Second-Order Phase Transition in Heisenberg Modelon Triangular Lattice with Competing Interactions

Ryo Tamura, Shu Tanaka, and Naoki KawashimaPhysical Review B 87, 214401 (2013)

Main ResultsWe studied the phase transition nature of a frustrated Heisenberg model on a distorted triangular lattice.

A second-order phase transition occurs.

- At the second-order phase transition point, Z2 symmetry (lattice re!ection symmetry) is broken.

- The universality class of the phase transition is the same as that of the 2D Ising model.

- Dissociation of Z2 vortices occurs at the second- order phase transition point.

SECOND-ORDER PHASE TRANSITION IN THE . . . PHYSICAL REVIEW B 87, 214401 (2013)

1

2

3

0.49 0.495 0.5

U4

T/J3

(c)

0

0.05

0.1

<m2 >

(b)

0

5

10

15

20

C

(a)

L=144L=216L=288

00.20.40.6

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

!L"-

2

(T-Tc)L1/#/J3

(f)

1

2

3

U4

(e)

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(n

v)

J3/T

Arrhenius law

(d)

FIG. 2. (Color online) Temperature dependence of equilibriumphysical quantities of the distorted J1-J3 model for J1/J3 =!0.4926 . . . and ! = 1.308 . . .. (a) Specific heat C. (b) Square ofthe order parameter "m2#. (c) Binder ratio U4. (d) Log of numberdensity of Z2 vortex nv versus J3/T . The dotted vertical line indicatesthe transition temperature Tc/J3 = 0.4950(5). (e) and (f) Finite-sizescaling of the Binder ratio U4 and that of the susceptibility " usingthe critical exponents of the 2D Ising model (# = 1 and $ = 1/4)and the transition temperature. Error bars are omitted for clarity sincetheir sizes are smaller than the symbol sizes.

In antiferromagnetic Heisenberg models on a triangularlattice, the dissociation of the Z2 vortices occurs at finitetemperature.13,27 In order to confirm the dissociation of theZ2 vortices in our model, we calculate the number density ofthe Z2 vortices nv by using the same manner as in Ref. 13. Aplot of ln nv versus J3/T in our model is shown in Fig. 2(d),and it is confirmed that nv obeys well the Arrhenius law belowTc. This result indicates that the dissociation of the Z2 vorticesoccurs at the second-order phase transition point.

To clarify the universality class of the phase transition, weperform the finite-size scaling using the following relations:

U4 $ f [(T ! Tc)L1/#], " $ L2!$g[(T ! Tc)L1/#], (6)

where the susceptibility " is defined as " := NJ3"m2#/T andf (·) and g(·) are scaling functions. The finite-size scalingresults using # = 1 and $ = 1/4 which are the criticalexponents of the 2D Ising model and the obtained Tc areshown in Figs. 2(e) and 2(f). Since all the data collapse ontoscaling functions, it is confirmed that the second-order phasetransition in our model belongs to the universality class of theIsing model.

Next, to obtain the relationship between ! and Tc, weconsider the case of J1/J3 = !0.7342 . . . which was usedin Ref. 21 by changing the value of !. For ! = 1, the modelexhibits a first-order phase transition with breaking of the C3symmetry at Tc/J3 = 0.4746(1).21 Here we study the nature ofthe phase transition of the distorted J1-J3 model with the openboundary condition. From the analysis of the GS as explainedbefore, a phase transition with breaking of the Z2 symmetryis expected to take place for 1 < ! < !0(= 2.8155 . . .) in thiscase. By analyzing the Binder ratio, we obtain ! dependence

0

0.2

0.4

0.6

1 1.5 2 2.5 3

Tc/

J 3

$

(a)

0.48

0.5

0.52

1 1.1 1.2

1.0

2.0

3.0

U4

(b)

L=108L=144L=180L=216

0.0

0.2

0.4

-4 -2 0 2 4

! L" -

2

(T-Tc)L1/#/J3

(c)

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

model for J1/J3 = !0.7342 . . .. The inset is an enlarged view. Theopen square indicates the transition temperature for ! = 1 where afirst-order phase transition with C3 symmetry breaking occurs.21 Thesolid circles represent transition temperatures at which a second-orderphase transition with Z2 symmetry breaking occurs. (b) and (c) Finite-size scaling of the Binder ratio U4 and that of the susceptibility " for! = 1.5 using # = 1 and $ = 1/4 which are the critical exponents ofthe 2D Ising model. Error bars are omitted for clarity since their sizesare smaller than the symbol sizes.

