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Presentation file using the workshop which was held at the University of Tokyo (March 26, 2014). ...

Presentation file using the workshop which was held at the University of Tokyo (March 26, 2014). The presentation was based on two papers:

- Physical Review B Vol. 87, 214401 (2013)

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401

(preprint: http://arxiv.org/abs/1209.2520)

(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)

- Physical Review E Vol. 88, 052138 (2013)

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138

(preprint: http://arxiv.org/abs/1308.2467)

(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)

2014年３月26日に東京大学で開催された「統計物理学の新しい潮流」での講演スライドです。この講演は、以下の２つの論文に関係するものです。

- Physical Review B Vol. 87, 214401 (2013)

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.87.214401

(preprint: http://arxiv.org/abs/1209.2520)

(A brief explanation: http://www.slideshare.net/shu-t/prb-87214401slideshare)

- Physical Review E Vol. 88, 052138 (2013)

http://journals.aps.org/pre/abstract/10.1103/PhysRevE.88.052138

(preprint: http://arxiv.org/abs/1308.2467)

(A brief explanation: http://www.slideshare.net/shu-t/interlayerinteraction-dependence-of-latent-heat-in-the-heisenberg-model-on-a-stacked-triangular-lattice-with-competing-interactions)

- Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering3年弱前 by Shu Tanaka
- Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice3年弱前 by Shu Tanaka
- Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions3年弱前 by Shu Tanaka

- Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions3年弱前 by Shu Tanaka
- Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions3年弱前 by Shu Tanaka
- ２次元可解量子系のエンタングルメント特性1年以上前 by Shu Tanaka

- Unconventional Phase Transitions

in Frustrated Systems

Shu Tanaka (The University of Tokyo)

Collaborators:

Ryo Tamura (NIMS)

Naoki Kawashima (ISSP)

2D case: PRB 87, 214401 (2013), 3D case: PRE 88, 052138 (2013). - Main results

To investigate unconventional phase transition

behavior in geometrically frustrated systems.

2D

3D

SO(3)xZ2

SO(3)xC3

- Z2 vortex dissociation

- 1st-order PT w/ SO(3)xC3

- 2nd-order PT w/ Z

breaking

2 breaking

(2-dim. Ising universality)

- incr

J

eases, decr

E

eases.

at the same temperature. - Conventional phase transitions

Ferromagnets

Antiferromagnets

In the ground state, all spin

pairs form stable spin

configurations.

Phase transition occurs.

Ordered phase

Disordered phase

Tc

Temperature

Order parameter

Type

1D

2D

3D

space

Ising

Z2

×

√

√

XY

U(1)

×

KT

√

Heisenberg

S2

×

×

√ - TAMURA, KAWASHIMA, YAMAMOTO, TASSEL, AND KAGEYAMA

PHYSICAL REVIEW B 84, 214408 (2011)

1

1

which is defined by Eq. (11). The finite-size scaling relations

L=12

L=12

θ

of d-dimensional systems are given by

θ

0.5

0.5

Gc(L/2) ∝ (tL1/ν),

(15)

sy

x=0.296875

Gc(L/4)

0

sy

x=0.53125

θ=0.494π

0

θ=0.112π

Gc(L/2) ∝ L−d+2−η (tL1/ν),

(16)

-0.5

me

mo

-0.5

me

mo

where

and

are scaling functions and t := T − Tc.37

We determine the transition temperature Tc as the crossing

-1

-1

-1

-0.5

0

0.5

1

-1

-0.5

0

0.5

1

point of Gc(L/2)/Gc(L/4) using L = 20–32 data and obtain

sx

sx

Tc/J = 1.085(5). This transition temperature is consistent

with the one found for

FIG. 8. (Color online) Snapshots of spin directions for

U

x =

4(π π π ) in Sec. IV A. The finite-size

0

scaling using the critical exponents of the three-dimensional

.296875 and 0.53125 when the lattice size is L = 12 in the ground

state. Coordinates (

Heisenberg universality class (ν = 0.704, η = 0.025)38 are

s

) define the orthogonal plane of the vector

x ,sy

m

shown in Fig. 10(a). Since the data are well fitted by

o × me.

scaling relations, we conclude that the higher-temperature

phase transition belongs to the three-dimensional Heisenberg

the mixed phase as the “random fan-out state.” This random

universality class in accord with the Harris criterion.

fan-out state is an exotic bulk spin ordering that explains the

Next, we investigate the lower-temperature phase transition

simultaneous appearance of (πππ) and (ππ0) wave vectors

from the (πππ) ordered phase to the mixed phase. The (πππ)

without any phase separation. In this model, the ferromagnetic

ordered phase is translationally symmetric with the O(3) spin

correlation between NNLs exists as explained in Sec. III.

rotation symmetry broken down the spin rotation symmetry

U(1). In the mixed phase, both the translational symmetry and

C. Universality classes of phase transitions

the U(1) spin rotation symmetry are broken. Thus we expect

that the transition to the mixed phase is characterized by the

Frustration: random spin systems

We study the universality classes of phase transitions of

breaking of U(1) spin rotation symmetry. In other words, we

our model. In the phase diagram (see Fig. 5), there are two

expect that the lower-temperature phase transition belongs to

types of phase boundaries. To make clear the universality

the three-dimensional XY universality class. To obtain the

classes of each phase transition, x is set to 3/16 = 0.1875 such

transition temperature and confirm the critical exponents, we

Ferromag that transition

netic

temperatures are separated sufficiently. For this

calculate the magnetization vector m(ππ0) defined by Eq. (8).

interactionparameter, the intermediate phase is the (πππ) ordered phase

The finite-size scaling relations are given by

(see the dotted arrow in Fig. 5).

First, we investigate the higher-temperature phase transition

|m(q)|4

from the paramagnetic

Ev

phase to the

en in the GS, (

U

π π π ) ordered phase.

4(q) = |m(q)|2 2 ∝ f (tL1/ν),

(17)

From the Harris criterion,36 we expect that the higher-

locally unstable spin state

temperature phase transition belongs to the three-dimensional

|m(q)|2

χ (q) = N

∝ L2−ηg(tL1/ν),

(18)

Antiferr

Heisenber

omagneticg universality class. This

appears due t is

o because

frustra the critical

T

tion.

interactionexponent α is negative in the three-dimensional Heisenberg

where f and g are scaling functions, and q = (ππ0). We

model, and thus the disorder should not affect the universality

determine the transition temperature as the crossing point of

class. To obtain the transition temperature and confirm the

U4(π π 0) using L = 20–32 data and obtain Tc/J = 0.4585(5).

critical exponents, we calculate the correlation function Gc(r ),

The finite-size scaling using the critical exponents of the three-

c

Slow relaxation

Novel order

dimensional XY universality class (ν

layered perovskite

= 0.672, η = 0.038)39

are shown in Fig. 10(b). Although we obtain a reasonably

(a)

(b)

SrFe1-xMnxO2

good fit, it is not good enough to detect the small difference

between XY critical exponents and the other critical exponents

θ

in a three-dimensional system. However, from the viewpoint

of spin rotation symmetry, we can deduce that the lower-

temperature phase transition belongs to the three-dimensional

XY universality class.

b

a

V. MEAN-FIELD CALCULATIONS

c

θ

Random Fan-Out State

In this section, to obtain an intuitive understanding of the

emergence mechanism of the mixed phase, we investigate

H. Takano and S. Miyashita, JPSJ 64, 423 (1995).

R. Tamura, N. Kawashima, H. Kageyama et al.,

the effect of random interlayer couplings by mean-field

E. Vincent, Lecture Notes in Physics 716 (2007),

FIG. 9. (Color online) (a) “Average” spin directions in the spin

configuration of the random fan-out state.

