Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy
Anisotropic antiferromagnets in an external magnetic field show a rich variety of different ground states meeting in transition lines and multicritical points. We study the dependence of the ground states of these systems in the three dimensional space on physical parameters such as exchange, sing...
Gespeichert in:
| Datum: | 2009 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
2009
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/120549 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy / T.-C. Dinh, R. Folk // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 689-696. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120549 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1205492017-06-13T03:05:40Z Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy Dinh, T.-C. Folk, R. Anisotropic antiferromagnets in an external magnetic field show a rich variety of different ground states meeting in transition lines and multicritical points. We study the dependence of the ground states of these systems in the three dimensional space on physical parameters such as exchange, single ion and cubic anisotropy. One identi es four different ground states: the paramagnetic (PM), the antiferromagnetic (AF), the spin flop (SF) and the biconical (BC) ground state. In the case of absence of a cubic anisotropy, the transition lines separating different ground states can be calculated analytically, otherwise they have to be calculated numerically. We also considered the behavior of the staggered magnetization which characterizes the different ground states. From its behavior the order of the transition from one state to the other is determined. But also the order of the transition changes along the transition lines when including the cubic anisotropy, especially at the reeentrant region where a transition from SF to BC and back to SF by increasing the external field H occurs. Multicritical points are found which are assumed to be tricritical or critical endpoints. The results obtained may be relevant for other systems since the antiferromagnetic model can be mapped to a lattice gas model where the biconical ground state is interpreted as supersolid phase. Recent renormalization group calculations show that such a phase would indicate the existence of a tetracritical point. Анiзотропнi антиферомегнетики у зовнiшнiх магнiтних полях проявляють велике рiзноманiття основних станiв, що межують на лiнiях фазових переходiв i в мультикритичних точках. Тут дослiджується залежнiсть основних станiв у таких тривимiрних системах вiд параметрiв обмiнної взаємодiї, одноiонної та кубiчної анiзотропiй. Iндентифiковано чотири рiзнi основнi стани: парамагнiтний (ПМ), антиферомагнiтний (АФ), з нахиленими спiнами (НС) та двоконусний (ДК). У випадку вiдсутностi кубiчної анiзотропiї лiнiї фазових переходiв, що роздiляють рiзнi основнi стани, можуть бути розрахованi аналiтично; в iншому випадку їх можна розрахувати лише чисельно. Розглянуто також поведiнку ступiнчастої намагнiченостi, типову для рiзних основних станiв. Аналiзуючи її, визначено рiд фазового переходу з одного стану до iншого. Проте рiд фазового переходу змiнюється також вздовж лiнiї переходу при включеннi кубiчної анiзотропiї, особливо в подвiйнiй областi , де при збiльшеннi зовнiшнього поля H вiдбувається спершу перехiд вiд НС до ДК, а далi - знову до НС. Знайденомультикритичнi точки, якi слiд думати будуть трикритичними або ж критичними кiнцевими точками. Отриманi результати можуть бути важливими i для iнших систем, оскiльки антиферомагнiтну модель можна вiдобразити у модель граткового газу, де двоконусний основний стан iнтерпретується як надплинний твердотiльний. Нещодавнi ренормгруповi розрахунки вказують, що така фаза мала б сигналiзувати про iснування чотирикритичної точки. 2009 Article Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy / T.-C. Dinh, R. Folk // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 689-696. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 75.10.Hk, 75.30.Kz, 67.80.bd DOI:10.5488/CMP.12.4.689 http://dspace.nbuv.gov.ua/handle/123456789/120549 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 |
Anisotropic antiferromagnets in an external magnetic field show a rich variety of different ground states
meeting in transition lines and multicritical points. We study the dependence of the ground states of these
systems in the three dimensional space on physical parameters such as exchange, single ion and cubic
anisotropy. One identi es four different ground states: the paramagnetic (PM), the antiferromagnetic (AF), the
spin flop (SF) and the biconical (BC) ground state. In the case of absence of a cubic anisotropy, the transition
lines separating different ground states can be calculated analytically, otherwise they have to be calculated
numerically. We also considered the behavior of the staggered magnetization which characterizes the different
ground states. From its behavior the order of the transition from one state to the other is determined. But also
the order of the transition changes along the transition lines when including the cubic anisotropy, especially