of transition temperatures as depicted in Fig. 3(a). An enlargedview near ! = 1 is shown in the inset of Fig. 3(a). Thisfigure indicates that the transition temperature near ! = 1smoothly connects to the transition temperature for ! = 1and the transition temperature goes continuously to zerowhen ! % !0. Figures 3(b) and 3(c) represent the finite-sizescaling of the Binder ratio and that of the susceptibility for! = 1.5 using # = 1, $ = 1/4, and Tc/J3 = 0.5521(1) as wellas the previous case. In this case, all the data collapse ontoscaling functions. Thus, we conclude that a second-orderphase transition with breaking of the Z2 symmetry occursand it belongs to the 2D Ising model universality class withincalculated !. However, at very close to ! = 1, the possibilitythat first-order phase transition occurs with breaking of the Z2symmetry cannot be denied. Unexpected phase transition fromonly underlying symmetry can occur in some cases.16,23,28 If afirst-order phase transition with breaking of the Z2 symmetryoccurs, a tricritical point should exist and to study its propertiessuch as universality class will be an important topic. From ourobservation, it is difficult to obtain the nature of the phasetransition near ! = 1 since the size dependence of physicalquantities are significant and we should calculate very largesystems with high accuracy.

In this paper we discovered an example where the second-order phase transition occurs accompanying the Z2 vortexdissociation at finite temperature. The model under consid-eration is the classical Heisenberg model on a triangularlattice with three types of interactions: (i) the uniaxiallydistorted nearest-neighbor ferromagnetic interaction alongaxis 1 (!J1), (ii) the nearest-neighbor ferromagnetic interactionalong axes 2 and 3 (J1), and (iii) the third nearest-neighborantiferromagnetic interaction (J3). In this model for 1 < ! <!0, the order parameter space is SO(3)&Z2. We found thesecond-order phase transition with spontaneous breaking ofthe Z2 symmetry in the region. The dissociation of the Z2vortices also occurs at the same temperature. The dissociation

214401-3

axis 1

axis 2 axis

3

BackgroundUnfrustrated systems (ferromagnet, bipartite antiferromagnet)

Ferromagnet Bipartite antiferromagnet

Model Order parameter spaceIsing Z2

XY U(1)Heisenberg S2

BackgroundFrustrated systems

Antiferromagnetic Ising model on triangle

Antiferromagnetic XY/Heisenberg modelon triangle

?triangular lattice kagome lattice pyrochlore lattice

BackgroundOrder parameter space in antiferromagnet on triangular lattice.

Model Order parameter space Phase transitionIsing --- ---

XY U(1) KT transitionHeisenberg SO(3) Z2 vortex dissociation

Motivation

To investigate the "nite-temperature properties in two-dimensional

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

- Phase transition occurs?

- Z2 vortex dissociation?

Model

si : Heisenberg spin (three components)

1st nearest-neighboraxis 1

3rd nearest-neighbor

H = �J1

�

�i,j�axis 1

si · sj + J1

�

�i,j�axis 2,3

si · sj + J3

�

��i,j��

si · sj

1st nearest-neighboraxis 2, 3

axis 1

axis 2 axis

3

� > 0, J3 > 0

Ground StateSpiral-spin con"guration si = R cos(k� · ri)� I sin(k� · ri)

Fourier transform of interactions

Find that minimizes the Fourier transform of interactions! k�

R, I are two arbitrary orthogonal unit vectors.

J(k)NJ3

=�J1

J3cos kx +

2J1

J3cos

kx

2cos�

3ky

2+ cos 2kx + 2 cos kx cos

�3ky

�4 < J1/J3 < 0

SO(3) x C3 & SO(3) x Z2(a) (c)

(b)axis 1

axis 2axis 3

(i) ferromagnetic

(ii) single-k spiral

(iii) double-k spiral

(iv) t

riple

-k s

pira

l

(ii) single-k spiral

4 independentsublattices

structure

structure

Fig. 1. (a) Triangular lattice with Lx ! Ly sites. (b) Enlarged view of the dotted hexagonal area in (a). Thethick and thin lines indicate !J1 and J1, respectively. The third nearest-neighbor interactions at the i-th site aredepicted. (c) Ground-state phase diagram of the model given by Eq. (1). Ground states can be categorized intofive types. More details in each ground state are given in the main text.

discussed the connection between frustrated continuous spin systems and a fundamental discrete spinsystem by using a locally defined parameter. The most famous example is the chiral phase transitionin the antiferromagnetic XY model on a triangular lattice. The relation between the phase transitionof the continuous spin system and 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 inFigs. 1(a) and (b) from a viewpoint 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 isgiven by

H = !J1

!