PRB In e

84 ach layer (ab plane),

, 214408 (2011)

calculations. For simplicity of notation, we study the system

N´eel order appears. Along the interlayer direction (c axis), the angle

where the intralayer interactions are FM. Under the gauge

θ between nearest-neighbor “average” spin pairs changes from π to

transformation at alternating sites, this model is equivalent to

0 with increasing x as shown in Fig. 7(a). The correlation between

our model given by Eq. (1) in the case of no external field. By

NNLs is FM. (b) Individual spins are randomly directed around the

applying the inverse of the gauge transformation to this model,

average direction.

we can obtain the same results as from the original model.

214408-6 - Frustration: geometrically frustrated systems

Ising model

Heisenberg model

Residual entropy

Single-q state

(macroscopically degenerated states)

(120-degree structure, spiral spin texture)

Antiferromagnets on triangle-based

lattice structures

Geometrical frustration - Unconventional behaviors in GFMs

Chirality and Z2 vortex

Reentrant phase transition

J. Phys. Soc. Jpn., Vol. 76, No. 10

LETTERS

S. TANAKA and S. MIYASHITA

0.16

1

oops

0.12

0.16

e L

(a)

0.12

Ferro

Para

Antiferro

Para

0.08

0.5

eathervan

0.08

Magnetization

f W

(c)

Magnetization

0.04

0.04

ber o

Temperature

um

N

(b)

0

S. Miyashita and H. Shiba, JPSJ 53, 1145 (1984).

0

5000

10000

0

Monte Carlo Step (MCS)

0

H. Kawamura and S. Miyashita, JPSJ 53, 4138 (1984).

H. Kitatani, S. Miyashita, and M. Suzuki, JPSJ 55, 865 (1986).

0

1

2

3

4

5

[×10+7]

0

1

2

3

4

5

[×10+7]

X. Hu, S. Miyashita, and M. Tachiki, PRL M

79onte Carlo Step (MCS)

, 3498 (1997).

Monte Carlo Step (MCS)

S. Miyashita, S. Tanaka, and M. Hirano, JPSJ 76, 083001 (2007).

R. Ishii et al.

R. Tamura, S. Tanaka, and N. Kawashima, PRB 87, 214401 (2013).

pffiffiffi

ffiffi

p ffi

Fig. 4.

(Color online) Relaxation of the magnetization and nloop at T ¼ 0:05J from (a) 3 Â 3 configuration, (b) q ¼ 0 configuration, and (c) random

configuration.

Successive phase transitions

Slow relaxation

T ! 0þ, we expect that nloop must be the maximum value

pffiffiffi

pffiffiffi

1

and the spin structure becomes the so-called

3 Â 3

structure.

Next, we study the relaxation of magnetization and nloop.

We ready the three types of initial configurations, i.e., (a) the

ffiffi

p ffi

pffiffiffi

T=0.04J

3 Â 3 structure, (b) the q ¼ 0 structure, and (c) a random

T=0.0425J

T=0.045J

structure. The configurations (a) and (b) are typical ground

eathervane loops 0.5

T=0.0475J

states of the present model, and the configuration (c)

f W

T=0.05J

corresponds to a state just after quench the temperature

T=0.055J

from a high temperature.

ber o

T=0.06J

um

T=0.065J

In Fig. 4, the relaxation processes at T ¼ 0:05J are

N

T=0.07J

plotted. In the cases (a) and (b), the magnetization is

T=0.08J

T=0.09J

maximum at t ¼ 0, and it relaxes very fast to uniformly

T=0.1J

magnetized ordered state. The relaxation of magnetization to

0

100

102

104

106

108

1010

the equilibrium is depicted in the inset. In contrast, in the

Monte Carlo Step (MCS)

S. Miy

case (c), i.e.,

ashita and H. Kawfrom a

amur random state,

a, JPSJ 54 it takes some time

, 3385 (1985).

to

ffiffi

p ffi

pffiffiffi

S. Miy

realize

ashita, JPSJ the

55 uniformly magnetized

, 3605 (1986).

state. Thus we regard

Fig. 5.

(Color online)

A. Kur Relaxation of n

oda and S. Miyashita, JPSJ 64, 4509 (1995).

loop from

3 Â 3 structure at

Fig. 1: (Color online) Crystal structures of Rb4Mn(MoO4)3 featuring (a) MnO

the relaxati

5 polyhedra, (b) equilateral triangular lattices

on time in the case (c) as the intrinsic relaxation

several temperatures. Dashed lines denote the fittling curves estimated

R. Ishii, S. Tanaka, S. Nakatsuji et al. EPL 94, 17001 (2011).

S. Tanaka and S. Miyashita, JPSJ 76, 103001 (2007).

of Mn2+ and MoO4 tetrahedra. Intralayer and interlayer distances betw time

een of

M the

n2+mag

ionetization

ns are

given by a = 6.099 ˚

Aand c/2 by

= eq. (2).

mag.

11.856 ˚

A, respectively, at 298 K. Phase diagrams of Rb4Mn(MoO4)3 for (The

c) relaxati

µ0H onc of n

and (d) µ

ab derived from the

loop is much slower.

0H

In the case (a) nloop

measurements indicated. The phase boundaries derived from Monte Carlo starts

sim with the

ulations max

forimum

D

value,

/J = 0. while

22

in

are the

i

case (c)

ndicatedit starts

by dashed

t

with zero. In all the cases, (a), (b), and (c), the numbers seem

nloopðtÞ ¼ AðTÞ exp À

þ n(eq)

lines.

loopðT Þ;

ð2Þ

to reach the same saturated value. This observation indicates

ðTÞ

that there exists stable equilibrium state for nloop. The

where ðTÞ denotes a characteristic relaxation time of the

relaxation time loop is also about the same, although in the

nloop and n(eq) is the equilibrium value of n

loop

loop. Because the

case (b) a non-monotonic process is observed in the early

first relaxation is much faster than the second one, the choise

state where the weathervane lines in the initial q

transitions

associated

with

respective

ordering

of

is achieved and establishes Rb

¼ 0 state

of start time of the second relaxation is irrelevant. Here we

4Mn(MoO4)3 as an ideal

are broken into short strips. Thus, we expect that there

fit the curves in the cases of T ¼ 0:07J, 0:065J, 0:06J,

the longitudinal and transverse spin components are model system for the 2D Heisenberg TAF described by

exists an intrinsic relaxation time for nloop. From the figure

0:055J, 0:05J, 0:0475J, 0:045J, 0:0425J, and 0:04J. The

expected [17,18]. Specifically on cooling, the system first

eq. (1)

we

with estimat

J = 1e. that

2 K and D = 0.26 K.

mag is about 104 MCS, and loop is of the

temperature dependence of and n(eq) are plotted in Fig. 6.

loop

order 107 MCS.

In Fig. 6(a), we find that nloop increases when the temper-

forms a collinear intermediate phase (IMP) with the

Single crystals with typical dimensions 1 × 1 × 0.5 mm3

In Fig. 5, we plot the relaxation process of nloop in the case

ature is lowered. We expect that this value continues to the

three-sublattice “uud ” structure [19] before transitioning

were synthesized by a flux method [24]. The structure

(a) at various temperatures below the critical temperature.

ground state value nloop ¼ 1. It should be noted that this

to 120◦ spin-order phase with a uniform vector chiral- was deter There

min ,

e we

d b find

y sitwo

ngl steps

e-cr in

ys relaxati

tal X on.

