unconventional phase transitions in frustrated systems (march, 2014)

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 resultsTo investigate unconventional phase transition behavior in geometrically frustrated systems.

2D 3D

SO(3)xZ2 SO(3)xC3- Z2 vortex dissociation- 2nd-order PT w/ Z2 breaking (2-dim. Ising universality)

at the same temperature.

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

- increases, decreases.J� �E

Conventional phase transitionsFerromagnets Antiferromagnets In the ground state, all spin

pairs form stable spin con!gurations.

Type Order parameter space 1D 2D 3D

Ising Z2 × √ √

XY U(1) × KT √

Heisenberg S2 × × √

Temperature

Ordered phase

Tc

Disordered phase

Phase transition occurs.

Frustration: random spin systems

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

Slow relaxation Novel order

TAMURA, KAWASHIMA, YAMAMOTO, TASSEL, AND KAGEYAMA PHYSICAL REVIEW B 84, 214408 (2011)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

s y s y

sx

!

mome

L=12

x=0.296875!=0.494"

x=0.53125!=0.112"

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1sx

L=12 !

mome

FIG. 8. (Color online) Snapshots of spin directions for x =0.296875 and 0.53125 when the lattice size is L = 12 in the groundstate. Coordinates (sx,sy) define the orthogonal plane of the vectormo ! me.

the mixed phase as the “random fan-out state.” This randomfan-out state is an exotic bulk spin ordering that explains thesimultaneous appearance of (!!! ) and (!!0) wave vectorswithout any phase separation. In this model, the ferromagneticcorrelation between NNLs exists as explained in Sec. III.

C. Universality classes of phase transitions

We study the universality classes of phase transitions ofour model. In the phase diagram (see Fig. 5), there are twotypes of phase boundaries. To make clear the universalityclasses of each phase transition, x is set to 3/16 = 0.1875 suchthat transition temperatures are separated sufficiently. For thisparameter, the intermediate phase is the (!!! ) ordered phase(see the dotted arrow in Fig. 5).

First, we investigate the higher-temperature phase transitionfrom the paramagnetic phase to the (!!! ) ordered phase.From the Harris criterion,36 we expect that the higher-temperature phase transition belongs to the three-dimensionalHeisenberg universality class. This is because the criticalexponent " is negative in the three-dimensional Heisenbergmodel, and thus the disorder should not affect the universalityclass. To obtain the transition temperature and confirm thecritical exponents, we calculate the correlation function Gc(rc),

a

b

c

(a) (b)

Random Fan-Out State!

!

FIG. 9. (Color online) (a) “Average” spin directions in the spinconfiguration of the random fan-out state. In each layer (ab plane),Neel order appears. Along the interlayer direction (c axis), the angle# between nearest-neighbor “average” spin pairs changes from ! to0 with increasing x as shown in Fig. 7(a). The correlation betweenNNLs is FM. (b) Individual spins are randomly directed around theaverage direction.

which is defined by Eq. (11). The finite-size scaling relationsof d-dimensional systems are given by

Gc(L/2)Gc(L/4)

" $(tL1/%), (15)

Gc(L/2) " L#d+2#&'(tL1/%), (16)

where $ and ' are scaling functions and t := T # Tc.37

We determine the transition temperature Tc as the crossingpoint of Gc(L/2)/Gc(L/4) using L = 20–32 data and obtainTc/J = 1.085(5). This transition temperature is consistentwith the one found for U4(!!! ) in Sec. IV A. The finite-sizescaling using the critical exponents of the three-dimensionalHeisenberg universality class (% = 0.704, & = 0.025)38 areshown in Fig. 10(a). Since the data are well fitted byscaling relations, we conclude that the higher-temperaturephase transition belongs to the three-dimensional Heisenberguniversality class in accord with the Harris criterion.

Next, we investigate the lower-temperature phase transitionfrom the (!!! ) ordered phase to the mixed phase. The (!!! )ordered phase is translationally symmetric with the O(3) spinrotation symmetry broken down the spin rotation symmetryU(1). In the mixed phase, both the translational symmetry andthe U(1) spin rotation symmetry are broken. Thus we expectthat the transition to the mixed phase is characterized by thebreaking of U(1) spin rotation symmetry. In other words, weexpect that the lower-temperature phase transition belongs tothe three-dimensional XY universality class. To obtain thetransition temperature and confirm the critical exponents, wecalculate the magnetization vector m(!!0) defined by Eq. (8).The finite-size scaling relations are given by

U4(q) = $|m(q)|4%$|m(q)|2%2

" f (tL1/%), (17)

( (q) = N$|m(q)|2%

T" L2#&g(tL1/%), (18)

where f and g are scaling functions, and q = (!!0). Wedetermine the transition temperature as the crossing point ofU4(!!0) using L = 20–32 data and obtain Tc/J = 0.4585(5).The finite-size scaling using the critical exponents of the three-dimensional XY universality class (% = 0.672, & = 0.038)39

are shown in Fig. 10(b). Although we obtain a reasonablygood fit, it is not good enough to detect the small differencebetween XY critical exponents and the other critical exponentsin a three-dimensional system. However, from the viewpointof spin rotation symmetry, we can deduce that the lower-temperature phase transition belongs to the three-dimensionalXY universality class.

