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Interlayer-interaction dependence of latent heat
in the Heisenberg model on a stacked triangular lattice
with competing interactions

Ryo Tamura and Shu Tanaka
Physical Review E 88, 052138 (2013)
ructure leads to exotic electronic properties such
ic phenomena [23–27], the anomalous Hall effect
alization of electronic wave functions [29]. Thus
s of frustrated systems have attracted attention
physics and condensed matter physics. Many
y frustrated systems such as stacked triangular
nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their
ave been investigated. In theoretical studies, the
een phase transition and order parameter space in
y frustrated systems has been considered [30–34].
mple of phase transition nature in geometrically
stems, properties of the Heisenberg model on
lattice have been theoretically studied for a
riangular antiferromagnetic systems are a typical
geometrically frustrated systems and have been
gated. The ground state of the ferromagnetic
model on a triangular lattice is a ferromagnetically
n structure. In this case, the order parameter space
ng-range order of spins does not appear at finite
because of the Mermin-Wagner theorem [35].
oes not exhibit any phase transitions. In contrast,
,37] reported that a topological phase transition
e Heisenberg model on a triangular lattice with
omagnetic nearest-neighbor interactions. In this
ong-range order of spins is prohibited by the
gner theorem and thus a phase transition driven
-range order of spins never occurs as well as
magnetic Heisenberg model. Since the ground

in two dimensions. In these systems, the order parameter
space is described by SO(3). The temperature dependence
of the vector chirality and that of the number density of Z2
vortices in the Heisenberg model on a kagome lattice were
also studied [38]. An indication of the Z2 vortex dissociation
has been observed in electron paramagnetic resonance and
electron spin resonance measurements [39–41].
Phase transition has been studied theoretically in stacked
triangular lattice systems as well as in two-dimensional
triangular lattice systems. In many cases, the phase transition
nature in three-dimensional systems differs from that in

Main Results

INTERLAYER-INTERACTION DEPEN

We studied the phase transition nature of a frustrated Heisenberg
(a) 10
model on a stacked triangular lattice.
0
10
0
10
0
10
0
10

axis 3

We found that a first-order phase
transition occurs. At the first-order
phase transition point, SO(3) and C3
symmetries are broken.

@nims.go.jp
m.s.u-tokyo.ac.jp

13/88(5)/052138(9)

0
10

axis 2
axis 1

FIG. 1. (Color online) Schematic picture of a stacked triangular
lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent
the nearest-neighbor and third-nearest-neighbor interactions in each
triangular layer and J⊥ is the interlayer interaction.

052138-1

©2013 American Physical Society

0
10
0
20
10
0
20
10

The transition temperature increases
but the latent heat decreases as the
interlayer interaction increases.

0
20
10
0

-2.6 -2.4 -2.2

-2

-1.8 -1.6 -1.4
Background
Unfrustrated systems (ferromagnet, bipartite antiferromagnet)
Ferromagnet

Bipartite antiferromagnet

Model
Ising

Order parameter space
Z2

XY
Heisenberg

U(1)
S2
Background
Frustrated systems
Antiferromagnetic Ising model
on triangle

Antiferromagnetic XY/Heisenberg model
on triangle

?
triangular lattice

kagome lattice

pyrochlore lattice
Background
Order parameter space in antiferromagnet on triangular lattice.

Model
Ising
XY

Order parameter space
--U(1)

Phase transition
--KT transition

Heisenberg

SO(3)

Z2 vortex dissociation
SO(3) x C3 & SO(3) x Z2
(ii) single-k spiral

(a)

(c)
structure

sp
ira

l

4 independent
sublattices

)t
rip
lek

structure

(iv

axis 3 axis 2
axis 1

(b)
(iii) double-k spiral

(ii) single-k spiral

(i) ferromagnetic

R. dotted hexagonal area in Soc. The
Fig. 1. (a) Triangular lattice with L x × Ly sites. (b) Enlarged view of theTamura and N. Kawashima, J. Phys. (a). Jpn., 77, 103002 (2008).
R. Tamura and N. Kawashima, J. i-th site are
thick and thin lines indicate λJ1 and J1 , respectively. The third nearest-neighbor interactions at thePhys. Soc. Jpn., 80, 074008 (2011).
R. Tamura, S. Tanaka, and N. be categorized into
depicted. (c) Ground-state phase diagram of the model given by Eq. (1). Ground states canKawashima, Phys. Rev. B, 87, 214401 (2013).
R. Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12.
five types. More details in each ground state are given in the main text.

J1-J3 model on triangular lattice
discussed the connection between frustrated continuous spin systems and a fundamental discrete spin
Order parameter space
Order of phase transition
system by using a locally defined parameter. The most famous example is the chiral phase transition
in the antiferromagnetic XY model on a triangular lattice. The relation between the phase transition
SO(3)xC3 that of the Ising model has been established [24, 25]. In this paper,
1st order
of the continuous spin system and
we study finite-temperature properties in the J1 -J3 model on a distorted triangular lattice depicted in
SO(3)xZ2
2nd order (Ising universality)
Figs. 1(a) and (b) from a viewpoint of the Potts model with invisible states.
Motivation

To investigate the phase transition behavior in three-dimensional
systems whose order parameter space is described by the direct
product between two groups A x B.
malous Hall effect
space is described by SO(3). The temperature dependence
nctions [29]. Thus
of the vector chirality and that of the number density of Z2
ttracted attention
vortices in the Heisenberg model on a kagome lattice were
er physics. Many
also studied [38]. An indication of the Z2 vortex dissociation
tacked triangular
has · sj observed ini electronJparamagnetic sresonance : and
H = J1
si been J3
s · sj
si · j
si Heisenberg spin
me antiferromagelectron spin resonance measurements [39–41].
(three components)
i,j 3
i,j
hesized and their i,j 1
Phase transition has been studied theoretically in stacked
etical studies, the
triangular lattice systems
well as in two-dimensional
1st nearest- 3rd nearest- as1st nearestarameter space in
triangular neighbor
lattice systems. In many cases, the phase transition
neighbor
neighbor
nsidered [30–34].
nature in three-dimensional systems differs from that in
e in geometrically
intralayer
intralayer
interlayer
enberg model on
ly studied for a
tems are a typical
ms and have been
he ferromagnetic
ferromagnetically
r parameter space
ot appear at finite
er theorem [35].
tions. In contrast,
l phase transition
gular lattice with
teractions. In this
axis 2
prohibited by the
transition driven
axis 1
ccurs as well as
axis 3

Model
Ground State
Spiral-spin configuration

si = R cos(k · ri )

I sin(k · ri )

R, I are two arbitrary orthogonal unit vectors.

