Phenomenological description of a spin chain system with geometrical frustration of couplings
The phenomenological model, using which we can in a relatively simple way calculate many magnetic, thermo-dynamic and dynamic characteristics of the spin chain material with the geometrical frustration of spin-spin couplings is proposed. The results of theoretical calculations well reproduce observe...
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irk-123456789-1762432021-02-05T01:26:55Z Phenomenological description of a spin chain system with geometrical frustration of couplings Zvyagin, A.A. Специальный выпуск К 80-летию со дня рождения А.И. Звягина The phenomenological model, using which we can in a relatively simple way calculate many magnetic, thermo-dynamic and dynamic characteristics of the spin chain material with the geometrical frustration of spin-spin couplings is proposed. The results of theoretical calculations well reproduce observed details of the low-temperature behavior of the magnetization, magnetic susceptibility, specific heat, magneto-acoustic characteristics, and some dynamical properties. In particular, the model permits to explain the double peak structure of the temperature dependencies of the magnetic susceptibility, specific heat, and the renormalization of the sound velocity, and can explain several features of the ESR frequency-field diagram. 2018 Article Phenomenological description of a spin chain system with geometrical frustration of couplings / A.A. Zvyagin // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1676-1683. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 75.10.Jm, 75.10.Pq, 75.40.Cx, 75.40.Jb http://dspace.nbuv.gov.ua/handle/123456789/176243 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Специальный выпуск К 80-летию со дня рождения А.И. Звягина Специальный выпуск К 80-летию со дня рождения А.И. Звягина |
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Специальный выпуск К 80-летию со дня рождения А.И. Звягина Специальный выпуск К 80-летию со дня рождения А.И. Звягина Zvyagin, A.A. Phenomenological description of a spin chain system with geometrical frustration of couplings Физика низких температур |
| description |
The phenomenological model, using which we can in a relatively simple way calculate many magnetic, thermo-dynamic and dynamic characteristics of the spin chain material with the geometrical frustration of spin-spin couplings is proposed. The results of theoretical calculations well reproduce observed details of the low-temperature behavior of the magnetization, magnetic susceptibility, specific heat, magneto-acoustic characteristics, and some dynamical properties. In particular, the model permits to explain the double peak structure of the temperature dependencies of the magnetic susceptibility, specific heat, and the renormalization of the sound velocity, and can explain several features of the ESR frequency-field diagram. |
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Article |
| author |
Zvyagin, A.A. |
| author_facet |
Zvyagin, A.A. |
| author_sort |
Zvyagin, A.A. |
| title |
Phenomenological description of a spin chain system with geometrical frustration of couplings |
| title_short |
Phenomenological description of a spin chain system with geometrical frustration of couplings |
| title_full |
Phenomenological description of a spin chain system with geometrical frustration of couplings |
| title_fullStr |
Phenomenological description of a spin chain system with geometrical frustration of couplings |
| title_full_unstemmed |
Phenomenological description of a spin chain system with geometrical frustration of couplings |
| title_sort |
phenomenological description of a spin chain system with geometrical frustration of couplings |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2018 |
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Специальный выпуск К 80-летию со дня рождения А.И. Звягина |
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http://dspace.nbuv.gov.ua/handle/123456789/176243 |
| citation_txt |
Phenomenological description of a spin chain system with geometrical frustration of couplings / A.A. Zvyagin // Физика низких температур. — 2018. — Т. 44, № 11. — С. 1676-1683. — Бібліогр.: 28 назв. — англ. |
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Физика низких температур |
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AT zvyaginaa phenomenologicaldescriptionofaspinchainsystemwithgeometricalfrustrationofcouplings |
| first_indexed |
2025-07-15T13:56:02Z |
| last_indexed |
2025-07-15T13:56:02Z |
| _version_ |
1837721466179682304 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11, pp. 1676–1683
Phenomenological description of a spin chain system
with geometrical frustration of couplings
A.A. Zvyagin
Max-Planck-Institut für Physik komplexer Systeme, 38 Nöthnitzer Str., Dresden 01187, Germany
B. Verkin Institute for Low Temperature Physics and Engeneering of the National Academy of Sciences of Ukraine
47 Nauky Ave., Kharkiv 61103, Ukraine
E-mail: zvyagin@ilt.kharkov.ua
Received April 18, 2017, published online September 25, 2017
The phenomenological model, using which we can in a relatively simple way calculate many magnetic, thermo-
dynamic and dynamic characteristics of the spin chain material with the geometrical frustration of spin-spin cou-
plings is proposed. The results of theoretical calculations well reproduce observed details of the low-temperature
behavior of the magnetization, magnetic susceptibility, specific heat, magneto-acoustic characteristics, and some
dynamical properties. In particular, the model permits to explain the double peak structure of the temperature de-
pendencies of the magnetic susceptibility, specific heat, and the renormalization of the sound velocity, and can ex-
plain several features of the ESR frequency-field diagram.
