二维磁性材料中畴壁动力学的数值模拟研究
|
摘要
磁性材料中的畴壁动力学过程及其相关物理现象是近年来理论与实验研究的热点之
特别是纳米尺度畴壁的动力学问题研究已成为实现新型磁电子器件的理论基础,因而受到广泛关注。本博士论文的主要目的是用数值模拟方法研究低维小尺度磁性材料中的畴壁动力学问题。为了抓住这类小尺度系统中畴壁运动的关键特性,我们引入带有微观结构和相互作用的格点模型,应用Monte Carlo和数值求解哈密顿方程等数值模拟方法,研究畴壁运动中的运动形态、相变现象、临界行为和普适类问题,并与相关实验结果进行比较,揭示新的物理现象。
在第一章中,我们首先介绍Landau-Lifshitz-Gilbert方程和Edwards-Wilkinson方程在描述低维小尺度畴壁动力学时的局限性,并由此提出应该建立微观格点模型来描述这一尺度上的畴壁问题。我们还介绍了短时动力学概念及其在研究临界行为时的重要应用。接下来我们简单回顾无序介质中畴壁动力学的研究进展,特别是本博士论文中所关心的相关课题。在最后,我们给出本文的研究动机和目的,以及后面几章的主要研究内容。
而在第二章中,我们以二维Ising模型为例,应用Monte Carlo方法,模拟序参数守恒模型(模型B)中畴壁在有序-无序相变温度下的动力学驰豫过程。我们在远离平衡态的宏观短时区域内研究系统的动力学标度行为,发现了模型B中动力学行为明显受到准随机粒子的影响,并且测量与之相关的新临界指数。当系统中的准随机粒子尚未达到均匀状态时,通过引入准随机粒子的空间扩散长度标度lr(t),我们揭示了磁化强度与Binder累积量满足的动力学标度行为,并测量与lr(t)相关的新指数γ和η。而当准随机粒子达到均匀状态后,标度函数中不再显含lr(t),我们发现此时磁化强度与Binder累积量表现出对格点大小L的内秉依赖,并测量了标度函数中与L相关的新指数σ和ρ。另一方面,我们在模型B老化现象的自关联函数中引入准随机粒子空间标度fr(t),由此提出一般化的标度函数假设,并用模拟数据支持了这一观点。
在第三章中,引入带弱无序相互作用的微观格点模型,在有限但较低温度下研究二维系统中畴壁的蠕动行为和零场弛豫过程。应用Monte Carlo方法,我们模拟在外场驱动下二维随机场Ising模型中畴壁的蠕动行为,重点强调系统中无序强度对于蠕动行为的影响。我们发现微观格点模型能够描述蠕动行为,并在不同无序强度下都证实了速度-外场的非线性关系,测量相应的蠕动指数μ。进一步的,通过研究零场弛豫过程,测量粗糙指数ζ和能垒指数Ψ,我们发现格点模型中畴壁动力学相关指数依赖于系统的无序强度,这与QEW方程理论描述的弹性弦模型差异很大。我们认为导致这两者差异的原因是畴壁在传播过程中产生的悬凸等微观构型。通过分析系统在不同无序强度下所变现出的微观构型,我们讨论了畴壁微观构型对蠕动行为普适类的影响。
在第四章中,我们通过构造含淬火无序的Ginzburg-Landau类型模型(Φ4理论),首次从多体的基本运动方程,即确定性的哈密顿运动方程出发,研究二维磁性系统中的畴壁钉扎相变现象。并且这一理论描述方法触及了统计物理和非平衡态统计物理的基本问题。应用短时动力学方法,我们仔细测定系统的钉扎相变点和相关的静态和动力学临界指数,这些结果表明哈密顿运动方程描述的钉扎相变现象属于新的普适类,与常见的等效随机运动方程和Monte Carlo动力学方法描述的普适类不同。特别是Φ4理论中静态指数β明显大于另外两种模型,而整体粗糙指数ζ又小于1。同时,我们测量得到的局域粗糙指数ζloc<1与实验中的结果可比较。
In recent years, the dynamics of domain wall has been a focus of theoretical and experimental studies. Especially, understanding the domain-wall dynamics in nano-scale magnetic materials is regarded as one of the key to realize future spintronic devices. Therefore, the purpose of this dissertation is to study the dynamics of domain wall in low-dimensional and small-scale materials, such as thin films and nanowires. To capture the crucial feature of the domain wall in these systems, we introduce lattice models with microscopic structures and interactions. Based on the Monte Carlo method and numerical solution of the Hamiltonian equations, we investigate the critical phenomenon, the scaling behavior and the universality class of the domain wall. With the short-time dynamic approach, we also reveal the nonequilibrium scaling form and determine critical exponents during the dynamics processes.
In Chapter1, we show that due to the limitations of Landau-Lifshitz-Gilbert equation and the Edwards-Wilkinson equation in describing the low-dimensional small-scale domain-wall dynam-ics, one need to build lattice models based on microscopic structures and interactions. We also introduce the concept of the short-time dynamics and how to use it as a research approach in s-tudying the critical phenomenon. We give a brief review of the studies on domain wall dynamics in disordered media, especially, the topics we concerned in the following chapters. In the end, we manifest our motivation and the main contents discussed in this dissertation.
In Chapter2, we simulate the critical domain-wall dynamics of model B with Monte Carlo methods, taking the two-dimensional Ising model as an example. In the macroscopic short-time regime, dynamic scaling forms are revealed. Due to the existence of the quasi-random walkers, the short-time regime is separated by a time scale ts, at which the quasi-random walkers reach a "homogeneous" state. To explore the scaling form in the t In Chapter3, with Monte Carlo simulations, we study the creep motion and the zero-field relaxation of a domain wall in the two-dimensional random-field Ising model. We observe the nonlinear field-velocity relation, and determine the creep exponent μ. To further investigate the universality class of the creep motion, we also measure the roughness exponent ζ and energy barrier exponent Ψ from the zero-field relaxation process. We find that all the exponents depend on the strength of disorder, which may be induced by microscopic structures of the domain wall.
In Chapter4, based on the Hamiltonian equation of motion of the04theory with quenched dis-order, we investigate the depinning phase transition of the domain-wall motion in two-dimensional magnets. With the short-time dynamic approach, we numerically determine the transition field, and the static and dynamic critical exponents. The results show that the fundamental Hamiltonian equation of motion belongs to a universality class very different from those effective equations of motion.
