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Phase Transitions

Competing field pulse induced dynamic transition in Ising Models

Arnab Chatterjee, Bikas K. Chakrabarti

Published May 1, 2004• 14 citations
Publisher

Abstract

We discuss dynamic magnetization-reversal phenomena in the Ising model under a finite-duration external magnetic field, which competes with the existing order for . The nature of the phase boundary is estimated from the mean-field (MF) equation of motion. The susceptibility and relaxation time diverge at the MF phase boundary. A Monte Carlo study also shows divergence of the relaxation time and order-parameter fluctuations at the phase boundary. The fourth-order cumulant shows two distinct behaviors. For low temperatures and pulse durations, the value of the cumulant at the crossing point for different system sizes is far less than that for the static transition in the same dimension. This suggests a new universality class for the dynamic transition. For higher temperatures and pulse durations, the transition falls in a MF-like weak-singularity universality class.

DOI
10.1080/01411590410001672576

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We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.

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We discuss dynamic magnetization-reversal phenomena in the Ising model under a finite-duration external magnetic field, which competes with the existing order for . The nature of the phase boundary is estimated from the mean-field (MF) equation of motion. The susceptibility and relaxation time diverge at the MF phase boundary. A Monte Carlo study also shows divergence of the relaxation time and order-parameter fluctuations at the phase boundary. The fourth-order cumulant shows two distinct behaviors. For low temperatures and pulse durations, the value of the cumulant at the crossing point for different system sizes is far less than that for the static transition in the same dimension. This suggests a new universality class for the dynamic transition. For higher temperatures and pulse durations, the transition falls in a MF-like weak-singularity universality class.
The universality class of the dynamic magnetization-reversal transition, induced by a competing field pulse, in an Ising model on a square lattice, below its static ordering temperature, is studied here using Monte Carlo simulations. Fourth-order cumulant of the order parameter distribution is studied for different system sizes around the phase boundary region. The crossing point of the cumulant (for different system sizes) gives the transition point and the value of the cumulant at the transition point indicates the universality class of the transition. The cumulant value at the crossing point for low temperature and pulse width range is observed to be significantly less than that for the static transition in the same two-dimensional Ising model. The finite-size scaling behavior in this range also indicates a higher correlation length exponent value. For higher temperature and pulse width range, the transition seems to fall in a mean-field-like universality class.
We report the results of mean field and the Monte Carlo study of the dynamic magnetization-reversal transition in the Ising model, brought about by the application of an external field pulse applied in opposition to the existing order before the application of the pulse. The parameters controlling the transition are the strength hp{h}_{p}hp​ and the duration $\ensuremath{\Delta}t$ of the pulse. In the mean field case, an approximate analytical expression is obtained for the phase boundary which agrees well with that obtained numerically in the small $\ensuremath{\Delta}t$ and large TTT limit. The order parameter of the transition has been identified, and is observed to vary continuously near the transition. The order parameter exponent $\ensuremath{\beta}$ was estimated both for the mean field $(\ensuremath{\beta}=1)$ and the Monte Carlo $(\ensuremath{\beta}=0.90\ifmmode\pm\else\textpm\fi{}0.02$ in two dimensions) cases. The transition shows a ``critical slowing down'' type behavior near the phase boundary with diverging relaxation time. The divergence was found to be logarithmic in the mean field case and exponential in the Monte Carlo case. The finite size scaling technique was employed to estimate the correlation length exponent $\ensuremath{\nu}$ $(=1.5\ifmmode\pm\else\textpm\fi{}0.3$ in two dimensions) in the Monte Carlo case.
The dynamical response to a pulsed magnetic field has been studied here both using Monte Carlo simulation and by solving numerically the mean-field dynamical equation of motion for the Ising model. The ratio Rp{\mathrm{R}}_{\mathrm{p}}Rp​ of the response magnetization half-width to the width of the external field pulse has been observed to diverge and pulse susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ (ratio of the response magnetization peak height and the pulse height) gives a peak near the order-disorder transition temperature Tc{\mathrm{T}}_{\mathrm{c}}Tc​ (for the unperturbed system). The Monte Carlo results for the Ising system on a square lattice show that Rp{\mathrm{R}}_{\mathrm{p}}Rp​ diverges at Tc{\mathrm{T}}_{\mathrm{c}}Tc​, with the exponent \ensuremath{\nu}z\ensuremath{\cong}2.0, while ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ shows a peak at Tce{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}Tce​, which is a function of the field pulse width \ensuremath{\delta}t. A finite-size (in time) scaling analysis shows that Tce{\mathrm{T}}_{\mathrm{c}}^{\mathrm{e}}Tce​=Tc{\mathrm{T}}_{\mathrm{c}}Tc​+C(\ensuremath{\delta}t${)}^{\mathrm{\ensuremath{-}}1\mathrm{/}\mathrm{x}}$, with x=\ensuremath{\nu}z\ensuremath{\cong}2.0. The mean-field results show that both the divergence of R and the peak in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$ occur at the mean-field transition temperature, while the peak height in ${\mathrm{\ensuremath{\chi}}}_{\mathrm{p}}$\ensuremath{\sim}(\ensuremath{\delta}t)y{)}^{\mathrm{y}})y, y\ensuremath{\cong}1 for small values of \ensuremath{\delta}t. These results also compare well with an approximate analytical solution of the mean-field equation of motion.