PACS: 75.10.Jm Quantized spin models, including quantum spin frustration;
75.10.Pq Spin chain models;
75.40.Cx Static properties;
75.40.Jb Dynamic properties.
Keywords: spin chain system, magnetic susceptibility, specific heat, sound velocity.
Quantum spin liquids are characterized by strong spin-
spin couplings. Emergent excitations of quantum spin
liquids are fractionalized quasiparticles. Among the large
variety of magnetic materials low dimensional spin sys-
tems and systems with frustration are known to demon-
strate properties of quantum spin liquids [1]. Frustrated
bonds, as well as quantum fluctuations, enhanced in low-
dimensional spin systems, often tend to destroy the mag-
netic ordering in quantum spin liquids. The progress in
preparation of magnetic materials with well defined one-
dimensional (1d) subsystems has motivated the interest in
their studies during last years. Another reason for the in-
vestigation of properties of quasi-1d spin systems is the
relatively rare possibility of comparison of the data of ex-
periments with the results of non-perturbative theories for
many-body models [2]. According to the Mermin–Wagner
theorem [3], totally 1d spin systems with isotropic spin-
spin interactions cannot reveal any magnetic ordering at
nonzero temperatures. However, for quasi-1d spin systems,
which 1d subsystems have gapless low-lying excitations,
the magnetization, magnetic susceptibility, specific heat,
etc. often manifest peculiarities, characteristic for phase
transitions to magnetically ordered states. Frustrated spin
systems also manifest peculiar properties [4], which distin-
guish them from standard magnetically ordered systems
and from ordinary paramagnets [5]. The system is called
frustrated if one cannot minimize its total energy (the mac-
roscopic state) by minimizing the energy of each pair in-
volving into the interaction. The frustration often produces
a high level of degeneracy of eigenstates. It is very difficult
to construct theories to explain the behavior of frustrated
spin systems, because the absence of the well-defined non-
degenerate ground state does not permit to use the power-
ful methods of the magnetism theory, as the mean field-
like approximations, or the spin wave approach. Therefore,
approximate models, even “exotic” ones, the monopole
theory for spin ices [6] being the prime example, are very
useful for the description of spin systems with geometrical
frustration.
For many quasi-1d spin systems and systems with the
geometrical frustration, in which (maximal) exchange pa-
rameters along the distinguished direction are relatively
small (1–40 K), the features of their behavior can be
governed by the external magnetic field. The latter can
induce quantum phase transitions, which manifestations at
nonzero, however low temperatures can be observed. In
© A.A. Zvyagin, 2017
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Phenomenological description of a spin chain system with geometrical frustration of couplings
low-temperature experiments high values of the magnetic
field (about 20 T for stationary fields and about 60 T for
pulse fields) can be used nowadays. Therefore, for many
quasi-1d spin systems it is possible to investigate experi-
mentally how the external magnetic field affects their pro-
perties. It is important for experimentalists, who study
magnetic properties of frustrated low-dimensional spin
systems, to use relatively simple theories, for comparison
of their results with the prediction of those theories.
