# Topological Defect Evolutions Guided by Varying the Initial Azimuthal Orientation
## 1 Introduction  
Topological defects are discontinuities or nontrivial regions in physical systems that cannot be eliminated through continuous symmetry operations [1]. They are distinguished by topology and play vital roles in various fields of physics, including condensed matter physics [2], cosmology [3], and quantum field theory [4], determining corresponding properties and behaviors. Topological defects inevitably arise during the phase transition of condensed matter because of a spontaneous symmetry breaking process [5]. Thus, it is meaningful to exploit the behaviors and mechanisms of topological defect transformations across the phase transitions. Liquid crystals (LCs) provide an ideal platform for such investigations owing to their unique combination of fluidity, stimuli-responsive optical anisotropy, and abundance of phases and topological defects [6]. The constituent rod-like molecules in the nematic phase (N) of LCs spontaneously orient along a dominant direction, referred to as the director \(n\), where \(n\) and \(-n\) are physically equivalent [7]. NLCs usually exhibit point defects, line defects, and wall defects. The topological charge \(s\) is a conserved quantity characterizing the number of rotations of the director around the defect core, and is considered as a key property of topological defects [7]. The value of \(s\) is infinite in theory; however, topological defects with \(\pm 1\) and \(\pm 1/2\) are usually observed experimentally. The spatial dimension is another feature of topological defects [8]. For example, point defects can be classified into bulk point defects (three-dimensional) and surface point defects (two-dimensional). The former is usuallyformed in spherical nematic droplets with perpendicular anchoring surfaces and uniformly aligned NLCs embedded with spherical particles of vertical alignment [9; 10; 11; 12]. The latter has been more extensively investigated due to its ability to freely generate and manipulate under external stimuli such as electric fields [13; 14], magnetic fields [15], topographically patterned mechanical scrubbing [16], and preprogrammed photopatterning [17]. Across the N-smectic A (SmA) phase transition, one-dimensional positional order is further introduced in addition to long-range directional order. Accordingly, the LC molecules arrange into parallel layers with nanometer thicknesses. Under hybrid anchoring conditions, these layers are periodically wrapped to generate toric focal conic domains (TFCDs) [18; 19]. TFCDs exhibit perfect rotational symmetry and can focus light like a microlens due to their radially gradient refractive index change [20]. Moreover, their specific director distribution enables the transportation of functional nanoparticles to the central singularity [21] and this unique topographic feature facilitates the manufacture of artificial superhydrophobic surfaces [22]. Recently, ordered TFCDs were converted from NLCs with predesigned point defects during the N-S phase transition [23; 24]. Up until now, the LC orientation around point defects has been restricted to the radial condition, hindering the behaviors of the corresponding structures. New architectures with unprecedented properties can be released if one can overcome the infinite rotational symmetry limitation. In this work, we investigate the topological defect evolution of LCs across the N-SmA phase transition in a hybrid cell via presetting different azimuthal orientations of a square alignment lattice. The square lattice of the radial alignment is imprinted on the substrate by photoalignment, while a perpendicular alignment is set on the superstrate. When the LC is filled with such a cell, a \(\pm 1\) point defect array is achieved in the N phase when it is slowly cooled from the isotropic phase. Via varying the initial azimuthal orientation of each alignment unit, the transformations of both the point defects and the surrounding director fields occur correspondingly.