特别是纳米尺度畴壁的动力学问题研究已成为实现新型磁电子器件的理论基础,因而受到广泛关注。本博士论文的主要目的是用数值模拟方法研究低维小尺度磁性材料中的畴壁动力学问题。为了抓住这类小尺度系统中畴壁运动的关键特性,我们引入带有微观结构和相互作用的格点模型,应用Monte Carlo和数值求解哈密顿方程等数值模拟方法,研究畴壁运动中的运动形态、相变现象、临界行为和普适类问题,并与相关实验结果进行比较,揭示新的物理现象。
在第一章中,我们首先介绍Landau-Lifshitz-Gilbert方程和Edwards-Wilkinson方程在描述低维小尺度畴壁动力学时的局限性,并由此提出应该建立微观格点模型来描述这一尺度上的畴壁问题。我们还介绍了短时动力学概念及其在研究临界行为时的重要应用。接下来我们简单回顾无序介质中畴壁动力学的研究进展,特别是本博士论文中所关心的相关课题。在最后,我们给出本文的研究动机和目的,以及后面几章的主要研究内容。
而在第二章中,我们以二维Ising模型为例,应用Monte Carlo方法,模拟序参数守恒模型(模型B)中畴壁在有序-无序相变温度下的动力学驰豫过程。我们在远离平衡态的宏观短时区域内研究系统的动力学标度行为,发现了模型B中动力学行为明显受到准随机粒子的影响,并且测量与之相关的新临界指数。当系统中的准随机粒子尚未达到均匀状态时,通过引入准随机粒子的空间扩散长度标度lr(t),我们揭示了磁化强度与Binder累积量满足的动力学标度行为,并测量与lr(t)相关的新指数γ和η。而当准随机粒子达到均匀状态后,标度函数中不再显含lr(t),我们发现此时磁化强度与Binder累积量表现出对格点大小L的内秉依赖,并测量了标度函数中与L相关的新指数σ和ρ。另一方面,我们在模型B老化现象的自关联函数中引入准随机粒子空间标度fr(t),由此提出一般化的标度函数假设,并用模拟数据支持了这一观点。
在第三章中,引入带弱无序相互作用的微观格点模型,在有限但较低温度下研究二维系统中畴壁的蠕动行为和零场弛豫过程。应用Monte Carlo方法,我们模拟在外场驱动下二维随机场Ising模型中畴壁的蠕动行为,重点强调系统中无序强度对于蠕动行为的影响。我们发现微观格点模型能够描述蠕动行为,并在不同无序强度下都证实了速度-外场的非线性关系,测量相应的蠕动指数μ。进一步的,通过研究零场弛豫过程,测量粗糙指数ζ和能垒指数Ψ,我们发现格点模型中畴壁动力学相关指数依赖于系统的无序强度,这与QEW方程理论描述的弹性弦模型差异很大。我们认为导致这两者差异的原因是畴壁在传播过程中产生的悬凸等微观构型。通过分析系统在不同无序强度下所变现出的微观构型,我们讨论了畴壁微观构型对蠕动行为普适类的影响。
在第四章中,我们通过构造含淬火无序的Ginzburg-Landau类型模型(Φ4理论),首次从多体的基本运动方程,即确定性的哈密顿运动方程出发,研究二维磁性系统中的畴壁钉扎相变现象。并且这一理论描述方法触及了统计物理和非平衡态统计物理的基本问题。应用短时动力学方法,我们仔细测定系统的钉扎相变点和相关的静态和动力学临界指数,这些结果表明哈密顿运动方程描述的钉扎相变现象属于新的普适类,与常见的等效随机运动方程和Monte Carlo动力学方法描述的普适类不同。特别是Φ4理论中静态指数β明显大于另外两种模型,而整体粗糙指数ζ又小于1。同时,我们测量得到的局域粗糙指数ζloc<1与实验中的结果可比较。
In recent years, the dynamics of domain wall has been a focus of theoretical and experimental studies. Especially, understanding the domain-wall dynamics in nano-scale magnetic materials is regarded as one of the key to realize future spintronic devices. Therefore, the purpose of this dissertation is to study the dynamics of domain wall in low-dimensional and small-scale materials, such as thin films and nanowires. To capture the crucial feature of the domain wall in these systems, we introduce lattice models with microscopic structures and interactions. Based on the Monte Carlo method and numerical solution of the Hamiltonian equations, we investigate the critical phenomenon, the scaling behavior and the universality class of the domain wall. With the short-time dynamic approach, we also reveal the nonequilibrium scaling form and determine critical exponents during the dynamics processes.
In Chapter1, we show that due to the limitations of Landau-Lifshitz-Gilbert equation and the Edwards-Wilkinson equation in describing the low-dimensional small-scale domain-wall dynam-ics, one need to build lattice models based on microscopic structures and interactions. We also introduce the concept of the short-time dynamics and how to use it as a research approach in s-tudying the critical phenomenon. We give a brief review of the studies on domain wall dynamics in disordered media, especially, the topics we concerned in the following chapters. In the end, we manifest our motivation and the main contents discussed in this dissertation.
In Chapter2, we simulate the critical domain-wall dynamics of model B with Monte Carlo methods, taking the two-dimensional Ising model as an example. In the macroscopic short-time regime, dynamic scaling forms are revealed. Due to the existence of the quasi-random walkers, the short-time regime is separated by a time scale ts, at which the quasi-random walkers reach a "homogeneous" state. To explore the scaling form in the t
In Chapter4, based on the Hamiltonian equation of motion of the04theory with quenched dis-order, we investigate the depinning phase transition of the domain-wall motion in two-dimensional magnets. With the short-time dynamic approach, we numerically determine the transition field, and the static and dynamic critical exponents. The results show that the fundamental Hamiltonian equation of motion belongs to a universality class very different from those effective equations of motion.
引文
[I]D. A. Allwood, Gang Xiong, M. D. Cooke, C. C. Faulkner, D. Atkinson, N. Vernier, R. P. Cowburn. Submicrometer ferromagnetic not gate and shift register[J]. Science,2002,296(5575):2003-2006.
[2]D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, R. P. Cowburn. Magnetic domain-wall logic[J]. Science,2005,309(5741):1688-1692.
[3]S.S.P. Parkin, M. Hayashi, L. Thomas. Magnetic domain-wall racetrack memory[J]. Science,2008, 320:190-194.
[4]S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, N. Ishiwata. Intrinsic threshold current density of domain wall motion in nanostrips with perpendicular magnetic anisotropy for use in low-write-current mrams, 2008.
[5]K.S. Lee, C.W. Lee, Y.J. Cho, S. Seo, D.H Kim, and S.B. Choe. Roughness exponent of domain interface in CoFe/Pt multilayer films[J]. IEEE T MAGN,2009,45:2548-2550.
[6]A. Kanda, A. Suzuki, F. Matsukura, and H. Ohno. Domain wall creep in (ga, mn) as[J]. Appl. Phys. Lett., 2010,97:032504.
[7]K.J. Kim, J.C. Lee, S.M. Ahn, K.S. Lee, C.W. Lee, Y.J. Cho, S. Seo, K.H. Shin, S.B. Choe, and H.W. Lee. Interdimensional universality of dynamic interfaces[J]. Nature,2009,458:740-742.
[8]A.B. Kolton, A. Rosso, and T. Giamarchi. Creep motion of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,94:047002-047005.
[9]H.K. Janssen, B. Schaub, B. Schmittmann. New universal short-time scaling behaviour of critical relax-ation processes[J]. Z. Phys. B,1989,73:539-549.
[10]B. Zheng. Generalized dynamic scaling for critical relaxations[J]. Phys. Rev. Lett.,1996,77:679-682.
[11]N.J. Zhou and B. Zheng. Nonequilibrium critical dynamics with domain wall and surface[J]. Phys. Rev. E,2008,77:051104-051114.
[12]Z.B. Li, L. Schulke, and B. Zheng. Dynamic monte carlo measurement of critical exponents[J]. Phys. Rev. Lett.,1995,74:3396-3398.
[13]H.J. Luo, L. Schulke, and B. Zheng. Dynamic approach to the fully frustrated xy model[J]. Phys. Rev. Lett.,1998,81:180.
[14]B.Zheng. Monte carlo simulations of short-time critical dynamics[J]. Int. J. Mod. Phys. B,1998,12:1419- 1484. Review article.
[15]B. Zheng, M. Schulz, and S. Trimper. Deterministic equations of motion and dynamic critical phenome-na[J]. Phys. Rev. Lett.,1999,82:1891.
[16]B. Zheng, F. Ren, and H. Ren. Corrections to scaling in two-dimensional dynamic xy and fully frustrated xy models[J]. Phys. Rev. E,2003,68:046120-046128.
[17]Y. Ozeki and N. Ito. Nonequilibrium relaxation analysis of fully frustrated xy models in two dimensions[J]. Phys. Rev. B,2003,68:054414.
