The interaction among topological defects can induce novel phenomena such as disclination pairs in liquid crystals and superconducting vortex lattices. Nanoscale topological vortices with swirling ferroelectric, magnetic, and structural antiphase relationships were found in multiferroic h-YMnO 3 . Herein, we report the discovery of intriguing, but seemingly irregular configurations of a zoo of topological vortices and antivortices. These configurations can be neatly analyzed in terms of graph theory and reflect the nature of self-organized criticality in complexity phenomena. External stimuli such as chemistry-driven or electric poling can induce the condensation and eventual annihilation of topological vortexantivortex pairs.

, superconductivity [6][7][8][9] , and discommensurations [10][11][12][13][14][15] . Intercalation of other transition metal ions between the MC 2 layers gives rise to distinct superstructures, leading to significant changes in crystallographic structures and physical properties. Fe-intercalated TaS 2 shows highly anisotropic ferromagnetism at low temperatures [16][17][18][19][20] . . Magnetic hysteresis curves of the crystals were obtained using a Quantum DesignMagnetic Property Measurement System, and the real Fe compositions were estimated from the saturation magnetic moments in the magnetic hysteresis curves with the assumption that each Supplementary Information, section S1). The distinct feature between the 2a×2a and √3a×√3a superstructures in Fe x TaS 2 is the different stacking sequence of the 2D supercells along the c-axis. Specifically, the 2a×2a superstructure consists of identically stacked 2D supercells (i.e., AA-type stacking), while the √3a×√3a superstructure contains shifted 2D supercells with AB-type stacking, as shown in Fig. 1b and Fig. 1f, -5 -respectively. These different stacking sequences result in the centrosymmetric P6 3 /mmc and noncentrosymmetric and chiral P6 3 22 space groups for the 2a×2a and √3a×√3a superstructures, respectively.We found complicated configurations of antiphase domains in the dark-field images of There is an extinction rule for the dark-field images of antiphase boundaries in the 2a×2a superstructure. For example, the antiphase boundary between the BB-type and CC-type antiphase domains appears in the S1=( /2 00) (Fig 2a) and S2=(0 1/2 0) (Fig. 2b) dark-field images, but disappears in the S3=(1/2 /2 0) dark-field image of Fig. 2c (see also Supplementary Information, section S3). Each antiphase boundary becomes invisible in a dark-field image taken using one out of three superlattice spots (namely, S1, S2, or S3), when no antiphase shift at the boundary exists along a certain superlattice modulation wave vector. This absence of antiphase shifts at the antiphase boundary leads to the extinction rule for the antiphase boundaries in superlattice dark-field images. This rule is summarized in Fig. 3, showing the local structures near boundaries between two antiphase domains. The boundaries are highlighted with yellow bands, and the three directions of the superlattice modulation wave vectors are denoted by S1, S2, and S3, respectively, as shown in Fig. 1a. The red, yellow, blue, and green circles correspond to -6 -AA-, BB-, CC-, and DD-type superstructures, respectively, which are associated with four possible origins of the 2a×2a Fe superstructure. It is evident that the superlattice modulation along only one out of three equivalent crystallographic directions does not show any antiphase shift; this is indicated by light green dashed lines (along the S1 direction), light blue dashed lines (along the S2 direction), and pink dashed lines (along the S3 direction). For example, the antiphase boundary between BB-type and CC-type (or AA-type and DD-type) antiphase domains ha...

Topological materials, including topological insulators, magnets with Skyrmions and ferroelectrics with topological vortices, have recently attracted phenomenal attention in the materials science community. Complex patterns of ferroelectric domains in hexagonalREMnO 3 (RE: rare earths) turn out to be associated with the macroscopic emergence of Z 2 ×Z 3 symmetry. The results of our depth profiling of crystals with a self-poling tendency near surfaces reveal that the partial dislocation (i.e., wall-wall) interaction, not the interaction between vortices and antivortices, is primarily responsible for topological condensation through the macroscopic breaking of the Z 2 -symmetry. 2Symmetries govern nature ubiquitously from the beauty of human faces [1] to the local gauge invariance of quantum field theory [2]. The spontaneous breaking of symmetry by a variable such as temperature gives rise to a phase transition. Dislocations are common topological defects in materials, which occur during symmetry breaking, and often effectively determine important fundamental crystal properties such as hardness and fatigue behavior, grain boundary development, and charge density wave discommensuration [3][4][5]. The Burgers vector characterizes each dislocation, and dislocation and anti-dislocation refer to two dislocations with oppositely directed Burgers vectors. Dislocations with Burgers vectors that are not translation vectors with integer times of the underlying lattice unit are called partial dislocations. For example, a charge density wave (CDW) discommensuration can be considered as a partial dislocation with a Burgers vector that is a fraction of a unit cell vector and a few of these discommensurations terminate at a full "CDW dislocation", corresponding to a topological defect with a unit-cell Burgers vector [6,7]. Dislocations can often interact with each other like particles in a dilute gas [8]. The overlap between the strain fields of adjacent dislocations can induce a paired interaction between the dislocations.Ferroelectric hexagonal-REMnO 3 (RE: rare earths) exhibit intriguing topological defects induced through a trimerization-type structural phase transition [9][10][11][12]. This structural transition leads to three structural antiphase domains (α, β, γ), each of which can support either of two directions (+,-) of ferroelectric polarization [13,14]. The six interlocked structural antiphase and ferroelectric domains of REMnO 3 meet in a cloverleaf arrangement that cycles through all six domain configurations [15,16]. Occurring in pairs, the cloverleaves can be viewed as vortices and antivortices with opposite cycles of domain configuration. We have observed two topologically-distinct types of large-scale vortex/antivortex domain patterns; type-I without any 3 preferred polarization direction, and type-II with a preferred polarization direction [17]. However, the physical nature of switching between type-I to type-II patterns has not been understood.Herein, we report depth profiling of the ferroelectric domain pat...

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