Fourier transform of interactions
J(k)
=
N

J1 cos(kx )
J3 cos(2kx )

2J1 cos

1
kx cos
2

3
ky
2

2J3 cos(kx ) cos( 3ky )

J cos(kz )

Find k that minimizes the Fourier transform of interactions!
We consider the case for J > 0 .

kz = 0
Ferromagnetic (J1>0)
Ground-state properties
Ferromagnetic (S2)
-1/4

Spiral-spin structure (SO(3)xC3)
0

J3 /J1
Ferromagnetic (J1>0)
Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)
J3 /J1

RYO TAMURA AND SHU TANAKA
-1/4
0

3-dim Heisenberg
(a)
universality class

(b)
Ferromagnetic (J1>0)
Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)

A

-1/4
(b) (b)

0

PHYSICAL REVIEW E E 88, 052138 (2013) 3 /J1
PHYSICAL REVIEW 88, 052138 (2013) J

???
1 1

(c)
(c)

0.80.8

0.60.6

0.40.4

0.20.2

0 0
-4 -4 -3 -2 -1 0 0 1 1
-3 -2 -1
Antiferromagnetic (J1<0)
Ground-state properties
Degenerated GSs
-1/9
order by disorder
Th. Jolicoeur et al., Phys. Rev. B, 42, 4800 (1990).

120-degree structure (SO(3))
0
chiral universality

J3 /J1

H. Kawamura, J. Phys. Soc. Jpn., 54, 3220 (1985).
H. Kawamura, J. Phys. Soc. Jpn., 61, 1299 (1992).
A. Pelissetto et al., Phys. Rev. B, 65, 020403 (2001).
P. Calabrese et al., Phys. Rev. B, 70, 174439 (2004).
A. K. Murtazaev and M. K. Ramazanov, Phys. Rev. B 76, 174421 (2007).
G. Zumbach, Phys. Rev. Lett. 71, 2421 (1993).
M. Tissier et al., Phys. Rev. Lett. 84, 5208 (2000).
M. Zelli et al., Phys. Rev. B 76, 224407 (2007).
V. T. Ngo and H. T. Diep, Phys. Rev. B, 78, 031119 (2008).

We focus on the case for ferromagnetic J1 and J3/J1 > -1/4 in which
the order parameter space is described by the direct product
between two groups.
Possible Scenarios
PHYSICAL REVIEW E space is AxB.
Let us consider the system whose order parameter 88, 052138 (2013)

(a) No symmetry is broken.
PHYSICAL REVIEW E 88, 052138 (2013)
(a)

Disordered phase

(a)

Disordered phase

(b)

Partially ordered phase
Disordered
(b) Only one of two symmetries is broken. phase
(b)

Partially ordered phase A is broken.
Disordered phase

Partially ordered phase

Disordered phase
A is broken.

Partially ordered phase B is broken.
Disordered phase

(c)
(i)

B is broken.

Ordered phase

Disordered phase

T
T
T
T

T
T
A is broken.
A is broken.
Partially ordered phase
Disordered phase
Partially ordered phase
Disordered phase

Possible Scenarios

T
T

B is broken.
(c) Both symmetries are broken.
B is broken.

(c-1) Both symmetries are broken simultaneously.
(c)
(c)
(i)
(i)

Ordered phase
Ordered phase

Disordered phase
Disordered phase

T
T

A and B are broken.
A and B are broken.

(c-2) Both symmetries are broken successively.
(ii)
(ii)

Ordered phase
Ordered phase

Partially ordered phase Disordered phase
Partially ordered phase Disordered phase

B is broken.
A is broken.
B is broken.
A is broken.
Ordered phase Partially ordered phase Disordered phase
Ordered phase Partially ordered phase Disordered phase
A is broken.
A is broken.

B is broken.
B is broken.

T
T

T
T
(b)

10

Internal Energy & Specific Heat

(a)
-2.3
5
30
40
-2.1
0.1
(d)
(b)
0 s ·s
H 20 J1
=
si · sj J3
si · sj J
-2.3 i 0.05j
-2.2
-2.1
25
J3 /J1 = 0.85355 · · · , J /J1 = 2
i,j 1
i,j 3
i,j
30
-2.2
-2.2
0
INTERLAYER-INTERACTION DEPENDENCE 30 45
OF LATENT . .
0
15
20
1.55
10
30
-2.1

-2.3
0
-2.1
40

Internal energy (a)
(c)
(b)

0.02
30
-2.2
0.01

20
-2.3
0
40 1.52
10

(a)
1.53

1.54

1.55

(b)

20 (e)
1.54
Specific heat
15
-2.3
40
30
1.53
10
10 0 0.1
0.00004
0.05
25
30
5
60
0 (f) 0 0
15
30
45
20
40
0
20
0.02
20
-2.3
-2.2
15
0
0
20000
40000
0.01
10
10
1.55

(e)

(a)

(d)
(b)
0.000

(c)

-2.1

60000

1.54
50
0
FIG. 4. (Color online) Temperature1.52
1.53
1.54
1.55
1.53 dependence of (a) intern
(c)
(c)
0
0.00004
0.000
20 per site E/J , (b) specific 0.020
0.02
heat C, and -2.2 order param
(c)
energy
-2.3
-2.1
1
60
C3 symmetry breaking of th
eter |µ|2 , which can detect the 0.01 (f)
0.01
FIG. 4. (Color online) Tempe
1.55
10
30
0

Phase transition occurs.
(b)

Order Parameter (C3)
30
H=

J1
i,j

1

si · sj

J3
i,j

3

si · sj

INTERLAYER-INTERACTION D
J

20

i,j

si · sj

J3 /J1 =

PHYSICAL REVIEW E 88, 052138 (2013)
(b)

0.85355 · · · , J /J1 = 2

-2.1
10

1

0.8

0.6

0.4

0.2

(c)

-2.2 Order parameter
0

0.02
-2.3
40
0.01
30
0

0
-4 -3 -2 -1

20

0

(c)
(a)
(b)

1.52
1

1.53

1.54

1.55

nd-state properties when the nearest-neighbor interaction J1 is ferromagnetic. (a) Position of k∗ ,
actions in the wave-vector space for J3 /J1 −1/4. The hexagon represents the first Brillouin Tamura and N. Kawashima, J. Phys. Soc. Jpn., 77, 103002 (2008).
R.
∗
figuration in each triangular layer is shown. (b) Position of k and the corresponding schematic Tamura and N. Kawashima, J. Phys. Soc. Jpn., 80, 074008 (2011).
10 55 . . .R.
layer when J3 /J1 < −1/4. The spin configurations are depicted for J3 /J1 = −0.853
c) The J3 /J1 dependence of k ∗ .