[18]E. Granato and D. Dominguez. Resistivity scaling and critical dynamics of fully frustrated josephson-junction arrays with on-site dissipation[J]. Phys. Rev. B,2005,71:094521.
[19]H. Liu, J.P. Lv, and Q.H. Chen. Ising-like phase transition in the fully frustrated xy model with weak disorder[J]. Europhys. Lett.,2008,84:66004.
[20]J.Q. Yin, B. Zheng, and S. Trimper. Dynamic monte carlo simulations of the three-dimensional random-bond Potts model[J]. Phys. Rev. E,2005,72:036122-036130.
[21]Yukiyasu Ozeki, Keita Ogawa. Nonequilibrium relaxation analysis of gauge glass models in two dimen-sions[J]. Phys. Rev. B,2005,71:220407.
[22]Yuan Chen, Zhi-Bing Li. Short-time dynamics of spin systems with long-range correlated quenched impurities[J]. Phys. Rev. B,2005,71:174433.
[23]P. Calabrese and A. Gambassi. Ageing properties of critical systems[J]. J. Phys. A:Math. Gen.,2005, 38:R133.
[24]X.W. Lei and B. Zheng. Short-time critical dynamics and aging phenomena in the two-dimensional XY model[J]. Phys. Rev. E,2007,75:040104-040107(R).
[25]Federico Roma, Daniel Dominguez. Nonequilibrium critical dynamics of the three-dimensional gauge glass[J]. Phys. Rev. B,2008,78:184431.
[26]H.K. Lee and Y. Okabe. Reweighting for nonequilibrium markov processes using sequential importance sampling methods[J]. Phys. Rev. E,2005,71:015102(R).
[27]Everaldo Arashiro, J. R. Drugowich de Felicio, Ulrich H. E. Hansmann. Short-time dynamics of polypep-tides[J]. The Journal of Chemical Physics,2007,126(4):045107.
[28]Shuangli Fan, Fan Zhong. Determination of the dynamic and static critical exponents of the two-dimensional three-state potts model using linearly varying temperature [J]. Phys. Rev. E,2007,76:041141.
[29]S.Z. Lin and B. Zheng. Short-time critical dynamics at perfect and imperfect surfaces[J]. Phys. Rev. E, 2008,78:011127-011137.
[30]Gabriel Baglietto, Ezequiel V. Albano. Finite-size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self-driven individuals[J]. Phys. Rev. E,2008, 78:021125.
[31]R.H. Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in monte carlo simulations[J]. Phys. Rev. Lett.,1987,58:86-88.
[32]F. Wang and D.P. Landau. Efficient multiple-range random walk algorithm to calculate the density of states[J]. Phys. Rev. Lett.,2001,86:2050-2053.
[33]U. Wolff. Collective monte carlo updating for spin systems[J]. Phys. Rev. Lett.,1989,62:361-364.
[34]A.J. Bray. Theory of phase-ordering kinetics[J]. Advances in Physics,1994,43(3):357-459.
[35]M. Jost and K.D. Usadel. Interface roughening in driven magnetic systems with quenched disorder[J]. Phys. Rev. B,1996,54:9314-9321.
[36]N.J. Zhou and B. Zheng. Non-equilibrium critical dynamics with domain interface[J]. Europhys. Lett., 2007,78:56001-56004.
[37]S.J. He, G.L.M.K.S. Kahanda, and P.Z. Wong. Roughness of wetting fluid invasion fronts in porous media[J]. Phys. Rev. Lett.,1992,69:3731-3734.
[38]S. Lemerle, J. Ferre, C. Chappert, V. Mathet, T. Giamarchi, and P. Le Doussal. Domain wall creep in an Ising ultrathin magnetic film[J]. Phys. Rev. Lett.,1998,80:849-852.
[39]T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo. Propagation of a magnetic domain wall in a submicrometer magnetic wire[J]. Science,1999,284:468.
[40]S. Moulinet, A. Rosso, W. Krauth, and E. Rolley. Width distribution of contact lines on a disordered substrate[J]. Phys. Rev. E,2004,69:035103-035106(R).
[41]M. Yamanouchi, J. Ieda, F. Matsukura, S.E. Barnes, S. Maekawa, and H. Ohno. Universality classes for domain wall motion in the ferromagnetic semiconductor (Ga,Mn)As[J]. Science,2007,317:1726-1729.
[42]M.Y. Im, L. Bocklage, P. Fischer, and G. Meier. Direct observation of stochastic domain-wall depinning in magnetic nanowires[J]. Phys. Rev. Lett.,2009,102:147204.
[43]A.B. Kolton, G. Schehr, and P.L. Doussal. Universal nonstationary dynamics at the depinning transition[J]. Phys. Rev. Lett.,2009,103:160602-160606.
[44]Daniel S. Fisher. Sliding charge-density waves as a dynamic critical phenomenon[J]. Phys. Rev. B,1985, 31:1396-1427.
[45]A.B. Kolton, A. Rosso, E.V. Albano, and T. Giamarchi. Short-time relaxation of a driven elastic string in a random medium[J]. Phys. Rev. B,2006,74:140201-140204(R).
[46]P. Le Doussal, K.J. Wiese, and P. Chauve. Two-loop functional renormalization group theory of the depinning transition[J]. Phys. Rev. B,2002,66:174201-174234.
[47]A. Rosso and W. Krauth. Origin of the roughness exponent in elastic strings at the depinning threshold[J]. Phys. Rev. Lett.,2001,87:187002-187005.
[48]N.J. Zhou, B. Zheng, and Y.Y. He. Short-time domain-wall dynamics in the random-field Ising model with a driving field[J]. Phys. Rev. B,2009,80:134425-134431.
[49]N.J. Zhou, B. Zheng, and D.P. Landau. Relaxation-to-creep transition of domain-wall motion in a two-dimensional random-field Ising model with an ac driving field[J]. Europhys. Lett.,2010,92:36001-36005.
[50]P. Chauve, T. Giamarchi, and P. Le Doussal. Creep and depinning in disordered media[J]. Phys. Rev. B, 2000,62:6241-6267.
[51]F. Cayssol, D. Ravelosona, C. Chappert, J. Ferre, J. P. Jamet. Domain wall creep in magnetic wires[J]. Phys. Rev. Lett.,2004,92:107202.
[52]V. Repain, M. Bauer, J.-P. Jamet, J. Ferr茅, A. Mougin, C. Chappert, H. Bernas. Creep motion of a magnetic wall:Avalanche size divergence[J]. EPL (Europhysics Letters),2004,68(3):460.
[53]P. Paruch, T. Giamarchi, and J.M. Triscone. Domain wall roughness in epitaxial ferroelectric PbZr0.2Ti0.aO3 thin films[J]. Phys. Rev. Lett.,2005,94:197601.
[54]A.B. Kolton, A. Rosso, and T. Giamarchi. Nonequilibrium relaxation of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,95:180604.
[55]J. D. Noh and H. Park. Relaxation dynamics of an elastic string in random media[J]. Phys. Rev. E,2009, 80:040102(R).
[56]Madan Rao, Amitabha Chakrabarti. Kinetics of domain growth in a random-field model in three dimen-sions[J]. Phys. Rev. Lett.,1993,71:3501-3504.
[57]Raja Paul, Sanjay Puri, Heiko Rieger. Domain growth in ising systems with quenched disorder[J]. Phys. Rev. E,2005,71:061109.
[58]Raja Paul, Gregory Schehr, Heiko Rieger. Superaging in two-dimensional random ferromagnets[J]. Phys. Rev. E,2007,75:030104.
[59]Malte Henkel, Michel Pleimling. Superuniversality in phase-ordering disordered ferromagnets[J]. Phys. Rev. B,2008,78:224419.