V. MEAN-FIELD CALCULATIONS

In this section, to obtain an intuitive understanding of theemergence mechanism of the mixed phase, we investigatethe effect of random interlayer couplings by mean-fieldcalculations. For simplicity of notation, we study the systemwhere the intralayer interactions are FM. Under the gaugetransformation at alternating sites, this model is equivalent toour model given by Eq. (1) in the case of no external field. Byapplying the inverse of the gauge transformation to this model,we can obtain the same results as from the original model.

214408-6

R. Tamura, N. Kawashima, H. Kageyama et al., PRB 84, 214408 (2011)

Ferromagnetic interaction

Antiferromagnetic interaction

Even in the GS, locally unstable spin state appears due to frustration.

layered perovskiteSrFe1-xMnxO2

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

Frustration: geometrically frustrated systemsIsing model Heisenberg model

Residual entropy(macroscopically degenerated states)

Single-q state(120-degree structure, spiral spin texture)

Antiferromagnets on triangle-based lattice structures

Geometrical frustration

Unconventional behaviors in GFMsChirality and Z2 vortex Reentrant phase transition

ParaAntiferroParaFerroTemperature

Slow relaxationT ! 0!, we expect that nloop must be the maximum valueand the spin structure becomes the so-called

!!!3p"

!!!3p

structure.Next, we study the relaxation of magnetization and nloop.

We ready the three types of initial configurations, i.e., (a) the!!!3p"

!!!3p

structure, (b) the q # 0 structure, and (c) a randomstructure. The configurations (a) and (b) are typical groundstates of the present model, and the configuration (c)corresponds to a state just after quench the temperaturefrom a high temperature.

In Fig. 4, the relaxation processes at T # 0:05J areplotted. In the cases (a) and (b), the magnetization ismaximum at t # 0, and it relaxes very fast to uniformlymagnetized ordered state. The relaxation of magnetization tothe equilibrium is depicted in the inset. In contrast, in thecase (c), i.e., from a random state, it takes some time torealize the uniformly magnetized state. Thus we regardthe relaxation time in the case (c) as the intrinsic relaxationtime of the magnetization !mag.

The relaxation of nloop is much slower. In the case (a) nloop

starts with the maximum value, while in the case (c) it startswith zero. In all the cases, (a), (b), and (c), the numbers seemto reach the same saturated value. This observation indicatesthat there exists stable equilibrium state for nloop. Therelaxation time !loop is also about the same, although in thecase (b) a non-monotonic process is observed in the earlystate where the weathervane lines in the initial q # 0 stateare broken into short strips. Thus, we expect that thereexists an intrinsic relaxation time for nloop. From the figurewe estimate that !mag is about 104 MCS, and !loop is of theorder 107 MCS.

In Fig. 5, we plot the relaxation process of nloop in the case(a) at various temperatures below the critical temperature.There, we find two steps in relaxation. The first relaxationcorresponds to initial local relaxation from the completeground state configuration. The relaxation time of thisprocess is very short and does not strongly depend on thetemperature. In the second relaxation, the re-constructionof the WLs takes place. The first and second relaxationscorrespond to energetical and entropical relaxations, respec-tively. It is noted that the two-step relaxation is a character-istics of the ordered state of the present model.

We fit all the second relaxation processes in the followingform

nloop$t% # A$T% exp &t

!$T%

" #! n(eq)

loop$T%; $2%

where !$T% denotes a characteristic relaxation time of thenloop and n(eq)

loop is the equilibrium value of nloop. Because thefirst relaxation is much faster than the second one, the choiseof start time of the second relaxation is irrelevant. Here wefit the curves in the cases of T # 0:07J, 0:065J, 0:06J,0:055J, 0:05J, 0:0475J, 0:045J, 0:0425J, and 0:04J. Thetemperature dependence of ! and n(eq)

loop are plotted in Fig. 6.In Fig. 6(a), we find that nloop increases when the temper-ature is lowered. We expect that this value continues to theground state value nloop # 1. It should be noted that thisquantity is not zero at the critical temperature, and it can notbe an order parameter although it is a good indicator of thedegree of the order of the structure. In Fig. 6(b), we find agood linearity, which indicates the Arrhenius law !$T% /e"!E. From the slope in Fig. 6(b), the energy barrier !E isestimated about 0:78J ' O$J%. It is expected that this energyis necessary when the WLs are reconnected locally.

We study slow relaxation of spin configuration in themagnetically ordered phase of the Ising-like Heisenbergkagome antiferromagnets. From the view point of theentropy, states with the larger nloop are preferable, although

0 [!10+7]0

0.04

0.08

0.12

0.16

0 5000 100000

0.04

0.08

0.12

0.16

Monte Carlo Step (MCS)

Mag

netiz

atio

n


Mag

netiz

atio

n

0 [!10+7]

0

0.5

1


Num

ber

of W

eath

erva

ne L

oops

(a)

(c)

(b)

54321 54321

Fig. 4. (Color online) Relaxation of the magnetization and nloop at T # 0:05J from (a)!!!3p"

!!!3p

configuration, (b) q # 0 configuration, and (c) randomconfiguration.

