Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain
We consider the antiferromagnetic Heisenberg model on a distorted diamond chain and use the localized-magnon picture adapted to a distorted geometry to discuss some of its high-field low-temperature properties. More specifically, in our study we assume that the partition function for a slightly dist...
Збережено в:
| Дата: | 2012 |
|---|---|
| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2012
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/120310 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain / O. Derzhko, J. Richter, O. Krupnitska // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43702:1-10 — Бібліогр.: 39 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120310 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1203102017-06-12T03:05:28Z Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain Derzhko, O. Richter, J. Krupnitska, O. We consider the antiferromagnetic Heisenberg model on a distorted diamond chain and use the localized-magnon picture adapted to a distorted geometry to discuss some of its high-field low-temperature properties. More specifically, in our study we assume that the partition function for a slightly distorted geometry has the same form as for ideal geometry, though with slightly dispersive one-magnon energies. We also discuss the relevance of such a description to azurite. Ми розглядаємо антиферомагнiтну модель Гайзенберга на деформованому ромбiчному ланцюжку i використовуємо картину локалiзованих магнонiв, пристосовану до деформованої геометрiї, щоб обговорити деякi низькотемпературнi властивостi моделi у сильних полях. Конкретнiше, у нашому дослiдженнi ми вважаємо, що статистична сума у випадку дещо деформованої геометрiї має таку ж форму як i у випадку iдеальної геометрiї, але з трошки дисперсними одномагнонними енергiями. Ми також обговорюємо застосовнiсть такого опису для азуриту. 2012 Article Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain / O. Derzhko, J. Richter, O. Krupnitska // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43702:1-10 — Бібліогр.: 39 назв. — англ. PACS: 75.10.Jm DOI:10.5488/CMP.15.43702 arXiv:1206.1441 http://dspace.nbuv.gov.ua/handle/123456789/120310 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We consider the antiferromagnetic Heisenberg model on a distorted diamond chain and use the localized-magnon picture adapted to a distorted geometry to discuss some of its high-field low-temperature properties. More specifically, in our study we assume that the partition function for a slightly distorted geometry has the same form as for ideal geometry, though with slightly dispersive one-magnon energies. We also discuss the relevance of such a description to azurite. |
| format |
Article |
| author |
Derzhko, O. Richter, J. Krupnitska, O. |
| spellingShingle |
Derzhko, O. Richter, J. Krupnitska, O. Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain Condensed Matter Physics |
| author_facet |
Derzhko, O. Richter, J. Krupnitska, O. |
| author_sort |
Derzhko, O. |
| title |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| title_short |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| title_full |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| title_fullStr |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| title_full_unstemmed |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| title_sort |
semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2012 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120310 |
| citation_txt |
Semiquantitative theory for high-field low-temperature properties of a distorted diamond spin chain / O. Derzhko, J. Richter, O. Krupnitska // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43702:1-10 — Бібліогр.: 39 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT derzhkoo semiquantitativetheoryforhighfieldlowtemperaturepropertiesofadistorteddiamondspinchain AT richterj semiquantitativetheoryforhighfieldlowtemperaturepropertiesofadistorteddiamondspinchain AT krupnitskao semiquantitativetheoryforhighfieldlowtemperaturepropertiesofadistorteddiamondspinchain |
| first_indexed |
2025-07-08T17:38:22Z |
| last_indexed |
2025-07-08T17:38:22Z |
| _version_ |
1837101278139777024 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 4, 43702: 1–10
DOI: 10.5488/CMP.15.43702
http://www.icmp.lviv.ua/journal
Semiquantitative theory for high-field
low-temperature properties of a distorted
diamond spin chain
O. Derzhko1,2,3, J. Richter3, O. Krupnitska2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Department for Theoretical Physics, Ivan Franko National University of Lviv,
12 Drahomanov St., 79005 Lviv, Ukraine
3 Institut für theoretische Physik, Universität Magdeburg, P.O. Box 4120, D–39016 Magdeburg, Germany
Received June 10, 2012, in final form September 6, 2012
We consider the antiferromagnetic Heisenberg model on a distorted diamond chain and use the localized-
magnon picture adapted to a distorted geometry to discuss some of its high-field low-temperature properties.
More specifically, in our study we assume that the partition function for a slightly distorted geometry has the
same form as for ideal geometry, though with slightly dispersive one-magnon energies. We also discuss the
relevance of such a description to azurite.
Key words: diamond spin chain, localized magnons, azurite
PACS: 75.10.Jm
1. Introduction
The concept of localized magnons was introduced some time ago [1, 2] and since then it has been
successfully used to examine the ground-state and low-temperature properties of a wide class of spin
models [3–6] (for a review see reference [7]). Most of the calculations refer to the so-called ideal lattice
geometry which implies a completely dispersionless lowest-energy one-magnon band. However, one can-
not expect that such conditions occur in real-life materials and, therefore, one has to go beyond the case of
ideal geometry dealing with a slightly dispersive lowest-energy one-magnon band, i.e., with almost local-
izedmagnons. Although a systematic quantitative theory of almost localized magnons has not been elabo-
rated so far, it is quite in order to mention here reference [8] that considers a distorted frustrated two-leg
spin ladder and references [9, 10] where an effective Hamiltonian is obtained for a distorted diamond
spin chain [in the context of the magnetic compounds Cu3Cl6(H2O)2·2H8C4SO2 and Cu3(CO3)2(OH)2].
