Standing magnetic wave on Ising ferromagnet: Nonequilibrium phase transition
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l Standing magnetic wave on Ising ferromagnet:Nonequilibrium phase transition.
AjayHalder ⋆ and MuktishAcharyya † Department of Physics,Presidency University86/1 College Street, Kolkata-73, India ⋆ ajay . rs @ presiuniv . ac . in † muktish . physics @ presiuniv . ac . in Abstract
The dynamical response of an Ising ferromagnet to a plane polarised standingmagnetic field wave is modelled and studied here by
Monte Carlo simulation intwo dimensions. The amplitude of standing magnetic wave is modulated along thedirection x . We have detected two main dynamical phases namely, pinned and oscillating spin clusters . Depending on the value of field amplitude the systemis found to undergo a phase transition from oscillating spin cluster to pinned asthe system is cooled down. The time averaged magnetisation over a full cycle ofmagnetic field oscillations is defined as the dynamic order parameter . The transitionis detected by studying the temperature dependences of the variance of the dynamicorder parameter, the derivative of the dynamic order parameter and the dynamicspecific heat. The dependence of the transition temperature on the magnetic fieldamplitude and on the wavelength of the magnetic field wave is studied at a singlefrequency. A comprehensive phase boundary is drawn in the plane described by thetemperature and field amplitude for two different wavelengths of the magnetic wave.The variation of instantaneous line magnetisation during a period of magnetic fieldoscillation for standing wave mode is compared to those for the propagating wavemode. Also the probability that a spin at any site, flips, is calculated. The abovementioned variations and the probability of spin flip clearly distinguish between thedynamical phases formed by propagating magnetic wave and by standing magneticwave in an Ising ferromagnet. Keywords: Standing wave, Ising Model, Metropolis rate, Monte-Carlo Simu-lation. . Introduction The study of the dynamical response of a thermodynamical system has become anactive field of research [1, 2] in recent years. Ferromagnetic system is one of some im-portant systems whose response to various kinds of driving force in equilibrium as wellas in non-equilibrium situations hold the key attention of many researchers for a longtime. A ferromagnetic system responses in a unique way to a time dependent magneticfield and studies of such dynamical responses revealed many interesting facts of somedynamical behaviour of the system. The nonequilibrium dynamic phase transition andthe hysteretic response are the main characteristic features of the ferromagnetic systemdriven by time dependent magnetic field. Some observations or studies regarding – (i) divergences of dynamic specific heat and relaxation time near transition point [3, 4], (ii) divergence of the relevant length scale near transition point [5], (iii) studies regardingexistence of tricritical point [6, 7], (iv) its relation with stochastic resonance [6], and thehysteresis loss [8] etc., establish that the dynamic phase transition is similar in many as-pects to the well known equilibrium thermodynamic phase transition. This fact is furthersupported by some experimental findings like– (i) detection of dynamic phase transitionin the ultra-thin Co film on Cu (001) system by surface magneto-optic Kerr effect [9, 10], (ii) direct excitation of propagating spin waves by focussed ultra short optical pulse [11], (iii) the transient behaviour of dynamically ordered phase in uniaxial cobalt film [12] etc.The surface and bulk transition [13] are found to be in different universality class in thedynamic transition of Ising ferromagnet driven by oscillating magnetic field. The surfacecritical behaviour is observed to differ from that of the bulk in these studies [13].Apart from the Ising model, nonequilibrium dynamic phase transition has also beenobserved in other magnetic models. The off-axial dynamic phase transition has beenobserved in the anisotropic classical Heisenberg model [14] and in the XY model [15].The multiple (surface and bulk) dynamic transition has been observed [16] in the classicalHeisenberg model. The dynamic transition has also been observed in the kinetic spin-3/2 Blume-Capel model [17] and in the Blume-Emery-Griffith model [18]. To study thedynamical phase transition in mixed spin systems also took much attention in modernresearch[19, 20, 21, 22, 23].Mainly, sinusoidally oscillating or randomly varying magnetic field, which are uniform over the space (lattice) at any instant of time has been used to study the nonequilib-rium dynamical phase transition and other characteristic behaviour in Ising magnetsand in various other magnetic models. The outcome of the above mentioned studies ofthe nonequilibrium phase transition has prompted the researchers towards the situationswhere magnetic excitations are also varied in space at any particular instant of