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic impurities
and a large concentration of impurities with similar values of the couplings to the host chain (a weak
disorder) do not change the linear-in-temperature low-T behavior of the thermal conductivity: Only the slope
of that behavior becomes smaller, comparing to the homogeneous case. The strong disorder in the distribution
of the impurity-host couplings produces more rapid temperature growth of the thermal conductivity,
compared to the linear in T dependence of the homogeneous chain and the chain with a weak disorder. Recent
experiments on the thermal conductivity in inhomogeneous quasi-one-dimensional quantum spin systems
manifest qualitative agreement with our results.
|
| first_indexed | 2025-12-07T18:28:10Z |
| format | Article |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 3, p. 273–277
Thermal conductivity of a quantum spin-1/2
antiferromagnetic chain with magnetic impurities
A.A. Zvyagin
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: zvyagin@ilt.kharkov.ua
Leibniz Institut für Festkörper- und Werkstoffforschung Dresden e.V. Postfach: 270116, D-01171 Dresden, Germany
Received October 15, 2007
We present an exact theory that describes how magnetic impurities change the behavior of the thermal
conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic im-
purities and a large concentration of impurities with similar values of the couplings to the host chain (a weak
disorder) do not change the linear-in-temperature low-T behavior of the thermal conductivity: Only the slope
of that behavior becomes smaller, comparing to the homogeneous case. The strong disorder in the distribu-
tion of the impurity-host couplings produces more rapid temperature growth of the thermal conductivity,
compared to the linear in T dependence of the homogeneous chain and the chain with a weak disorder. Re-
cent experiments on the thermal conductivity in inhomogeneous quasi-one-dimensional quantum spin sys-
tems manifest qualitative agreement with our results.
PACS: 75.10.Jm Quantized spin models;
71.10–w Theories and models of many-electron systems;
72.10.Di Scattering by phonons, magnons, and other nonlocalized excitations (including Kondo
effect);
72.15.Eb Electrical and thermal conduction in crystalline metals and alloys.
Keywords: thermal conductivity, spin chain, magnetic impurities.
Quasi-one-dimensional quantum spin systems have at-
tracted much attention of researchers in recent years.
There are several reasons for that growth of interest. First,
during the last decade many real compounds with the
properties of quasi-one-dimensional quantum spin chains
were synthesized and studied experimentally. On the
other hand, namely quantum spin chains provide physi-
cists with the rare opportunity to compare the experimen-
tal observations with exact theoretical results, since many
exact results have been obtained for one-dimensional
quantum spin chains. And, last but not least, one-dimen-
sional spin models are actively used in many modern ap-
plications, like in the theory of nano-structures,
mesoscopic devices, and, even in the theory of quantum
computations.
One of the main models that is frequently considered
by physicists, is the Heisenberg spin-1/2 chain with
antiferromagnetic interactions, solved exactly by Bethe
in 1931 [1]. Since that time many results have been ob-
tained for the model; for recent review see, e.g., Ref. 2.
However, in most cases theories were developed for ho-
mogeneous spin chains, despite the fact that in nature one
more frequently encounters inhomogeneous situations,
i.e., real spin chains contain defects, for instance, mag-
netic impurities. Impurities in quantum systems can dras-
tically change the behavior of the system, especially in
low-dimensional cases. For example, it is well known that
in the one-dimensional case impurities produce localiza-
tion of wave functions, changing the behavior from me-
tallic to insulating. This is why the study of models with
impurities, especially of systems with strong correlations
between particles, is one of the most important and diffi-
cult tasks of the modern theoretical physics. The study of
the thermal conductivity in systems of weakly coupled
quantum spin chains has become popular in recent years,
because the thermal conductivity can be a strong experi-
mental instrument to investigate magnetic properties of
materials.
© A.A. Zvyagin, 2008
In this paper we study theoretically how magnetic im-
purities can affect the thermal conductivity in quantum
spin chains. For our purpose we choose an integrable spin
chain, because in such chains one can introduce any num-
ber of magnetic impurities, and because for that case one
can obtain exact results. Earlier studies of integrable spin
chains with magnetic impurities manifested a good agree-
ment of theoretical results with experimental data for
quasi-one-dimensional inhomogeneous compounds.