A famous solid-state example of a model compound for a frustrated diamond Heisenberg spin-chain
system is the natural mineral azurite Cu3(CO3)2(OH)2 [11–13]. For another possible experimental candi-
dates see references [14, 15]. High-field magnetization curves for an azurite single crystal have beenmea-
sured below 4.2 K up to about 40 T, see references [11, 12]. The magnetization curve has a clear plateau
at one third of the saturation magnetization. Moreover, the magnetization curve has a very steep part
(although not a perfect vertical jump) between one third of the saturation magnetization and the satura-
tion magnetization. An appropriate magnetic model for azurite [11–13] is a spin-1/2 distorted Heisenberg
diamond chain with four different antiferromagnetic exchange constants J1, J2, J3, and Jm , see figure 1
and sections 2, 4. The ideal geometry supporting the localized-magnon states occurs for the case J1 = J3,
Jm = 0, J2 > 2J1. However, the set of exchange constants obtained from the first-principle density func-
tional computations reads [13]
J1 = 15.51 K, J2 = 33 K, J3 = 6.93 K, Jm = 4.62 K. (1.1)
© O.Derzhko, J.Richter, O.Krupnitska, 2012 43702-1
http://dx.doi.org/10.5488/CMP.15.43702
http://www.icmp.lviv.ua/journal
O.Derzhko, J.Richter, O.Krupnitska
m
m
J
J
J
J
m
m
mm
J
J
,3
,2
,1
1
3 1
3
+1,1
+1,2
+1,3
2
m
Figure 1. (Color online) The distorted diamond spin chain considered in this paper. The ideal diamond
spin chain corresponds to J1 = J3, Jm = 0, J2 > 2J1.
Although the parameter set (1.1) does not satisfy the ideal geometry conditions, it is not very far from
an ideal set. This statement is supported by the measured magnetization curve that resembles the one
predicted by a localized-magnon picture, see [3–7]. Therefore, one may expect that an appropriate mod-
ification of the localized-magnon picture would be capable of describing the high-field low-temperature
properties of azurite.
It should bementioned here that the diamond spin chain with J1 = J3, Jm = 0 is a representative of the
models with local conservation laws, cf., e.g., references [16–20]. Local conservation laws provide a very
special mechanism for trapping the magnons. Therefore, the ideal diamond chain cannot be considered
as a generic spin model with localized magnons. On the other hand, this is probably the most suitable
well known solid-state realization of a localized-magnon system.
Bearing in mind this motivation, in the present paper we discuss one simple route to the high-field
low-temperature thermodynamics of a distorted diamond spin chain which is based on the localized-
magnon picture. After recalling in section 2 the basic points of the standard consideration which is valid
for ideal geometry [3–7], we introduce in section 3 a plausible form of the partition function of the dis-
torted diamond spin chain for small deviations from ideal geometry and calculate thermodynamic quan-
tities in this case. In section 4 we apply this consideration to azurite. We conclude in section 5 with a brief
summary and prospects for further studies.
2. Localized magnons on an ideal diamond chain
In our study we consider the spin-1/2 Heisenberg antiferromagnet with the Hamiltonian
H =
∑
(i j )
Ji j si ·s j −hSz , Sz =
∑
i
sz
i (2.1)
on a N -site diamond chain [17, 18, 21–25], see figure 1. We use standard notations in equation (2.1) and
imply periodic boundary conditions. It is convenient to label the lattice sites with a pair of indeces, where
the first number enumerates the cells (m = 1, . . . ,N = N /3) and the second one enumerates the position
of the site within the cell, see figure 1. Therefore, spin Hamiltonian (2.1) on the distorted diamond-chain
lattice reads
H =
N
∑
m=1
[
J2sm,1 ·sm,2 + J3sm,1 ·sm,3 + J1sm,2 ·sm,3 + J1sm,3 ·sm+1,1 + J3sm,3 ·sm+1,2
+Jm sm,3 ·sm+1,3 −h
(
sz
m,1 + sz
m,2 + sz
m,3
)]
. (2.2)
Note that the spin Hamiltonian commutes with Sz . Therefore, we may consider the subspaces with dif-
ferent values of Sz = N /2, N /2−1, . . . separately. Moreover, we may assume for brevity at first h = 0 and
then trivially add the contribution of the Zeeman term.
We begin with the ideal geometry case. If J1 = J3, Jm = 0, the one-magnon (i.e., Sz = 3N /2−1) en-
ergy bands εi (κ), i = 1,2,3, κ = 2πn/N , n = −N /2,−N /2+1, . . . ,N /2−1 (we assume without loss of
generality that N is even) follow from the equation
det
−J1 − J2
2
−εi (κ) J2
2
J1
2
(
1+eiκ
)
J2
2
−J1 − J2
2
−εi (κ) J1
2
(
1+eiκ
)
J1
2
(
1+e−iκ
) J1
2
(
1+e−iκ
)
−2J1 −εi (κ)
= 0 (2.3)
43702-2
Distorted diamond spin chain at high fields and low temperatures
and have the form
ε1(κ) =−J1 − J2, ε2,3(κ) =−
3J1
2
∓
J1
2
p
5+4cosκ . (2.4)
For J1/J2 < 1/2 (from now on we assume that this inequality holds) the flat band ε1(κ) = ε1 becomes
the lowest-energy one. The states from the flat band can be visualized as singlets located on the vertical
bonds J2, see figure 1.