The observation of the 1/3 magnetization plateau [7] in
the low-temperature magnetization behavior of the natural
mineral azurite Cu3(CO3)2(OH)2 has attracted the interest
of researchers to that compound. Similar magnetization
plateaux were observed in many frustrated spin systems [8],
hence those plateaux often serve as the “benchmark” of
frustrated magnetic materials. The structure of azurite
(a monoclinic crystal structure with the space group 12 / ,P c
where the magnetic ions Cu2+ are arranged in chains build
of diamond-shaped units, running along b direction of the
crystal) implies the quasi-1d spin system with weakly cou-
pled diamond spin chains (with mostly antiferromagnetic
spin-spin interactions [9]). Formally diamond spin chains
have the geometric frustration [9]. However, the character-
istics of azurite do not show a high level of degeneracy,
hence, the weak inter-chain coupling produces the magnet-
ic ordering at low temperatures TN 1.85 K [10], while
some other features of the low-temperature magnetic be-
havior are characteristic for geometrically frustrated spin
systems. The inelastic neutron scattering did not observe a
strong dispersion perpendicular to the chain direction [10]
even at < .NT T Therefore, azurite at low enough tempera-
tures, however for > ,NT T can be considered as the good
representative of quantum spin liquids having, on the one
hand, a strong spin-spin coupling along chains, and, on the
other hand, frustrated spin-spin bonds, together with the
fractional emergent excitations, see below.
There were many attempts to describe theoretically the
properties of azurite, see, e.g., [7,10–14], etc. However,
most of them either explain analytically only few features
of the behavior of azurite characteristics, or use numerical
calculations like the exact diagonalization, or the DMRG.
The disadvantage of the latter is in the necessity of the
production of new calculations with non-clear results to
obtain results for a new set of parameters, and hence, long
times of computer calculations. This is why, the objection
of the present study is to construct an analytic theory (con-
venient for comparison with experiments), which can be
used to describe many features of the behavior of azurite
from the same grounds. In this contribution we propose the
relatively simple analytic approach, which models such a
quasi-1d frustrated spin system, revealing many properties
of quantum spin liquids. The results of our approach mani-
fest a very good agreement with the data of experiments
related to magnetic, thermodynamic, and dynamic proper-
ties of azurite.
Let us start with the description of the magnetic field
dependence of the magnetization of azurite. It manifests
one of the most interesting features, the magnetization
plateau at approximately 1/3 of the total magnetization in
saturation 0M [7]. The magnetization grows with the ex-
ternal magnetic field H for 1< ,H H where 1H is the first
critical value of the field, and for 2 < < sH H H , where 2H
is the second critical value, and sH is the value of the field,
at which the magnetization is saturated. On the other hand,
the magnetization is almost constant as a function of the
field ((1/3)M0 for 1 2< <H H H and M0 for > sH H ) at
low temperatures. The magnetization of the 1d spin system
can have magnetization plateaux according to the generali-
zation of the Lieb–Schulz–Mattis theorem [2,15,16], in-
cluding the case of three critical values of the field. For
example, three critical values of the magnetic field is the
characteristic feature for the behavior of the trimerized
exactly solvable quantum spin chain [2,17]. The Hamiltoni-
an of the exactly solvable trimerized spin 1/2 chain is [2,17]
3 0 ,0 ,1 1 ,1 ,2= (1/2) [( n n n n
n
J S S J S S+ − + −+ +∑
2 ,2 1,0 H.c.)n nJ S S+ −
++ + −
0 ,0 1 ,1 2 ,22 ( )] ,z z z
B n n nH g S g S g S− µ + + (1)
where 0,1,2J determine the values of the exchange integrals
between neighboring spins along the chain, ,0,1,2nS± ≡
,0,1,2 ,0,1,2 ,yx
n nS iS≡ ± 0,1,2g are the values of the g-factor for
each of three non-equivalent magnetic centers, and Bµ is
the Bohr magneton. After the generalized Jordan–Wigner
transformation, the Fourier transform, and the unitary trans-
formation [2,17] the Hamiltonian can be presented as the
sum of three diagonal quadratic forms of spinless fermion
operators
2
†
3 , , 0 1 2,
=0
= [ ( )/2] ,k j k j Bk j
j k
b b L H g g gε − µ + +∑∑ (2)
where ,k jb ( †
,k jb ) destroys (creates) the spinless fermion of
the branch = 0, 1, 2j with the quasimomentum k, and 3L is
the length of the chain (we use units in which the inter-cite
distance is unity). For the simplest case 0 1 2= = =g g g g
the dispersion relations can be presented as
2 2 2
0 1 2
,
2 2= cos ,
3 3k j B
J J J jg H
+ + π ε µ − α +
(3)
where = 1, 2, 3,j and
0 1 2
2 2 2 3/2
0 1 2
3 3 cos( )1= arccos .
