Phase diagram of ferromagnetic XY model with nematic coupling on a triangular lattice

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Phase diagram of ferromagnetic XY model with nematic coupling on a triangular lattice

K. Qi

^a,b

, M.H. Qin

^a,ⁿ

, X.T. Jia

^c

, J.-M. Liu

^b,ⁿⁿ

aLaboratory for Advanced Materials, South China Normal University, Guangzhou 510006, China

bLaboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

cSchool of Physics and Chemistry, Henan Polytechnic University, Jiaozuo 454000, China

a r t i c l e i n f o

Article history:

Received 11 December 2012 Available online 9 April 2013 Keywords:

XYmodel

Kosterlitz–Thouless transition Monte Carlo simulation

a b s t r a c t

The phase diagram of a ferromagneticXYmodel with a nematic coupling (coupling strengthx) on a triangular lattice is studied by means of Monte Carlo simulation. The algebraic-magnetic order associated with Kosterlitz–Thouless (KT) transition is observed over the wholexrange. In the largexregion, the phase transition from the algebraic-magnetic order to the algebraic-nematic order occurs at TI. In addition, this phase transition can be scaled with the two-dimensional Ising critical exponents, demonstrating that the present system belongs to the universality class of Ising transition atTI.

1. Introduction

The two dimensional (2D)XYmodel has been well investigated for several decades due to its application in magnetic systems with planar anisotropy, quantum liquids and superconductors. As early as in 1966, it was proved that the 2DXYmodel cannot sustain long- range order even with trivial thermalﬂuctuations[1]. Alternatively, the so-called algebraic-magnetic (aM) order with Kosterlitz– Thouless (KT) transition may ensue[2,3]. After that, lots of work about theXYmodel have been reported[4–18].

On the other hand, nontrivial orders such as chiral order and nematic order in magnets, are drawing more and more attentions due to their relevancy with real magnetic materials as well as the contribution to the development of statistical mechanics. For example, a phase with coexisting nematic and vector spin chirality orders has been observed in the antiferromagneticXYmodel with a nematic (biquadratic) coupling on the triangular lattice[7]. Later on, the same phase is also reported in our earlier work where a frustratedXYmodel on the square lattice has been studied with the Monte Carlo method[8]. In fact, the ferromagneticXY model with a nematic coupling on square lattice has been studied as early as in 1989 [5]. The variations of temperature and the nematic coupling strength lead to three phases: a high-temperature disordered phase and two low temperature phases, namely, aM phase and algebraic-nematic (aN) phase. At non-zero temperatures, spin waves destroy the long-range order of the ground state, leaving power-law decay of the spin correlations. The high-temperature

phase is entered respectively via the transition associated with an integer vortex pair excitation in the aM phase and an half-integer vortex pairs one in the aN phase[9]. At the same time, it is stated that the phase transition from disordered phase to aN phase is driven by the domain wall in which the free energy is expected to decrease with increasing temperature.

In fact, the consideration of the nematic coupling terms is mostly due to the fact that they can be large for magnetic ions with large spin[19]. For example, it is identiﬁed that the nematic coupling and the ferromagnetic coupling between the nearest neighbors may play an important role in triangular lattice system NiGa2S4, as revealed most recently[20]. In this work, a ferromag- neticXYmodel with a nematic coupling (coupling strengthx) on a triangular lattice is studied by means of Monte Carlo simulation.

Besides its contribution to the development of statistical mechanics, the study may be helpful to understand the experimental phenomena observed in NiGa2S4. As far as we know, few works on such a system have been reported. It will be demon- strated that a general KT transition from the algebraically correlated phase to the paramagnetic phase occurs when temperature raises up to a critical value. For the region in which the nematic coupling is dominated, a transition from the aM phase to the aN phase occurs at the critical temperatureTIwhich is much lower thanTKT. In addition, the transition atTIhas the same universality of scaling as the two-dimensional (2D) Ising transition, which is similar to earlier report[5].