Many-magnon (i.e., Sz = 3N /2−2, . . . ,3N /2−N ) ground states can be constructed by filling the verti-
cal bonds (traps) bymagnons. These independent localized-magnon states dominate the low-temperature
properties of the spin system in a magnetic field h around the saturation value hsat =−ε1 = J1 + J2. The
partition function of the diamond spin chain in this regime reads [3–7]
Z (T,h, N ) = exp
(
−
EFM − h
2
N
T
)
N
∑
n=0
gN (n)en(hsat−h)/T (2.5)
(we set kB = 1) with EFM = (J2/4+ J1)N and gN (n) =N !/(n!(N −n)!). Therefore,
Z (T,h, N ) = exp
(
−
EFM − h
2
N
T
)
[
1+e(hsat−h)/T
]N
(2.6)
and the free energy per cell reads
f (T,h) = lim
N →∞
−T ln Z (T,h, N )
N
=
EFM − h
2 N
N
−T ln
[
1+e(hsat−h)/T
]
. (2.7)
The magnetization per cell can be obtained from equation (2.7) according to the standard relation
m(T,h) =−
∂ f (T,h)
∂h
=
3
2
−
e(hsat−h)/T
1+e(hsat−h)/T
. (2.8)
Some further calculations of the high-field low-temperature thermodynamic quantities can be found
in [3–7].
3. Almost localized magnons on a distorted diamond chain
Now we assume a small “distortion” |J1 − J3|/J2 ≪ 1, Jm/J2 ≪ 1. Our aim is to construct an effective
description of the low-temperature thermodynamics of the distorted diamond spin chain (2.2) around
h1 = (J1 + J3)/2+ J2.
We begin with the one-magnon energy bands εi (κ), which follows now from the equation
det
−J − J2
2 −εi (κ)
J2
2
J
2
(
1+eiκ
)
− δ
2
(
1−eiκ
)
J2
2 −J − J2
2 −εi (κ) J
2
(
1+eiκ
)
+ δ
2
(
1−eiκ
)
J
2
(
1+e−iκ
)
− δ
2
(
1−e−iκ
)
J
2
(
1+e−iκ
)
+ δ
2
(
1−e−iκ
)
−2J − Jm (1−cosκ)−εi (κ)
= 0.
(3.1)
Here, we have introduced the following notations
J =
J1 + J3
2
, δ=
J1 − J3
2
. (3.2)
Equation (3.1) yields a cubic equation for εi (κ),
−
[
J +
J2
2
+εi (κ)
]2 [
2J +2Jm sin2 κ
2
+εi (κ)
]
+ J2
(
J 2 cos2 κ
2
−δ2 sin2 κ
2
)
(3.3)
+ [2J + J2 +2εi (κ)]
(
J 2 cos2 κ
2
+δ2 sin2 κ
2
)
+
J 2
2
4
[
2J +2Jm sin2 κ
2
+εi (κ)
]
= 0.
43702-3
O.Derzhko, J.Richter, O.Krupnitska
In the ideal geometry limit J = J1, δ= 0, Jm = 0, equation (3.3) becomes
−
[
J1 +
J2
2
+εi (κ)
]2
[2J1 +εi (κ)]+2[J1 + J2 +εi (κ)] J 2
1 cos2 κ
2
+
J 2
2
4
[2J1 +εi (κ)]= 0. (3.4)
Obviously, ε1(κ) = ε1 =−J1 − J2 (2.4) satisfies equation (3.4).
We pass to the distorted case assuming δ/J2 to be small. Inserting
ε1(κ) =−J − J2 +ε(2)
1 δ2 + . . . (3.5)
into equation (3.3) and collecting the terms of the order δ2 we find
ε(2)
1 =
2J2 sin2 κ
2
J2(J − J2)+2J 2 cos2 κ
2
+2J2 Jm sin2 κ
2
. (3.6)
Thus, up to the terms of the order δ2, we have
ε1(κ) = −J − J2 +
1
1− J
J2
− (1+cosκ) J2
J2
2
− (1−cosκ)
Jm
J2
(J1 − J3)2
4J2
(−1+cosκ)+ . . . (3.7)
= −h1 +
1
1− J
J2
− Jm
J2
− J2
J2
2
+
(
Jm
J2
− J2
J2
2
)
cosκ
(J1 − J3)2
4J2
(−1+cosκ)+ . . . .
Clearly, the flat band ε1 becomes dispersive if δ , 0. Interestingly, as it follows from equation (3.7) and
can be expected from the coupling geometry in figure 1, the exchange coupling Jm affects the flat band
only if J1 , J3 (i.e., Jm cannot spoil the flat band alone). Furthermore, assuming in addition that J/J2,
Jm/J2 are also small, we can rewrite equation (3.7) in the form
ε1(κ) ≈−h1 −
(J1 − J3)2
4J2
+
(J1 − J3)2
4J2
cosκ . (3.8)
In figure 2 we compare ε1(κ) obtained from equations (3.3), (3.7), and (3.8) for two particular sets of
parameters J1 = 0.85, J2 = 3, J3 = 1.15, Jm = 0 and Jm = 0.2, cf. equation (1.1).