at the reeentrant region where a transition from SF to BC and back to SF by increasing the external field
H occurs. Multicritical points are found which are assumed to be tricritical or critical endpoints. The results
obtained may be relevant for other systems since the antiferromagnetic model can be mapped to a lattice
gas model where the biconical ground state is interpreted as supersolid phase. Recent renormalization group
calculations show that such a phase would indicate the existence of a tetracritical point. |
| format |
Article |
| author |
Dinh, T.-C. Folk, R. |
| spellingShingle |
Dinh, T.-C. Folk, R. Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy Condensed Matter Physics |
| author_facet |
Dinh, T.-C. Folk, R. |
| author_sort |
Dinh, T.-C. |
| title |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| title_short |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| title_full |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| title_fullStr |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| title_full_unstemmed |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| title_sort |
ground states of anisotropic antiferromagnets with single ion and cubic anisotropy |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120549 |
| citation_txt |
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy / T.-C. Dinh, R. Folk // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 689-696. — Бібліогр.: 15 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT dinhtc groundstatesofanisotropicantiferromagnetswithsingleionandcubicanisotropy AT folkr groundstatesofanisotropicantiferromagnetswithsingleionandcubicanisotropy |
| first_indexed |
2025-07-08T18:07:02Z |
| last_indexed |
2025-07-08T18:07:02Z |
| _version_ |
1837103083917672448 |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 4, pp. 689–696
Ground states of anisotropic antiferromagnets with
single ion and cubic anisotropy
T.-C. Dinh, R. Folk
Institut für Theoretische Physik, Johannes Kepler Universität Linz, 4040 Linz, Austria
Received June 15, 2009, in final form July 1, 2009
Anisotropic antiferromagnets in an external magnetic field show a rich variety of different ground states
meeting in transition lines and multicritical points. We study the dependence of the ground states of these
systems in the three dimensional space on physical parameters such as exchange, single ion and cubic
anisotropy. One identifies four different ground states: the paramagnetic (PM), the antiferromagnetic (AF), the
spin flop (SF) and the biconical (BC) ground state. In the case of absence of a cubic anisotropy, the transition
lines separating different ground states can be calculated analytically, otherwise they have to be calculated
numerically. We also considered the behavior of the staggered magnetization which characterizes the different
ground states. From its behavior the order of the transition from one state to the other is determined. But also
the order of the transition changes along the transition lines when including the cubic anisotropy, especially
at the reeentrant region where a transition from SF to BC and back to SF by increasing the external field
H occurs. Multicritical points are found which are assumed to be tricritical or critical endpoints. The results
obtained may be relevant for other systems since the antiferromagnetic model can be mapped to a lattice
gas model where the biconical ground state is interpreted as supersolid phase. Recent renormalization group
calculations show that such a phase would indicate the existence of a tetracritical point.
Key words: antiferromagnetism, quantum lattice gas, ground states
PACS: 75.10.Hk, 75.30.Kz, 67.80.bd
1. Introduction
Anisotropic systems in an external field not coupled to the order parameters (OP) characterizing
the condensed phases show interesting phase diagrams depending on the values of the anisotropy
and the external field. One example is the anisotropic antiferromagnet in an external magnetic
field [1]. Other ones are crystal systems undergoing a displacive phase transition under external
pressure [2]. The possible variety of phases at finite temperature is also determined by the topology
of the ground state phase diagram since the phases corresponding to different ground states extend
to finite temperatures. However, at finite temperature new phases not present at T = 0 may arise
[3]. Due to the variety of possible ground states or phases multicritical points of different kind
are possible at finite T . Using the magnetic language these systems show antiferromagnetic (AF),
spinflop (SF), biconical (BC) and paramagnetic (PM) ground states and phases. The transition
lines between these phases may meet in one point: (i) the bicritical point, where three phases
coexist (AF, SP, and PM) and (ii) the tetracritical point, where four phases (AF, BC, SP, and
PM) coexist. Going beyond the mean field theory [3] the nature of the multicritical points has been
studied within renormalization group theory [4] (in a one loop order calculation). These results have
been questioned by higher loop calculations [5,6]. Recently both using simulation methods [7,8]
and renormalization group theory [9,10] the nature of the multicritical point has been reconsidered.