appropriate way,

FIG. 4. (Color online) Temp
energy per site
C3 symmetry is broken. E/J1 , (b) spec

energy is observed at a certain temperature. In addition, the
0
-2.1

Energy Histogram

YER-INTERACTION DEPENDENCE ·OF -2.2
LATENT . . .
H= J
s ·s
J
s ·s
J
s s
1

i

i,j

j

3

i

i,j

1

j

E/kB T

E(L) : width between
two peaks (a)

(b)

j

J3 /J1 =

i,j

3

30

P (E; T ) = D(E)e
D(E) : density of states

i

25
20

-2.3
40

0.1
0.05
0

0

0.85355 · · · , J /J1 = 2

(d)

30
15

30

45

20

15
10

10

0

5
0
-2.3
1.55

0.02

-2.2

0.01

-2.1

First-order phase transition occurs.
0
(e)
5

15

0
-2.3

SO(3) × C3 . -2.2was confirmed
It
-2.1
10
the first-order phase transition
1.55
10
Heisenberg model on a stack
H5
= J1
si · sj J3
si · sj J 1.54 (e) · sj
si
a
J /J = 0.85355
, J /J = 2
i,j
i,j nearest-neighbor· · ·interactio
0 i,j
0
1.53
-2.3
-2.2
-2.1(c)
0
0.00004
space is SO(3), a single0.00008 i
peak
0.02
Tc(L)
Max of specific heat
dependence of the specific heat
1.55
60
(e)
0.01
40 (f)
finite-temperature phase transi
1.54
20
state and magnetic ordered sta
0
1.53
0
0
0.00004
0.00008
20000
40000
60000
1.52
1.53
1.54
1.55
is 0broken. Then, in our mode
60
break at the first-order phase tr
(f)
( E)2 Ld
40
Tc (L) = (Color online) Temperature dependence ofpeakinternal
aL d + Tc
heat Cmax (L)single c2(a) corresp
has a
4T
20 FIG. 4.
0
heat (1986).
energy per site M. S. S. Challa,,D. P.(b) and K. Binder, Phys.transition. To confirm this w
E/J1 Landau, specific Rev. B, 34, 1841C, and (c) order param0
20000
40000
60000
2
C3 symmetry the structure fac
eter |µ| , which can detect the dependence ofbreaking of the
20

Finite-size Scaling

1

3

3

1

1

model with J3 /J1 = −0.853 55 . . . and J⊥ /J1 = 2 for L = 24,32,40.
1
mperature dependence distribution of the internal energy P (E; Tc (L)). The
of (a) internal
(d) Probability
S(k) :=
s
ecific heat C, andthe lattice-size dependence of the width between bimodal
N i,j
inset shows (c) order paramthe C3 symmetry breaking of of Tctransition occurs. . (f) Plot
Plot the
peaks E(L)/J1 . (e) phase (L)/J1 as a function of L−3
First-order
3
. . . and J /J = 2 for L = 24,32,40.
phase transition in

factor S(k∗ ) increases. The structure factors at kz = 0 in the
first Brillouin zone at several temperatures for L = 40 are also
shown in Fig. 5(b). As mentioned in Sec. II, the spiral-spin
structure represented by k is the same as that represented by
−k in the Heisenberg models. Figure 5(b) confirms that one
si · sj J3
si · sj J
si · sj
distinct wave vector is chosen from three types of ordered

Order Parameter (SO(3))

(9)

H=

J1

(10)

i,j

nsition temperature
mit. The coefficient
stant. Figures 4(e)
)/J1 and Cmax (L),
is a nonzero value
shows an almost
n of L3 . However,
btain the transition
dynamic limit with
ize effect. Next we
the width between
hown in Fig. 4(d).
ented by E(L) =
are the averages of
re phase and that in
he thermodynamic
δ function and then
Fig. 4(d) shows the
he width enlarges as
hat the latent heat is

1

i,j

(a)

J3 /J1 =

i,j

3

0.85355 · · · , J /J1 = 2

0.5
0.4
0.3
0.2
0.1
0

Order parameter
0

0.5

1

1.5

2

(b)
10 -1
10 -2
10 -3
10 -4
10 -5

FIG. 5. (Color online) (a) Temperature dependence of the largest
value of structure factors S(k∗ ) calculated by six wave vectors in
Eq. (4) for J3 /J1 = −0.853 55 . . . and J⊥ /J1 = 2. Error bars are
omitted for clarity since their sizes are smaller than the symbol size.

SO(3) symmetry is broken at the phase
transition temperature.
Dependence on Interlayer Interaction
H=

which is
rst-order

a phase
nterlayer
[62–65],
he model
For large
hase and
occurs at
arameter
tion with
the result
systems,
when the
mperature
he J1 -J3
der phase
urs when
simplest
ymmetry
del [76].
mensions
nder that
previous

TION

J1
i,j

1

si · sj

J3

si · sj

PHYSICAL REVIEW E 88, 052138 (2013)

i,j

3

(a)

-1.5
-2

J

0.50

0.25

0.75

1.00

1.25

-2.5

1.50

1.75

2.00

2.25

2.50

(b)

increases

0.75
1.00

1.25

1.50

0.25

0.50

0.75

1.00

1.25

1.50

1.75

0
10

2.50

2.00 2.25 2.50

1

0
10
0
10

1.75 2.00

2.25

0
0.85

1.5

0
10

0.50

20

(b)

0
10

0.25

J /J1

(a) 10

0
10

-3
40

si · sj
INTERLAYER-INTERACTION DEPENDENCE OF LA
J3 /J1 = 0.85355 · · · , J /J1 = 2
i,j

0.8

0

0.75

20

0.7

10
(c)

0.15

0.25

0.50

0.1

0.75

1.00

1.25

1.50

(d)
1.75

2.00

2.25

0.05

2.50

0.2

0.50 0.75

1.00

1.25

1.50

1.75

2.00

2.25

(e)

2.50

20
0.04

0
20
10

0.1
0

0.08

0
10

0
0.25

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0

(c)
-2.6 -2.4 -2.2

-2

-1.8 -1.6 -1.4

0

0

1
Dependence on Interlayer Interaction

H = J1
si · sj J
CTION DEPENDENCE OF LATENT . . . 3
i,j 1
i,j

-1.8 -1.6 -1.4

(b)
Transition
temperature
1.5

1

J

si · s REVIEW E 88, 052138 (2013)
PHYSICAL j
J3 /J1 = 0.85355 · · · , J /J1 = 2
i,j