[60]Hyunhang Park, Michel Pleimling. Aging in coarsening diluted ferromagnets[J]. Phys. Rev. B,2010, 82:144406.
[61]P. W. ANDERSON, Y. B. KIM. Hard superconductivity:Theory of the motion of abrikosov flux lines[J]. Rev. Mod. Phys.,1964,36:39-43.
[62]T. Tybell, P. Paruch, T. Giamarchi, and J.M. Triscone. Domain wall creep in epitaxial ferroelectric pbzr0.2ti0.8O3 thin films[J]. Phys. Rev. Lett.,2002,89:097601.
[63]L B Ioffe, V M Vinokur. Dynamics of interfaces and dislocations in disordered media[J]. Journal of Physics C:Solid State Physics,1987,20(36):6149.
[64]T. Nattermann. Interface roughening in systems with quenched random impurities [J]. EPL (Europhysics Letters),1987,4(11):1241.
[65]M. V. Feigel' man, V. B. Geshkenbein, A. I. Larkin, V. M. Vinokur. Theory of collective flux creep[J]. Phys. Rev. Lett.,1989,63:2303-2306.
[66]T. Nattermann, Y. Shapir, and I. Vilfan. Interface pinning and dynamics in random systems[J]. Phys. Rev. B,1990,42:8577-8586.
[67]J.Y. Jo, S.M. Yang, T.H. Kim, H.N. Lee, J.G. Yoon, S. Park, Y. Jo, M.H. Jung, and T.W. Noh. Nonlin-ear dynamics of domain-wall propagation in epitaxial ferroelectric thin films[J]. Phys. Rev. Lett.,2009, 102:045701.
[68]J.-C. Lee, K.-J. Kim, J. Ryu, K.-W. Moon, S.-J. Yun, G.-H. Gim, K.-S. Lee, K.-H. Shin, H.-W. Lee, and S.-B. Choe. Universality classes of magnetic domain wall motion[J]. Phys. Rev. Lett.,2011,107:067201.
[69]P. Chauve, T. Giamarchi, P. Le Doussal. Creep via dynamical functional renormalization group[J]. EPL (Europhysics Letters),1998,44(1):110.
[70]T. Shibauchi, L. Krusin-Elbaum, V. M. Vinokur, B. Argyle, D. Weller, B. D. Terris. Deroughening of a 1d domain wall in an ultrathin magnetic film by a correlated defect[J]. Phys. Rev. Lett.,2001,87:267201.
[71]A.B. Kolton, A. Rosso, and T. Giamarchi. Creep motion of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,94:047002.
[72]P.C. Hohenberg and B.I. Halperin. Theory of dynamic critical phenomena[J]. Rev. Mod. Phys.,1977, 49:435-479.
[73]B. Zheng. Monte carlo simulations and numerical solutions of short-time critical dynamics[J]. Physica, 2000, A283:80.
[74]B. Zheng and S. Trimper. Comment on "Universal Fluctuations in Correlated Systems"[J]. Phys. Rev. Lett.,2001,87:188901.
[75]Kyozi Kawasaki. Diffusion constants near the critical point for time-dependent ising models. i[J]. Phys. Rev.,1966,145:224-230.
[76]A. J. Bray. Theory of phase-ordering kinetics[J]. Advances in Physics,2002,51(2):481-587.
[77]Ezequiel V. Albano, Gustavo Saracco. Dynamic behavior of anisotropic nonequilibrium driving lattice gases[J]. Phys. Rev. Lett.,2002,88:145701.
[78]Sergio Caracciolo, Andrea Gambassi, Massimiliano Gubinelli, Andrea Pelissetto. Critical behavior of the two-dimensional randomly driven lattice gas[J]. Phys. Rev. E,2005,72:056111.
[79]Thomas H. R. Smith, Oleg Vasilyev, Douglas B. Abraham, Anna Maciolek, Matthias Schmidt. Interfaces in driven ising models:Shear enhances confinement[J]. Phys. Rev. Lett.,2008,101:067203.
[80]Sergio Caracciolo, Andrea Gambassi, Massimiliano Gubinelli, Andrea Pelissetto. Comment on "dynamic behavior of anisotropic nonequilibrium driving lattice gases"[J]. Phys. Rev. Lett.,2004,92:029601.
[81]E. V. Albano, G. Saracco. Albano and saracco reply:[J]. Phys. Rev. Lett.,2004,92:029602.
[82]F. J. Alexander, D. A. Huse, S. A. Janowsky. Dynamical scaling and decay of correlations for spinodal decomposition at tc[i]. Phys. Rev. B,1994,50:663-667.
[83]Satya N. Majumdar, David A. Huse, Boris D. Lubachevsky. Growth of long-range correlations after a quench in conserved-order-parameter systems[J]. Phys. Rev. Lett.,1994,73:182-185.
[84]Claude Godreche, Florent Krzakala, Federico Ricci-Tersenghi. Non-equilibrium critical dynamics of the ferromagnetic ising model with kawasaki dynamics[J]. Journal of Statistical Mechanics:Theory and Experiment,2004,2004(04):P04007.
[85]Clement Sire. Autocorrelation exponent of conserved spin systems in the scaling regime following a critical quench[J]. Phys. Rev. Lett.,2004,93:130602.
[86]Y.Y. He, B. Zheng, and N.J. Zhou. Logarithmic correction to scaling in domain-wall dynamics at kosterlitz-thouless phase transitions[J]. Phys. Rev. E,2009,79:021107-021114.
[87]T. Nattermann, V. Pokrovsky, and V.M. Vinokur. Hysteretic dynamics of domain walls at finite tempera-tures[J]. Phys. Rev. Lett.,2001,87:197005-197008.
[88]W. Kleemann. Universal domain wall dynamics in disordered ferroic materials[J]. Annu. Rev. Mater. Res.,2007,37:415-448.
[89]Satya N. Majumdar, David A. Huse. Growth of long-range correlations after a quench in phase-ordering systems[J]. Phys. Rev. E,1995,52:270-284.
[90]Chuck Yeung, Madan Rao, Rashmi C. Desai. Bounds on the decay of the autocorrelation in phase ordering dynamics[J]. Phys. Rev. E,1996,53:3073-3077.
[91]J. F. Marko, G. T. Barkema. Phase ordering in the ising model with conserved spin[J]. Phys. Rev. E,1995, 52:2522-2534.
[92]Tomoyuki Nagaya, Jean-Marc Gilli. Experimental study of spinodal decomposition in a Id conserved order parameter system[J]. Phys. Rev. Lett.,2004,92:145504.
[93]P.J. Metaxas, J.P. Jamet, A. Mougin, M. Cormier, J. Ferre, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps. Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy[J]. Phys. Rev. Lett.,2007,99:217208-217213.
[94]Y.H. Shin, I. Grinberg, I.W. Chen, and A.M. Rappe. Nucleation and growth mechanism of ferroelectric domain-wall motion[J]. Nature,2007,449:881-886.
[95]A. Dourlat, V. Jeudy, A. Lemaitre, and C. Gourdon. Field-driven domain-wall dynamics in (Ga,Mn)As films with perpendicular anisotropy[J]. Phys. Rev. B,2008,78:161303-161306(R).
[96]U. Nowak and K. D. Usadel. Influence of temperature on the depinning transition of driven interfaces[J]. Europhys. Lett.,1998,44:634-640.
[97]O. Duemmer and W. Krauth. Critical exponents of the driven elastic string in a disordered medium[J]. Phys. Rev. E,2005,71:061601-061604.
[98]A.B. Kolton, A. Rosso, T. Giamarchi, and W. Krauth. Dynamics below the depinning threshold in disor-dered elastic systems[J]. Phys. Rev. Lett.,2006,97:057001-057004.
