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The purpose of this work is to identify the universality class of the nonequilibrium phase transition in the two-dimensional kinetic Ising ferromagnet driven by propagating magnetic field wave. To address this issue, the finite size analysis of the nonequilibrium phase transition, in two-dimensional Ising ferromagnet driven by plane propagating magnetic wave, is studied by Monte Carlo simulation. It is observed that the system undergoes a nonequilibrium dynamic phase transition from a high temperature dynamically symmetric (
*propagating*) phase to a low temperature dynamically symmetry-broken (
*pinned*) phase as the system is cooled below the transition temperature. This transition temperature is determined precisely by studying the fourth-order Binder Cumulant of the dynamic order parameter as a function of temperature for different system sizes (
*L*). From the finite size analysis of dynamic order parameter
and the dynamic susceptibility
, we have estimated the critical exponents
and
(measured from the data read at the critical temperature obtained from Binder cumulant), and
(measured from the peak positions of dynamic susceptibility). Our results indicate that such driven Ising ferromagnet belongs to the same universality class of the two-dimensional equilibrium Ising ferromagnet (where
and
), within the limits of statistical errors.

The driven Ising ferromagnet shows interesting nonequilibrium phase transitions [

The nonequilibrium phase transitions in other magnetic models (e.g., Blume-Capel, Blume-Emery-Griffiths models etc.) driven by oscillating (in time but uniform over the space) magnetic field have been studied [

The another kind of external drive may be the magnetic field with spatio-temporal variation. The prototypes of these spatio-temporal drives are propagating or standing magnetic field waves. In the last few years, a number of investigations, on the nonequilibrium phase transitions in Ising ferromagnet driven by propagating and standing magnetic field wave, are done [

However, the detailed finite size analyses were not yet performed to know the universality class of this nonequilibrium phase transition observed in Ising ferromagnet driven by propagating magnetic field wave. This is the key issue of the present study.

In this paper, we have investigated the nonequilibrium behaviour and the finite size effect of spin- 1 2 Ising ferromagnet under the influence of propagating magnetic wave by Monte Carlo methods. The paper is organized as follows: The model and the MC simulation technique are discussed in Section II, the numerical results are reported in Section III and the paper ends with a summary in Section IV.

The time dependent Hamiltonian of a two dimensional driven Ising ferromagnet is represented by,

H ( t ) = − J Σ Σ ′ s z ( x , y , t ) s z ( x ′ , y ′ , t ) − Σ h z ( x , y , t ) s z ( x , y , t ) . (1)

Here s z ( x , y , t ) = ± 1 , is the Ising spin variable at lattice site ( x , y ) at time t. The summation Σ ′ extends over the nearest neighbour sites ( x ′ , y ′ ) of a given site ( x , y ) . J ( > 0 ) is the ferromagnetic spin-spin interaction strength between the nearest neighbour pairs of Ising spins. For simplicity, we have considered the value of J to be uniform over the whole lattice. The externally applied driving magnetic field, is denoted by h z ( x , y , t ) , at site ( x , y ) at time t. h z ( x , y , t ) has the following form for propagating magnetic wave,

h z ( x , y , t ) = h 0 cos { 2 π ( f t − x λ ) } (2)

Here h 0 and f represent the field amplitude and the frequency respectively of the propagating magnetic wave, whereas λ represents the wavelength of the wave. The wave propagates along the X-direction through the lattice.

An L × L square lattice of Ising spins is taken here as a model system. The boundary conditions applied at both directions are periodic which preserve the translational invariances in the system. Using Monte Carlo Metropolis single spin flip algorithm with parallel updating rule [

W ( ( s z ) i → ( s z ) f ) = min [ exp ( − Δ E k B T ) , 1 ] (3)

where Δ E is the energy change due to spin flip from i-th state to f-th state and k B is the Boltzman constant. In a chosen configuration, the probability of flipping of each spin is calculated from the above rule. Then prepared a list of L 2 such values of probability of flipping. On the other hand, a list of L 2 random fraction (collected from a uniformly distributed random numbers) is prepared, keeping in mind that each random fraction is associated to the probability of flipping of each spin. The spins are flipped simultaneously where the probability of flipping exceeds (or equal to) the random fraction. This is so called parallel updating of spins. Such parallel updating of L 2 spin states in an L × L square lattice constitute the unit time step and is called Monte Carlo Step per Spin (MCSS). The applied magnetic field and the temperature are measured in the units of J and J / k B respectively. The choices of such units of applied magnetic field and the temperatures are very common in the literatures [

The nonequilibrium behaviour of the two dimensional Ising ferromagnet is studied here in L × L square lattices of different sizes (L) where a propagating magnetic wave is passing through the system. The frequency (f) of magnetic field oscillation, wavelength ( λ ) of the magnetic wave and the amplitude ( h 0 ) of field strength are kept constant throughout the present study. These constant values are respectively f = 0.01 ( MCSS ) − 1 , λ = 16 lattice units and h 0 = 0.3 J . For f = 0.01 , 100 MCSS is required for a complete time cycle.

