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Our paper entitled “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a...

Our paper entitled “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" was published in Physical Review E. This work was done in collaboration with Dr. Ryo Tamura (NIMS).

http://pre.aps.org/abstract/PRE/v88/i5/e052138

NIMSの田村亮さんとの共同研究論文 “Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions" が Physical Review E に掲載されました。

http://pre.aps.org/abstract/PRE/v88/i5/e052138

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

Interlayer-interaction dependence of latent heat in the Heisenberg model

on a stacked triangular lattice with competing interactions

Ryo Tamura1,* and Shu Tanaka2,†

1International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan

2Department of Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

(Received 12 August 2013; published 26 November 2013)

We study the phase transition behavior of a frustrated Heisenberg model on a stacked triangular lattice by Monte

Carlo simulations. The model has three types of interactions: the ferromagnetic nearest-neighbor interaction J1 and

antiferromagnetic third nearest-neighbor interaction J3 in each triangular layer and the ferromagnetic interlayer

interaction J⊥. Frustration comes from the intralayer interactions J1 and J3. We focus on the case that the order

parameter space is SO(3)×C3. We find that the model exhibits a first-order phase transition with breaking of the

SO(3) and C3 symmetries at finite temperature. We also discover that the transition temperature increases but

the latent heat decreases as J⊥/J1 increases, which is opposite to the behavior observed in typical unfrustrated

three-dimensional systems.

DOI: 10.1103/PhysRevE.88.052138

PACS number(s): 75.10.Hk, 64.60.De, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

state of the model is the 120◦ structure, the order parameter

space is SO(3), which is the global rotational symmetry of

Geometrically frustrated systems often exhibit a charac-

spins. Thus the point defect, i.e., the

teristic phase transition, such as successive phase transitions,

Z2 = π1 [SO(3)] vortex

defect, can exist in the model. Then the topological phase

order by disorder, and a reentrant phase transition, and an

transition occurs by dissociating the

unconventional ground state, such as the spin liquid state

Z2 vortices at finite

temperature [31,36,37]. The dissociation of

[1–20]. In frustrated continuous spin systems, the ground

Z2 vortices is

one of the characteristic properties of geometrically frustrated

state is often a noncollinear spiral-spin structure [21,22]. The

systems when the ground state is a noncollinear spin structure

spiral-spin structure leads to exotic electronic properties such

in two dimensions. In these systems, the order parameter

as multiferroic phenomena [23–27], the anomalous Hall effect

space is described by SO(3). The temperature dependence

[28], and localization of electronic wave functions [29]. Thus

of the vector chirality and that of the number density of

the properties of frustrated systems have attracted attention

Z2

vortices in the Heisenberg model on a kagome lattice were

in statistical physics and condensed matter physics. Many

also studied [38]. An indication of the

vortex dissociation

geometrically frustrated systems such as stacked triangular

has

Ma

been observed in in R

Z2

electron

esults

paramagnetic resonance and

antiferromagnets (see Fig. 1), stacked kagome antiferromag-

electron spin resonance measurements [39–41].

nets, and spin-ice systems have been synthesized and their

Phase transition has been studied theoretically in stacked

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

properties have been W

inv e studied the phase tr

estigated. In theoretical studies, the

triangular lattice systems as well as in two-dimensional

relation between phase transition and order parameter space in

ansition nature of a frustrated Heisenberg

geometrically frustrated systems has been considered [30–34].

triangular lattice systems. In many cases, the phase transition

As an example of model on a stacked tr

phase transition nature in geometrically

nature

iangular la

in three-dimensional

ttic

systems

e

dif .

fers from that in

(a) 10

In Sec. IV, we investigated the interlayer interaction effect

frustrated systems, properties of the Heisenberg model on

(b)

a triangular lattice have been theoretically studied for a

0

on the nature of phase transitions. We confirmed that the

long time. Triangular antiferromagnetic systems are a typical

10

first-order phase transition occurs for 0.25

J

example of geometrically frustrated systems and have been

1.5

⊥/J1

2.5 and

well investigated. The ground state of the ferromagnetic

0

J3/J1 = −0.853 55 . . ., which was used in Sec. III. We could

Heisenberg model on a triangular lattice is a ferromagnetically

10

collinear spin structure. In this case, the order parameter space

not determine the existence of the first-order phase transition

is

0

S2. The long-range order of spins does not appear at finite

for

temperature because of the Mermin-Wagner theorem [35].

10

J⊥/J1 < 0.25 or J⊥/J1 > 2.5 by Monte Carlo simulations.

The model does not exhibit any phase transitions. In contrast,

0

In the parameter ranges, the width of two peaks in the probabil-

Refs. [31,36,37] reported that a topological phase transition

10

occurs in the Heisenberg model on a triangular lattice with

1

ity distribution of the internal energy cannot be estimated easily

only antiferromagnetic nearest-neighbor interactions. In this

0

because of the finite-size effect. It is a remaining problem to

model the long-range order of spins is prohibited by the

axis 2

10

Mermin-Wagner theorem and thus a phase transition driven

axis 1

determine whether a second-order phase transition occurs for

by the long-range order of spins never occurs as well as

axis 3

0

in the ferromagnetic Heisenberg model. Since the ground

We found that a first-order phase

10

large J⊥/J1 as in the J1-J2 Heisenberg model on a stacked

FIG. 1. (Color online) Schematic picture of a stacked triangular

lattice with

0

triangular lattice [62]. As the ratio

L

sites. Here

J

x × Ly × Lz

J1 and J3 respectively represent

transition occurs. At the first-order

0.5

⊥/J1 increases, the first-

*tamura.ryo@nims.go.jp

the nearest-neighbor and third-nearest-neighbor interactions in each

20

order phase transition temperature monotonically increases

†shu-t@chem.s.u-tokyo.ac.jp

triangular layer and J⊥ is the interlayer interaction.

phase transition point, SO(3) and C3

10

but the latent heat decreases. This is opposite to the behavior

1539-3755/2013/88(5)/052138(9)

052138-1

©2013 American Physical Society

symmetries are broken.

0

0.08

observed in typical unfrustrated three-dimensional systems

20

that exhibit a first-order phase transition at finite temperature.

10

For example, the q-state Potts model with ferromagnetic

The transition temperature increases

0

0.04

intralayer and interlayer interactions (q

3) is a fundamental

20

model that exhibits a temperature-induced first-order phase

but the latent heat decreases as the

10

(c)

transition with q-fold symmetry breaking [76]. From a mean-

interlayer interaction increases.

0

field analysis of the ferromagnetic Potts model [76,83], as the

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0 0

1

2

interlayer interaction increases, both the transition temperature

and the latent heat increase. The same behavior was observed

FIG. 7. (Color online) (a) Interlayer-interaction J⊥/J1 depen-

in the Ising-O(3) model on a stacked square lattice [77]. As

dence of the probability distribution of internal energy P (E; Tc(L))

just described, in general, if the parameter that can stabilize

when the specific heat becomes the maximum value for L = 24.

the ground state becomes large, the transition temperature

(b) The J⊥/J1 dependence of Tc(L)/J1 at which the specific

increases and the latent heat increases [76,77,83]. Furthermore,

heat becomes the maximum value for L = 16−40. (c) The J⊥/J1

in conventional systems, both the transition temperature and

dependence of the width between bimodal peaks of the energy

distribution

the latent heat are expressed by the value of an effective

E(L)/J1. Error bars in all figures are omitted for clarity

since their sizes are smaller than the symbol size.

interaction obtained by a characteristic temperature such as

the Curie-Weiss temperature. However, in our model, the

Curie-Weiss temperature does not characterize the first-order

the thermodynamic limit corresponds to the latent heat. Thus

phase transition, as will be shown in the Appendix. Thus our

Fig. 7(c) suggests that the latent heat decreases as J⊥/J1

result is an unusual behavior. The investigation of the essence

increases in the thermodynamic limit.

of the obtained results is a remaining problem.

V. DISCUSSION AND CONCLUSION

ACKNOWLEDGMENTS

In this paper, the nature of the phase transition of the

R.T. was partially supported by a Grand-in-Aid for Sci-

Heisenberg model on a stacked triangular lattice was studied by

entific Research (C) (Grant No. 25420698) and National

Monte Carlo simulations. In our model, there are three kinds of

Institute for Materials Science. S.T. was partially supported

interactions: the ferromagnetic nearest-neighbor interaction J1

by a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).

and antiferromagnetic third nearest-ne - Background

Unfrustrated systems (ferromagnet, bipartite antiferromagnet)

Ferromagnet

Bipartite antiferromagnet

Model

Order parameter space

Ising

Z2

XY

U(1)

Heisenberg

S2 - Background

Frustrated systems

Antiferromagnetic Ising model

Antiferromagnetic XY/Heisenberg model

on triangle

on triangle

?

triangular lattice

kagome lattice

pyrochlore lattice - Background

Order parameter space in antiferromagnet on triangular lattice.

Model

Order parameter space

Phase transition

Ising

---

---

XY

U(1)

KT transition

Heisenberg

SO(3)

Z2 vortex dissociation - SO(3) x C3 & SO(3) x Z2

(ii) single-k spiral

(a)

(c)

structure

4 independent

al

sublattices

spir

structure

ple-k

axis 3 axis 2

tri

v)

(i

axis 1

(b)

(iii) double-k spiral

(ii) single-k spiral

(i) ferromagnetic

Fig. 1. (a) Triangular lattice with Lx × Ly sites. (b) Enlarged view of the

R. dotted

Tamur

hexagonal

a and N. Kaw

area

ashima, J in

. Phys (a).

. Soc The

. Jpn., 77, 103002 (2008).

thick and thin lines indicate λJ1 and J1, respectively. The third nearest-neighbor

R. Tamurinteractions

a and N. Kaw

at the

ashima, J i-th

. Phys site

. Soc. J are

pn., 80, 074008 (2011).

depicted. (c) Ground-state phase diagram of the model given by Eq. (1).

R. T Ground

amura, S. T states

anak

can be

a, and N. Kaw categorized

ashima, Phys

into

. Rev. B, 87, 214401 (2013).

five types. More details in each ground state are given in the main te

R. xt.

Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12.

J1-J3 model on triangular lattice

discussed the

Or connection

der par between

amet frustrated

er spac continuous

e

spin

Or systems and a fundamental

der of phase tr discrete spin

ansition

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)xC

of the continuous spin system and

3

1st order

that of the Ising model has been established [24,25]. In this paper,

we study finite-temperature properties in the J1-J3 model on a distorted triangular lattice depicted in

SO(3)xZ

Figs. 1(a) and (b) from a viewpoint

2

2nd order (Ising universality)

of the Potts model with invisible states.

2. Model and Ground State Phase Diagram

We consider the classical Heisenberg model on a distorted triangular lattice. The Hamiltonian is

given by

H = λJ1

si · sj + J1

si · sj + J3

si · sj,

(1)

i, j axis1

i, j axis2,3

i, j

where the first term represents the nearest-neighbor interactions along axis 1, the second term denotes

the nearest-neighbor interactions along axes 2 and 3, and the third term is the third nearest-neighbor

interactions [see Fig. 1(b)]. The variable si is the three-dimensional vector spin of unit length. The

parameter λ(> 0) represents a uniaxial distortion along axis 1. Here we consider the case that the third

nearest-neighbor interaction J3 is antiferromagnetic (J3 > 0). The ground state of the model given

by Eq. (1) is represented by the wave vector k∗ at which the Fourier transform of interactions J(k) is

minimized. In this case, J(k) is given by

√

J(k)

λJ1

2J

k

3k

√

cos k

1 cos x cos

y

3k

NJ =

x +

y,

(2)

3

J3

J3

2

2 + cos 2kx + 2 cos kx cos

where N(= Lx × Ly) is the number of spins. Here the lattice constant is set to unity. It should be noted

that the spin structures denoted by k and −k are the same in the Heisenberg models. Figure 1 (c)

depicts the ground-state phase diagram, which shows five types of ground states, depending on the

2 - 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. - PHYSICAL REVIEW E 88, 052138 (2013)

Interlayer-interaction dependence of latent heat in the Heisenberg model

on a stacked triangular lattice with competing interactions

Ryo Tamura1,* and Shu Tanaka2,†

1International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan

2Department of Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

(Received 12 August 2013; published 26 November 2013)

We study the phase transition behavior of a frustrated Heisenberg model on a stacked triangular lattice by Monte

Carlo simulations. The model has three types of interactions: the ferromagnetic nearest-neighbor interaction J1 and

antiferromagnetic third nearest-neighbor interaction J3 in each triangular layer and the ferromagnetic interlayer

interaction J⊥. Frustration comes from the intralayer interactions J1 and J3. We focus on the case that the order

parameter space is SO(3)×C3. We find that the model exhibits a first-order phase transition with breaking of the

SO(3) and C3 symmetries at finite temperature. We also discover that the transition temperature increases but

the latent heat decreases as J⊥/J1 increases, which is opposite to the behavior observed in typical unfrustrated

three-dimensional systems.

DOI: 10.1103/PhysRevE.88.052138

PACS number(s): 75.10.Hk, 64.60.De, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

state of the model is the 120◦ structure, the order parameter

space is SO(3), which is the global rotational symmetry of

Geometrically frustrated systems often exhibit a charac-

spins. Thus the point defect, i.e., the

teristic phase transition, such as successive phase transitions,

Z2 = π1 [SO(3)] vortex

defect, can exist in the model. Then the topological phase

order by disorder, and a reentrant phase transition, and an

transition occurs by dissociating the

unconventional ground state, such as the spin liquid state

Z2 vortices at finite

temperature [31,36,37]. The dissociation of

[1–20]. In frustrated continuous spin systems, the ground

Z2 vortices is

one of the characteristic properties of geometrically frustrated

state is often a noncollinear spiral-spin structure [21,22]. The

systems when the ground state is a noncollinear spin structure

spiral-spin structure leads to exotic electronic properties such

in two dimensions. In these systems, the order parameter

as multiferroic phenomena [23–27], the anomalous Hall effect

space is described by SO(3). The temperature dependence

[28], and localization of electronic wave functions [29]. Thus

of the vector chirality and that of the number density of

the properties of frustrated systems have attracted attention

Model

Z2

vortices in the Heisenberg model on a kagome lattice were

in statistical physics and condensed matter physics. Many

also studied [38]. An indication of the

geometrically frustrated systems such as stacked triangular

Z2 vortex dissociation

has been observed in electron paramagnetic resonance and

antiferromagnets (see Fig. 1), stacked kagome antiferromag-

H = J1

si · sj J3

si · sj J

si · sj

s

electron spin resonance measurements [39–41].

i : Heisenberg spin

nets, and spin-ice systems have been synthesized and their i,j 1

(three components)

Phase transition

i,j has

3

been studied theoretically

i,j

in stacked

properties have been investigated. In theoretical studies, the

triangular lattice systems as well as in two-dimensional

relation between phase transition and order parameter space in

1st nearest-

3rd nearest-

1st nearest-

geometrically frustrated systems has been considered [30–34].

triangular lattice systems. In many cases, the phase transition

neighbor

neighbor

neighbor

As an example of phase transition nature in geometrically

nature in three-dimensional systems differs from that in

intralayer

intralayer

interlayer

frustrated systems, properties of the Heisenberg model on

a triangular lattice have been theoretically studied for a

long time. Triangular antiferromagnetic systems are a typical

example of geometrically frustrated systems and have been

well investigated. The ground state of the ferromagnetic

Heisenberg model on a triangular lattice is a ferromagnetically

collinear spin structure. In this case, the order parameter space

is S2. The long-range order of spins does not appear at finite

temperature because of the Mermin-Wagner theorem [35].

