Introducing Projective Crystal Symmetry

Crystalline materials depend on symmetries like translations, rotations, and reflections. Traditional symmetries combine two symmetry operations to produce a result that exactly matches the group's rules. However, quantum states exhibit “phase ambiguity,” meaning that states that differ only by a phase factor are physically identical. This detail comes from quantum physics. This allows projective symmetry descriptions to be broader.

These projective representations characterise a crystal's symmetry group in quantum projective crystal symmetry. By combining two symmetry operations, an additional phase factor can change the standard rules of symmetry algebra. This theory was first proposed by Wigner a century ago, but its effects on crystalline materials have just recently been studied.

This framework expands material classification and discovers new topological phases by revealing electron behaviour. In condensed matter physics and artificial systems like photonic and acoustic crystals, this new discipline has spurred imaginative theoretical and practical research.

Physical Origin and Realisation

Projective symmetry's additional phase elements can be physically realised via static gauge flux setups, not only mathematically. Think of a crystal lattice particle jumping around a loop. If this loop contains a gauge flux (like magnetic flux), the particle gains phase. This process creates a projective algebra, changing the algebraic relationship between symmetry operators like translation and mirror reflection.

Solid-state materials require strong magnetic fields to generate big fluxes, however artificial crystals like photonic crystals, acoustic settings, and electric circuit arrays may manufacture massive gauge fluxes. Several projective crystal symmetry predictions have been confirmed in these artificial systems. Recently, magnetic and twisted Moiré layered materials have been proposed as solid-state solutions for these phenomena.

MSNS: Momentum-Space Nonsymmetry

Momentum-space The distinctive and major impact of projective crystal symmetry is nonsymmorphic symmetry (MSNS). In traditional physics, fractional lattice translation-based nonsymmorphic symmetries like screw axes and glide planes were thought to exist only in real space. Symmetries were always symmorphic in momentum space, meaning they were just rotations or reflections.

This cognition is completely transformed by projections. In momentum space, a real-space symmetry may become nonsymmorphic. When projected, a simple mirror image can behave as a gliding mirror in momentum space. This means it moves and reflects a momentum vector via a reciprocal lattice vector. The discovery of momentum-space glide mirrors and screw axes is revolutionary.

Redefining Momentum Space and Finding New Topologies MSNS substantially alters the Brillouin zone, momentum space's fundamental region.

The Brillouin zone is usually a two-dimensional torus. The fundamental domain of MSNS is a Klein bottle, a twisting non-orientable surface. This modification predicts new phases like the Klein bottle insulator, requiring a new topological categorisation for insulators.

Three-dimensional platycosms have more potential. No longer limited to a 3D torus, the fundamental domain can be used to generate any of the 10 platycosms, ten flat, compact 3D manifolds. Nine of them, including the Didicosm and numerous amphicosms, have complex topologies other than the torus. Every “Brillouin platycosm” has various topological phases and classifications, making the ground more fertile for innovative materials with strange properties.

Broader implications and future directions

In materials research, projective crystal symmetry presents interesting new possibilities. New matter states, such as the projective translation-shielded Möbius insulator, have been anticipated. The combining of projective representations with internal symmetries like chiral or time-reversal yields even more fascinating physics. By dramatically changing the algebra of symmetry operators, projective symmetry can reverse topological classifications of systems into spinful and spinless categories.