Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
Theoretical description of magnetic properties of antiferromagnetic state, induced by the longitudinal magnetic eld, in the Van Vleck singlet magnet with single-ion anisotropy of easy-plane type and ion spin S = 1 is proposed. It is shown that the quantum phase transition to the antiferroma...
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Інститут фізики конденсованих систем НАН України
2008
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| Цитувати: | Quantum phase transition: Van Vleck antiferromagnet in a magnetic field / I.M. Ivanova, V.M. Kalita, V.O. Pashkov, V.M. Loktev // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 509-522. — Бібліогр.: 42 назв. — англ. |
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Ivanova, I.M. Kalita, V.M. Pashkov, V.O. Loktev, V.M. 2017-06-06T13:39:12Z 2017-06-06T13:39:12Z 2008 Quantum phase transition: Van Vleck antiferromagnet in a magnetic field / I.M. Ivanova, V.M. Kalita, V.O. Pashkov, V.M. Loktev // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 509-522. — Бібліогр.: 42 назв. — англ. 1607-324X PACS: 75.10.-b, 75.10.Jm, 75.30.Gw, 75.30.Kz, 75.50.Ee DOI:10.5488/CMP.11.3.509 https://nasplib.isofts.kiev.ua/handle/123456789/119337 Theoretical description of magnetic properties of antiferromagnetic state, induced by the longitudinal magnetic eld, in the Van Vleck singlet magnet with single-ion anisotropy of easy-plane type and ion spin S = 1 is proposed. It is shown that the quantum phase transition to the antiferromagnetic state is connected with the spontaneous appearance of spin polarization in the easy plane. Spin polarization of the ground nondegenerated state proves to be the order parameter of such a transition and the Landau thermodynamic approach can be employed for its (transition) description. The magnetic properties which include the eld behavior of magnetization and magnetic susceptibility of the antiferromagnetic phase in the elds of different directions are studied. An attempt is made to qualitatively compare the obtained results with the available experimental data. Проведено теоретичний опис iндукованого магнiтним полем квантового фазового переходу у антиферомагнiтий стан у ван-флекiвському антиферомагнетику з легкоплощинним типом однойонної анiзотропiї i величиною спiнiв магнiтних йонiв S = 1. Показано, що при магнiтному фазовому переходi параметром порядку є величина спiнової поляризацiї основного стану, а для опису цього переходу застосовна термодинамiчна теорiя фазових переходiв Ландау. Вивченi магнiтнi властивостi iндукованої магнiтним полем антиферомагнiтної фази, проаналiзованi польовi залежностi намагнiченостi i магнiтної сприйнятливостi. Зроблена спроба якiсного порiвняння отриманих результатiв з наявними експериментальними даними. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Quantum phase transition: Van Vleck antiferromagnet in a magnetic field Квантовий фазовий перехiд: ван-флекiвський антиферомагнетик у магнiтному полi Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field |
| spellingShingle |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field Ivanova, I.M. Kalita, V.M. Pashkov, V.O. Loktev, V.M. |
| title_short |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field |
| title_full |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field |
| title_fullStr |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field |
| title_full_unstemmed |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field |
| title_sort |
quantum phase transition: van vleck antiferromagnet in a magnetic field |
| author |
Ivanova, I.M. Kalita, V.M. Pashkov, V.O. Loktev, V.M. |
| author_facet |
Ivanova, I.M. Kalita, V.M. Pashkov, V.O. Loktev, V.M. |
| publishDate |
2008 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Квантовий фазовий перехiд: ван-флекiвський антиферомагнетик у магнiтному полi |
| description |
Theoretical description of magnetic properties of antiferromagnetic state, induced by the longitudinal magnetic
eld, in the Van Vleck singlet magnet with single-ion anisotropy of easy-plane type and ion spin S = 1
is proposed. It is shown that the quantum phase transition to the antiferromagnetic state is connected with
the spontaneous appearance of spin polarization in the easy plane. Spin polarization of the ground nondegenerated
state proves to be the order parameter of such a transition and the Landau thermodynamic
approach can be employed for its (transition) description. The magnetic properties which include the eld
behavior of magnetization and magnetic susceptibility of the antiferromagnetic phase in the elds of different
directions are studied. An attempt is made to qualitatively compare the obtained results with the available
experimental data.
Проведено теоретичний опис iндукованого магнiтним полем квантового фазового переходу у антиферомагнiтий стан у ван-флекiвському антиферомагнетику з легкоплощинним типом однойонної анiзотропiї i величиною спiнiв магнiтних йонiв S = 1. Показано, що при магнiтному фазовому переходi параметром порядку є величина спiнової поляризацiї основного стану, а для опису цього переходу застосовна термодинамiчна теорiя фазових переходiв Ландау. Вивченi магнiтнi властивостi iндукованої магнiтним полем антиферомагнiтної фази, проаналiзованi польовi залежностi намагнiченостi i магнiтної сприйнятливостi. Зроблена спроба якiсного порiвняння отриманих результатiв з наявними експериментальними даними.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/119337 |
| citation_txt |
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field / I.M. Ivanova, V.M. Kalita, V.O. Pashkov, V.M. Loktev // Condensed Matter Physics. — 2008. — Т. 11, № 3(55). — С. 509-522. — Бібліогр.: 42 назв. — англ. |
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2025-11-26T04:51:38Z |
| last_indexed |
2025-11-26T04:51:38Z |
| _version_ |
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| fulltext |
Condensed Matter Physics 2008, Vol. 11, No 3(55), pp. 509–522
Quantum phase transition: Van Vleck antiferromagnet in
a magnetic field
I.M.Ivanova1, V.M.Kalita2 , V.O.Pashkov3, V.M.Loktev4
1 National Technical University of Ukraine “KPI”, Prosp. Pobedy 37, Kyiv 03056, Ukraine
2 Institute of Physics of the National Academy of Science of Ukraine, Prosp. Nauki 46, Kyiv 03680, Ukraine
3 National Aviation University, Kosmonavta Komarova Ave. 1, Kyiv 03058, Ukraine
4 Bogolyubov Institute for Theoretical Physics of the National Academy of Science of Ukraine, Metrolohichna
Str. 14, Kyiv 03680, Ukraine
Received May 7, 2008, in final form July 7, 2008
Theoretical description of magnetic properties of antiferromagnetic state, induced by the longitudinal magnetic
field, in the Van Vleck – singlet – magnet with single-ion anisotropy of “easy-plane” type and ion spin S = 1
is proposed. It is shown that the quantum phase transition to the antiferromagnetic state is connected with
the spontaneous appearance of spin polarization in the easy plane. Spin polarization of the ground non-
degenerated state proves to be the order parameter of such a transition and the Landau thermodynamic
approach can be employed for its (transition) description. The magnetic properties which include the field
behavior of magnetization and magnetic susceptibility of the antiferromagnetic phase in the fields of different
directions are studied. An attempt is made to qualitatively compare the obtained results with the available
experimental data.