3 ( )
J J J k
J J J
α −
+ +
(4)
One can see that all thermodynamic properties of the ex-
actly solvable trimerized spin-1/2 chain are described by
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1677
A.A. Zvyagin
three non-interacting fermionic Hamiltonians, which pa-
rameters depend on the g-factors and exchange integrals of
the spin chain, and on the value of the external magnetic
field. The ground state for any fermion system is organized
by the filling of all states with negative energies (the Fermi
sea). Excitations are related either to the creation of parti-
cles with positive energies with respect to the Fermi ener-
gy, or to the creation of holes below the Fermi energy. It is
easy to see that for = 0H one branch of the spectrum
,0,1,2kε is gapless, and two others are gapped. The magnetic
field (we consider 0H ≥ here and below) plays the role of
the effective chemical potential for each branch. With the
growth of the value of the magnetic field H, the Fermi level
of the gapless branch goes down, so that above the critical
value of the field 1H the gapless branch becomes gapped.
In the region 1 2H H H≤ ≤ all branches are gapped. Then,
when the value of the magnetic field continues to grow, at
the critical value 2H the second branch becomes gapless.
Finally for > sH H the second branch again becomes
gapped. The third branch of the spectrum is gapped for any
value of the field H. For > sH H the total ground state
magnetization of the trimerized exactly solvable spin 1/2
chain (normalized by 3 )BLgµ has the value 1, while for
1 2< <H H H it is 1/3 (for those values of the field all
branches are gapped). The ground state magnetization
grows with H only for the values of the field 10 H H≤ ≤
and 2 3,H H H≤ ≤ where some of the eigenstates have
negative energies. Thus, the behavior of the magnetization
of the trimerized exactly solvable spin chain is reminiscent
of the one for azurite [7].
We can see that at least three non-equivalent magnetic
centers are present in azurite. Moreover, according to the
neutron scattering, the dispersion exists mostly along one
direction in azurite [10], at least at intermediate tempera-
tures .NT T≥ This is why, to describe thermodynamic
characteristics of azurite we propose to use the phenome-
nological model with the simple Hamiltonian
3
†
, ,,
1
= [ (1/2)]p k j k jk j
j k
b b
=
ε −∑∑ ,
where the energies are
,1 1= cos( ) ,k B Ag H J kε µ −
2 2
,2,3 2
1= 2 cos
2k B B C B Cg H J J J J kε µ ± + + . (5)
Here , ,A B CJ are phenomenological exchange constants,
and 1,2g are effective g-factors to be determined from the
fitting of the characteristics of the model to the values,
obtained from the experiments on azurite. The properties
of the phenomenological model are similar to the exactly
solvable trimerized model. There are three branches of
fermion excitations, and the external magnetic field plays
the role of the effective chemical potential for fermions in
each branch. The illustration of the behavior of the spectra
of the model is shown in Fig. 1. The parameters are taken
from the fitting of the experimental results for the magneti-
zation of azurite for H directed perpendicular to the axis b,
see below. In fact, this phenomenological model describes
the behavior of two independent spin-1/2 chains: the ho-
mogeneous chain and the dimerized one [2], both with the
XX0 magnetic anisotropy. Notice that it is only the phe-
nomenological model. We cannot exactly derive this model
from the microscopical structure of azurite like the ap-
proach of Refs. 10–14. However, the estimations of the
values of the parameters of the phenomenological model,
see below, can be related to the estimations of the ex-
change integrals of azurite using similar considerations, as
in Refs. 7, 10–14. Also, the magnetic anisotropy of the
inter-ion type, present in XX0 spin chains with the excita-
tion spectra Eqs. (5), is not related to the real magnetic
anisotropy of azurite. Here we use just the phenomenologi-
cal theory, similar to the approach of Ref. 18, where the
strongly anisotropic XXZ spin-1/2 chain model together
with the fixed value of the applied effective field was suc-
cessfully used to describe the properties of the isotropic (or
weakly anisotropic) two-leg spin ladder. The dispersion in
our model is along one space direction, which also in
agreement with the behavior of azurite [10]. The phenom-
enological model is reminiscent of (but not equal to) the
monomer–dimer model [9,10]. The main difference is in
the presence of the third branch of excitations, totally ab-
sent for the monomer–dimer model.