-ra The

y difirst

ff relaxati

raction on

quantity

and is

is not zero at the critical temperature, and it can not

corresponds to initial local relaxation from the complete

be an order parameter although it is a good indicator of the

ity. A recent theoretical study has indicated another described by the space group P 63/mmc symmetry (R1 =

ground state configuration. The relaxation time of this

degree of the order of the structure. In Fig. 6(b), we find a

phase transition in the IMP, separating the lower-

2.88%). Powder neutron diffraction (PND) measurements

process is very short and does not strongly depend on the

good linearity, which indicates the Arrhenius law ðTÞ /

temperature “uud ” phase and a higher-temperature

were perfotemperat

rmed ure.

on In

BTthe

1 a second

t NIS relaxati

T, an on,

d c the

onfi re-constru

rmed t ction

his str e ÁE

uc- . From the slope in Fig. 6(b), the energy barrier ÁE is

of the WLs takes place. The first and second relaxations

estimated about 0:78J

collinear phase with three different sublattice moments.

ture and its stability down to 1.5 K. Thus, Rb

$ OðJÞ. It is expected that this energy

4Mn(MoO4)3

correspond to energetical and entropical relaxations, respec-

is necessary when the WLs are reconnected locally.

However, the latter phase is only stable in the purely 2D

contains an equilateral triangular lattice of Mn2+ ions.

tively. It is noted that the two-step relaxation is a character-

We study slow relaxation of spin configuration in the

limit, and thus with finite interlayer coupling, the “uud ”

Each Mn2+istics of

ion the

is ordered

lo

state

cated in of

a the pres

MnO ent

5 pmodel.

olyhedron and magnet

has

ically ordered phase of the Ising-like Heisenberg

phase should become dominant throughout the collinear

a high-spin We

t3 fit all the second relaxation processes in the following

kagome´ antiferromagnets. From the view point of the

2ge2

g state, which represents a S = 5/2 Heisen-

form

entropy, states with the larger nloop are preferable, although

IMP [19]. For the easy-axis case, unlike the Heisenberg

berg spin. The dominant intralayer coupling J should

103001-3

and XY , experiments have generally confirmed the

result from the superexchange path Mn-O-O-Mn involving

theoretical predictions. Specifically, successive transi-

two oxygen atoms. The interlayer interaction is expected

tions and/or a 1/3 magnetization plateau have been to be negligible because two Rb+ ions and two MoO4

observed for TAFs with easy-axis anisotropy such as

tetrahedra yield a large separation between neighboring

VCl2 [20], ACrO2 (A = Li, Cu) [21,22], and a metallic planes.

TAF GdPd2Al3 [23]. However, neither a detailed study of

DC magnetization (M ) was measured by a commercial

the phase diagram under external field nor a quantitative

SQUID magnetometer above 1.8 K, and by a Faraday

comparison between experiment and theory has so far

method for 0.37 K < T < 2 K [25]. Specific heat, CP , was

been possible, because of large values of J which preclude

measured by a thermal relaxation method do - Phase transition in 2D GFMs

H = J1

si · sj

J3

si · sj

2D triangular lattice

i,j

i,j

NiGa

3

2S4

The 1st n.n. interaction

The 3rd n.n. interaction

J1: Antiferro

Degenerated GSs

120-degree structure (SO(3))

Order by disorder -1/9

0

J3/J1

Z

J

2 vortex dissociation

1: Ferro

Ferromagnetic (S2)

Spiral-spin structure (SO(3)xC3)

-1/4

0

J3/J1

1st-order PT w/ 3-fold symmetry breaking and

Z2 vortex dissociation occur.

R. Tamura and N. Kawashima,

S. Nakatsuji, Y. Nambu, Y. Maeno et al.,

JPSJ 77, 103002 (2008), JPSJ 80, 074008 (2011).

Science 309, 1697 (2005). - Main results

To investigate unconventional phase transition

behavior in geometrically frustrated systems.

2D

3D

SO(3)xZ2

SO(3)xC3

- Z2 vortex dissociation

- 1st-order PT w/ SO(3)xC3

- 2nd-order PT w/ Z

breaking

2 breaking

(2-dim. Ising universality)

- incr

J

eases, decr

E

eases.

at the same temperature. - Model

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3 - Model

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

4 types of ground states for ferromagnetic J1

1. Ferromagnetic state (S2)

N. D. Mermin and H. Wagner, PRL 17, 1133 (1966).

No phase transition occurs at finite T (Mermin-Wagner theorem).

2. Single-q spiral state (SO(3))

H. Kawamura and S. Miyashita, JPSJ 53, 4138 (1984).

Z2 vortex dissociation occurs at finite T.

3. double-q spiral state (SO(3)xZ2)

4. triple-q spiral state (SO(3)xC

R. Tamura and N. Kawashima, JPSJ 77, 103002 (2008).

3)

R. Tamura and N. Kawashima, JPSJ 80, 074008 (2011).

1st-order PT and Z2 vortex dissociation occur at the same T. - Ground state phase diagram

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

(ii) single-k spiral

SO(3)xC3

structure

4 independent

sublattices

structure

(iv) triple-k spiral

(iii) double-k spiral

SO(3)xZ2

(ii) single-k spiral

(i) ferromagnetic - Model

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

J1/J3 = 0.4926 · · · , = 1.308 · · ·

axis 2

axis 3

axis 1

Order parameter space: SO(3)xZ2 - SECOND-ORDER PHASE TRANSITION IN THE

Physical quantities

. . .

PHYSICAL REVIEW B 87, 214401 (2013)

20SECOND-ORDER PHASE T

s

RANSITION IN T s

0.6

L=144

H =

J1

i · sj

J1

i ·

HE sj J

. . . 3

si · sj

3.0

PHYSICAL REVIEW B 87, 214401 (2013)

-2.0

(d)

(a)

(b)

15

L=216

i,j

axis 1

i,j

axis 2,3

i,j 3

L=288

10

)

4

C

-2.2

v

J1/J3 = 0.4926 · · · , = 1.308 · · ·

2.0

U

L=108

20

5

ln(n

0.4

0.6

L=144

-2.4

L=144

3.0

Arrhenius

L=180

L=216

-2.0

3

0.52

(d)

(a)

(b)

L=216

0 specific hea

15

t

(a) L=288

/J

1.0

-2.6

law

0.1

T c

order paramet

10

er

(b)

)

0.5

-2.2

0.4

4

C

v

2.0

U

L=108

2.00

2.02

2.04

2.06

2.08

0.2

>

5

0.4

L=144

axis 2

0.48

η-2

ln(n

2

-2.4

0.2

0.05

L=180

axis 3

J

Arrhenius

3/T

3

0.52

χL

<m

L=216

0

3

axis 1(a)

la

/J

1.0

1

1.1

1.2

-2.6

w

T c

0.0

(c)

0.1

0

0.5

Order parameter detecting Z

0.4

0

4

2 (b)

U

2 breaking

1

1.5

2

2.5

3

-4

-2

0

2

4

0.2

3 Binder ratio

2.00

2.02

2.04

2.06

2.08

>

η-2

1

(t)

(e)

(t)

λ

0.48

(T-T

2

:= s(t)

s(t)

m :=

/N

c)L1/ν/J30.2

0.05

1

· s(t)

2

3

, J3/T

χL

4

<m

(f)

t

2

0.6

U

3

1

1.1

1.2

(c)

η-2

0.4

FIG. 3. (Color online) (a) Phase diagram of the distorted J

0.0 1-J3

Binder ratio U4 := m4

0

0

χL 0.2

4

m2 2

model for J

2

1/J3 = −0.7342 . . .. The inset is an enlarged view. The

U

1

1.5

2

2.5

3

-4

-2

0

2

4

1

(c)

3

0

open square indicates the transition temperature for λ = 1 where a

Crossing point

(e)

0.49

0.495

0.5

-1.5 -1.0 -0.51 0

0.5 1.0 1.5

first-order phase transition with

λ

(T-T

C

c)L1/ν/J3

3 symmetry breaking occurs.21 The

T/J

4

3

(T-Tc)L1/ν/J

0.6

3

solid circ

(f) les represent transition temperatures at which a second-order

2

U

η-2

0.4

phase transition with Z

FIG. 3. (Color online) (a) Phase diagram of the distorted

2 symmetry breaking occurs. (b) and (c) Finite-

J1-J3

FIG. 2. (Color online) Temperature dependence

χL

of equilibrium

0.2

size scaling of the Binder

model ratio

for U

J1 4 and

/J3 t

=hat

− o

0 f. the susceptibility

7342 . . .. The inχ

sefor

t is an enlarged view. The

physical quantities

1

of the distorted J(c)

1-J3 model for J1/J3

0

=

λ = 1.5 using ν = 1 an

open dsη = 1/

quare 4 which ar

indicates e the

the critical exp

transition onents of

temperature for λ = 1 where a

−0.4926 . . . and λ

0.49= 1.308 . . .. (a) Specific heat C. (b) Square of

0.495

0.5

-1.5 -1.0 -0.5 0

the

0.5 2D

1.0 Ising

1.5 model. Error bars are omitted for clarity since their sizes

first-order phase transition with C

the order parameter

3 symmetry breaking occurs.21 The

m2 . (c) Binder ratio U4. (d) Log of number

are smaller than the symbol sizes.