Key words: quantum magnetic phase transition, single-ion anisotropy, spin polarization, Van-Vleck
antiferromagnet
PACS: 75.10.-b, 75.10.Jm, 75.30.Gw, 75.30.Kz, 75.50.Ee
1. Introduction
Quantum phase transition is the transition which occurs at temperature T = 0, when some
external “governing” parameter is changed. As a rule, these are the transitions of “order-order”
type, which corresponds to the zero entropy of both phases. Transition point is called quantum
transition point [1] and now there are many examples of such transitions. It will be shown that
one of the interesting cases of the quantum phase transitions in the Van Vleck antiferromagnets is
the magnetic transition in the external field.
Indeed, it is a well-known fact that magnetization of classical (or, which is the same, weakly
anisotropic) antiferromagnets (AFMs) at low temperatures (far below the Neel one TN) is con-
nected only with sublattice magnetization turn [2]. From this fact, it is usually supposed that their
magnitude remains constant, and only their directions change under the effect of the external mag-
netic field. The character and peculiarities of magnetization process (spin-flip, spin-flop as well as
orientation phase transitions of the Ist order) of such AFMs depend on the following parameters:
value and direction of the external magnetic field, anisotropy, and intersublattice exchange [3–6].
For example, field behavior of magnetization in “easy-plane” two-sublattice dihalogenids NiCl2 or
CoCl2 of iron group [7–11] is well satisfied by quasi-classical approach, although these magnets
are markedly different. The “easy-plane” single-ion anisotropy (SIA) in NiCl2 is much lower than
the exchange – each ion orbital moment in crystal field is almost completely frozen. At the same
time, there is only partial freezing of the orbital moment in CoCl2, and “easy-plane” SIA is ap-
proximately half the exchange (in order of its magnitude [8]). Field dependence of the induced
magnetostriction in these AFMs [12–14] also agrees with a conception of the tilt of sublattice
magnetizations, keeping the absolute value.
c© I.M.Ivanova, V.M.Kalita, V.O.Pashkov, V.M.Loktev 509
I.M.Ivanova et al.
However, there is a family of crystals among AFMs, where SIA exceeds the inter-ion exchange
[15,16]. These are the so-called Van Vleck, or singlet antiferromagnets (SAFMs). They have no
magnetic ordering at any temperature, up to T = 0. Such materials include, in particular, hexagonal
crystals of ABX3 type, where A is the ion of alkali metal (A = Cs,Rb), B is the transition
metal (B = Fe), X is the halogenyde (X = Cl,Br). In these crystals, magnetic moments induced
by external field on paramagnetic ions B2+, form, on the one hand, antiferromagnetic chains
along C3 axis, and, on the other hand, triangular structures in basic plane (see reviews [1,17–
19]). There are also some other compounds classed with SAFMs, including the so-called DTN,
the chemical formula of which is NiCl2–4SC(NH2)2 [20–23]. It also has (AFM) chains Ni-Cl-Cl-Ni
along a “hard” magnetic axis, although the mean spin on each site is equal to zero at field absence,
since the parameters of both intra- and intersublattice exchange are exceeded by SIA. It should
be emphasized that DTN can be considered to belong to a group of two-sublattice SAFMs, which
have a crystal structure other than that of NiCl2, and another character of exchange interactions,
which are much lower in the value than the single-ion anisotropy [22,23].
The magnetization process in SAFMs is fundamentally different from that taking place in
classical Neel AFMs [25–28]. Firstly, they have no magnetic ordering at the absence of the external
magnetic field. Hence, they have no magnetic sublattices. Secondly, magnetic, or in particular
AFM, ordering in SAFMs may appear spontaneously by way of the quantum (in definition of
reference [24]) phase transition induced by the magnetic field [1,17–23].
The weak dependence of magnetic susceptibility on external magnetic field, thus, represents
some sort of peculiarity of AFM phase. As a result, the observed magnetization actually follows
the linear field behavior [22,23,29,30]. In other words, this behavior of magnetization of the system
(but not the proper sublattice magnetization) in the AFM phase turns out to be similar to the
magnetization induced by the external field in Neel AFMs. This could be understandable if the
transition to the AFM phase were a phase transition of the Ist order. Then, at the transition point,
the sublattices could magnetize, due to jump (in the presence of corresponding susceptibility sin-
gularity), and with further field growth the sublattice magnetic vectors would turn only. However,
the experiment shows that transformation of the non-magnetic (singlet) state into the AFM state
takes place continuously, i.e., this magnetic transformation is a phase transition of the IInd order
[22,23,29,30]. The latter demonstrates that sublattice magnetizations appear and change from their
initial zero value to maximal value also continuously. Therefore, the classical approach with the
constant module of average sublattice spin for SAFMs is not applicable fundamentally.
As it follows from the above, there still remain some unresolved issues in the description of
induced magnetic phase transition into the AFM phase, and its magnetic characteristics (field
dependencies of sublattice and of the system as a whole, magnetization, magnetic susceptibility,
magnetostriction) in Van Vleck systems.