The model does not exhibit any phase transitions. In contrast,

Refs. [31,36,37] reported that a topological phase transition

occurs in the Heisenberg model on a triangular lattice with

only antiferromagnetic nearest-neighbor interactions. In this

model the long-range order of spins is prohibited by the

axis 2

Mermin-Wagner theorem and thus a phase transition driven

axis 1

by the long-range order of spins never occurs as well as

axis 3

in the ferromagnetic Heisenberg model. Since the ground

FIG. 1. (Color online) Schematic picture of a stacked triangular

lattice with L

sites. Here

x × Ly × Lz

J1 and J3 respectively represent

*tamura.ryo@nims.go.jp

the nearest-neighbor and third-nearest-neighbor interactions in each

†shu-t@chem.s.u-tokyo.ac.jp

triangular layer and J⊥ is the interlayer interaction.

1539-3755/2013/88(5)/052138(9)

052138-1

©2013 American Physical Society - 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)

1

3

=

J

N

1 cos(kx)

2J1 cos 2kx cos 2 ky

J3 cos(2kx)

2J3 cos(kx) cos( 3ky) J cos(kz)

Find tha

k

t minimizes the Fourier transform of interactions!

We consider the case for .

J > 0

k = 0

z - Ferromagnetic (J1>0)

Ground-state properties

Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)

J3/J1

-1/4

0 - Ferromagnetic (J1>0)

Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)

J3/J1

RYO -1/

T 4

AMURA AND 0

SHU TANAKA

PHYSICAL REVIEW E 88, 052138 (2013)

3-dim Heisenberg

(a)

(b)

universality class

1

(c)

0.8

0.6

0.4

0.2

0-4 -3 -2 -1 0 1

FIG. 3. (Color online) Explanation of ground-state properties when the nearest-neighbor interaction J1 is ferromagnetic. (a) Position of k∗,

which minimizes the Fourier transform of interactions in the wave-vector space for J3/J1

−1/4. The hexagon represents the first Brillouin

zone. A schematic of a ferromagnetic spin configuration in each triangular layer is shown. (b) Position of k∗ and the corresponding schematic

of spiral-spin configurations in each triangular layer when J3/J1 < −1/4. The spin configurations are depicted for J3/J1 = −0.853 55 . . .

corresponding to k∗ = π/2 and then θ = 90◦. (c) The J3/J1 dependence of k∗.

√

represented by k∗ = (0,2π/ 3,0) in an appropriate way,

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

shown in Fig. 2 in Ref. [78]. Then the order parameter space

specific heat has a divergent single peak at the temperature.

is not well defined in this case.

These behaviors indicate the existence of a finite-temperature

Our purpose is to investigate the phase transition behavior

phase transition. As will be shown in Sec. IV, the uniform

when the order parameter space is described by the direct

magnetic susceptibility can be used as an indicator of the phase

product between two groups. Hereafter we focus on the

transition. To investigate the way of ordering, the temperature

parameter region J

dependence of an order parameter is considered. The order

3/J1 < −1/4 in the case of ferromagnetic

J

parameter

1. Throughout the paper, we use the interaction ratio J3/J1 =

µ that can detect the C3 symmetry breaking is

−0.853 55 . . . so that the ground state is represented by

defined by

k∗ = π/2 in Eq. (4). In this case, along one of three axes,

the relative angle between nearest-neighbor spin pairs is 90◦,

µ := ε1e1 + ε2e2 + ε3e3,

(7)

while along the other axes, the relative angle is 45◦ in the

1

ground-state spin configuration [see Fig. 3(b)]. When the

ε :

s

η =

i · sj ,

(8)

period of the lattice is set to 8, the commensurate spiral-spin

N i,j 1 axis η

configuration appears in the ground state. Then, in order to

avoid the incompatibility due to the boundary effect, the linear

where the subscript η (η = 1,2,3) assigns the axis (see

dimension

Fig. 1). The vectors e are unit vectors along the axis

L = 8n (n ∈ N ) is used and the periodic boundary

η

η

√

conditions in all directions are imposed.

in each triangular layer, i.e., e1 = (1,0), e2 = (−1/2, 3/2),

√

and e3 = (−1/2, − 3/2). The temperature dependence of

III. FINITE-TEMPERATURE PROPERTIES

|µ|2 is shown in Fig. 4(c). The order parameter abruptly

OF THE STACKED MODEL

increases around the temperature at which the specific heat

has a divergent peak. These results conclude that the phase

In this section, we investigate the finite-temperature prop-

transition is accompanied by the C3 symmetry breaking.

erties of the Heisenberg model on a stacked triangular

To decide the order of the phase transition, we calculate the

lattice with competing interactions given by Eq. (1) with

probability distribution of the internal energy at T , P (E; T ) =

J3/J1 = −0.853 55 . . . and J⊥/J1 = 2. Using Monte Carlo

D(E) exp(−NE/T ), where D(E) is the density of states.

simulations with the single-spin-flip heat-bath method and the

When a system exhibits a first-order phase transition, the

overrelaxation method [79,80], we calculate the temperature

energy distribution P (E; T ) should be a bimodal structure

dependence of physical quantities. Figures 4(a) and 4(b)

at temperature Tc(L) for system size L. Here Tc(L) is the

show the internal energy per site E and specific heat C for

temperature at which the specific heat becomes the maximum

L = 24,32,40. The specific heat at temperature T is given by

value Cmax(L). To obtain Tc(L) and Cmax(L), we perform

the reweighting method [81]. Figure 4(d) shows P (E; Tc(L))

E2 − E 2

C = N

,

(6)

for system sizes L = 24,32,40. As stated above, the bimodal

T 2

structure in the energy distribution suggests a first-order phase

where O denotes the equilibrium value of the physical

transition.

quantity O. Here the Boltzmann constant is set to unity. As

To confirm whether the first-order phase transition behavior

the system size increases, a sudden change in the internal

remains in the thermodynamic limit, we perform two types

052138-4 - Ferromagnetic (J1>0)

Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)

RYOR TAMURA

YO T

AND

AMURA

SHU

AND

TAN

SHU TAKA

ANAKA

PHYSICAL REVIEW

PHYSICAL

E

REVIEW 88

E , 0

88,52

0 1

5 3

2 8

13(2

8 0

( 1

2 3

0 )

1 J

3) 3/J1

-1/4

0

???

(a) (a)

(b) (b)

1 1

(c)(c)

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0

-4 --3

4 - -2

3 - -1

2 - 0

1

1

0 1

FIG. 3.

FIG.(Color

3.

online)

(Color

Explanation

online)

of

Explanationground-state

of g

properties

round-state p

when

roperties w the

hen nearest-neighbor

the

interaction

nearest-neighbor interaction

J J1 is ferromagnetic. (a) Position of k∗,

1 is ferromagnetic. (a) Position of k∗,

which

whm

icin

h im

mi iz

nies

mi th

zees Fto

h u

e ri

F eoru tr

ri a

ernstrfo

a r

n m

sfoorfminotferia

ntcetiroan

ctsi ion

ns thie

n tw

h aeve

w -avveec-tvoerctsp

oracse

pa fcoerforJ J3/J1 −1/4. The hexagon represents the first Brillouin

3/J1

−1/4. The hexagon represents the first Brillouin

zone.

z A

ones.ch

A em

sc ahteic

m o

atfi a

c foefrarofm

er arg

o n

m eatigcn s

etpiicns cpoin

n fi

c g

o u

n rfiagtiuorn

atiin

on ea

incheatrcihantrgiu

a lnagru llaayrelrayiser sh

is ow

sh no.w(b

n. )(Pbo)si

P toio

si nti o

o fn ko∗f and

k∗ the

and corresponding

the

schematic

corresponding schematic

of spiral-spin

of

configurations

spiral-spin

in

configurations each

in

triangular

each

layer

triangular

when

layer when

J J3/J1 < −1/4. The spin configurations are depicted for J3/J1 = −0.853 55 . . .

3/J1 < −1/4. The spin configurations are depicted for J3/J1 = −0.853 55 . . .

corresponding to

corresponding to k∗

π/2 and then θ

90◦. (c) The J

k∗ = π/

=2 and then θ = 90=◦. (c) The J 3/J1 dependence of k∗.

3/J1 dependence of k∗.

√

represented by k∗

(0 √

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

represented by k∗

3

= (0 ,2π/

,0) in an appropriate way,

,

= 2π/ 3,0) in an appropriate way,

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

specific heat has a divergent single peak at the temperature.

shown

sho in

wnFig.

in 2 in

Fig. 2Ref.

in [78

Ref. ].[ Then

78].

the

Then order

the

parameter

order

space

parameter space specific heat has a divergent single peak at the temperature.

These behaviors indicate the existence of a finite-temperature

is not

is well

not defined

well

in

definedthis

in case.

this case.

These behaviors indicate the existence of a finite-temperature

phase transition. As will be shown in Sec. IV, the uniform

Our purpose

Our

is to

purpose isinv

to estig

inv ate

estig the

ate phase

the

transition

phase

beha

transition

vior

behavior phase transition. As will be shown in Sec. IV, the uniform

magnetic susceptibility can be used as an indicator of the phase

when the

when order

the

parameter

order

space

parameter

is

space described

is

by

described the

by direct

the direct magnetic susceptibility can be used as an indicator of the phase

product between two groups. Hereafter we focus on the

transition. To investigate the way of ordering, the temperature

product between two groups. Hereafter we focus on the

transition. To investigate the way of ordering, the temperature

parameter region

dependence of an order parameter is considered. The order

parameter region

J

J

3/J1 < −1/4 in the case of ferromagnetic dependence of an order parameter is considered. The order

3/J1 < −1/4 in the case of ferromagnetic

J

parameter

1. Throughout the paper, we use the interaction ratio J3/J1 =

µ that can detect the C3 symmetry breaking is

J

parameter

1. Throughout the paper, we use the interaction ratio J3/J1 =

µ that can detect the C3 symmetry breaking is

−0.853 55 . . . so that the ground state is represented by

defined by

−0.853 55 . . . so that the ground state is represented by

defined by

k∗ = π/2 in Eq. (4). In this case, along one of three axes,

k∗ = π/2 in Eq. (4). In this case, along one of three axes,

the relative angle between nearest-neighbor spin pairs is 90◦,

µ := ε1e1 + ε2e2 + ε3e3,

(7)

the relative angle between nearest-neighbor spin pairs is 90◦,

µ := ε1e1 + ε2e2 + ε3e3,

(7)

while along the other axes, the relative angle is 45◦ in the

while along the other axes, the relative angle is 45◦ in the

1

ground-state spin configuration [see Fig. 3(b)]. When the

ε :

s

η

1

ground-state spin configuration [see Fig. 3(b)]. When the

=

i · sj ,

(8)

ε :

N

s

η

period of the lattice is set to 8, the commensurate spiral-spin

=

i · sj ,

(8)

period of the lattice is set to 8, the commensurate spiral-spin

N

i,j 1 axis η

i,j 1 axis η

configuration appears in the ground state. Then, in order to

configuration appears in the ground state. Then, in order to

avoid the incompatibility due to the boundary effect, the linear

where the subscript η (η = 1,2,3) assigns the axis (see

avoid the incompatibility due to the boundary effect, the linear

where the subscript η (η

Fig. 1). The vectors e = 1,2,3) assigns the axis (see

dimension

are unit vectors along the axis

L = 8n (n ∈ N ) is used and the periodic boundary

η

η

dimension

Fig. 1). The vectors e are unit vectors along the axis

√

L = 8n (n

conditions in all

∈ N ) is used and the periodic boundary

η

η

directions are imposed.

in each triangular layer, i.e., e1 = (1,0), e2 = (−1/ √

2, 3/2),

conditions in all directions are imposed.

in each triangular layer,

√ i.e., e1

and e

3/2).

= (1,0), e2

The

= (

temperature −1/2, 3/2),

3 = (−1/2, √

−

dependence of

and e3 = (−1/2, − 3/2). The temperature dependence of

III. FINITE-TEMPERATURE PROPERTIES

|µ|2 is shown in Fig. 4(c). The order parameter abruptly

III. FINITE-TEMPERATURE PROPERTIES

|µ|2 is shown in Fig. 4(c). The order parameter abruptly

OF THE STACKED MODEL

increases around the temperature at which the specific heat

OF THE STACKED MODEL

increases around the temperature at which the specific heat

has a divergent peak. These results conclude that the phase

In this section, we investigate the finite-temperature prop- has a divergent peak. These results conclude that the phase

transition is accompanied by the C3 symmetry breaking.

In this section, we investigate the finite-temperature prop-

erties of the Heisenberg model on a stacked triangular transition is accompanied by the C

To decide the order of the pha 3 symmetry breaking.

se transition, we calculate the

erties of

lattice the

with Heisenberg

competing model on

interactions agisvtack

en ed

by triangular

Eq. (1) with

To decide

probabilitythe ord

distriber of

utionthe

of ph

thease trans

internal ition

ener , w

gy e

at calculate the

T , P (E; T ) =

lattice with competing interactions given by Eq. (1) with

J

probability distribution of the internal energy at T , P (E; T )

3/J1 = −0.853 55 . . . and J

D(E) exp(−NE/T ), where D(E) is the density of st =

⊥/J1 = 2. Using Monte Carlo

ates.