Let us now fit the behavior of the low-temperature
magnetization of azurite [7] using our phenomenological
model. It gives for the magnetization per site
Fig. 1. (Color online) Three branches of eigenvalues (black (1),
red (2) and blue (3) lines) of the phenomenological model with JA =
= 1.31 meV, JB = 0.42 meV, and JC = 8.07 meV. Solid lines: H = 0;
dashed lines gµBH = 14 T.
1678 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Phenomenological description of a spin chain system with geometrical frustration of couplings
3
,
0 =1 0
1= tanh
6 2
k j
Bj
M dk
M k T
π ε
π
∑ ∫ , (6)
where Bk is the Boltzmann constant, and 0M is the maximal
value of the magnetization. Figure 2 shows the result of such
a fit for the magnetic field directed perpendicular to the direc-
tion of the chains b. We used the values for the exchange
integrals ( / ) = 11p
BAJ gµ T, ( / ) = 3.5p
BBJ gµ T, and
( / ) = 64.5p
BCJ gµ T (the index p signals about the orienta-
tion of the external magnetic field perpendicular to the chain
direction) with the effective average g-factor for azurite taken
from the ESR measurements 1 2= =g g = 2.06g [19] at the
values of the temperature = 0.08T K, = 1.3T K, and
= 4.2T K. One can see the excellent agreement of the data
of experiment with the results of calculations for the phe-
nomenological model. Notice that the magnetic ordering at
= NT T almost does not affect the higher-energy branches of
the phenomenological model (their contribution to the order-
ing is exponentially small). On the other hand, the difference
in the behavior of the magnetization of the magnetically or-
dered antiferromagnet (see below) for 1H H≤ does not differ
drastically from the behavior of the 1d spin chain for the
temperature T of order of the Néel temperature.
On the other hand, for the magnetic field applied along
the direction of the chains b the results of the fit are present-
ed in Fig. 3. We used the following set of the parameters
( / ) = 15l
A BJ gµ T, ( / ) = 0.7l
B BJ gµ T, and ( / ) = 59l
C BJ gµ T
(the index l indicates the orientation of the external magnetic
field along the chain direction). We also see the very good
agreement of the results of calculations for the phenomeno-
logical model and the data of experiments [7]. Different
values of the parameters of the model can be considered as
follows. In reality there is a magnetic anisotropy in azurite
(to remind, the symmetry of its crystal structure is mono-
clinic, which implies at least a bi-axial magnetic anisotropy;
however for the behavior of some characteristics of azurite it
is enough to consider only the uniaxial inter-ion anisotropy).
Hence, for example, from the values of l
AJ and p
AJ we can
extract the parameters of the effective isotropic exchange
and the magnetic anisotropy = 17.9AJ K and = 2,76z
AJ K,
respectively, which implies that the magnetic anisotropy of
that branch is about 15% of the isotropic exchange. Similar
estimations can be obtained for the effective isotropic ex-
change and magnetic anisotropy of the two other branches
of the phenomenological model, yielding the effective iso-
tropic exchange = 2.9BJ K and = 85.2CJ K with the ef-
fective magnetic anisotropy for the parameters BJ and CJ
equal to 67% and 4.5%, respectively. Notice that relatively
large value of the anisotropy for the parameter z
BJ is con-
nected with the larger uncertainty in the definition of the
second critical field for azurite for H || b. It is worth men-
tioning: About twice larger estimates for the values of cou-
pling constants in our approach, comparing with the earlier
estimations [10–14], are related to the fact that we use the
free fermion model (equivalent to the XX0 spin chain),
for which the critical value of the field is / ,BJ gµ while
for the isotropic Heisenberg antiferromagnetic chain,
used in [10–14], the critical value of the field is 2 / .BJ gµ
Our next step is to calculate the temperature behavior of
the magnetic susceptibility of the phenomenological mo-
Fig. 2. (Color online) The behavior of the magnetization per site of
azurite with the external magnetic field applied perpendicular to b
(red squares [7] T = 0.08 K) and theoretical calculations for the
phenomenological model with the parameters ( / ) = 11 T,p
BAJ gµ
( / ) = 3.5 T,p
BBJ gµ and ( / ) = 64.5 Tp
BCJ gµ and the effective aver-
age g-factors g = 2.06 [19]. The black solid line (1) describes the
case for T = 0.08 K, the dotted green line (2) describes T = 1.3 K,
and the dashed blue line (3) T = 4.2 K.