100 102 104 106 108 10100

0.5

1

Num

ber

of W

eath

erva

ne lo

ops


T=0.0425JT=0.04J

T=0.045J

T=0.0475JT=0.05JT=0.055JT=0.06JT=0.065JT=0.07J

T=0.1JT=0.09JT=0.08J

Fig. 5. (Color online) Relaxation of nloop from!!!3p"

!!!3p

structure atseveral temperatures. Dashed lines denote the fittling curves estimatedby eq. (2).

J. Phys. Soc. Jpn., Vol. 76, No. 10 LETTERS S. TANAKA and S. MIYASHITA

103001-3

A. Kuroda and S. Miyashita, JPSJ 64, 4509 (1995).S. Tanaka and S. Miyashita, JPSJ 76, 103001 (2007).

S. Miyashita and H. Shiba, JPSJ 53, 1145 (1984).H. Kawamura and S. Miyashita, JPSJ 53, 4138 (1984).X. Hu, S. Miyashita, and M. Tachiki, PRL 79, 3498 (1997).R. Tamura, S. Tanaka, and N. Kawashima, PRB 87, 214401 (2013).

H. Kitatani, S. Miyashita, and M. Suzuki, JPSJ 55, 865 (1986).S. Miyashita, S. Tanaka, and M. Hirano, JPSJ 76, 083001 (2007).

Successive phase transitionsR. Ishii et al.

Fig. 1: (Color online) Crystal structures of Rb4Mn(MoO4)3 featuring (a) MnO5 polyhedra, (b) equilateral triangular latticesof Mn2+ and MoO4 tetrahedra. Intralayer and interlayer distances between Mn

2+ ions are given by a= 6.099 Aand c/2 =11.856 A, respectively, at 298K. Phase diagrams of Rb4Mn(MoO4)3 for (c) µ0H ! c and (d) µ0H ! ab derived from themeasurements indicated. The phase boundaries derived from Monte Carlo simulations for D/J = 0.22 are indicated by dashedlines.

transitions associated with respective ordering ofthe longitudinal and transverse spin components areexpected [17,18]. Specifically on cooling, the system firstforms a collinear intermediate phase (IMP) with thethree-sublattice “uud” structure [19] before transitioningto 120! spin-order phase with a uniform vector chiral-ity. A recent theoretical study has indicated anotherphase transition in the IMP, separating the lower-temperature “uud” phase and a higher-temperaturecollinear phase with three di!erent sublattice moments.However, the latter phase is only stable in the purely 2Dlimit, and thus with finite interlayer coupling, the “uud”phase should become dominant throughout the collinearIMP [19]. For the easy-axis case, unlike the Heisenbergand XY , experiments have generally confirmed thetheoretical predictions. Specifically, successive transi-tions and/or a 1/3 magnetization plateau have beenobserved for TAFs with easy-axis anisotropy such asVCl2 [20], ACrO2 (A=Li, Cu) [21,22], and a metallicTAF GdPd2Al3 [23]. However, neither a detailed study ofthe phase diagram under external field nor a quantitativecomparison between experiment and theory has so farbeen possible, because of large values of J which precludeaccess to the high-field regime and/or magnetostriction,which complicates quantitative comparison betweentheory and experiment. Here, we report a comprehen-sive study of the crystal structure, spin structure andthermo-dynamic properties of the quasi-2D HeisenbergTAF Rb4Mn(MoO4)3. This material exhibits successivezero-field phase transitions and a 1/3 magnetizationplateau, as a result of its easy-axis anisotropy. The rela-tively small exchange constant, J , allows us for the firsttime to determine the complete phase diagrams underfield both parallel and perpendicular to the easy axis.In a rare case for geometrically frustrated magnetism,quantitative agreement between experiment and theory

is achieved and establishes Rb4Mn(MoO4)3 as an idealmodel system for the 2D Heisenberg TAF described byeq. (1) with J = 1.2K and D= 0.26K.Single crystals with typical dimensions 1! 1! 0.5mm3

were synthesized by a flux method [24]. The structurewas determined by single-crystal X-ray di!raction and isdescribed by the space group P63/mmc symmetry (R1 =2.88%). Powder neutron di!raction (PND) measurementswere performed on BT1 at NIST, and confirmed this struc-ture and its stability down to 1.5K. Thus, Rb4Mn(MoO4)3contains an equilateral triangular lattice of Mn2+ ions.Each Mn2+ ion is located in a MnO5 polyhedron and hasa high-spin t32ge

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

berg spin. The dominant intralayer coupling J shouldresult from the superexchange path Mn-O-O-Mn involvingtwo oxygen atoms. The interlayer interaction is expectedto be negligible because two Rb+ ions and two MoO4tetrahedra yield a large separation between neighboringplanes.DC magnetization (M) was measured by a commercial

SQUID magnetometer above 1.8K, and by a Faradaymethod for 0.37K<T < 2K [25]. Specific heat, CP , wasmeasured by a thermal relaxation method down to 0.4Kunder fields up to 9T. Pulsed-field measurements ofM were performed up to 27T. Classical Monte Carlosimulations (MCs) were carried out using the standardMetropolis method. The error bars of the MCs results aresmaller than the symbol sizes used in all figures.For an overview, we first present the T -H phase

diagrams in figs. 1(c) and (d). The qualitative nature ofthe phase diagram for µ0H " c is fully consistent withthe pioneer theoretical works in refs. [17] and [18]. Onthe other hand, for µ0H " ab, this is the first report onthe phase diagram for a quasi-2D TAF with an easy-axis–type anisotropy. For these two field directions, atotal of six phases (A)–(F) are identified. Each symbol

17001-p2

S. Miyashita and H. Kawamura, JPSJ 54, 3385 (1985).S. Miyashita, JPSJ 55, 3605 (1986).R. Ishii, S. Tanaka, S. Nakatsuji et al. EPL 94, 17001 (2011).