J3/J1 = −0.853

simulations 55

with . . .

the and

s

J⊥/J1 = 2.

ingle-spin-flip Using Monte

heat-bath

Carlo

method and the D(E) ex

Whenp(−

a N

s E/T

ystem ),

e where

xhibits D

a (E) is th

first-order e dens

phase ity of stat

transition,es.

the

simulations

overrela with

xatio the

n m s

e ingle-spin-flip

thod [79,80], heat-bath

we

method

calculate the

and the

temperature When

ener a

gy system

distrib exhibits

ution P ( a

E; first-order

T ) should phase

be a bitransition,

modal stru the

cture

overrelaxation

dependencemet

of hod

ph [79,80

ysical ], we calculate

quantities.

the

Figures temperature

4(a) and 4(b) ener

atgy distribution

temperature T P (E; T ) should be a bimodal structure

c(L) for system size L. Here Tc(L) is the

dependence

show the of physical

internal ener quantities.

gy per site Figures

E and

4(a) and

specific

4(b)

heat C for at temperature

temperature T

at c(L) f

which or s

the y

s stem s

pecific ize

heatL. Here

becomesTc(L

the ) is the

maximum

show

L the internal energy per site E and specific heat C for

= 24,32,40. The specific heat at temperature T is given by temperature

value C at which the specific heat becomes the maximum

max(L). To obtain Tc(L) and Cmax(L), we perform

L = 24,32,40. The specific heat at temperature T is given by

value

the C

remax(L). T

weightingo obtain

method [ Tc

81(L

]. ) and

Figure Cmax - Antiferromagnetic (J1<0)

Ground-state properties

Degenerated GSs

120-degree structure (SO(3))

J3/J1

-1/9

0

order by disorder

chiral universality

Th. Jolicoeur et al., Phys. Rev. B, 42, 4800 (1990).

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

RYO TAMURA AND SHU TANAKA

Let us consider the system whose order par

PHYSICAL

amet

REVIEW er spac

E 88, e is AxB

052138 .(2013)

(a) No symmetry is broken.

tw

R o-dimensional

YO TAMURA

systems.

AND SHU T In

AN the

AKA Heisenberg model on a

PHYSICAL REVIEW E 88, 052138 (2013)

(a)

stacked triangular lattice with the antiferromagnetic nearest-

Disordered phase

T

neighbor

tw

intralayer

o-dimensional

interaction

systems. In J1

the and the nearest-neighbor

Heisenberg model on a

(a)

interlayer

stacked interaction

triangular

J

Disordered phase

⊥,

latticethe gr

with oun

the d state is a 120◦ structure

antiferromagnetic

in

nearest-

(b)

T

each triangular

neighbor

layer

intralayer . Thus the order

interaction J1 parameter

and the

space is SO(3)

Partially ordered phase

Disordered phase

nearest-neighbor

(b) Only one of two symmetries is broken.

T

as in the

interlayertwo-dimensional

interaction J⊥, th case.

e grouT

n w

d o

st types

ate is a of

12 contradictory

0◦ structure in

(b)

results

each

have been

triangular

reported.

layer. Thus

In

the

one,

order

a second-order

parameter space is phase

Partially ordered phase A is broken.

SO(3)

Disordered phase

T

transition

as in the belonging

tw

to the

o-dimensional universality

case. Two class

types called

of

the chiral

Partially ordered phase

Disordered phase

contradictory

A is broken.

T

universality

results hav class,

e

which

been

relates

reported. to

In the SO(3)

one, a s symmetry,

econd-order occurs

phase

Partially ordered phase B is broken.

[3,42–46].

transition In the other

belonging ,

to a first-order

the univ

phase

ersality

transition

class called occurs

the

at

chiral

Disordered phase

T

finite

univ temperature

ersality class, [47–50

which ]. In either

relates to

case,

the

the

SO(3) phase transition

symmetry, occurs

(c)

B is broken.

nature

[3,42–in

46 the

]. Heisenber

In the other,gamodel on a

first-order stack

phaseed triangular

transition

lattice

occurs at

(i)

differs

finite from that on

temperature [ a

47tw

– o-dimensional

50]. In either

triangular

case, the

lattice.

Ordered phase

Disordered phase

phase transition

(c)

T

Recently

nature in ,

the another kind

Heisenberg

of characteristic

model on a stacked phase transition

triangular lattice

(i)

Ordered phase A and B are broken.

Disordered phase

nature

differs has

f

been

rom that found

on a in

tw Heisenberg models

o-dimensional

on

triangular a triangular

lattice.

T

lattice with

Recently, further

another interactions

kind of

[33,34,51

characteristic–53]. The

phase

order

transition

A and B are broken.

parameter

nature

space

has been is described

found in

by the direct

Heisenberg

product

models on between

a

the

triangular

(ii)

Ordered phase

Partially ordered phase Disordered phase

global

lattice rotational

with

symmetry

further

of spins

interactions [ SO(3)

33,34, and

51– discrete

53]. The lattice

order

T

rotational

parameter symmetry

space is , which

described depends

by the

on

direct the ground

product

state.

between In

the

(ii)

Ordered phase B is broken.

A is

Partially ordered phase broken.

Disordered phase

these m

global odels, a

rotationalphase transition

symmetry of s with

pins

the

SO(3) discrete

and

symmetry

discrete lattice

T

Ordered phase

Partially ordered phase Disordered phase

breaking occurs

rotational

at finite

symmetry,

temperature.

which depends In the

on

J

the 1-J3 Heisenber

ground state. g

In

B is broken.

A is broken.

T

model

these on

m a triangular

odels, a

lattice,

phase t

the ground

ransition with state

the d is the

iscretespiral-spin

symmetry

Ordered phase A is broken.

B is

Partially ordered phase broken.

Disordered phase

structure

breaking where C

occurs 3

at lattice

finite rotational symmetry

temperature. In the J is

1- brok

J3

en due to

Heisenberg

T

the competition

model on a

between

triangular

the

lattice, ferromagnetic

the ground

nearest-neighbor

state is the spiral-spin

A is broken.

B is broken.

interaction

FIG. 2. (Color online) Schematic of the phase transition nature

structure w J1 and

here C3 antiferromagnetic

lattice rotational

third nearest-neighbor

symmetry is broken due to

interaction

in systems where the order parameter space is the direct product

the

J3 [51

competition ,52]. In

between this

the case,

f

the order

erromagnetic parameter space

nearest-neighbor

is SO(3)

between

FIG. tw

2. o groups

(Color online) Schematic of the phase transition nature

×

A and B: (a) Neither symmetry is broken,

C

interaction

and antiferromagnetic third nearest-neighbor

3.

J This

1

model exhibits a first-order phase transition

in systems where the order parameter space is the direct product

with breaking

interaction J3 of

[ the

(b) either A or B is broken but the other is not broken, and

51, C

52]. In this case, the order parameter space

3 symmetry. In addition, the dissociation

between two groups

and

of

(c) both A and B are brok

A en.

is

B: (a) Neither symmetry is broken,

Z

SO(3)

. This model exhibits a first-order phase transition

2 vorti

× ce

C3s that comes from the SO(3) symmetry occurs at

(b) either A or B is broken but the other is not broken, and

the first-order

with breaking phase

of the transition

C3

temperature.

symmetry. In

A

addition, similar

the

phase

dissociation

(c) both A and B are broken.

transition

of Z2 vor with

tices the

that discrete

comes f s

r ymmetry

om the SObreaking

(3) sym also

metry has

occ been

urs at

space is U(1) × Z2. Contradictory results were reported as

found

the

in Heisenber

first-order

g

phase models on

transition square and he

temperature. xagonal

A s

lattices

imilar phase

for the Heisenberg model on a stacked triangular lattice as

with further

transition

interactions

with the

[54

discrete –

s 56]. To

ymmetry consider

breakinga microscopic

also has been

mentioned

space is

abo

U(1) v

×e.Z Reference

2. C

[43]

ontradictory rreporte

esults d tha

were t a secon

reported da-s

mechanism

found in

of the fi

Heisenber rst-order

g models phase

on s transition

quare and w

he ith the d

xagonal iscrete

lattices

order

for

phase

the

transition

Heisenberg

occurs

model onata finite

stack temperature.

ed triangular Howev

lattice er,

as

symmetry

with

breaking

further

in frustrated

interactions [54–56 continuous

]. To

spin

consider a systems, a

microscopic

the authors

mentioned in

aboRef.

ve. [60] con

Referenceclu

[ de

43]drth

e a

p t

ortae fi

d rtst

h -aotrdaersepchoanse

d-

generalized

mechanism Potts

of

model,

the fi

called

rst-order

the

phase Potts

t

model

ransition w with

ith

in

the visible

discrete

transition

order

occurs.

phase

In either

transition

case,

occurs ata phase

finite transition occurs

temperature. Ho only

wever,

states, has

symmetry been studied

breaking in [57–59].

frustrated continuous spin systems, a

once

the in the

authors model.

in Ref. Another

[60] con ecxample

luded this

at aa first-order

first-order phase

phase

As shown

generalized

before,

Potts

the

model, phase

called

transition

the Potts

nature

model withinin three-

visible

transition

transition in the

occurs. antiferromagnetic

In either case, a

Heisenber

phase

g

transitionmodel

occurs on a

only

dimensional

states, has

systems

been

dif

studied [ fers

57– from

59].

that in two-dimensional

face-

oncecen

in tere

the d-cubic

model. lattice [61

Another ].

e The

xampleorder

is a parameter

first-order space

phase

systems

As

ev

sho en

wn when indi

before,

vidual

the

order

phase

parameter

transition

spaces

nature in - RYO TAMURA AND SHU TANAKA

PHYSICAL REVIEW E 88, 052138 (2013)

RYO TAMURA AND SHU TANAKA

PHYSICAL REVIEW E 88, 052138 (2013)

two-dimensional systems. In the Heisenberg model on a

two-dimensional systems. In the Heisenberg model on a

(a)

stacked triangular lattice with the antiferromagnetic nearest-

(a)

Disordered phase

stacked triangular lattice with the antiferromagnetic nearest-

Disordered phase

T

neighbor intralayer interaction

T

J1 and the nearest-neighbor

neighbor intralayer interaction J1 and the nearest-neighbor

interlayer interaction J

interlayer interaction ⊥, the ground state is a 120◦ structure in

J

(b)

each triangular layer. Thus

⊥, the ground state is a 120◦ structure in

the order parameter space is SO(3)

(b)

Partially ordered phase

Disordered phase

each triangular layer. Thus the order parameter space is SO(3)

Partially ordered phase

Disordered phase

T

as in the two-dimensional case. Two types of contradictory

T

as in the two-dimensional case. Two types of contradictory

results have been reported. In one, a second-order phase

A is broken.

results have been reported. In one, a second-order phase

A is broken.

transition belonging to the universality class called the chiral

Partially ordered phase

Disordered phase

transition belonging to the universality class called the chiral

Possible S

Partially ordered phase

cenarios

Disordered phase

T

universality class, which relates to the SO(3) symmetry, occurs

T

universality class, which relates to the SO(3) symmetry, occurs

(c) Both symmetries a

B is broken.

[3,42–46]. In the other, a first-order phase transition occurs at

re brok

B en.

is broken.

[3,42–46]. In the other, a first-order phase transition occurs at

finite temperature [47–50]. In either case, the phase transition

finite temperature [47–50]. In either case, the phase transition

(c-1(c)

) Both symmetries are broken simultaneously.

nature in the Heisenberg model on a stacked triangular lattice

(c)

nature in the Heisenberg model on a stacked triangular lattice

(i)

(i)

differs from that on a two-dimensional triangular lattice.

Ordered phase

Disordered phase

differs from that on a two-dimensional triangular lattice.

Ordered phase

Disordered phase

TT

Recently, another kind of characteristic phase transition

Recently, another kind of characteristic phase transition

A and B are broken.

nature has been found in Heisenberg models on a triangular

A and B are broken.

nature has been found in Heisenberg models on a triangular

lattice with further interactions [33,34,51–53]. The order

lattice with further interactions [33,34,51–53]. The order

(c-2) Both symmetries are broken successively.

parameter space is described by the direct product between the

(ii)

parameter space is described by the direct product between the

(ii)

Ordered phase

Partially ordered phase Disordered phase

global rotational symmetry of spins SO(3) and discrete lattice

Ordered phase

Partially ordered phase Disordered phase

T

global rotational symmetry of spins SO(3) and discrete lattice

T

rotational symmetry, which depends on the ground state. In

rotational symmetry, which depends on the ground state. In

B is broken.

A is broken.

B is broken.

A is broken.

these models, a phase transition with the discrete symmetry

these models, a phase transition with the discrete symmetry

Ordered phase

Partially ordered phase Disordered phase

breaking occurs at finite temperature. In the

Ordered phase

Partially ordered phase Disordered phase

J

breaking occurs at finite temperature. In the 1-J3 Heisenberg

T

J1-J3 Heisenberg

T

model on a triangular lattice, the ground state is the spiral-spin

model on a triangular lattice, the ground state is the spiral-spin

A is broken.

B is broken.

structure where

A is broken.

B is broken.

C

structure where 3 lattice rotational symmetry is broken due to

C3 lattice rotational symmetry is broken due to

the competition between the ferromagnetic nearest-neighbor

the competition between the ferromagnetic nearest-neighbor

interaction

FIG. 2. (Color online) Schematic of the phase transition nature

J

interaction 1 and antiferromagnetic third nearest-neighbor

FIG. 2. (Color online) Schematic of the phase transition nature

J1 and antiferromagnetic third nearest-neighbor

interaction

in systems where the order parameter space is the direct product

J

interaction

in systems where the order parameter space is the direct product

3 [51,52]. In this case, the order parameter space

J3 [51,52]. In this case, the order parameter space

is SO(3)

between two groups

is SO(3)×

A and B: (a) Neither symmetry is broken,

C

between two groups

3. This model exhibits a first-order phase transition

×

A and B: (a) Neither symmetry is broken,

C3. This model exhibits a first-order phase transition

with breaking of the

(b) either A or B is broken but the other is not broken, and

C

with breaking of the

(b) either

3 symmetry. In addition, the dissociation

A or B is broken but the other is not broken, and

C3 symmetry. In addition, the dissociation

of

(c) both A and B are broken.

Z

of

(c) both A and B are broken.