T/J

(T-T

density of

3

solid circles represent transition temperatures at which a second-order

Z

c)L1/ν/J3

2 vortex nv versus J3/T . The dotted vertical line indicates

the transition temperature

phase transition with

T

Z

c/J3 = 0.4950(5). (e) and (f) Finite-size

2 symmetry breaking occurs. (b) and (c) Finite-

scaling of the Binder

FIG. 2. ratio

(Color online) Temperature dependence of equilibrium

U

size scaling of the Binder ratio

4 and that of the susceptibility χ using

of transition temperatures as depicted in Fig. 3(a).

UA

4 and that of the susceptibility

n enlarged

χ for

the critical expo

physicalnents of the

quantities 2DofIsing

the model (ν =

distorted 1Ja1n-dJ3η = 1/4)

model for J

view

1/Jnear

3 =

λ = 1.5 using ν = 1 and η = 1/4 which are the critical exponents of

and the transition

−0.4926 temperature.

. . . and λ Error

= 1. bars

308 . a.re

.. omitted

(a) Sp f

e or

ci clarity

fic heastince

λ = 1 is shown in the inset of Fig. 3(a). This

C. (b) Square of

figure indicates that

the 2 the

D I transition

sing model. temperature

Error bars a near

re omitted for clarity since their sizes

their sizes

the are sma

order ller than the

parameter sym

m b

2 o.l s(icz)es.

λ = 1

Binder ratio U4. (d) Log smoothly

of numberconnects

are sto the

maller transition

than the temperature

symbol sizes. for λ = 1

density of Z2 vortex nv versus J3/T . The dotted vertical lin and

e ind the

icatetransition

s

temperature goes continuously to zero

In antiferromagnetic Heisenberg models on a triangular

the transition temperature Tc/J3 = 0.4950(5). (e) and (f) when

Finite-size

lattice, the dissociation of the

λ → λ0. Figures 3(b) and 3(c) represent the finite-size

Z2 vortices occurs at finite

scaling of the Binder ratio U4 and that of the

scaling

susceptibility χ

of

using the Binder ratio and that of the susceptibility for

temperature.13,27 In order to confirm the dissociation of the

of transition temperatures as depicted in Fig. 3(a). An enlarged

the critical exponents of the 2D Ising model (ν = 1 an λ

d =

η

1

= .5

1/us

4)ing ν = 1, η = 1/4, and Tc/J3 = 0.5521(1) as well

Z2 vortices in our model, we calculate the number density of

view near

and the transition temperature. Error bars are omitted for as the

clarity s previous

ince

case. In this case,

λ = 1 is shown in the inset of Fig. 3(a). This

all the data collapse onto

the Z2 vortices nv by using the same manner as in Ref. 13. A

figure indicates that the transition temperature near

their sizes are smaller than the symbol sizes.

scaling functions. Thus, we conclude that a second-order

λ = 1

plot of ln nv versus J3/T in our model is shown in Fig. 2(d),

phase transition with breaking

smoothly

of the

connects to the transition temperature for λ = 1

and it is confirmed that

Z2 symmetry occurs

nv obeys well the Arrhenius law below

and it belongs to the

and 2D Ising

the

model

transitionuniversality class

temperature within

goes continuously to zero

Tc. This resu

In lt indicates that the di

antiferromagneticssociation of

Heisenber th

g e Z2 vort

modelsices

on a triangular

calculated

when

3(b) and 3(c) represent the finite-size

occurs at the second-order

lattice, the

phase transition

dissociation of the point.

λ. However, a

λ t very close to λ

→ λ0. Figures= 1, the possibility

Z2 vortices occursthat

at first-order

finite

phase transition occurs with breaking of the

To clarify the universality class of the phase transition, we

Z

scaling of the Binder ratio and that

2

of the susceptibility for

temperature.13,27 In order to confirm the

symmetry

dissociation of thecannot be denied. Unexpected phase transition from

perform the finite-size scaling using the following relations:

λ = 1.5 using ν = 1, η = 1/4, and Tc/J3 = 0.5521(1) as well

Z2 vortices in our model, we calculate the number only

dens underlying

ity of

symmetry can occur in some cases.16,23,28 If a

as the previous case. In this case, all the data collapse onto

U the Z vortices n by using the same manner as in

first-order

Ref. 13. Aphase transition with breaking of the Z

4 ∝ f [(T

2 − Tc)L1/ν],v χ ∝ L2−ηg[(T − Tc)L1/ν], (6)

2 symmetry

occurs, a tricritical point

scaling should

f

exist and

unctions.

to study

Thus,

its

we properties

conclude that a second-order

plot of ln nv versus J3/T in our model is shown in Fig. 2(d),

where the susceptibility χ is defined as χ := NJ

phase transition with breaking of the Z

3 m2 /T and

such as universality class will be an important topic. From our

and it is confirmed that

2 symmetry occurs

nv obeys well the Arrhenius law below

f (·) and g(·) are scaling functions. The finite-size scaling

observation, it is dif

and ficult

it

to obtain

belongs to the

the nature

2D

of

Ising the phase

model universality class within

resultsT using

c. This result indicates that the dissociation of the Z

ν = 1 and η = 1/4 which are the critical

transition

2 vorticesnear λ = 1 since

calculated th

λ. e s

H iz

o e

w d

e e

v p

e e

r,nd

a e

t nc

v e

er o

y fc p

l h

o y

s s

e icta

o lλ = 1, the possibility

exponents of

occurs th

ate 2D

the Ising model

second-orderand the

phase obtained

transition point.

Tc are

quantities are significant and we should calculate very large

that first-order phase transition occurs with breaking of the Z2

shown in Figs.

To cl2(e)

arif and

y th 2(f)

e u .

ni S

v in

ercsealailtlythcel d

asastao cfotllhaepspeh o

a n

s teotran systems

sition, with

we - Z2 vortex dissociation

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

J1/J3 = 0.4926 · · · , = 1.308 · · ·

No phase transition w/ SO(3)

-2.0

Z2 vortex density

breaking occurs at finite T.

(Mermin-Wagner theorem)

) -2.2

v

Point defect: 1(SO(3)) = Z2

ln(n -2.4

Arrhenius law

Z

-2.6

2 vortex dissociation can occur

at finite T.

2.00

2.02

2.04

2.06

2.08 Z2 vortex dissociation occurs

J3/T

at the 2nd-order PT point (Tc). - Finite size scaling

H =

J1

si · sj

J1

si · sj

J3

si · sj

i,j

axis 1

i,j

axis 2,3

i,j 3

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

J1/J3 = 0.4926 · · · , = 1.308 · · ·

3

Binder ratio

Finite size scaling relations

4

2

U

U4

f (T

Tc)L1/

L2

g (T

T

1

c)L1/

= 1

0.6

, = 1/4

2D Ising universality class

-2

= 1

0.4

, = 1/4

L 0.2

0

Susceptibility

Z2 vortex dissociation does not

-1.5 -1.0 -0.5 0

0.5 1.0 1.5 aﬀect the phase transition nature.