Phase transition in 2D GFMsH = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj

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

J3/J10-1/4

Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3)

J1: FerroJ3/J10-1/9

Degenerated GSs 120-degree structure (SO(3))

Order by disorder

J1: Antiferro

2D triangular latticeNiGa2S4

S. Nakatsuji, Y. Nambu, Y. Maeno et al., Science 309, 1697 (2005).

1st-order PT w/ 3-fold symmetry breaking and Z2 vortex dissociation occur.

R. Tamura and N. Kawashima,JPSJ 77, 103002 (2008), JPSJ 80, 074008 (2011).

Z2 vortex dissociation


2D 3D





Model

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�

�i,j��axis 2,3

�si · �sj � J3

�

�i,j�3

�si · �sj

The 1st n.n. interactionalong axes 2 and 3

The 3rd n.n. interactionThe 1st n.n. interactionalong axis 1

Model

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj



4 types of ground states for ferromagnetic J1

1. Ferromagnetic state (S2)

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

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

4. triple-q spiral state (SO(3)xC3)

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

Z2 vortex dissociation occurs at !nite T.

1st-order PT and Z2 vortex dissociation occur at the same T.

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

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

R. Tamura and N. Kawashima, JPSJ 77, 103002 (2008).R. Tamura and N. Kawashima, JPSJ 80, 074008 (2011).

Ground state phase diagram

SO(3)xC3

SO(3)xZ2

(i) ferromagnetic

(ii) single-k spiral

(iii) double-k spiral

(iv) t

riple-

k spir

al

(ii) single-k spiral

4 independentsublattices

structure

structure

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj



Model

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj

axis 1

axis 2 axis

3



Order parameter space: SO(3)xZ2

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

Physical quantitiesSECOND-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

speci!c heatorder parameter

Binder ratio

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj

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

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)




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



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)





214401-3

�(t) := s(t)1 ·

�s(t)2 � s(t)

3

�, m :=

�

t

�(t)/N

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

Order parameter detecting Z2 breaking

U4 :=�m4��m2�2Binder ratio

Crossing point

Z2 vortex dissociation

-2.6

-2.4

-2.2

-2.0

2.00 2.02 2.04 2.06 2.08

ln(nv)

J3/T

Arrhenius law

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj



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

No phase transition w/ SO(3) breaking occurs at !nite T.(Mermin-Wagner theorem)

Point defect: �1(SO(3)) = Z2

Z2 vortex dissociation can occurat !nite T.

Z2 vortex density

Z2 vortex dissociation occursat the 2nd-order PT point (Tc).

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

� = 1, � = 1/4

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj



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

Binder ratio

Susceptibility

Finite size scaling relations�U4 � f

�(T � Tc)L1/�

�

� � L2��g�(T � Tc)L1/�

�

2D Ising universality class� = 1, � = 1/4

Z2 vortex dissociation does not affect the phase transition nature.

Phase diagram

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)




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



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)L

1/#/J3

(c)





214401-3

H = ��J1

�

�i,j��axis 1

�si · �sj � J1

�


�si · �sj � J3

�

�i,j�3

�si · �sj



J1/J3 = �0.7342 · · ·

SO(3)xC31st-order PT w/ C3 breaking & Z2 vortex dissociation occur at the same T. SO(3)xZ2 SO(3)

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

JPSJ 80, 074008 (2011).

SO(3)xZ22nd-order PT w/ Z2 breaking & Z2 vortex dissociation occur at the same T.2D Ising universality

SO(3)Z2 vortex dissociation occur at !nite T.

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


2D 3D





Model

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

The 3rd n.n. interactionintralayer

The 1st n.n. interactioninterlayer

The 1st n.n. interactionintralayer

Ground state

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

�/2

�

�/2 ��/2

�/2 �/2

�/2�

Order parameter space: SO(3)xC3




Internal energy and speci!c heat

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

J3/J1 = �0.85355 · · · , J�/J1 = 2INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . . PHYSICAL REVIEW E 88, 052138 (2013)

0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1

FIG. 4. (Color online) Temperature dependence of (a) internalenergy per site E/J1, (b) specific heat C, and (c) order param-eter !|µ|2", which can detect the C3 symmetry breaking of themodel with J3/J1 = #0.853 55 . . . and J$/J1 = 2 for L = 24,32,40.(d) Probability distribution of the internal energy P (E; Tc(L)). Theinset shows the lattice-size dependence of the width between bimodalpeaks !E(L)/J1. (e) Plot of Tc(L)/J1 as a function of L#3. (f) Plotof Cmax(L) as a function of L3. Lines are just visual guides and errorbars in all figures are omitted for clarity since their sizes are smallerthan the symbol size.