2 vortices that comes from the SO(3) symmetry occurs at

Z2 vortices that comes from the SO(3) symmetry occurs at

the first-order phase transition temperature. A similar phase

the first-order phase transition temperature. A similar phase

transition with the discrete symmetry breaking also has been

transition with the discrete symmetry breaking also has been

space is U(1)

space is U(1) × Z2. Contradictory results were reported as

× Z

found in Heisenberg models on square and hexagonal lattices

2. Contradictory results were reported as

found in Heisenberg models on square and hexagonal lattices

for the Heisenberg model on a stacked triangular lattice as

for the Heisenberg model on a stacked triangular lattice as

with further interactions [54–56]. To consider a microscopic

with further interactions [54–56]. To consider a microscopic

mentioned above. Reference [43] reported that a second-

mentioned above. Reference [43] reported that a second-

mechanism of the first-order phase transition with the discrete

mechanism of the first-order phase transition with the discrete

order phase transition occurs at finite temperature. However,

order phase transition occurs at finite temperature. However,

symmetry breaking in frustrated continuous spin systems, a

symmetry breaking in frustrated continuous spin systems, a

the authors in Ref. [60] concluded that a first-order phase

the authors in Ref. [60] concluded that a first-order phase

generalized Potts model, called the Potts model with invisible

generalized Potts model, called the Potts model with invisible

transition occurs. In either case, a phase transition occurs only

transition occurs. In either case, a phase transition occurs only

states, has been studied [57–59].

states, has been studied [57–59].

once in the model. Another example is a first-order phase

once in the model. Another example is a first-order phase

As shown before, the phase transition nature in three-

As shown before, the phase transition nature in three-

transition in the antiferromagnetic Heisenberg model on a

transition in the antiferromagnetic Heisenberg model on a

dimensional systems differs from that in two-dimensional

dimensional systems differs from that in two-dimensional

face-centered-cubic lattice [61]. The order parameter space

face-centered-cubic lattice [61]. The order parameter space

systems even when individual order parameter spaces are

systems even when individual order parameter spaces are

of the model is SO(3)

of the model is SO(3) × Z3. Moreover, in many cases, a

the same. Here let us review the phase transition behavior

× Z3. Moreover, in many cases, a

the same. Here let us review the phase transition behavior

phase transition occurs only once in systems with the order

phase transition occurs only once in systems with the order

in three-dimensional systems where the order parameter space

in three-dimensional systems where the order parameter space

parameter space described by the direct product between two

parameter space described by the direct product between two

is described by the direct product between two groups. Before

is described by the direct product between two groups. Before

groups when two symmetries break at the phase transition

groups when two symmetries break at the phase transition

we show some examples that have already been reported in

we show some examples that have already been reported in

temperature [32,43,60–67]. Next we show an example of

temperature [32,43,60–67]. Next we show an example of

a number of specific models, we consider generally what

a num - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

INTERLAYER-INTERACTION D

first-order

EPENDENCE Ophase

F LA transition

TENT . . .

with the C

PHYSICAL REVIEW E 88, 052138 (2013)

0.1

(d)

3 symmetry breaking at

25

finite temperature.

0.05

-2.2

0

We further investigate the way of spin ordering. As

-2.1

0

15

30

45

20

30

first-order phase transition with the C

mentioned

0.1

above, the

(d) order parameter space of the system is

3 symmetry breaking at

finite temperature.

-2.3

(a)

15

25

0.05

-2.2

SO(3)

40

INTERLAYER-INTERA

INTERLA

CTION D

YER-INTERA

EPENDENCE OF LACTION

TENT . D

. . EPENDENCE

×

0

We

PHYSICAL

furthe

REVIEW rE in

88v

, e

0s5ti

2 g

1 a

3 te

8 (2th

0 e

13) way of spin ordering. As

0

15

3O

0 F

C3. It was confirmed that the

4 LA

5

TENT

C3 symmetry breaks at

. . .

PHYSICAL REVIEW E 88, 052138 (2013)

(b)

10

20

the first-order phase transition point. In the antiferromagnetic

mentioned above, the order parameter space of the system is

Int

30

ernal Energy & S

5

-2.3 pecific Heat (a)

15

Heisenberg model on a stacked triangular lattice with only

SO(3)

as confirmed that the

-2.1

-2.1

40

30

30

first-order phase transition w first-order

×

ith the C3 phase

C3. It wtransition

symmetry

with

breaking atthe C C3 symmetry breaks at

0.1

0.1

(b)

(d)

a nearest-neighbor interaction where the order parameter

0

10

(d)

3 symmetry breaking at

H 20

= J

s

s

s

the first-order phase transition point. In the antiferromagnetic

1

i · sj

J3

i · sj

J

-2.3 i · sj

25

finite temperature.

0.05

-2.2

-2.1

25

finite temperature.

0.05

space is SO(3), a single peak is observed for the temperature

i,j 1

i,j 3

30

i,j

J3/J1 = 0.85355 · · · , J /J1 = 2

-2.2

Heisenberg model on a stacked triangular lattice with only

INTERLAYER-INTERACTION DEPENDENCE

-2.2

0

We further investigate the way of spin ordering. As

0

15

O

30 F 4LA

5

TENT

5

. . .

0

We further investigate the way of spin ordering. As

0

15

30

4

20

dependence 5

PHYSICAL REVIEW E

of the specific heat [42,43]. The 88, 052138 (2013)

10

1.55

20

peak indicates the

(e)

a nearest-neighbor interaction where the order parameter

0

mentioned above, the order mentioned

parameter abov

space e,

of the order

s

parameter

ystem is

space of the system is

20

Internal energy (a)

15

Specific heat

-2.3 finite-temperature

-2.2

phase

-2.1

transition between the paramagnetic

-2.3

-2.3

1.54

(a)

15

space is SO(3), a single peak is observed for the temperature

0

SO(3) × C3. It was confirme SO(3)

d that the

symmetry breaks at

-2.1

40

30

40

first-order

state and phase transition

magnetic

w

ordered ith the

state

C

× C3

C . It was confirmed that the C3 symmetry breaks at

where the SO(3) symmetry

1.53

0.1

(c)

(b)

10

10

(d)

3 symmetry breaking at

dependence of the specific heat [42,43]. The peak indicates the

0

0.00004

(b)0.00008 10

1.55

the first-order phase

the

transition

first-order

point. In the phase transition point.

antiferromagnetic

In the antiferromagnetic

0.02

25

finite

is

(e)

temperature.

broken. Then, in our model, the SO(3) symmetry should

0.05

finite-temperature phase transition between the paramagnetic

-2.2

30

30

60

5

5

1.54

Heisenberg model on a s

Heisenber

tacked

g model

triangular

on

lattice a stack

with

ed

only triangular lattice with only

0

0

We

break f

at urth

the er investi

first-order gate

phasethe way

transition of sp

point in or

since der

thein

s g. As

pecific

(f) 0 15 30 45

0.01

20

state and magnetic ordered state where the SO(3) symmetry

40

1.53

a nearest-neighbor interaction where the order parameter

0

(c)

a nearest-neighbor interaction where the order parameter

20

20

0 0 mentioned

heat has a abo

s

0.00004 ve,

ingle the order

peak

0.00008

parameter

corresponding space

to the of the system

first-order

is

phase

20

0.02-2.3

-2.2

-2.1

is broken. Then, in our model, the SO(3) symmetry should

-2.3

(a)

15

-2.3

-2.2

-2.1

0

space is SO(3), a single peak is observed for the temperature

0

SO(3)

space is SO(3), a single peak is observed for the temperature

transition. To confirm this we calculate the temperature

40

60

× C3. It was confirmed that the C3 symmetry breaks at

1.52

1.53

1.54

break at the first-order phase transition point since the specific

10

(b)

1.55

0

20000

40000

60000

10

10

0.01

1.55

1.55

40

the

(f)

first-order

dependence

dependence phase

of

of the

the ss transition

tructure

pecific f

heatpoint.

actor

[ of

42, In

43]. the

dependence

spin

Theantiferromagnetic

of the s

peakpecific heat

indicates [42

the ,43]. The peak indicates the

(e)

(e)

heat has a single peak corresponding to the first-order phase

20

finite-temperature phase

finite-temperature

transition between the phase transition

paramagnetic

between the paramagnetic

30

5

1.54

1.54

Heisenberg model on a stacked triangular lattice with only

0

0

0

transition. To confirm this we calculate the temperature

FIG. 4. (Color online) Temperature dependence

0

state and magnetic ordered state where the SO(3) symmetry

1.53 1.52

1.53

of (a)

1.54

internal

1

state and magnetic

1.55

0

20000

40000

60000

)

ordered state where the SO(3) symmetry

1.53

(c)

(c)

a nearest-neighbor interaction

0

0.00004

0.00008

s where the order parameter

S(k) :=

dependence of ,

the structure (11)

20

0

0

0.00004

0.00008

energy

0.02 per site

i · sj e−ik·(ri −rj

factor of spin

E/J1, (b) specific -2.3

0.02

heat C, and -2.2

(c) order p -2.1

aram-

space

is

is

brok SO(3),

en.

a

Then, single

in N

our peak is

is

model, observ

brok

the

ed

en. for

SO(3) s the

Then, in temperature

our

ymmetry model,

should the SO(3) symmetry should

Phase tra

60

nsition occurs.

60

eter |

i,j

µ|2 , which can detect the C3 symmetry breaking of the

break at the first-order phase break at the

transition

first-order

point since phase

the s

transition

pecific

point since the specific

(f)

1

10

0.01

1.55

0.01

FIG. 4. (Color online) Temperature dependence

(f)dependence of

of the

(a) specific

internal heat [42,43]. The peak indicates the

model with

ik

40

·(r

)

J

(

heat has a single peak corresponding to the k) :

first-order

s

i −rj

(11)

3/J1 = −0.853 55 . . . and (e)

40

J

which is the magnetic order heat has a

parameter single

forSs peak

= corresponding

piral-spin states.

i ·

phase sj e−to the first-order

,

phase

⊥/J1 = 2 for L = 24,32,40.

1.54

energy

20 per site E/J

20

finite-temperature phase transition between the paramagnetic

(d) P

1, (b) specific heat C, and (c) order param-

N

0

robability

0

distribution of the

0

internal energy P (E; T

0

c(L)). The 0

When the

transition. magnetic

To

ordered

confirm this transition.

state

we

To

described

calculate confirm

by

thek∗

this

where

i,j

we

the

temperature calculate the temperature

eter

1.52

1.53

1.54

1.55

1.52

|µ

0 |2 ,

1.53

which

20000

1.54

can det

40000

1.55

ect the

60000

0

C3 state and

20000

symmetry magnetic

40000

breaking ordered

60000

of the

state where the SO(3) symmetry

1.53

inset shows the lattice-size d (c)

ependence of the width between bimodal

model with

SO(3) symmetry

dependence of

is

the s broken

tructure f dependence

appears,

actor of spinof the

bec sotructure

mes a f

fi actor

nite of spin

0.02

0

0.00004

0.00008

J3/J1 = −0.853 55 . . . and J is broken. Then, in our model, the

which SO(3)

S(k∗)

is the symmetry

magnetic

should

peaks

order parameter for spiral-spin states.

E(L)/J

⊥/J1 = 2 for L = 24,32,40.

1. (e) Plot of Tc(L)

(d) 60

/J1 as a function of L−3. (f) Plot

Probability distribution of the

break

va

internal lu

e e

nerat

igythe

n t

Ph

( first-order

eE; th

T ermod

c(L)) - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

first-order phase transition with the C

0.1

(d)

3 symmetry breaking at

25

finite temperature.

0.05

-2.2

0

We further investigate the way of spin ordering. As

0

15

30

45

20

mentioned above, the order parameter space of the system is

-2.3

(a)

15

SO(3)

40

× C3. It was confirmed that the C3 symmetry breaks at

(b)

10

the first-order phase transition point. In the antiferromagnetic

Order Paramet

30 er (C3)

5

Heisenberg model on a stacked triangular lattice with only

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

H = J1

si · sj J3

si · sj J

si · sj

a nearest-neighbor interactio

PHYSICAL n where

REVIEW E the

88, 0o5rd

21e3r8 (p2a0ra

13m

) eter

20

0

i,j 1

i,j 3

i,j

J3/J1 = 0.85355 · · · , J /J1 = 2

-2.3

-2.2

-2.1

space is SO(3), a single peak is observed for the temperature

RYO TAMURA AND SHU TANAKA

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

first-order phase transition with the

symmetry breaking at

10

1.55

C

0.1

dependence of the specific heat [42,343]. The peak indicates the

(d)

(a)

(b)

1

(e)

finite temperature.

(c)

1.54

25

0.05

finite-temperature phase transition between the paramagnetic

-2.2 Order parameter

We further investigate the way of spin ordering. As

0.8

0

0 0

15

30

45

20

state and magnetic ordered state where the SO(3) symmetry

1.53

(c)

0.02

0

0.00004

0.00008

is broken.

mentioned

Then,

above,

in

the our

order model, the

parameter SO(3)

space of symmetry

the system sihould

s

0.6

-2.3

(a)

15

SO(3)

was confirmed that the

symmetry breaks at

40

60

break

× at

C the

3. It first-order phase transition

C3 point since the specific

0.4

0.01

(b)

40

10

(f)

heat

the

has a

first-order single

phase peak corresponding

transition point. In

to

the the first-order phase

antiferromagnetic

0.2

20

0

30

5

Heisenberg model on a stacked triangular lattice with only

0

transition. To confirm this we calculate the temperature

0

0

20000

40000

60000

-4 -3 -2 -1 0 1

1.52

1.53

1.54

1.55

a nearest-neighbor interaction where the order parameter

20

0

dependence of the structure factor of spin

-2.3

-2.2

-2.1

FIG. 3. (Color online) Explanation of ground-state properties when the nearest-neighbor interaction

space is SO(3), a single peak is observed for the temperature

J1 is ferromagnetic. (a) Position of k∗,

which minimizes the Fourier transform of interactions in the wave-vector space for J3/J1

−1/4. The hexagon represents the first BrillouinR. Tamura and N. Kawashima, J. Phys. Soc. Jpn., 77, 103002 (2008).

zone. A schematic of a ferromagnetic spin configuration in each triangular layer is shown. (b) Position of k∗ and the corresponding schematic

FIG.

10

4.

R. Tamur (Color

a and N. Kawashima, J online)

. Phys. Soc. Jpn., 80

Temperature

, 074008 (2011).

1.55

dependence of (a) internal dependence of the specific1heat [42,43]. The peak indicates the

of spiral-spin configurations in each triangular layer when J

)

3/J1 < −1/4. The spin configurations are depicted for J3/J1 = −0.853 55 . . .

(e)

s

corresponding to

S(k) :=

,

(11)

k∗ = π/2 and then θ = 90◦. (c) The J3/J1 dependence of k∗.

