Dear Colleagues,

Symmetry and its breaking are important components in the scientific knowledge of nature. Their identification and classification and the description of their manifestation are some of the paths of knowledge in any scientific discipline, from physics, chemistry, and biology up to physiology.

The first to try and apply the laws of symmetry to the knowledge of nature were, perhaps, Ancient Greek naturalist philosophers. Plato's philosophy was based on the belief that there is an absolute truth that can be achieved through intelligent thinking. He expounded the Pythagorean doctrine of regular polyhedra, which, thereafter, became known as Platonic solids. Plato associated these bodies with the atom forms of the basic elements of nature. What does this represent but a desire to apply the laws of symmetry to the knowledge of nature?

On the other hand, we know that time is a fleeting, elusive phenomenon resulting from the permanent motion of matter in space. The Ancient Greek philosopher Zeno of Elea was devoted to considering this phenomenon and proposed a series of paradoxes, opening its fundamental nature. Surprisingly, we often ignore Zeno's paradoxes when computing infinitesimals. As a result, we are faced with singularities which we try to heroically overcome.

At present, we know that baryonic matter exists and makes constant movements in 3D space (x, y, z). We can define the geometrical parameters (t, x, y, z) pointing to each position at each moment of a material body using 4D spacetime. A material body has an inertial mass, m. Consequently, one can define the dynamical variables: energy and three momentum components. It turns out that both spaces—geometrical space (time and space) and dynamical space (energy and momentum—are connected to each other through the continuous symmetry of a physical system by the conservation laws of energy and the momentum. To put it simply, the Noether theorem in its most general formulation states: "If a physical system has continuous symmetry, then there will be corresponding quantities in it that retain their values over time". This is true for both classical and quantum systems. In the latter case, energy and the momenta accurate for the Planck constant are equal to de Broglie's frequency and the wave numbers of the particle under consideration.

It should be noted that along with matter there is antimatter. A law of symmetry acts between these two antipodes: CPT symmetry. According to it, the behavior of any physical system, including the entire Universe, should not change with the simultaneous replacement of all particles with antiparticles (a change of charge to the opposite), an inversion of parity, and time.

But the above picture is only a small part of a complete pattern weaved into the whole Universe. Baryonic matter is seen only as a fleeting foam against a vast ocean of dark matter and dark energy. We do not know what it is. There are vague guesses that this dark matter/energy appears to be a superfluid Bose–Einstein condensate. Because of Meissner's pushing out effect, any interactions with this dark entity are doomed to fail. Nevertheless, scientists have not given up trying to understand this dark essence. The laws of symmetry play a leading role in this cognition.

Symmetry breaking gives a series of problems arising at the phase transition from one phase state of matter to the other. The transition is accompanied by a change in the symmetry of matter. Near the transition point, such amazing phenomena are observed as the lifetime and correlation lengths of fluctuations tend to infinity. An example of the phase transition is the lambda point (about 2.17 K), below which liquid helium (helium I) goes into a state of superfluidity (helium II). In this case, a new state, the Bose–Einstein condensate, arises. Another example is the bifurcation point, above which Faraday waves appear when shaking the fluid poured in a bath. Near this point, the surprising behavior of drops bouncing along the fluid is observed. These classical drops demonstrate behavior similar to quantum particles.

One more surprising manifestation of symmetry breaking occurs at the edge of chaos. The edge of chaos is an area where chaotic activity is too weak to reproduce high entropy, but still strong enough to generate complex self-sustained patterns via a nonlinear dynamical system. The edge of chaos borders a well-ordered region where the diversity of the reproduced patterns disappears. One can say that the edge of chaos is where life manifests itself to the fullest. We also note that the visible Universe evolving as a fleeting foam against a vast ocean of dark matter and dark energy manifests its life on the edge of chaos. Therefore, it is quite natural for scientists to compare the filamentous structure of the universe with the organization of the nervous tissue of the brain. There are many questions here: How do symmetries manifest themselves on the edge of chaos? Where does symmetry come from in a self-sustaining pattern if the latter is reproduced by chaos? Why are the self-sustained patterns robust?

Dr. Valeriy Sbitnev
Prof. Dr. Markus Scholle
Guest Editors

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Title: Emergent Phenomena with Broken Time-Reversal Symmetry
Authors: Sang-Wook Cheong; Fei-Ting Huang
Affiliation: Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University
Abstract: Time-reversal symmetry (T-symmetry) in crystalline solids can be disturbed via magnetic phase transitions, leading to a myriad of emergent phenomena. These include the Magneto-optical Kerr effect (MOKE), Directional nonreciprocity, Diagonal linear magnetoelectric effect, and Off-diagonal linear magnetoelectric effect. Commonly held views suggest that MOKE manifests only in magnets with a substantial net magnetic moment. Linear magnetoelectricity is discernible in magnets where both T-symmetry and space-inversion symmetries are broken. Meanwhile, directional nonreciprocity is typically observed in magnets boasting toroidal magnetic moments or in chiral crystallographic solids under external magnetic fields. Using the symmetry operational similarity (SOS) framework, we elucidate the precise symmetry prerequisites, which encompass broken T-symmetry, and pinpoint the pertinent magnetic point groups for MOKE, Directional nonreciprocity, Diagonal linear magnetoelectric effect, and Off-diagonal linear magnetoelectric effect. Our insights provide invaluable direction for unearthing novel materials showcasing these effects. Furthermore, the SOS methodology holds promise as a revolutionary tool for symmetry-driven material exploration and innovation.