(T-Tc)L1/ /J3 - Phase diagram

H =

J1

si · sj

J1

si · sj

J3

si · sj

SECOND-ORDER PHASE TRANSITION IN THE

i,j

i,j

i,j

. . .

axis 1

PHYSICAL

axis 2,3

REVIEW B

3

87, 214401 (2013)

The 1st n.n. interaction

The 1st n.n. interaction

The 3rd n.n. interaction

along axis 1

along axes 2 and 3

20

0.6

L=144

3.0

-2.0

(d)

(a)

J1/J3 (b)

= 0.7342 · · ·

15

L=216

L=288

10

)

4

C

-2.2

v

2.0

USO(3)xZ2

L=108

5

ln(n

0.4

L=144

-2.4

2nd-order PT w/ Z2

Arrhenius

L=180

3

0.52

breaking &

L=216

0

(a)

/J

1.0

-2.6

law

Z2 vortex dissociation

0.1

T c

(b)

0.5

0.4

occur at the same T.

2.00

2.02

2.04

R. T

2.06

amur

2.08

a and N. Kawashima,

0.2

>

JPSJ 77, 103002 (2008).

0.48

η-2 2D Ising universality

2

0.2

0.05

J3/T JPSJ 80, 074008 (2011).

χL

<m

3

1

1.1

1.2

SO(3)xC

0.0

(c)

3

SO(3)

0

0

4

2

U

1st-order PT w/ C

Z

tex dissociation

3

1

1.5

2

2.5

3

2 vor -4

-2

0

2

4

3

breaking &

occur at finite T.

1

(e)

λ

(T-Tc)L1/ν/J3

Z2 vortex dissociation

H. Kawamura and S. Miyashita,

4

occur a (f)

t the same T.

2

0.6

U

SO(3)xZ

SO(3)

JPSJ 53, 4138 (1984).

η-2

0.4

FIG. 3. (Color online)

2

(a) Phase diagram of the distorted J1-J3

χL 0.2

model for J1/J3 = −0.7342 . . .. The inset is an enlarged view. The

1

(c)

0

open square indicates the transition temperature for λ = 1 where a

0.49

0.495

0.5

-1.5 -1.0 -0.5 0

0.5 1.0 1.5

first-order phase transition with C3 symmetry breaking occurs.21 The

T/J3

(T-Tc)L1/ν/J3

solid circles represent transition temperatures at which a second-order

phase transition with Z2 symmetry breaking occurs. (b) and (c) Finite-

FIG. 2. (Color online) Temperature dependence of equilibrium

size scaling of the Binder ratio U4 and that of the susceptibility χ for

physical quantities of the distorted J1-J3 model for J1/J3 =

λ = 1.5 using ν = 1 and η = 1/4 which are the critical exponents of

−0.4926 . . . and λ = 1.308 . . .. (a) Specific heat C. (b) Square of

the 2D Ising model. Error bars are omitted for clarity since their sizes

the order parameter m2 . (c) Binder ratio U4. (d) Log of number

are smaller than the symbol sizes.

density of Z2 vortex nv versus J3/T . The dotted vertical line indicates

the transition temperature Tc/J3 = 0.4950(5). (e) and (f) Finite-size

scaling of the Binder ratio U4 and that of the susceptibility χ using

of transition temperatures as depicted in Fig. 3(a). An enlarged

the critical exponents of the 2D Ising model (ν = 1 and η = 1/4)

view near

and the transition temperature. Error bars are omitted for clarity since

λ = 1 is shown in the inset of Fig. 3(a). This

figure indicates that the transition temperature near

their sizes are smaller than the symbol sizes.

λ = 1

smoothly connects to the transition temperature for λ = 1

and the transition temperature goes continuously to zero

In antiferromagnetic Heisenberg models on a triangular

when

lattice, the dissociation of the

λ → λ0. Figures 3(b) and 3(c) represent the finite-size

Z2 vortices occurs at finite

scaling of the Binder ratio and that of the susceptibility for

temperature.13,27 In order to confirm the dissociation of the

λ = 1.5 using ν = 1, η = 1/4, and Tc/J3 = 0.5521(1) as well

Z2 vortices in our model, we calculate the number density of

as the previous case. In this case, all the data collapse onto

the Z2 vortices nv by using the same manner as in Ref. 13. A

scaling functions. Thus, we conclude that a second-order

plot of ln nv versus J3/T in our model is shown in Fig. 2(d),

phase transition with breaking of the

and it is confirmed that

Z2 symmetry occurs

nv obeys well the Arrhenius law below

and it belongs to the 2D Ising model universality class within

Tc. This result indicates that the dissociation of the Z2 vortices

calculated

occurs at the second-order phase transition point.

λ. However, at very close to λ = 1, the possibility

that first-order phase transition occurs with breaking of the

To clarify the universality class of the phase transition, we

Z2

symmetry cannot be denied. Unexpected phase transition from

perform the finite-size scaling using the following relations:

only underlying symmetry can occur in some cases.16,23,28 If a

U

first-order phase transition with breaking of the Z

4 ∝ f [(T − Tc)L1/ν ],

χ ∝ L2−ηg[(T − Tc)L1/ν], (6)

2 symmetry

occurs, a tricritical point should exist and to study its properties

where the susceptibility χ is defined as χ := NJ3 m2 /T and

such as universality class will be an important topic. From our

f (·) and g(·) are scaling functions. The finite-size scaling

observation, it is difficult to obtain the nature of the phase

results using ν = 1 and η = 1/4 which are the critical

transition near λ = 1 since the size dependence of physical

exponents of the 2D Ising model and the obtained Tc are

quantities are significant and we should calculate very large

shown in Figs. 2(e) and 2(f). Since all the data collapse onto

systems with high accuracy.

scaling functions, it is confirmed that the second-order phase

In this paper we discovered an example where the second-

transition in our model belongs to the universality class of the

order phase transition occurs accompanying the Z2 vortex

Ising model.

dissociation at finite temperature. The model under consid-

Next, to obtain the relationship between λ and Tc, we

eration is the classical Heisenberg model on a triangular

consider the case of J1/J3 = −0.7342 . . . which was used

lattice with three types of interactions: (i) the uniaxially

in Ref. 21 by changing the value of λ. For λ = 1, the model

distorted nearest-neighbor ferromagnetic interaction along

exhibits a first-order phase transition with breaking of the C3

axis 1 (λJ1), (ii) the nearest-neighbor ferromagnetic interaction

symmetry at Tc/J3 = 0.4746(1).21 Here we study the nature of

along axes 2 and 3 (J1), and (iii) the third nearest-neighbor

the phase transition of the distorted J1-J3 model with the open

antiferromagnetic interaction (J3). In this model for 1 < λ <

boundary condition. From the analysis of the GS as explained

λ0, the order parameter space is SO(3)×Z2. We found the

before, a phase transition with breaking of the Z2 symmetry

second-order phase transition with spontaneous breaking of

is expected to take place for 1 < λ < λ0(= 2.8155 . . .) in this

the Z2 symmetry in the region. The dissociation of the Z2

case. By analyzing the Binder ratio, we obtain λ dependence

vortices also occurs at the same temperature. The dissociation

214401-3 - Main results

To investigate unconventional phase transition

behavior in geometrically frustrated systems.