C

energy per site

i · sj e−ik·(ri −rj

E/J

finite-temperature phase transition between the paramagnetic

3 symmetry is broken.

1, (b) specific

1.54heat C, and (c) order param-

N

√

represented by k∗ = (0

0

i,j

,2π/ 3,0) in an appropriate way,

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

eter

state and magnetic ordered state where the SO(3) symmetry

shown in Fig. 2 in Ref. [78]. Then the order parameter space

specific heat has a divergent single peak at the temperature.

|µ|2 , which can detect the C

1.53 3 symmetry breaking of the

(c)

is not well defined in this case.

These behaviors indicate the existence of a finite-temperature

0.02

0

0.00004

0.00008

is broken. Then, in our model, the SO(3) symmetry should

Our purpose is to investigate the phase transition behavior

phase transition. As will be shown in Sec. IV, the uniform

model with J3/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,32,40.

which is the magnetic order parameter for spiral-spin states.

when the order parameter space is described by the direct

magnetic susceptibility can be used as an indicator of the phase

60

product between two groups. Hereafter we focus on the

transition. To investigate the way of ordering, the temperature

(d) Probability distribution of the internal energy P (E; T

break at the first-order phase transition point since the specific

0.01

(f)

c(L)). The

When the magnetic ordered state described by k∗ where the

parameter region J

dependence of an order parameter is considered. The order

3/J1 < −1/4 in the case of ferromagnetic

40

J

parameter

heat has a single peak corresponding to the first-order phase

1. Throughout the paper, we use the interaction ratio J3/J1 =

µ that can detect the C3 symmetry breaking is

inset shows the lattice-size dependence of the width between bimodal

−0

20

SO(3) symmetry is broken appears,

.853 55 . . . so that the ground state is represented by

defined by

S(k∗) becomes a finite

k∗ = π/2 in Eq. (4). In this case, along one of three axes,

peaks 0

0

transition. To confirm this we calculate the temperature

the relative angle between nearest-neighbor spin pairs is 90◦,

µ := ε

E(L)/J

1e1 + ε2e2 + ε3e3,

(7)

1. (e) Plot of Tc(L)/J1 as a function of L−3. (f) Plot

1.52

1.53

1.54

1.55

0

20000

40000

60000

value in the thermodynamic limit. Figure 5(a) shows the

while along the other axes, the relative angle is 45◦ in the

1

dependence of the structure factor of spin

ground-state spin configuration [see Fig. 3(b)]. When the

of

ε :

s

C

η =

i · sj ,

(8)

max(L) as a function of L3. Lines are just visual guides and error

temperature dependence of the largest value of structure factors

period of the lattice is set to 8, the commensurate spiral-spin

N i,j 1 axis η

configuration appears in the ground state. Then, in order to

bars in all figures are omitted for clarity since their sizes are smaller

avoid the incompatibility due to the boundary effect, the linear

where the subscript η (η = 1,2,3) assigns the axis (see

S(k∗) calculated by1 six wave vectors in Eq. (4). Here S(k∗)

FIG. 4. (Color online) Temperature dependence of (a) internal

dimension

Fig. 1). The vectors e are unit vectors along the axis

)

L = 8n (n ∈ N ) is used and the periodic boundary

η

η

than √the symbol size.

S(k) :=

s

j ,

(11)

conditions in all directions are imposed.

in each triangular layer, i.e., e1 = (1,0), e2 = (−1/2, 3/2),

√

energy per site

becomes zero in the thermodynamic

i · sj e−ik·(ri −rlimit above the first-

and e

E/J

3 = (−1/2, −

3/2). The temperature dependence of

1, (b) specific heat C, and (c) order param-

N

III. FINITE-TEMPERATURE PROPERTIES

|µ|2 is shown in Fig. 4(c). The order parameter abruptly

eter

order phase transition temperature.

i,j

The structure factor S(k∗)

OF THE STACKED MODEL

increases around the temperature at which the specific heat

|µ|2 , which can detect the C3 symmetry breaking of the

has a divergent peak. These results conclude that the phase

of analysis.

model with

One is the finite-size scaling and the other is

In this section, we investigate the finite-temperature prop-

transition is accompanied by the C

J3/J1 = −0.853 55 . . . and J

becomes a nonzero value at the first-order phase transition

3 symmetry breaking.

which is the magnetic order parameter for spiral-spin states.

erties of the Heisenberg model on a stacked triangular

To decide the order of the phase transition, we calculate the

⊥/J1 = 2 for L = 24,32,40.

lattice with competing interactions given by Eq. (1) with

probability distribution of the internal energy at

a T,P

n (E

(d) a ;

Pi T)

v =

e analy

robability sis of

distribu the

tion pr

of oba

the bility d

internal is

e tri

ner b

g u

y tio

P (n

E;P

T (E; T

c(L)). c(

T L

he)).

temperature.

When the

Moreo

magnetic

ver, as

ordered temperature

state

decreases,

described by k∗

the structure

where the

J3/J1 = −0.853 55 . . . and J⊥/J1 = 2. Using Monte Carlo

D(E) exp(−NE/T ), where D(E) is the density of states.

simulations with the single-spin-flip heat-bath method and the

When a system exhibits a first-order phase transition, the

T - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

first-order phase transition with the C3 symmetry breaking at

Energy Histogram

0.1

(d)

25

finite temperature.

0.05

INTERLAYER-INTERACTION DEPENDENCE OF LA

-2.2 TENT . . .

0

We further investigate the way of spin ordering. As

0

15

3 PHYSICAL

0

45

REVIEW E 88, 052138 (2013)

s

s

s

20

H = J1

i · sj

J3

i · sj

J

i · sj

i,j

i,j

i,j

J

mentioned above, the order parameter space of the system is

1

3

3/J1 =

0.85355 · · · , J /J1 = 2

-2.3

(a)

15

SO(3) × C

-2.1

30

40

first-order phase transition with the C

3. It was confirmed that the C3 symmetry breaks at

0.1

(d)

3 symmetry breaking at

(b)

10

the first-order phase transition point. In the antiferromagnetic

P (E; T ) = D(E)e E/kBT

25

finite temperature.

0.05

D(E) : density of states

30

Heisenberg model on a stacked triangular lattice with only

-2.2

5

0

We further investigate the way of spin ordering. As

0

15

30

45

20

a nearest-neighbor interaction where the order parameter

E(L) : width between

20

mentioned 0abo

-2.3 ve, the order

-2.2

parameter

-2.1

space of the system is

two peaks

space is SO(3), a single peak is observed for the temperature

-2.3

(a)

15

SO(3)

40

× C3. It was confirmed that the C3 symmetry breaks at

10

1.55

dependence of the specific heat [42,43]. The peak indicates the

(b)

10

the first-order(e)phase transition point. In the antiferromagnetic

1.54

finite-temperature phase transition between the paramagnetic

30

5

0

Heisenberg model on a stacked triangular lattice

state and

with only

magnetic ordered state where the SO(3) symmetry

1.53

(c)

0.02

a nearest-ne0ighbor intera

0.00004 ction whe

0.00008 re th

ise ord

brok er

en. param

Then, et

in er

our model, the SO(3) symmetry should

20

0-2.3

-2.2

-2.1

space is SO(3),

60

a single peak is observed for

breakthe

at temperature

the first-order phase transition point since the specific

0.01

40 (f)

heat has a single peak corresponding to the first-order phase

10

1.55

dependence of the specific heat [42,43]. The peak indicates the

First-

20

(e)

order phase transition occurs.

0

transition. To confirm this we calculate the temperature

1.54

finite-temperature

0

phase transition between the paramagnetic

1.52

1.53

1.54

1.55

0

20000

40000

60000

0

state and magnetic ordered state where the SO(3) s

dependence ymmetry

of the structure factor of spin

1.53

(c)

0.02

0

0.00004

0.00008

is broken. Then, in our model, the SO(3) symmetry should

FIG. 4. (Color online) Temperature dependence of (a) internal

1

)

60

break at the first-order phase transition point since the s

S pecific

(k) :=

si · sj e−ik·(ri−rj ,

(11)

0.01

energy per site

40 (f)

E/J1, (b) specific heat C, and (c) order param-

N

eter

heat has a single peak corresponding to the first-order phase

i,j

20

|µ|2 , which can detect the C3 symmetry breaking of the

0

0

model with J3/J1 = −0.transition.

853 55 . . .

T

and o

J

confirm this we calculate the temperature

which is the magnetic order parameter for spiral-spin states.

1.52

1.53

1.54

1.55

0

20000

40000

60000

⊥/J1 = 2 for L = 24,32,40.

(d) Probability distribu dependence

tion of the

of

internal the

e

s

ner tructure

gy P (E; T fcactor

(L)). Tof

he spinWhen the magnetic ordered state described by k∗ where the

inset shows the lattice-size dependence of the width between bimodal

SO(3) symmetry is broken appears, S(k∗) becomes a finite

FIG. 4. (Color online) Temperature dependence of

peaks (a)E internal

(

1

L)/J1. (e) Plot of Tc(L)/J1 as a function of L−3. (f) Plot

S(k) :=

s

)

value ,in the the (11)

rmodynamic limit. Figure 5(a) shows the

energy per site

i · sj e−ik·(ri −rj

E/J1, (b) specific heat C, and (c

of)Cord

max e

( r

L) p

aasra

a m-

function of L3. Lines are just v

N

isual guides and error

temperature dependence of the largest value of structure factors

eter |

i,j

µ|2 , which can detect the C

bars in all figures are omitted for clarity since their sizes are smaller

3 symmetry breaking of the

S(k∗) calculated by six wave vectors in Eq. (4). Here S(k∗)

model with J

than the symbol size.

3/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,32,40.

which is the magnetic order parameter for spiral-spin

becomes zero states.

in the thermodynamic limit above the first-

(d) Probability distribution of the internal energy P (E; Tc(L)). The

When the magnetic ordered state described by

order

k∗

phasewhere the

transition temperature. The structure factor S(k∗)

inset shows the lattice-size dependence of the width be

of tween bim

analysis. odal

One is SO(3)

the

symmetry

finite-size s

is

caling brok

and en

the appears,

other is S(k∗) beco

becomes maes a fini

nonzero tevalue at the first-order phase transition

peaks E(L)/J1. (e) Plot of Tc(L)/J1 as a function

a of

na L−3

ive .a (f)

nal Plot

ysis of t va

he lu

pr eob ianbiltihte

y dtihsetrrim

buotd

ioynnaPm

( iEc;Tlcim

(Li)t).. Figure 5(a) sho

temperature. ws the

Moreover, as temperature decreases, the structure

of Cmax(L) as a function of L3. Lines are just visual guides

The s and

caling error

relationstemperature

for the

dependence

first-order phase of the largest

transition in value

fa of

cto structure

r S(k∗) infactors

creases. The structure factors at kz = 0 in the

bars in all figures are omitted for clarity since their s

d izes are smal

-dimensionaller

systems [82] are given by

first Brillouin zone at several temperatures for

S(k∗) calculated by six wave vectors in Eq. (4). Here S(k∗)

L = 40 are also

than the symbol size.

becomes zero in the thermodynamic limit

sho abo

wn v

in e the

Fig.

first-

5(b). As mentioned in Sec. II, the spiral-spin

Tc(L) = aL−d + T

order phase transition

c,

(9)

temperature. The structure

structure fractor

epresented by k is the same as that represented by

S(k∗)

of analysis. One is the finite-size scaling and the other is

becomes

−k in the Heisenberg models. Figure 5(b) confirms that one

( a nonzero

E)2Ld

value at the first-order phase transition

a naive analysis of the probability distribution

distinct wave vector is chosen from three types of ordered

P (E; T

C

,

(10)

c(L)).

temperature.

max(L) ∝

Moreover, as temperature decreases, the structure

4T 2

The scaling relations for the first-order phase transition in

factor

c

S(k∗) increases. The structure factors at kz = 0 in the

where T

d-dimensional systems [82] are given by

c and

E are, respectively, the transition temperature

first Brillouin zone at several temperatures for

(a) 0.5

L = 40 are also

and the latent heat in the thermodynamic limit. The coefficient

shown in Fig. 5(b). As mentioned in Sec. II, the

0.4

spiral-spin

of the first term in Eq. (9), a, is a constant. Figures 4(e)

Tc(L) = aL−d + Tc,

(9)

structure represented by k is the same as that represented

0.3

by

and 4(f) show the scaling plots for Tc(L)/J1 and Cmax(L),

respectively. Figure −k

4(e) in the Heisenber

indicates that

g models. Figure 5(b) confirms that one

(

T

E)2Ld

c is a nonzero value

0.2

C

in the

distinct

thermodynamic limit. wave

Figure vector

4(f)

is

sho chosen

ws an a from

lmost three types of ordered

max(L) ∝

,

(10)

4T 2c

linear dependence of C

0.1

max(L) as a function of L3. However,

where T

using the finite-size scaling, we cannot obtain the transition

c and

E are, respectively, the transition temperature

(a) 0.5

0 0

0.5

1

1.5

2

and the latent heat in the thermodynamic limit. The coefficient

temperature and latent heat in the thermodynamic limit with

(b)

of the first term in Eq. (9),

0.4

high accuracy because of the strong finite-size effect. Next we

a, is a constant. Figures 4(e)

and 4(f) show the scaling plots for

directly calculate the size dependence of the width between

T

10-1

c(L)/J1 and Cmax(L),

0.3

respectively. Figure 4(e) indicates that

bimodal peaks of the energy distribution shown in Fig. 4(d).

T

10-2

c is a nonzero value

0.2

The width for the system size

in the thermodynamic limit. Figure 4(f) shows an almost

L is represented by

E(L) =

10-3

E

linear dependence of

+(L) − E−(L), where E+(L) and E−(L) are the averages of

C

0.1

10-4

max(L) as a function of L3. However,

the Gaussian func - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

first-order phase transition with the C

0.1

(d)

3 symmetry breaking at

INTERLAYER-INTERACTION DEPENDENCE OF LATENT

finite temperature.

. . .

25

0.05

PHYSICAL REVIEW E 88, 052138 (2013)

-2.2

0

We further investigate the way of spin ordering. As

0

15

30

45

20

mentioned above, the order parameter space of the system is

-2.1

30

-2.3

(a)

first-order

15

phase transition with the C

0.1

(d)

3 symmetry breaking at

SO(3)

40

× C3. It was confirmed that the C3 symmetry breaks at

25

(b)

finite

10

temperature.

