Subspace symmetry for Bloch electrons

We demonstrate that the electronic Hamiltonian of a solid may not strictly share the exact symmetry of its host lattice, even though the wave functions generally do. Based on this observation, we argue that the conventional space-group description of a solid's energy bands may be incomplete; the Hamiltonian itself can possess more fundamental properties than the wave function, challenging the traditional understanding of the role of symmetry. We introduce this emergent symmetry as subspace symmetry, which, in certain cases, remains exact across the full Hilbert space. General proofs of its existence are provided, along with calculations on a 1D chain model, showing that topology can arise within otherwise trivial band structures. Finally, we explore its connections to other symmetries and present a concrete example of a ``fake'' altermagnet. Subspace symmetry offers new insights into our understanding of symmetry, topology, and condensed matter physics.