Nowwe return to equation (2.6) which takes into account the contribution of the many-magnon states
to the thermodynamics for ideal geometry when the lowest-energy one-magnon band is completely flat.
Evidently, equation (2.6) can be considered as
Z (T,h, N ) = exp
(
−
EFM − h
2 N
T
)
∏
κ
{
1+e[−ε1(κ)−h]/T
}
, (3.9)
Figure 2. (Color online) ε1(κ) obtained from equation (3.3) (line), equation (3.7) (large squares), and equa-
tion (3.8) (small squares) for the set of parameters J1 = 0.85, J2 = 3, J3 = 1.15, Jm = 0 (left hand panel),
and Jm = 0.2 (right hand panel).
43702-4
Distorted diamond spin chain at high fields and low temperatures
where−ε1(κ) =−ε1 = hsat for the flat-band case. Our basic assumption for the distorted case is as follows:
We assume that the partition function for a slightly distorted diamond spin chain still has the form given
in equation (3.9), though with the one-magnon energies given in equation (3.7). Preserving the structure
of the partition function, we adopt the hard-monomer rule, though facing now the hard monomers with
slightly dispersive energies. Thus, we have
Z (T,h, N ) = exp
(
−
EFM − h
2
N
T
)
∏
κ
{
1+e[−ε1(κ)−h]/T
}
(3.10)
with ε1(κ) given in equation (3.7). As a result, the free energy per cell reads
f (T,h) = lim
N →∞
−T ln Z (T,h, N )
N
=
EFM − h
2
N
N
−
T
2π
π
∫
−π
dκ ln
{
1+e[−ε1(κ)−h]/T
}
(3.11)
with ε1(κ) given in equation (3.7). A further improved (but more complicated) result will be obtained if
one utilizes for ε1(κ) the corresponding solution of cubic equation (3.3), see section 4.
It is worth noting that the second term in equation (3.11) with ε1(κ) given by equation (3.8) corre-
sponds to the free energy per site (up to an unimportant constant h/2) of the spin-1/2 X X chain in a
transverse field [26–28] defined by the Hamiltonian
H=−h
N
∑
m=1
τz
m +J
N
∑
m=1
(
τx
mτx
m+1 +τ
y
mτ
y
m+1
)
(3.12)
with
h=−h+h1 +
(J1 − J3)2
4J2
, J=
(J1 − J3)2
4J2
. (3.13)
This finding can be compared with the effective Hamiltonian for a distorted diamond spin chain derived
in references [9, 10] within the second-order perturbation theory in |J1|/J2, |J3|/J2. This effective Hamil-
tonian also corresponds to the spin-1/2 X X chain in a transverse field (3.12) with the parameters
h= h−h1 −
(J1 − J3)2
4J2
, J=
(J1 − J3)2
4J2
, (3.14)
see equations (7) and (8) of reference [10].
Furthermore, within the adopted approach it is easy to obtain the magnetization. Using (3.11) one
gets
m(T,h) =
3
2
−
1
2π
π
∫
−π
dκ
e[−ε1(κ)−h]/T
1+e[−ε1(κ)−h]/T
= 1−
1
4π
π
∫
−π
dκtanh
−ε1(κ)−h
2T
. (3.15)
Here, ε1(κ) is given by equation (3.7) [or by the corresponding solution of cubic equation (3.3)]. If one
assumes for ε1(κ) the simpler formula (3.8), the second term in the r.h.s. of equation (3.15) corresponds
to the behavior of the (transverse) magnetization of the spin-1/2 X X chain in a transverse field.
In figure 3 we compare the exact diagonalization data with our predictions from the approximate
analytical theory. Exact diagonalization data refer to finite systems of N = 18 sites. The ground-state
magnetization curve for finite systems consists of the steps which become smeared out as the tempera-
ture increases. Although analytical predictions according to equation (3.15) refer to thermodynamically
large systems with N →∞, we may reproduce the finite-N magnetization replacing the integral by the
sum, i.e.,
∫π
−π dκ(. . .)/(2π) →
∑
κ(. . .)/N . Comparing the exact diagonalization data and approximate an-
alytical calculations, one concludes the following. Equation (3.8) (it corresponds to the effective spin-1/2
X X chain in a transverse field introduced in references [9, 10]) only qualitatively reproduces the ex-
act diagonalization data for |J1 − J3|/J2 = 0.12, 0.10; the value of the saturation field is underestimated,
the end field for the 1/3 plateau equals h1 and is overestimated. Equation (3.7) works much better for
43702-5
O.Derzhko, J.Richter, O.Krupnitska
|J1− J3|/J2 = 0.12, 0.10 and provides a good agreement with the exact diagonalization data above and just
below the saturation field. However, the end field for the 1/3 plateau again equals h1 and around h1 both
approximations (3.7) and (3.8) exhibit similar shortcomings. For a smaller value of |J1 − J3|/J2 = 0.06, the
agreement between both approximations and with the exact diagonalization data becomes better. This is
not surprising since equation (3.8) corresponds to the second-order perturbation theory in |J1|/J2, |J3|/J2,
see references [9, 10].
