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Our paper entitled “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with ...

Our paper entitled “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" was published in Physical Review B. This work was done in collaboration with Dr. Ryo Tamura (NIMS) and Professor Naoki Kawashima (ISSP).

http://prb.aps.org/abstract/PRB/v87/i21/e214401

NIMSの田村亮さん、物性研の川島直輝教授との共同研究論文 “Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions" が Physical Review B に掲載されました。

http://prb.aps.org/abstract/PRB/v87/i21/e214401

- Second-Order Phase Transition in Heisenberg Model

on Triangular Lattice with Competing Interactions

Ryo Tamura, Shu Tanaka, and Naoki Kawashima

Physical Review B 87, 214401 (2013) - Main Results

SECOND-ORDER PHASE TRANSITION IN THE

We studied the phase transition nature of a frustrated Heisenber

. . .

PHYSICAL REVIEW B 87, 214401 (2013)

g

model on a distorted triangular lattice. 20

0.6

L=144

3.0

-2.0

(d)

(a)

(b)

15

L=216

L=288

10

)

4

C

-2.2

v

2.0

U

L=108

5

ln(n

0.4

L=144

-2.4

Arrhenius

L=180

3

0.52

L=216

0

(a)

/J

1.0

-2.6

law

0.1

T c

(b)

0.5

0.4

2.00

2.02

2.04

2.06

2.08

0.2

>

0.48

η-2

2

0.2

0.05

J3/T

χL

<m

3

1

1.1

1.2

0.0

(c)

0

0

4

2

U

1

1.5

2

2.5

3

-4

-2

0

2

4

axis 2

3

axis 3

1

(e)

λ

(T-Tc)L1/ν/J3

axis 1

4

(f)

2

0.6

U

η-2

0.4

FIG. 3. (Color online) (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

A second-order phase transition occurs. 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-

- At the second-order phase transition poin

FIG. 2.t, Z

(Color online) Temperature dependence of equilibrium

2

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

symmetry (lattice reflection symmetry) is br

−0.4926 . . . oken.

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 universality class of the phase transition is

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 same as that of the 2D Ising model

th .ecritical 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

- Dissociation of Z

smoothly connects to the transition temperature for λ = 1

2 vortices occurs at the second-

and the transition temperature goes continuously to zero

In antiferromagnetic Heisenberg models on a triangular

order phase transition point.

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 - 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 - Motivation

To investigate the finite-temperature properties in two-dimensional

systems whose order parameter space is SO(3)xZ2.

- Phase transition occurs?

- Z2 vortex dissociation? - Model

H = J1

si · sj + J1

si · sj + J3

si · sj

> 0, J3 > 0

i,j axis 1

i,j axis 2,3

i,j

1st nearest-

1st nearest-

3rd nearest-

neighbor

neighbor

neighbor

axis 1

axis 2, 3

si: Heisenberg spin

(three components)

axis 2

axis 3

axis 1 - 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)

2

3

= J1 cos

J

k

k

1 cos kx cos

y + cos 2k

N J

x +

x + 2 cos kx cos

3ky

3

J3

J3

2

2

Find tha

k

t minimizes the Fourier transform of interactions!

4 < J1/J3 < 0 - 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). Ground states

R. T

can

amur

be

a and S. cate

T

gorized

anaka, Phys

into

. Rev. E, 88, 052138 (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 - SECOND-ORDER PHASE TRANSITION IN THE . . .

PHYSICAL REVIEW B 87, 214401 (2013)

Specific hea

SECOND-ORDER P t, order pa

HASE T

r

RANSITION a

IN met

THE . .er

.

PHYSICAL REVIEW B 87, 214401 (2013)

20

0.6

L=144

= J

s

s

s

3.0

H -2.0 1

i · sj + J1

(d)

i · sj + J3

i · sj

(a)

(b)

15

L=216

L=288

i,j axis 1

i,j axis 2,3

i,j

20

10

)

0.6

4

C

-2.2

v

L=144

J

3.0

1/J3 =

0.4926 · · · , = 1.308 · · ·

2.0

U

L=108

L=216

-2.0

(d)

(a)

(b)

5

15

ln(n

0.4

L=144

-2.4

L=288

Arrhenius

L=180

3

0.52

L=216

0 specific hea

10

t

(a)

) -2.2

/J

1.0

4

C

v

2.0

-2.6

law

U

L=108

0.1

T c

order parameter

(b)

0.5

0.4

0.4

5

ln(n

L=144

-2.4

2.00

2.02

2.04

2.06

2.08

Arrhenius

0.2

L=180

3

0.52

>

0.48

η-2

/J

1.0

L=216

2

axis 2

0

(a)

la

0.2

0.05

w

axis 3

J

-2.63/T

χL

T c

<m

0.1

3

axis 1 (b)

