Kibble-Zurek mechanism

The Kibble-Zurek Mechanism ( KZM ), named after Tom Kibble and Wojciech Zurek , describes, for example, in cosmology and solid-state physics the non-equilibrium dynamics of critical fluctuations and the occurrence of topological defects when a system has a finite, non-zero cooling rate is cooled by a continuous phase transition .

Central idea

Based on the formalism of the spontaneous symmetry breaking of the Higgs field , Tom Kibble was the first to formulate this idea for a primordial two-component scalar field: If during the expansion and cooling of the very early universe (shortly after the Big Bang ) a two-component scalar field from the isotropic and homogeneous high temperature phase changes into the symmetry-broken state of the low-temperature phase, the order parameter for regions of the universe that are not causally related does not necessarily have to assume the same value everywhere. Not causally related means that there are regions that are so far apart that (given the age of the universe) they could not even "communicate" at the speed of light. It follows that the symmetry breaking cannot have occurred globally. In regions that are not causally related, the order parameter will generally assume different values ​​in the symmetry-broken regions and these regions will be separated from one another by domain walls. Depending on the dimensionality of the system and the order parameter, further topological defects can occur, such as monopoles, vortices or textures. Magnetic monopoles have long been considered hot candidates to be remnants of the defects of the Higgs field. The fact that no residuals of such defects have been found within the event horizon of the observable universe is one of the main reasons (in addition to the general isotropy of the cosmic background radiation and the flatness of the observed space-time), why an inflationary expansion of the universe shortly after the Big Bang is assumed today. During the exponentially rapid expansion within the first 10 ⁻³⁰  s after the Big Bang, all possible defects in space have been thinned out so that they lie behind the event horizon. For the two-component primordial scalar field, the name inflaton has meanwhile become established.

Meaning in condensed matter

The blue curve shows the divergence of the correlation times as a function of the control parameter (e.g. temperature difference to the phase transition). For linear cooling rates, the red curve shows the time until the transition is reached as a function of the control parameter. At the intersections the system falls out of balance and becomes non-adiabatic.

Wojciech Zurek worked out that the same considerations apply to the phase transition from liquid helium to superfluid helium. The analogy to the Higgs field is that the order parameter is also two-component; superfluid helium is characterized by a macroscopic quantum mechanical wave function with a global phase; the two components are magnitude and phase or real and imaginary parts of the complex wave function. The topological defects in superfluid helium are the normal fluid threads in which the coherent, macroscopic wave function disappears. In the broken-symmetry phase, they represent residuals of the high-symmetry phase.

In general, the energy differences between ordered and disordered phases disappear for continuous phase transitions, so that the fluctuations between the two phases at the transition point are arbitrarily large. This so-called critical behavior not only means that spatial correlation lengths diverge, but also that temporal correlations of the fluctuations between the two phases become arbitrarily slow. If the phase transition is now cooled at a non-vanishing (e.g. linear) cooling rate, the time to reach the phase transition will at some point be shorter than the correlation times of the critical fluctuations. From this point onwards the fluctuations are too slow to follow the cooling rate: the system has fallen out of balance. During this "downtime" a fingerprint of the critical fluctuations has been taken and the longest length scale of the domain size is determined, which determines the later evolution of the system. For fast cooling rates, the system will fall out of balance early and the domains will be small. For slow cooling rates, the system only falls out of balance late, the length scales of the critical fluctuations and thus the domains become large.

Derivation of the domain size

Considering a system that has a continuous phase transition as a function of the dimensionless control parameter, e.g. B. the reduced temperature , if the transition temperature is, then the theory of critical phenomena says that the correlation lengths and the correlation times with the critical exponent given by the universality class diverge algebraically and is the dynamic exponent that is the temporal with the spatial Linked to fluctuations. ${\displaystyle \epsilon ={\frac {T-T_{c}}{T_{c}}}}$ ${\displaystyle \epsilon _{c}=0}$ ${\displaystyle T_{c}}$ ${\displaystyle \xi }$ ${\displaystyle \tau }$ ${\displaystyle \nu }$ ${\displaystyle z}$

