Commun. Theor. Phys. (Beijing, China) 46 (2006) pp. 663 667 c International Academic Publishers Vol. 46, No. 4, October 15, 2006 Monte Carlo Study of Planar Rotator Model with Weak Dzyaloshinsky Moriya Interaction SUN Yun-Zhou, LIU Hui-Ping, and YI Lin Department of Physics and State Key Laboratory of Laser Technology, Huazhong University of Science and Technology, Wuhan 430074, China (Received December 19, 2005; Revised March 21, 2006) Abstract With the help of an improvement Monte Carlo method, the Berezinskii Kosterlitz Thouless phase transition arising in two-dimensional planar rotator model with weak Dzyaloshinsky Moriya (DM) interaction is investigated. The effects of the DM interaction on specific heat, susceptibility, and magnetization are simulated. The critical temperature of transitions is determined by the so-called Binder cumulant and the susceptibility of finite-size scaling. We find that the chiral Z 2 symmetry reduced by the DM interactions plays an important role in a two-dimensional XY spin system, typically, the critical temperature is sensitive to weak DM spin couplings. PACS numbers: 07.05.Tp, 21.60.Ka, 67.57.Lm, 75.10.Hk Key words: Monte Carlo method, Berezinskii Kosterlitz Thouless phase transition, Dzyaloshinsky Moriya interaction, XY model The role of frustrating interactions in low-dimensional systems is a very important aspect in the modern field theory of magnetism in solids and molecular clusters. In particular, the presence of finite temperature phase transitions in two-dimensional systems with continuous spinrotational symmetry, which are ruled out by a naive interpretation of the Mermin and Wigner theorem, has been clearly documented during the last years and continues to attract much interest since the variety of phenomena that could be generated at low temperatures. [1,2] For recent years, there has been increasing interest in the study of the effect of anisotropy on the critical behavior, arising in magnetic systems. As an example, Dzyaloshinsky Moriya (DM) interaction between spins is such an important type of anisotropy. It has been known that DM interaction plays an important role in the study of spin glasses as well as the emergence of weak ferromagnetism observed in the low temperature orthorhombic phase of lamellar copper oxide superconductors. [3,4] On the other hand, the DM interaction has also been found to violate the crystal symmetry of quantum spin liquid system SrC u2 (BO 3 ) 2, provided magnetic field is high. [5] On the appearance of anisotropic interactions, a number of transitions occur and the system displays complex thermodynamic and magnetic characteristics that have to simulate numerically. The Berezinskii Kosterlitz Thouless (BKT) phase transition that is caused by the unbinding of vortexantivortex pairs is found to be common in a twodimensional XY model. [6 9] The DM interaction can induce an easy-plane anisotropy, from which a BKT transition is thereby expected. [10 12] In the past years, many methods such as mean-field theory, high temperature series expansion and the Monte Carlo (MC) simulation are proved to be efficient for the study of such transition phenomenon. With the development of nonlocal updated algorithms, the Monte Carlo methods are verified to be a good way for the study of various complex problems such as phase transitions. For spin systems, the MC method is usually combined with the finite-size scaling (FSS), cluster algorithms turn out to be particularly successful to the study of critical properties. The applications of the multiple and single cluster variants to different models have demonstrated that Swenden Wang (SW) multiple [9,13] and Wolff single cluster variants [14] are both good tools to simulate two-dimensional systems. In this letter, we use metropolis method combined with SW cluster algorithm to study the critical problems of planar rotator model with weak DM interaction on an L L periodic square lattice. According to the simulated results, we find that the DM effects on the phase transition is of great importance even if the DM strength is weak. The model Hamiltonian under considerations reads H = J S i S j D ( S i S j ) ij ij = J ij cos(θ i θ j ϕ), (1) where S i are two-component classical vectors of unit length, while the DM vector D is considered to be along the z-axial direction. For simplicity, J = J 1 + d2, d = D/J, and ϕ = cos 1 (J/ J). The symbols, J and D, are positive and denote the strengths of the nearest neighbors ferromagnetic coupling and DM interaction respectively. indicates the sum over the nearest neighbors. Here θ i and θ j are the spin rotation angles of the i-th and j-th sites, relative to x-axial direction, respectively. To simplify the model, we introduce a rotational transformation that takes the form θ i = θ 0 i π 2 (1 + σ i), (2) The project supported by Natural Science Foundation of Hubei Province of China under Grant No. 2003ABA004 E-mail: liuhuiping@hust.edu.cn
664 SUN Yun-Zhou, LIU Hui-Ping, and YI Lin Vol. 46 where σ i = ±1, and θi 0 denotes a trial angle at the site. Inserting Eq. (2) into Eq. (1), one has H = J ij cos(θ 0 i θ 0 j ϕ)σ i σ j = ij J ij σ i σ j, (3) where J ij = J cos(θi 0 θ0 j ϕ) is an effective Ising coupling between spins. In the MC simulations, we first take a standard Metropolis algorithm to determine whether one Metropolis step is accepted or not. Subsequently, we run SW code followed by Metropolis progress, in which σ i = 1 is chosen as the initial spin configuration. For each lattice we perform N 1.2 10 6 measurements after 1 10 4 MC steps for enough thermal equilibration. In order to avoid a correlation, measurements were taken every 4 8 MC steps. For each run, we record the time series of the energy density e = E/V and order parameter magnetization density m = M/V, where lattice size V = L L. The magnetization M chosen as an order parameter in the MC simulations can be expressed as [15] M = ( V ) 2 ( V ) 2 cos θi 0 + sin θi 0. (4) i=1 i=1 The associated susceptibility is given by χ = (M)2 M 2. (5) V k B T The average energy per site and the specific heat can be determined by e = 1 N J ij, (6) NV n=1 ij C V = E2 E 2 V k B T 2, (7) where k B is the Boltzmann constant and T indicates temperature. In our paper, the temperature, the specific heat, the energy per spin, the order parameter, and the susceptibility are measured in unites of J/k B, k B, J, gµ B S, and (gµ B S) 2 /J respectively, where g is the Lande factor, µ B is the Bohr magneton. The spin is fixed at S = 1 for simplicity. Fig. 1 The specific heat of different lattice sizes measured with d = 0.05. Errors are comparable to the symbol sizes. Fig. 2 (a) Variation of magnetization density, specific heat, and energy with temperature when L = 24, d = 0.0 and 0.08. (b) Magnetization density versus temperature for various sizes at d = 0.02. Figure 1 shows the specific heat for d = 0.05 with system size is L = 8, 16, 24, 32, where the maximum value of C V is independent of the lattice size for large enough lattice. The pseudo-transition temperature T c (L) of C V maximum has nothing to do with KT transition temperature, which is the characteristic of KT transition. The energy, order parameter magnetization density, and specific heat per spin of d = 0.08, compared with the system with-
No. 4 Monte Carlo Study of Planar Rotator Model with Weak Dzyaloshinsky Moriya Interaction 665 out DM interaction, are shown in Fig. 2(a). The major difference is that the transition temperature and the specific heat are lowered when the DM interaction appears. The order parameter magnetization density tends to zero in the high temperature phase, and with the increase of size, it has a sharp drop off as pointed in Fig. 2(b). Note that the energy has also a rather weak dependence on the size as the specific heat. Strikingly, there is difference near the pseudo-transition temperature due to the DM effects. and below T c. In this sense, the susceptibility is scaled in terms of a power form of size L, yielding χ L 2 η, (12) where η represents the exponent of spin correlation below T c. Fig. 4 Variation of the susceptibility of finite-size scaling with size, where d = 0.05. Fig. 3 Least square fit of Binder cumulant, where d = 0 and L = 8, 16, 24, 32. During the MC algorithm proceeds, we observe the change of the plane magnetization components