For a classicalXYspin model on a triangular lattice, we consider the following Hamiltonian which includes the nematic coupling interaction:

H¼−J1∑

½i;j

cosðθijÞ−J2∑

½i;j

cosð2θijÞ; ð1Þ

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Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2013.03.036

nCorresponding author. Tel.:+86 13632457166.

nnCorresponding author.

E-mail addresses:[email protected] (M.H. Qin), [email protected] (J.-M. Liu).

Journal of Magnetism and Magnetic Materials 340 (2013) 127–130

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where θij is the angle difference θi−θj between the nearest neighbors [i,j]. J1¼1−x is the strength of the ferromagnetic coupling, andJ2¼xis the nematic coupling strength. For deﬁnition of the energy parametersJ1andJ2, the Boltzmann constant and the lattice constant are set to unity.

Unlike the model studied in Ref. [7], our model does not contain any chiral orders due to the lack of the frustration ingredient. In the largeJ2/J1region where the nematic interaction is much stronger than the ferromagnetic one, the spins between

the nearest neighbors prefer either parallel or antiparallel with each other at the equal probabilities at low temperature, forming the possible nematic order, same as the earlier report[5].

The Monte Carlo simulation is performed on a 2DLL(L¼18, 27, 36, 45, 54, and 72) triangular lattice with period boundary conditions using the standard Metropolis algorithm [21]. The initial spin conﬁguration at high temperature (T) is totally disordered. Typically, the initial 310⁵ Monte Carlo steps are discarded for the equilibrium consideration and another 210⁵ Monte Carlo steps are retained for statistic averaging of the simulation.

The phase diagram in the x−Tplane for the model stated in Eq.(1) is shown inFig. 1. The two curves mark the boundaries between three different phases, which are the aM phase, aN phase and paramagnetic (PM) phase. An integer vortex-mediated KT transition marking the PM–aM boundary splits into a half-integer vortex-mediated KT transition which marks the PM–aN boundary, plus a transition which separates the aM order from the aN order.

It is noticed that in the most cases the critical temperatures of the KT transition and Ising transition are relatively higher than the corresponding ones[5]. This phenomenon can be easily under- stood from the point that for the systems with the same ferromagnetic coupling, one with higher coordination number shows the higher critical temperature. It is noted that for triangular system one spin interacts with six nearest neighbors rather than four for square system. So, the algebraically correlated order in triangular system is so robust and its destruction needs relatively high temperature.

0.4 0.8

0.0 0.2 0.6 1.0

0.0 0.4 0.8 1.2 1.6

T_KT T_I

T

x aM

PM

aN

Fig. 1.Calculated phase diagram for the model in Eq.(1). The high-temperature paramagnetic phase is denoted by PM, the phases with algebraic correlations in magnetic and nematic order by aM and aN respectively. The statistical errors of all the symbols are given in theTdirection.

0.0 1.0 2.0

0.0 0.8 1.6 2.4 3.2

0.0 0.5 1.0 2.0

0.5 1.5 0 1.5 2.5

1 2 3 4 5

0.00 0.02 0.04 0.06

1.26 1.28 1.30 1.32 1.34

0.00 0.02 0.04 0.06

1.38 1.44 1.50 L

18 27 36 45 54 72

Υ

T

x = 0.4

x = 0.9

x = 0.9 Υ = (3.81/π) T

L 18 27 36 45 54 72

T

Υ = (6.41/π) T

T

L^-1 L^-1

Fig. 2.Helicity modulus Y according to Eq.(2) for various sizesL(a) atx¼0.4 and (c)x¼0.9. The straight line isð2=πÞð ffiffiffi p3

=2Þð1þ3xÞT. The crossing temperatures of this line and Y for eachL⁻¹are shown in (b) forx¼0.4 and (d)x¼0.9 with the extrapolation toL⁻¹¼0.