Therefore, we expect that the calculated thermal conduc-
tivity for the integrable model will reveal properties, ge-
neric for inhomogeneous quantum spin chains.
We investigate the thermal conductivity of the
integrable spin-1/2 antiferromagnetic quantum chain
with spin-1/2 magnetic impurities. The Hamiltonian of
the system has the form [2,3]: � � �� �h imp , where the
host part is �h j j jH� � �, 1, with H j j j j, � ��1 1S S (the
host exchange constant J is set to unity). The impurity
part of the Hamiltonian has the special form, necessary
for exact integrability of the model. Suppose the concen-
tration of impurities is such that impurities are not nearest
neighbors. Then for the impurity situated between sites m
and m�1of the host the integrable Hamiltonian reads
�imp imp imp imp� � ��J H Hm m(( ), , 1
� �� �� �2
1 1H i H Hm m m m, , ,[ , ] )imp imp , (1)
where J imp � �1 12/ ( )� , and [., .] denotes the commutator.
The lattice Hamiltonian Eq. (1) describes the impurity
spin coupled to the host with the strength J imp ,
parametrized by the constant �. One can see that the limit
�� 0 corresponds to the simple inclusion of an additional
site coupled with the bulk interaction to the system. On
the other hand, for �� � one obtains an impurity spin to-
tally decoupled from the host. The Hamiltonian also con-
tains additional terms which renormalize the coupling be-
tween the neighboring sites of the host, and three-spin
terms. However, it has been shown [2,3] that in the long-
wave limit such a lattice form of the impurity Hamiltonian
yields the well-known form of the contact impurity-host
interaction, similar to the one of, e.g., the Kondo problem
of a magnetic impurity in a metal. The contact impurity
coupling in that limit is also determined by the same con-
stant �. The advantage of the consideration of the in-
tegrable Hamiltonian � is obvious: One can independ-
ently incorporate any number of magnetic impurities into
the host chain. Each impurity is characterized by its own
coupling to the host, i.e., by own � j . Finally, we like to
note that all considered impurities are elastic scatterers,
i.e., each excitation only changes its phase when scatter-
ing off an impurity, but is not reflected. However, the
same property holds for the standard Kondo impurity in a
free electron host. Equally important to mention is that we
study a lattice model, hence all two-particle scattering
processes, in particular from one Fermi point to the other
(backscattering), are taken into account in the model.
To find the thermodynamic characteristics of our
one-dimensional quantum spin Hamiltonian at finite tem-
perature we use the so-called «quantum transfer matrix»
approach, see, e.g., [3,4]. Let R u
i i
i i
�
�1 ( ) be the standard
R-matrix of the Heisenberg spin-1/2 chain [4]. Indices � i
and i denote states of the spin at site i, and
denotes
states in the auxiliary space. The «standard» transfer ma-
trices (row-to-row from the viewpoint of statistical 2D
problem) ��
( )u have the form of the trace over the auxil-
iary space of the product of R-matrices
� � ��
�
( ,{ } ) ( , )u R ui
L
i
L
i
i i
i i
�
�
� ���1
1
1 , (2)
where L is the length of the quantum chain and � i are the
inhomogeneity parameters, which are shifts of the spec-
tral parameter. The R-matrices satisfy the Yang–Baxter
equations, hence the transfer matrices with different spec-
tral parameters commute. Conservation laws of the
integrable model can be constructed as derivatives of the
logarithm of the transfer matrix as
A
u
un
n
u�
�
�
�
�
�
�
�
� �const ln ( )|� 0. (3)
The Hamiltonian of the model corresponds (up to a nones-
sential constant term) to the n � 2 case with const �1. One
can introduce R-matrices of different type, related to the
initial one by an anticlockwise and clockwise rotation
R u R u
R u R u
�
�
�
�
�
�
�
�
( ) ( ) ,
~
( ) ( ) .