In Sec. IV, we investigated the interlayer interaction effect
on the nature of phase transitions. We confirmed that the
first-order phase transition occurs for 0.25 J⊥ /J1 2.5 and
J3 /J1 = −0.853 55 . . ., which was used in Sec. III. We could
not determine the existence of the first-order phase transition
for J⊥ /J1 < 0.25 or J⊥ /J1 > 2.5 by Monte Carlo simulations.
In the parameter ranges, the width of two peaks in the probability distribution of the internal energy cannot be estimated easily
because of the finite-size effect. It is a remaining problem to
determine whether a second-order phase transition occurs for
large J⊥ /J1 as in the J1 -J2 Heisenberg model on a stacked
triangular lattice [62]. As the ratio J⊥ /J1 increases, the firstorder phase transition temperature monotonically increases
but the latent heat decreases. This is opposite to the behavior
observed in typical unfrustrated three-dimensional systems
that exhibit a first-order phase transition at finite temperature.
For example, the q-state Potts model with ferromagnetic
intralayer and interlayer interactions (q 3) is a fundamental
model that exhibits a temperature-induced first-order phase
transition with q-fold symmetry breaking [76]. From a meanfield analysis of the ferromagnetic Potts model [76,83], as the
interlayer interaction increases, both the transition temperature

As the interlayer interaction
increases, ...
transition temperature
increases.

0.5

0.08

latent heat decreases.

0.04

0

3

si · sj

(c)
Latent heat
0

1

2
ructure leads to exotic electronic properties such
ic phenomena [23–27], the anomalous Hall effect
alization of electronic wave functions [29]. Thus
s of frustrated systems have attracted attention
physics and condensed matter physics. Many
y frustrated systems such as stacked triangular
nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their
ave been investigated. In theoretical studies, the
een phase transition and order parameter space in
y frustrated systems has been considered [30–34].
mple of phase transition nature in geometrically
stems, properties of the Heisenberg model on
lattice have been theoretically studied for a
riangular antiferromagnetic systems are a typical
geometrically frustrated systems and have been
gated. The ground state of the ferromagnetic
model on a triangular lattice is a ferromagnetically
n structure. In this case, the order parameter space
ng-range order of spins does not appear at finite
because of the Mermin-Wagner theorem [35].
oes not exhibit any phase transitions. In contrast,
,37] reported that a topological phase transition
e Heisenberg model on a triangular lattice with
omagnetic nearest-neighbor interactions. In this
ong-range order of spins is prohibited by the
gner theorem and thus a phase transition driven
-range order of spins never occurs as well as
magnetic Heisenberg model. Since the ground

in two dimensions. In these systems, the order parameter
space is described by SO(3). The temperature dependence
of the vector chirality and that of the number density of Z2
vortices in the Heisenberg model on a kagome lattice were
also studied [38]. An indication of the Z2 vortex dissociation
has been observed in electron paramagnetic resonance and
electron spin resonance measurements [39–41].
Phase transition has been studied theoretically in stacked
triangular lattice systems as well as in two-dimensional
triangular lattice systems. In many cases, the phase transition
nature in three-dimensional systems differs from that in

Conclusion

INTERLAYER-INTERACTION DEPEN

PHYSICAL REVIEW E 88, of a frustrated
We studied the phase transition nature 052138 (2013) Heisenberg
(a) 10
model on a stacked triangular lattice.

(a)

(b)

0
10

Disordered phase

T

0
10

Partially ordered phase

Disordered phase

13/88(5)/052138(9)

(c)
(i)

T

axis 2

A is broken.

axis 3

We found ordereda first-order phase phase
that phase
Partially
Disordered
transition occurs. At the first-order
phase transition point, SO(3) and C3
B is broken.
symmetries are broken.

axis 1

FIG. 1. (Color online) Schematic picture of a stacked triangular
lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent
the nearest-neighbor and third-nearest-neighbor interactions in each
triangular layer and J⊥ is the interlayer interaction.

@nims.go.jp
m.s.u-tokyo.ac.jp

0
10

052138-1

0
10
0
10
0
10

T

0
20
10

©2013 American Physical Society

0
20
10

Ordered phase

Disordered phase
A and B are broken.

0

T

20
10
0

-2.6 -2.4 -2.2

-2

-1.8 -1.6 -1.4
ructure leads to exotic electronic properties such
ic phenomena [23–27], the anomalous Hall effect
alization of electronic wave functions [29]. Thus
s of frustrated systems have attracted attention
physics and condensed matter physics. Many
y frustrated systems such as stacked triangular
nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their
ave been investigated. In theoretical studies, the
een phase transition and order parameter space in
y frustrated systems has been considered [30–34].
mple of phase transition nature in geometrically
stems, properties of the Heisenberg model on
lattice have been theoretically studied for a
riangular antiferromagnetic systems are a typical
geometrically frustrated systems and have been
gated. The ground state of the ferromagnetic
model on a triangular lattice is a ferromagnetically
n structure. In this case, the order parameter space
ng-range order of spins does not appear at finite
because of the Mermin-Wagner theorem [35].
oes not exhibit any phase transitions. In contrast,
,37] reported that a topological phase transition
e Heisenberg model on a triangular lattice with
omagnetic nearest-neighbor interactions. In this
ong-range order of spins is prohibited by the
gner theorem and thus a phase transition driven
-range order of spins never occurs as well as
magnetic Heisenberg model. Since the ground

in two dimensions. In these systems, the order parameter
space is described by SO(3). The temperature dependence
of the vector chirality and that of the number density of Z2
vortices in the Heisenberg model on a kagome lattice were
also studied [38]. An indication of the Z2 vortex dissociation
has been observed in electron paramagnetic resonance and
electron spin resonance measurements [39–41].
Phase transition has been studied theoretically in stacked
triangular lattice systems as well as in two-dimensional
triangular lattice systems. In many cases, the phase transition
nature in three-dimensional systems differs from that in

Conclusion

INTERLAYER-INTERACTION DEPEN

We studied the phase transition nature of a frustrated Heisenberg
(a) 10
model on a stacked triangular lattice.
0
10
0
10
0
10
0
10

axis 3

We found that a first-order phase
transition occurs. At the first-order
phase transition point, SO(3) and C3
symmetries are broken.