It is well known that there is a correspondence between such magnetic model and the quantum
lattice gas [11] which allows to transpose the results obtained for the magnetic system to systems
like He4 with superfluid and supersolid phases [12]. The question of the existence of such phases
is of general interest.
Symmetry considerations are essential for the existence and the nature of ground states and
phases. This is also the case for the systems treated here. It has been found that interaction terms
of cubic symmetry have a strong effect on the phase diagram [13]. Here we present the results at
c© T.-C. Dinh, R. Folk 689
T.-C. Dinh, R. Folk
T = 0 which clarifies the order of the transition lines between different ground states. Moreover,
we extend the models studied so far by combining several types of anisotropy.
2. Single ion anisotropy
Let us first consider the well studied case of an antiferromagnet with the Hamiltonian H
H=J
NA,NB
∑
〈ij〉
(∆ (SixSjx+SiySjy)+SizSjz)−H
NA
∑
i
Siz+
NB
∑
j
Sjz
+D
NA
∑
i
S2
iz+
NB
∑
j
S2
jz
. (1)
It describes NA and NB interacting classical spins ~Si on two sublattices A and B, where NA = NB .
The anisotropy of the positive exchange interaction J is characterised by the parameter ∆ with
∆ = 0 for a magnet where only the z-components of the three dimensional spins interact and
∆ = 1 for an isotropic magnet. The spins are subjected to an external magnetic field H and a
single-ion anisotropy D.
Figure 1. Left: The phase diagram for the XXZ-Model including the single-ion-anisotropy. Right:
The order parameter of the four ground states configuration can be calculated analytically.
The case where beside the exchange anisotropy only a single-ion anisotropy is present can be
solved analytically [14] at T = 0. A BC ground state exists for a certain region of positive D and
external magnetic field H for ∆ = 0. This region becomes smaller for increasing ∆ and diminishes
for an isotropic antiferromagnet with ∆ = 1. The transition lines between the AF to the PM and
between the AF and the SF ground state are of the first order. The other transitions are of the
second order. Figure 1 shows the whole ground state diagram with the parameters chosen so that a
BC ground state appears. Here we divide the parameters H, D, F by the number of next neighbors
(z = 6 for simple cubic system), so in order to compare with the results in [13,15] one has to
multiply all the three parameters in this paper with 6. Also the OP for the complicated BC ground
state can be calculated exactly which reads:
cos θA,B(H, D, ∆)=
H ±
√√
(1−2D)2−∆2((2D+1)2+H2−∆2)+2H(4D2+∆2−1)√
(1−2D)2−∆2
−
√
(1−2D)2−∆2
4D
. (2)
An extension to finite temperature within mean field theory has been discussed in [3]. As
already explained in [11] and used in [14] a quantal lattice gas can be mapped on an anisotropic
antiferromagnet with a Hamiltonian of form (1). The correspondence of different ground states is
given in table 1.
In the H − T ground state diagram the ground states at T = 0 extend as phases to higher
temperature. Since for high enough temperature and large enough external magnetic field the PM
phase should be the stable one, all other phases are confined to finite regions in the phase diagram.
690
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy
Table 1. Correspondence between the phases of the classical magnetic system and those of the
quantum lattice gas (QLG). Also shown are the order parameters, the alternating magnetization
in the field direction N‖ and perpendicular to it N⊥, and the components of the magnetization
M‖ and M⊥ .
Phase N‖ N⊥ M‖ M⊥ QLG
AF 6= 0 0 0 0 solid
SF 0 6= 0 6= 0 0 superfluid
BC 6= 0 6= 0 6= 0 6= 0 super solid
PM 0 0 6= 0 0 fluid
Thus it happens that the phase transition lines intersect each other in multicritical points. This
might be a bicritical point when the AF, the SF and the PM phase meet, or a tetracritical point
where the AF, the SF, the BC and the PM phase meet. It should be noted that this is even possible
if the BC phase does not extend to T = 0 [3].
