Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular...

Interlayer-interaction dependence of latent heat in the Heisenberg model on a stacked triangular lattice

with competing interactions

Ryo Tamura and Shu TanakaPhysical Review E 88, 052138 (2013)

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

PHYSICAL REVIEW E 88, 052138 (2013)

Interlayer-interaction dependence of latent heat in the Heisenberg modelon a stacked triangular lattice with competing interactions

Ryo Tamura1,* and Shu Tanaka2,†1International Center for Young Scientists, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-shi, Ibaraki 305-0047, Japan

2Department of Chemistry, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan(Received 12 August 2013; published 26 November 2013)

We study the phase transition behavior of a frustrated Heisenberg model on a stacked triangular lattice by MonteCarlo simulations. The model has three types of interactions: the ferromagnetic nearest-neighbor interaction J1 andantiferromagnetic third nearest-neighbor interaction J3 in each triangular layer and the ferromagnetic interlayerinteraction J!. Frustration comes from the intralayer interactions J1 and J3. We focus on the case that the orderparameter space is SO(3)"C3. We find that the model exhibits a first-order phase transition with breaking of theSO(3) and C3 symmetries at finite temperature. We also discover that the transition temperature increases butthe latent heat decreases as J!/J1 increases, which is opposite to the behavior observed in typical unfrustratedthree-dimensional systems.

DOI: 10.1103/PhysRevE.88.052138 PACS number(s): 75.10.Hk, 64.60.De, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

Geometrically frustrated systems often exhibit a charac-teristic phase transition, such as successive phase transitions,order by disorder, and a reentrant phase transition, and anunconventional ground state, such as the spin liquid state[1–20]. In frustrated continuous spin systems, the groundstate is often a noncollinear spiral-spin structure [21,22]. Thespiral-spin structure leads to exotic electronic properties suchas multiferroic phenomena [23–27], the anomalous Hall effect[28], and localization of electronic wave functions [29]. Thusthe properties of frustrated systems have attracted attentionin statistical physics and condensed matter physics. Manygeometrically frustrated systems such as stacked triangularantiferromagnets (see Fig. 1), stacked kagome antiferromag-nets, and spin-ice systems have been synthesized and theirproperties have been investigated. In theoretical studies, therelation between phase transition and order parameter space ingeometrically frustrated systems has been considered [30–34].

As an example of phase transition nature in geometricallyfrustrated systems, properties of the Heisenberg model ona triangular lattice have been theoretically studied for along time. Triangular antiferromagnetic systems are a typicalexample of geometrically frustrated systems and have beenwell investigated. The ground state of the ferromagneticHeisenberg model on a triangular lattice is a ferromagneticallycollinear spin structure. In this case, the order parameter spaceis S2. The long-range order of spins does not appear at finitetemperature because of the Mermin-Wagner theorem [35].The model does not exhibit any phase transitions. In contrast,Refs. [31,36,37] reported that a topological phase transitionoccurs in the Heisenberg model on a triangular lattice withonly antiferromagnetic nearest-neighbor interactions. In thismodel the long-range order of spins is prohibited by theMermin-Wagner theorem and thus a phase transition drivenby the long-range order of spins never occurs as well asin the ferromagnetic Heisenberg model. Since the ground

*[email protected]†[email protected]

state of the model is the 120# structure, the order parameterspace is SO(3), which is the global rotational symmetry ofspins. Thus the point defect, i.e., the Z2 = !1 [SO(3)] vortexdefect, can exist in the model. Then the topological phasetransition occurs by dissociating the Z2 vortices at finitetemperature [31,36,37]. The dissociation of Z2 vortices isone of the characteristic properties of geometrically frustratedsystems when the ground state is a noncollinear spin structurein two dimensions. In these systems, the order parameterspace is described by SO(3). The temperature dependenceof the vector chirality and that of the number density of Z2vortices in the Heisenberg model on a kagome lattice werealso studied [38]. An indication of the Z2 vortex dissociationhas been observed in electron paramagnetic resonance andelectron spin resonance measurements [39–41].

Phase transition has been studied theoretically in stackedtriangular lattice systems as well as in two-dimensionaltriangular lattice systems. In many cases, the phase transitionnature in three-dimensional systems differs from that in

axis 1

axis 2

axis 3

FIG. 1. (Color online) Schematic picture of a stacked triangularlattice with Lx " Ly " Lz sites. Here J1 and J3 respectively representthe nearest-neighbor and third-nearest-neighbor interactions in eachtriangular layer and J! is the interlayer interaction.

