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The two sexes are defined by the sizes of the gametes they produce, anisogamy being the state with two differing gamete sizes (hence, females and males). The origin of this divergence has received much research interest, both theoretically and empirically. The gamete dynamics (GD) theory is a widely accepted theoretical explanation for anisogamy, and green algae have been an important empirical testing ground for the theory. However, some green and brown algae produce parthenogenetic gametes (gametes that can develop without fusing with another gamete), in contrast to an assumption in GD theory that unfused gametes do not develop. Here, we construct a GD model accounting for parthenogenetic gametes. We find that under conditions of panmixia and highly efficient fertilization (i.e., conditions of classical GD models from 1972 onward), the results remain largely unaltered by parthenogametes. However, under gamete-limited conditions anisogamy evolves less easily in the new model, and a novel result emerges: whereas previous models typically predict the evolution of either anisogamy or small isogamy, the current model shows that large isogamy can evolve when parthenogenetic gametes evolve under conditions of inefficient fertilization. Our analyses uncover unexplored complications relating to sex ratios under this relatively uncharted gametic system. We discuss limitations these complications impose on our models and suggest avenues for future research. We compare model results to algae with parthenogenetic gametes in nature.

Bulk properties of quantum phases should be independent of a specific choice of boundary conditions as long as the boundary respects the symmetries. Based on this physically reasonable requirement, we discuss the Lieb-Schultz-Mattis-type ingappability in two-dimensional quantum magnets under a boundary condition that makes evident a quantum anomaly underlying the lattice system. In particular, we direct our attention to those on the checkerboard lattice, which are closely related to frustrated quantum magnets on the square lattice and on the Shastry-Sutherland lattice. Our discussion is focused on the adiabatic U(1) flux insertion through a closed path in a boundary condition twisted by a spatial rotation and a reflection. Two-dimensional systems in this boundary condition are effectively put on a nonorientable space, namely the Klein bottle. We show that the translation symmetry on the Klein-bottle space excludes the possibility of the unique and gapped ground state. Taking advantage of the flux insertion argument, we also discuss the ground-state degeneracy on magnetization plateaus of the Heisenberg antiferromagnet on the checkerboard lattice.

Efimov physics is renowned for the self-similar spectrum featuring the universal ratio of one eigenenergy to its neighbor. Even more esoteric is the numerically unveiled fact that every Efimov trimer is accompanied by a pair of tetramers. Here we demonstrate that this hierarchy of universal few-body clusters has a topological origin by identifying the numbers of universal three- and four-body bound states with the winding numbers of the renormalization-group limit cycle in theory space. The finding suggests a topological phase transition in mass-imbalanced few-body systems which should be tested experimentally.

In cell divisions, relative size of daughter cells should play fundamental roles in gametogenesis and embryogenesis. Differences in gamete size between the two mating types underlie sexual selection. Size of daughter cells is a key factor to regulate cell divisions during cleavage. In cleavage, the form of cell divisions (equal/unequal in size) determines the developmental fate of each blastomere. However, strict validation of the form of cell divisions is rarely demonstrated. We cannot distinguish between equal and unequal cell divisions by analysing only the mean size of daughter cells, because their means can be the same. In contrast, the dispersion of daughter cell size depends on the forms of cell divisions. Based on this, we show that gametogenesis in the marine green alga, Monostroma angicava, exhibits equal size cell divisions. The variance and the mean of gamete size (volume) of each mating type measured agree closely with the prediction from synchronized equal size cell divisions. Gamete size actually takes only discrete values here. This is a key theoretical assumption made to explain the diversified evolution of isogamy and anisogamy in marine green algae. Our results suggest that germ cells adopt equal size cell divisions during gametogenesis.

The Efimov effect is the only experimentally realized universal phenomenon that exhibits the renormalization-group limit cycle with the three-body parameter parametrizing a family of universality classes. Recent experiments in ultracold atoms have unexpectedly revealed that the three-body parameter itself is universal when measured in units of an effective range. By performing an exact functional renormalization-group analysis with various finite-range interaction potentials, we demonstrate that the onset of the renormalization-group flow into the limit cycle is universal, regardless of short-range details, which connects the missing link between the two universalities of the Efimov physics. A close connection between the topological property of the limit cycle and few-body physics is also delineated., 5 pages, 4 figures