[99]B. Bako, D. Weygand, M. Samaras, W. Hoffelner, and M. Zaiser. Dislocation depinning transition in a dispersion-strengthened steel [J]. Phys. Rev. B,2008,78:144104-144108.
[100]A.B. Kolton, A. Rosso, and T. Giamarchi. Nonequilibrium relaxation of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,95:180604-180607.
[101]L. Roters, S. Liibeck, and K. D. Usadel. Creep motion in a random-field Ising model[J]. Phys. Rev. E, 2001,63:026113-026117.
[102]N.J. Zhou and B. Zheng. Dynamic effect of overhangs and islands at the depinning transition in two-dimensional magnets[J]. Phys. Rev. E,2010,82:031139-031147.
[103]W. Kleemann, J. Rhensius, O. Petracic, J. Ferre, J.P. Jamet, and H. Bernas. Modes of periodic domain-wall motion in ultrathin ferromagnetic layers[J], Phys. Rev. Lett.,2007,99:097203-097206.
[104]N. A. Pertsev, A. Petraru, H. Kohlstedt, R. Waser, I. K. Bdikin, D. Kiselev, and A. L. Kholkin. Dynam-ics of ferroelectric nanodomains in BaTiO3 epitaxial thin films via piezoresponse force microscopy[J]. Nanotechnology,2008,19:375703.
[105]A.B. Kolton, A. Rosso, T. Giamarchi, and W. Krauth. Creep dynamics of elastic manifolds via exact transition pathways[J]. Phys. Rev. B,2009,79:184207-184219.
[106]Cecile Monthus and Thomas Garel. Driven interfaces in random media at finite temperature:Existence of an anomalous zero-velocity phase at small external force [J]. Phys. Rev. E,2008,78:041138-041147.
[107]H. N. Yang and T. M. Lu. Inconsistency between height-height correlation and power-spectrum functions of scale-invariant surfaces for roughness exponent α～1 [J]. Phys. Rev. E,1995,51:2479.
[108]J. F. Fernandez, G. Grinstein, Y. Imry, S. Kirkpatrick. Numerical evidence for dc=2 in the random-field ising model[J]. Phys. Rev. Lett.,1983,51:203-206.
[109]Scott R. Anderson. Growth and equilibration in the two-dimensional random-field ising model[J]. Phys. Rev. B,1987,36:8435-8446.
[110]L. Roters, A. Hucht, S. Liibeck, U. Nowak, and K.D. Usadel. Depinning transition and thermal fluctuations in the random-field ising model[J]. Phys. Rev. E,1999,60:5202-5207.
[111]Daniel S. Fisher. Interface fluctuations in disordered systems:5-f expansion and failure of dimensional reduction[J]. Phys. Rev. Lett.,1986,56:1964-1967.
[112]Hong Ji, Mark O. Robbins. Transition from compact to self-similar growth in disordered systems:Fluid invasion and magnetic-domain growth[J]. Phys. Rev. A,1991,44:2538-2542.
[113]X P Qin, B Zheng, N J Zhou. Universality class of the depinning transition in the two-dimensional ising model with quenched disorder[J]. Journal of Physics A:Mathematical and Theoretical,2012, 45(11):115001.
[114]David A. Huse, Christopher L. Henley. Pinning and roughening of domain walls in ising systems due to random impurities[J]. Phys. Rev. Lett.,1985,54:2708-2711.
[115]Mehran Kardar. Roughening by impurities at finite temperatures[J]. Phys. Rev. Lett.,1985,55:2923-2923.
[116]R.H. Dong, B. Zheng, and N.J. Zhou. Critical domain-wall dynamics of model b[J]. Phys. Rev. E,2009, 79:051125-051128.
[117]R. H. Dong, B. Zheng, N. J. Zhou. Creep motion of a domain wall in the two-dimensional random-field ising model with a driving field[J]. EPL (Europhysics Letters),2012,98(3):36002.
[118]Joseph Ford. The fermi-pasta-ulam problem:Paradox turns discovery[J]. Physics Reports,1992, 213(5):271-310.
[119]Dominique Escande, Holger Kantz, Roberto Livi, Stefano Ruffo. Self-consistent check of the valid-ity of gibbs calculus using dynamical variables[J]. Journal of Statistical Physics,1994,76:605-626. 10.1007/BF02188677.
[120]Mickael Antoni, Stefano Ruffo. Clustering and relaxation in hamiltonian long-range dynamics[J]. Phys. Rev. E,1995,52:2361-2374.
[121]Lando Caiani, Lapo Casetti, Marco Pettini. Hamiltonian dynamics of the two-dimensional lattice mod-el[J]. Journal of Physics A:Mathematical and General,1998,31(15):3357.
[122]Lando Caiani, Lapo Casetti, Cecilia Clementi, Giulio Pettini, Marco Pettini, Raoul Gatto. Geometry of dynamics and phase transitions in classical lattice φ4 theories[J]. Phys. Rev. E,1998,57:3886-3899.
[123]Xavier Leoncini, Alberto D. Verga, Stefano Ruffo. Hamiltonian dynamics and the phase transition of the XY model[J]. Phys. Rev. E,1998,57:6377-6389.
[124]B. Zheng. Deterministic equations of motion and phase ordering dynamics[J]. Phys. Rev. E,2000,61:153.
[125]B. Zheng. Deterministic equations of motion and phase ordering dynamics[J]. Phys. Rev. E,2000,61:153-156.
[126]M. Jost and K.D. Usadel. Interface roughening in driven magnetic systems with quenched disorder. In Chaos and Fractals in Chemical Engineering. World Scientific,1997.
[127]M. Jost, J. Heimel and T. Kleinefeld. Domain-wall roughening in Co_(28)Pt_(72) alloy films[J]. Phys. Rev. B,1998,57:5316-5319.
[128]P. Cordoba-Torres, T. J. Mesquita, I. N. Bastos, R. P. Nogueira. Complex dynamics during metal dissolu-tion:From intrinsic to faceted anomalous scaling[J]. Phys. Rev. Lett.,2009,102:055504.
[2]D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, R. P. Cowburn. Magnetic domain-wall logic[J]. Science,2005,309(5741):1688-1692.
[3]S.S.P. Parkin, M. Hayashi, L. Thomas. Magnetic domain-wall racetrack memory[J]. Science,2008, 320:190-194.
[4]S. Fukami, T. Suzuki, N. Ohshima, K. Nagahara, N. Ishiwata. Intrinsic threshold current density of domain wall motion in nanostrips with perpendicular magnetic anisotropy for use in low-write-current mrams, 2008.
[5]K.S. Lee, C.W. Lee, Y.J. Cho, S. Seo, D.H Kim, and S.B. Choe. Roughness exponent of domain interface in CoFe/Pt multilayer films[J]. IEEE T MAGN,2009,45:2548-2550.
[6]A. Kanda, A. Suzuki, F. Matsukura, and H. Ohno. Domain wall creep in (ga, mn) as[J]. Appl. Phys. Lett., 2010,97:032504.
[7]K.J. Kim, J.C. Lee, S.M. Ahn, K.S. Lee, C.W. Lee, Y.J. Cho, S. Seo, K.H. Shin, S.B. Choe, and H.W. Lee. Interdimensional universality of dynamic interfaces[J]. Nature,2009,458:740-742.
[8]A.B. Kolton, A. Rosso, and T. Giamarchi. Creep motion of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,94:047002-047005.
[9]H.K. Janssen, B. Schaub, B. Schmittmann. New universal short-time scaling behaviour of critical relax-ation processes[J]. Z. Phys. B,1989,73:539-549.
[10]B. Zheng. Generalized dynamic scaling for critical relaxations[J]. Phys. Rev. Lett.,1996,77:679-682.