052138-11539-3755/2013/88(5)/052138(9) ©2013 American Physical Society

We found that a !rst-order phase transition occurs. At the !rst-order phase transition point, SO(3) and C3 symmetries are broken.

The transition temperature increases but the latent heat decreases as the interlayer interaction increases.

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

0

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0

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20

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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

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

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, S. Tanaka, and N. Kawashima, Phys. Rev. B, 87, 214401 (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

Motivation

To investigate the phase transition behavior in three-dimensional

systems whose order parameter space is described by the direct

product between two groups A x B.

Model







I. INTRODUCTION






axis 1

axis 2

axis 3



H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj si : Heisenberg spin (three components)

1st nearest-neighborintralayer

3rd nearest-neighborintralayer

1st nearest-neighborinterlayer

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

J(k)N

=� J1 cos(kx)� 2J1 cos�

12kx

�cos

��3

2ky

�

� J3 cos(2kx)� 2J3 cos(kx) cos(�

3ky)� J� cos(kz)

Fourier transform of interactions

We consider the case for .J� > 0 k�z = 0

Find that minimizes the Fourier transform of interactions! k�

R, I are two arbitrary orthogonal unit vectors.

Ferromagnetic (J1>0)

J3/J1

0-1/4

Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3)Ground-state properties


J3/J1

0-1/4

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

RYO TAMURA AND SHU TANAKA PHYSICAL REVIEW E 88, 052138 (2013)

(a) (b)

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1

(c)

FIG. 3. (Color online) Explanation of ground-state properties when the nearest-neighbor interaction J1 is ferromagnetic. (a) Position of k!,which minimizes the Fourier transform of interactions in the wave-vector space for J3/J1 ! "1/4. The hexagon represents the first Brillouinzone. A schematic of a ferromagnetic spin configuration in each triangular layer is shown. (b) Position of k! and the corresponding schematicof spiral-spin configurations in each triangular layer when J3/J1 < "1/4. The spin configurations are depicted for J3/J1 = "0.853 55 . . .

corresponding to k! = !/2 and then " = 90#. (c) The J3/J1 dependence of k!.

represented by k! = (0,2!/$

3,0) in an appropriate way,shown in Fig. 2 in Ref. [78]. Then the order parameter spaceis not well defined in this case.

Our purpose is to investigate the phase transition behaviorwhen the order parameter space is described by the directproduct between two groups. Hereafter we focus on theparameter region J3/J1 < "1/4 in the case of ferromagneticJ1. Throughout the paper, we use the interaction ratio J3/J1 ="0.853 55 . . . so that the ground state is represented byk! = !/2 in Eq. (4). In this case, along one of three axes,the relative angle between nearest-neighbor spin pairs is 90#,while along the other axes, the relative angle is 45# in theground-state spin configuration [see Fig. 3(b)]. When theperiod of the lattice is set to 8, the commensurate spiral-spinconfiguration appears in the ground state. Then, in order toavoid the incompatibility due to the boundary effect, the lineardimension L = 8n (n % N ) is used and the periodic boundaryconditions in all directions are imposed.

III. FINITE-TEMPERATURE PROPERTIESOF THE STACKED MODEL

In this section, we investigate the finite-temperature prop-erties of the Heisenberg model on a stacked triangularlattice with competing interactions given by Eq. (1) withJ3/J1 = "0.853 55 . . . and J&/J1 = 2. Using Monte Carlosimulations with the single-spin-flip heat-bath method and theoverrelaxation method [79,80], we calculate the temperaturedependence of physical quantities. Figures 4(a) and 4(b)show the internal energy per site E and specific heat C forL = 24,32,40. The specific heat at temperature T is given by

C = N'E2( " 'E(2

T 2, (6)

where 'O( denotes the equilibrium value of the physicalquantity O. Here the Boltzmann constant is set to unity. Asthe system size increases, a sudden change in the internal

energy is observed at a certain temperature. In addition, thespecific heat has a divergent single peak at the temperature.These behaviors indicate the existence of a finite-temperaturephase transition. As will be shown in Sec. IV, the uniformmagnetic susceptibility can be used as an indicator of the phasetransition. To investigate the way of ordering, the temperaturedependence of an order parameter is considered. The orderparameter µ that can detect the C3 symmetry breaking isdefined by

µ := #1e1 + #2e2 + #3e3, (7)

#$ := 1N

!

