Kibble–Zurek mechanism
The Kibble–Zurek mechanism (KZM) describes the non-equilibrium dynamics and the formation of topological defects in a system which is driven through a continuous phase transition at finite rate.
It is named after Tom W. B. Kibble, who pioneered the study of domain structure formation through cosmological phase transitions in the early universe, and Wojciech H. Zurek, who related the number of defects it creates to the critical exponents of the transition and to its rate—to how quickly the critical point is traversed.
[1][2] If a two-component scalar field switches from the isotropic and homogeneous high-temperature phase to the symmetry-broken stage during cooling and expansion of the very early universe (shortly after Big Bang), the order parameter necessarily cannot be the same in regions which are not connected by causality.
Regions are not connected by causality if they are separated far enough (at the given age of the universe) that they cannot "communicate" even with the speed of light.
It was debated for quite a while if magnetic monopoles might be residuals of defects in the symmetry-broken Higgs field.
This is one of the main reasons (beside the isotropy of the cosmic background radiation and the flatness of spacetime) why nowadays an inflationary expansion of the universe is postulated.
During the exponentially fast expansion within the first 10−30 second after Big-Bang, all possible defects were diluted so strongly that they lie beyond the event horizon.
Today, the two-component primordial scalar field is usually named inflaton.
[4][5][6] The analogy between the Higgs field and superfluid helium is given by the two-component order parameter; superfluid helium is described via a macroscopic quantum mechanical wave function with global phase.
In helium, two components of the order parameter are magnitude and phase (or real and imaginary part) of the complex wave function.
Defects in superfluid helium are given by vortex lines, where the coherent macroscopic wave function disappears within the core.
If a system is cooled at any non-zero rate (e.g. linearly) through a continuous phase transition, the time to reach the transition will eventually become shorter than the correlation time of the critical fluctuations.
At this time, the fluctuations are too slow to follow the cooling rate; the system has fallen out of equilibrium and ceases to be adiabatic.
A "fingerprint" of critical fluctuations is taken at this fall-out time and the longest-length scale of the domain size is frozen out.
For very fast cooling rates, the system will fall out of equilibrium very early and far away from the transition.
[footnote 1] The inverse of this length scale can be used as an estimate of the density of topological defects, and it obeys a power law in the quench rate.
Consider a system that undergoes a continuous phase transition at the critical value
is the dynamic exponent which relates spatial with temporal critical fluctuations.
The Kibble–Zurek mechanism describes the nonadiabatic dynamics resulting from driving a high-symmetry (i.e. disordered) phase
The distance to the transition is on one hand side the time to reach the transition as function of cooling rate (red curve) and for linear cooling rates at the same time the difference of the control parameter to the critical point (blue curve).
[7][8] In 2005, it was shown that KZM describes as well the dynamics through a quantum phase transition.
[9][10][11][12] In 2008 spontaneous vortices were observed in the formation of atomic Bose-Einstein condensates, consistent with the Kibble-Zurek mechanism.
[13] The mechanism also applies in the presence of inhomogeneities,[14] ubiquitous in condensed matter experiments, to both classical,[15][16][17] quantum phase transitions[18][19] and even in optics.
[22] A system, where structure formation can be visualized directly is given by a colloidal mono-layer which forms a hexagonal crystal in two dimensions.
The latter are nothing else but the monopoles of the high-symmetry phase within the six-fold director field of crystal axes.
A special feature of Kosterlitz–Thouless transitions is the exponential divergence of correlation times and length (instead of algebraic ones).
The figure shows a comparison of the Kibble–Zurek scaling with algebraic and exponential divergences.
The data illustrate, that the Kibble–Zurek mechanism also works for transitions of the Kosterlitz–Thoules universality class.