Below we will proceed from an assumption that in such magnets the intrinsic spontaneous
magnetic (or AFM) moment is equal to zero. So, without the external magnetic field, they lack
the magnetic ordering temperature. Seemingly, the fact that the magnetic (dipole) moment or, in
other words, magnetization (spin) on the site is equal to zero is indicative not only of the absence
of any magnetic ordering, but, most probably, of any magnetic contributions to physical properties
of the corresponding systems. However, this is not the case, because the absence of ordinary –
exchange-induced – spin ordering does not exclude the presence of the ordering of another type,
e. g., the quadrupole one. The latter, in one or the other way, is characteristic of all SAFMs, which,
in turn, are a special case in the magnetic crystals with more specific – nematic – type of spin
ordering1 [31].
It should be noted that some studies (e. g. [32–35]), describe the phase transition between
the singlet and induced AFM states by using the representation of Bose-Einstein condensation
of magnons. Indeed, the appearance of magnetization in finite magnetic fields can be formally
described in terms of some magnetic excitations condensation. But in reality no true condensation
1As pointed in the recent study [31], the nematic (in the vary sense of this term) ordering in strongly anisotropic
magnets is absent. However, the ground state of such magnet, that is singlet, does not differ, in physical sense, from
the previous one which is caused by the large biquadratic exchange [27,28].
510
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
of quasiparticles occurs in the observed systems, because, and it will be shown below, one should
consider rearrangement of the ground state only, and, hence, virtual, rather than real magnons [36].
Below we consider the model of strongly anisotropic, two-sublattice AFM with ion bare spin
S = 1. In the framework of quantum approach an attempt will be made to describe the crystal
magnetization, magnetic susceptibility and magnetostriction at magnetically induced phase tran-
sition from the initial singlet state to the spin-ordered state. To calculate physical characteristics
of the system, we will use the total energy E, that is the sum of relevant contributions:
E = Eexch +Ean +Eh , (1)
where Eexch is the exchange energy; Ean is the magnetic anisotropy energy and Eh is Zeeman
energy.
This work is dedicated to prof. I.V. Stasyuk, whose research of the diagram methods of cal-
culation for Hubbard operators helped in understanding the magnetic properties of magnets with
strong SIA.
2. Ground state of the model
According to above-said, for simplicity let us limit our consideration to the bilinear anisotropic
(intra- and intersublattice) exchange interaction, single-ion “easy-plane” anisotropy and Zeeman
contribution. In this case, the simplest model Hamiltonian of a system, which defines the first three
contributions, Eexch, Ean and Eh, in equation (1), can be written down as follows:
H =
1
2
∑
nα,mβ
Jnαmβ
Snα
Smβ
+
1
2
∑
nα,mβ
JZ
nαmβ
SZ
nα
SZ
mβ
+D
∑
nα
(
SZ
nα
)2 − h
∑
nα
Snα
, (2)
where α, β = 1, 2 are the magnetic sublattice indices, the numbers of which in the considered
system was chosen to be equal to 2; vectors n and m specify the spins position in magnetic
sublattices described by spin operators Snα; constant D > 0 that reflects an “easy-plane” magnetic
structure; magnetic field h is determined in energy units, hence h = µBgH; H is the magnetic
field. Crystallographic symmetry axis OZ is perpendicular to the “easy” plane. It is at h ‖ OZ
that the magnetic field induces the phase transition to the AFM state. Transverse field h ⊥ OZ
in two-sublattice SAFM does not induce any phase transitions. Parameter Jnαmβ
characterizes
the value of an isotropic part of the exchange interaction, and JZ
nαmβ
characterizes the exchange
anisotropy, which, in principle, can be of either “easy-axis” or “easy-plane” type. However, we will
assume that the inter-ion anisotropy, like SIA, relates to the same, i.e “easy-plane”, type.
The convenience of these restrictions is conditioned by the fact that in such a situation both
sublattices become symmetric relatively to the external field, what allows to reduce twice the
number of equations.
Analysis of possible eigenstates of Hamiltonian (2) at h ‖ OZ will be provided, using self-
consistent field approximation that corresponds to the spin fluctuation neglecting and to the change
of average of spin operators products at different sites by product of averages. In this case the
ground state energyEgr, normalizing on one cell (for nearest both inter- and intrasublattice different
spins) is equal to:
Egr =
1
2
∑
αβ
Jαβzαβsαsβ +
1
2
∑
αβ
JZ
αβzαβs
Z
αs
Z
β +D
∑
α
QZZ
α − h‖
∑
α
sZ
α , (3)
where sα is the quantum-mechanical average of the spin of the α-th sublattice in the ion ground
state; zαβ is the number of the nearest neighbors from the same (zαα) and another (zαβ ≡ z12)
sublattices. Introduced also are the averages for components of spin quadrupole moment QZZ
α =
〈ψ(0)
α |(SZ
nα)2|ψ(0)
α 〉 [38–40], where ψ
(0)
α is the wave function of the ground state of the ion from
α-sublattice. Note that for AFM the intersublattice exchange is J12z12 ≡ I > 0. At the same time,
the value J11z11 = J22z22 ≡ J of intrasublattice exchange can be of any sign, which is assumed for
511
I.M.Ivanova et al.
simplicity as furthering to the ordering, J < 0. The exchange anisotropy, in this case, satisfies the
conditions of its “easy-plane” type: JZ
12z12 ≡ 4I < 0 and JZ
11z11 = JZ
22z22 ≡ 4J > 0.