@nims.go.jp
m.s.u-tokyo.ac.jp

13/88(5)/052138(9)

0
10

axis 2
axis 1

FIG. 1. (Color online) Schematic picture of a stacked triangular
lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent
the nearest-neighbor and third-nearest-neighbor interactions in each
triangular layer and J⊥ is the interlayer interaction.

052138-1

©2013 American Physical Society

0
10
0
20
10
0
20
10

The transition temperature increases
but the latent heat decreases as the
interlayer interaction increases.

0
20
10
0

-2.6 -2.4 -2.2

-2

-1.8 -1.6 -1.4
Thank you !

Ryo Tamura and Shu Tanaka
Physical Review E 88, 052138 (2013)

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Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions

  • 1. Interlayer-interaction dependence of latent heat in the Heisenberg model on a stacked triangular lattice with competing interactions Ryo Tamura and Shu Tanaka Physical Review E 88, 052138 (2013)
  • 2. ructure leads to exotic electronic properties such ic phenomena [23–27], the anomalous Hall effect alization of electronic wave functions [29]. Thus s of frustrated systems have attracted attention physics and condensed matter physics. Many y frustrated systems such as stacked triangular nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their ave been investigated. In theoretical studies, the een phase transition and order parameter space in y frustrated systems has been considered [30–34]. mple of phase transition nature in geometrically stems, properties of the Heisenberg model on lattice have been theoretically studied for a riangular antiferromagnetic systems are a typical geometrically frustrated systems and have been gated. The ground state of the ferromagnetic model on a triangular lattice is a ferromagnetically n structure. In this case, the order parameter space ng-range order of spins does not appear at finite because of the Mermin-Wagner theorem [35]. oes not exhibit any phase transitions. In contrast, ,37] reported that a topological phase transition e Heisenberg model on a triangular lattice with omagnetic nearest-neighbor interactions. In this ong-range order of spins is prohibited by the gner theorem and thus a phase transition driven -range order of spins never occurs as well as magnetic Heisenberg model. Since the ground in two dimensions. In these systems, the order parameter space is described by SO(3). The temperature dependence of the vector chirality and that of the number density of Z2 vortices in the Heisenberg model on a kagome lattice were also studied [38]. An indication of the Z2 vortex dissociation has been observed in electron paramagnetic resonance and electron spin resonance measurements [39–41]. Phase transition has been studied theoretically in stacked triangular lattice systems as well as in two-dimensional triangular lattice systems. In many cases, the phase transition nature in three-dimensional systems differs from that in Main Results INTERLAYER-INTERACTION DEPEN We studied the phase transition nature of a frustrated Heisenberg (a) 10 model on a stacked triangular lattice. 0 10 0 10 0 10 0 10 axis 3 We found that a first-order phase transition occurs. At the first-order phase transition point, SO(3) and C3 symmetries are broken. @nims.go.jp m.s.u-tokyo.ac.jp 13/88(5)/052138(9) 0 10 axis 2 axis 1 FIG. 1. (Color online) Schematic picture of a stacked triangular lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent the nearest-neighbor and third-nearest-neighbor interactions in each triangular layer and J⊥ is the interlayer interaction. 052138-1 ©2013 American Physical Society 0 10 0 20 10 0 20 10 The transition temperature increases but the latent heat decreases as the interlayer interaction increases. 0 20 10 0 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4
  • 3. Background Unfrustrated systems (ferromagnet, bipartite antiferromagnet) Ferromagnet Bipartite antiferromagnet Model Ising Order parameter space Z2 XY Heisenberg U(1) S2
  • 4. Background Frustrated systems Antiferromagnetic Ising model on triangle Antiferromagnetic XY/Heisenberg model on triangle ? triangular lattice kagome lattice pyrochlore lattice
  • 5. Background Order parameter space in antiferromagnet on triangular lattice. Model Ising XY Order parameter space --U(1) Phase transition --KT transition Heisenberg SO(3) Z2 vortex dissociation
  • 6. SO(3) x C3 & SO(3) x Z2 (ii) single-k spiral (a) (c) structure sp ira l 4 independent sublattices )t rip lek structure (iv axis 3 axis 2 axis 1 (b) (iii) double-k spiral (ii) single-k spiral (i) ferromagnetic R. dotted hexagonal area in Soc. The Fig. 1. (a) Triangular lattice with L x × Ly sites. (b) Enlarged view of theTamura and N. Kawashima, J. Phys. (a). Jpn., 77, 103002 (2008). R. Tamura and N. Kawashima, J. i-th site are thick and thin lines indicate λJ1 and J1 , respectively. The third nearest-neighbor interactions at thePhys. Soc. Jpn., 80, 074008 (2011). R. Tamura, S. Tanaka, and N. be categorized into depicted. (c) Ground-state phase diagram of the model given by Eq. (1). Ground states canKawashima, Phys. Rev. B, 87, 214401 (2013). R. Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12. five types. More details in each ground state are given in the main text. J1-J3 model on triangular lattice discussed the connection between frustrated continuous spin systems and a fundamental discrete spin Order parameter space Order of phase transition system by using a locally defined parameter. The most famous example is the chiral phase transition in the antiferromagnetic XY model on a triangular lattice. The relation between the phase transition SO(3)xC3 that of the Ising model has been established [24, 25]. In this paper, 1st order of the continuous spin system and we study finite-temperature properties in the J1 -J3 model on a distorted triangular lattice depicted in SO(3)xZ2 2nd order (Ising universality) Figs. 1(a) and (b) from a viewpoint of the Potts model with invisible states.
  • 7. Motivation To investigate the phase transition behavior in three-dimensional systems whose order parameter space is described by the direct product between two groups A x B.