'i,j(1)axis $

si · sj , (8)

where the subscript $ ($ = 1,2,3) assigns the axis (seeFig. 1). The vectors e$ are unit vectors along the axis $

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

3/2),and e3 = ("1/2, "

$3/2). The temperature dependence of

'|µ|2( is shown in Fig. 4(c). The order parameter abruptlyincreases around the temperature at which the specific heathas a divergent peak. These results conclude that the phasetransition is accompanied by the C3 symmetry breaking.

To decide the order of the phase transition, we calculate theprobability distribution of the internal energy at T , P (E; T ) =D(E) exp("NE/T ), where D(E) is the density of states.When a system exhibits a first-order phase transition, theenergy distribution P (E; T ) should be a bimodal structureat temperature Tc(L) for system size L. Here Tc(L) is thetemperature at which the specific heat becomes the maximumvalue Cmax(L). To obtain Tc(L) and Cmax(L), we performthe reweighting method [81]. Figure 4(d) shows P (E; Tc(L))for system sizes L = 24,32,40. As stated above, the bimodalstructure in the energy distribution suggests a first-order phasetransition.

To confirm whether the first-order phase transition behaviorremains in the thermodynamic limit, we perform two types

052138-4

3-dim Heisenberg universality class


J3/J1

0-1/4

Ferromagnetic (S2) Spiral-spin structure (SO(3)xC3)RYO TAMURA AND SHU TANAKA PHYSICAL REVIEW E 88, 052138 (2013)

(a) (b)

0

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0.8

1

-4 -3 -2 -1 0 1

(c)








C = N'E2( " 'E(2

T 2, (6)



µ := #1e1 + #2e2 + #3e3, (7)

#$ := 1N

!

'i,j(1)axis $

si · sj , (8)



3/2),and e3 = ("1/2, "





052138-4


(a) (b)

0

0.2

0.4

0.6

0.8

1

-4 -3 -2 -1 0 1

(c)








C = N'E2( " 'E(2

T 2, (6)



µ := #1e1 + #2e2 + #3e3, (7)

#$ := 1N

!

'i,j(1)axis $

si · sj , (8)



3/2),and e3 = ("1/2, "





052138-4

???

Antiferromagnetic (J1<0)

J3/J1

0-1/9

Degenerated GSs 120-degree structure (SO(3))Ground-state properties

We focus on the case for ferromagnetic J1 and J3/J1 > -1/4 in which the order parameter space is described by the direct product between two groups.

chiral universalityorder by disorderH. Kawamura, J. Phys. Soc. Jpn., 54, 3220 (1985).H. Kawamura, J. Phys. Soc. Jpn., 61, 1299 (1992).A. Pelissetto et al., Phys. Rev. B, 65, 020403 (2001).P. Calabrese et al., Phys. Rev. B, 70, 174439 (2004).A. K. Murtazaev and M. K. Ramazanov, Phys. Rev. B 76, 174421 (2007).G. Zumbach, Phys. Rev. Lett. 71, 2421 (1993).M. Tissier et al., Phys. Rev. Lett. 84, 5208 (2000).M. Zelli et al., Phys. Rev. B 76, 224407 (2007).V. T. Ngo and H. T. Diep, Phys. Rev. B, 78, 031119 (2008).

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

Possible ScenariosLet us consider the system whose order parameter space is AxB.

(a) No symmetry is broken.


two-dimensional systems. In the Heisenberg model on astacked triangular lattice with the antiferromagnetic nearest-neighbor intralayer interaction J1 and the nearest-neighborinterlayer interaction J!, the ground state is a 120" structure ineach triangular layer. Thus the order parameter space is SO(3)as in the two-dimensional case. Two types of contradictoryresults have been reported. In one, a second-order phasetransition belonging to the universality class called the chiraluniversality class, which relates to the SO(3) symmetry, occurs[3,42–46]. In the other, a first-order phase transition occurs atfinite temperature [47–50]. In either case, the phase transitionnature in the Heisenberg model on a stacked triangular latticediffers from that on a two-dimensional triangular lattice.