3. Cubic anisotropy
Uniaxial antiferromagnets may also have cubic anisotropy. Here we consider an anisotropic
antiferromagnet with a cubic anisotropy F instead of the single-ion anisotropy D. For such a
system the Hamiltonian reads
H = J
NA,NB
∑
〈ij〉
(∆ (SixSjx + SiySjy) + SizSjz) − H
NA
∑
i
Siz +
NB
∑
j
Sjz
+ F
NA
∑
i
(
S4
ix + S4
iy + S4
iz
)
+
NB
∑
j
(
S4
jx + S4
jy + S4
jz
)
. (3)
TR2
PM
AF
BC1
-12 -10 -8 -6 -4 -2
0.0
0.5
1.0
1.5
2.0
F
H
D=0, D=0.8
PM
SF
BC2AF
BC1
T1
T2
CE1
CE2TR1
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
0
0.5
1
1.5
2
F
D=0, D=0.8
H
Figure 2. Ground states minimizing the energy, equation (3), in the magnetic field H and cubic
anisotropy F plane at an exchange anisotropy ∆ = 0.8 and with zero single ion anisotropy D.
The ground states are denoted as follows: AF antiferromagnetic, PM paramagnetic, SF spin flop
; BC1,2 biconical oriented along the diagonal or the axis in the x − y-plane. Shown are special
points where either more than two ground states meet or where the order of the transition line
changes from the first (dashed line) to the second order (solid line). These are the triple points
TR1,2, the tricritical points T1,2 and the critical end points CE1,2.
The effect of such a type of the cubic anisotropy has been studied recently by [13,15]. It has
been shown that two types of BC ground states are possible (BC1 for F < 0 and BC2 for F >
0), depending whether the perpendicular component of the alternating magnetization lies in the
691
T.-C. Dinh, R. Folk
diagonals or along the axes in the xy-plane. Moreover, it has been found that most of the transition
lines between the ground states become first order. There was also found a reentrant region for
the SF ground state becoming unstable for increasing the external magnetic field with respect to
the BC state, which for even larger field again becomes unstable with respect to the SF phase (see
figure 2). In this reentrant region the transition line has been found to be of second order.
SF
BC1
SF
1.05 1.10 1.15 1.20 1.25
0.80
0.85
0.90
0.95
1.00
1.05
H
Θ A
,Θ
B
F=-0.155, D=0, D=0.8
Figure 3. A 3D and a 2D diagram showing the occurrence of the reentrant region by increasing
the magnetic field H.
We verify the occurrence of the reentrant region [13] by calculating the order parameter of the
ground states which take place in the vicinity of this region (see figure 3). The 2D figure shows
that the SF-BC1 occurs by increasing the applying field H . In case of further increasing the field,
the SF reappears again.
-0.16 -0.14 -0.12 -0.10 -0.08 -0.06
1.0
1.1
1.2
1.3
1.4
1.5
1.6
F
H
D=0, D=0.8
T1
CE1
TR1
HHD, D, FL = 1 + D + 2 D + 4 F
-0.15 -0.10 -0.05 0.00
1.2
1.3
1.4
1.5
1.6
1.7
1.8
F
H
D=0; D=0.8
Figure 4. Left: Detail from the right part of figure 1 on a larger scale around the reentrant
region. Solid lines are of the second order, dashed lines of the first order. The second order
reentrant line between the biconical and spin flop ground state starting at the tricritical point
TR1 ends in the critical end point CE1 on the transition line to the paramagnetic ground
state. On the transition line between the paramagnetic and spin flop ground state one finds the
tricritical point T1. Right: Comparison of the transition line between the SF and PM ground
state calculated on the assumption of a second order transition (straight line) and the transition
line calculated numerically by minimizing the energy. Indeed when the transition curve deviates
from the second order line the transition becomes the first order.