  • 8. malous Hall effect space is described by SO(3). The temperature dependence nctions [29]. Thus of the vector chirality and that of the number density of Z2 ttracted attention vortices in the Heisenberg model on a kagome lattice were er physics. Many also studied [38]. An indication of the Z2 vortex dissociation tacked triangular has · sj observed ini electronJparamagnetic sresonance : and H = J1 si been J3 s · sj si · j si Heisenberg spin me antiferromagelectron spin resonance measurements [39–41]. (three components) i,j 3 i,j hesized and their i,j 1 Phase transition has been studied theoretically in stacked etical studies, the triangular lattice systems well as in two-dimensional 1st nearest- 3rd nearest- as1st nearestarameter space in triangular neighbor lattice systems. In many cases, the phase transition neighbor neighbor nsidered [30–34]. nature in three-dimensional systems differs from that in e in geometrically intralayer intralayer interlayer enberg model on ly studied for a tems are a typical ms and have been he ferromagnetic ferromagnetically r parameter space ot appear at finite er theorem [35]. tions. In contrast, l phase transition gular lattice with teractions. In this axis 2 prohibited by the transition driven axis 1 ccurs as well as axis 3 Model
  • 9. Ground State Spiral-spin configuration si = R cos(k · ri ) I sin(k · ri ) R, I are two arbitrary orthogonal unit vectors. Fourier transform of interactions J(k) = N J1 cos(kx ) J3 cos(2kx ) 2J1 cos 1 kx cos 2 3 ky 2 2J3 cos(kx ) cos( 3ky ) J cos(kz ) Find k that minimizes the Fourier transform of interactions! We consider the case for J > 0 . kz = 0
  • 10. Ferromagnetic (J1>0) Ground-state properties Ferromagnetic (S2) -1/4 Spiral-spin structure (SO(3)xC3) 0 J3 /J1
  • 11. Ferromagnetic (J1>0) Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3) J3 /J1 RYO TAMURA AND SHU TANAKA -1/4 0 3-dim Heisenberg (a) universality class (b)
  • 12. Ferromagnetic (J1>0) Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3) A -1/4 (b) (b) 0 PHYSICAL REVIEW E E 88, 052138 (2013) 3 /J1 PHYSICAL REVIEW 88, 052138 (2013) J ??? 1 1 (c) (c) 0.80.8 0.60.6 0.40.4 0.20.2 0 0 -4 -4 -3 -2 -1 0 0 1 1 -3 -2 -1
  • 13. Antiferromagnetic (J1<0) Ground-state properties Degenerated GSs -1/9 order by disorder Th. Jolicoeur et al., Phys. Rev. B, 42, 4800 (1990). 120-degree structure (SO(3)) 0 chiral universality J3 /J1 H. Kawamura, J. Phys. Soc. Jpn., 54, 3220 (1985). H. Kawamura, J. Phys. Soc. Jpn., 61, 1299 (1992). A. Pelissetto et al., Phys. Rev. B, 65, 020403 (2001). P. Calabrese et al., Phys. Rev. B, 70, 174439 (2004). A. K. Murtazaev and M. K. Ramazanov, Phys. Rev. B 76, 174421 (2007). G. Zumbach, Phys. Rev. Lett. 71, 2421 (1993). M. Tissier et al., Phys. Rev. Lett. 84, 5208 (2000). M. Zelli et al., Phys. Rev. B 76, 224407 (2007). V. T. Ngo and H. T. Diep, Phys. Rev. B, 78, 031119 (2008). We focus on the case for ferromagnetic J1 and J3/J1 > -1/4 in which the order parameter space is described by the direct product between two groups.
  • 14. Possible Scenarios PHYSICAL REVIEW E space is AxB. Let us consider the system whose order parameter 88, 052138 (2013) (a) No symmetry is broken. PHYSICAL REVIEW E 88, 052138 (2013) (a) Disordered phase (a) Disordered phase (b) Partially ordered phase Disordered (b) Only one of two symmetries is broken. phase (b) Partially ordered phase A is broken. Disordered phase Partially ordered phase Disordered phase A is broken. Partially ordered phase B is broken. Disordered phase (c) (i) B is broken. Ordered phase Disordered phase T T T T T T
  • 15. A is broken. A is broken. Partially ordered phase Disordered phase Partially ordered phase Disordered phase Possible Scenarios T T B is broken. (c) Both symmetries are broken. B is broken. (c-1) Both symmetries are broken simultaneously. (c) (c) (i) (i) Ordered phase Ordered phase Disordered phase Disordered phase T T A and B are broken. A and B are broken. (c-2) Both symmetries are broken successively. (ii) (ii) Ordered phase Ordered phase Partially ordered phase Disordered phase Partially ordered phase Disordered phase B is broken. A is broken. B is broken. A is broken. Ordered phase Partially ordered phase Disordered phase Ordered phase Partially ordered phase Disordered phase A is broken. A is broken. B is broken. B is broken. T T T T
  • 16. (b) 10 Internal Energy & Specific Heat (a) -2.3 5 30 40 -2.1 0.1 (d) (b) 0 s ·s H 20 J1 = si · sj J3 si · sj J -2.3 i 0.05j -2.2 -2.1 25 J3 /J1 = 0.85355 · · · , J /J1 = 2 i,j 1 i,j 3 i,j 30 -2.2 -2.2 0 INTERLAYER-INTERACTION DEPENDENCE 30 45 OF LATENT . . 0 15 20 1.55 10 30 -2.1 -2.3 0 -2.1 40 Internal energy (a) (c) (b) 0.02 30 -2.2 0.01 20 -2.3 0 40 1.52 10 (a) 1.53 1.54 1.55 (b) 20 (e) 1.54 Specific heat 15 -2.3 40 30 1.53 10 10 0 0.1 0.00004 0.05 25 30 5 60 0 (f) 0 0 15 30 45 20 40 0 20 0.02 20 -2.3 -2.2 15 0 0 20000 40000 0.01 10 10 1.55 (e) (a) (d) (b) 0.000 (c) -2.1 60000 1.54 50 0 FIG. 4. (Color online) Temperature1.52 1.53 1.54 1.55 1.53 dependence of (a) intern (c) (c) 0 0.00004 0.000 20 per site E/J , (b) specific 0.020 0.02 heat C, and -2.2 order param (c) energy -2.3 -2.1 1 60 C3 symmetry breaking of th eter |µ|2 , which can detect the 0.01 (f) 0.01 FIG. 4. (Color online) Tempe 1.55 10 30 0 Phase transition occurs.
  • 17. (b) Order Parameter (C3) 30 H= J1 i,j 1 si · sj J3 i,j 3 si · sj INTERLAYER-INTERACTION D J 20 i,j si · sj J3 /J1 = PHYSICAL REVIEW E 88, 052138 (2013) (b) 0.85355 · · · , J /J1 = 2 -2.1 10 1 0.8 0.6 0.4 0.2 (c) -2.2 Order parameter 0 0.02 -2.3 40 0.01 30 0 0 -4 -3 -2 -1 20 0 (c) (a) (b) 1.52 1 1.53 1.54 1.55 nd-state properties when the nearest-neighbor interaction J1 is ferromagnetic. (a) Position of k∗ , actions in the wave-vector space for J3 /J1 −1/4. The hexagon represents the first Brillouin Tamura and N. Kawashima, J. Phys. Soc. Jpn., 77, 103002 (2008). R. ∗ figuration in each triangular layer is shown. (b) Position of k and the corresponding schematic Tamura and N. Kawashima, J. Phys. Soc. Jpn., 80, 074008 (2011). 10 55 . . .R. layer when J3 /J1 < −1/4. The spin configurations are depicted for J3 /J1 = −0.853 c) The J3 /J1 dependence of k ∗ . appropriate way, FIG. 4. (Color online) Temp energy per site C3 symmetry is broken. E/J1 , (b) spec energy is observed at a certain temperature. In addition, the 0