0.5

1

1.1

1.2

0.4

0.0

(c)

2.00

2.02

2.04

2.06 0 2.08

0.2

0

4

>

2

U

(t)

(t)

1

1.5

2

2.5

0.48 3

-4

-2

0

η-2 2

4

:= s(t)

s(t)

m :=

/N

0.2

3 2Binder ra

0.05 tio

1

· s(t)

2

3

, J3/T

χL

<m

1

(e)

t

λ

(T-Tc)L1/ν/J3

3

1

1.1

1.2

0.0

(c)

4

(f)

2

0.6

0

U

0

4

η-2

0.4

2

FIG. 3. (Color online) (a) Phase diagram of the distorted J1-J3

Phase tr

U ansition with

1

1.5

2

2.5

3

-4

-2

0

2

4

3

χL 0.2

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

(e)

1

(c)

Z

λ

(T-T

02 symmter

1

y break

open

ing

square indicates the transition temperature for λ = 1 where a

c)L1/ν/J3

0.49

4

0.495

0.5

-1.5 -1.0 -0.5 0

0.5 1.0 1.5

0.6

first-order

(f) phase transition with C3 symmetry breaking occurs.21 The

2

U

occurs.

T/J

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

3

η-2 (T-T

J

c)L1/ν/J

0.4

3

solid circles represent transition temperatures at which a second-order

1-J3

χL 0.2

phase transition with Z

model for

. The inset is an enlarged view. The

2 symmetry

J1 breaking

/J3 = −0occurs.

.7342 (b)

. . . and (c) Finite-

FIG. 2. (Color

1

online) Temperature dependence

(c)

0 of equilibrium

size scaling of the Binder

open s ratio U

quare 4 and that

indicates of the

the susceptibility

transition

χ for

temperature for λ = 1 where a

physical quantities of the distorted J

0.49

0.495

0.5 1-J3 model for J

-1.5 -1.0 1/J3

-0.5 =0

λ

0.5 = 1

1.0.5 us

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

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

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

T/J

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

(T-T

solid circles represent transition temperatures at which a second-order

the order parameter

3

c)L1/ν/J3

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

are smaller than the symbol sizes.

density of

phase transition with Z

Z

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

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

the transition temperature

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

T

size scaling of the Binder ratio U

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

4 and that of the susceptibility χ for

scaling of the Binder

physical

ratio

quantities of the distorted

U

J

J

4 and that of the susceptibility

1-J3 model for

χ using

of transition

1/J3 = temperatures

λ = 1.5 using

as depicted

ν = 1 and

in Fig.

η 3(a)

= 1 ./4 which are the critical exponents of

An enlarged

the critic

−al

0. expon

4926 e.n.t.s of th

and e

λ 2D

= I

1 s.ing

308 m

. .o.d.el(a(ν

) =

Sp 1

ecain

fidc ηh =

ea 1

t /4)

C. (b) Square of

view near

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

and the transition

the ordertemperature.

parameter Error

m2 b.ars

(c a

) re

Bomitted

inder r faor

ti clarity

o U

since

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

4. (d) Log figure

of nu indicates

mber

that

are s the transition

maller than the temperature

symbol

near

sizes.

their sizes are smaller than the symbol sizes.

λ = 1

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

e indicatesconnects to the transition temperature for λ = 1

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

Finite-size

temperature goes continuously to zero

In antiferromagnetic Heisenberg models on a triangular

scaling of the Binder ratio U4 and that of the

when

susceptibility χ using

lattice, the dissociation of the

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

Z

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

2 vortices occurs at finite

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

d η = 1 of

/4) the Binder ratio and that of the susceptibility for

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

view near

and the transition temperature. Error bars are omitted for

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

λ =

clarity1.

s 5 us

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

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

figure indicates that the transition temperature near

their sizes are smaller than the symbol sizes.

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

λ = 1

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

scaling functions. Thus,

smoothlywe conclude

connects

that

to thea second-order

transition temperature for λ = 1

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

phase transition with

and breaking

the

of the

transition temperature goes continuously to zero

and it is confirmed

In

that

Z2 symmetry occurs

nv obeys well

antiferromagnetic

the Arrhenius

Heisenberg

law belo

models w

on a triangular

and it belongs to the 2D

when λ Ising

→ model

λ0. Figuni

ur v

e ersality

s 3(b) class

and within

3(c) represent the finite-size

Tc. This result

lattice, indic

the ates that the dis

dissociation soc

of iatio

then of

Z the Z

2 vor 2

tivo

cert

s ice

o s

ccurscalculated

at finite

occurs at the second-order phase transition point.