Let us impose for spins of each sublattice their proper (rotating) coordinate systems ξαηαζα
such, that αth sublattice average spin is always oriented along ζα axis, what means that this axis
is the quantization one for this spin sublattice, and ξα axis is lain in Zζα plane. Then the correct
wave function of the ground spin state of the α-th sublattice in such a coordinate system, as it is
well known, will have the following form2 [37]:
ψ(0)
α = cosφα | 1〉 + sinφα | −1〉, (4)
where | ±1〉 and | 0〉 are eigenfunctions of operator Sζ
nα
in bracket representation. Next, it can be
calculated, using (4) the quantum-mechanical spin and quadrupole averages:
s = cos 2φ, Qζζ = 1, Qξξ =
1
2
(1 + sin 2φ) , Qηη =
1
2
(1 − sin 2φ) . (5)
Sublattice indices are omitted in the expressions (5), because, as already noted, with the chosen
field direction, the evident dependence of observables on index α is absent.
The use of functions (4) makes it possible to obtain the energy (3) at h ‖ OZ as follows:
Egr = I cos2 2φ cos 2θ − |J | cos2 2φ+ JZ cos2 2φ cos2 θ + 2D
[
cos2 θ +
sin2 θ
2
(1 + sin 2φ)
]
− 2h‖ cos θ cos 2φ, (6)
where JZ ≡ 4J −4I and θ is the angle between the sublattice magnetization and OZ axis.
The field behavior of mean spin (and its direction) for each sublattice in the field should be
found in order to determine magnetization, magnetic susceptibility and subsequently striction.
The same calculations should also be made for spin quadrupole moment. As reported in references
[38,39], solution of the problem of spin configuration in the magnetic field suggests minimization
of expression (6) by all available unknown quantities: the geometric angle θ and (see equation (4))
the angle φ of quantum states mixture. This method of finding the observables, being completely
an equivalent to the solution of quantum self-consistent problem, is more convenient and more
consistent, because it allows to perform generalization on the case of finite temperatures [27,28].
The equations for both required angles are:
∂Egr
∂φ
= −2
(
I cos 2θ − |J | + JZ cos2 θ
)
sin 4φ+ 2D sin2 θ cos 2φ+ 4h‖ cos θ sin 2φ = 0, (7)
∂Egr
∂θ
= − (2I + JZ) sin 2θ cos2 2φ−D sin 2θ (1 − sin 2φ) + 2h‖ sin θ cos 2φ = 0. (8)
As it is known from reference [36], the set of equations (7) and (8) has two solutions for the case
of the absence of the external magnetic field: non-magnetic one, s = 0, that exists at D > 2(I+ |J |)
and “magnetic” one at D 6 2(I + |J |), with which the reduced value of single-site mean spin
s =
√
1 − D2
4 (I + |J |)2
< 1 (9)
is associated.
The initial ground state of the system should be the singlet state, s = 0, so that the quantum
phase transition (at the magnetic field h ‖ OZ) from this state to magnetically ordered state occurs.
So, assume that the above explicit inequality, i.e. D > 2(I + |J |), is satisfied. With this ratio of
the model parameters, the ground state of the system is really nonmagnetic, and the ordering in
the absence of magnetic field cannot be realized at any temperatures [39]. In other words, ratio (9)
2It should be denoted, that ground state functions, given in [20] for laboratory (crystallographic) coordinates
system, can not be considered as eigen ones, until the self-consistent problem is solved (see [38]).
512
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
determines the condition of singletness of magnet ground state, which is Van Vleck state. Solution
s = 0 is satisfied in the interval h‖ < hs =
√
1 − (I + |J |/D).
As the field grows, the finite value, s 6= 0, of the mean spin on the site appears. It can be
derived from equation (8) that the expression for the average spin orientation relatively to the
crystallographic axis is:
cos θ =
h‖ cos 2φ
D (1 − sin 2φ) + (2I + JZ) cos2 2φ
. (10)
From equations (7) and (10) it can be seen that in large fields, where h‖ = hflip (where hflip ≡
D + 2I + JZ) the state, in which the spins of both sublattices are directed along “hard” (θ = 0)
axis, is established. Then, spin projection on the field direction will be maximum and equal to
s = S = 1. For h‖ < hflip, the spins of sublattices are orientated at a finite angle 0 < θ 6 π/2 to
the “hard” axis.
3. Thermodynamic analysis
Using formulas (5) and substituting (10) into the equation (6), the ground state energy in the
form of functional can be obtained:
Egr = − (I + |J |) s2 +D
(
1 −
√
1 − s2
)
− h2s2
D
(
1 +
√
1 − s2
)
+ (2I + JZ) s2
, (11)
which depends on the spin polarization s only. The expansion of this energy over the small s gives:
Egr =
hs
D
(
hs − h‖
)
s2 +
D
8
(
1 +
2h2
s (2I + JZ)
D3
)
s4 , (12)
where hs = D
√
1 − 2(I + |J |)/D is the critical field of the appearance of magnetization.
In the expansion (12), which refers to the field region h‖ → hs, one can restrict to the terms
that are not higher than of 4th power by s. Actually this expansion for the ground state energy is
similar to the free energy expression in Landau theory of phase transitions. However, in equation
(12) the ground state spin polarization corresponds to the order parameter, and the leading value,
that results in the phase transition, is not the temperature, but the magnetic field. It can be also
seen from equation (12) that at h‖ < hs the coefficients at s2 and s4 are positive, and so the
ground state of spin system will be Van Vleck non-magnetic single-ion state. At the point h‖ = hs
the sign of coefficient at s2 changes, and in the fields h‖ > hs the spin polarization (of still non-
degenerate ground state) spontaneously appears. The value of polarization can be readily found
by minimization of Egr (12):
∂Egr
∂s
= 2s
[
hs
D
(
hs − h‖
)
+
D
4
(
1 +
2h2
s (2I + JZ)
D3
)
s2
]
= 0. (13)
From equation (13) it follows that near the quantum critical point h‖ > hs this polarization
(or simply the ground state spin)
s
(
h‖
)
=
√
4hs
(
h‖ − hs
)
D2 + 2h2
s (2I + JZ) /D
(14)
fundamentally depends on the field.