�
� (4)
The transfer matrix � �( ,{ } )u i
L
�1 can be constructed in a
way similar to the case of �. Then we substitute
u J NT� � / (J �1), where N is the Trotter number. We
find
[ ( ) ( )] / ( / )/� �u u T NN 2 1� � �exp ( )� � . (5)
Hence, the partition function of the quantum 1D system is
identical to the partition function of an inhomogeneous
classical vertex model with alternating rows on a square
lattice of size L N� ,
Z u u
N
N�
��
lim [ ( ) ( )] /Tr � � 2 . (6)
The interactions on the 2D lattice are four-spin inter-
actions with coupling parameters depending on ( )NT �1
and interaction parameters � j where j is the number of
the column to which the considered vertex of the lattice
belongs. The corresponding column-to-column transfer
matrices are referred to as «quantum transfer matrices»
274 Fizika Nizkikh Temperatur, 2008, v. 34, No. 3
A.A. Zvyagin
(where an external magnetic field h is included by means
of twisted boundary conditions)
� � �
�
QTM j
h T
i
N
u R u i
i i
i i( , ) (
/
/
� �� � � �
�
�
e 1
2 1 2 1
2 1 2
1
2
j )�
� ��~
( )R u i
i i
i i
j�
�
2 2
2 2 1 . (7)
In general all «quantum transfer matrices» corresponding
to the L many columns are different. However, all these
operators can be proven to commute pairwise. Therefore,
the free energy per lattice site of our system can be calcu-
lated from just the largest eigenvalues of the «quantum
transfer matrices» (corresponding to only one eigenstate).
The partition function of the model is determined by
the following nonlinear integral equation for the «energy
density» functions a, a, A a� �1 and A a� �1 in relation
to the spectral parameter u:
[ ( ) ln ( ) ( ) ln ( )]� � � � � � �k u A k u i i A dv v v v v� �
� � �ln ( )
cosh
a u
T u
h
T
Fv
2
, (8)
with vF � � / 2 being the Fermi velocity of low-energy ex-
citations of the chain (spinons) (in what follows we con-
sider the case h � 0), with infinitesimally small �, and the
kernel function being defined as
k x d
i x
( )
cosh ( )
( | | )
� �
� �1
2
2
�
�
��
�� �e
. (9)
The equation for a follows from Eq. (8) with the formal
changes h h� � , i i� � , a a� and vice versa.
In the linear response theory the Kubo formulas [5]
yield the thermal conductivity � relating the thermal cur-
rent to the temperature gradient �T as � E T� �� , where
� � � ��( ) ( )
/
� � � � ��
�
� �
1
0 0
1
T
dt d t ii t
E E T
T
e � � , (10)
and brackets denote thermal average. It was pointed out
[6,7] that for the homogeneous quantum spin chain, i.e.,
in the limit � j � 0, the operator of the thermal current is
equal to the operator of the third conservation current.
Following the logic of Refs. 6, 7, we denote the operator
of the thermal current, �J E , for the integrable model of a
spin chain with magnetic impurities (i.e., for possible
nonzero � j ) as A3 (up to a nonessential constant term)
with const � i [7]. Then, by construction, the operator of
the thermal current for the integrable model with mag-
netic impurities commutes with the Hamiltonian. It is
easy to check that such a definition of the thermal current
respects the continuity equation for the considered mo-
del; cf. [7]. Hence, one finds
� �
� �
�( )
( )
|� �
� �
�
� �i
T i
E T�
2
2 0 . (11)
Therefore the real part of the conductivity can be written
as
Re � � �� �( ) ( ) /� � �� E T T2 2. (12)
This means that the thermal conductivity in our integrable
model of the antiferromagnetic quantum spin-1/2 chain
with magnetic impurities is infinite at zero frequency, as
for the homogeneous model, without impurities. Natu-
rally, it is the consequence of the exact integrability of the
model considered, i.e., of the infinite number of conser-
vation laws. To calculate � �� E T
2 for our model we use the
trick, proposed in [7]. We define some modified partition
function
Z T f J E� � �Tr exp [ ( / ) � ]1 � . (13)
Using that partition function one can easily find that
� � �� E T 0, and
� � �
�
�
�
�
��
�
�
�� �� E T f
f
Z2
2
0ln | . (14)
The eigenvalue of the modified partition function per site
( )u is given by
ln ( )
( ) ln[ ( ) ( )]
cosh ( )
u
e u
T
A v A v dv
u v
� �
��0 1
2�
, (15)
where e0 is the ground-state energy (defined for the sys-
tem with the partition function Z). The eigenfunction of
the modified partition function of the total chain with im-
purities is
tot � ��
j
ju( / )�� 2 , (16)
where the product is taken over all the sites (for sites
without impurities we get ( )u � 0 ). Equations (8) are
easily solved numerically for arbitrary temperatures. For
the spin chain with the disordered ensemble of magnetic
impurities the random distribution of the values � j can be
described by a distribution function P j( )� .