time. Prop-agating magnetic field wave is an example of such spatially as well as temporally variedmagnetic field applied to the ferromagnetic system. This kind of variation of magneticfield is closely related to the situation where an electromagnetic wave passes through a2agnetic system. Actually, the varying ( in time as well as in space ) magnetic field wavecoupled with the spins of the ferromagnetic system affects the dynamic nature of thesystem.The nonequilibrium phase transition in Ising ferromagnet swept by propagating mag-netic field wave is studied[24]. Similar observations are obtained in the random field Isingmodel (RFIM) swept by propagating magnetic field wave [25]. A pinned phase and a phase of coherent motion of spin clusters have been observed. In RFIM the nonequilib-rium phase transition has been studied at zero temperature and is tuned by quenchedrandom (field) disorder [25].Pinned phase and propagating phase (phase of coherent motion of spin clusters) arealso observed in the two dimensional Ising ferromagnet swept by propagating magneticfield wave [26]. The transition is detected by studying the variance of the dynamic orderparameter, the derivative of the dynamic order parameter, and the dynamic specific heat which show sharp peak or dip near transition temperature. In the propagating phasespin clusters form a definite pattern which move coherently with the magnetic field wave,whereas in the pinned phase the spin clusters do not move coherently in time. Thedynamic phase transition is observed to depend upon the amplitude and wave lengthof the propagating magnetic wave. The phase boundary is found to shrink towards thelow temperature for shorter wavelengths. The relevant length scale also diverges nearthe transition. A dynamic symmetry breaking breathing and spreading transitions [27]are also recently found in Ising ferromagnet irradiated by spherical magnetic wave. Thenonequilibrium behaviour of the random field Ising ferromagnet, at zero temperature,driven by standing magnetic field wave [28] has been studied recently by Monte Carlosimulation in two dimensions using uniform, bimodal and Gaussain distributions of thequenched random fields. Depending on the values of the amplitude of standing magneticfield wave and the strength of quenched random field three distinct nonequilibrium phasesnamely, pinned, oscillating spin clusters and random are observed. These phases, thoughhave similarities, are different from those found in case of propagating magnetic wave.There has been much amount of studies done in the nonequilibrium dynamic phasetransition using the Ising ferromagnet and still considerable amount of work is going onto understand other characteristic behaviours related to such dynamic phase transitions.But in all these studies in a two dimensional Ising ferromagnet boundary conditions hasbeen kept periodic , to preserve the translational invariance.It would be interesting to know how the Ising ferromagnet, driven by standing magneticwave behaves at finite temperatures and how the difference with the propagating magneticwave can be characterised and quantified. How does the boundary affect the dynamicphase transitions?In the present study we have shown the effects of Standing magnetic wave on Isingferromagnet. The paper is organised as follows: The model and the MC simulation3echnique are discussed in Sec. II, the numerical results are reported in Sec. III and thepaper ends with a summary in Sec. IV. II. Model and Simulation
The Hamiltonian ( time dependent ) of a two dimensional Ising ferromagnet, having uni-form nearest neighbour spin-spin interaction in presence of an external standing magneticwave is represented by, H ( t ) = − J ΣΣ ′ s z ( x, y, t ) s z ( x ′ , y ′ , t ) − Σ h z ( x, y, t ) s z ( x, y, t ) (1)where s z ( x, y, t ) is the Ising spin variable ( ±
1) at lattice site ( x, y ) at time t . The sum-mation Σ ′ extends over the nearest neighbour sites ( x ′ , y ′ ) of given site ( x, y ). J ( > ferromagnetic Spin-Spin interaction strength between the nearest neighbours. It isconsidered to be uniform over the whole lattice for simplicity. h z ( x, y, t ) is the magneticfield at site ( x, y ) at time t , which has the following form of Standing wave, h z ( x, y, t ) = h sin (2 πf t ) cos (2 πx/λ ) (2)The h , f and λ represent respectively the field amplitude, the frequency and the wavelength of the standing magnetic wave. Here, the magnetic wave is assumed as linearly polarised along the direction parallel to the spins ( s z ). The modulation in amplitude of the standingmagnetic field wave is considered along the x direction only. It is worthy to mention thatthe magnetic field wave considered here is externally applied magnetic field wave and ithas no connection with the usual spin wave formed in real ferromagnets.An L × L square lattice of Ising spins is considered with open boundary conditions applied at both directions. Also antinodes of standing magnetic field wave are taken atx-boundaries. Monte Carlo Metropolis single spin flip algorithm is used for simulationof the dynamics. The initial spin configuration corresponds to high temperature randomdisordered state in which 50% of the lattice sites have spin state (+1) and the other 50%have ( − x, y ) is updated with the Metropolisprobability [29] at temperature T , given by, W ( s z → − s z ) = M in [ exp ( − ∆ E/k B T ) ,