  • 18. -2.1 Energy Histogram YER-INTERACTION DEPENDENCE ·OF -2.2 LATENT . . . H= J s ·s J s ·s J s s 1 i i,j j 3 i i,j 1 j E/kB T E(L) : width between two peaks (a) (b) j J3 /J1 = i,j 3 30 P (E; T ) = D(E)e D(E) : density of states i 25 20 -2.3 40 0.1 0.05 0 0 0.85355 · · · , J /J1 = 2 (d) 30 15 30 45 20 15 10 10 0 5 0 -2.3 1.55 0.02 -2.2 0.01 -2.1 First-order phase transition occurs. 0 (e)
  • 19. 5 15 0 -2.3 SO(3) × C3 . -2.2was confirmed It -2.1 10 the first-order phase transition 1.55 10 Heisenberg model on a stack H5 = J1 si · sj J3 si · sj J 1.54 (e) · sj si a J /J = 0.85355 , J /J = 2 i,j i,j nearest-neighbor· · ·interactio 0 i,j 0 1.53 -2.3 -2.2 -2.1(c) 0 0.00004 space is SO(3), a single0.00008 i peak 0.02 Tc(L) Max of specific heat dependence of the specific heat 1.55 60 (e) 0.01 40 (f) finite-temperature phase transi 1.54 20 state and magnetic ordered sta 0 1.53 0 0 0.00004 0.00008 20000 40000 60000 1.52 1.53 1.54 1.55 is 0broken. Then, in our mode 60 break at the first-order phase tr (f) ( E)2 Ld 40 Tc (L) = (Color online) Temperature dependence ofpeakinternal aL d + Tc heat Cmax (L)single c2(a) corresp has a 4T 20 FIG. 4. 0 heat (1986). energy per site M. S. S. Challa,,D. P.(b) and K. Binder, Phys.transition. To confirm this w E/J1 Landau, specific Rev. B, 34, 1841C, and (c) order param0 20000 40000 60000 2 C3 symmetry the structure fac eter |µ| , which can detect the dependence ofbreaking of the 20 Finite-size Scaling 1 3 3 1 1 model with J3 /J1 = −0.853 55 . . . and J⊥ /J1 = 2 for L = 24,32,40. 1 mperature dependence distribution of the internal energy P (E; Tc (L)). The of (a) internal (d) Probability S(k) := s ecific heat C, andthe lattice-size dependence of the width between bimodal N i,j inset shows (c) order paramthe C3 symmetry breaking of of Tctransition occurs. . (f) Plot Plot the peaks E(L)/J1 . (e) phase (L)/J1 as a function of L−3 First-order 3 . . . and J /J = 2 for L = 24,32,40.
  • 20. phase transition in factor S(k∗ ) increases. The structure factors at kz = 0 in the first Brillouin zone at several temperatures for L = 40 are also shown in Fig. 5(b). As mentioned in Sec. II, the spiral-spin structure represented by k is the same as that represented by −k in the Heisenberg models. Figure 5(b) confirms that one si · sj J3 si · sj J si · sj distinct wave vector is chosen from three types of ordered Order Parameter (SO(3)) (9) H= J1 (10) i,j nsition temperature mit. The coefficient stant. Figures 4(e) )/J1 and Cmax (L), is a nonzero value shows an almost n of L3 . However, btain the transition dynamic limit with ize effect. Next we the width between hown in Fig. 4(d). ented by E(L) = are the averages of re phase and that in he thermodynamic δ function and then Fig. 4(d) shows the he width enlarges as hat the latent heat is 1 i,j (a) J3 /J1 = i,j 3 0.85355 · · · , J /J1 = 2 0.5 0.4 0.3 0.2 0.1 0 Order parameter 0 0.5 1 1.5 2 (b) 10 -1 10 -2 10 -3 10 -4 10 -5 FIG. 5. (Color online) (a) Temperature dependence of the largest value of structure factors S(k∗ ) calculated by six wave vectors in Eq. (4) for J3 /J1 = −0.853 55 . . . and J⊥ /J1 = 2. Error bars are omitted for clarity since their sizes are smaller than the symbol size. SO(3) symmetry is broken at the phase transition temperature.
  • 21. Dependence on Interlayer Interaction H= which is rst-order a phase nterlayer [62–65], he model For large hase and occurs at arameter tion with the result systems, when the mperature he J1 -J3 der phase urs when simplest ymmetry del [76]. mensions nder that previous TION J1 i,j 1 si · sj J3 si · sj PHYSICAL REVIEW E 88, 052138 (2013) i,j 3 (a) -1.5 -2 J 0.50 0.25 0.75 1.00 1.25 -2.5 1.50 1.75 2.00 2.25 2.50 (b) increases 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 1.75 0 10 2.50 2.00 2.25 2.50 1 0 10 0 10 1.75 2.00 2.25 0 0.85 1.5 0 10 0.50 20 (b) 0 10 0.25 J /J1 (a) 10 0 10 -3 40 si · sj INTERLAYER-INTERACTION DEPENDENCE OF LA J3 /J1 = 0.85355 · · · , J /J1 = 2 i,j 0.8 0 0.75 20 0.7 10 (c) 0.15 0.25 0.50 0.1 0.75 1.00 1.25 1.50 (d) 1.75 2.00 2.25 0.05 2.50 0.2 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 (e) 2.50 20 0.04 0 20 10 0.1 0 0.08 0 10 0 0.25 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0 (c) -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 0 0 1
  • 22. Dependence on Interlayer Interaction H = J1 si · sj J CTION DEPENDENCE OF LATENT . . . 3 i,j 1 i,j -1.8 -1.6 -1.4 (b) Transition temperature 1.5 1 J si · s REVIEW E 88, 052138 (2013) PHYSICAL j J3 /J1 = 0.85355 · · · , J /J1 = 2 i,j In Sec. IV, we investigated the interlayer interaction effect on the nature of phase transitions. We confirmed that the first-order phase transition occurs for 0.25 J⊥ /J1 2.5 and J3 /J1 = −0.853 55 . . ., which was used in Sec. III. We could not determine the existence of the first-order phase transition for J⊥ /J1 < 0.25 or J⊥ /J1 > 2.5 by Monte Carlo simulations. In the parameter ranges, the width of two peaks in the probability distribution of the internal energy cannot be estimated easily because of the finite-size effect. It is a remaining problem to determine whether a second-order phase transition occurs for large J⊥ /J1 as in the J1 -J2 Heisenberg model on a stacked triangular lattice [62]. As the ratio J⊥ /J1 increases, the firstorder phase transition temperature monotonically increases but the latent heat decreases. This is opposite to the behavior observed in typical unfrustrated three-dimensional systems that exhibit a first-order phase transition at finite temperature. For example, the q-state Potts model with ferromagnetic intralayer and interlayer interactions (q 3) is a fundamental model that exhibits a temperature-induced first-order phase transition with q-fold symmetry breaking [76]. From a meanfield analysis of the ferromagnetic Potts model [76,83], as the interlayer interaction increases, both the transition temperature As the interlayer interaction increases, ... transition temperature increases. 0.5 0.08 latent heat decreases. 0.04 0 3 si · sj (c) Latent heat 0 1 2