In the same vicinity, h‖ > hs, of the quantum critical point, the angle θ between vector s and
axis OZ is determined by the expression:
cos θ =
hs
2D
√
4hs
(
h‖ − hs
)
D2 + 2h2
s (2I + JZ) /D
. (15)
513
I.M.Ivanova et al.
Thus, it is found, that at h‖ = hs the spin polarization spontaneously arises as field grows in
the very “easy” plane, because at h‖ − hs → 0 the angle θ → π/2. In other words, it turns out
that at the moment of its appearance, vector s(h‖ > hs) is perpendicular to the longitudinal field:
s ⊥ H ‖ OZ. Further magnetic field growth leads not only to the decrease of θ, as it follows from
equations (14 – 15), but also to a simultaneous increase of spin polarization, that is the bigger its
value is, the more it flattens against the “hard” axis.
On the whole, the induced tilt of the magnetic sublattices, and thereafter the magnetization
of SAFM includes two processes: the classical rotation of spins (sublattice magnetizations) and
purely quantum (due to the change of the angle φ) growth of single-site polarization s(h‖). Both
processes also take place at T = 0. The AFM magnetization (normalizing on one magnetic atom)
is described by the evident product:
m‖ ≡ m
(
h‖
)
= s
(
h‖
)
cos θ =
2h2
s
(
h‖ − hs
)
D3 + 2h2
s (2I + JZ)
. (16)
As a result, one arrives to an unexpected result: the observed magnetization near the critical
field of quantum transition from singlet to spin-polarized state depends linearly – as in classical
AFMs – upon the external magnetic field, that induces the vary transition. From this comes another
rather remarkable conclusion: at such a phase transition the magnetic susceptibility of a system
should have a jump.
Thus, in the framework of the approach that is similar to the Landau thermodynamic approach,
it was demonstrated that the spin polarization is the only order parameter for quantum phase
transition from Van Vleck phase to the AFM phase induced by magnetic field h ‖ OZ. Despite
the fact, that calculations were made for the case T = 0, the required polarization proves to be
essentially dependent on the external field. It should be reminded, that in classical AFMs, ion spin
polarization is fixed at T = 0 and it is not evaluated in the field, while in Van Vleck system it
appears as a consequence (in terminology of reference [24]) of quantum phase transition [27,28].
Next, attention should be drawn to the following analogy, that SIA, reducing the average spin,
plays a role of “disordering” factor, and in this sense it can be compared with entropy. It (SIA)
leads to the mixture (or linear combination) of quantum states that results in the absence of spin
polarization of ions in their ground state. Exchange and magnetic fields, on the contrary, resist
this, “magnetizing” the system and causing a spontaneous (or forced) spin polarization, which,
at the moment of its appearance, is directed perpendicularly to the magnetic field.
The studied quantum phase transition between Van Vleck (also ordered, inessential) and AFM
states is, as it was seen, the consequence of competition of different interactions (exchange, Zeeman
and spin-orbital, that lies at the heart of SIA). Therefore, such a quantum transformation can be
naturally identified as the magnetic phase transition of “displacement” but not of “order-disorder”
type. As distinct from the last one, the transition of displacement type is not the transition in the
system of spins, which fluctuates “up” and “down” between degenerated (or almost degenerated)
quantum states, because the ground state of quantum paramagnets is always non-degenerated and
its polarization is the direct consequence of rearrangement of this state in the external field.
Note that the applicability of phenomenological theory, which is based on the expansion (11)
is confined by the fields h‖ > hs in the vicinity of critical point hs. In the field region h‖ � hs
the magnetization process should be analyzed with more exact expressions both for ground state
energy and for the equations that define the spin configurations. However, the latter can be easily
found numerically.
4. The magnetization and magnetic susceptibility
Substituting expression (5) into equation (11), the following equation, which describes the spin
polarization as the function of longitudinal field, can be obtained:
s
(
D − 2 (I + |J |)
√
1 − s2 −
D
(
1 +
√
1 − s2
)
h2
‖
(D
(
1 +
√
1 − s2
)
+ (2I + JZ) s2)2
)
= 0. (17)
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Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
Note that this equation refers both to the fields h‖ < hs of the existence of Van Vleck phase,
where (see (12)) hs =
√
D2 − 2D(I + |J |), the nonmagnetic, s = 0, state is stable, and to the
region hs 6 h‖ 6 hflip of the AFM phase (up to the point hflip of its flipping). It is obvious, that
at the point (see (10) ) h‖ = hflip, which corresponds to θ = 0, the polarization arrives at its
maximum value s = 1 on the site.
Figure 1. Magnetization m‖ (solid curves), spin s, quadrupole moment Q and value of cos θ
versus field. The curves for s, Q and cos θ are calculated at |J |/D = 0.455 and at condition
I = JZ = 0. Magnetization curves 1-5 are for the following parameters: curve 1 is for |J |/D =
0.455, I = JZ = 0; curve 2 is for |J |/D = 0.35, I = JZ = 0; curve 3 is for |J |/D = 0.35,
I = 0, JZ/D = 0.5; curve 4 is for |J |/D = 0, I/D = 0.35, JZ = 0; curve 5 is for |J |/D = 0.15,
I/D = 0.2, JZ/D = 0.5.
Using equation (17), the behavior s(h‖) in the region hs 6 h‖ 6 hflip can be found, and
from equation (10) – the angle θ can also be found. Then, it is not difficult to define the field
dependence of the quadrupole QZZ
α in equation (3). In the framework of such an approach, the
field dependencies of magnetization were calculated (see figure 1).
Curve 1 in figure 1 was plotted for the case where intrasublattice exchange prevails, while
intersubalttice and anisotropic ones are omitted. At chosen parameters, the magnetic sublattices
are fixed artificially, because this extreme case actually corresponds to the two independent AFMs,
The field behavior for s, QZZ
α ≡ Q and cos θ is also shown in figure 1 for these parameters. It can
be seen that at the point h‖ = hs the average site spin really spontaneously appears and exists in
the region h‖ > hs. With further field growth, the value of s is increasing, guided by velocity of the
change of angle θ. This velocity, however, becomes higher, when the field approaches the flipping
field and, correspondingly, s→ 1. In such a case the intrasublattice exchange (due to its isotropy)
has no effect on spin saturation, and so the value hflip of this critical field is completely defined by
the single-ion anisotropy.