"i, j#axis 1

si · s j + J1

!

"i, j#axis 2,3

si · s j + J3

!

""i, j##si · s j, (1)

where the first term represents the nearest-neighbor interactions along axis 1, the second term denotesthe nearest-neighbor interactions along axes 2 and 3, and the third term is the third nearest-neighborinteractions [see Fig. 1(b)]. The variable si is the three-dimensional vector spin of unit length. Theparameter !(> 0) represents a uniaxial distortion along axis 1. Here we consider the case that the thirdnearest-neighbor interaction J3 is antiferromagnetic (J3 > 0). The ground state of the model givenby Eq. (1) is represented by the wave vector k$ at which the Fourier transform of interactions J(k) isminimized. In this case, J(k) is given by

J(k)NJ3

=!J1

J3cos kx +

2J1

J3cos

kx

2cos

%3ky

2+ cos 2kx + 2 cos kx cos

%3ky, (2)

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

R. Tamura and N. Kawashima, J. Phys. Soc. Jpn., 77, 103002 (2008).R. Tamura and N. Kawashima, J. Phys. Soc. Jpn., 80, 074008 (2011).

R. Tamura and S. Tanaka, Phys. Rev. E, 88, 052138 (2013).R. Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12.

Order parameter space Order of phase transition

SO(3)xC3 1st order

SO(3)xZ2 2nd order (Ising universality)

J1-J3 model on triangular lattice

Speci!c heat, order parameterSECOND-ORDER PHASE TRANSITION IN THE . . . PHYSICAL REVIEW B 87, 214401 (2013)

1

2

3

0.49 0.495 0.5

U4

T/J3

(c)

0

0.05

0.1

<m2 >

(b)

0

5

10

15

20

C

(a)

L=144L=216L=288

00.20.40.6

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

!L"-

2

(T-Tc)L1/#/J3

(f)

1

2

3U

4

(e)

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(n

v)

J3/T

Arrhenius law

(d)

FIG. 2. (Color online) Temperature dependence of equilibriumphysical quantities of the distorted J1-J3 model for J1/J3 =!0.4926 . . . and ! = 1.308 . . .. (a) Specific heat C. (b) Square ofthe order parameter "m2#. (c) Binder ratio U4. (d) Log of numberdensity of Z2 vortex nv versus J3/T . The dotted vertical line indicatesthe transition temperature Tc/J3 = 0.4950(5). (e) and (f) Finite-sizescaling of the Binder ratio U4 and that of the susceptibility " usingthe critical exponents of the 2D Ising model (# = 1 and $ = 1/4)and the transition temperature. Error bars are omitted for clarity sincetheir sizes are smaller than the symbol sizes.

In antiferromagnetic Heisenberg models on a triangularlattice, the dissociation of the Z2 vortices occurs at finitetemperature.13,27 In order to confirm the dissociation of theZ2 vortices in our model, we calculate the number density ofthe Z2 vortices nv by using the same manner as in Ref. 13. Aplot of ln nv versus J3/T in our model is shown in Fig. 2(d),and it is confirmed that nv obeys well the Arrhenius law belowTc. This result indicates that the dissociation of the Z2 vorticesoccurs at the second-order phase transition point.

To clarify the universality class of the phase transition, weperform the finite-size scaling using the following relations:

U4 $ f [(T ! Tc)L1/#], " $ L2!$g[(T ! Tc)L1/#], (6)

where the susceptibility " is defined as " := NJ3"m2#/T andf (·) and g(·) are scaling functions. The finite-size scalingresults using # = 1 and $ = 1/4 which are the criticalexponents of the 2D Ising model and the obtained Tc areshown in Figs. 2(e) and 2(f). Since all the data collapse ontoscaling functions, it is confirmed that the second-order phasetransition in our model belongs to the universality class of theIsing model.