0.05

the first-order phase transition point. In the antiferromagnetic

-2.2

0

We further investigate the way of spin ordering. As

0

30

15

30

45

20

5

Heisenberg model on a stacked triangular lattice with only

mentioned above, the order parameter space of the system is

a nearest-neighbor interaction where the order parameter

-2.3

(a)

15

20

SO(3)

0

40

-2.3

× C3. It w

-2.2 as confirm

-2.1 ed that the C3 symmetry breaks at

space is SO(3), a single peak is observed for the temperature

(b)

10

Finite-size Sc

thealing

first-order phase transition point. In the antiferromagnetic

10

1.55

dependence of the specific heat [42,43]. The peak indicates the

30

5

Heisenber

(e)

g model on a stacked triangular lattice with only

H = J1

si · sj J3

si · sj J 1.54 si · sj

finite-temperature phase transition between the paramagnetic

0 i,j 1

i,j 3

ai,j nearesJt3-n

/J1 ei

= gh

0. bor

85355 · · i

· n

, t

J er

/J a

1 c

= ti

2 on where the order parameter

20

0

state and magnetic ordered state where the SO(3) symmetry

-2.3

-2.2

-2.1(c)

space

1.53 0 is SO(3), a

0.00004 single peak

0.00008 is observed for the temperature

0.02

is broken. Then, in our model, the SO(3) symmetry should

Tc(L)

Max of specific heat

10

1.55

dependence of the specific heat [42,43]. The peak indicates the

60

(e)

break at the first-order phase transition point since the specific

0.01

(f)

1.54

finite-temperature

40

phase transition between the paramagnetic

heat has a single peak corresponding to the first-order phase

0

state

20

and magnetic ordered state where the SO(3) symmetry

1.53

(c)

0

0

transition. To confirm this we calculate the temperature

0.02

0

0.00004

0.00008

1.52

1.53

1.54

1.55

is brok

0

en. Then,

20000

in

40000

our

60000 model, the SO(3) symmetry should

dependence of the structure factor of spin

60

break at the first-order phase transition point since the specific

0.01

40 (f)

( E)2Ld

FIG.

T

( )

c(L 4.

) =(Color

aL d + online)

Tc

T

heat

emperature has

Cmax aL single

dependence

peak

of

4T (a)

2

corresponding

internal

to the first-order phase

1

20

c

s

)

0

S(k) :=

i · sj e−ik·(ri −rj ,

(11)

0energy per site E/J1, (b) specifictransition.

heat C, an T

d o(c confirm

) order pathis

ram- we calculate the temperature

N

1.52

1.53

1.54

1.55

0

20000

40000

M. S. S. C 60000

halla, D. P. Landau, and K. Binder, Phys. Rev. B, 34, 1841 (1986).

eter |

i,j

µ|2 , which can detect the dependence

C3

of

symmetry the

b

structure

reaking of

factor

the

of spin

model with J3/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,32,40.

which is the magnetic order parameter for spiral-spin states.

FIG. 4. (Color online) Temperature

(d) dependence

Probability of (a)

distribuinternal

1

tion of the internal energy S(

P k

( )

E;:=

Tc(L)). Thes

)

When the ,

magnetic

(11)

ordered state described by k∗ where the

energy per site

i · sj e−ik·(ri −rj

E/J1, (b) specific hienastetC

s ,h a

o n

w d

s t (

h c

e )laotr

tidcer-s p

i a

z r

e adm

e -

pendence of the width between N

bimodal

SO(3) symmetry is broken appears, S(k∗) becomes a finite

eter |

i,j

µ|2 , which can detect the C3 symmetry

peaks

breaking of the

E(L)/J1. (e) Plot of Tc(L)/J1 as a function of L−3. (f) Plot

First-order phase transition occurs.

value in the thermodynamic limit. Figure 5(a) shows the

model with J3/J1 = −0.853 55 . . . and J

of ⊥/J

C

1

max(= 2 for L

L) as a

= 24,

function32

of,40.

L3. L

which

ines are

is

just the

v

magnetic

isual guides a order

nd errorparameter for spiral-spin states.

temperature dependence of the largest value of structure factors

(d) Probability distribution of the internal

bars ienner

allgy

figP

u (

r E

es ; T

ar ce(L

o ))

m .it T

te h

d efor cl When

arity si the

nce magnetic

their sizes a ordered

re smalle state

r

described by k∗ where the

inset shows the lattice-size dependence of t

S(k∗) calculated by six wave vectors in Eq. (4). Here S(k∗)

than he w

the i

s dth be

ymboltween

size. bimodal

SO(3) symmetry is broken appears, S(k∗) becomes a finite

peaks

becomes zero in the thermodynamic limit above the first-

E(L)/J1. (e) Plot of Tc(L)/J1 as a function of L−3. (f) Plot

value in the thermodynamic limit. Figure 5(a) shows the

of

order phase transition temperature. The structure factor

C

S(k∗)

max(L) as a function of L3. Lines are just visual guides and error

temperature dependence of the largest value of structure factors

bars in all figures are omitted for clarity

of since their

analysis. sizes

One are

is sma

the ller

finite-size scaling and the other is

becomes a nonzero value at the first-order phase transition

S(

a naive analysis of the probabilit k∗) calculated by six wave vectors in Eq. (4). Here S(

y distribution

k∗)

than the symbol size.

P (E; Tc(L)).

temperature. Moreover, as temperature decreases, the structure

becomes zero in the thermodynamic limit above the first-

The scaling relations for the first-order phase transition in

factor S(k∗) increases. The structure factors at k

order phase transition temperature. The structure factor

z = 0 in the

S(k∗)

d-dimensional systems [82] are given by

first Brillouin zone at several temperatures for

of analysis. One is the finite-size scaling and the other is

L = 40 are also

becomes a nonzero value at the first-order phase transition

shown in Fig. 5(b). As mentioned in Sec. II, the spiral-spin

a naive analysis of the probability distribution P (E; Tc(L)).

temperature. Moreover, as temperature decreases, the structure

T

structure represented by k is the same as that represented by

The scaling relations for the first-order phase transition

c(L) = aL−d + T

in

c,

(9)

factor S(k∗) increases. The structure factors at kz = 0 in the

−k in the Heisenberg models. Figure 5(b) confirms that one

d-dimensional systems [82] are given by

( first

E) Brillouin

2Ld

zone at several temperatures for L

distinct wave = 40 are also

C

vector is chosen from three types of ordered

max(L) ∝

shown in, Fig. 5(b). As mentioned

(10)

in Sec. II, the spiral-spin

4T 2

T

c

c(L) = aL−d + Tc,

(9)

structure represented by k is the same as that represented by

where Tc and E are, respectively, the transition temperature

−k in the Heisenberg models. Figure

(a) 5(b)

0.5

confirms that one

( E)2Ld

and the latent heat in the thermodynamic limit. The coefficient

C

distinct wave vector is chosen from three

0.4

types of ordered

max(L) ∝

,

(10)

4of the first term in Eq. (9),

T 2

a, is a constant. Figures 4(e)

c

and 4(f) show the scaling plots for T

0.3

where

c(L)/J1 and Cmax(L),

Tc and

E are, respectively, the transition temperature

(a) 0.5

respectively. Figure 4(e) indicates that T

and the latent heat in the thermodynamic limit. The coefficient

c is a nonzero value

0.2

in the thermodynamic limit. Figure 4(f) shows an almost

of the first term in Eq. (9),

0.4

a, is a constant. Figures 4(e)

linear dependence of C

0.1

and 4(f) show the scaling plots for

max(L) as a function of L3. However,

Tc(L)/J1 and Cmax(L),

0.3

using the finite-size scaling, we cannot obtain the transition

respectively. Figure 4(e) indicates that

0

Tc is a nonzero value

temperature and latent heat in the

0

0.5

1

1.5

2

0.2

thermodynamic limit with

in the thermodynamic limit. Figure 4(f) shows an almost

high accuracy because of the strong finite-size effect. Next we

(b)

linear dependence of C

0.1

max(L) as a function of L3. However,

directly calculate the size dependence of the width between

10-1

using the finite-size scaling, we cannot obtain the transition

0

bimodal p - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

first-order phase transition with the C

0.1

(d)

3 symmetry breaking at

25

finite temperature.

0.05

-2.2

0

We further investigate the way of spin ordering. As

0

15

30

45

20

mentioned above, the order parameter space of the system is

-2.3

(a)

15

SO(3)

40

× C3. It was confirmed that the C3 symmetry breaks at

(b)

10

the first-order phase transition point. In the antiferromagnetic

30

5

Heisenberg model on a stacked triangular lattice with only

a nearest-neighbor interaction where the order parameter

20

0-2.3

-2.2

-2.1

space is SO(3), a single peak is observed for the temperature

10

1.55

dependence of the specific heat [42,43]. The peak indicates the

(e)

1.54

finite-temperature phase transition between the paramagnetic

0

state and magnetic ordered state where the SO(3) symmetry

1.53

(c)

0.02

0

0.00004

0.00008

is broken. Then, in our model, the SO(3) symmetry should

60

break at the first-order phase transition point since the specific

0.01

40 (f)

heat has a single peak corresponding to the first-order phase

20

0

0

transition. To confirm this we calculate the temperature

1.52

1.53

1.54

1.55

0

20000

40000

60000

dependence of the structure factor of spin

FIG. 4. (Color online) Temperature dependence of (a) internal

1

S(k) :=

s

),

(11)

energy per site

i · sj e−ik·(ri −rj

E/J1, (b) specific heat C, and (c) order param-

N

eter |

i,j

µ|2 , which can detect the C3 symmetry breaking of the

model with J3/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,32,40.

which is the magnetic order parameter for spiral-spin states.

(d) Probability distribution of the internal energy P (E; Tc(L)). The

When the magnetic ordered state described by k∗ where the

inset shows the lattice-size dependence of the width between bimodal

SO(3) symmetry is broken appears, S(k∗) becomes a finite

peaks E(L)/J1. (e) Plot of Tc(L)/J1 as a function of L−3. (f) Plot

value in the thermodynamic limit. Figure 5(a) shows the

of Cmax(L) as a function of L3. Lines are just visual guides and error

temperature dependence of the largest value of structure factors

bars in all figures are omitted for clarity since their sizes are smaller

S(k∗) calculated by six wave vectors in Eq. (4). Here S(k∗)

than the symbol size.

becomes zero in the thermodynamic limit above the first-

order phase transition temperature. The structure factor S(k∗)

of analysis. One is the finite-size scaling and the other is

becomes a nonzero value at the first-order phase transition

a naive analysis of the probability distribution P (E; Tc(L)).

temperature. Moreover, as temperature decreases, the structure

The scaling relations for the first-order phase transition in

factor S(k∗) increases. The structure factors at kz = 0 in the

d-dimensional systems [82] are given by

Order P

first Brillouin

a

zone r

at a

sev met

eral

er

temperatures

(

for SO

L = 40 (

are 3)

also

shown in Fig. 5(b). As mentioned in Sec. II, the spiral-spin )

Tc(L) = aL−d + Tc,

(9)

structure represented by k is the same as that represented by

−k in the Heisenberg models. Figure 5(b) confirms that one

( E)2Ld

H = J1

si · sj J3

si · sj J

si · sj

C

distinct wave vector is chosen from three types of ordered

max(L) ∝

,

(10)

4T 2

i,j

i,j

i,j

J

c

1

3

3/J1 =

0.85355 · · · , J /J1 = 2

where Tc and E are, respectively, the transition temperature

(a) 0.5

and the latent heat in the thermodynamic limit. The coefficient

of the first term in Eq. (9),

0.4

a, is a constant. Figures 4(e)

and 4(f) show the scaling plots for Tc(L)/J1 and Cmax(L),

0.3

respectively. Figure 4(e) indicates that Tc is a nonzero value

0.2

in the thermodynamic limit. Figure 4(f) shows an almost

linear dependence of C

0.1

max(L) as a function of L3. However,

Order parameter

using the finite-size scaling, we cannot obtain the transition

0

temperature and latent heat in the thermodynamic limit with

0

0.5

1

1.5

2

high accuracy because of the strong finite-size effect. Next we

(b)

directly calculate the size dependence of the width between

10-1

bimodal peaks of the energy distribution shown in Fig. 4(d).

10-2

The width for the system size L is represented by E(L) =

10-3

E+(L) − E−(L), where E+(L) and E−(L) are the averages of

10-4

the Gaussian function in the high-temperature phase and that in

the low-temperature phase, respectively. In the thermodynamic

10-5

limit, each Gaussian function becomes the δ function and then

FIG. 5. (Color online) (a) Temperature dependence of the largest

E(L) converges to

E [82]. The inset of Fig. 4(d) shows the

SO(3 v)alue of structure factors

symmetry is broken at the phase

S(k∗) calculated by six wave vectors in

size dependence of the width E(L)/J1. The width enlarges as

Eq. (4) for J3/J1 = −0.853 55 . . . and J⊥/J1 = 2. Error bars are

the system size increases, which indicates that the latent heat

tr is

ansition temperature.

omitted for clarity since their sizes are smaller than the symbol size.

a nonzero value in the thermodynamic limit. The results shown

(b) Structure factors at kz = 0 in the first Brillouin zone at several

in Fig. 4 conclude that the model given by Eq. (1) exhibits the

temperatures for L = 40.

052138-5 - Dependence on Interlayer Interaction

H = J1

si · sj J3

si · sj J

si

INTERLA · sj

YER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

RYO TAMURA AND SHU TANAKA

PHYSICAL REVIEW E 88, 052138 (2013)

i,j 1

i,j 3

i,j

J3/J1 = 0.85355 · · · , J /J1 = 2

vectors below the first-order phase transition point, which is

(a) 10

In Sec. IV, we investigated the interlayer interaction effect

-1.5

(a)

(b)

further evidence of the C3 symmetry breaking at the first-order

0

on the nature of phase transitions. We confirmed that the

phase transition temperature.

0.25 0.50

-2

0.75

1.00

1.25

10

first-order phase transition occurs for 0

Before we end this section, let us mention a phase

.25

J

1.50

⊥/J1

2.5 and

-2.5

1.75

1.5

transition nature in the

2.00

0

J

J

1-J2 Heisenberg model with interlayer

2.25

3/J1 = −0.853 55 . . ., which was used in Sec. III. We could

2.50

10

interaction J

-3

⊥ on a stacked triangular lattice. In Refs. [62–65],

not determine the existence of the first-order phase transition

0.25

the authors studied the phase transition behavior of the model

(b)

0

40

for

when

10

J

J

J /J

⊥/J1 < 0.25 or J⊥/J1 > 2.5 by Monte Carlo simulations.