2D

3D

SO(3)xZ2

SO(3)xC3

- Z2 vortex dissociation

- 1st-order PT w/ SO(3)xC3

- 2nd-order PT w/ Z

breaking

2 breaking

(2-dim. Ising universality)

- incr

J

eases, decr

E

eases.

at the same temperature. - Model

H = J1

si · sj

J3

si · sj

J

si · sj

i,j

i,j 3

i,j

The 1st n.n. interaction The 3rd n.n. interaction The 1st n.n. interaction

intralayer

intralayer

interlayer - Ground state

H = J1

si · sj

J3

si · sj

J

si · sj

i,j

i,j 3

i,j

The 1st n.n. interaction The 3rd n.n. interaction The 1st n.n. interaction

intralayer

intralayer

interlayer

/2

/2

/2

/2

/2

/2

Order parameter space: SO(3)xC3 - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

-2.1

30

INTERLAYER-INTERACTION D

first-order

EPENDENCE phase

OF 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

(a)

finite temperature.

Internal ener

-2.3

15

25

SO(3)

0.05

40

gy and specific heat

INTERLAYER-INTERA

INTERLA

CTION D -2.2

YER-INTERA

EPENDENCE OF LACTION

TENT

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

. D

. . EPENDENCE

0

We further investigate the way of spin ordering. As

0

15

O

30 F 4LA

5

TENT . . .

PHYSICAL REVIEW E 88, 05 PHYSICAL

2138 (2013) 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

30

H = J1

si · sj

J3

si · sj

J

si · sj

5

-2.3

(a)

15

Heisenberg model on a stacked triangular lattice with only

SO(3)

i,j

i,j

× C

3

i,j

-2.1

-2.1

30

40

30

first-order phase transition w first-order

ith the C

a nearest-neighbor interaction wher 3. It was confirmed that the C3 symmetry breaks at

3 phase transition

symmetry

with

breaking atthe C

0.1

0.1

e the order parameter

0

(d)

(b)

10

(d)

3 symmetry breaking at

20

The 1st n.n. interaction The 3rd n.n. interaction The 1st n.n. interaction

the first-order phase transition point. In the antiferromagnetic

25-2.3

-2.2

-2.1

finite temperature.

0.05

25

finite temperature.

0.05

intralayer

intralayer

interlayer

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

30

Heisenberg model on a stacked triangular lattice with only

INTERLA

-2.2

YER-INTERACTION D -2.2

5

0

We further investigate the We

wa f

y uroth

f er

spiin

n ve

o srti

d g

e a

ri te

ng. the

As way of spin ordering. As

J

EPENDENCE OF LATENT . . .

0

0

15

30

45

0

15

30

4

dependence 5of the s PHYSICAL

pecific heat [ REVIEW

42,43].

E

The 88, 0

peak 52138 (20

indicates 13)

10

the

3/J

20

1.55 1 = 0.85355 · · · , J /J1 = 2

20

(e)

a nearest-neighbor interaction where the order parameter

20

0

mentioned above, the order mentioned

parameter abov

space e,

of the order

s

parameter

ystem is

space of the system is

-2.3finite-temperature

-2.2

phase

-2.1

transition between the paramagnetic

-2.3 Internal energy

(a)

-2.3

1.54

(a)

15 Specific heat

15

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

0

SO(3)

C

state and 3. It was confirme SO(3)

-2.1

d that the

symmetry breaks at

40

30

40

magnetic ordered state where the SO(3) symmetry

1.53

first-order

× phase transition with the C

× C3

C . It was confirmed that the C3 symmetry breaks at

0.1

(b)

(c)

10

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

0

10

0.00004

(d)

3 symmetry breaking at

(b)0.00008 10

1.55

0.02

the first-order phase

the

transition

first-order

point. In the phase transition point.

antiferromagnetic

In the antiferromagnetic

is

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

25

finite temperature.

0.05

finite-temperature phase transition between the paramagnetic

-2.2

30

30

5

60

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

theinsg. As

pecific

0.01

(f) 0 15 30 45

20

40

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

1.53

(c)

a nearest-neighbor interacti a

on nea

w r

hesrt-

e ne

t i

h g

e hb

o o

r r

de i

r nte

p r

a a

r c

a ti

m o

e n

ter where the order parameter

0 heat has a single

0.00004

peak corresponding

0.00008

to the first-order phase

20

20

0

0

mentioned above, the order parameter space of the system is

20

0.02

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

-2.3

(a)

15-2.3

-2.2

-2.1

-2.3

-2.2

-2.1

0

0

SO(3)

space is SO(3), a single

space

peak is

is SO(3),

observed for a single

the

peak is

temperatureobserved 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

1.55

0

20000

40000

60000

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

10

(b)

10

10

1.55

0.01

1.55

40 the

(f)

first-order

dependence

dependence phase

of the s

of the s transition

pecific heat

tructure f point.

[42,

actor of In

43]. the

dependence

The

spin antiferromagnetic

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

Phase transition occurs

0

0

FIG. 4. (Color online) Temperature dependence

0

transition. To confirm this we calculate the temperature

1.52

1.53

of (a)

1.54

internal

1.55

0 state and

20000

magnetic

40000

ordered

1

state

60000

state and magnetic

where the )

ordered

SO(3) s

state

ymmetry where the SO(3) symmetry

1.53

1.53

at finite T.

(c)

(c)

a nearest-neig

S h

( b

k o

) r: interaction

s where the or,der parameter

=

(11)

20

0 0

0.00004

0.00008

0

0.00004

0.00008

i · sj e−ik·(ri −rj

dependence of the structure factor of spin

0.02

energy per site 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 our

N 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

eter |

i,j

µ|2 , which can detect the 60

60

C3 symmetry breaking of the

break at the first-order phase break at the

transition

first-order

point since phase

the

1 s

transition

pecific

point since the specific

10

0.01

1.55

0.01 (f)

FIG. 4. (Color online) Temperature dependence

(f)dependence of

of the

(a) specific

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

40

model with

−ik·(r

)

J

) :

s

i −rj

(11)

3/J1 = −0.853 55 . . . and (e)

40

J⊥/J1 = 2 for L = 24,32,40.

heat has

which is a single

the

peak

magnetic order heat has

corresponding a

parameter s

to ingle

the

for s

S(k peak

= corresponding

first-order

piral-spin phase

states.

i · sj e to the first-order

,

phase

1.54

20

energy per site E/J

20

finite-temperature phase transition between the paramagnetic

(d)

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

N

0

P 0robability distribution of the

0

internal energy

0

P (E; Tc(L)). The 0 transition.

When the To confirm

magnetic

this

ordered transition.

we

state

To

calculate

described confirm

the

by k∗

this

where

i,j

we

temperature

the calculate the temperature

1.52

1.53

1.54

1.55 eter

1.52

0|µ|2 , 1.53

20000

which 1.54

40000

can det 1.55

60000

ect the 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

dependence

SO(3) s

of the

ymmetry s

istructure

broken f dependence

actor of

appears, spin

S(kof

∗) the

becstructure

omes a f

fi actor

nite of spin

0.02

0

0.00004

0.00008

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

which SO(3)

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)/J

(d) 601 as a function of L−3. (f) Plot

Proba - 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

INTERLA

20

YER-INTERA

0-2.3

-2.2

CTION

-2.1

DEPENDENCE

space is SO(3), a single peak is

OF

observed fLA

or the TENT

temperature . . .

PHYSICAL REVIEW E 88, 052138 (2013)

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 -2.1

40 (f)

30

first-order phase transition with the C

heat has a s

0.1

ingle peak corresponding to the first-order phase

20

(d)

3 symmetry breaking at

0

0

transition. To confirm this we calculate the temperature

1.52

1.53

1.54

1.55

0

20000

40000

60000

25

dependence of the structure factor of spin

finite temperature.

0.05

-2.2

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

0 1

S(k) :=

s

)

i · sj e−ik·(ri −rj ,

(11)

We further investigate the way of spin ordering. As

0

15

30

45

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

20

N

eter |

i,j

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

mentioned above, the order parameter space of the system is

model with J

32,40.

