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Symmetry breaking is ubiquitous in nature and represents the key mechanism behind rich diversity of patterns exhibited by nature. One commonly introduces an order parameter field to describe onset of qualitatively new ordering in a system on varying a relevant control parameter driving a symmetry breaking transition. In case of continuous symmetry breaking an order parameter consists of two qualitatively different components: an amplitude and gauge field. The latter component enables energy degeneracy and reveals how symmetry is broken. Inherent degeneracy could in general lead to nearby regions exhibiting significantly different gauge fields. Resulting frustrations can nucleate topological defects (TDs) [1]. These represent topologically stable localized nonlinear order parameter solutions. Their key property is a discrete topological charge, which is a conserved quantity. One commonly refers to TDs with positive (negative) charge as defects (antidefects). In general a nearby pair defectantidefect tends to annihilate each order because presence of TDs is in general energetically costly. The total topological charge in a system is conserved for fixed boundary conditions. Consequently number of topological defects could be varied either my merging of relevant TDs or formations of {defect,antidefect} pairs.

There is strong interest in different fields of physics to find conditions where configurations rich in TDs could be stabilized in a stable or strongly metastable phases. Famous examples represent Abrikosov lattices [2] of TDs in superconductors and Skyrmions [3] in nuclear physics. In particular, the latter example suggests that TDs might represent “particles” if fields are fundamental constituents of nature.

A convenient systems to study patterns exhibiting TDs are various liquid crystalline (LC) phases [4]. Due to their softness, fluidity and optical anisotroy&transparency TDs could be relatively easily experimentally generated and observed in them. Therefore, in LCs various theoretical predictions could be relatively easily tested. In this lecture we present our theory explaining supercooling driven glassy behaviour in systems exhibiting continuous symmetry breaking. Note that glass behaviour is still mysterious and several features remain unanswered despite intensive research in the respective field. In case of LCs we combine Kibble-Zurek (KZ) mechanism [5,6] and Imry-Ma (IM) theorem [7] to reproduce some characteristic features of general supercooling-driven glass behaviour [8]. The KZ mechanism was originally introduced in cosmology to explain coarsening dynamics in the Higgs field in the early Universe [5]. Furthermore, IM theorem [7] was introduced in magnetism to explain impact of random field-type disorder on magnetic ordering. We show that combination of these well known mechanisms one could stabilise patterns of TDs, which macroscopically yield several glass-type characteristics [8].

[1] N.D. Mermin, The topological theory of defects in ordered media. Rev, Mod. Phys. 51, 591-648 (1976).

[2] A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Sol. 2(3), 199-208 (1957).

[3] T. H. Skyrme, A non-linear field theory, Proc. R. Soc. Lond. Ser. A 260, 127–138 (1961).

[4] S.R. Renn, T.C. Lubensky, Abrikosov dislocation lattice in a model of the cholesteric to smectic-A transition, Phys. Rev. A 38, 2132-2147 (1988).

[5] T.W.B. Kibble, Topology of cosmic domains and strings, J. Phys. A: Math. Gen. 9, 1387-1389 (1976).

[6] W.H. Zurek, Cosmological experiments in condensed matter, Nature 317, 505-508 (1985).

[7] Y. Imry, S.K. Ma, Random-field instability of the ordered state of continuous symmetry, Phys. Rev. Lett. 35, 1399-1401 (1975).

[8] C.A. Angell, Liquid fragility and the glass transition in water and aqueous solutions, Chem. Rev. 102, 2627- 2650 (2002).