1 and J2 are antiferromagnetic interactions. For large

1 increases

J

In the parameter ranges, the width of two peaks in the probabil-

2/J1, a phase transition between the paramagnetic phase and

0

0.50

ordered incommensurate spiral-spin structure phase occurs at

10

20

1

ity distribution of the internal energy cannot be estimated easily

0.75

finite temperature. In the parameter region, the order parameter

1.00

0

1.25

because of the finite-size effect. It is a remaining problem to

space is SO(3)

1.50

× C

1.75 2.00

3 and a second-order phase transition with

2.25 2.50

10

threefold symmetry occurs [62], which differs from the result

determine whether a second-order phase transition occurs for

0

0

obtained in this section. However, in frustrated spin systems,

10

large

0.85

0.25

0.50

0.75

1.00

1.25

1.50

1.75 2.00 2.25 2.50

J⊥/J1 as in the J1-J2 Heisenberg model on a stacked

a different phase transition nature happens even when the

0

triangular lattice [62]. As the ratio

symmetry that is broken at the phase transition temperature

0.8

0.5

J⊥/J1 increases, the first-

is the same as for other models. For example, in the J

20

order phase transition temperature monotonically increases

1-J3

0.75

Heisenberg model on a triangular lattice, a first-order phase

10

but the latent heat decreases. This is opposite to the behavior

0.7

transition with threefold symmetry breaking occurs when

(c)

0

0.08

observed in typical unfrustrated three-dimensional systems

J

0.15

3/J1 < −1/4 and J1 > 0. It is well known that the simplest

0.25

0.50

(d)

model that exhibits a phase transition with threefold symmetry

0.75

1.00

0.1

1.25

1.50

20

that exhibit a first-order phase transition at finite temperature.

1.75

2.00

breaking is the three-state ferromagnetic Potts model [76].

2.25

2.50

0.05

10

For example, the q-state Potts model with ferromagnetic

The three-state ferromagnetic Potts model in two dimensions

0

0

0.04

intralayer and interlayer interactions (

exhibits a second-order phase transition. It is no wonder that

q

3) is a fundamental

0.25

0.50 0.75

1.00

1.25

1.50

1.75

(e)

our obtained result differs from the results in the previous

0.2

2.00

2.25

2.50

20

model that exhibits a temperature-induced first-order phase

study [62].

0.1

10

(c)

transition with q-fold symmetry breaking [76]. From a mean-

0

0

0

field analysis of the ferromagnetic Potts model [76,83], as the

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

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0

1

2

IV. DEPENDENCE ON INTERLAYER INTERACTION

interlayer interaction increases, both the transition temperature

In this section, we study interlayer-interaction dependence

FIG. 6. (Color online) Interlayer-interaction J

and the latent heat increase. The same behavior was observed

⊥/J1 dependence

of the phase transition behavior. Here we set the interaction

of (a) internal energy per site E/J

FIG. 7. (Color online) (a) Interlayer-interaction

1, (b) specific heat C, (c) uniform

J⊥/J1 depen-

in the Ising-O(3) model on a stacked square lattice [77]. As

ratio J3/J1 = −0.853 55 . . . at which the ground state is

magnetic susceptibility χ, (d) order parameter |µ|2 , which can

dence of the probability distribution of internal energy P (E; T

represented by

c(L))

k∗ = π/2 in Eq. (4), as in the previous

detect the

just described, in general, if the parameter that can stabilize

C3 symmetry breaking, and (e) largest value of structure

section. In the previous section, we considered the case that

when the specific heat becomes the maximum value for

factors

L = 24.

S(k∗) calculated by six wave vectors in Eq. (4) for L = 24.

the ground state becomes large, the transition temperature

J

(b) The

⊥/J1 = 2. We found that the first-order phase transition

Error bars in all figures are omitted for clarity since their sizes are

J⊥/J1 dependence of Tc(L)/J1 at which the specific

increases and the latent heat increases [76,77,83]. Furthermore,

with the C3 symmetry breaking occurs and breaking of the

smaller than the symbol size.

heat becomes the maximum value for L = 16−40. (c) The J

SO(3) symmetry at the first-order phase transition point was

⊥/J1

in conventional systems, both the transition temperature and

dependence of the width between bimodal peaks of the energy

confirmed.

transition temperature. As stated in Sec. III, it can be used

the latent heat are expressed by the value of an effective

Figure 6 shows the temperature dependence of phys-

as an indicator of the first-order phase transition. Note that

distribution E(L)/J1. Error bars in all figures are omitted for clarity

interaction obtained by a characteristic temperature such as

ical quantities for L = 24 with several interlayer inter-

the magnetic susceptibility of the model with J⊥ differs from

since their sizes are smaller than the symbol size.

actions 0.25

J

the Curie-Weiss temperature. However, in our model, the

⊥/J1

2.5, setting J3/J1 = −0.853 55 . . ..

that with −J⊥. However, the sudden change in χ at the

Figure 6(a) shows the internal energy as a function of temper-

first-order phase transition temperature is also observed when

Curie-Weiss temperature does not characterize the first-order

ature, which displays that the temperature at which the sudden

the interlayer interaction is antiferromagnetic. We obtain the

the thermodynamic limit corresponds to the latent heat. Thus

phase transition, as will be shown in the Appendix. Thus our

change of the internal energy appears increases as J⊥/J1

Curie-Weiss temperature from the magnetic susceptibility for

Fig. 7(c) suggests that the latent heat decreases as

increases. In other words, Fig. 6(a) indicates that the first-

several

J

J

⊥/J1

result is an unusual behavior. The investigation of the essence

⊥/J1, including the case of antiferromagnetic J⊥,

order phase transition temperature monotonically increases

which will be shown in the Appendix. In addition, Figs. 6(d)

increases in the thermodynamic limit.

of the obtained results is a remaining problem.

as a function of J⊥/J1. In addition, the energy difference

and 6(e) confirm that phase transitions always accompany the

between the high-temperature phase and low-temperature

C3 lattice rotational symmetry breaking and breaking of the

phase decreases as J⊥/J1 increases. These behaviors are

global rotational symmetry of spin, the SO(3) symmetry, for

V. DISCUSSION AND CONCLUSION

ACKNOWLEDGMENTS

supported by the temperature dependence of the specific heat

the considered J⊥/J1, respectively.

shown in Fig. 6(b). Furthermore, in the specific heat, no

Next, in order to consider the J⊥/J1 dependence of the

peaks, except the first-order phase transition temperature, are

latent heat, we calculate the probability distribution of the

In this paper, the nature of the phase transition of the

R.T. was partially supported by a Grand-in-Aid for Sci-

observed by changing the value of J

Heisenberg model on a stacked triangular lattice was studied by

entific Research (C) (Grant No. 25420698) and National

⊥/J1. Figure 6(c) shows

internal energy P (E; Tc(L)) for several values of J⊥/J1 shown

the uniform magnetic susceptibility χ, which is calculated by

in Fig. 7(a). The width between bimodal peaks decreases

Monte Carlo simulations. In our model, there are three kinds of

Institute for Materials Science. S.T. was partially supported

as J⊥/J1 increases. Furthermore, we calculate interlayer-

N J

interactions: the ferromagnetic nearest-neighbor interaction J

by a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).

1 |m|2

1

interaction dependences of T< - Dependence on Interlayer Interaction

s

s

s

INTERLAYER-INTERACTION DEPENDENCE

H = J1 OF LAiTENT

· sj

J

. . . 3

i · sj

J

PHYSICAL

i · sREVIEW

j

E 88, 052138 (2013)

i,j 1

i,j 3

i,j

J3/J1 = 0.85355 · · · , J /J1 = 2

(a) 10

(b)

In Sec. IV, we investigated the interlayer interaction effect

Transition

0

on the nature of phase transitions. We confirmed that the

temperature

As the interlayer interaction

10

first-order phase transition occurs for 0.25

J⊥/J1

2.5 and

0

1.5

J increases, ...

10

3/J1 = −0.853 55 . . ., which was used in Sec. III. We could

not determine the existence of the first-order phase transition

0

10

for J⊥/J1 < 0.25 or J⊥/J1 > 2.5 by Monte Carlo simulations.

0

In the parameter ranges, the width of two peaks in the probabil-

10

1

ity distribution of the internal energy cannot be estimated easily

0

10

because of the finite-size effect. It is a remaining problem to

transition temperature

0

determine whether a second-order phase transition occurs for

10

large J⊥/J1 as in the J1-J2 Heisenberg model on a stacked

0

triangular

inc lattice [62]. As

reases. the ratio

0.5

J⊥/J1 increases, the first-

20

order phase transition temperature monotonically increases

10

but the latent heat decreases. This is opposite to the behavior

0

0.08

observed in typical unfrustrated three-dimensional systems

20

that exhibit a first-order phase transition at finite temperature.

10

For example, the q-state Potts model with ferromagnetic

0

0.04

intralayer

lat and interlayer interactions

ent heat de (

creases.

q

3) is a fundamental

20

model that exhibits a temperature-induced first-order phase

10

(c)

transition with q-fold symmetry breaking [76]. From a mean-

Latent heat

0

field analysis of the ferromagnetic Potts model [76,83], as the

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0 0

1

2

interlayer interaction increases, both the transition temperature

and the latent heat increase. The same behavior was observed

FIG. 7. (Color online) (a) Interlayer-interaction J⊥/J1 depen-

in the Ising-O(3) model on a stacked square lattice [77]. As

dence of the probability distribution of internal energy P (E; Tc(L))

just described, in general, if the parameter that can stabilize

when the specific heat becomes the maximum value for L = 24.

the ground state becomes large, the transition temperature

(b) The J⊥/J1 dependence of Tc(L)/J1 at which the specific

increases and the latent heat increases [76,77,83]. Furthermore,

heat becomes the maximum value for L = 16−40. (c) The J⊥/J1

in conventional systems, both the transition temperature and

dependence of the width between bimodal peaks of the energy

distribution

the latent heat are expressed by the value of an effective

E(L)/J1. Error bars in all figures are omitted for clarity

since their sizes are smaller than the symbol size.

interaction obtained by a characteristic temperature such as

the Curie-Weiss temperature. However, in our model, the

Curie-Weiss temperature does not characterize the first-order

the thermodynamic limit corresponds to the latent heat. Thus

phase transition, as will be shown in the Appendix. Thus our

Fig. 7(c) suggests that the latent heat decreases as J⊥/J1

result is an unusual behavior. The investigation of the essence

increases in the thermodynamic limit.

of the obtained results is a remaining problem.

V. DISCUSSION AND CONCLUSION

ACKNOWLEDGMENTS

In this paper, the nature of the phase transition of the

R.T. was partially supported by a Grand-in-Aid for Sci-

Heisenberg model on a stacked triangular lattice was studied by

entific Research (C) (Grant No. 25420698) and National

Monte Carlo simulations. In our model, there are three kinds of

Institute for Materials Science. S.T. was partially supported

interactions: the ferromagnetic nearest-neighbor interaction J1

by a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).

and antiferromagnetic third nearest-neighbor interaction J3 in

The computations in the present work were performed on

each triangular layer and the ferromagnetic nearest-neighbor

computers at the Supercomputer Center, Institute for Solid

interlayer interaction J⊥. When J3/J1 < −1/4, the ground

State Physics, University of Tokyo.

state is a spiral-spin structure in which the C3 symmetry is

broken as in the case of two-dimensional J1-J3 Heisenberg

APPENDIX: INTERLAYER-INTERACTION DEPENDENCE

model on a triangular lattice [51,52]. Then the order parameter

OF THE CURIE-WEISS TEMPERATURE

space in the case is described by SO(3) × C3.

In Sec. III, we studied the finite-temperature properties

In this section, we obtain the Curie-Weiss temperature for

of the system with J3/J1 = −0.853 55 . . . and J⊥/J1 = 2.

several J⊥/J1, including the case of the antiferromagnetic

We found that a first-order phase transition takes place

interlayer interaction. Here we also use the interaction ratio

at finite temperature. The temperature dependence of the

J3/J1 = −0.853 55 . . ., which was used in Secs. III and IV.

order parameter indicates that the C3 symmetry breaks at

As mentioned in Sec. II, the phase transition behavior of

the transition temperature, which is the same feature as in

the model with J⊥ is the same as that with −J⊥, which

the two-dimensional case [51,52]. We also calculated the

is proved by the local gauge transformation. However, the

temperature dependence of the structure factor at the wave

Curie-Weiss temperature for J⊥ differs from that for −J⊥.

vector representing the ground state, which is the magnetic

Figure 8(a) shows the inverse of the magnetic susceptibility

order parameter for spiral-spin states. The result shows that

χ −1 as a function of temperature in the high-temperature

the SO(3) symmetry breaks at the transition temperature.

region for L = 24. In general, the temperature dependence of

052138-7 - PHYSICAL REVIEW E 88, 052138 (2013)

Interlayer-interaction dependence of latent heat in the Heisenberg model

on a stacked triangular lattice with competing interactions

Ryo Tamura1,* and Shu Tanaka2,†

1International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan

2Department of Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

(Received 12 August 2013; published 26 November 2013)

We study the phase transition behavior of a frustrated Heisenberg model on a stacked triangular lattice by Monte

Carlo simulations. The model has three types of interactions: the ferromagnetic nearest-neighbor interaction J1 and

antiferromagnetic third nearest-neighbor interaction J3 in each triangular layer and the ferromagnetic interlayer

interaction J⊥. Frustration comes from the intralayer interactions J1 and J3. We focus on the case that the order

parameter space is SO(3)×C3. We find that the model exhibits a first-order phase transition with breaking of the

SO(3) and C3 symmetries at finite temperature. We also discover that the transition temperature increases but

the latent heat decreases as J⊥/J1 increases, which is opposite to the behavior observed in typical unfrustrated

three-dimensional systems.