From curve 1, that refers to the case D− 2|J | � D, it follows, that the fields hflip and hs differ
quite weakly (hflip/hs ≈ 3), although in the experiment [22,23] their ratio reaches 6, and from the
data of the study [30] this ratio is about 8. Besides, the field dependence of magnetization for the
considered case I = 0 reveals, as can be seen, large nonlinearity, while the experimental data for
all the above mentioned compounds are evidence of nearly linear dependence of magnetization on
magnetizing force.
It should be noted that the case, in which inequality (D − 2|J |)/D � 1 satisfies, is physically
available, but it cannot be justified from the experimental point of view. To demonstrate this in
figure 1, the curve 2 is plotted, for which the difference between parameters in intrasublattice
exchange and SIA is chosen not less, but bigger than for curve 1. This choice really leads to
the increase of the field hs and results in the decrease of the ratio hflip/hs, which indicates that
attempting to interpret the experimental magnetization, the intersublattice exchange cannot be
neglected.
It is interesting, that when I/D → 0, equation (17) has an exact solution, using which the
515
I.M.Ivanova et al.
ground state energy can be written in the form of a function of magnetic field:
Egr =
1
4D2|J |
(
h2
‖ − h2
s
)2
. (18)
Then the magnetization (normalizing on one magnetic ion again) takes the following form:
m‖ = −∂Egr
∂h‖
=
h‖
2D2|J |
(
h2
‖ − h2
s
)
. (19)
The last dependence is described by curves 1 and 2 in figure 1. From equations (18) and
(19), quite a big field nonlinearity of magnetization in the antiferromagnetic phase can bee seen.
Expression (19), for fields h‖ → hs can be also presented in the form of equation (16), when
2|J | → D.
Now let us consider the opposite limiting case, when intersublattice exchange is the biggest.
It can be seen that even if one preserves the exchange (that formally gives the same value of hs),
which effects the spin from another sublattice, the change of magnetization (curve 4 in figure 1)
occurs. The AFM (intersublattice) exchange, unlike the intrasublattice one, leads to the growth of
the field hflip, because in this case the external field will overcome the effect of the same anisotropy,
on the one hand, and the effect of exchange field that prevents parallel orientation of sublattice
spins, on the other hand.
Curve 3 already shows the nonlinearity decrease in m(h‖), as if it was rectified by intersublattice
exchange (or by its anisotropic part). At the same time, AFM exchange together with the external
magnetic field (in the region h‖ > hs), while resisting the anisotropy, leads to the establishment of
spontaneous polarization. In the large fields, when polarization tends to its maximum value, the
behavior of exchange in SAFM does not differ from that in classical AFMs: it simply resists the
parallel configuration of both sublattice spins.
Figure 2. Magnetic susceptibility χ‖ versus field. The numbers of the curves correspond with
parameters, that were used in plotting the lines with the same numbers as in figure 1.
Curves 3 and 5 demonstrate the effect of easy-plane exchange anisotropy. Actually this anisotropy
does not change the position of critical field hs, but it also does not “desire” the establishment
of collinear state, when s1 → s2 ‖ OZ. At the same time the account of exchange anisotropy of
easy-plane type being taken into account, it permits to obtain such a behavior of magnetization
that is close to linear and is observed in the studies [22,23,29,30]. For clarifying how well the linear
dependence corresponds to m(h‖), the magnetic susceptibility χ‖ = dm‖/dh is shown in figure 2
for the same parameters as in figure 1.
Since magnetization is nothing more but m‖ = s cos θ, where s = s(h‖), the longitudinal
magnetic susceptibility of SAFMs is naturally to be represented in the form of the two above
516
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
mentioned terms, i.e., the classical χcl and the quantum χquan, so that
χ‖ = χcl + χquan , χcl = s sin θ
∂θ
∂h
, χquan = cos θ
∂s
∂h
. (20)
As can be seen from figure 1, near hs the biggest growth reveals the spin polarization s, so in
the fields h‖ → hs the “quantum” contribution will dominate in the magnetic susceptibility. And
when the value of spin polarization saturates (s(h‖ → hflip) → 1), the susceptibility will be mainly
controlled by classical term (see equations (20)).
Figure 2 shows that, when intrasublattice exchange is really the largest, then magnetic suscep-
tibility grows, increasing in the field region hs 6 h‖ 6 hflip four times. If intersublattice exchange
and/or exchange anisotropy “switches” on, then field dependence of differential magnetic suscep-
tibility χ‖ ≡ χ(h‖) becomes essentially weaker. Nevertheless, it can be seen from the curves, shown
in figure 1 that nonlinearity of function m(h‖) in the antiferromagnetic phase at the chosen pa-
rameters remains quite noticeable. The case, when within the boundaries of this phase, the value
Figure 3. Longitudinal magnetization m‖ and magnetic susceptibility χ‖ versus field at |J |/D =
0.05, I/D = 0.3, JZ/D = 1 and 1.5. The functions s(h‖), cos θ(h‖) and Q(h‖) are shown only
for |J |/D = 0.05, I/D = 0.3, JZ/D = 1.
of χ(h‖) changes (30–50%), is shown in figure 3, which meets the model parameters |J |/D = 0.05,
I/D = 0.3, JZ/D = 1 or JZ/D = 1.5. In other words, the exchange anisotropy is comparable
or even exceeds the SIA. At such ratios between the parameters, the field hflip almost five times
exceeds the field hs (one should note that the experimentally observed ratio is hflip/hs ≈ 6 [22,23]).