Fig. 3. (Color online) The behavior of the magnetization per site
of azurite with the external magnetic field applied parallel to b
(red squares [7] T = 1.5 K ) and theoretical calculations for the
phenomenological model with the parameters ( / ) = 15l
A BJ gµ T,
( / ) = 0.7l
B BJ gµ T, and ( / ) = 59l
C BJ gµ T. The black solid
line (1) describes the case for T = 1.5 K, the dotted green line (2)
describes T = 2 K, and the dashed blue line (3) T = 4.2 K.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1679
A.A. Zvyagin
del and to compare it with the data of the experiments in
azurite
22 3
,
=1 0
( )
= cosh .
12 2
k jB
B Bj
g
dk
k T k T
−π ε µ
χ π
∑ ∫ (7)
The temperature dependence of the magnetic susceptibility
of azurite manifests for > NT T the two-maximum be-
havior [7], which is supposed to be one of the characteris-
tic features of frustrated magnetic systems [5]. To describe
the behavior of the susceptibility for the parameters of the
model we use the values obtained for the fitting of the low-
temperature magnetization. The results of such calculations
are presented in Fig. 4 for the magnetic field directed along
b. Here we used the parameters = 11.9AJ K, = 1.05 KBJ
and = 88CJ K, related to the ones in Fig. 3. Notice that we
used the renormalization of the parameters (shift of the
position by approximately 0.46 and the renormalization of
the maximum value by approximately 0.45) caused by the
difference in the behavior of the magnetic susceptibility of
the XX0 chain and the XXZ antiferromagnetic spin 1/2 chain
calculated in the framework of the exact Bethe ansatz solu-
tion using the quantum transfer matrix approach [2,20].
We have included to the total susceptibility the contribu-
tion from low-temperature part, ,NT T≤ i.e., the magnetic
susceptibility of the magnetically ordered phase (for the
experiment in azurite, see Ref. 10), which was calculated
in the framework of the mean field approximation. Here
we have supposed that the spin chains with the first branch
spectra are weakly coupled, which produces antiferromag-
netic ordering with two magnetic sublattices; on the other
hand, the dimerized spin chains with the (gapped) second
and the third branches of excitations yield only exponen-
tially small contribution to the ordering and remain effec-
tively one-dimensional at T TN. It turns out that the phe-
nomenological model describes very well the two-
maximum structure of the temperature dependence of the
magnetic susceptibility of azurite at > NT T (it was not
reproduced well in the previous calculations of the proper-
ties of azurite). The deeper minimum, present in the theo-
retical result for the magnetic susceptibility, with respect to
the data of experiments in azurite [7,10] is related to the
free fermion (XX0) character of our phenomenological mo-
del, compared with the more realistic XXZ-like behavior
[2,20], for which the highs in the temperature behavior of
χ are not so sharp as in the XX0 chain.
Then, in the framework of the phenomenological model
we calculate the magnetic contribution to the specific heat
23
,
2 2
,=1 0
1=
( /2 )12 cosh
k j
m
k j BjB
c dk
k Tk T
π ε
επ
∑ ∫ . (8)
The results of calculations are presented in Fig. 5 (we used
the same set of parameters as in Figs. 3–4). The low-
temperature peak is related to the magnetic ordering at
= 1.85NT K. The contribution for NT T≤ is calculated in
the framework of the mean field approximation, in a way
similar to the calculation of the contribution to the magnet-
Fig. 4. (Color online) The behavior of the magnetic susceptibility
of azurite with the external magnetic field applied parallel to b
(red squares [7]) and theoretical calculations for the phenomeno-
logical model with the parameters (JA = 11.9 K, JB = 1.05 K, and
JC = 88 K (the solid black line). The low-temperature peak is
related to the magnetic ordering at TN = 1.85 K, calculated in the
framework of the mean field approximation.
Fig. 5. (Color online) The behavior of the specific heat of azurite
at H = 0 (red squares [7]) and theoretical calculations for the phe-
nomenological model with the same parameters as in Figs. 3 and
4 (the solid black line (1)). The low-temperature peak is related to
the magnetic ordering at TN = 1.85 K, calculated in the frame-
work of the mean field approximation. The blue dashed line (2) is
the contribution of phonons [10].
