The Kibble–Zurek mechanism in a subcritical bifurcation

Miranda, M. A.; Laroze, David; González-Viñas, Wenceslao

doi:10.1088/0953-8984/25/40/404208

Cited by 9 publications

(7 citation statements)

References 79 publications

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“…By contrast, whether the initial state is equilibrium or not is irrelevant as the system can quickly equilibrate once ζ s is small. This has been confirmed by a lot of experiments and numerical simulations [23][24][25][43][44][45][46][47] . On the other hand, when ε 0 = 0 and the initial state is the equilibrium state there, it has been shown by an adiabatic perturbation method that the scaling of topological defects is consistent with the KZ scaling 39,48,49 .…”

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confidence: 60%

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Kibble-Zurek mechanism beyond adiabaticity: Finite-time scaling with critical initial slip

Huang

Yin

et al. 2016

Phys. Rev. B

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The Kibble-Zurek mechanism demands an initial adiabatic stage before an impulse stage to have a frozen correlation length that generates topological defects in a cooling phase transition. Here we study such a driven critical dynamics but with an initial condition that is near the critical point and that is far away from equilibrium. In this case, there is no initial adiabatic stage at all and thus adiabaticity is broken. However, we show that there again exists a finite length scale arising from the driving that divides the evolution into three stages. A relaxation-finite-time scaling-adiabatic scenario is then proposed in place of the adiabatic-impulse-adiabatic scenario of the original KibbleZurek mechanism. A unified scaling theory, which combines finite-time scaling with critical initial slip, is developed to describe the universal behavior and is confirmed with numerical simulations of a two-dimensional classical Ising model. The Kibble-Zurek mechanism (KZM)1-5 describes topological defect formation in driven critical dynamics in a variety of systems, ranging from classical 6-28 to quantum phase transitions [29][30][31][32][33][34][35][36][37][38][39][40] . Kibble first proposed it in cosmology 1,2 by identifying a frozen correlation lengtĥ ξ that renders spatially distant regions causally independent during the cooling of the universe from the big bang. Then, Zurek brought this proposal to condensed matter physics and offered a method to compute the density of defects formed 3,4 . As a system cannot always follow adiabatically the cooling of a finite rate R due to critical slowing down near the critical point, its evolution from a temperature T 0 , sufficiently higher than the critical temperature T c , can be divided into three sequential stages, an initial adiabatic stage, an impulse stage, and a final adiabatic stage below T c . In the initial adiabatic regime, the correlation length ξ and the correlation time ζ s grow as |ε| −ν and ξ z , respectively, as the distance to the critical point, ε ≡ T − T c , is reduced, where ν and z are the correlation-length and the dynamic critical exponents, respectively 41 . The boundaries between the stages are then determined by the frozen instantt at which the time interval before the transition, t c − t = ε/R, equals ζ s 3,4 , where t c = ε 0 /R is the time at ε = 0. This leads to t c −t ∼ R −z/rT 3,4 , where r T = z + 1/ν is a rate exponent 42 . Upon assuming evolutionless in the middle impulse stage,t then determinesξ ∼ R −1/rT and thus the defect density n ∼ R d/rT , the KZ scaling 3,4 .Crucial in the derivation is the existence of the initial adiabatic stage that gives rise toξ. It results from the large ε 0 = T 0 − T c and thus small ζ s . By contrast, whether the initial state is equilibrium or not is irrelevant as the system can quickly equilibrate once ζ s is small. This has been confirmed by a lot of experiments and numerical simulations [23][24][25][43][44][45][46][47] . On the other hand, when ε 0 = 0 and the initial state is the equilibrium state there,...