In the same figure 3 the dependencies of s(h‖), cos θ(h‖) and Q(h‖) are shown for parameters
|J |/D = 0.05, I/D = 0.3 and JZ/D = 1. The behavior of Q(h‖) almost coincides (see figure 1)
with the field dependence m(h‖). Moreover, it follows from figure 3 that exchange anisotropy, even
comparable with SIA, does not fully linearize the function m(h‖). As it was mentioned, this fact
can be explained by the existence of two different regions in the magnetization of SAFM.
In the first of these regions, near hs, the quantum process, as it was pointed out, is determinant
and the magnetization is defined basically by the appearance and growth of s(h‖). In the second
region, in the vicinity of h‖ 6 hflip, the classical rotation of sublattice spins towards the field
direction becomes more important, at essentially less (but still present) role the vary spin value
change plays. It is obvious that in this region, the susceptibility depends much weaker on the
value of magnetic field. So, it can be supposed, that the flipping of tilted AFM sublattices, or the
transition of Van Vleck system between the induced two- and one-sublattice magnetically ordered
states occurs as an orientation quantum phase transition in ordinary AFM, when the variation of
sublattice magnetization directions is in fact the only process taking place.
However, even for this field transition, the quasiclassical approach does not give the correct
solution to m(h‖) in spin nematic. Indeed, one could suppose that near the flipping field, when
517
I.M.Ivanova et al.
s(h‖ → hflip) ≈ 1, the quasiclassical magnetic energy in the ground state takes the form:
Egr = I cos 2θ + (JZ +D) cos2 θ − 2h‖ cos θ = 0. (21)
Then, from equation (21) there immediately follows the equation
dEgr
dh‖
= 2
[
− (2I + JZ +D) cos θ + h‖
]
sin θ = 0, (22)
which shows that in the vicinity of h‖ → hflip the magnetization of the tilted (θ 6= 0) phase is
proportional to the field: m‖ = χ̃‖h‖, where
χ̃‖ =
1
D + 2I + JZ
≡ 1
hflip
= const. (23)
It can be seen that in the field hflip, the magnetization (on one spin) is m‖ = 1. However,
the susceptibility (23) differs from the exact ratio (20) and gives physically incorrect behavior of
magnetization. It is connected with the fact, that in the region h‖ → hflip it appears that m‖
depends linearly on the magnetic field, and asymptotically tends to zero at h‖ → 0. As for the
plots, shown in figures 1 and 3, it is easy to see that function m(h‖), although it behaves linearly
by field, but nevertheless it depends on the magnetic field in an indirect proportion.
An important conclusion follows from this: the quasiclassical approach (21), based on the sub-
stitution of quadrupole moment QZZ by the average spin Zth projection square, appears to be
unapplicable even in such a field region, where spin polarization almost reaches its saturation value
s→ 1.
5. The magnetization in transversal magnetic field
As it was mentioned, the phase transition to the AFM state does not occur at h ⊥ OZ, although
the magnetic field magnetizes the system.
Let us suppose that due to AFM exchange in the easy plane, two sublattices are formed. Then,
their spins lie in this plane and are identically tilted to the field. In this case, the ground state
energy is:
Egr = I cos 2ϕ cos2 2φ− |J | cos2 2φ+D (1 + sin 2φ) − 2h⊥ cosϕ cos 2φ, (24)
where ϕ is the angle between vector s1 (or vector s2) and field h, and the angle between s1 and s2
is twice larger, 2ϕ.
The spin configuration will be defined, as usual, by minimizing the energy (24). As a result, it
is the set of equations (compare equations (7) and (8)):
∂Egr
∂ϕ
= −2I cos2 2φ sin 2ϕ+ 2h⊥ sinϕ cos 2φ, (25)
∂Egr
∂φ
= −2 (I cos 2φ− |J |) sin 4φ+ 2D cos 2φ+ 4h⊥ cosϕ sin 2φ = 0. (26)
Equation (25) has two solutions. For the first of them, ϕ = 0 and it corresponds to one-
sublattice magnetization, when the polarization of magnetic ions is directed along the field. The
second solution cosϕ = h⊥/(2I cos 2φ) provides the existence of two sublattices. The latter, taking
into account equation (5), can be rewritten in the usual form:
cosϕ = h⊥/2Is. (27)
The denominator of equation (27) is the intersublattice exchange field, and this expression is
similar to the expression for the magnetic sublattice tilt angle in classical AFMs [2,3]. Nevertheless,
it should be noted that the spin in equation (27) is not equal to its maximum value.
518
Quantum phase transition: Van Vleck antiferromagnet in a magnetic field
Substituting (27) in (26), one arrives at the equation:
2 [2 (I + |J |) sin 2φ+D] cos 2φ = 0. (28)
It follows from equation (28) that spin polarization for AFM state at h ⊥ OZ should be equal
to (9). However, the model parameters, accepted above, are such that the denominator under
the root in equation (9) is larger than 1, and non-polarized singlet is the ground state of ions.
Thus, the solution (27) is possible only for initially polarized ground state, or when the AFM (not
singlet) phase is realized in the system even at h⊥ = 0. But if without field the polarization is
s = 0, then from the set of equations (25)–(26) a fundamentally different result follows: the critical
field of polarization appearance in the transversal geometry is the h⊥ = 0. The distinction from
“longitudinal” case, for which the critical field is finite, is easy to explain. At any fields h⊥ 6= 0 the
ground state with SZ = 0 (in crystal coordinate system) is immediately admixed with the ionic
exited states, which have SZ = ±1. In the case of longitudinal field, there is a threshold for such
an admixture. Then, taking into account that the transversal field does not induce AFM phase,
one can obtain:
Egr = (I + |J |) s2 +D
(
1 −
√
1 − s2
)
− 2h⊥s. (29)
Note that at h ⊥ OZ the spin polarization is always equal to magnetization, which is directed
along h, i.e. m⊥ = s. Minimizing energy (29), one readily arrives at the equation
∂Egr
∂s
= 2 (I + |J |) s+D
s√
1 − s2
− 2h⊥ = 0 (30)
which permits to define the dependence of spin polarization upon the transversal field.
