A phase transition is an example of a “topological defect” in the space of parameters of a quantum or classical many-body system. In this paper, we consider phase diagram topological defects of higher codimension. These have the property that equilibrium states undergo some kind of nontrivial winding as one moves around the defect. We show that such topological defects exist even in classical statistical mechanical systems, and describe their general structure in this context. We then introduce the term “diabolical critical point” (DCP), which is a higher-codimension analog of a continuous phase transition, with the proximate phases of matter replaced by the nontrivial winding of the proximate equilibrium states. We propose conditions under which a system can have a stable DCP. We also discuss some examples of stable DCPs in $(1+1)$-dimensional quantum systems.

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• Quantum field theory
• Quantum field theory (low energy)
• Quantum phase transitions
• Spontaneous symmetry breaking
• Topological defects
• Quantum many-body systems