Since we have chosen the values of L in the multiple of 16, the wavelength λ = 16 is a reasonable choice. In this case, the smallest system will contain a full wave. The frequency, f = 0.01 , is chosen to have the adequate number of cycles, of the propagating magnetic field, to get a reasonable average value. The choice of the amplitude h 0 = 0.3 J is just to keep the nonequilibrium phase transition in the higher temperature range.

The finite size effect is studied by taking into account four different system sizes (within the limited computational facilities available) such as L = 16, 32, 48 and 128. The system (for any fixed value of L) has been cooled down in small steps ( Δ T = 0.005 J / k B ) from high temperature phase, i.e. the dynamical disordered phase, to reach any dynamical steady state at temperature T. The dynamical quantities are calculated when the system has achieved the nonequilibrium steady state. For this we have kept the system in constant temperature for a sufficiently long time; 12,000 (for L = 128 ) to 32,000 (for L = 16 ) cycles of magnetic oscillations and discarding the initial (or transient) 1000 cycles and taking average over the remaining cycles. We have detected a dynamical phase transition from high temperature symmetric propagating (spin bands) phase to low temperature symmetry-broken pinned phase. The dynamic Order parameter for the phase transition is defined as the average magnetisation per spin over a full cycle of external magnetic field, i.e.

Q = f × ∮ M ( t ) d t , (4)

where M ( t ) is the value of instantaneous magnetisation per spin at time t which can be obtained as

M ( t ) = 1 L 2 ∑ i = 1 L 2 s z ( x , y , t ) (5)

At very high temperature, the flipping probability of the spin, is quite high alongwith the oscillation of the magnetic field. As a result the value of the instantaneous magnetisation is almost close to zero. Consequently, by definition, the value of the dynamic order parameter is very small, thus identifying the dynamically disordered propagating phase ( Q = 0 ) (see

the spin gets reduced; also the magnetic field strength may not be adequate to flip the spins and the spins are locked or pinned in a particular orientation giving rise to a large and nearly steady value of average magnetisation. This phase is identified as the dynamically ordered or pinned phase ( Q ≠ 0 ) (see

The dynamical critical point is determined with high precision by studying the thermal variation of fourth order Binder cumulant ( U L ( T ) = 1 − 〈 Q 4 〉 L 3 〈 Q 2 〉 L 2 ) of dynamic order parameter Q for different system sizes (L).

Binder cumulants for different lattice sizes have a common intersection. Now it is known from the behaviour of the kinetic Ising model that the scaled variance of the dynamical order parameter may be regarded as the susceptibility of the system, which can be defined as follows:

χ L Q = L 2 ( 〈 Q 2 〉 − 〈 Q 〉 2 ) . (6)

〈 Q 〉 L ∝ L − β / ν (7)

χ L Q ∝ L γ / ν . (8)

It has to be noted here that though we do not have any value measured at the critical temperature which has been determined (as common intersection) from the Binder cumulant versus temperature curves for different L, the values of Q & χ Q have been read out from the respective graphs which represent the average values at any temperature. Moreover, the detailed investigations done previously [

In this study, we have mainly focused our attention on the finite size analysis and the critical aspects of the dynamic phase transition near the dynamic transition temperature of an L × L square type Ising ferromagnet driven by propagating magnetic wave. We have taken four different sizes of square lattice (L = 16, 32, 64 and 128). We have simulated the results using Monte Carlo methods using the Metropolis single spin flip algorithm with parallel updating rules. Our findings suggest that, within the limits of statistical errors obtained in this study, the estimated values of the critical exponents near the dynamic transition temperature are very close to those for the two-dimensional equilibrium Ising ferromagnet. As concluding remarks, we state that the nonequilibrium

phase transition, observed in the two-dimensional Ising ferromagnet driven by magnetic field wave, belongs to the same universality class of equilibrium two-dimensional Ising equilibrium ferro-para phase transition. Recently, the nonequilibrium phase transition in the kinetic Ising model via the violation of principle of detailed balance was studied (Manoj Kumar and ChandanDasgupta, IISc, Bangalore) and estimated the exponents in close agreement with the present observations.

MA thanks Chandan Dasgupta for helpful discussion and acknowledges financial support through FRPDF grant provided by Presidency University.

The authors declare no conflicts of interest regarding the publication of this paper.

Halder, A. and Acharyya, M. (2019) Universality Class of the Nonequilibrium Phase Transition in Two-Dimensional Ising Ferromagnet Driven by Propagating Magnetic Field Wave. Applied Mathematics, 10, 568-577. https://doi.org/10.4236/am.2019.107040