through calculating the following term M = (M x, M y ) = S i. (8) i This offers us the important information of susceptibility components, χ α = M α 2 M α 2. (9) V k B T Subsequently, the average of χ x and χ y over the MC configurations yields χ = 1 2 (χx + χ y ). (10) The critical temperature is obtained by calculating the Binder cumulant as follows: [16,17] U L = 1 (M 2 x + M 2 y ) 2 2 M 2 x + M 2 y 2. (11) The Binder cumulant, U L, is expected to be approximately independent of lattice sizes at the critical point. According to the crossing point of U L at various sizes, a critical temperature T c can be obtained. Figure 3 shows the variation of U L with size L of model without DM interaction. In our simulations, the critical temperature is T c 0.92 at the crossing point. This value is comparable to the exact value 0.89 to 0.9 for the relevant large samples. [18 21] In Ref. [22], an efficient and precise way to find T c was provided, based on the scaling relation near For the planar rotator, transition temperature is located at the temperature where η = 1/4. Finite-size scaling of the χ data can be used to determine η(t ) by the slope of the plot ln(χ/l 2 ) versus ln(l). As an example, we show the linear fit of finite-size scaling of the susceptibility data with different temperature at d = 0.05 in Fig. 4. It is interesting that weak DM interaction can only change the basic symmetries slightly in the transition, therefore we find that the condition η = 1/4 at T c is valid for the present model. A curve of η(t ) at d = 0.0 is shown in Fig. 5(a), where the transition temperature T c 0.911. This is in agreement with the BKT temperature. [18 21] Because the correction term in Eq. (12) is neglected, the calculated value is a little higher than exact estimated value 0.894. When measured at T = 0.894, the value of η is 0.237, which is consistent with the value in Ref. [21]. Figure 5 also shows η(t ) at d = 0.02, 0.05, 0.08, separately, the transition temperatures follow as T c = 0.908, 0.896, and 0.865, corresponding to d = 0.02, 0.05, and 0.08 respectively. The simulations of transition temperature are carried and exhibit the variation of χ/l (2 η) with T, similar to the results in Ref. [22] as shown in Fig. 6. The result of T c is in excellent agreement with the calculations using by η(t ) method. The change of T c with DM interaction is also shown in the inset of Fig. 7. Strikingly, the critical temperature is varied with the DM interaction. Alternative method to find transition temperature is as pointed in Ref. [23] [25]. From Eq. (5), the maximum of susceptibility at the pseudo-transition point T c (L) is obtained. According to the finite-size scaling relation, we have π 2 T c (L) T c + 4c(ln L) 2. (13)
666 SUN Yun-Zhou, LIU Hui-Ping, and YI Lin Vol. 46 Fig. 5 (a) Variation of η with temperature when d = 0.00, 0.05, and 0.08; (b) d = 0.02. Fig. 6 Variation of χ/l (2 η) with T. (a) d = 0.0; (b) d = 0.02. Fig. 7 Calculation of critical temperature from the finite-size scaling relation (13). The inset displays the change of T c with d. Furthermore, we plot T c (L) as a function of (ln L) 2 and determine its value from the crossing point of the linear fit straight line with the y axis simply. It is noted that this method needs more MC computer time to get T c (L) exactly, while the maximum of the order susceptibility is not easy to find since the so-called critical slowing down. So the T c obtained in this way is not very exact, provided the size is too small. For instance, T c is 0.941 at d = 0.0, comparative higher than the exact value 0.894 as shown in Fig. 7. In conclusion, the critical properties of planar rotator model with weak DM interaction are studied at various reduced DM interactions with the help of the multi- MC method. The associated temperatures of the BKT transition are calculated by using different methods. It is found that thermodynamic and magnetic properties and the phase transition are affected since the DM interaction and the reduced Z 2 chirality. Acknowledgments We are indebted to professor Wang Jian-Sheng for his stimulating discussions.
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