1680 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11
Phenomenological description of a spin chain system with geometrical frustration of couplings
ic susceptibility, see above. The higher-temperature part of
the specific heat for azurite is determined by phonons [10];
the estimation of the contribution of phonons is also shown
in Fig. 5. We have also taken into account the difference of
the behavior of the specific heat of the phenomenological
free fermion model (XX0 spin chains) with respect to the
more realistic XXZ kind, from the comparison of the exact
Bethe ansatz results [2,20]. We can see that the phenome-
nological model reproduces well two highs in the tempera-
ture dependence (for T > TN) of the specific heat, related to
the contribution of the first and second and third branches
of eigenstates of the phenomenological model. Again,we
used the renormalization (the position of the maxima, and
their values) caused by the difference between the free
fermion model (equivalent to XX0 spin 1/2 chain) and
more realistic XXZ case. Notice that the weak magnetic
field reduces the values of the highs of the magnetic spe-
cific heat in our phenomenological model, also in agree-
ment with the data of experiments in azurite [10].
Then, according to [21] it is possible to calculate the
temperature dependence of the relative change /∆v v of the
sound veloclty v, caused by the spin-phonon coupling (the
exchange striction) in the framework of the phenomeno-
logical model. Neglecting the inhomogeneous contribu-
tions, we obtain
1 2= ( ) ,C T C∆
− χ χ +
v
v
(9)
where 1C and 2C are the parameters, related to the gradient
of the exchange constants in the spin system (depenent on
the direction and the polarization of the sound wave [21]).
The results are shown in Fig. 6. One can see that the be-
havior of the calculated sound velocity is similar to the
one, observed in azurite for the temperature behavior of
the 22C elastic modulus [22], connected with the one of
/∆v v.
Finally, in the framework of the phenomenological
model we calculate the frequency-magnetic field phase
diagram, related to the observed electron spin resonance
(ESR) signals in azurite [19,23,24]. The ESR dynamical
characteristics of azurite are close to the ones of the neu-
tron scattering [10,25] at k = 0. The calculations of the
resonance frequencies of the ESR in the framework of the
phenomenological model for the 1d situation for > NT T
are performed using the approach similar to Refs. 2, 26. On
the other hand, for < NT T we used the standard approach
[27] for the magnetically ordered two-sublattice anti-
ferromagnet (to remind, we have supposed that the gapless
branch of the phenomenological model plays the major
role in the magnetic ordering due to the weak inter-chain
coupling, yielding two-sublattice antiferromagnet, see
above). The parameters of the exchange and the anisotropy
fields for that two-sublattice ordering are extracted from
the estimations of the exchange constants and the anisotro-
py for the magnetic field behavior of the magnetization,
see above. The results of calculations are presented in
Fig. 7 (we used the values of g-factors from the ESR ex-
periments in azurite [19,23]). We have to distinguish seve-
ral cases. First, for H || b the calculations in the framework
of the phenomenological theory give two ESR lines (the
green long-dashed and dashed-dotted lines), related to the
Fig. 6. The temperature dependence of the relative change of the
sound velocity at H = 0 calculated for the phenomenological model
with the same parameters as in Figs. 3–5. The low-temperature
feature is related to the magnetic ordering at TN = 1.85 K, calcula-
ted in the framework of the mean field approximation.
Fig. 7. (Color online) The magnetic field dependence of the reso-
nance frequencies calculated for the phenomenological model
with the same parameters as in Figs. 1–6. The green dotted-
dashed (1) and long-dashed (2) lines correspond to the contribu-
tion of the second and the third branches of the phenomenological
model. All other lines are connected with the first branch of the
model. The red dashed line (3) is related to the ESR signal for
T > TN and with the AFMR mode for T < TN (see below).