The nonequilibrium dynamic phase transition, in the kinetic Ising model in the presence of an oscillating magnetic field has been studied both by Monte Carlo simulation and by solving numerically the mean-field dynamic equation of motion for the average magnetization. In both cases, the Debye ``relaxation'' behavior of the dynamic order parameter has been observed and the ``relaxation time'' is found to diverge near the dynamic transition point. The Debye relaxation of the dynamic order parameter and the power law divergence of the relaxation time have been obtained from a very approximate solution of the mean-field dynamic equation. The temperature variation of appropriately defined ``specific heat'' is studied by the Monte Carlo simulation near the transition point. The specific heat has been observed to diverge near the dynamic transition point.
We have studied, using Monte Carlo (MC) simulation for ferromagnetic Ising systems in one to four dimensions and solving numerically the mean-field (MF) equation of motion, the nature of the response magnetization m(t) of an Ising system in the presence of a periodically varying external field [h(t)=h0{\mathit{h}}_{0}h0​cos(\ensuremath{\omega}t)]. From these studies, we determine the m-h loop or hysteresis loop area A(=\ensuremath{\oint}mdh) and the dynamic order parameter Q(=\ensuremath{\oint}mdt) and investigate their variations with the frequency (\ensuremath{\omega}) and amplitude (h0{\mathit{h}}_{0}h0​) of the applied external magnetic field and the temperature (T) of the system. The variations in A are fitted to a scaling form, assumed to be valid over a wide range of parameter (\ensuremath{\omega},h0{\mathit{h}}_{0}h0​,T) values, and the best-fit exponents are obtained in all three dimensions (D=2,3,4). The scaling function is Lorentzian in the MF case and is exponentially decaying, with an initial power law, for the MC cases. The dynamic phase boundary (in the h0{\mathit{h}}_{0}h0​-T plane) is found to be frequency dependent and the transition (from Q\ensuremath{\ne}0 for low T and h0{\mathit{h}}_{0}h0​ to Q=0 for high T and h0{\mathit{h}}_{0}h0​) across the boundary crosses over from a discontinuous to a continuous one at a tricritical point. These boundaries are determined in various cases.We find that the response can be generally expressed as m(t)=P(\ensuremath{\omega}(t-${\mathrm{\ensuremath{\tau}}}_{\mathrm{eff}}$)) where P denotes a periodic function with the same frequency \ensuremath{\omega} of the perturbing field and ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$(h0{\mathit{h}}_{0}h0​,\ensuremath{\omega},T) denotes the effective delay. We established that this effective delay ${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ of the response is the crucial term and it practically determines all the above observations for A, Q, etc. Investigating the nature of the in-phase (\ensuremath{\chi}\ensuremath{'}) and the out-of-phase (\ensuremath{\chi}\ensuremath{'}\ensuremath{'}) susceptibility, defined as \ensuremath{\chi}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)cos(\ensuremath{\varphi}) and \ensuremath{\chi}\ensuremath{'}\ensuremath{'}=(m0{\mathit{m}}_{0}m0​/h0{\mathit{h}}_{0}h0​)sin(\ensuremath{\varphi}); \ensuremath{\varphi}=\ensuremath{\omega}${\mathrm{\ensuremath{\tau}}}_{\mathit{e}\mathit{f}\mathit{f}}$ [and m0{\mathit{m}}_{0}m0​ is the amplitude of m(t)], we find that the loop area A is directly given by \ensuremath{\chi}\ensuremath{'}\ensuremath{'} and also the temperature variation of \ensuremath{\chi}\ensuremath{'}\ensuremath{'}, at fixed \ensuremath{\omega} and h0{\mathit{h}}_{0}h0​, gives a prominent peak at the dynamic transition point. We have also studied the behavior of the response magnetization by the application of a short-duration (compared with the relaxation time) pulsed magnetic field.Here, we observed (both in the MC and MF cases) that the width ratio (of the half-width and the width of the response magnetization and of the pulsed field, respectively) and the susceptibility (the ratio of excess magnetization over its equilibrium value and the height of the pulsed field) both show sharp peaks at the order-disorder (ferromagnetic-paramagnetic) transition point. We have also studied similar response behavior of Ising systems in the presence of time-varying longitudinal and transverse fields, solving numerically the mean-field equation of motion. We have again studied here the nature of the dynamic phase transition and the behavior of the ac susceptibility (both longitudinal and transverse) across the dynamic phase boundary. For a short-duration pulse of the transverse field, the width ratio and the pulse susceptibility are again seen to diverge at the order-disorder transition point.
We explore the critical properties of the recently discovered finite-time dynamical phase transition in the nonequilibrium relaxation of Ising magnets after a temperature quench. The transition is characterized by a sudden switch in the relaxation dynamics and it occurs at a sharp critical time. While previous works have focused either on mean-field interactions or on investigating the critical time, we analyze the critical fluctuations at the phase transition in the nearest-neighbor Ising model on a square lattice using Monte Carlo simulations. By means of a finite-size scaling analysis, we extract the critical exponents for the transition. In two spatial dimensions, we find that the exponents are consistent with those of the two-dimensional Ising universality class when the system is initially in the vicinity of the critical point. For initial temperatures below the critical one, however, the critical exponents differ from the Ising-exponents, indicating a distinct dynamical critical phenomenon.