In order to find out the transition points where a change in the order of the transition takes
place we have to calculate the nonzero order parameter component along the border lines for the
case treated in [13]. Along the transition line between the BC1 and the PM ground state the
parallel staggered magnetization N‖ goes to zero at the point where the SF–PM and the BC1–SF
transition lines meet (see figure 2). The behavior of the OP is compatible with a power law with
692
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy
a correction term
|N⊥,‖| = A
∣
∣
∣
∣
|F (H) − F ?|
F ?
∣
∣
∣
∣
0.5 (
1 + B
∣
∣
∣
∣
F (H) − F ?
F ?
∣
∣
∣
∣
)
, (4)
where F ? is the value of the cubic anisotropy at the special point, and F (H) the value of the cubic
anisotropy along the transition line. A, B and F ? are fit parameters. This point is a critical end
point (CE1). From this critical endpoint the second order reentrant line starts. This line ends in a
point where the AF, the BC1 and the SF ground state meet. This point turns out to be a triple
point (TR1) since the second order character of the reentrant line is changed to the first order (at
T2) before it reaches the triple point. The value of N‖ becomes nonzero on the reentrant line at
T2 and this point might be a tricritical point (see figure 5).
We also find a change of the order along the transitions line between the SF and the PM ground
state. At T1 the transition becomes the second order for larger values of F (H). This is just at
the point where the transition line begins to deviate from the analytically calculated second order
transition line (see figure 5).
-0.190 -0.185 -0.180 -0.175 -0.170 -0.165
0.0
0.1
0.2
0.3
0.4
0.5
FHHSF-BC1L
ÈN parÈ
D=0; D=0.8
-0.190 -0.185 -0.180 -0.175 -0.170 -0.165
0.0
0.1
0.2
0.3
0.4
F HHBC1-PML
ÈN parÈ
D=0; D=0.8
Figure 5. Left: Change of the parallel order parameter |N‖| going along the first order transition
lines between the BC1 and PM ground states. The dots are the calculated values and the solid
line is a fit according to equation (4). The zero value of the parallel OP is reached at the critical
end point CE1. Right: Similar the change of the perpendicular order parameter |N⊥| going
along the first order transition lines between the SF and PM ground state. The zero value of
the perpendicular OP is reached at the tricritical point T1.
Conclusively we summarize all the critical points in a table (see table 2)
Table 2. Summary of all the special points discovered in the ground state diagram for D = 0
and ∆ = 0.8. There exist tricritical points (T), tripel points (TR) and the critical end points
(CE).
Critical Points Coordinates (F, H) Critical Points Coordinates (F, H)
T1 (−0.087747, 1.49) TR2 (−12, 1)
T2 (−0.167755, 1) CE1 (−0.164823, 1.26)
TR1 (−0.6, 1.6) CE2 (0, 0.6)
4. Single ion and cubic anisotropy
Let us now consider the general case when the exchange, single-ion and cubic anisotropy are
present
H = J
NA,NB
∑
〈ij〉
(∆ (SixSjx + SiySjy) + SizSjz) − H
NA
∑
i
Siz +
NB
∑
j
Sjz
693
T.-C. Dinh, R. Folk
+ D
NA
∑
i
S2
iz +
NB
∑
j
S2
jz
+ F
NA
∑
i
(
S4
ix + S4
iy + S4
iz
)
+
NB
∑
j
(
S4
jx + S4
jy + S4
jz
)
. (5)
Then the planar ground state regions extend to three dimensional regions and whole lines of mul-
ticritical points appear. In figure 7 we give as an example the case where the single ion anisotropy
has been chosen nonzero (∆ = 0.8) and a sketch of the three dimensional ground state diagram.
An analytic result for the transition line between the SF ground state and the PM ground state
can be calculated assuming a second order transition, which of course is just correct in certain
section of the transition line (see figure 4). The relation reads
H(∆, D, F ) = 2D + 4F + ∆ + 1. (6)
Also in zero magnetic field (H = 0) the stable ground states as a function of the anisotropies can
be calculated analytically. The transition between the AF and the BC2 ground state starts at a
positive value of the cubic anisotropy F
F (D, ∆) = (1 − 2D − ∆)/4. (7)
The transition occurs just only in the region D < 1−∆
2 and is continuous (see figure 6), where the
OP parameter of the biconical ground state (BC2) can be given.