Recently, another kind of characteristic phase transitionnature has been found in Heisenberg models on a triangularlattice with further interactions [33,34,51–53]. The orderparameter space is described by the direct product between theglobal rotational symmetry of spins SO(3) and discrete latticerotational symmetry, which depends on the ground state. Inthese models, a phase transition with the discrete symmetrybreaking occurs at finite temperature. In the J1-J3 Heisenbergmodel on a triangular lattice, the ground state is the spiral-spinstructure where C3 lattice rotational symmetry is broken due tothe competition between the ferromagnetic nearest-neighborinteraction J1 and antiferromagnetic third nearest-neighborinteraction J3 [51,52]. In this case, the order parameter spaceis SO(3)#C3. This model exhibits a first-order phase transitionwith breaking of the C3 symmetry. In addition, the dissociationof Z2 vortices that comes from the SO(3) symmetry occurs atthe first-order phase transition temperature. A similar phasetransition with the discrete symmetry breaking also has beenfound in Heisenberg models on square and hexagonal latticeswith further interactions [54–56]. To consider a microscopicmechanism of the first-order phase transition with the discretesymmetry breaking in frustrated continuous spin systems, ageneralized Potts model, called the Potts model with invisiblestates, has been studied [57–59].

As shown before, the phase transition nature in three-dimensional systems differs from that in two-dimensionalsystems even when individual order parameter spaces arethe same. Here let us review the phase transition behaviorin three-dimensional systems where the order parameter spaceis described by the direct product between two groups. Beforewe show some examples that have already been reported ina number of specific models, we consider generally whathappens in systems where the order parameter space isdescribed by the direct product between two groups A # B.In these systems there are the following possible scenariosof whether two symmetries A and B are broken at finitetemperature, which are summarized in Fig. 2: (a) Neithersymmetry is broken, (b) either A or B is broken but theother is not broken, and (c) both A and B are broken. Inthree-dimensional systems, since a breaking of continuoussymmetry at finite temperature is not prohibited in contrastto two-dimensional systems, the most possible scenario isthat in Fig. 2(c). In the case of Fig. 2(c), two scenarioscan be considered: (i) Two symmetries A and B are brokensimultaneously and (ii) A and B are broken successively.An example of case (i) is the phase transition behaviorin the antiferromagnetic XY model. The order parameter

T

T

T

T

T

T

(a)

(b)

(c)(i)

(ii)

Disordered phase

Disordered phasePartially ordered phase


Partially ordered phase

Disordered phase

Disordered phase

Ordered phase

Ordered phase

Partially ordered phase Disordered phaseOrdered phase

A is broken.

B is broken.

A and B are broken.

A is broken.B is broken.

B is broken.A is broken.

FIG. 2. (Color online) Schematic of the phase transition naturein systems where the order parameter space is the direct productbetween two groups A and B: (a) Neither symmetry is broken,(b) either A or B is broken but the other is not broken, and(c) both A and B are broken.

space is U(1) # Z2. Contradictory results were reported asfor the Heisenberg model on a stacked triangular lattice asmentioned above. Reference [43] reported that a second-order phase transition occurs at finite temperature. However,the authors in Ref. [60] concluded that a first-order phasetransition occurs. In either case, a phase transition occurs onlyonce in the model. Another example is a first-order phasetransition in the antiferromagnetic Heisenberg model on aface-centered-cubic lattice [61]. The order parameter spaceof the model is SO(3) # Z3. Moreover, in many cases, aphase transition occurs only once in systems with the orderparameter space described by the direct product between twogroups when two symmetries break at the phase transitiontemperature [32,43,60–67]. Next we show an example of(ii) where the successive phase transitions occur. The richphase diagram of the Bose-Hubbard model has been inves-tigated by many kinds of methods [68–74]. At a certainparameter region, the ordered phase is the supersolid phasein which the U(1) phase symmetry and a symmetry X definedby a commensurate wave vector are broken. Then the orderparameter space is U(1) # X in the parameter region. In theparameter region except for the tricritical point, successivephase transitions were observed [71,73,74]. Furthermore,which phase transition occurs at higher temperature dependson the parameter. Recently, successive phase transitions thatrelate to two symmetries were also found in the site-randomHeisenberg model on a three-dimensional lattice [75]. As justdescribed, a variety of phase transition natures appears inthree-dimensional systems having the order parameter spaceA # B.

