**Effective Topological Charge Cancellation Mechanism**

Samo Kralj^{1,2}

^{1}*FNM, University of Maribor, Koro**š**ka 160, 2000 Maribor, Slovenia*

^{2}*Jožef Stefan Institute, Jamova 39,1000 Ljubljana, Slovenia*

Topological defects (TDs) appear almost unavoidably in continuous symmetry breaking phase transitions [1]. Topological origin makes their key features independent of systems’ microscopic details and therefore TDs display many universalities. In general, TDs have strong impact on material properties and play signiﬁcant role in several technological applications. Furthermore, investigations of TDs in relevant fields are interesting for fundamental science. For example, relatively large scientific community believes that *fields *represent fundamental entity of nature and not “particles” [2]. If this is the case then TDs might effectively play role of “particles”, as the pioneering work of Skyrme suggests [3].

Among others it is of strong interest to ﬁnd simple and robust mechanisms controlling positioning and local number of TDs [4]. In the lecture we will introduce the universal *Eﬀective Topological Charge Cancellation* (ECTT) mechanism on curved 2D manifolds, which we have recently developed [5,6,7]. As demonstrative systems we will consider thin nematic liquid crystalline films using a Landau-type approach in terms of the tensor nematic order parameter. The ETCC mechanism efficiently predicts localized positional assembling tendency of TDs and formation of pairs {*defect, antidefect*} on curved surfaces. By exploiting analogy of electric field driven generation of pairs { *positron,electron*} we will express critical condition for curvature driven formation of pairs {*defect, antidefect*} on closed structures exhibiting spherical and toroidal topology. We will also discuss role of *intrinsic* in *extrinsic* energy terms [7]. Note that the latter terms are commonly neglected in theoretical studies of TDs on frozen surfaces [4]. Namely, majority of studies express elastic penalties using the covariant derivatives [4], which automatically rule out the *extrinsic* contributions. We will demonstrate that omission of such terms is in general not justified.

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

[2] A. Hobson, There are no particles, there are only fields, *Am. J. Phys.* **81**, 211 (2013).

[3] T. Skyrme, A unified field theory of mesons and baryons, *Nucl. Phys.* **31**, 556 (1962).

[4] A.M. Turner *et al.*, Vortices on curved surfaces, *Rev. Mod. Phys.* **82**, 1301 (2010).

[5] S. Kralj *et al.*, Curvature control of valence on nematic shells, *Soft Matter* **7**, 670 (2011).

[6] D. Jesenek *et al.*, Defect unbinding on a toroidal nematic shell, *Soft Matter* **11**, 2434 (2015).

[7] L. Mesarec *et al.*, Effective Charge Cancellation Mechanism, *Scientific Reports* (2016), doi: 10.1038/srep27117.