[11]N.J. Zhou and B. Zheng. Nonequilibrium critical dynamics with domain wall and surface[J]. Phys. Rev. E,2008,77:051104-051114.
[12]Z.B. Li, L. Schulke, and B. Zheng. Dynamic monte carlo measurement of critical exponents[J]. Phys. Rev. Lett.,1995,74:3396-3398.
[13]H.J. Luo, L. Schulke, and B. Zheng. Dynamic approach to the fully frustrated xy model[J]. Phys. Rev. Lett.,1998,81:180.
[14]B.Zheng. Monte carlo simulations of short-time critical dynamics[J]. Int. J. Mod. Phys. B,1998,12:1419- 1484. Review article.
[15]B. Zheng, M. Schulz, and S. Trimper. Deterministic equations of motion and dynamic critical phenome-na[J]. Phys. Rev. Lett.,1999,82:1891.
[16]B. Zheng, F. Ren, and H. Ren. Corrections to scaling in two-dimensional dynamic xy and fully frustrated xy models[J]. Phys. Rev. E,2003,68:046120-046128.
[17]Y. Ozeki and N. Ito. Nonequilibrium relaxation analysis of fully frustrated xy models in two dimensions[J]. Phys. Rev. B,2003,68:054414.
[18]E. Granato and D. Dominguez. Resistivity scaling and critical dynamics of fully frustrated josephson-junction arrays with on-site dissipation[J]. Phys. Rev. B,2005,71:094521.
[19]H. Liu, J.P. Lv, and Q.H. Chen. Ising-like phase transition in the fully frustrated xy model with weak disorder[J]. Europhys. Lett.,2008,84:66004.
[20]J.Q. Yin, B. Zheng, and S. Trimper. Dynamic monte carlo simulations of the three-dimensional random-bond Potts model[J]. Phys. Rev. E,2005,72:036122-036130.
[21]Yukiyasu Ozeki, Keita Ogawa. Nonequilibrium relaxation analysis of gauge glass models in two dimen-sions[J]. Phys. Rev. B,2005,71:220407.
[22]Yuan Chen, Zhi-Bing Li. Short-time dynamics of spin systems with long-range correlated quenched impurities[J]. Phys. Rev. B,2005,71:174433.
[23]P. Calabrese and A. Gambassi. Ageing properties of critical systems[J]. J. Phys. A:Math. Gen.,2005, 38:R133.
[24]X.W. Lei and B. Zheng. Short-time critical dynamics and aging phenomena in the two-dimensional XY model[J]. Phys. Rev. E,2007,75:040104-040107(R).
[25]Federico Roma, Daniel Dominguez. Nonequilibrium critical dynamics of the three-dimensional gauge glass[J]. Phys. Rev. B,2008,78:184431.
[26]H.K. Lee and Y. Okabe. Reweighting for nonequilibrium markov processes using sequential importance sampling methods[J]. Phys. Rev. E,2005,71:015102(R).
[27]Everaldo Arashiro, J. R. Drugowich de Felicio, Ulrich H. E. Hansmann. Short-time dynamics of polypep-tides[J]. The Journal of Chemical Physics,2007,126(4):045107.
[28]Shuangli Fan, Fan Zhong. Determination of the dynamic and static critical exponents of the two-dimensional three-state potts model using linearly varying temperature [J]. Phys. Rev. E,2007,76:041141.
[29]S.Z. Lin and B. Zheng. Short-time critical dynamics at perfect and imperfect surfaces[J]. Phys. Rev. E, 2008,78:011127-011137.
[30]Gabriel Baglietto, Ezequiel V. Albano. Finite-size scaling analysis and dynamic study of the critical behavior of a model for the collective displacement of self-driven individuals[J]. Phys. Rev. E,2008, 78:021125.
[31]R.H. Swendsen and Jian-Sheng Wang. Nonuniversal critical dynamics in monte carlo simulations[J]. Phys. Rev. Lett.,1987,58:86-88.
[32]F. Wang and D.P. Landau. Efficient multiple-range random walk algorithm to calculate the density of states[J]. Phys. Rev. Lett.,2001,86:2050-2053.
[33]U. Wolff. Collective monte carlo updating for spin systems[J]. Phys. Rev. Lett.,1989,62:361-364.
[34]A.J. Bray. Theory of phase-ordering kinetics[J]. Advances in Physics,1994,43(3):357-459.
[35]M. Jost and K.D. Usadel. Interface roughening in driven magnetic systems with quenched disorder[J]. Phys. Rev. B,1996,54:9314-9321.
[36]N.J. Zhou and B. Zheng. Non-equilibrium critical dynamics with domain interface[J]. Europhys. Lett., 2007,78:56001-56004.
[37]S.J. He, G.L.M.K.S. Kahanda, and P.Z. Wong. Roughness of wetting fluid invasion fronts in porous media[J]. Phys. Rev. Lett.,1992,69:3731-3734.
[38]S. Lemerle, J. Ferre, C. Chappert, V. Mathet, T. Giamarchi, and P. Le Doussal. Domain wall creep in an Ising ultrathin magnetic film[J]. Phys. Rev. Lett.,1998,80:849-852.
[39]T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo. Propagation of a magnetic domain wall in a submicrometer magnetic wire[J]. Science,1999,284:468.
[40]S. Moulinet, A. Rosso, W. Krauth, and E. Rolley. Width distribution of contact lines on a disordered substrate[J]. Phys. Rev. E,2004,69:035103-035106(R).
[41]M. Yamanouchi, J. Ieda, F. Matsukura, S.E. Barnes, S. Maekawa, and H. Ohno. Universality classes for domain wall motion in the ferromagnetic semiconductor (Ga,Mn)As[J]. Science,2007,317:1726-1729.
[42]M.Y. Im, L. Bocklage, P. Fischer, and G. Meier. Direct observation of stochastic domain-wall depinning in magnetic nanowires[J]. Phys. Rev. Lett.,2009,102:147204.
[43]A.B. Kolton, G. Schehr, and P.L. Doussal. Universal nonstationary dynamics at the depinning transition[J]. Phys. Rev. Lett.,2009,103:160602-160606.
[44]Daniel S. Fisher. Sliding charge-density waves as a dynamic critical phenomenon[J]. Phys. Rev. B,1985, 31:1396-1427.
[45]A.B. Kolton, A. Rosso, E.V. Albano, and T. Giamarchi. Short-time relaxation of a driven elastic string in a random medium[J]. Phys. Rev. B,2006,74:140201-140204(R).
[46]P. Le Doussal, K.J. Wiese, and P. Chauve. Two-loop functional renormalization group theory of the depinning transition[J]. Phys. Rev. B,2002,66:174201-174234.
[47]A. Rosso and W. Krauth. Origin of the roughness exponent in elastic strings at the depinning threshold[J]. Phys. Rev. Lett.,2001,87:187002-187005.
[48]N.J. Zhou, B. Zheng, and Y.Y. He. Short-time domain-wall dynamics in the random-field Ising model with a driving field[J]. Phys. Rev. B,2009,80:134425-134431.
[49]N.J. Zhou, B. Zheng, and D.P. Landau. Relaxation-to-creep transition of domain-wall motion in a two-dimensional random-field Ising model with an ac driving field[J]. Europhys. Lett.,2010,92:36001-36005.
[50]P. Chauve, T. Giamarchi, and P. Le Doussal. Creep and depinning in disordered media[J]. Phys. Rev. B, 2000,62:6241-6267.
[51]F. Cayssol, D. Ravelosona, C. Chappert, J. Ferre, J. P. Jamet. Domain wall creep in magnetic wires[J]. Phys. Rev. Lett.,2004,92:107202.