1] (3)where ∆ E is the change in energy due to spin flip and k B is the Boltzman constant. L random updating of spin states in an L × L square lattice constitute the unit time stepcalled Monte Carlo Step per Spin (MCSS). The values of the applied magnetic field andthe temperature are measured in the units of J and J/k B , respectively. Any dynamicalstate is reached by cooling the system slowly in small steps, from the high-temperaturestate, which is the dynamically disordered state. The values of different dynamical pa-rameters at any temperature are calculated after the system achieved steady state and4nitial transient states are discarded. The system is kept at constant temperature for asufficiently long time and the average values of those parameters are taken throughoutthe time for consideration of the steady state dynamical behaviour.5 II. Results
In the present study a square lattice of size ( L = 100) is considered. The frequency ofstanding wave is taken through the study as f = 0 . M CSS − . Different field amplitude( h ) and wavelength ( λ ) of the standing magnetic field wave are considered to study thedependence of transition temperature as dependent on these parameters. Total lengthof simulation is 2 × MCSS for each temperature value. The steady state dynamicalbehaviour is studied here after discarding initial (5 × MCSS) transient data for eachtemperature value. The measured quantities are thus obtained by averaging over 15 × MCSS. Since, f = 0 . M CSS − , a full cycle requires 100 M CSS . So, in 15 × M CSS ,we have 15 × no. of cycles. All dynamical quantities are calculated by averaging over15 × cycles. Temperature is cooled in small steps of 0 . J/k B , i.e. ∆ T = 0 . J/k B ,here. This particular choice is a compromise between the computational time and theprecision in measuring the transition temperature.Two distinct phases namely, Pinned and
Oscillating spin clusters are identified in thesteady state. The pinned phase is such a phase where all the spins are almost paral-lel and remain parallel (along a fixed direction either upward or downward) due to thesmall value of the probability of spin flip. The pinned state is formed below a certaintransition temperature called the dynamic transition temperature , whereas the oscillatingspin clusters phase is formed above this temperature. In the low temperature and forsmall values of the amplitude of standing magnetic field wave, the probalibility of spinflip becomes very small, which leads to the dynamical pinned phase . These phases areshown in fig.1. In the pinned phase most of the spins are in some preferred direction i.e.either upward or downward but in oscillating spin clusters phase approximately half ofthe total spins are up and the others are down. The oscillating spin clusters phase has adefinite pattern of spins forming bands parallel to y axis. Alternate bands of up and downspins having the bandwidth λ/ / f and it bcemoes again aband of up spins after a further time interval 1 / f . In this way the spin band oscillates,forming a standing wave, instead of showing a propagation observed in earlier studies[24].For sufficiently high values of temperature and the amplitude of the standing magneticfield wave, due to the higher rate of spin flip, the system of spins effectively follow thespatio-temporal variation of applied magnetic standing wave, eventually leading to an oscillating spin bands phase.The dynamic order parameter Q for such transition is defined as the time averagedmagnetisation per site over a full cycle of the standing magnetic field oscillations , i.e. Q = fL I Z m ( x, t ) dx dt. m ( x, t ), defined as, m ( x, t ) = 1 L Z s ( x, y, t ) dy, is the average instantaneous line magnetisation per site at lattice coordinate ( x ), s ( x, y, t )being the instantaneous spin variable at lattice point ( x, y ). In the pinned phase, theorder parameter Q has non-zero value because of arrangement of spins throughout thewhole lattice whereas in the oscillating spin-clusters phase it is zero because spin-clustersare arranged in alternate values ( ± Q becomes non-zero(at lower temperature) from a zero value (at higher temperature) defining the dynamictransition.The temperature variations of the dynamic order parameter Q , the derivative of Q i.e. dQdT , the L × variance of Q i.e. L h ( δQ ) i and the dynamic specific heat C = dEdT , where E = f I {− J ΣΣ ′ s z ( x, y, t ) s z ( x ′ , y ′ , t ) } dt is the average dynamic cooperative energy per spin state of the system (without consid-ering the field energy). All the above mentioned dynamical quantities are studied for twodifferent values of field amplitude ( h = 0 . .
