Weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="double-struck">Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> supertopology

Crystal symmetries can enforce all bands of a material to be topological, a property that is commonly referred to as “supertopology.” Here, we determine the symmetry-enforced <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> supertopologies of nonmagnetic centrosymmetric materials with weak and strong spin-orbit coupling (SOC). For weak (i.e., negligible) SOC, crystal symmetries can enforce Dirac nodal lines protected by a <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>π</c:mi></c:math>-Berry phase, while for strong SOC, crystal symmetries can give rise to nontrivial weak <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi mathvariant="double-struck">Z</d:mi><d:mn>2</d:mn></d:msub></d:math> topologies in two-dimensional subplanes of the three-dimensional Brillouin zone. We catalog all centrosymmetric space groups whose symmetries enforce these <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi mathvariant="double-struck">Z</f:mi><f:mn>2</f:mn></f:msub></f:math> supertopologies. Suitable material realizations are identified and experimental signatures of the supertopologies, such as quantum spin Hall states, are discussed.