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supporting

confidence: 60%

“…As ζ i is shorter than ζ d in short times, it dominates the dynamics and the stage is thus relaxational similar to the critical initial slip, while the external driving in only a perturbation. Note that this relaxation stage has nothing to do with the free relaxation regime 23,26,27 that follows the final adiabatic stage and that has no driving at all.…”

mentioning

confidence: 99%

Kibble-Zurek mechanism beyond adiabaticity: Finite-time scaling with critical initial slip

Huang

Yin

et al. 2016

Phys. Rev. B

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“…Experiments testing KZM scaling were the focus of our review. The Kibble-Zurek scaling was also tested in classical and quantum phase transitions in a variety of computer experiments 22,31,86,100,112,113,127,[150][151][152][153][154][155][156][157][158][159][160][161][162][163][164][165][166] and analytical works, 46,60,61,[167][168][169][170] and found to hold essentially whenever it was expected to apply. Laboratory experiments are, of course, more difficult.…”

Section: Discussionmentioning

confidence: 99%

Universality of Phase Transition Dynamics: Topological Defects from Symmetry Breaking

del Campo,

Zurek

2013

Preprint

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In the course of a non-equilibrium continuous phase transition, the dynamics ceases to be adiabatic in the vicinity of the critical point as a result of the critical slowing down (the divergence of the relaxation time in the neighborhood of the critical point). This enforces a local choice of the broken symmetry and can lead to the formation of topological defects. The Kibble-Zurek mechanism (KZM) was developed to describe the associated nonequilibrium dynamics and to estimate the density of defects as a function of the quench rate through the transition. During recent years, several new experiments investigating formation of defects in phase transitions induced by a quench both in classical and quantum mechanical systems were carried out. At the same time, some established results were called into question. We review and analyze the Kibble-Zurek mechanism focusing in particular on this surge of activity, and suggest possible directions for further progress.

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“…15 Since then, the Kibble-Zurek (KZ) power-law scaling predictions were confirmed experimentally and utilized in a variety of systems, including liquid crystals, 3,16 colloidal monolayers, 17 ion crystals, 18 Bose-Einstein condensates, 19 superfluids 10,12,20 and cold atomic clouds. 21 KZ-scalings were also suggested to govern the defect formation in Rayleigh-Bénard convection, [22][23][24] although, in contrast to theory, 25 experimentally measured exponents appeared to depend on non-universal system properties. Moreover, since previous theory and experiments focused primarily on phase transitions in planar Euclidean spaces, much less is known about the existence of KZ-type scaling laws in more complex geometries and topologies.…”

Section: Introductionmentioning

confidence: 94%

Defect formation dynamics in curved elastic surface crystals

Stoop

Dunkel

2018

Soft Matter

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Topological defects shape the material and transport properties of physical systems. Examples range from vortex lines in quantum superfluids, defect-mediated buckling of graphene, and grain boundaries in ferromagnets and colloidal crystals, to domain structures formed in the early universe. The Kibble-Zurek (KZ) mechanism describes the topological defect formation in continuous non-equilibrium phase transitions with a constant finite quench rate. Universal KZ scaling laws have been verified experimentally and numerically for second-order transitions in planar Euclidean geometries, but their validity for non-thermal transitions in curved and topologically nontrivial systems still poses open questions. Here, we use recent experimentally confirmed theory to investigate topological defect formation in curved elastic surface crystals formed by stress-quenching a bilayer material. For both spherical and toroidal crystals, we find that the defect densities follow KZ-type power laws. Moreover, the nucleation sequences agree with recent experimental observations for spherical colloidal crystals. Our results suggest that curved elastic bilayers provide an experimentally accessible macroscopic system to study universal properties of non-thermal phase transitions in non-planar geometries.

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The Kibble–Zurek mechanism in a subcritical bifurcation

Cited by 9 publications

References 79 publications

Kibble-Zurek mechanism beyond adiabaticity: Finite-time scaling with critical initial slip

Kibble-Zurek mechanism beyond adiabaticity: Finite-time scaling with critical initial slip

Universality of Phase Transition Dynamics: Topological Defects from Symmetry Breaking

Defect formation dynamics in curved elastic surface crystals

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