Figure 3. (Color online) Magnetization curve
m(T,h)/3 vs h for the distorted diamond chain
[J1 = 0.85, J2 = 2.5 (upper panel), J2 = 3 (middle
panel), J2 = 5 (lower panel), J3 = 1.15, Jm = 0] at
T = 0.001 (thick lines, squares) and T = 0.01 (thin
line, triangles). Exact diagonalization data for N =
18: lines; approximate analytical theorywhich uses
equation (3.7): large empty (N → ∞) and filled
(N = 6) symbols; approximate analytical theory
which uses equation (3.8): small empty (N → ∞)
and filled (N = 6) symbols.
We conclude this section by making some gen-
eral remarks concerning the suggested approxima-
tion (3.10) based on the discussed results. Appar-
ently, this educated ansatz which originates from
the localized-magnon theory works well when the
number of magnons is small (i.e., around the satu-
ration field) provided the one-magnon energies are
reproduced correctly [compare ε1(κ) obtained from
equations (3.3), (3.7), and (3.8) and shown in the left
hand panel of figure 2]. If the number of magnons
becomes large (i.e., when approaching the end field
for the 1/3 plateau) a simple hard-core rule fails to
describe the system since the incompletely local-
ized magnons may exhibit more complicated inter-
actions.
4. Magnetization curves for azurite
The natural mineral azurite Cu3(CO3)2(OH)2 has
been a subject of intensive experimental and the-
oretical studies recently. After the discovery of a
plateau at 1/3 of the saturation value at the low-
temperature magnetization curve [11, 12], there
were other experiments concerning the magnetic
properties of azurite, e.g., measurements of the
magnetic susceptibility, the specific heat, the struc-
ture of the 1/3 plateau determined by nuclear mag-
netic resonance, inelastic neutron scattering on
the 1/3 plateau etc., see reference [10] and refer-
ences therein. Applying different theoretical tools,
it was demonstrated that a generalized diamond
spin chain is consistent with these experiments
and thus, azurite Cu3(CO3)2(OH)2 may be viewed as
a model substance for a frustrated diamond spin
chain, see reference [10] and references therein. As
mentioned in section 1, the magnetic properties of
azurite Cu3(CO3)2(OH)2 can be described by a dis-
torted diamond Heisenberg spin chain with a set
of exchange couplings given in equation (1.1) and
the gyromagnetic ratio g = 2.06, see references [9,
10, 13]. The reduced field h in equation (2.1) or
equation (2.2) is related to the physical field H by
h = gµBH withµB ≈ 0.67171 K/T in the units where
kB = 1, see reference [9].
In what follows we discuss the high-field part
of the low-temperature magnetization curve [11, 12] which exhibits almost a direct transition from the
plateau at 1/3 of the saturation value to the saturation value at fields slightly above 30 T and tempera-
43702-6
Distorted diamond spin chain at high fields and low temperatures
tures about 0.1 K. This characteristic feature of the magnetization curve may be viewed as a remnant of
localized magnons which dominate the high-field low-temperature properties of the ideal diamond spin
chain, see figure 1. Herein belowwe use the approximate analytical description based on equation (3.10).
Another approach to the calculation of magnetization which is based on variational mean-field-like treat-
ment with the help of Gibbs-Bogolyubov inequality has been reported recently in reference [29].
Bearing in mind the localized-magnon picture emerging for the ideal diamond spin chain, we may
expect for azurite that the lowest-energy states having different Sz = N /2, . . . , N /2−N at a magnetic
field around the saturation field, have almost the same value of energy. This will obviously produce a very
steep part at the low-temperature magnetization around the saturation field. However, due to a non-ideal
geometry, these lowest-energy states are not localized magnons (yielding a perfect jump in the ground-
state magnetization curve), but almost localized magnons and their effect on high-field low-temperature
thermodynamics can be estimated using equation (3.10).
Before applying the approach based on equation (3.10) to the model of azurite we have to make the
following remarks. First of all we note that according to equation (1.1) (J1 − J3)/J2 = 0.26, that is not so
small in comparison with the cases reported in figure 3 (recall we had |J1 − J3|/J2 = 0.12, 0.10, 0.06 for
the upper, middle and lower panels, respectively). Moreover, now Jm /J2 = 0.14 , 0. Clearly, under these
conditions we may question the accuracy of the approach based on equation (3.10). Anyway, after insert-
ing the Hamiltonian parameters for azurite into equation (3.15) with ε1(κ) following from equation (3.3)
(huge empty symbols), equation (3.7) (large empty symbols) or equation (3.8) (small empty symbols) we
have obtained the results shown in figure 4, which can be compared with experimental data of refer-
ence [12], see figure 2 in this paper. We also report the corresponding exact diagonalization data for
N = 18.
Figure 4. (Color online) Magnetization curve m(T,h)/3 vs H for the distorted diamond chain with a set
of exchange constants given in equation (1.1) and the value of the gyromagnetic ratio g = 2.06 at T = 0.08
K (left hand panel) and T = 1.3 K (right hand panel). Exact diagonalization data for N = 18: lines; approx-
imate analytical theory which uses equation (3.3): huge empty symbols; approximate analytical theory
which uses equation (3.7): large empty symbols; approximate analytical theory which uses equation (3.8):
small empty symbols.