The lines, related to the magnetic ordering at T < TN (the AFMR
modes, the black (4) and blue (5) solid lines and the red dashed
line), are calculated in the framework of the mean field approxi-
mation.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 11 1681
A.A. Zvyagin
gapped branches of the phenomenological model. Similar
lines were observed in the ESR [19,23,24], as well as in the
inelastic neutron scattering [10] both for > NT T [19,23] and
for < NT T [10,24]. Notice that the experiments [10,19,23,24]
observed different slopes (effective g-factors in the ESR)
for H < H1 15 T and for 1> ,H H which was related in
[10] to the change of the local mean fields from the antifer-
romagnetic to the polarized state. The other ESR line (sim-
ilar line was observed [19,23,24] both for > NT T and for
< )NT T is presented as the dashed red line; it is due to the
first branch of the phenomenological model for > NT T
(the antiferromagnetic spin chain). For < NT T that ESR
line survives as the mode of the antiferromagnetic reso-
nance (AFMR) for the anisotropic two-sublattice magnetic
model for the magnetic field, directed perpendicular to the
distinguished by the magnetic anisotropy axis. Notice that
the weak bi-axial anisotropy, present in azurite, produces a
small gap (of order of 30 GHz [23], see also the inelastic
neutron scattering results [25]), which is not taken into
account in our simplified phenomenological model. How-
ever, it is easy to include such a bi-axial anisotropy in the
framework of the free fermion model [2]. That small bi-
axial magnetic anisotropy can slightly renormalize pre-
sented above results for other characteristics of azurite; all
main features, however remain if that anisotropy is smaller
than the uniaxial one. At < NT T our theory also predicts
the AFMR mode (the blue solid line), caused by the order-
ing related to the gapless branch of the phenomenological
model. Similar mode was observed in the ESR experiments
in azurite [19,23,24]. On the other hand, for the direction
of the magnetic field H perpendicular to the b axis, the
gapped ESR modes caused by the gapped branches of the
phenomenological model (the green long-dashed and dash-
ed-dotted lines) survive (both for > NT T and < ).NT T On
the other hand, the gapless branch of the phenomenological
model yields the red dashed line in Fig. 7 for > ,NT T and
two AFMR modes, which exist at < ,NT T shown as the
black and blue solid lines in Fig. 7. The line, similar to the
upper (black) line was observed in Refs. 19,23,24. Regard-
ing the neutron scattering: In fact the authors of Ref. 25
used the approach, similar to our phenomenological free
fermion model (the gapless first branch), to explain theo-
retically results for the low-energy dynamical structure
factor of azurite, which well agree with the observed in
experiment data.
In summary, we have proposed the simple phenomeno-
logical model to calculate various characteristics of the
spin chain system with the geometrical frustration, using
azurite as the main example for comparison. The proposed
phenomenological model describes very well the behavior
of the magnetization, the magnetic susceptibility, the mag-
netic contribution to the specific heat, the exchange-stric-
tion caused renormalization of the sound velocity, and the
dynamical characteristics: some features of the ESR and
neutron scattering. The advantage of the model is in its
simplicity; also it permits to calculate many characteristics
of the spin chain compound from the same grounds, with-
out long-time numerical calculations. On the other hand,
the model does not take into account bound states, present
in spin-1/2 chains with small ratios of the magnetic anisot-
ropy to the isotropic exchange integrals, known from the
exact Bethe ansatz results [2]. Notice that we do not dis-
cuss the origin of the magnetic anisotropy in azurite: We
introduce it by hand, fitting the data, observed in experi-
ments. The symmetry of azurite allows the presence of the
Dzyaloshinskii–Moriya interactions, so that the difference
in the magnetic behavior parallel and perpendicular to the
chain axis b can be caused microscopically by the latter, as
well as by the inter-spin anisotropic exchange interactions.
We expect that our results can be helpful for experiment-
alists, who study magnetic properties of azurite. Also,
similar models can be constructed for other spin chain ma-
terials with possible frustration of spin-spin bonds. For
example, our approach (with different values of exchange
constants) can be applied to the theoretical description of
thermodynamic, magnetic, and dynamic properties of
A3Cu3AlO2(SO4)4 (A = K, Rb, and Cs) [28]. That makes
our approach generic, and important for studies of quasi-
one-dimensional quantum spin liquids, especially due to
the recent progress in fabrication of new magnetic materi-
als with the properties of spin liquids.
I thank S. Zvyagin and R. Moessner for helpful discus-
sions, and G.A. Zvyagina for the support in production of
figures. The support from the Deutsche Forschungsgemein-
schaft via SFB1143, and the Institute for Chemistry of
the V.N. Karazin Kharkov National University is acknow-
ledged.
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