of analysis. One is the finite-size scaling and the other isa naive analysis of the probability distribution P (E; Tc(L)).The scaling relations for the first-order phase transition ind-dimensional systems [82] are given by

Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)

where Tc and !E are, respectively, the transition temperatureand the latent heat in the thermodynamic limit. The coefficientof the first term in Eq. (9), a, is a constant. Figures 4(e)and 4(f) show the scaling plots for Tc(L)/J1 and Cmax(L),respectively. Figure 4(e) indicates that Tc is a nonzero valuein the thermodynamic limit. Figure 4(f) shows an almostlinear dependence of Cmax(L) as a function of L3. However,using the finite-size scaling, we cannot obtain the transitiontemperature and latent heat in the thermodynamic limit withhigh accuracy because of the strong finite-size effect. Next wedirectly calculate the size dependence of the width betweenbimodal peaks of the energy distribution shown in Fig. 4(d).The width for the system size L is represented by !E(L) =E+(L) # E#(L), where E+(L) and E#(L) are the averages ofthe Gaussian function in the high-temperature phase and that inthe low-temperature phase, respectively. In the thermodynamiclimit, each Gaussian function becomes the " function and then!E(L) converges to !E [82]. The inset of Fig. 4(d) shows thesize dependence of the width !E(L)/J1. The width enlarges asthe system size increases, which indicates that the latent heat isa nonzero value in the thermodynamic limit. The results shownin Fig. 4 conclude that the model given by Eq. (1) exhibits the

first-order phase transition with the C3 symmetry breaking atfinite temperature.

We further investigate the way of spin ordering. Asmentioned above, the order parameter space of the system isSO(3) & C3. It was confirmed that the C3 symmetry breaks atthe first-order phase transition point. In the antiferromagneticHeisenberg model on a stacked triangular lattice with onlya nearest-neighbor interaction where the order parameterspace is SO(3), a single peak is observed for the temperaturedependence of the specific heat [42,43]. The peak indicates thefinite-temperature phase transition between the paramagneticstate and magnetic ordered state where the SO(3) symmetryis broken. Then, in our model, the SO(3) symmetry shouldbreak at the first-order phase transition point since the specificheat has a single peak corresponding to the first-order phasetransition. To confirm this we calculate the temperaturedependence of the structure factor of spin

S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)

which is the magnetic order parameter for spiral-spin states.When the magnetic ordered state described by k' where theSO(3) symmetry is broken appears, S(k') becomes a finitevalue in the thermodynamic limit. Figure 5(a) shows thetemperature dependence of the largest value of structure factorsS(k') calculated by six wave vectors in Eq. (4). Here S(k')becomes zero in the thermodynamic limit above the first-order phase transition temperature. The structure factor S(k')becomes a nonzero value at the first-order phase transitiontemperature. Moreover, as temperature decreases, the structurefactor S(k') increases. The structure factors at kz = 0 in thefirst Brillouin zone at several temperatures for L = 40 are alsoshown in Fig. 5(b). As mentioned in Sec. II, the spiral-spinstructure represented by k is the same as that represented by#k in the Heisenberg models. Figure 5(b) confirms that onedistinct wave vector is chosen from three types of ordered

(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1

FIG. 5. (Color online) (a) Temperature dependence of the largestvalue of structure factors S(k') calculated by six wave vectors inEq. (4) for J3/J1 = #0.853 55 . . . and J$/J1 = 2. Error bars areomitted for clarity since their sizes are smaller than the symbol size.(b) Structure factors at kz = 0 in the first Brillouin zone at severaltemperatures for L = 40.

052138-5

INTERLAYER-INTERACTION DEPENDENCE OF LATENT . . . PHYSICAL REVIEW E 88, 052138 (2013)

0

0.01

0.02

1.52 1.53 1.54 1.55

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30

40-2.3

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-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

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0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

Internal energy Speci!c heat

Phase transition occursat "nite T.


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5




Order parameter (C3 and SO(3))

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

J3/J1 = �0.85355 · · · , J�/J1 = 2


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

C3 symmetry breaks at Tc.

Order parameter (C3)


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

Order parameter (SO(3))

SO(3) symmetry breaks at Tc.




Energy histogram

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

J3/J1 = �0.85355 · · · , J�/J1 = 2


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

P (E;T ) = D(E)e�E/kBT

D(E) : density of states

�E(L) : width between two peaks

Bimodal distribution

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




Finite size scaling

H = �J1

�

�i,j�

�si · �sj � J3

�

�i,j�3

�si · �sj � J��

�i,j��

�si · �sj

J3/J1 = �0.85355 · · · , J�/J1 = 2


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5


0

0.01

0.02

1.52 1.53 1.54 1.55

0

10

20

30

40-2.3

-2.2

-2.1

1.53

1.54

1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

(f)

(b)

(c)

0

0.05

0.1

0 15 30 45

0

5

10

15

20

25

30

-2.3 -2.2 -2.1



Tc(L) = aL#d + Tc, (9)

Cmax(L) % (!E)2Ld

4T 2c

, (10)




S(k) := 1N

!

i,j

!si · sj "e#ik·(ri#rj ), (11)


(a)

(b)

0

0.1

0.2

0.3

0.4

0.5

0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

Max of speci"c heatTc(L)

Tc(L) = aL�d + Tc Cmax(L) � (�E)2Ld

4T 2c

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

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




Interlayer interaction dependence !xing J3/J1RYO TAMURA AND SHU TANAKA PHYSICAL REVIEW E 88, 052138 (2013)

vectors below the first-order phase transition point, which isfurther evidence of the C3 symmetry breaking at the first-orderphase transition temperature.