[52]V. Repain, M. Bauer, J.-P. Jamet, J. Ferr茅, A. Mougin, C. Chappert, H. Bernas. Creep motion of a magnetic wall:Avalanche size divergence[J]. EPL (Europhysics Letters),2004,68(3):460.
[53]P. Paruch, T. Giamarchi, and J.M. Triscone. Domain wall roughness in epitaxial ferroelectric PbZr0.2Ti0.aO3 thin films[J]. Phys. Rev. Lett.,2005,94:197601.
[54]A.B. Kolton, A. Rosso, and T. Giamarchi. Nonequilibrium relaxation of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,95:180604.
[55]J. D. Noh and H. Park. Relaxation dynamics of an elastic string in random media[J]. Phys. Rev. E,2009, 80:040102(R).
[56]Madan Rao, Amitabha Chakrabarti. Kinetics of domain growth in a random-field model in three dimen-sions[J]. Phys. Rev. Lett.,1993,71:3501-3504.
[57]Raja Paul, Sanjay Puri, Heiko Rieger. Domain growth in ising systems with quenched disorder[J]. Phys. Rev. E,2005,71:061109.
[58]Raja Paul, Gregory Schehr, Heiko Rieger. Superaging in two-dimensional random ferromagnets[J]. Phys. Rev. E,2007,75:030104.
[59]Malte Henkel, Michel Pleimling. Superuniversality in phase-ordering disordered ferromagnets[J]. Phys. Rev. B,2008,78:224419.
[60]Hyunhang Park, Michel Pleimling. Aging in coarsening diluted ferromagnets[J]. Phys. Rev. B,2010, 82:144406.
[61]P. W. ANDERSON, Y. B. KIM. Hard superconductivity:Theory of the motion of abrikosov flux lines[J]. Rev. Mod. Phys.,1964,36:39-43.
[62]T. Tybell, P. Paruch, T. Giamarchi, and J.M. Triscone. Domain wall creep in epitaxial ferroelectric pbzr0.2ti0.8O3 thin films[J]. Phys. Rev. Lett.,2002,89:097601.
[63]L B Ioffe, V M Vinokur. Dynamics of interfaces and dislocations in disordered media[J]. Journal of Physics C:Solid State Physics,1987,20(36):6149.
[64]T. Nattermann. Interface roughening in systems with quenched random impurities [J]. EPL (Europhysics Letters),1987,4(11):1241.
[65]M. V. Feigel' man, V. B. Geshkenbein, A. I. Larkin, V. M. Vinokur. Theory of collective flux creep[J]. Phys. Rev. Lett.,1989,63:2303-2306.
[66]T. Nattermann, Y. Shapir, and I. Vilfan. Interface pinning and dynamics in random systems[J]. Phys. Rev. B,1990,42:8577-8586.
[67]J.Y. Jo, S.M. Yang, T.H. Kim, H.N. Lee, J.G. Yoon, S. Park, Y. Jo, M.H. Jung, and T.W. Noh. Nonlin-ear dynamics of domain-wall propagation in epitaxial ferroelectric thin films[J]. Phys. Rev. Lett.,2009, 102:045701.
[68]J.-C. Lee, K.-J. Kim, J. Ryu, K.-W. Moon, S.-J. Yun, G.-H. Gim, K.-S. Lee, K.-H. Shin, H.-W. Lee, and S.-B. Choe. Universality classes of magnetic domain wall motion[J]. Phys. Rev. Lett.,2011,107:067201.
[69]P. Chauve, T. Giamarchi, P. Le Doussal. Creep via dynamical functional renormalization group[J]. EPL (Europhysics Letters),1998,44(1):110.
[70]T. Shibauchi, L. Krusin-Elbaum, V. M. Vinokur, B. Argyle, D. Weller, B. D. Terris. Deroughening of a 1d domain wall in an ultrathin magnetic film by a correlated defect[J]. Phys. Rev. Lett.,2001,87:267201.
[71]A.B. Kolton, A. Rosso, and T. Giamarchi. Creep motion of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,94:047002.
[72]P.C. Hohenberg and B.I. Halperin. Theory of dynamic critical phenomena[J]. Rev. Mod. Phys.,1977, 49:435-479.
[73]B. Zheng. Monte carlo simulations and numerical solutions of short-time critical dynamics[J]. Physica, 2000, A283:80.
[74]B. Zheng and S. Trimper. Comment on "Universal Fluctuations in Correlated Systems"[J]. Phys. Rev. Lett.,2001,87:188901.
[75]Kyozi Kawasaki. Diffusion constants near the critical point for time-dependent ising models. i[J]. Phys. Rev.,1966,145:224-230.
[76]A. J. Bray. Theory of phase-ordering kinetics[J]. Advances in Physics,2002,51(2):481-587.
[77]Ezequiel V. Albano, Gustavo Saracco. Dynamic behavior of anisotropic nonequilibrium driving lattice gases[J]. Phys. Rev. Lett.,2002,88:145701.
[78]Sergio Caracciolo, Andrea Gambassi, Massimiliano Gubinelli, Andrea Pelissetto. Critical behavior of the two-dimensional randomly driven lattice gas[J]. Phys. Rev. E,2005,72:056111.
[79]Thomas H. R. Smith, Oleg Vasilyev, Douglas B. Abraham, Anna Maciolek, Matthias Schmidt. Interfaces in driven ising models:Shear enhances confinement[J]. Phys. Rev. Lett.,2008,101:067203.
[80]Sergio Caracciolo, Andrea Gambassi, Massimiliano Gubinelli, Andrea Pelissetto. Comment on "dynamic behavior of anisotropic nonequilibrium driving lattice gases"[J]. Phys. Rev. Lett.,2004,92:029601.
[81]E. V. Albano, G. Saracco. Albano and saracco reply:[J]. Phys. Rev. Lett.,2004,92:029602.
[82]F. J. Alexander, D. A. Huse, S. A. Janowsky. Dynamical scaling and decay of correlations for spinodal decomposition at tc[i]. Phys. Rev. B,1994,50:663-667.
[83]Satya N. Majumdar, David A. Huse, Boris D. Lubachevsky. Growth of long-range correlations after a quench in conserved-order-parameter systems[J]. Phys. Rev. Lett.,1994,73:182-185.
[84]Claude Godreche, Florent Krzakala, Federico Ricci-Tersenghi. Non-equilibrium critical dynamics of the ferromagnetic ising model with kawasaki dynamics[J]. Journal of Statistical Mechanics:Theory and Experiment,2004,2004(04):P04007.
[85]Clement Sire. Autocorrelation exponent of conserved spin systems in the scaling regime following a critical quench[J]. Phys. Rev. Lett.,2004,93:130602.
[86]Y.Y. He, B. Zheng, and N.J. Zhou. Logarithmic correction to scaling in domain-wall dynamics at kosterlitz-thouless phase transitions[J]. Phys. Rev. E,2009,79:021107-021114.
[87]T. Nattermann, V. Pokrovsky, and V.M. Vinokur. Hysteretic dynamics of domain walls at finite tempera-tures[J]. Phys. Rev. Lett.,2001,87:197005-197008.
[88]W. Kleemann. Universal domain wall dynamics in disordered ferroic materials[J]. Annu. Rev. Mater. Res.,2007,37:415-448.
[89]Satya N. Majumdar, David A. Huse. Growth of long-range correlations after a quench in phase-ordering systems[J]. Phys. Rev. E,1995,52:270-284.
[90]Chuck Yeung, Madan Rao, Rashmi C. Desai. Bounds on the decay of the autocorrelation in phase ordering dynamics[J]. Phys. Rev. E,1996,53:3073-3077.