  • 23. ructure leads to exotic electronic properties such ic phenomena [23–27], the anomalous Hall effect alization of electronic wave functions [29]. Thus s of frustrated systems have attracted attention physics and condensed matter physics. Many y frustrated systems such as stacked triangular nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their ave been investigated. In theoretical studies, the een phase transition and order parameter space in y frustrated systems has been considered [30–34]. mple of phase transition nature in geometrically stems, properties of the Heisenberg model on lattice have been theoretically studied for a riangular antiferromagnetic systems are a typical geometrically frustrated systems and have been gated. The ground state of the ferromagnetic model on a triangular lattice is a ferromagnetically n structure. In this case, the order parameter space ng-range order of spins does not appear at finite because of the Mermin-Wagner theorem [35]. oes not exhibit any phase transitions. In contrast, ,37] reported that a topological phase transition e Heisenberg model on a triangular lattice with omagnetic nearest-neighbor interactions. In this ong-range order of spins is prohibited by the gner theorem and thus a phase transition driven -range order of spins never occurs as well as magnetic Heisenberg model. Since the ground in two dimensions. In these systems, the order parameter space is described by SO(3). The temperature dependence of the vector chirality and that of the number density of Z2 vortices in the Heisenberg model on a kagome lattice were also studied [38]. An indication of the Z2 vortex dissociation has been observed in electron paramagnetic resonance and electron spin resonance measurements [39–41]. Phase transition has been studied theoretically in stacked triangular lattice systems as well as in two-dimensional triangular lattice systems. In many cases, the phase transition nature in three-dimensional systems differs from that in Conclusion INTERLAYER-INTERACTION DEPEN PHYSICAL REVIEW E 88, of a frustrated We studied the phase transition nature 052138 (2013) Heisenberg (a) 10 model on a stacked triangular lattice. (a) (b) 0 10 Disordered phase T 0 10 Partially ordered phase Disordered phase 13/88(5)/052138(9) (c) (i) T axis 2 A is broken. axis 3 We found ordereda first-order phase phase that phase Partially Disordered transition occurs. At the first-order phase transition point, SO(3) and C3 B is broken. symmetries are broken. axis 1 FIG. 1. (Color online) Schematic picture of a stacked triangular lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent the nearest-neighbor and third-nearest-neighbor interactions in each triangular layer and J⊥ is the interlayer interaction. @nims.go.jp m.s.u-tokyo.ac.jp 0 10 052138-1 0 10 0 10 0 10 T 0 20 10 ©2013 American Physical Society 0 20 10 Ordered phase Disordered phase A and B are broken. 0 T 20 10 0 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4
  • 24. ructure leads to exotic electronic properties such ic phenomena [23–27], the anomalous Hall effect alization of electronic wave functions [29]. Thus s of frustrated systems have attracted attention physics and condensed matter physics. Many y frustrated systems such as stacked triangular nets (see Fig. 1), stacked kagome antiferromagin-ice systems have been synthesized and their ave been investigated. In theoretical studies, the een phase transition and order parameter space in y frustrated systems has been considered [30–34]. mple of phase transition nature in geometrically stems, properties of the Heisenberg model on lattice have been theoretically studied for a riangular antiferromagnetic systems are a typical geometrically frustrated systems and have been gated. The ground state of the ferromagnetic model on a triangular lattice is a ferromagnetically n structure. In this case, the order parameter space ng-range order of spins does not appear at finite because of the Mermin-Wagner theorem [35]. oes not exhibit any phase transitions. In contrast, ,37] reported that a topological phase transition e Heisenberg model on a triangular lattice with omagnetic nearest-neighbor interactions. In this ong-range order of spins is prohibited by the gner theorem and thus a phase transition driven -range order of spins never occurs as well as magnetic Heisenberg model. Since the ground in two dimensions. In these systems, the order parameter space is described by SO(3). The temperature dependence of the vector chirality and that of the number density of Z2 vortices in the Heisenberg model on a kagome lattice were also studied [38]. An indication of the Z2 vortex dissociation has been observed in electron paramagnetic resonance and electron spin resonance measurements [39–41]. Phase transition has been studied theoretically in stacked triangular lattice systems as well as in two-dimensional triangular lattice systems. In many cases, the phase transition nature in three-dimensional systems differs from that in Conclusion INTERLAYER-INTERACTION DEPEN We studied the phase transition nature of a frustrated Heisenberg (a) 10 model on a stacked triangular lattice. 0 10 0 10 0 10 0 10 axis 3 We found that a first-order phase transition occurs. At the first-order phase transition point, SO(3) and C3 symmetries are broken. @nims.go.jp m.s.u-tokyo.ac.jp 13/88(5)/052138(9) 0 10 axis 2 axis 1 FIG. 1. (Color online) Schematic picture of a stacked triangular lattice with Lx × Ly × Lz sites. Here J1 and J3 respectively represent the nearest-neighbor and third-nearest-neighbor interactions in each triangular layer and J⊥ is the interlayer interaction. 052138-1 ©2013 American Physical Society 0 10 0 20 10 0 20 10 The transition temperature increases but the latent heat decreases as the interlayer interaction increases. 0 20 10 0 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4
  • 25. Thank you ! Ryo Tamura and Shu Tanaka Physical Review E 88, 052138 (2013)