Next, to obtain the relationship between ! and Tc, weconsider the case of J1/J3 = !0.7342 . . . which was usedin Ref. 21 by changing the value of !. For ! = 1, the modelexhibits a first-order phase transition with breaking of the C3symmetry at Tc/J3 = 0.4746(1).21 Here we study the nature ofthe phase transition of the distorted J1-J3 model with the openboundary condition. From the analysis of the GS as explainedbefore, a phase transition with breaking of the Z2 symmetryis expected to take place for 1 < ! < !0(= 2.8155 . . .) in thiscase. By analyzing the Binder ratio, we obtain ! dependence

0

0.2

0.4

0.6

1 1.5 2 2.5 3

Tc/

J 3

$

(a)

0.48

0.5

0.52

1 1.1 1.2

1.0

2.0

3.0

U4

(b)

L=108L=144L=180L=216

0.0

0.2

0.4

-4 -2 0 2 4

! L" -

2

(T-Tc)L1/#/J3

(c)

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

model for J1/J3 = !0.7342 . . .. The inset is an enlarged view. Theopen square indicates the transition temperature for ! = 1 where afirst-order phase transition with C3 symmetry breaking occurs.21 Thesolid circles represent transition temperatures at which a second-orderphase transition with Z2 symmetry breaking occurs. (b) and (c) Finite-size scaling of the Binder ratio U4 and that of the susceptibility " for! = 1.5 using # = 1 and $ = 1/4 which are the critical exponents ofthe 2D Ising model. Error bars are omitted for clarity since their sizesare smaller than the symbol sizes.

of transition temperatures as depicted in Fig. 3(a). An enlargedview near ! = 1 is shown in the inset of Fig. 3(a). Thisfigure indicates that the transition temperature near ! = 1smoothly connects to the transition temperature for ! = 1and the transition temperature goes continuously to zerowhen ! % !0. Figures 3(b) and 3(c) represent the finite-sizescaling of the Binder ratio and that of the susceptibility for! = 1.5 using # = 1, $ = 1/4, and Tc/J3 = 0.5521(1) as wellas the previous case. In this case, all the data collapse ontoscaling functions. Thus, we conclude that a second-orderphase transition with breaking of the Z2 symmetry occursand it belongs to the 2D Ising model universality class withincalculated !. However, at very close to ! = 1, the possibilitythat first-order phase transition occurs with breaking of the Z2symmetry cannot be denied. Unexpected phase transition fromonly underlying symmetry can occur in some cases.16,23,28 If afirst-order phase transition with breaking of the Z2 symmetryoccurs, a tricritical point should exist and to study its propertiessuch as universality class will be an important topic. From ourobservation, it is difficult to obtain the nature of the phasetransition near ! = 1 since the size dependence of physicalquantities are significant and we should calculate very largesystems with high accuracy.

In this paper we discovered an example where the second-order phase transition occurs accompanying the Z2 vortexdissociation at finite temperature. The model under consid-eration is the classical Heisenberg model on a triangularlattice with three types of interactions: (i) the uniaxiallydistorted nearest-neighbor ferromagnetic interaction alongaxis 1 (!J1), (ii) the nearest-neighbor ferromagnetic interactionalong axes 2 and 3 (J1), and (iii) the third nearest-neighborantiferromagnetic interaction (J3). In this model for 1 < ! <!0, the order parameter space is SO(3)&Z2. We found thesecond-order phase transition with spontaneous breaking ofthe Z2 symmetry in the region. The dissociation of the Z2vortices also occurs at the same temperature. The dissociation

214401-3

Phase transition with Z2 symmtery breaking occurs.

speci!c heatorder parameter

Binder ratio

H = �J1

�

�i,j�axis 1

si · sj + J1

�

�i,j�axis 2,3

si · sj + J3

�

��i,j��

si · sj

J1/J3 = �0.4926 · · · ,� = 1.308 · · ·

axis 1

axis 2 axis

3

SECOND-ORDER PHASE TRANSITION IN THE . . . PHYSICAL REVIEW B 87, 214401 (2013)

1

2

3

0.49 0.495 0.5

U4

T/J3

(c)

0

0.05

0.1

<m2 >

(b)

0

5

10

15

20

C

(a)

L=144L=216L=288

00.20.40.6

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

!L"-

2

(T-Tc)L1/#/J3

(f)

1

2

3

U4

(e)

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(n

v)

J3/T

Arrhenius law

(d)

FIG. 2. (Color online) Temperature dependence of equilibriumphysical quantities of the distorted J1-J3 model for J1/J3 =!0.4926 . . . and ! = 1.308 . . .. (a) Specific heat C. (b) Square ofthe order parameter "m2#. (c) Binder ratio U4. (d) Log of numberdensity of Z2 vortex nv versus J3/T . The dotted vertical line indicatesthe transition temperature Tc/J3 = 0.4950(5). (e) and (f) Finite-sizescaling of the Binder ratio U4 and that of the susceptibility " usingthe critical exponents of the 2D Ising model (# = 1 and $ = 1/4)and the transition temperature. Error bars are omitted for clarity sincetheir sizes are smaller than the symbol sizes.

