Crystallographic point and space group symmetries in correlated many-body wave functions

Martin Dunn, D. K. Watson, John G. Loeser

J. W. Clark, MANFRED L. RISTIG

Ising, Heisenberg and Hubbard models in relation to insulating and metallic ferro- and antiferro-magnets studies of lattice spin systems using series expansions application of linked-cluster expansions to quantum Hamiltonian lattice systems critical properties of 1-D spin 1/2 antiferro-magnetic Heisenberg model the Z(2) lattice gauge vacuum and the transverse Ising model crystallographic point and space group symmetries in correlated many-body wave functions Hamiltonian lattice calculations on gauge and chiral meson field theories.

Hermann G. Kümmel

T. R. Liu, Zheng Zhang, Y. X. Zhao

Crystallographic groups are conventionally studied in real space to characterize crystal symmetries. Recent work has recognized that when these symmetries are realized projectively, momentum space inherently accommodates nonsymmorphic symmetries, thereby evoking the concept of \textit{momentum-space crystallographic groups} (MCGs). Here, we reveal that the cohomology of MCGs encodes fundamental data of crystalline topological band structures. Specifically, the collection of second cohomology groups, $H^2(Γ_F,\mathbb{Z})$, for all MCGs $Γ_F$, provides an exhaustive classification of Abelian crystalline topological insulators, serving as an effective approximation to the full crystalline topological classification. Meanwhile, the third cohomology groups $H^3(Γ_F,\mathbb{Z})$ across all MCGs exhaustively classify all possible twistings of point-group actions on the Brillouin torus, essential data for twisted equivariant K-theory. Furthermore, we establish the isomorphism $H^{n+1}(Γ_F,\mathbb{Z})\cong H^n\big(Γ_F,\operatorname{\mathcal{F}}(\mathbb{R}^d_F,U(1))\big)$ for $ n\ge 1$, where $\operatorname{\mathcal{F}}(\mathbb{R}^d_F,U(1))$ denotes the space of continuous $U(1)$-valued functions on the $d$D momentum space $\mathbb{R}^d_F$. The case $n=1$ yields a complete set of topological invariants formulated in purely algebraic terms, which differs fundamentally from the conventional formulation in terms of differential forms. The case $n=2$, analogously, provides a fully algebraic description for all such twistings. Thus, the cohomological theory of MCGs serves as a key technical framework for analyzing crystalline topological phases within the general setting of projective symmetry.

Walter Greiner, Ludwig Neise, Horst Stöcker B

Andrew Stewart Caldin

FINDING: The search results are dominated by crystallographic symmetry lectures (2D plane groups, 3D lattices, point groups) and one lattice QCD paper; no direct video on quantum nonlocality with C4 square lattice measurement settings was retrieved. The crystallographic material confirms the mathematical framework of combining translational symmetry with point groups to enumerate plane groups (17) and space groups (230). | MATH: Plane group enumeration: 17 wallpaper groups; point group C4 (4-fold rotation, order 4) acting on a square lattice (Z²) yields p4, p4m, p4g plane groups. Lattice QCD: heavy-quark mass from Fermilab method uses lattice spacing a, quark mass m_q via meson mass M = m_q1 + m_q2 + E_binding, with one-loop perturbative corrections; no explicit constants given. | CONNECTION: C4 point group is the symmetry of the square lattice — its character table has irreducible representations A, B, E (1D real, 1D complex conjugate pair, 2D). The rotation angle 90° = π/2, whose cos Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com