At low energies the behavior of each impurity depends
on the energy scale [2]
T jK F j! �v exp ( | | )� � , (17)
which plays the same role as the Kondo temperature for a
magnetic impurity in a metal. The magnetic impurity be-
haves asymptotically free for T T jK"" , while it is
strongly coupled to the spin chain for T T jK## . In other
words, � j measures the shift off the Kondo resonance
(higher values of | |� j correspond to lower values on the
Kondo scale) of the impurity level with host spin excita-
tions, similar to the standard picture of the Kondo effect
Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities
Fizika Nizkikh Temperatur, 2008, v. 34, No. 3 275
in a metallic host. It has been shown [8] that the low-en-
ergy thermodynamic characteristics of the considered
quantum spin chain with magnetic impurities are deter-
mined, in fact, by the distribution of those effective Kon-
do temperatures. At low temperatures one can replace the
functions a, a, A and A by the scaling functions, intro-
duced as
a x a x T TjK$ % $ $( ) [ ln ( / )]� (18)
(� being some constant), etc. [3,7]. For those functions
integral equations are transformed in such a way that for
that new set of scaling functions the only asymptotic be-
havior of A $ and A $ enters. We obtain that each site of
the chain contributes to the low-temperature thermal cur-
rent as
� � & �� jE T
jKT T
O T2
3
4
3
�
( ) , (19)
where for sites without impurities one gets T jK F� v . It
means that the contribution from each site to the thermal
conductivity is linear in temperature at low T ,
Re � � �
�
j
jKT T
& ( )
3
. (20)
For a single impurity one has P T T TjK jK K( ) ( )' �� .
Hence, at low temperatures a single magnetic impurity
does not change the linear in T dependence of the thermal
conductivity of the spin chain.
Consider ensembles of magnetic impurities with a
weak disorder (i.e., for ensembles with random distribu-
tions of the impurity-host couplings, for which those cou-
plings, and, hence, their effective Kondo temperatures,
are similar to each other). Let those Kondo temperatures
are close to some TK . Hence, the total thermal conductiv-
ity for integrable spin chains with weakly disordered
magnetic impurities can be estimated as
Re tot� � � �
�
� &� dT P T
T T
jK jK j
K( ) ( )
3
. (21)
This formula implies that for integrable spin systems with
weakly disordered ensembles of magnetic impurities the
thermal conductivity is finite. It is proportional to T at
low temperatures, but the slope is smaller than the one for
the homogeneous integrable chain, because for magnetic
impurities, antiferromagnetically coupled to the chain
(and we consider here only such impurities) the effective
Kondo temperature is smaller than the Fermi veloc-
ity of spinons (vF J� � / 2), i.e., one has TK F' �v
� � � #exp ( ( ) / )� J J J Fimp imp v . Notice that in our mo-
del 0 ( (J Jimp , and the case without impurities corre-
sponds to J Jimp � . Obviously, one should expect a
similar behavior for the thermal conductivity of a spin
chain with a single antiferromagnetic impurity (or with a
small concentration of such impurities).