λ. However,

scalingat v

of ery c

the lose to λ

Binder = 1,

ratio the possibility

and that of the susceptibility for

temperature.13,27 In order to confirm the

that

dissociation first-order

of the

phase transition occurs with breaking of the

To clarify the universality class of the phase transition, we

Z2

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 symmetry

density ofcannot be denied. Unexpected phase transition from

perform the finite-size scaling using the following relations:

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

the Z

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

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

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

U

first-order phase transition with breaking of the Z

4 ∝ f

plot [(T

of −

ln T

ncv)L1/ν

ver ]

s ,us χ

J3 ∝ L2−η

/T in g[(T

our − Tc)L1/ν

model is ],

sho(6)

wn in Fig. 2(d),

2 symmetry

occurs, a tricritical point

phase should exist

transition and

withto study its properties

breaking of the

and it is confirmed that

Z2 symmetry occurs

n

where the susceptibility

v obeys well the Arrhenius law below

χ is defined as χ := NJ

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

3 m2 /T and

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

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

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

observation, it is difficult

calculatedtoλ obtain

. How the

evernature

, at ve of

ry the

clo phase

se to λ = 1, the possibility

results using

occurs at the second-order phase transition point.

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

transition near λ =

that 1 since the

first-order size de

phase pendence o

transition f physic

occurs al

with breaking of the Z2

exponents of

To tchle

ari2fD

y t Is

h ieng

un m

ivoedreslalaintd

y t

c h

l e

as o

s bota

f itnhed

e pThase tran - Number density of Z2 vortex

H = J1

si · sj + J1

si · sj + J3

si · sj

i,j axis 1

i,j axis 2,3

i,j

J1/J3 = 0.4926 · · · , = 1.308 · · ·

-2.0

No phase transition with SO(3)

) -2.2

symmetry breaking occurs at

v

finite temperatures.

(Mermin-Wagner theorem)

ln(n -2.4

Arrhenius law

-2.6

2.00

2.02

2.04

2.06

2.08

J3/T

Dissociation of Z2 vortices occurs at the

second-order phase transition temperature. - 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

J1/J3 = 0.4926 · · · , = 1.308 · · ·

4

2

U

1

= 1, = 1/4

0.6

-2

0.4

L 0.2

0

-1.5 -1.0 -0.5 0

0.5 1.0 1.5

(T-Tc)L1/ /J3

2D Ising universality class !! - Phase diagram

0.6

0.4

1st order phase

3

0.52

/J

transition

cT

0.5

R. Tamura and N. Kawashima,

J. Phys. Soc. Jpn., 77, 103002 (2008).

0.2

R. Tamura and N. Kawashima,

J. Phys. Soc. Jpn., 80, 074008 (2011).

0.48

R. Tamura, S. Tanaka, and N. Kawashima,

to appear in Proceedings of APPC12.

1

1.1

1.2

0 1 1.5 2 2.5 3

no phase transition

2nd order phase

transition - Conclusion

SECOND-ORDER PHASE TRANSITION IN THE

We studied the phase transition nature of a frustrated Heisenber

. . .

PHYSICAL REVIEW B 87, 214401 (2013)

g

model on a distorted triangular lattice. 20

0.6

L=144

3.0

-2.0

(d)

(a)

(b)

15

L=216

L=288

10

)

4

C

-2.2

v

2.0

U

L=108

5

ln(n

0.4

L=144

-2.4

Arrhenius

L=180

3

0.52

L=216

0

(a)

/J

1.0

-2.6

law

0.1

T c

(b)

0.5

0.4

2.00

2.02

2.04

2.06

2.08

0.2

>

0.48

η-2

2

0.2

0.05

J3/T

χL

<m

3

1

1.1

1.2

0.0

(c)

0

0

4

2

U

1

1.5

2

2.5

3

-4

-2

0

2

4

axis 2

3

axis 3

1

(e)

λ

(T-Tc)L1/ν/J3

axis 1

4

(f)

2

0.6

U

η-2

0.4

FIG. 3. (Color online) (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

A second-order phase transition occurs. 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-

- At the second-order phase transition poin

FIG. 2.t, Z

(Color online) Temperature dependence of equilibrium

2

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

symmetry (lattice reflection symmetry) is br

−0.4926 . . . oken.

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 universality class of the phase transition is

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 same as that of the 2D Ising model

th .ecritical 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

- Dissociation of Z

smoothly connects to the transition temperature for λ = 1

2 vortices occurs at the second-

and the transition temperature goes continuously to zero

In antiferromagnetic Heisenberg models on a triangular

order phase transition point.

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 - Thank you !

Ryo Tamura, Shu Tanaka, and Naoki Kawashima

Physical Review B 87, 214401 (2013)