The field behavior for m⊥, which is obtained from equation (30) is shown in figure 5. It is seen
that if intrasublattice exchange dominates in the system, then the magnetization increases rapidly,
and if intersublattice exchange is “added”, then the magnetization slows down.
Despite the fact that at h ⊥ OZ the average spins are also oriented perpendicularly to Z,
spin quadrupole moment QZZ versus field reveals the behavior (see figure 5) similar to the spin
polarization. Magnetization rate decreases as the field grows and the magnetic susceptibility has a
maximum at h⊥ → 0. The normalized magnetic susceptibility χ
(0)
⊥ = χ(h⊥ = 0) is also plotted in
figure 4.
Figure 4. Field behavior of m(h⊥) (lines 1–4), QZZ(h⊥) (line 5) and χ⊥ for h ⊥ OZ. The line 1
is calculated for parameters |J |/D = 0.455, I = 0, the line 2 is for |J |/D = 0.35, I = 0, the
line 3 is for |J |/D = 0, I = 0.35, lines 4–6 are for |J |/D = 0.05, I = 0.3.
Using equation (30), the expression for magnetic susceptibility at h ⊥ OZ can be obtained; it
has the form:
χ⊥ ≡ χ (h⊥) =
1
2 (I + |J |) +D (1− s2)
−3/2
. (31)
519
I.M.Ivanova et al.
It is easy to see that in large fields, when s→ 1, transversal susceptibility χ⊥ → 0. In the opposite
limit h⊥ → 0, the magnetic susceptibility is equal to:
χ
(0)
⊥ =
2
D + 2 (I + |J |) . (32)
It also follows from equation (32), that when the intrasublattice exchange increases, the value
of χ
(0)
⊥ grows and, on the contrary, at the increase of intersublattice exchange it decreases. This is
a usual situation in physics of phase transitions, because the growth of intrasublattice exchange at
I = 0 can result in ferromagnetic state with susceptibility singularity characteristic of such type
of transition (it goes to the infinity at the transition point). At the same time, the transition to
the AFM state is not accompanied by the abnormal growth of magnetic susceptibility. Indeed,
the point of phase transition to the AFM phase corresponds to the equality D = 2(I + |J |). At
its substitution in equation (32) the value χ
(0)
⊥ = 1/2I is directly obtained. The same will be the
value of magnetic susceptibility in the AFM phase, the magnetization of which is determined by
the expression (27).
6. Conclusion
Thus, it was obtained that the phase transition in SAFM to magnetically ordered state, induced
by magnetic field, is the quantum phase transition. The spin polarization of the magnetic ion
ground state is the order parameter of this phase transition. The Landau theory can be used for its
description. The considered transition is a consequence of the competition of different interactions,
and, what is important, it appears in the field that is perpendicular to the easy plane. Such a field
does not reduce the symmetry in this plane, leaving all directions in it equivalent. Conservation of
degeneracy for directions of sublattice magnetizations in the “easy” plane is the crucial symmetrical
condition for the phase transition to the AFM state with spontaneous magnetizations lying in this
plane.
Also, it is shown that in the magnetic field induced AFM phase, the spin polarization (magneti-
zation) of the sublattice changes continuously from zero, reaching its maximum at the spin flipping
point. In contrast to classical Neel AFM, in the magnetic phase of Van Vleck (singlet) AFM, the
value of the sublattice magnetization strongly depends on the field. The same field dependence
has an angle that defines the deviation of sublattice magnetization from the field direction. At the
same time, the magnetization of a system as a whole, being weekly dependent on the field, shows
almost linear field behavior (allowing the exchange anisotropy).
Finally, it is necessary to make a methodical remark. The above results were obtained in an
approximation of self-consistent field. It was assumed that more accurate calculations would not
give any qualitative results. However, they might have a quantitative effect. A separate paper will
be devoted to the quantitative comparison of calculations and the available experimental data.
We are grateful to Prof. S.M. Ryabchenko, who drew our attention to experimental studies
[22,23], as well as to the problem of magnetostriction in singlet magnets.
This work is partially supported by Grant of Ministry of Education and Science of Ukraine
(No. 25.2/043).
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Квантовий фазовий перехiд: ван-флекiвський
антиферомагнетик у магнiтному полi
I.М.Iванова1, В.М.Калiта2, В.О.Пашков3, В.М.Локтєв4
1 Нацiональний технiчний унiверситет України “КПI”, пр. Перемоги 37, Київ, 03056 Україна
2 Iнститут фiзики НАН України, пр. Науки 46, Київ, 03680 Украина
3 Нацiональний авiацiйний iнститут, пр. Космонавта Комарова. 1, Київ, 03058 Україна
4 Iнститут теоретичної фiзики iм. М.М. Боголюбова НАН України, вул. Метрологiчна 14-б, Київ, 03143
Україна
Отримано 7 травня 2008 р., в остаточному виглядi – 7 липня 2008 р.
Проведено теоретичний опис iндукованого магнiтним полем квантового фазового переходу у ан-
тиферомагнiтий стан у ван-флекiвському антиферомагнетику з легкоплощинним типом однойонної
анiзотропiї i величиною спiнiв магнiтних йонiв S = 1. Показано, що при магнiтному фазовому пере-
ходi параметром порядку є величина спiнової поляризацiї основного стану, а для опису цього пе-
реходу застосовна термодинамiчна теорiя фазових переходiв Ландау. Вивченi магнiтнi властивостi
iндукованої магнiтним полем антиферомагнiтної фази, проаналiзованi польовi залежностi намагнi-
ченостi i магнiтної сприйнятливостi. Зроблена спроба якiсного порiвняння отриманих результатiв з
наявними експериментальними даними.
Ключовi слова: квантовий магнiтний фазовий перехiд, однойонна анiзотропiя, спiнова
поляризацiя, ван-флекiвський антиферомагнетик
PACS: 75.10.-b, 75.10.Jm, 75.30.Gw, 75.30.Kz, 75.50.Ee
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