0) and two different values of wavelength( λ = 25 & 50) of the standing magnetic field wave (see fig.2 and fig.3). The derivativesare calculated numerically using three-point central difference formula [30]. All thesequantities are calculated statisically over 1500 different samples (i.e. cycles of standingwave). Transition is detected by the sharp peaks (for L h ( δQ ) i and C ) or dip (for dQdT )in the temperature variations of the corresponding quantities.It is evident from all the figures that the transition temperature decreases with in-crease in the field amplitude. The nature of transition looks similar to that observed inthe case of propagating magnetic field wave [26]. But there are differences in differentphases formed in both the cases. As can be seen in fig.4 that the instantaneous line mag-netisation at lattice sites ( x ), which lie between any two consecutive nodes of standingwave, oscillates coherently with different amplitudes. Whereas it propagates along withthe propagating wave. At nodes of standing magnetic wave, the amplitude of oscillationof instantaneous line magnetisation is minimum (zero) and at antinodes it is maximum.Thus, the instantaneous line magnetisation at different positions ( x ) forms loops betweenany two consecutive nodes of standing magnetic field wave. In such a case of standingmagnetic wave, spins inside a loop oscillate coherently where two nearest loops oscillatein opposite phase. At loop boundaries i.e. at nodes spins feel minimum effect of themagnetic field and thus their dynamics are more thermally driven. On the other handthe dynamics of the spins at the antinodes are governed by the field. This is shown infig.5. The probability that a spin, at any site ( x ) in the lattice will flip, depends onthe temperature and the local magnetic field strength. Since at nodes the magnetic field7trength is minimum the probability of spin flip is high. The peaks at nodes (fig.5) showthat the average probability of spin flip over a full period of magnetic oscillation is quitelarge at these sites as compared to other lattice sites. Again in case of Ising ferromagnet,driven by propagating magnetic field wave, the probability of spin flip is quite low at alllattice sites, since they all feel the same magnetic field strength over a full period. Thischaracteristic difference distinguishes between the dynamical phases in Ising ferromagnetdriven by standing magnetic wave and propagating magnetic wave.Now collecting all the values of the transition temperatures T d corresponding to differ-ent values of the magnetic field amplitudes h for a particular wavelength λ , a comprehen-sive phase boundary may be drawn. Fig.6. shows the phase boundary for two differentwavelengths λ = 25 & 50 . As can be seen from the diagrams that the phase boundaryshrinks towards the low field and low temperature values for shorter wavelength, which isconsistent with the results obtained previously with propagating and standing magneticwave using periodic boundary conditions.
IV. Summary:
The dynamical response of a two dimensional Ising ferromagnet, having open bound-aries , to the standing magnetic field wave is modelled and studied here. MonteCarlotechnique is used for simulating the observed result. In steady state, two distinct phases;namely pinned and oscillating spin clusters are observed. The pinned phase, with asym-metric and static arrangement of spins is the dynamically ordered phase having non-zerovalue of average magnetisation. The oscillating spin clusters phase consists of many par-allel band shaped spin clusters and has zero average magnetisation. As the system iscooled from high temperature to low temperature, the dynamic order parameter becomesnon-zero below a certain transition temperature. The dynamic transition seems to be ofcontinuous nature and the dynamic transition temperature ( T d ) depends on the valuesof the amplitude ( h ) and the wavelength ( λ ) of the standing magnetic field wave at asingle frequency. It should be mentioned here that the earlier studies[6], on the dynamictransition, in the Ising ferromagnet driven by oscillating (in time but uniform over space)magnetic field, reported the presence of discontinuos transition and located a tricriticalpoint on the phase diagram. Later on, the studies[7] on the distribution of dynamic orderparameter with much improved statistics showed the absence of any discontinuous tran-sition. So, to identify any tricritical behaviour (or discontinuous transition, if any), oneshould study this with much improved statistics, which is beyond the scope of our com-putational facilities. Here, we do not make any such comment on the presence/absenceof any tricritical point.Phase boundaries are drawn for two different wavelengths in T d versus h plane. Thephase boundary is observed to shrink towards asymetric phase for shorter wavelength.8nlike the propagating phase where instantaneous line magnetisation oscillates with thesame amplitude at all lattice sites, the same oscillates with different amplitudes at differentlattice sites along the standing wave. The spins flip more frequently at the nodes of thestanding wave. With open boundary condition applied to the lattice the system achievedsteady state after longer time as compared to periodic boundaries applied to the lattice[26]. Apart from this, the nature of transition is found similar to the earlier studies withperiodic boundaries applied in the case of propagating wave.It would be interesting to see the effects of Standing wave on the highly anisotropicferromagnetic thin film (Co/Ni system) experimentally by time resolved magneto-opticKerr (TRMOKE) effect.Controlling the dynamics of a group of spins by external magnetic field having aspatio-temporal variation is quite important in the branch of spintronics, magnonics inmodern condensed matter physics[31]. This present study is a simple statistical mechan-ical approach of achieving various dynamical modes of Ising ferromagnet irradiated bya standing magnetic wave, just to have a preliminary notion about the behaviour of aferromagnetic sample placed in intense optical pattern.9 . References:
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FIG.1.