At the low temperature T = 0.08 K (left hand panel in figure 4) the obtained results are in a rea-
sonable agreement with experimental data, see figure 2 in reference [12], and the exact diagonalization
data. More precisely, the approximate analytical result which uses ε1(κ) from equation (3.3) agrees per-
fectly with the exact diagonalization data above 33.5 T, though yields the 1/3 plateau already below 32 T,
whereas the exact diagonalization prediction is about 31 T. Magnetization curves with ε1(κ) from equa-
tion (3.7) or equation (3.8) reproduce the exact diagonalization data only qualitatively, yielding either a
larger or a smaller value of the saturation field. Moreover, slightly above 32 T, approximate analytical
results based on equations (3.3), (3.7), (3.8) coincide. At the temperature T = 1.3 K (right hand panel in
figure 4) our calculations show almost no traces of the step-like part nicely seen at T = 0.08 K (left hand
panel in figure 4). Furthermore, at this temperature (T = 1.3 K) all approximate results are closer to each
other and to the exact diagonalization data.
43702-7
O.Derzhko, J.Richter, O.Krupnitska
5. Conclusions
To summarize, we have considered the high-field low-temperature properties of a distorted diamond
spin chain using a localized-magnon picture. The free energy f (T,h) relevant in this regime is given in
equation (3.11) with ε1(κ) determined from cubic equation (3.3),
ε3
i (κ)+aε2
i (κ)+bεi (κ)+c = 0, (5.1)
a = 4J + J2 +2Jm sin2 κ
2
,
b =−2
(
J 2 cos2 κ
2
+δ2 sin2 κ
2
)
+2(2J + J2)
(
J + Jm sin2 κ
2
)
+ J (J + J2) ,
c =−(2J + J2)
(
J 2 cos2 κ
2
+δ2 sin2 κ
2
)
− J2
(
J 2 cos2 κ
2
−δ2 sin2 κ
2
)
+2J (J + J2)
(
J + Jm sin2 κ
2
)
,
i.e.,
ε1(κ) =−2
√
−
p
3
cos
α−π
3
−
a
3
, p =−
a2
3
+b , q =
2a3
27
−
ab
3
+c , cosα=−
q
2
√
− p3
27
. (5.2)
For small δ/J2 instead of equation (5.2) we can take ε1(κ) from equation (3.7). Furthermore, assuming
that J/J2, Jm/J2 are also small, we arrive at equation (3.8). Although equation (5.2) for ε1(κ) provides the
best results, equation (3.8) for ε1(κ) (valid for J/J2 ≪ 1, Jm/J2 ≪ 1) permits a very transparent interpre-
tation of the thermodynamics in terms of the emergent spin-1/2 X X chain in a transverse field (3.12) and
links our results to the ones obtained earlier within a completely different approach [9, 10].
The effective description resembles, to some extent, the spin-1/2 transverse X X chain theory: Hard-
core bosons (spins 1/2) mimic the hard-monomer rule whereas a nonzero X X exchange coupling is
related to a small dispersion of the former flat one-magnon band. The elaborated description can re-
produce the basic features of the high-field magnetization process at low temperatures even quantita-
tively. It might be interesting to consider other properties such as the low-temperature entropy, specific
heat or the magnetocaloric effect around the saturation field within the suggested scheme, cf. also refer-
ence [10]. From reference [4] we know that deviations from ideal geometry may produce an interesting
low-temperature behavior, e.g., of the entropy around the saturation field.
It seems quite evident that such a description can be applied to other spin systems of the hard-
monomer universality class [3–7], e.g., the dimer-plaquette chain [19, 20] or the two-dimensional square-
kagome lattice [30, 31]. Another interesting question concerns the applicability of such an approximate
approach to distorted spin systems of other universality classes [3–7], in particular, of the hard-dimer
universality class. Moreover, it might be interesting to use a similar scheme to analyze the high-field
low-temperature thermodynamics of the frustrated triangular spin-tube compound considered in ref-
erence [32] extending the ideal geometry description of references [33, 34]. Last but not least, we may
mention a consideration of distorted electron models [35–38].
Finally it should be stressed that in spite the fact that the suggested approach provides a semiquanti-
tative description of the high-field low-temperature properties of a distorted diamond spin chain, it is not
clear how it can be systematically improved. From this point of view, another systematic approach, e.g., a
perturbation theory (though not with respect to small |J1|/J2, |J3|/J2 but with respect to small |J1− J3|/J2),
is required. Moreover, to achieve a better agreement with experimental data for azurite, one apparently
has to take the three-dimensional coupling geometry of this compound into account.
Acknowledgements
The numerical calculations were performed using J. Schulenburg’s spinpack [39]. The present study
was supported by the DFG (project RI615/21-1). O. D. acknowledges the kind hospitality of the University
of Magdeburg in the spring of 2012. O. D. would like to thank the Abdus Salam International Centre for
Theoretical Physics (Trieste, Italy) for partial support of these studies through the Senior Associate award.
43702-8
Distorted diamond spin chain at high fields and low temperatures
References
1. Schnack J., Schmidt H.-J., Richter J., Schulenburg J., Eur. Phys. J. B, 2001, 24, 475; doi:10.1007/s10051-001-8701-6.
2. Schulenburg J., Honecker A., Schnack J., Richter J., Schmidt H.-J., Phys. Rev. Lett., 2002, 88, 167207;
doi:10.1103/PhysRevLett.88.167207.