Before we end this section, let us mention a phasetransition nature in the J1-J2 Heisenberg model with interlayerinteraction J! on a stacked triangular lattice. In Refs. [62–65],the authors studied the phase transition behavior of the modelwhen J1 and J2 are antiferromagnetic interactions. For largeJ2/J1, a phase transition between the paramagnetic phase andordered incommensurate spiral-spin structure phase occurs atfinite temperature. In the parameter region, the order parameterspace is SO(3) " C3 and a second-order phase transition withthreefold symmetry occurs [62], which differs from the resultobtained in this section. However, in frustrated spin systems,a different phase transition nature happens even when thesymmetry that is broken at the phase transition temperatureis the same as for other models. For example, in the J1-J3Heisenberg model on a triangular lattice, a first-order phasetransition with threefold symmetry breaking occurs whenJ3/J1 < #1/4 and J1 > 0. It is well known that the simplestmodel that exhibits a phase transition with threefold symmetrybreaking is the three-state ferromagnetic Potts model [76].The three-state ferromagnetic Potts model in two dimensionsexhibits a second-order phase transition. It is no wonder thatour obtained result differs from the results in the previousstudy [62].

IV. DEPENDENCE ON INTERLAYER INTERACTION

In this section, we study interlayer-interaction dependenceof the phase transition behavior. Here we set the interactionratio J3/J1 = #0.853 55 . . . at which the ground state isrepresented by k$ = !/2 in Eq. (4), as in the previoussection. In the previous section, we considered the case thatJ!/J1 = 2. We found that the first-order phase transitionwith the C3 symmetry breaking occurs and breaking of theSO(3) symmetry at the first-order phase transition point wasconfirmed.

Figure 6 shows the temperature dependence of phys-ical quantities for L = 24 with several interlayer inter-actions 0.25 ! J!/J1 ! 2.5, setting J3/J1 = #0.853 55 . . ..Figure 6(a) shows the internal energy as a function of temper-ature, which displays that the temperature at which the suddenchange of the internal energy appears increases as J!/J1increases. In other words, Fig. 6(a) indicates that the first-order phase transition temperature monotonically increasesas a function of J!/J1. In addition, the energy differencebetween the high-temperature phase and low-temperaturephase decreases as J!/J1 increases. These behaviors aresupported by the temperature dependence of the specific heatshown in Fig. 6(b). Furthermore, in the specific heat, nopeaks, except the first-order phase transition temperature, areobserved by changing the value of J!/J1. Figure 6(c) showsthe uniform magnetic susceptibility " , which is calculated by

" = NJ1%|m|2&T

, m = 1N

!

i

si , (12)

where m is the uniform magnetization. The uniform magneticsusceptibility has the sudden change at the first-order phase

0

0.1

0.2

0

0.05

0.1

0.15

0.7

0.75

0.8

0.85

0

20

40

-3

-2.5

-2

-1.5

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

(a)

(b)

(c)

(d)

(e)

0.25 0.50 0.751.00

1.251.50

1.75 2.00 2.252.50

0.25

0.50

0.751.00

1.25 1.50 1.75 2.00 2.25 2.50

0.25

0.25

0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

0.500.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

FIG. 6. (Color online) Interlayer-interaction J!/J1 dependenceof (a) internal energy per site E/J1, (b) specific heat C, (c) uniformmagnetic susceptibility " , (d) order parameter %|µ|2&, which candetect the C3 symmetry breaking, and (e) largest value of structurefactors S(k$) calculated by six wave vectors in Eq. (4) for L = 24.Error bars in all figures are omitted for clarity since their sizes aresmaller than the symbol size.

transition temperature. As stated in Sec. III, it can be usedas an indicator of the first-order phase transition. Note thatthe magnetic susceptibility of the model with J! differs fromthat with #J!. However, the sudden change in " at thefirst-order phase transition temperature is also observed whenthe interlayer interaction is antiferromagnetic. We obtain theCurie-Weiss temperature from the magnetic susceptibility forseveral J!/J1, including the case of antiferromagnetic J!,which will be shown in the Appendix. In addition, Figs. 6(d)and 6(e) confirm that phase transitions always accompany theC3 lattice rotational symmetry breaking and breaking of theglobal rotational symmetry of spin, the SO(3) symmetry, forthe considered J!/J1, respectively.