The situation becomes very different for integrable spin
chains with strongly disordered ensembles of magnetic
impurities. For those ensembles the distribution function
for effective Kondo temperatures is wide. Hence, for a
fixed temperature, for a large fraction of the impurities
their effective Kondo temperatures are lower than the
temperature of the system, and, hence, they remain un-
screened. This effect produces the divergency of the mag-
netic susceptibility and the Sommerfeld coefficient of the
specific heat for such chains [2]. It implies that the con-
sidered spin chain manifests the Griffiths’-phase-like be-
havior [9]. Let us consider the realistic distribution of
Kondo temperatures for magnetic impurities in the spin
chain [2,8,10], which starts from the term
P T G TjK jK( ) ( )! � �) ) 1 (22)
() # 1), valid till some energy scale G for the lowest values
of T jK . This distribution was shown to pertain to real dis-
ordered quantum spin chains [10,11]. Averaging the ther-
mal conductivity with this distribution, we obtain
Re ( )
( )
� � �
�
)
)
)tot '
�
�T
G
2
3 1
. (23)
Hence, for integrable spin chains with ensembles of
strongly disordered magnetic impurities the thermal con-
ductivity is also finite, but it grows with T at low tempera-
tures much faster, than for the homogeneous chain, or for
the chain with weakly disordered magnetic impurities.
We can solve nonlinear integral equations analytically
also at high temperatures, using the asymptotic form of
«energy density» functions a, a, A and A, cf. [7]. At high
temperatures for our integrable chain with magnetic im-
purities the thermal conductivity decays with the temper-
ature as
Re ( ) ( )� � � �'
3
2 2T
. (24)
Obviously, at high temperatures, larger than the coupling
constants, the system behaves as a gas of noninteracting
spins 1/2, and the result does not depend on the parame-
ters of the impurity-host couplings, as must be the case.
Hence, the thermal conductivity of the integrable spin
chain with magnetic impurities, as a function of tempera-
ture, grows with T at low temperatures, manifests a maxi-
mum, and then decays with T , as the homogeneous chain.
The main differences, caused by magnetic impurities, are
present at low temperatures.
In summary, in the framework of the integrable quan-
tum spin model we have calculated the thermal conduc-
tivity for a spin chain with magnetic impurities. Previous
theoretical studies of thermodynamic characteristics for
the model considered [2,8,10] have shown that the model
exhibits generic features of disordered quantum spin sys-
276 Fizika Nizkikh Temperatur, 2008, v. 34, No. 3
A.A. Zvyagin
tems, e.g., the Griffiths’-phase-like behavior, and the re-
sults of those studies well agree with the data of experi-
ments on quasi-one-dimensional antiferromagnetic
quantum spin-1/2 systems with magnetic impurities, see,
e.g., [11,12]. Our investigation has shown that the pres-
ence of magnetic impurities in an integrable quantum
chain does not destroy the most exciting property of an
integrable spin chain — the infinite thermal conductivity
at zero frequency. The �-function dependence of the ther-
mal conductivity, obtained in this work (as well as for
spin chains without impurities), is related to the infinite
number of integrals of motion in the integrable model
considered, which, in turn, is based on the special choise
of the interaction (only elastic scatterings, absence of re-
flections). In a more realistic situation one should, natu-
rally, take into account the finite value of the width of the
frequency dependence of the thermal conductivity, which
is present there because of inelastic processes and reflec-
tions off impurities. However, we expect that our model
reproduces the generic features of the thermal conductiv-
ity of a quantum spin chain with impurities.