3. Zhitomirsky M.E., Tsunetsugu H., Phys. Rev. B, 2004, 70, 100403(R); doi:10.1103/PhysRevB.70.100403.
4. Derzhko O., Richter J., Phys. Rev. B, 2004, 70, 104415; doi:10.1103/PhysRevB.70.104415.
5. Zhitomirsky M.E., Tsunetsugu H., Prog. Theor. Phys. Suppl., 2005, 160, 361; doi:10.1143/PTPS.160.361.
6. Derzhko O., Richter J., Eur. Phys. J. B, 2006, 52, 23; doi:10.1140/epjb/e2006-00273-y.
7. Derzhko O., Richter J., Honecker A., Schmidt H.-J., Fiz. Nizk. Temp., 2007, 33, 982 [Low Temp. Phys., 2007, 33, 745;
doi:10.1063/1.2780166].
8. Fouet J.-B., Mila F., Clarke D., Youk H., Tchernyshyov O., Fendley P., Noack R.M., Phys. Rev. B, 2006, 73, 214405;
doi:10.1103/PhysRevB.73.214405.
9. Honecker A., Läuchli A., Phys. Rev. B, 2001, 63, 174407; doi:10.1103/PhysRevB.63.174407.
10. Honecker A., Hu S., Peters R., Richter J., J. Phys.: Condens. Matter, 2011, 23, 164211;
doi:10.1088/0953-8984/23/16/164211.
11. Kikuchi H., Fujii Y., Chiba M., Mitsudo S., Idehara T., Tonegawa T., Okamoto K., Sakai T., Kuwai T., Ohta H., Phys.
Rev. Lett., 2005, 94, 227201; doi:10.1103/PhysRevLett.94.227201.
12. Kikuchi H., Fujii Y., Chiba M., Mitsudo S., Idehara T., Tonegawa T., Okamoto K., Sakai T., Kuwai T., Kindo K.,
Matsuo A., Higemoto W., Nishiyama K., Horvatić M., Bertheir C., Prog. Theor. Phys. Suppl., 2005, 159, 1;
doi:10.1143/PTPS.159.1.
13. Jeschke H., Opahle I., Kandpal H., Valenti R., Das H., Saha-Dasgupta T., Janson O., Rosner H., Brühl A., Wolf B.,
Lang M., Richter J., Hu S., Wang X., Peters R., Pruschke T., Honecker A., Phys. Rev. Lett., 2011, 106, 217201;
doi:10.1103/PhysRevLett.106.217201.
14. Mo X., Etheredge K.M.S., Hwu S.-J., Huang Q., Inorg. Chem., 2006, 45, 3478; doi:10.1021/ic060292q.
15. Mole R.A., Stride J.A., Henry P.F., Hoelzel M., Senyshyn A., Alberola A., Garcia C.J.G., Raithby P.R., Wood P.T., Inorg.
Chem., 2011, 50, 2246; doi:10.1021/ic101897a.
16. Gelfand M.P., Phys. Rev. B, 1991, 43, 8644; doi:10.1103/PhysRevB.43.8644.
17. Niggemann H., Uimin G., Zittartz J., J. Phys.: Condens. Matter, 1997, 9, 9031; doi:10.1088/0953-8984/9/42/017.
18. Niggemann H., Uimin G., Zittartz J., J. Phys.: Condens. Matter, 1998, 10, 5217; doi:10.1088/0953-8984/10/23/021.
19. Ivanov N.B., Richter J., Phys. Lett. A, 1997, 232, 308; doi:10.1016/S0375-9601(97)00374-5.
20. Schulenburg J., Richter J., Phys. Rev. B, 2002, 65, 054420; doi:10.1103/PhysRevB.65.054420.
21. Takano K., Kubo K., Sakamoto H., J. Phys.: Condens. Matter, 1996, 8, 6405; doi:10.1088/0953-8984/8/35/009.
22. OkamotoK., Tonegawa T., KaburagiM., J. Phys.: Condens. Matter, 2003, 15, 5979; doi:10.1088/0953-8984/15/35/307.
23. Čanová L., Strečka J., Jaščur M., J. Phys.: Condens. Matter, 2006, 18, 4967; doi:10.1088/0953-8984/18/20/020.
24. Verkholyak T., Strečka J., Jaščur M., Richter J., Eur. Phys. J. B, 2011, 80, 433; doi:10.1140/epjb/e2011-10681-5.
25. Verkholyak T., Strečka J., Jaščur M., Richter J., Acta Phys. Pol. A, 2010, 118, 978.
26. Lieb E., Schultz T., Mattis D., Ann. Phys. (N.Y.), 1961, 16, 407; doi:10.1016/0003-4916(61)90115-4.
27. Katsura S., Phys. Rev., 1962, 127, 1508; doi:10.1103/PhysRev.127.1508.
28. Katsura S., Phys. Rev., 1963, 129, 2835; doi:10.1103/PhysRev.129.2835.4.
29. Ananikian N., Lazaryan H., Nalbandyan M., Eur. Phys. J. B, 2012, 85, 223; doi:10.1140/epjb/e2012-30289-5.
30. Siddharthan R., Georges A., Phys. Rev. B, 2002, 65, 014417; doi:10.1103/PhysRevB.65.014417.
31. Richter J., Schulenburg J., Tomczak P., Schmalfuß D., Condens. Matter Phys., 2009, 12, 507;
doi:10.5488/CMP.12.3.507.