θA,B = arccos
(
±1 + 2F − 2D − ∆
6F
)
. (8)
This 2nd order transition continues to be also valid when applying the magnetic field. Increasing
the single ion isotropy ( D > 1−∆
2 ) the Spin-Flop is preferred to the BC2 and the transition takes
place at
F (D, ∆) =
−1 + 2D + ∆
2
. (9)
If the cubic anisotropy is zero, no transition to the BC1 ground state takes place.
AF
BC2
SF
0.00 0.02 0.04 0.06 0.08 0.10
-0.10
-0.05
0.00
0.05
0.10
0.15
0.20
F
D
Single Ion and Cubic Anisotropy; H=0; D=0.8
Figure 6. Left: The analytically calculated ground state diagram for H = 0. Right: The contin-
uous transition from AF to BC2. The two polar angles, from which the OP can be calculated,
of the two different sublattices are shown.
For negative cubic anisotropy in zero magnetic field the transition from the AF to the SF state
takes place at
D(F, ∆) =
1
2
(1 − ∆) (10)
and is independent of F .
694
Ground states of anisotropic antiferromagnets with single ion and cubic anisotropy
5. Conclusion and outlook
A system of an anisotropic antiferromagnet shows lots of interesting ground states and critical
points. It can also be mapped into a quantum lattice system where the biconical ground state is
interpreted as the supersolid state. We have shown that for the simple case where just the exchange
(∆) and the single-ion anisotropy (D) are considered, one can analytically calculate all the ground
states. In the case with cubic anisotropy the resulting Hamiltonian is more difficult to be solved,
so that we have to calculate the ground states numerically. In this system two kinds of biconical
ground states appear, the BC1 for F < 0 and BC2 for F > 0.
Most of the transition lines are of the first order, but in some cases the order of the transition also
changes along the transition line. So, lots of interesting critical points appear ( critical endpoint,
tricritical point). For example, along the transition line between BC1 and SF for F < 0 a reentrance
region can be shown where the order of the transition changes from the second to the first order
by increasing the absolute value of the cubic anisotropy.
However, in the absence of the field H we can derive the transition surfaces analytically, because
the term with the applied field (H) is the part of the Hamiltonion which breaks the symmetry and,
therefore, causes this Hamiltonian unsolvable. The BC2 ground state does also exist at H = 0 for
a large value of F .
The topology of the ground state diagram at T = 0 determines the possible multicritical points
at finite temperatures. Without cubic anisotropy bicritical or tetracritical points are possible where
two lines of the second order phase transitions with order parameters of dimension n = 1 and n = 2
meet. Cubic anisotropy may change this picture and multicritical point where two lines with n = 1
and n = 3 or n = 3 and n = 2 might meet (see figures 2,7 for positive values of F ). Cubic anisotropy
changes most of the transitions to the first order, however on the transition line separating the BC1
from the SF ground state, part of this line starting at a critical end point is of the second order
type. Moreover, part of the transition line separating the spinflop state from the paramagnetic
state is of the second order.
PM
BC1
SF
AF
BC2
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4
0.0
0.5
1.0
1.5
2.0
F
H
D=0.05, D=0.8
PM
AF
SF
BC2
BC1
Figure 7. Left: Ground states minimizing the energy, equation (5), in the magnetic field H and
cubic anisotropy F plane at an exchange anisotropy ∆ = 0.8 and with single ion anisotropy
D = 0.05 . Right: A 3D-phase diagram H(F,D) with fix single-ion-anisotropy ∆ = 0.8.
Acknowledgements
We thank W. Selke for valuable discussions. This work was supported by the Fonds zur
Förderung der wissenschaftlichen Forschung under Project No. P19583–N20.
695
T.-C. Dinh, R. Folk
References
1. Shapira Y. Multicritical Phenomena. Plenum, New York and London, 1983, p. 35;for a more recent
review see Aharony A., J. Stat, 2003, 110, 659.