${\displaystyle \xi \sim \lambda ^{-\nu },\qquad \tau \sim \lambda ^{-z\nu },}$

If the control parameter varies linearly in time with the cooling rate , equating the correlation times to the time when the transition is reached provides the time when the system is out of balance. In the sketch, this is the point of intersection between the blue and red curves: In the sketch, the distance to the transition is the time between reaching the transition as a function of the control parameter (red curve) and, for linear cooling rates, the difference in the control parameter (e.g. B. Temperature) to the critical point (or the transition temperature), given by the blue curve ${\displaystyle t}$ ${\displaystyle v}$ ${\displaystyle \epsilon (t)=vt}$ ${\displaystyle {\bar {t}}}$

${\displaystyle {\bar {t}}=[\epsilon ({\bar {t}})]^{-z\nu }\Rightarrow {\bar {t}}\sim v^{-z\nu /(1+z\nu )}.}$

The correlation length then gives the mean size of the domains when the system becomes non-adiabatic, ${\displaystyle {\bar {\xi }}}$

${\displaystyle {\bar {\xi }}\equiv \xi [\lambda ({\bar {t}})]\sim v^{-\nu /(1+z\nu )}.}$

The reciprocal of the correlation lengths gives the defect density if is the dimension of the system. ${\displaystyle \rho \sim {\bar {\xi }}^{-d}}$ ${\displaystyle d}$

Experimental reviews

Exponential divergence of the correlation times of a Kosterlitz-Thouless transition . On the left is the domain structure of a 2D monolayer of colloids at a very high cooling rate when the system fell out of balance. On the right, the structure is shown at a late point in time (after further coarsening) when the system was cooled at medium rates.

Some experiments on quite different systems have been carried out to verify the Kibble-Zurek mechanism. This includes liquid crystals in which the defect structures can be made clearly visible using polarization microscopy. However, the phase transition in liquid crystals is weak first order, so that the system does not ideally fit the theory. Further systems are superfluid He3, superconducting systems, multiferroic systems, quantum systems, ion crystals and Bose-Einstein condensates . The last two systems are not free from inhomogeneities, such as e.g. B. are temperature gradients. Extensive review articles on the significance and limitation of these experiments have been written by T. Kibble (status 2007) and A. del Campo (status 2014).

Example in two dimensions

Domain size as a function of the cooling rate in a colloidal monolayer. The control parameter in this system is given by the strength of the interaction . ${\displaystyle \Gamma }$

One system in which structure formation can be observed directly are colloidal monolayers, which form a hexagonal crystal in the low temperature phase. The phase transition follows the so-called KTHNY theory in which translational and rotational symmetry are broken by two Kosterlitz-Thouless transitions . The associated topological defects are dislocations and disclinations in 2D. In the sense of the Kibble-Zurek mechanism, the latter are nothing more than the monopoles of the high symmetry phase in the sixfold directional field of the crystal axes. A special feature of the Kosterlitz-Thouless transition is the exponential divergence of the correlation lengths and correlation times. At linear cooling rates, this leads to a transcendent equation that can be solved numerically. The figure shows the comparison of the Kibble-Zurek scaling with algebraic and those with exponential divergences. The measurement data show that the Kibble-Zurek mechanism can also be applied to phase transitions of the Kosterlitz-Thouless universality class.

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It has long been argued that the existence of grain boundaries necessarily indicates a phase transition of the first order, if different domains or crystallites have formed after homogeneous or heterogeneous nucleation of nuclei and after their growth. This is one of the reasons why the melt theory in two dimensions ( KTHNY theory ) has long been debated after the first computer simulations of phase transitions in two dimensions found both grain boundaries and phase coexistence. The Kibble-Zurek mechanism shows that in the event of spontaneous symmetry breaking, the symmetry cannot instantaneously change globally. The mechanism also leads to domains and defect structures for continuous phase transitions if the system is sufficiently large. Sufficiently large means that the system size (in the shortest direction) is greater than the correlation length if the system is out of balance.