Next, in order to consider the J!/J1 dependence of thelatent heat, we calculate the probability distribution of theinternal energy P (E; Tc(L)) for several values of J!/J1 shownin Fig. 7(a). The width between bimodal peaks decreasesas J!/J1 increases. Furthermore, we calculate interlayer-interaction dependences of Tc(L)/J1 and #E(L)/J1 for L =16–40, which are shown in Figs. 7(b) and 7(c). As J!/J1increases, Tc(L)/J1 monotonically increases and #E(L)/J1decreases for each system size. In addition, #E(L)/J1increases as the system size increases. Here #E(L)/J1 in

052138-6

J�/J1 increases


0

10

20

-2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4

0

10

20

0

10

20

0 10

0 10

0 10

0 10

0 10

0 10

0 10

0

0.04

0.08

0 1 2

0.5

1

1.5

(a) (b)

(c)

FIG. 7. (Color online) (a) Interlayer-interaction J!/J1 depen-dence of the probability distribution of internal energy P (E; Tc(L))when the specific heat becomes the maximum value for L = 24.(b) The J!/J1 dependence of Tc(L)/J1 at which the specificheat becomes the maximum value for L = 16"40. (c) The J!/J1

dependence of the width between bimodal peaks of the energydistribution !E(L)/J1. Error bars in all figures are omitted for claritysince their sizes are smaller than the symbol size.

the thermodynamic limit corresponds to the latent heat. ThusFig. 7(c) suggests that the latent heat decreases as J!/J1increases in the thermodynamic limit.

V. DISCUSSION AND CONCLUSION

In this paper, the nature of the phase transition of theHeisenberg model on a stacked triangular lattice was studied byMonte Carlo simulations. In our model, there are three kinds ofinteractions: the ferromagnetic nearest-neighbor interaction J1and antiferromagnetic third nearest-neighbor interaction J3 ineach triangular layer and the ferromagnetic nearest-neighborinterlayer interaction J!. When J3/J1 < "1/4, the groundstate is a spiral-spin structure in which the C3 symmetry isbroken as in the case of two-dimensional J1-J3 Heisenbergmodel on a triangular lattice [51,52]. Then the order parameterspace in the case is described by SO(3) # C3.

In Sec. III, we studied the finite-temperature propertiesof the system with J3/J1 = "0.853 55 . . . and J!/J1 = 2.We found that a first-order phase transition takes placeat finite temperature. The temperature dependence of theorder parameter indicates that the C3 symmetry breaks atthe transition temperature, which is the same feature as inthe two-dimensional case [51,52]. We also calculated thetemperature dependence of the structure factor at the wavevector representing the ground state, which is the magneticorder parameter for spiral-spin states. The result shows thatthe SO(3) symmetry breaks at the transition temperature.

In Sec. IV, we investigated the interlayer interaction effecton the nature of phase transitions. We confirmed that thefirst-order phase transition occurs for 0.25 ! J!/J1 ! 2.5 andJ3/J1 = "0.853 55 . . ., which was used in Sec. III. We couldnot determine the existence of the first-order phase transitionfor J!/J1 < 0.25 or J!/J1 > 2.5 by Monte Carlo simulations.In the parameter ranges, the width of two peaks in the probabil-ity distribution of the internal energy cannot be estimated easilybecause of the finite-size effect. It is a remaining problem todetermine whether a second-order phase transition occurs forlarge J!/J1 as in the J1-J2 Heisenberg model on a stackedtriangular lattice [62]. As the ratio J!/J1 increases, the first-order phase transition temperature monotonically increasesbut the latent heat decreases. This is opposite to the behaviorobserved in typical unfrustrated three-dimensional systemsthat exhibit a first-order phase transition at finite temperature.For example, the q-state Potts model with ferromagneticintralayer and interlayer interactions (q " 3) is a fundamentalmodel that exhibits a temperature-induced first-order phasetransition with q-fold symmetry breaking [76]. From a mean-field analysis of the ferromagnetic Potts model [76,83], as theinterlayer interaction increases, both the transition temperatureand the latent heat increase. The same behavior was observedin the Ising-O(3) model on a stacked square lattice [77]. Asjust described, in general, if the parameter that can stabilizethe ground state becomes large, the transition temperatureincreases and the latent heat increases [76,77,83]. Furthermore,in conventional systems, both the transition temperature andthe latent heat are expressed by the value of an effectiveinteraction obtained by a characteristic temperature such asthe Curie-Weiss temperature. However, in our model, theCurie-Weiss temperature does not characterize the first-orderphase transition, as will be shown in the Appendix. Thus ourresult is an unusual behavior. The investigation of the essenceof the obtained results is a remaining problem.

ACKNOWLEDGMENTS

R.T. was partially supported by a Grand-in-Aid for Sci-entific Research (C) (Grant No. 25420698) and NationalInstitute for Materials Science. S.T. was partially supportedby a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).The computations in the present work were performed oncomputers at the Supercomputer Center, Institute for SolidState Physics, University of Tokyo.

APPENDIX: INTERLAYER-INTERACTION DEPENDENCEOF THE CURIE-WEISS TEMPERATURE

In this section, we obtain the Curie-Weiss temperature forseveral J!/J1, including the case of the antiferromagneticinterlayer interaction. Here we also use the interaction ratioJ3/J1 = "0.853 55 . . ., which was used in Secs. III and IV.As mentioned in Sec. II, the phase transition behavior ofthe model with J! is the same as that with "J!, whichis proved by the local gauge transformation. However, theCurie-Weiss temperature for J! differs from that for "J!.Figure 8(a) shows the inverse of the magnetic susceptibility""1 as a function of temperature in the high-temperatureregion for L = 24. In general, the temperature dependence of

052138-7


0

10

20

-2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4

0

10

20

0

10

20

0 10

0 10

0 10

0 10

0 10

0 10

0 10

0

0.04

0.08

0 1 2

0.5

1

1.5

(a) (b)

(c)

FIG. 7. (Color online) (a) Interlayer-interaction J!/J1 depen-dence of the probability distribution of internal energy P (E; Tc(L))when the specific heat becomes the maximum value for L = 24.(b) The J!/J1 dependence of Tc(L)/J1 at which the specificheat becomes the maximum value for L = 16"40. (c) The J!/J1

dependence of the width between bimodal peaks of the energydistribution !E(L)/J1. Error bars in all figures are omitted for claritysince their sizes are smaller than the symbol size.

the thermodynamic limit corresponds to the latent heat. ThusFig. 7(c) suggests that the latent heat decreases as J!/J1increases in the thermodynamic limit.