DOI: 10.1103/PhysRevE.88.052138

PACS number(s): 75.10.Hk, 64.60.De, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

state of the model is the 120◦ structure, the order parameter

space is SO(3), which is the global rotational symmetry of

Geometrically frustrated systems often exhibit a charac-

spins. Thus the point defect, i.e., the

teristic phase transition, such as successive phase transitions,

Z2 = π1 [SO(3)] vortex

defect, can exist in the model. Then the topological phase

order by disorder, and a reentrant phase transition, and an

transition occurs by dissociating the

unconventional ground state, such as the spin liquid state

Z2 vortices at finite

temperature [31,36,37]. The dissociation of

[1–20]. In frustrated continuous spin systems, the ground

Z2 vortices is

one of the characteristic properties of geometrically frustrated

state is often a noncollinear spiral-spin structure [21,22]. The

systems when the ground state is a noncollinear spin structure

spiral-spin structure leads to exotic electronic properties such

in two dimensions. In these systems, the order parameter

as multiferroic phenomena [23–27], the anomalous Hall effect

space is described by SO(3). The temperature dependence

[28], and localization of electronic wave functions [29]. Thus

of the vector chirality and that of the number density of

the properties of frustrated systems have attracted attention

Z2

vortices in the Heisenberg model on a kagome lattice were

in statistical physics and condensed matter physics. Many

also studied [38]. An indication of the

geometrically frustrated systems such as stacked triangular

Conclusion

Z2 vortex dissociation

has been observed in electron paramagnetic resonance and

antiferromagnets (see Fig. 1), stacked kagome antiferromag-

electron spin resonance measurements [39–41].

nets, and spin-ice systems have been synthesized and their

Phase transition has been studied theoretically in stacked

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

properties have been

RYO TAMURA AND SHU TANAKA

W

inv e studied the phase tr

estigated. In theoretical studies, the

triangular lattice

PHYSICAL systems as well as

REVIEW in

E two-dimensional

88, 052138 (2013)

relation between phase transition and order parameter space in

ansition nature of a frustrated Heisenberg

geometrically frustrated systems has been considered [30–34].

triangular lattice systems. In many cases, the phase transition

As an example of model on a stacked tr

phase transition nature in geometrically

nature

iangular la

in three-dimensional

ttic

systems

e

dif .

fers from that in

(a) 10

In Sec. IV, we investigated the interlayer interaction effect

frustrated systems, properties of the Heisenberg model on

(b)

a triangular lattice have been theoretically studied for a

0

on the nature of phase transitions. We confirmed that the

two-dimensional systems. In the Heisenberg model on a long time. Triangular antiferromagnetic systems are a typical

10

(a)

first-order phase transition occurs for 0.25

J

example of geometrically frustrated systems and have been Disordered phase

1.5

⊥/J1

2.5 and

stacked triangular lattice with the antiferromagnetic nearest- well investigated. The ground state of the ferromagnetic

0

J3/J1 = −0.853 55 . . ., which was used in Sec. III. We could

Heisenberg model on a triangular lattice is a ferromagnetically

T 10

collinear spin structure. In this case, the order parameter space

not determine the existence of the first-order phase transition

neighbor intralayer interaction J1 and the nearest-neighbor is

0

S2. The long-range order of spins does not appear at finite

for

temperature because of the Mermin-Wagner theorem [35].

10

J⊥/J1 < 0.25 or J⊥/J1 > 2.5 by Monte Carlo simulations.

interlayer interaction J

The model does not exhibit any phase transitions. In contrast,

⊥, the ground state is a 120◦ structure in

(b)

0

In the parameter ranges, the width of two peaks in the probabil-

Refs. [31,36,37] reported that a topological phase transition

Partially ordered phase

Disordered phase

10

each triangular layer. Thus the order parameter space is SO(3) occurs in the Heisenberg model on a triangular lattice with

1

ity distribution of the internal energy cannot be estimated easily

T

only antiferromagnetic nearest-neighbor interactions. In this

0

because of the finite-size effect. It is a remaining problem to

model the long-range order of spins is prohibited by the

axis 2

as in the two-dimensional case. Two types of contradictory

10

Mermin-Wagner theorem and thus a phase transition driven

A is broken.

axis 1

determine whether a second-order phase transition occurs for

by the long-range order of spins never occurs as well as

axis 3

0

results have been reported. In one, a second-order phase in the ferromagnetic Heisenberg model. Since the ground

We found that a first-order phase

10

large J⊥/J1 as in the J1-J2 Heisenberg model on a stacked

FIG. 1. (Color online) Schematic picture of a stacked triangular

transition belonging to the universality class called the chiral

Partially ordered phase

Disordered phase

lattice with

0

triangular lattice [62]. As the ratio

L

sites. Here

J

x × Ly × Lz

J1 and J3 respectively represent

transition occurs. At the first-order

T

0.5

⊥/J1 increases, the first-

*tamura.ryo@nims.go.jp

the nearest-neighbor and third-nearest-neighbor interactions in each

20

order phase transition temperature monotonically increases

†shu-t@chem.s.u-tokyo.ac.jp

triangular layer and J⊥ is the interlayer interaction.

universality class, which relates to the SO(3) symmetry, occurs

phase transition point

B , SO(3) and C

is broken.

3

10

but the latent heat decreases. This is opposite to the behavior

1539-3755/2013/88(5)/052138(9)

052138-1

©2013 American Physical Society

[3,42–46]. In the other, a first-order phase transition occurs at

symmetries are broken.

0

0.08

observed in typical unfrustrated three-dimensional systems

20

finite temperature [47–50]. In either case, the phase transition

that exhibit a first-order phase transition at finite temperature.

(c)

10

For example, the q-state Potts model with ferromagnetic

nature in the Heisenberg model on a stacked triangular lattice

(i)

0

0.04

intralayer and interlayer interactions (q

3) is a fundamental

differs from that on a two-dimensional triangular lattice.

Ordered phase

Disordered phase

T 20

model that exhibits a temperature-induced first-order phase

10

Recently, another kind of characteristic phase transition

(c)

transition with q-fold symmetry breaking [76]. From a mean-

A and B are broken.

0

field analysis of the ferromagnetic Potts model [76,83], as the

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0 0

1

2

nature has been found in Heisenberg models on a triangular

interlayer interaction increases, both the transition temperature

lattice with further interactions [33,34,51–53]. The order

and the latent heat increase. The same behavior was observed

FIG. 7. (Color online) (a) Interlayer-interaction J

(ii)

⊥/J1 depen-

parameter space is described by the direct product between the

in the Ising-O(3) model on a stacked square lattice [77]. As

Ordered phase

Partially ordered phase Disordered phase

dence of the probability distribution of internal energy P (E; Tc(L))

just described, in general, if the parameter that can stabilize

global rotational symmetry of spins SO(3) and discrete lattice

w

T hen the specific heat becomes the maximum value for L = 24.

the ground state becomes large, the transition temperature

(b) The J

rotational symmetry, which depends on the ground state. In

⊥/J1 dependence of Tc(L)/J1 at which the specific

B is - PHYSICAL REVIEW E 88, 052138 (2013)

Interlayer-interaction dependence of latent heat in the Heisenberg model

on a stacked triangular lattice with competing interactions

Ryo Tamura1,* and Shu Tanaka2,†

1International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan

2Department of Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

(Received 12 August 2013; published 26 November 2013)

We study the phase transition behavior of a frustrated Heisenberg model on a stacked triangular lattice by Monte

Carlo simulations. The model has three types of interactions: the ferromagnetic nearest-neighbor interaction J1 and

antiferromagnetic third nearest-neighbor interaction J3 in each triangular layer and the ferromagnetic interlayer

interaction J⊥. Frustration comes from the intralayer interactions J1 and J3. We focus on the case that the order

parameter space is SO(3)×C3. We find that the model exhibits a first-order phase transition with breaking of the

SO(3) and C3 symmetries at finite temperature. We also discover that the transition temperature increases but

the latent heat decreases as J⊥/J1 increases, which is opposite to the behavior observed in typical unfrustrated

three-dimensional systems.

DOI: 10.1103/PhysRevE.88.052138

PACS number(s): 75.10.Hk, 64.60.De, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

state of the model is the 120◦ structure, the order parameter

space is SO(3), which is the global rotational symmetry of

Geometrically frustrated systems often exhibit a charac-

spins. Thus the point defect, i.e., the

teristic phase transition, such as successive phase transitions,

Z2 = π1 [SO(3)] vortex

defect, can exist in the model. Then the topological phase

order by disorder, and a reentrant phase transition, and an

transition occurs by dissociating the

unconventional ground state, such as the spin liquid state

Z2 vortices at finite

temperature [31,36,37]. The dissociation of

[1–20]. In frustrated continuous spin systems, the ground

Z2 vortices is

one of the characteristic properties of geometrically frustrated

state is often a noncollinear spiral-spin structure [21,22]. The

systems when the ground state is a noncollinear spin structure

spiral-spin structure leads to exotic electronic properties such

in two dimensions. In these systems, the order parameter

as multiferroic phenomena [23–27], the anomalous Hall effect

space is described by SO(3). The temperature dependence

[28], and localization of electronic wave functions [29]. Thus

of the vector chirality and that of the number density of

the properties of frustrated systems have attracted attention

Z2

vortices in the Heisenberg model on a kagome lattice were

in statistical physics and condensed matter physics. Many

also studied [38]. An indication of the

geometrically frustrated systems such as stacked triangular

Conclusion

Z2 vortex dissociation

has been observed in electron paramagnetic resonance and

antiferromagnets (see Fig. 1), stacked kagome antiferromag-

electron spin resonance measurements [39–41].

nets, and spin-ice systems have been synthesized and their

Phase transition has been studied theoretically in stacked

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

properties have been W

inv e studied the phase tr

estigated. In theoretical studies, the

triangular lattice systems as well as in two-dimensional

relation between phase transition and order parameter space in

ansition nature of a frustrated Heisenberg

geometrically frustrated systems has been considered [30–34].

triangular lattice systems. In many cases, the phase transition

As an example of model on a stacked tr

phase transition nature in geometrically

nature

iangular la

in three-dimensional

ttic

systems

e

dif .

fers from that in

(a) 10

In Sec. IV, we investigated the interlayer interaction effect

frustrated systems, properties of the Heisenberg model on

(b)

a triangular lattice have been theoretically studied for a

0

on the nature of phase transitions. We confirmed that the

long time. Triangular antiferromagnetic systems are a typical

10

first-order phase transition occurs for 0.25

J

example of geometrically frustrated systems and have been

1.5

⊥/J1

2.5 and

well investigated. The ground state of the ferromagnetic

0

J3/J1 = −0.853 55 . . ., which was used in Sec. III. We could

Heisenberg model on a triangular lattice is a ferromagnetically

10

collinear spin structure. In this case, the order parameter space

not determine the existence of the first-order phase transition

is

0

S2. The long-range order of spins does not appear at finite

for

temperature because of the Mermin-Wagner theorem [35].

10

J⊥/J1 < 0.25 or J⊥/J1 > 2.5 by Monte Carlo simulations.

The model does not exhibit any phase transitions. In contrast,

0

In the parameter ranges, the width of two peaks in the probabil-

Refs. [31,36,37] reported that a topological phase transition

10

occurs in the Heisenberg model on a triangular lattice with

1

ity distribution of the internal energy cannot be estimated easily

only antiferromagnetic nearest-neighbor interactions. In this

0

because of the finite-size effect. It is a remaining problem to

model the long-range order of spins is prohibited by the

axis 2

10

Mermin-Wagner theorem and thus a phase transition driven

axis 1

determine whether a second-order phase transition occurs for

by the long-range order of spins never occurs as well as

axis 3

0

in the ferromagnetic Heisenberg model. Since the ground

We found that a first-order phase

10

large J⊥/J1 as in the J1-J2 Heisenberg model on a stacked

FIG. 1. (Color online) Schematic picture of a stacked triangular

lattice with

0

triangular lattice [62]. As the ratio

L

sites. Here

J

x × Ly × Lz

J1 and J3 respectively represent

transition occurs. At the first-order

0.5

⊥/J1 increases, the first-

*tamura.ryo@nims.go.jp

the nearest-neighbor and third-nearest-neighbor interactions in each

20

order phase transition temperature monotonically increases

†shu-t@chem.s.u-tokyo.ac.jp

triangular layer and J⊥ is the interlayer interaction.

phase transition point, SO(3) and C3

10

but the latent heat decreases. This is opposite to the behavior

1539-3755/2013/88(5)/052138(9)

052138-1

©2013 American Physical Society

symmetries are broken.

0

0.08

observed in typical unfrustrated three-dimensional systems

20

that exhibit a first-order phase transition at finite temperature.

10

For example, the q-state Potts model with ferromagnetic

The transition temperature increases

0

0.04

intralayer and interlayer interactions (q

3) is a fundamental

20

model that exhibits a temperature-induced first-order phase

but the latent heat decreases as the

10

(c)

transition with q-fold symmetry breaking [76]. From a mean-

interlayer interaction increases.

0

field analysis of the ferromagnetic Potts model [76,83], as the

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0 0

1

2

interlayer interaction increases, both the transition temperature

and the latent heat increase. The same behavior was observed

FIG. 7. (Color online) (a) Interlayer-interaction J⊥/J1 depen-

in the Ising-O(3) model on a stacked square lattice [77]. As

dence of the probability distribution of internal energy P (E; Tc(L))

just described, in general, if the parameter that can stabilize

when the specific heat becomes the maximum value for L = 24.

the ground state becomes large, the transition temperature

(b) The J⊥/J1 dependence of Tc(L)/J1 at which the specific

increases and the latent heat increases [76,77,83]. Furthermore,

heat becomes the maximum value for L = 16−40. (c) The J⊥/J1

in conventional systems, both the transition temperature and

dependence of the width between bimodal peaks of the energy

distribution

the latent heat are expressed by the value of an effective

E(L)/J1. Error bars in all figures are omitted for clarity

since their sizes are smaller than the symbol size.

interaction obtained by a characteristic temperature such as

the Curie-Weiss temperature. However, in our model, the

Curie-Weiss temperature does not characterize the first-order

the thermodynamic limit corresponds to the latent heat. Thus

phase transition, as will be shown in the Appendix. Thus our

Fig. 7(c) suggests that the latent heat decreases as J⊥/J1

result is an unusual behavior. The investigation of the essence

increases in the thermodynamic limit.

of the obtained results is a remaining problem.

V. DISCUSSION AND CONCLUSION

ACKNOWLEDGMENTS

In this paper, the nature of the phase transition of the

R.T. was partially supported by a Grand-in-Aid for Sci-

Heisenberg model on a stacked triangular lattice was studied by

entific Research (C) (Grant No. 25420698) and National

Monte Carlo simulations. In our model, there are three kinds of

Institute for Materials Science. S.T. was partially supported

interactions: the ferromagnetic nearest-neighbor interaction J1

by a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).

and antiferromagnetic third nearest-neighbor interaction J3 in

The computations in the - Thank you !

Ryo Tamura and Shu Tanaka

Physical Review E 88, 052138 (2013)