Dynamical lattice morphology for frequency f = 0 .
01 & field amplitude h = 0 .
6. (a) oscillatingspin clusters phase (temp. T = 2 . T = 1 . · ) symboldenotes the up spin states. Q T (a) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ -5-4.5-4-3.5-3-2.5-2-1.5-1-0.500.5 0 0.5 1 1.5 2 2.5 3 3.5 dQdT T (b) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ L h ( δQ ) i T (c) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ C v T (d) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ FIG.2.
Temperature ( T ) variations of (a) Q , (b) dQdT , (c) L h ( δQ ) i and (d) C for two different valuesof standing magnetic field amplitude h . Here Q is the order parameter, L is the lattice size and C isthe specific heat. Symbols ( • ) & ( ∗ ) represent h = 0 . h = 1 . . M CSS − and 25 lattice units. The size of thelattice is 100 × Q T (a) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ -4-3.5-3-2.5-2-1.5-1-0.50 0 0.5 1 1.5 2 2.5 3 3.5 dQdT T (b) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ L h ( δQ ) i T (c) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ C v T (d) •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ FIG.3.
Temperature ( T ) variations of (a) Q , (b) dQdT , (c) L h ( δQ ) i and (d) C v for two different valuesof standing magnetic field amplitude h . Here Q is the order parameter, L is the lattice size and C v isthe specific heat. Symbols ( • ) & ( ∗ ) represent h = 0 . h = 1 . . M CSS − and 50 lattice units. The size of thelattice is 100 × m ( x, t ) x (a) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ -1-0.500.51 0 10 20 30 40 50 60 70 80 90 100 m ( x, t ) x (b) ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦ FIG.4.
Periodic variation of instantaneous line magnetisation m ( x, t ) at different lattice sites ( x ); ( a ) for standing wave , ( b ) for propagating wave in disordered phase, temperature T = 2 .
00 in units of
J/k B .Different symbols represent different times (+) at 199900 MCSS, ( ⋄ ) at 199925 MCSS, ( ∗ ) at 199950MCSS, ( ◦ ) at 199975 MCSS, where time period of magnetic field oscillation is 100 MCSS. The standingwave is along x axis and the propagating wave propagates along x axis. Here the values field amplitude h , frequency f and wavelength λ are 0 . J, . M CSS − & 25 lattice units respectively. .10.120.140.160.180.20.220.240.26 0 10 20 30 40 50 60 70 80 90 100 P s ( x ) x ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ FIG.5.
Probability P s ( x ) of spin flips at different lattice sites along x axis for standing magnetic wave ( • )and for propagating magnetic wave ( ∗ ) respectively in disordered phase, temperature T = 2 .
00 in unitsof
J/k B . The standing wave is along x axis and the propagating wave propagates along x axis. Here thevalues of field amplitude h , frequency f and wavelength λ are 0 . J, . M CSS − & 25 lattice units respectively. 00.511.522.53 0 0.5 1 1.5 2 2.5 h T d ( Q = 0) ( Q = 0) •••••••••••••• ∗∗ ∗ ∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ FIG.6.
Phase diagram (dynamic transition temperature T d vs. field amplitude h ) for two differentwavelength ( λ = 25 ( • ) & 50 ( ∗ )) of the standing magnetic field wave. The frequency of the standingwave is f = 0 . M CSS − ..