K. Qi et al. / Journal of Magnetism and Magnetic Materials 340 (2013) 127–130 128

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The determination ofTKTis made with the helicity modulusY, also called the spin-wave stiffness [22,23]. Under this circum- stance,Ycan be deﬁned by

Y¼ J₁ 2L² ∑

½i;j

cosθij

* +

þ2J₂ L² ∑

½i;j

cos 2θij

* +

− 1 TL² J1∑

½i;jxijsinθijþ2J2∑

½i;jxijsin2θij

!2

* +

ð2Þ

Herexij¼xi−xjis the separation of two coordinate sites. For a given lattice sizeL, the critical temperatureTKTcan be determined by the crossing ofY(T) with the straight lineð2=πÞð ffiffiffi

p3

=2ÞðJ1þ4J2Þ T¼ ð2=πÞð ffiffiffi

p3

=2Þð1þ3xÞT. The helicity modulus for L¼18–72 at x¼0.4 and 0.9 are shown inFig. 2(a) and (c), and the corresponding crossing points are shown inFig. 2(b) and (d) respectively. The extrapolations to L-∞ using the polynomialﬁts yield the estimated values ofTKTwhich is 1.254(5) forx¼0.4 and 1.379(5) for x¼0.9. This method is effective in giving a good estimate ofTKT

and a more sophisticated method taking into account the loga- rithmic correction gives a similar result[24].

The critical temperature of the transition from the aM phase to the aN phase can be easily estimated from the low-temperature speciﬁc-heat peak, as stated in our earlier work[8]. Speciﬁc heatC as a function ofTatx¼0.4,x¼0.73 andx¼0.9 forL¼36 are plotted inFig. 3. It is indicated that the one single peak at smallxseparates to two independent peaks which gradually detach from each other with the increasing of x. In the low x region (xo0.65), no transition from the aM phase to the aN phase occurs, leaving the single peak in theC–Tcurves, as shown inFig. 3(a). On the other hand, the low-temperature sharp peaks at 0.96(2) forx¼0.73 and 0.36(2) for x¼0.9 clearly mark the nematic phase transitions, as shown inFig. 3(b) and (c).

InFig. 4(a), we show a snapshot of the nematic order atT¼0.4 for x¼0.95. The spins become generally parallel or antiparallel with each other, forming the so-called nematic order. At last, the critical exponent of the nematic transition is estimated with the dependence of the specific heat peak upon the absolute value of the difference between the critical temperature and its neighbor ones, i.e., Cpeak∝|Tc−T|^−α. In Fig. 4(b), we plot the specific heats under different temperatures aroundCpeakforL¼36 atx¼0.9. The linearfit givesα≈0.02 which is almost same as that of 2D Ising model, i.e.,α¼0. Taking into account the simulation errors, it is reasonable to argue that the universality class of this phase transition is that of 2D Ising transition as reported in earlier works [5,7].

To sum up, the phase diagram of ferromagneticXYmodel with nematic coupling (x) on a triangular lattice is studied in details with Monte Carlo method. The phase diagram exhibits three phases including the algebraic-magnetic phase, the algebraic- nematic phase and the paramagnetic phase. In the large region ofx(x≥0.65), an Ising transition from the aM phase to aN phase is observed in addition to the usual KT transition. This work is a complementary one to the study of 2D XY models, and may be helpful to understand the experimental phenomena observed in NiGa2S4.

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.5 1.0 1.5 2.0 2.5

1.4

0.0 0.7 2.1 2.8

0.5 1.0 1.5 2.0

x = 0.4

CV x = 0.73

T_I = 0.96(2)

x = 0.9

T TI = 0.36(2)

Fig. 3.Speciﬁc heatCas a function ofTforL¼36 at (a)x¼0.4, (b)x¼0.73 and (c)x¼0.9.

-6.0 -5.6 -5.2 0.4

0.5 0.6 0.7

lnCV

ln(|T-T_I|) x = 0.9

Fig. 4.(a) A snapshot of the nematic order atT¼0.4 atx¼0.95. (b) A scaling plot of speciﬁc heats under differentTaround critical temperature for lattice sizeL¼36 atx¼0.9.

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Acknowledgment

This work was supported by the Natural Science Foundation of China (11204091, 11274094, and 11234005), the National Key Projects for Basic Research of China (2009CB623303), China Postdoctoral Science Foundation funded project (2012T50684 and 20100480768), and the Priority Academic Program Develop- ment of Jiangsu Higher Education Institutions, China.

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