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

(d) P -2.33/J1 = −0.853 55 . . . and J⊥/J1 = 2 for L = 24,

robability distribution of the internal energy

(a)

P (E; Tc(L)). The

When 15

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,

SO(3)

40

S(k∗) becomes a finite

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

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

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

C

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

(f) Plot

of

temperature dependence of the largest value of structure factors

b O

ars ir

max

n der paramet

all figures are omitted for cl er (

arity sin C

error

10

the first-order phase transition point. In the antiferromagnetic

ce 3

th and SO(3))

eir sizes are smaller

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

than the 30

symbol size.

5

becomes zero in the thermodynamic limit above the first-

Heisenberg model on a stacked triangular lattice with only

INTERLAYER-INTERACTION DEPENDENCE

order phase transition

OF

temperature.

LA

The s

TENT

tructure factor S(k∗)

of analysis. One H

is

=

the

J

. . .

PHYSICAL REVIEW E

finite-size scaling and the other is

88, 052138 (2013)

1

si · sj

J3

si ·

becomes s

a j

J

nonzero value at

s

the i · sj

first-order phase transition

a nearest-neighbor interaction where the order parameter

a naive 20

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

0

temperature. Moreover, as temperature decreases, the structure

The scaling relations for the

i,j

first-order phase transition in i,j 3

factor S( -2.3

k∗) increases. Ti,j

he structu-2.2

re factors at k

e

z = 0 in th -2.1

d-dimensional systems [82] are given by

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

first Brillouin zone at several temperatures for L = 40 are also

-2.1

The 1st n.n. interaction The 3r

30

shown

d n.n. int in

eracFig.

tion 5(b)

T . As mentioned

he 1st n.n. inter in

ac Sec.

tion II, the spiral-spin

first-order phase transition with the

symmetry breaking at

10

Tc(L) = aL−d + Tc,

(9)

intralayer

intralayer 1.55

C

structure r 0.1

epresented by k is the same as that represented by

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

interlayer

(d)

−k in the

(e)

Heisenberg models. Figure 5(b) confirms that one

( E)2Ld

finite temperature.

C

distinct wave vector is chosen from three types of ordered

max(L) ∝

,

(10)

J 25

0.05

finite-temperature phase transition between the paramagnetic

3/J1

1.54 = 0.85355 · · · , J /J1 = 2

0

-2.2

4T 2c

0

We further investigate the way of spin ordering. As

where Tc and E are, respectively, the transition temperature

(a) 0.5

0

15

30

45

20

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

1.53

and the latent heat in the

Order paramet

thermodynamic limit. The coef (c)

er (C

ficient

of

0.02

3)

0.4

0

0.00004

0.00008

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

is broken.

mentioned

Then,

above,

in

the our

order model, the

parameter SO(3)

space of symmetry

the system sihould

s

and

-2.3

4(f) show the scaling plots for T

(a)

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

15 0.3

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

SO(3)

was confirmed that the

symmetry breaks at

40

60

0.2

break

× at

C the

3. It first-order phase transition

C3 point since the specific

in

0.01

the thermodynamic limit. Figure 4(f) shows an (b)

almost

linear dependence of C

0.1

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

40

10

(f)

Order parameter (SO(3))

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

0

heat

the

has a

first-order single

phase peak corresponding

transition point. In

to

the the first-order phase

antiferromagnetic

20

0

0.5

1

1.5

2

0

30

temperature and latent heat in the thermodynamic limit with

(b) 5

Heisenberg model on a stacked triangular lattice with only

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

0

transition. To confirm this we calculate the temperature

directly

1.52

calculate the size 1.53

dependence of 1.54

the width

1.55

between

10-1 0

20000

40000

60000

a nearest-neighbor interaction where the order parameter

20

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

010-2

dependence of the structure factor of spin

The width for the system size L is represented by

E(L) =

-2.3

10-3

-2.2

-2.1

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

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

10-4

the

FIG.

Gaussian function in the high-temperature phase and that in

the lo 10

4.

w-temperature

(Color

phase, respectiv online)

ely. In the

Temperature

thermodynamic

1.5510-5

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

limit, each Gaussian function becomes the δ function and then

FIG. 5.(e)

(Color online) (a) Temperature dependence of the largest

S(k) :=

s

),

(11)

ener

E C

(L) gy

convergper

es to

s

E [ ite

i · sj e−ik·(ri −rj

82].

E/

The inset J

of Fig. 4(d) shows the

value of structure factors S(k∗) calculated by six wave vectors in

finite-temperature phase transition between the paramagnetic

3 symmetry br

1, (b

eaks a )

t s

T pecific

1.54heat C, and

SO(3) symmetr (c

y br) order

eaks at p

T aram-

c.

c.

N

size dependence of the width E(L)/J

eter

1. The width enlarges as

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

the system

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

a nonzero v |

0

i,j

s µ

ize

alue i|2 ,

increases, wh

whichich c

indicates an

that the de

latenttec

heat tis the C

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

1.53 3 symmetry breaking of the

n the thermodynamic limit. The result (c)

s shown

(b) Structure factors at k

model

z = 0 in the first Brillouin zone at several

in

0.02

Fig. 4

with

0

0.00004

0.00008

conclude that J

the model given by Eq. (1) exhibits the

temperatures for L = 40.

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

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

w - INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

PHYSICAL REVIEW E 88, 052138 (2013)

Energy histogram -2.1

30

first-order phase transition with the C

0.1

(d)

3 symmetry breaking at

H = J

finite temperature.

1

si · sj

J3

si · sj

J

si · sj 25

0.05

i,j

-2.2

i,j 3

i,j

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . .

0

We further investigate the way of spin ordering. As

0

15

3 PHYSICAL

0

45

REVIEW E 88, 052138 (2013)

20

The 1st n.n. interaction The 3rd n.n. interaction The 1st n.n. interaction

mentioned above, the order parameter space of the system is

intralayer -2.3 intralayer

int

(a)

erlayer

15

SO(3) × C

-2.1

30

40

J

3. It was confirmed that the C3 symmetry breaks at

3/J1 =

0.85355 · · · , J /J

first-order 1 = 2

phase transition with the C

0.1

(d)

3 symmetry breaking at

(b)

10

the first-order phase transition point. In the antiferromagnetic

25

P (E; T ) finite

= D(E)e temperature.

E/k

0.05

BT

30

Heisenberg model on a stacked triangular lattice with only

-2.2

5

0

D(E) : densit We

y of stat f

esurther investigate the way of spin ordering. As

0

15

30

45

20

a nearest-neighbor interaction where the order parameter

20

E(L) mentioned 0

: width between tw abo

-2.3 v

o peaks e, the order

-2.2

parameter

-2.1

space of the system is

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

Bimodal distribution (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

1st-or

(c)

der PT w/ SO(3)xC

0.02

a nearest-ne0igh3 bor 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

breaking occurs.

-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

20

(e)

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 r - 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

Finit

25 e siz

(b)

finite

10

temperature.

0.05 e scaling

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

H = J1

si · sj

J3

si · s

0 j

J

si · sj

20

SO(3)

40

-2.3

× C3. It w

-2.2 as confirm

-2.1 ed that the C3 symmetry breaks at

i,j

i,j

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

3

i,j

(b)

10

the first-order phase transition point. In the antiferromagnetic

10

The 1st n.n. interaction The 3r

1.55

d n.n. interaction The 1st n.n. interaction

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

30

5

intralayer

intralayer Heisenber

(e) int g

erlay model

er

on a stacked triangular lattice with only

1.54

finite-temperature phase transition between the paramagnetic

0

Ja nearest-neighbor interaction where the order parameter

0

3/J1 =

0.85355 · · · , J /J1 = 2

20

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 aLsingle

dependence 4 peak

ofT (a)

2

corresponding

internal

to the first-order phase

1

20

c

s

)

0

S(k) :

M. S. S. Challa, D. P. Landau, and K. Binder, PRB 34, 1841 (1986).

=

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

60000

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

1st-or

= −0.853 55 . . . and J

der PT w/ SO(3)xC3

⊥/J1

break = 2 for L

ing oc = 24,32,40.

curs.