[91]J. F. Marko, G. T. Barkema. Phase ordering in the ising model with conserved spin[J]. Phys. Rev. E,1995, 52:2522-2534.
[92]Tomoyuki Nagaya, Jean-Marc Gilli. Experimental study of spinodal decomposition in a Id conserved order parameter system[J]. Phys. Rev. Lett.,2004,92:145504.
[93]P.J. Metaxas, J.P. Jamet, A. Mougin, M. Cormier, J. Ferre, V. Baltz, B. Rodmacq, B. Dieny, and R. L. Stamps. Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy[J]. Phys. Rev. Lett.,2007,99:217208-217213.
[94]Y.H. Shin, I. Grinberg, I.W. Chen, and A.M. Rappe. Nucleation and growth mechanism of ferroelectric domain-wall motion[J]. Nature,2007,449:881-886.
[95]A. Dourlat, V. Jeudy, A. Lemaitre, and C. Gourdon. Field-driven domain-wall dynamics in (Ga,Mn)As films with perpendicular anisotropy[J]. Phys. Rev. B,2008,78:161303-161306(R).
[96]U. Nowak and K. D. Usadel. Influence of temperature on the depinning transition of driven interfaces[J]. Europhys. Lett.,1998,44:634-640.
[97]O. Duemmer and W. Krauth. Critical exponents of the driven elastic string in a disordered medium[J]. Phys. Rev. E,2005,71:061601-061604.
[98]A.B. Kolton, A. Rosso, T. Giamarchi, and W. Krauth. Dynamics below the depinning threshold in disor-dered elastic systems[J]. Phys. Rev. Lett.,2006,97:057001-057004.
[99]B. Bako, D. Weygand, M. Samaras, W. Hoffelner, and M. Zaiser. Dislocation depinning transition in a dispersion-strengthened steel [J]. Phys. Rev. B,2008,78:144104-144108.
[100]A.B. Kolton, A. Rosso, and T. Giamarchi. Nonequilibrium relaxation of an elastic string in a random potential[J]. Phys. Rev. Lett.,2005,95:180604-180607.
[101]L. Roters, S. Liibeck, and K. D. Usadel. Creep motion in a random-field Ising model[J]. Phys. Rev. E, 2001,63:026113-026117.
[102]N.J. Zhou and B. Zheng. Dynamic effect of overhangs and islands at the depinning transition in two-dimensional magnets[J]. Phys. Rev. E,2010,82:031139-031147.
[103]W. Kleemann, J. Rhensius, O. Petracic, J. Ferre, J.P. Jamet, and H. Bernas. Modes of periodic domain-wall motion in ultrathin ferromagnetic layers[J], Phys. Rev. Lett.,2007,99:097203-097206.
[104]N. A. Pertsev, A. Petraru, H. Kohlstedt, R. Waser, I. K. Bdikin, D. Kiselev, and A. L. Kholkin. Dynam-ics of ferroelectric nanodomains in BaTiO3 epitaxial thin films via piezoresponse force microscopy[J]. Nanotechnology,2008,19:375703.
[105]A.B. Kolton, A. Rosso, T. Giamarchi, and W. Krauth. Creep dynamics of elastic manifolds via exact transition pathways[J]. Phys. Rev. B,2009,79:184207-184219.
[106]Cecile Monthus and Thomas Garel. Driven interfaces in random media at finite temperature:Existence of an anomalous zero-velocity phase at small external force [J]. Phys. Rev. E,2008,78:041138-041147.
[107]H. N. Yang and T. M. Lu. Inconsistency between height-height correlation and power-spectrum functions of scale-invariant surfaces for roughness exponent α～1 [J]. Phys. Rev. E,1995,51:2479.
[108]J. F. Fernandez, G. Grinstein, Y. Imry, S. Kirkpatrick. Numerical evidence for dc=2 in the random-field ising model[J]. Phys. Rev. Lett.,1983,51:203-206.
[109]Scott R. Anderson. Growth and equilibration in the two-dimensional random-field ising model[J]. Phys. Rev. B,1987,36:8435-8446.
[110]L. Roters, A. Hucht, S. Liibeck, U. Nowak, and K.D. Usadel. Depinning transition and thermal fluctuations in the random-field ising model[J]. Phys. Rev. E,1999,60:5202-5207.
[111]Daniel S. Fisher. Interface fluctuations in disordered systems:5-f expansion and failure of dimensional reduction[J]. Phys. Rev. Lett.,1986,56:1964-1967.
[112]Hong Ji, Mark O. Robbins. Transition from compact to self-similar growth in disordered systems:Fluid invasion and magnetic-domain growth[J]. Phys. Rev. A,1991,44:2538-2542.
[113]X P Qin, B Zheng, N J Zhou. Universality class of the depinning transition in the two-dimensional ising model with quenched disorder[J]. Journal of Physics A:Mathematical and Theoretical,2012, 45(11):115001.
[114]David A. Huse, Christopher L. Henley. Pinning and roughening of domain walls in ising systems due to random impurities[J]. Phys. Rev. Lett.,1985,54:2708-2711.
[115]Mehran Kardar. Roughening by impurities at finite temperatures[J]. Phys. Rev. Lett.,1985,55:2923-2923.
[116]R.H. Dong, B. Zheng, and N.J. Zhou. Critical domain-wall dynamics of model b[J]. Phys. Rev. E,2009, 79:051125-051128.
[117]R. H. Dong, B. Zheng, N. J. Zhou. Creep motion of a domain wall in the two-dimensional random-field ising model with a driving field[J]. EPL (Europhysics Letters),2012,98(3):36002.
[118]Joseph Ford. The fermi-pasta-ulam problem:Paradox turns discovery[J]. Physics Reports,1992, 213(5):271-310.
[119]Dominique Escande, Holger Kantz, Roberto Livi, Stefano Ruffo. Self-consistent check of the valid-ity of gibbs calculus using dynamical variables[J]. Journal of Statistical Physics,1994,76:605-626. 10.1007/BF02188677.
[120]Mickael Antoni, Stefano Ruffo. Clustering and relaxation in hamiltonian long-range dynamics[J]. Phys. Rev. E,1995,52:2361-2374.
[121]Lando Caiani, Lapo Casetti, Marco Pettini. Hamiltonian dynamics of the two-dimensional lattice mod-el[J]. Journal of Physics A:Mathematical and General,1998,31(15):3357.
[122]Lando Caiani, Lapo Casetti, Cecilia Clementi, Giulio Pettini, Marco Pettini, Raoul Gatto. Geometry of dynamics and phase transitions in classical lattice φ4 theories[J]. Phys. Rev. E,1998,57:3886-3899.
[123]Xavier Leoncini, Alberto D. Verga, Stefano Ruffo. Hamiltonian dynamics and the phase transition of the XY model[J]. Phys. Rev. E,1998,57:6377-6389.
[124]B. Zheng. Deterministic equations of motion and phase ordering dynamics[J]. Phys. Rev. E,2000,61:153.
[125]B. Zheng. Deterministic equations of motion and phase ordering dynamics[J]. Phys. Rev. E,2000,61:153-156.
[126]M. Jost and K.D. Usadel. Interface roughening in driven magnetic systems with quenched disorder. In Chaos and Fractals in Chemical Engineering. World Scientific,1997.
[127]M. Jost, J. Heimel and T. Kleinefeld. Domain-wall roughening in Co_(28)Pt_(72) alloy films[J]. Phys. Rev. B,1998,57:5316-5319.
[128]P. Cordoba-Torres, T. J. Mesquita, I. N. Bastos, R. P. Nogueira. Complex dynamics during metal dissolu-tion:From intrinsic to faceted anomalous scaling[J]. Phys. Rev. Lett.,2009,102:055504.