Classifying phases of matter is a central problem in many-body physics. Phases are usually distinguished by how they break symmetries or what topology they carry. In recent years, however, it has become clear that the phase diagram has a finer structure: it contains closed paths that cannot be continuously contracted to a point without crossing a gapless region. Such paths are called noncontractible loops, or S¹-families. They have been studied for systems with individual unbroken or spontaneously broken symmetries, but the case where unbroken and broken symmetries interact nontrivially remained open. This work fills that gap.
The authors use a duality method that establishes a one-to-one correspondence between the phase diagrams of two systems. For example, the Kramers-Wannier duality relates a phase with spontaneous Z₂ symmetry breaking to a symmetric phase, exchanging domain walls and charges. This allows transferring the classification of noncontractible loops from one side to the other. For systems with anomalies and semidirect products of symmetries, the authors construct explicit unitary evolutions that generate noncontractible loops and verify their algebraic properties on lattice models.
For systems with anomalous Z₂×Z₂ symmetry, the authors find that S¹-families of phases with spontaneous breaking of one of the Z₂ symmetries are classified by the group Z₄. This is because the domain wall of one symmetry carries half the charge of the other. For systems with dihedral symmetry D₂ₙ and phases breaking a Z₂ subgroup, the classification of S¹-families turns out to be D₂ₙ for odd n and D₈ or Z₂×Z₂ for even n, depending on which subgroup remains unbroken. For non-invertible SPT phases with Rep(D₂ₙ) symmetry, the authors show that the classification of S¹-families depends on whether the base point is trivial or nontrivial, contrasting with ordinary SPT phases.
This work demonstrates that mixed anomalies and semidirect products of symmetries lead to richer classifications of noncontractible loops than one might expect from independent symmetries. It deepens our understanding of the topological structure of phase diagrams and provides a tool for finding inevitable gapless points. Unlike neighboring works that consider either unbroken or fully broken symmetries, this is the first systematic study of partial breaking with nontrivial interaction between sectors.
The proposed approach can be generalized to broader classes of non-invertible symmetries, such as G×Rep(G), and to higher-dimensional systems. It may be possible to find new types of noncontractible loops associated with exotic symmetry structures and to construct their lattice realizations. This could lead to new topological invariants for quantum phases.
The results could impact the theory of quantum phase transitions and topological phases, as well as the understanding of anomalies in condensed matter physics.
The next step will be to classify S¹-families for more general non-invertible symmetries and to construct explicit models in two-dimensional systems.
This work is directly related to the problem of classifying quantum phases with exotic symmetries, including non-invertible ones, and to the question of what topological structures can exist in phase diagrams.
Noncontractible loops in the phase diagram mean that reaching a gapless phase requires fine-tuning only a finite number of parameters, even if there are infinitely many parameter axes.