Our model permits one to include any number of mag-
netic impurities in the spin chain. Therefore, it permitted
us to obtain results for small concentration of impurities,
when each impurity behaves as a single one, and for large
concentration of impurities. The important conclusion
from our study is that the presence of magnetic impurities
mostly affects the low-temperature behavior of the zero-
frequency thermal conductivity. Single magnetic impuri-
ties and a large concentration of impurities with similar
values of the couplings to the host chain (a weak disorder)
do not change the linear-in-temperature low-T behavior
of the thermal conductivity, but the slope of that behavior
becomes smaller, compared to the homogeneous case. On
the other hand, the strong disorder in the distribution of
the impurity-host couplings (when the chain manifests
the Griffiths’-phase-like behavior) produces essential
changes in the temperature behavior of the low-T thermal
conductivity, with the growth with temperature becom-
ing more rapid as compared to the linear temperature de-
pendence of the homogeneous chain and the chain with a
weak disorder. We believe that such a behavior of the
thermal conductivity for integrable quantum spin chains
with impurities has generic features. Similar situation
was recently observed in quasi-one-dimensional compo-
unds Sr Ca Cu O14 24 41�x x [13], Sr Ca CuO1 2�x x [14], and
Sr Cu Zn O14 24 41�x x [15], where the introduction of Ca or
Zn impurities reduced the slopes of the low-temperature
linear-in-T behavior of the magnetic thermal conductiv-
ity. It is possible that ions of Zn and Ca, introduced in
those compounds, produce local changes in the spin-spin
antiferromagnetic coupling between Cu ions. Hence, the
situation can be modelled by the magnetic spin-1/2 chain
with impurities. Then experiment [15] has revealed that
for systems with doped Zn the low-temperature thermal
conductivity is linear in temperature, as in the undoped
cases. However, the slopes of the linear temperature
dependence of the thermal conductivity really became
smaller with the doping impurities, which qualitatively
agree with our theory. The results of our theory can
be also applied, e.g., to quasi-one-dimensional antifer-
romagnetic systems with magnetic impurities, like
(Sr,Ca,La) 14Cu 24O 41, BaCu2(Si 1�xGe x)2O7 [12], or or-
ganic materials with the properties of inhomogeneous
quantum spin chains, see, e.g., [11].
The financial support from Deutsche Forschungs-
gemeinschaft (Grant No 436 UKR 17/21/06) is acknow-
ledged. Stimulating discussions with B. B�chner, C. Hess,
S.-L. Drechsler, V. Kataev, and P. Ribeiro are greatly ap-
preciated.
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104407 (2006).
Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities
Fizika Nizkikh Temperatur, 2008, v. 34, No. 3 277
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| id | nasplib_isofts_kiev_ua-123456789-116849 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T18:28:10Z |
| publishDate | 2008 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Zvyagin, A.A. 2017-05-16T19:12:30Z 2017-05-16T19:12:30Z 2008 Thermal conductivity of a quantum spin-1/2
 antiferromagnetic chain with magnetic impurities / A.A. Zvyagin // Физика низких температур. — 2008. — Т. 34, № 3. — С. 273–277. — Бібліогр.: 15 назв. — англ. 0132-6414 PACS: 75.10.Jm;71.10–w;72.10.Di;72.15.Eb https://nasplib.isofts.kiev.ua/handle/123456789/116849 We present an exact theory that describes how magnetic impurities change the behavior of the thermal
 conductivity for the integrable Heisenberg antiferromagnetic quantum spin-1/2 chain. Single magnetic impurities
 and a large concentration of impurities with similar values of the couplings to the host chain (a weak
 disorder) do not change the linear-in-temperature low-T behavior of the thermal conductivity: Only the slope
 of that behavior becomes smaller, comparing to the homogeneous case. The strong disorder in the distribution
 of the impurity-host couplings produces more rapid temperature growth of the thermal conductivity,
 compared to the linear in T dependence of the homogeneous chain and the chain with a weak disorder. Recent
 experiments on the thermal conductivity in inhomogeneous quasi-one-dimensional quantum spin systems
 manifest qualitative agreement with our results. The financial support from Deutsche Forschungsgemeinschaft (Grant No 436 UKR 17/21/06) is acknowledged. Stimulating discussions with B. Buchner, C. Hess,
 S.-L. Drechsler, V. Kataev, and P. Ribeiro are greatly appreciated en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низкоразмерные и неупорядоченные системы Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities Article published earlier |
| spellingShingle | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities Zvyagin, A.A. Низкоразмерные и неупорядоченные системы |
| title | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| title_full | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| title_fullStr | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| title_full_unstemmed | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| title_short | Thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| title_sort | thermal conductivity of a quantum spin-1/2 antiferromagnetic chain with magnetic impurities |
| topic | Низкоразмерные и неупорядоченные системы |
| topic_facet | Низкоразмерные и неупорядоченные системы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/116849 |
| work_keys_str_mv | AT zvyaginaa thermalconductivityofaquantumspin12antiferromagneticchainwithmagneticimpurities |