2. Bruce A.D., Aharony A., Phys. Rev. B, 1975, 11, 478.
3. Liu K.-S., Fisher M.E., J. Low Temp. Phys., 1973, 10, 655.
4. Nelson D., Kosterlitz J.M., Fisher M.E., Phys. Rev. Lett., 1974, 33, 813.
5. Prudnikov V.V., Prudnikov P.V., Fedorenko A.A., JETP Lett., 1998, 68, 950.
6. Calabrese P., Pelissetto A., Vicari E., Phys. Rev., 2003, 67, 054505.
7. Selke W., Holtschneider M., Leidl R., Wessel S., Bannasch G. Computer Simulation Studies in Con-
densed Matter Physics XXI, eds. D. P. Landau, S.P. Lewis, and H. B. Schüttler. Springer-Verlag,
Heidelberg, Berlin, 2008.
8. Holtschneider M. Thesis RWTH Aachen, 2007.
9. Folk R., Holovatch Yu., Moser G., Phys. Rev. E, 2008, 78, 041124.
10. Folk R., Holovatch Yu., Moser G., Phys. Rev. E, 2008, 78, 041125.
11. Fisher M.E., Rep. Prog. Phys., 1967, 30, 615.
12. Peters D., McCulloch I.P., Selke W., Phys. Rev. B, 2009, 79, 132406.
13. Bannasch G., Selke W., Eur. Phys. J. B, 2009, 69, No. 3, 432 [Preprint arXiv:cond-mat/0807.1019].
14. Matsuda H., Tsuneto T., Prog. Theor. Phys., 1970, 40, 411.
15. Bannasch G. Diploma Thesis RWTH Aachen, 2008.
Основнi стани анiзотропних антиферомагнетикiв з
одно-iонною та кубiчною анiзотропiєю
Т.-Ц. Дiнг, Р. Фольк
Iнститут теоретичної фiзики, Унiверситет iм. Йогана Кеплера, 4040 Лiнц, Австрiя
Отримано 15 червня 2009 р., в остаточному виглядi – 1 липня 2009 р.
Анiзотропнi антиферомегнетики у зовнiшнiх магнiтних полях проявляють велике рiзноманiття основ-
них станiв, що межують на лiнiях фазових переходiв i в мультикритичних точках. Тут дослiджується
залежнiсть основних станiв у таких тривимiрних системах вiд параметрiв обмiнної взаємодiї, одно-
iонної та кубiчної анiзотропiй. Iндентифiковано чотири рiзнi основнi стани: парамагнiтний (ПМ), анти-
феромагнiтний (АФ), з нахиленими спiнами (НС) та двоконусний (ДК). У випадку вiдсутностi кубiчної
анiзотропiї лiнiї фазових переходiв, що роздiляють рiзнi основнi стани, можуть бути розрахованi
аналiтично; в iншому випадку їх можна розрахувати лише чисельно. Розглянуто також поведiнку
ступiнчастої намагнiченостi, типову для рiзних основних станiв. Аналiзуючи її, визначено рiд фазо-
вого переходу з одного стану до iншого. Проте рiд фазового переходу змiнюється також вздовж
лiнiї переходу при включеннi кубiчної анiзотропiї, особливо в подвiйнiй областi , де при збiльшен-
нi зовнiшнього поля H вiдбувається спершу перехiд вiд НС до ДК, а далi - знову до НС. Знайдено
мультикритичнi точки, якi слiд думати будуть трикритичними або ж критичними кiнцевими точками.
Отриманi результати можуть бути важливими i для iнших систем, оскiльки антиферомагнiтну мо-
дель можна вiдобразити у модель граткового газу, де двоконусний основний стан iнтерпретується
як надплинний твердотiльний. Нещодавнi ренормгруповi розрахунки вказують, що така фаза мала
б сигналiзувати про iснування чотирикритичної точки.
Ключовi слова: антиферомагнетизм, квантовий гратковий газ, основнi стани
PACS: 75.10.Hk, 75.30.Kz, 67.80.bd
696
|