V. DISCUSSION AND CONCLUSION

In this paper, the nature of the phase transition of theHeisenberg model on a stacked triangular lattice was studied byMonte Carlo simulations. In our model, there are three kinds ofinteractions: the ferromagnetic nearest-neighbor interaction J1and antiferromagnetic third nearest-neighbor interaction J3 ineach triangular layer and the ferromagnetic nearest-neighborinterlayer interaction J!. When J3/J1 < "1/4, the groundstate is a spiral-spin structure in which the C3 symmetry isbroken as in the case of two-dimensional J1-J3 Heisenbergmodel on a triangular lattice [51,52]. Then the order parameterspace in the case is described by SO(3) # C3.

In Sec. III, we studied the finite-temperature propertiesof the system with J3/J1 = "0.853 55 . . . and J!/J1 = 2.We found that a first-order phase transition takes placeat finite temperature. The temperature dependence of theorder parameter indicates that the C3 symmetry breaks atthe transition temperature, which is the same feature as inthe two-dimensional case [51,52]. We also calculated thetemperature dependence of the structure factor at the wavevector representing the ground state, which is the magneticorder parameter for spiral-spin states. The result shows thatthe SO(3) symmetry breaks at the transition temperature.

In Sec. IV, we investigated the interlayer interaction effecton the nature of phase transitions. We confirmed that thefirst-order phase transition occurs for 0.25 ! J!/J1 ! 2.5 andJ3/J1 = "0.853 55 . . ., which was used in Sec. III. We couldnot determine the existence of the first-order phase transitionfor J!/J1 < 0.25 or J!/J1 > 2.5 by Monte Carlo simulations.In the parameter ranges, the width of two peaks in the probabil-ity distribution of the internal energy cannot be estimated easilybecause of the finite-size effect. It is a remaining problem todetermine whether a second-order phase transition occurs forlarge J!/J1 as in the J1-J2 Heisenberg model on a stackedtriangular lattice [62]. As the ratio J!/J1 increases, the first-order phase transition temperature monotonically increasesbut the latent heat decreases. This is opposite to the behaviorobserved in typical unfrustrated three-dimensional systemsthat exhibit a first-order phase transition at finite temperature.For example, the q-state Potts model with ferromagneticintralayer and interlayer interactions (q " 3) is a fundamentalmodel that exhibits a temperature-induced first-order phasetransition with q-fold symmetry breaking [76]. From a mean-field analysis of the ferromagnetic Potts model [76,83], as theinterlayer interaction increases, both the transition temperatureand the latent heat increase. The same behavior was observedin the Ising-O(3) model on a stacked square lattice [77]. Asjust described, in general, if the parameter that can stabilizethe ground state becomes large, the transition temperatureincreases and the latent heat increases [76,77,83]. Furthermore,in conventional systems, both the transition temperature andthe latent heat are expressed by the value of an effectiveinteraction obtained by a characteristic temperature such asthe Curie-Weiss temperature. However, in our model, theCurie-Weiss temperature does not characterize the first-orderphase transition, as will be shown in the Appendix. Thus ourresult is an unusual behavior. The investigation of the essenceof the obtained results is a remaining problem.

ACKNOWLEDGMENTS

R.T. was partially supported by a Grand-in-Aid for Sci-entific Research (C) (Grant No. 25420698) and NationalInstitute for Materials Science. S.T. was partially supportedby a Grand-in-Aid for JSPS Fellows (Grant No. 23-7601).The computations in the present work were performed oncomputers at the Supercomputer Center, Institute for SolidState Physics, University of Tokyo.

APPENDIX: INTERLAYER-INTERACTION DEPENDENCEOF THE CURIE-WEISS TEMPERATURE

In this section, we obtain the Curie-Weiss temperature forseveral J!/J1, including the case of the antiferromagneticinterlayer interaction. Here we also use the interaction ratioJ3/J1 = "0.853 55 . . ., which was used in Secs. III and IV.As mentioned in Sec. II, the phase transition behavior ofthe model with J! is the same as that with "J!, whichis proved by the local gauge transformation. However, theCurie-Weiss temperature for J! differs from that for "J!.Figure 8(a) shows the inverse of the magnetic susceptibility""1 as a function of temperature in the high-temperatureregion for L = 24. In general, the temperature dependence of

052138-7

Transition temperature

Latent heat

As the interlayer interaction increases,transition temperature increases but latent heat decreases.

ConclusionWe investigated unconventional phase transition behavior in geometrically frustrated systems.

2D 3D





Thank you for your attention!!

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