052138-2

(b) Only one of two symmetries is broken.





T

T

T

T

T

T

(a)

(b)

(c)(i)

(ii)

Disordered phase




Disordered phase

Disordered phase

Ordered phase

Ordered phase


A is broken.

B is broken.

A and B are broken.





052138-2





T

T

T

T

T

T

(a)

(b)

(c)(i)

(ii)

Disordered phase




Disordered phase

Disordered phase

Ordered phase

Ordered phase


A is broken.

B is broken.

A and B are broken.





052138-2

Possible Scenarios(c) Both symmetries are broken.

(c-1) Both symmetries are broken simultaneously.





T

T

T

T

T

T

(a)

(b)

(c)(i)

(ii)

Disordered phase




Disordered phase

Disordered phase

Ordered phase

Ordered phase


A is broken.

B is broken.

A and B are broken.





052138-2

(c-2) Both symmetries are broken successively.

Internal Energy & Speci!c Heat


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


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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)

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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)

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30

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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

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0 0.5 1 1.5 2

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10 -4

10 -3

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052138-5

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 2

Internal energy Speci!c heat


0

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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)

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052138-5

Phase transition occurs.


(a) (b)

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-4 -3 -2 -1 0 1

(c)








C = N'E2( " 'E(2

T 2, (6)



µ := #1e1 + #2e2 + #3e3, (7)

#$ := 1N

!

'i,j(1)axis $

si · sj , (8)



3/2),and e3 = ("1/2, "





052138-4

Order Parameter (C3)


0

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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)

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052138-5


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30

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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 symmetry is broken.

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 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).

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

0

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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

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0 0.5 1 1.5 2

10 -5

10 -4

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052138-5

First-order phase transition occurs.

P (E;T ) = D(E)e�E/kBT

D(E) : density of states


0

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(a)

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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

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 2

�E(L) : width between two peaks

Finite-size Scaling


0

0.01

0.02

1.52 1.53 1.54 1.55

0

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20

30

40-2.3

-2.2

-2.1

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0 0.00004 0.00008

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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

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052138-5


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1.54

1.55

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0 20 40 60

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(a)

(d)

(e)

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0.1

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0

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15

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25

30

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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, Phys. Rev. B, 34, 1841 (1986).

First-order phase transition occurs.

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 2

Order Parameter (SO(3))


0

0.01

0.02

1.52 1.53 1.54 1.55

0

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40-2.3

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-2.1

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1.55

0 0.00004 0.00008

0 20 40 60

0 20000 40000 60000

(a)

(d)

(e)

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(b)

(c)

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0.1

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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

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0 0.5 1 1.5 2

10 -5

10 -4

10 -3

10 -2

10 -1


052138-5

Order parameter

SO(3) symmetry is broken at the phase transition temperature.

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 2

Dependence on Interlayer Interaction


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

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40

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(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

H = �J1

�

�i,j�1

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)

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ACKNOWLEDGMENTS




052138-7

J�/J1 increases

Dependence on Interlayer InteractionINTERLAYER-INTERACTION DEPENDENCE OF LATENT . . . PHYSICAL REVIEW E 88, 052138 (2013)

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ACKNOWLEDGMENTS




052138-7

H = �J1

�

�i,j�1

si · sj � J3

�

�i,j�3

si · sj � J��

�i,j��

si · sj

J3/J1 = �0.85355 · · · , J�/J1 = 2

Transition temperature

Latent heat

As the interlayer interaction increases, ...

transition temperature increases.

latent heat decreases.

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







I. INTRODUCTION






axis 1

axis 2

axis 3





0

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(c)








ACKNOWLEDGMENTS




052138-7





T

T

T

T

T

T

(a)

(b)

(c)(i)

(ii)

Disordered phase




Disordered phase

Disordered phase

Ordered phase

Ordered phase


A is broken.

B is broken.

A and B are broken.





052138-2

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







I. INTRODUCTION






axis 1

axis 2

axis 3




The transition temperature increases but the latent heat decreases as the interlayer interaction increases.


0

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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)








ACKNOWLEDGMENTS




052138-7

Thank you !

Ryo Tamura and Shu TanakaPhysical Review E 88, 052138 (2013)

Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular...

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