Footnotes

1. ↑ As an analogy one can imagine the phase transition from non-magnetic to ferromagnetic: Below the Curie temperature, the material will be ferromagnetic; The direction in which the field points is not determined a priori. Domains with different orientations of the magnetic field are called Weiss domains. However, since macroscopic magnetization costs energy, the domain walls appear in this example for energetic reasons.
2. ↑ The maximum signal speed in condensed matter is not given by the speed of light, but by the speed of sound (or second sound in the case of superfluid helium).

Individual evidence

1. ↑ TWB Kibble: Topology of cosmic domains and strings . In: J. Phys. A: Math. Gen. tape 9 , no. 8 , 1976, p. 1387-1398 , doi : 10.1088 / 0305-4470 / 9/8/029 .
2. ↑ TWB Kibble: Some implications of a cosmological phase transition . In: Phys. Rep. Band 67 , no. 1 , 1980, p. 183-199 , doi : 10.1016 / 0370-1573 (80) 90091-5 .
3. ↑ AH Guth: Inflationary universe: A possible solution to the horizon and flatness problems . In: Phys. Rev. D. Volume 23 , no. 2 , 1981, p. 347-356 , doi : 10.1103 / PhysRevD.23.347 .
4. ↑ WH Zurek: Cosmological experiments in superfluid helium? In: Nature . tape 317 , no. 6037 , 1985, pp. 505-508 , doi : 10.1038 / 317505a0 .
5. ^ WH Zurek: Cosmic Strings in Laboratory Superfluids and the Topological Remnants of Other Phase Transitions . In: Acta Phys. Pole. B . tape 24 , 1993, pp. 1301 ( edu.pl ).
6. ↑ WH Zurek: Cosmological experiments in condensed matter systems . In: Phys. Rep. Band 276 , no. 4 , 1996, pp. 177-221 , doi : 10.1016 / S0370-1573 (96) 00009-9 .
7. ^ I. Chuang et al.: Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals . In: Science . tape 251 , 1991, pp. 1336-1342 , doi : 10.1126 / science.251.4999.1336 .
8. ^ C. Bauerle et al.: Laboratory simulation of cosmic string formation in the early Universe using superfluid 3He . In: Nature . tape 382 , 1996, pp. 332-334 .
9. ^ R. Carmi et al .: Observation of Spontaneous Flux Generation in a Multi-Josephson Junction Loop . In: Phys. Rev. Lett. tape 84 , 2000, pp. 4966-4969 .
10. ↑ SC Chae et al .: Direct observation of the proliferation of ferroelectric loop domains and vortex-antivortex pairs . In: Phys. Rev. Lett. tape 108 , 2012, p. 167603 .
11. ↑ XY Xu ua: Quantum simulation of Landau-Zener model dynamics supporting the Kibble-Zurek mechanism . In: Phys. Rev. Lett. tape 112 , 2014, p. 035701 .
12. ↑ S. Ulm et al.: Observation of the Kibble-Zurek scaling law for defect formation in ion crystals . In: Nat. Comm. tape 4 , 2013, p. 2290 .
13. ↑ K. Pyka et al.: Topological defect formation and spontaneous symmetry breaking in ion Coulomb crystals . In: Nat. Comm. tape 4 , 2013, p. 2291 .
14. ↑ G. Lamporesi et al: Spontaneous creation of Kibble-Zurek solitons in a Bose-Einstein condensate . In: Nat. Phys. tape 9 , 2013, p. 656 .
15. ^ TBW Kibble: Phase-transition dynamics in the lab and the universe . In: Physics Today . tape 60 , 2007, p. 47-52 , doi : 10.1063 / 1.2784684 .
16. ^ A. del Campo et al .: Universality of phase transition dynamics: Topological defects from symmetry breaking . In: International Journal of Modern Physics A . tape 29 , 2014, p. 1430018 , doi : 10.1142 / S0217751X1430018X .
17. ^ S. Deutschländer et al: Kibble-Zurek mechanism in colloidal monolayers . In: Proc. Natl. Acad. Sci. A . tape 112 , 2015, p. 6925-6930 , doi : 10.1073 / pnas.1500763112 .
18. ↑ S. Alder et al.: Phase Transition in Elastic Disks . In: Phys. Rev. Band 127 , no. 2 , 1962, pp. 359-361 , doi : 10.1103 / PhysRev.127.359 .