In antiferromagnetic Heisenberg models on a triangularlattice, the dissociation of the Z2 vortices occurs at finitetemperature.13,27 In order to confirm the dissociation of theZ2 vortices in our model, we calculate the number density ofthe Z2 vortices nv by using the same manner as in Ref. 13. Aplot of ln nv versus J3/T in our model is shown in Fig. 2(d),and it is confirmed that nv obeys well the Arrhenius law belowTc. This result indicates that the dissociation of the Z2 vorticesoccurs at the second-order phase transition point.

To clarify the universality class of the phase transition, weperform the finite-size scaling using the following relations:

U4 $ f [(T ! Tc)L1/#], " $ L2!$g[(T ! Tc)L1/#], (6)

where the susceptibility " is defined as " := NJ3"m2#/T andf (·) and g(·) are scaling functions. The finite-size scalingresults using # = 1 and $ = 1/4 which are the criticalexponents of the 2D Ising model and the obtained Tc areshown in Figs. 2(e) and 2(f). Since all the data collapse ontoscaling functions, it is confirmed that the second-order phasetransition in our model belongs to the universality class of theIsing model.

Next, to obtain the relationship between ! and Tc, weconsider the case of J1/J3 = !0.7342 . . . which was usedin Ref. 21 by changing the value of !. For ! = 1, the modelexhibits a first-order phase transition with breaking of the C3symmetry at Tc/J3 = 0.4746(1).21 Here we study the nature ofthe phase transition of the distorted J1-J3 model with the openboundary condition. From the analysis of the GS as explainedbefore, a phase transition with breaking of the Z2 symmetryis expected to take place for 1 < ! < !0(= 2.8155 . . .) in thiscase. By analyzing the Binder ratio, we obtain ! dependence

0

0.2

0.4

0.6

1 1.5 2 2.5 3

Tc/

J 3

$

(a)

0.48

0.5

0.52

1 1.1 1.2

1.0

2.0

3.0

U4

(b)

L=108L=144L=180L=216

0.0

0.2

0.4

-4 -2 0 2 4

! L" -

2

(T-Tc)L1/#/J3

(c)

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

model for J1/J3 = !0.7342 . . .. The inset is an enlarged view. Theopen square indicates the transition temperature for ! = 1 where afirst-order phase transition with C3 symmetry breaking occurs.21 Thesolid circles represent transition temperatures at which a second-orderphase transition with Z2 symmetry breaking occurs. (b) and (c) Finite-size scaling of the Binder ratio U4 and that of the susceptibility " for! = 1.5 using # = 1 and $ = 1/4 which are the critical exponents ofthe 2D Ising model. Error bars are omitted for clarity since their sizesare smaller than the symbol sizes.

of transition temperatures as depicted in Fig. 3(a). An enlargedview near ! = 1 is shown in the inset of Fig. 3(a). Thisfigure indicates that the transition temperature near ! = 1smoothly connects to the transition temperature for ! = 1and the transition temperature goes continuously to zerowhen ! % !0. Figures 3(b) and 3(c) represent the finite-sizescaling of the Binder ratio and that of the susceptibility for! = 1.5 using # = 1, $ = 1/4, and Tc/J3 = 0.5521(1) as wellas the previous case. In this case, all the data collapse ontoscaling functions. Thus, we conclude that a second-orderphase transition with breaking of the Z2 symmetry occursand it belongs to the 2D Ising model universality class withincalculated !. However, at very close to ! = 1, the possibilitythat first-order phase transition occurs with breaking of the Z2symmetry cannot be denied. Unexpected phase transition fromonly underlying symmetry can occur in some cases.16,23,28 If afirst-order phase transition with breaking of the Z2 symmetryoccurs, a tricritical point should exist and to study its propertiessuch as universality class will be an important topic. From ourobservation, it is difficult to obtain the nature of the phasetransition near ! = 1 since the size dependence of physicalquantities are significant and we should calculate very largesystems with high accuracy.