32. Ivanov N.B., Schnack J., Schnalle R., Richter J., Kögerler P., Newton G.N., Cronin L., Oshima Y., Nojiri H., Phys.
Rev. Lett., 2010, 105, 037206; doi:10.1103/PhysRevLett.105.037206.
33. Maksymenko M., Derzhko O., Richter J., Acta Phys. Pol. A, 2011, 119, 860.
34. Maksymenko M., Derzhko O., Richter J., Eur. Phys. J. B, 2011, 84, 397; doi:10.1140/epjb/e2011-20706-8.
35. Derzhko O., Honecker A., Richter J., Phys. Rev. B, 2007, 76, 220402; doi:10.1103/PhysRevB.76.220402.
36. Derzhko O., Honecker A., Richter J., Phys. Rev. B, 2009, 79, 054403; doi:10.1103/PhysRevB.79.054403.
37. Derzhko O., Richter J., Honecker A., Maksymenko M., Moessner M., Phys. Rev. B, 2010, 81, 014421;
doi:10.1103/PhysRevB.81.014421.
38. Derzhko O., Maksymenko M., Richter J., Honecker A., Moessner M., Acta Phys. Pol. A, 2010, 118, 736.
39. http://www-e.uni-magdeburg.de/jschulen/spin/
43702-9
http://dx.doi.org/10.1007/s10051-001-8701-6
http://dx.doi.org/10.1103/PhysRevLett.88.167207
http://dx.doi.org/10.1103/PhysRevB.70.100403
http://dx.doi.org/10.1103/PhysRevB.70.104415
http://dx.doi.org/10.1143/PTPS.160.361
http://dx.doi.org/10.1140/epjb/e2006-00273-y
http://dx.doi.org/10.1063/1.2780166
http://dx.doi.org/10.1103/PhysRevB.73.214405
http://dx.doi.org/10.1103/PhysRevB.63.174407
http://dx.doi.org/10.1088/0953-8984/23/16/164211
http://dx.doi.org/10.1103/PhysRevLett.94.227201
http://dx.doi.org/10.1143/PTPS.159.1
http://dx.doi.org/10.1103/PhysRevLett.106.217201
http://dx.doi.org/10.1021/ic060292q
http://dx.doi.org/10.1021/ic101897a
http://dx.doi.org/10.1103/PhysRevB.43.8644
http://dx.doi.org/10.1088/0953-8984/9/42/017
http://dx.doi.org/10.1088/0953-8984/10/23/021
http://dx.doi.org/10.1016/S0375-9601(97)00374-5
http://dx.doi.org/10.1103/PhysRevB.65.054420
http://dx.doi.org/10.1088/0953-8984/8/35/009
http://dx.doi.org/10.1088/0953-8984/15/35/307
http://dx.doi.org/10.1088/0953-8984/18/20/020
http://dx.doi.org/10.1140/epjb/e2011-10681-5
http://dx.doi.org/10.1016/0003-4916(61)90115-4
http://dx.doi.org/10.1103/PhysRev.127.1508
http://dx.doi.org/10.1103/PhysRev.129.2835.4
http://dx.doi.org/10.1140/epjb/e2012-30289-5
http://dx.doi.org/10.1103/PhysRevB.65.014417
http://dx.doi.org/10.5488/CMP.12.3.507
http://dx.doi.org/10.1103/PhysRevLett.105.037206
http://dx.doi.org/10.1140/epjb/e2011-20706-8
http://dx.doi.org/10.1103/PhysRevB.76.220402
http://dx.doi.org/10.1103/PhysRevB.79.054403
http://dx.doi.org/10.1103/PhysRevB.81.014421
http://www-e.uni-magdeburg.de/jschulen/spin/
O.Derzhko, J.Richter, O.Krupnitska
Напiвкiлькiсна теорiя низькотемпературних властивостей
деформованого ромбiчного спiнового ланцюжка
у сильних полях
Олег Держко1,2,3, Йоганес Рiхтер3, Олеся Крупнiцька2
1 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького, 1, 79011 Львiв, Україна
2 Кафедра теоретичної фiзики Львiвського нацiонального унiверситету iм. Iвана Франка,
вул. Драгоманова, 12, 79005 Львiв, Україна
3 Iнститут теоретичної фiзики, Унiверситет Магдебурга, D-39016 Магдебург, Нiмеччина
Ми розглядаємо антиферомагнiтну модель Гайзенберга на деформованому ромбiчному ланцюжку i вико-
ристовуємо картину локалiзованих магнонiв, пристосовану до деформованої геометрiї, щоб обговорити
деякi низькотемпературнi властивостi моделi у сильних полях. Конкретнiше, у нашому дослiдженнi ми
вважаємо, що статистична сума у випадку дещо деформованої геометрiї має таку ж форму як i у випад-
ку iдеальної геометрiї, але з трошки дисперсними одномагнонними енергiями. Ми також обговорюємо
застосовнiсть такого опису для азуриту.
Ключовi слова: ромбiчний спiновий ланцюжок, локалiзованi магнони, азурит
43702-10
Introduction
Localized magnons on an ideal diamond chain
Almost localized magnons on a distorted diamond chain
Magnetization curves for azurite
Conclusions
|