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

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 dependen - INTERLAYER-INTERA

INTERLA

CTION

YER-INTERA

DEPENDENCE

D

OF

O LATENT

LA

. . .

PHYSICAL REVIEW

REVIEWEE88,

88,05

0 2

5 1

2 3

1 8

3 (

8 2(0

2 1

0 3

1 )3)

RYO TAMURA AND SHU TANAKA

Interlayer int

PHYSICAL

erac

REVIEW E tion dependence

88, 052138 (2013)

fixing J3/J1

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

(a) 10

In Sec. IV,

IV we

w in

i v

n e

v s

e tsitg

i a

g taetd

e th

t e

h in

i t

netrelraly

a e

y re initnetrearc

a tcitoin

onefefefc

e tct

-1.5

(a) (a) 10

(b)

further evidence of the

Transition

C3 symmetry breaking at the first-order

0

temperature

on the nature of phase transitions. We

W confirmed that

that the

the

phase transition temperature.

0.25 0.50

-2

0.75

0

1.00

1.25

10

Before we end this section, let us mention a phase

1.50

first-order phase transition occurs

transition

f

occurs or

f 0

or .

025

.

JJ⊥/J1

2.5 and

-2.5

1.75

1.5

⊥/J1

2.5 and

1.5

transition nature in the

2.00

J

0

1-J2 Heisenberg model with interlayer

2.25

J /J

0.853 55 . . ., which was used in Sec. III. We could

2.50

3/J

3

1 = −0

− .853 55 . . ., which was used in Sec. III. We could

interaction

10

J

-3

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

0.25

not determine the existence

e

of the first-order phase transition

transition

the authors studied the phase transition behavior of the model

(b)

40

0

when

10

for

J

10

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

1 and J2 are antiferromagnetic interactions. For large

J /J1 increases

⊥/J

⊥

1 < 0.25 or J⊥/J

⊥

11 > 2.5 by Monte Carlo simulations.

J2/J1, a phase transition between the paramagnetic phase and

0.50

0

In the parameter ranges, the width

the

of

width tw

of o

tw peaks

o

in

peaks the

in

probabil-

the probabil-

ordered incommensurate spiral-spin structure phase occurs at

10

20

0.75

1

ity distribution

distrib

of

ution the

of

internal

the

ener

internal

gy

ener cannot

gy

be

cannot estimated

be

easily

estimated easily

finite temperature. In the parameter region, the order parameter

1.00

1.25

0

space is SO(3)

1.50

×

because of the finite-size effect.

ef

ItI is a remaining problem

problemtoto

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

s

phase transition occurs

occursfor

for

0

0

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

0.85

0.25

0.50

0.75

1.00

1.25

1.50

1.75 2.00 2.25 2.50

10

large

lar

J /J as in the J -J Heisenberg model on a stacked

a different phase transition nature happens even when the

⊥/J

⊥

1 as in the J1-

1 J22 Heisenberg model on a stacked

symmetry that is broken at the phase transition temperature

0.8

0

triangular lattice [62

[ ].

62 As the ratio

r

increases, the first-

0.5

J

0.5

⊥

J /J

⊥ 1

/J1 increases, the first-

is the same as for other models. For example, in the J1-J3

0.75

20

order phase transition temperature monotonically increases

increases

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

0.7

10

but

bu th

t e

h la

l t

a e

t n

e t

n he

h a

e ta de

d c

e r

c e

r a

e s

a e

s s

e .s Th

T i

hsi isi op

o p

p o

p s

o istiet to

t th

t e

h be

b h

e av

h i

avoir

or

transition with threefold symmetry breaking occurs when

(c)

0

0.08

observed

observ

in typical unfrustrated three-dimensional systems

J

0.15

systems

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

1.75

20

that exhibit

e

a first-order phase transition at finite temperature.

2.00

temperature.

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

2.25

2.50

0.05

10

For

Fo exa

ex m

a p

m l

p e

l ,e th

t e

h q-state

q

Potts model with ferromagnetic

ferromagnetic

The three-state ferromagnetic Potts model in two dimensions

0

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

0

0.04

intralayer and interlayer interactions (q

(q

3) is a fundamental

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

e

a temperature-induced first-order

first-order phase

phase

study [62].

0.1

10

(c)

Latent heat

transition with q-fold

q

symmetry

s

breaking [76

[ ].

76 From aamean-

(c)

mean-

0

0

0

field analysis of the ferromagnetic

f

Potts model [76

[ ,

7683

, ],

83],as

asthe

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0

the

IV. DEPENDENCE ON INTERLAYER INTERACTION

-2.6 -2.4 -2.2

-2 -1.8 -1.6 -1.4

0

1

2

interlayer interaction increases, both

increases,

the

both

transition

the

temperature

transition temperature

In this section, we study interlayer-interaction dependence

FIG. 6. (Color online) Interlayer-interaction J⊥/J1 dependence

and the latent

l

heat increase. The same

s

behavior

beha

was

w observed

observed

of the phase transition behavior. Here we set the interaction

of (a) internal energy per site E/J

As the interlayer interaction increases,

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

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

Interlayer

J

ratio

⊥/J1 depen-

J

⊥/J1

in the Ising-O(3) model on a stack

s

ed

tack square lattice [77

[ ].

77].As

As

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

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

represented by

de

d n

e c

n e

c of

o th

t e

h pr

p o

r b

o a

b b

a i

b liiltiy

t di

d s

i t

s rtirb

i u

b t

u ito

i n

o of

o int

in e

t r

e n

r a

n l

a en

e e

n r

e g

r y

g P (E

( ;

E T (

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

detect

trthe

T L))

C

c(

c L)

ansition temperature increases but latent heat decreases.

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

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

stabilize

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

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

wh

w e

h n

e th

t e

h sp

s e

p c

e i

c fi

i c

fi he

h a

e t

a be

b c

e o

c m

o e

m s

e the

th ma

m x

a i

x m

i u

m m

u va

v l

a u

l e

u fo

f r

o L = 24.

the ground state becomes large,

lar

the transition temperature

temperature

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

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

(b) The

T

J⊥

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

with the

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

C

increases and

increases

the

and

latent

the

heat

latent

increases

heat

[

increases 76

[ ,

7677

, ,

7783

, ].

83 Furthermore,

]. Furthermore,

3 symmetry breaking occurs and breaking of the

smaller than the symbol size.

he

h a

e t

a be

b c

e o

c m

o e

m s

e the

th ma

m x

a i

x m

i u

m m

u va

v l

a u

l e

u fo

f r

o

16

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

L =

−40. (c) The J

−40. (c) The ⊥

J /J1

⊥/J

in conv

con entional

v

systems, both the transition temperature

temperatureand

and

confirmed.

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

de

d p

e e

p n

e d

n e

d n

e c

n e

c of

o the

th wi

w d

i t

d h

t be

b t

e w

t e

w e

e n

e bi

b m

i o

m d

o a

d l

a pe

p a

e k

a s

k of

o the

th en

e e

n r

e g

r y

g

the latent heat are expressed

e

by the value

v

of an eff

efecti

f

v

ecti e

ve

Figure 6 shows the temperature dependence of phys-

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

distribution

distrib

E(

E L

( )

L /J

) 1

/J .

1 Error bars

b

in all

a figures are omitted for - Conclusion

We investigated unconventional phase transition

behavior in geometrically frustrated systems.

2D

3D

SO(3)xZ2

SO(3)xC3

- Z2 vortex dissociation

- 1st-order PT w/ SO(3)xC3

- 2nd-order PT w/ Z

breaking

2 breaking

(2-dim. Ising universality)

- incr

J

eases, decr

E

eases.

at the same temperature. - Thank you for your attention!!

2D case: PRB 87, 214401 (2013), 3D case: PRE 88, 052138 (2013).