In this paper we discovered an example where the second-order phase transition occurs accompanying the Z2 vortexdissociation at finite temperature. The model under consid-eration is the classical Heisenberg model on a triangularlattice with three types of interactions: (i) the uniaxiallydistorted nearest-neighbor ferromagnetic interaction alongaxis 1 (!J1), (ii) the nearest-neighbor ferromagnetic interactionalong axes 2 and 3 (J1), and (iii) the third nearest-neighborantiferromagnetic interaction (J3). In this model for 1 < ! <!0, the order parameter space is SO(3)&Z2. We found thesecond-order phase transition with spontaneous breaking ofthe Z2 symmetry in the region. The dissociation of the Z2vortices also occurs at the same temperature. The dissociation

214401-3

�(t) := s(t)1 ·

�s(t)2 � s(t)

3

�, m :=

�

t

�(t)/N

Number density of Z2 vortex

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(nv)

J3/T

Arrhenius law

Dissociation of Z2 vortices occurs at the second-order phase transition temperature.

H = �J1

�

�i,j�axis 1

si · sj + J1

�

�i,j�axis 2,3

si · sj + J3

�

��i,j��

si · sj

J1/J3 = �0.4926 · · · ,� = 1.308 · · ·

No phase transition with SO(3) symmetry breaking occurs at "nite temperatures.(Mermin-Wagner theorem)

Finite size scaling

00.20.40.6

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

rLd-2

(T-Tc)L1/i/J3

1

2

3

U4

2D Ising universality class !!

� = 1, � = 1/4

H = �J1

�

�i,j�axis 1

si · sj + J1

�

�i,j�axis 2,3

si · sj + J3

�

��i,j��

si · sj

J1/J3 = �0.4926 · · · ,� = 1.308 · · ·

Phase diagram

0

0.2

0.4

0.6

1 1.5 2 2.5 3

T c/J

3

h

0.48

0.5

0.52

1 1.1 1.2

1st order phase transitionR. Tamura and N. Kawashima, J. Phys. Soc. Jpn., 77, 103002 (2008).R. Tamura and N. Kawashima, J. Phys. Soc. Jpn., 80, 074008 (2011).R. Tamura, S. Tanaka, and N. Kawashima, to appear in Proceedings of APPC12.

2nd order phase transition

no phase transition

ConclusionWe studied the phase transition nature of a frustrated Heisenberg model on a distorted triangular lattice.

A second-order phase transition occurs.

- At the second-order phase transition point, Z2 symmetry (lattice re!ection symmetry) is broken.

- The universality class of the phase transition is the same as that of the 2D Ising model.

- Dissociation of Z2 vortices occurs at the second- order phase transition point.

SECOND-ORDER PHASE TRANSITION IN THE . . . PHYSICAL REVIEW B 87, 214401 (2013)

1

2

3

0.49 0.495 0.5

U4

T/J3

(c)

0

0.05

0.1

<m2 >

(b)

0

5

10

15

20

C

(a)

L=144L=216L=288

00.20.40.6

-1.5 -1.0 -0.5 0 0.5 1.0 1.5

!L"-

2

(T-Tc)L1/#/J3

(f)

1

2

3

U4

(e)

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(n

v)

J3/T

Arrhenius law

(d)

FIG. 2. (Color online) Temperature dependence of equilibriumphysical quantities of the distorted J1-J3 model for J1/J3 =!0.4926 . . . and ! = 1.308 . . .. (a) Specific heat C. (b) Square ofthe order parameter "m2#. (c) Binder ratio U4. (d) Log of numberdensity of Z2 vortex nv versus J3/T . The dotted vertical line indicatesthe transition temperature Tc/J3 = 0.4950(5). (e) and (f) Finite-sizescaling of the Binder ratio U4 and that of the susceptibility " usingthe critical exponents of the 2D Ising model (# = 1 and $ = 1/4)and the transition temperature. Error bars are omitted for clarity sincetheir sizes are smaller than the symbol sizes.

In antiferromagnetic Heisenberg models on a triangularlattice, the dissociation of the Z2 vortices occurs at finitetemperature.13,27 In order to confirm the dissociation of theZ2 vortices in our model, we calculate the number density ofthe Z2 vortices nv by using the same manner as in Ref. 13. Aplot of ln nv versus J3/T in our model is shown in Fig. 2(d),and it is confirmed that nv obeys well the Arrhenius law belowTc. This result indicates that the dissociation of the Z2 vorticesoccurs at the second-order phase transition point.

To clarify the universality class of the phase transition, weperform the finite-size scaling using the following relations:

U4 $ f [(T ! Tc)L1/#], " $ L2!$g[(T ! Tc)L1/#], (6)

where the susceptibility " is defined as " := NJ3"m2#/T andf (·) and g(·) are scaling functions. The finite-size scalingresults using # = 1 and $ = 1/4 which are the criticalexponents of the 2D Ising model and the obtained Tc areshown in Figs. 2(e) and 2(f). Since all the data collapse ontoscaling functions, it is confirmed that the second-order phasetransition in our model belongs to the universality class of theIsing model.

Next, to obtain the relationship between ! and Tc, weconsider the case of J1/J3 = !0.7342 . . . which was usedin Ref. 21 by changing the value of !. For ! = 1, the modelexhibits a first-order phase transition with breaking of the C3symmetry at Tc/J3 = 0.4746(1).21 Here we study the nature ofthe phase transition of the distorted J1-J3 model with the openboundary condition. From the analysis of the GS as explainedbefore, a phase transition with breaking of the Z2 symmetryis expected to take place for 1 < ! < !0(= 2.8155 . . .) in thiscase. By analyzing the Binder ratio, we obtain ! dependence

0

0.2

0.4

0.6

1 1.5 2 2.5 3

Tc/

J 3

$

(a)

0.48

0.5

0.52

1 1.1 1.2

1.0

2.0

3.0

U4

(b)

L=108L=144L=180L=216

0.0

0.2

0.4

-4 -2 0 2 4

! L" -

2

(T-Tc)L1/#/J3

(c)

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

model for J1/J3 = !0.7342 . . .. The inset is an enlarged view. Theopen square indicates the transition temperature for ! = 1 where afirst-order phase transition with C3 symmetry breaking occurs.21 Thesolid circles represent transition temperatures at which a second-orderphase transition with Z2 symmetry breaking occurs. (b) and (c) Finite-size scaling of the Binder ratio U4 and that of the susceptibility " for! = 1.5 using # = 1 and $ = 1/4 which are the critical exponents ofthe 2D Ising model. Error bars are omitted for clarity since their sizesare smaller than the symbol sizes.

of transition temperatures as depicted in Fig. 3(a). An enlargedview near ! = 1 is shown in the inset of Fig. 3(a). Thisfigure indicates that the transition temperature near ! = 1smoothly connects to the transition temperature for ! = 1and the transition temperature goes continuously to zerowhen ! % !0. Figures 3(b) and 3(c) represent the finite-sizescaling of the Binder ratio and that of the susceptibility for! = 1.5 using # = 1, $ = 1/4, and Tc/J3 = 0.5521(1) as wellas the previous case. In this case, all the data collapse ontoscaling functions. Thus, we conclude that a second-orderphase transition with breaking of the Z2 symmetry occursand it belongs to the 2D Ising model universality class withincalculated !. However, at very close to ! = 1, the possibilitythat first-order phase transition occurs with breaking of the Z2symmetry cannot be denied. Unexpected phase transition fromonly underlying symmetry can occur in some cases.16,23,28 If afirst-order phase transition with breaking of the Z2 symmetryoccurs, a tricritical point should exist and to study its propertiessuch as universality class will be an important topic. From ourobservation, it is difficult to obtain the nature of the phasetransition near ! = 1 since the size dependence of physicalquantities are significant and we should calculate very largesystems with high accuracy.

In this paper we discovered an example where the second-order phase transition occurs accompanying the Z2 vortexdissociation at finite temperature. The model under consid-eration is the classical Heisenberg model on a triangularlattice with three types of interactions: (i) the uniaxiallydistorted nearest-neighbor ferromagnetic interaction alongaxis 1 (!J1), (ii) the nearest-neighbor ferromagnetic interactionalong axes 2 and 3 (J1), and (iii) the third nearest-neighborantiferromagnetic interaction (J3). In this model for 1 < ! <!0, the order parameter space is SO(3)&Z2. We found thesecond-order phase transition with spontaneous breaking ofthe Z2 symmetry in the region. The dissociation of the Z2vortices also occurs at the same temperature. The dissociation

214401-3

axis 1

axis 2 axis

3

Thank you !

Ryo Tamura, Shu Tanaka, and Naoki KawashimaPhysical Review B 87, 214401 (2013)