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In search of diabolical critical points

Naren Manjunath, Dominic V. Else

TL;DR

This work introduces phase diagram topological defects and diabolical critical points (DCPs) as higher-codimension generalizations of phase transitions, applicable to both classical and quantum many-body systems. It develops a fiber-bundle framework to describe nontrivial windings of ground-state manifolds around defects and formulates compatibility conditions for DCPs in quantum settings, linking emergent symmetries and anomalies. The authors present concrete 1D examples—Thouless pump and a higher Chern-number S^3 family—that host stable DCPs, illustrating how multiple relevant operators and an emergent symmetry can stabilize critical points with codimension $N$. These results extend the notion of criticality beyond conventional phase transitions and open avenues for exploring DCPs via conformal bootstrap and related approaches in higher dimensions and more complex topologies.

Abstract

A phase transition is an example of a ``topological defect'' in the space of parameters of a quantum or classical many-body systems. In this paper, we consider phase diagram topological defects of higher codimension. These have the property that equilibrium states undergo some kind of non-trivial 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 non-trivial 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.

In search of diabolical critical points

TL;DR

This work introduces phase diagram topological defects and diabolical critical points (DCPs) as higher-codimension generalizations of phase transitions, applicable to both classical and quantum many-body systems. It develops a fiber-bundle framework to describe nontrivial windings of ground-state manifolds around defects and formulates compatibility conditions for DCPs in quantum settings, linking emergent symmetries and anomalies. The authors present concrete 1D examples—Thouless pump and a higher Chern-number S^3 family—that host stable DCPs, illustrating how multiple relevant operators and an emergent symmetry can stabilize critical points with codimension . These results extend the notion of criticality beyond conventional phase transitions and open avenues for exploring DCPs via conformal bootstrap and related approaches in higher dimensions and more complex topologies.

Abstract

A phase transition is an example of a ``topological defect'' in the space of parameters of a quantum or classical many-body systems. In this paper, we consider phase diagram topological defects of higher codimension. These have the property that equilibrium states undergo some kind of non-trivial 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 non-trivial 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.
Paper Structure (18 sections, 20 equations, 2 figures)

This paper contains 18 sections, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical examples from spontaneous symmetry breaking
  3. Ising-SSB family
  4. General mathematical formulation of classical SSB families
  5. SSB family with a continuous ground state manifold
  6. Diabolical critical points: Properties and Examples
  7. Compatibility properties: quantum case
  8. Review: Parameterized families of quantum many-body states
  9. The compatibility result for DCPs
  10. Examples of phase diagrams with stable DCPs
  11. Thouless pump
  12. Family with higher Chern number
  13. Discussion
  14. Acknowledgements
  15. Computations for classical SSB family over $S^2$
  16. ...and 3 more sections

Figures (2)

  • Figure 1: (a) The simplest case of a diabolical critical point (DCP) occurs for a parameter space $\mathbb{R}^N$ containing a family over $S^{N-1}$. In this case the DCP has codimension $N$ and is a single point. Here we show the case $N=2$ (the DCP is marked in black, and a loop indicating the $S^1$ family is shown in red) (b) We can include additional parameters that change the parameter space to $\mathbb{R}^n$ for $n>N$ but there is still a DCP in parameter space if the singular surface of the defect has codimension $N$. (c) If the singular surface has codimension less than $N$, it is not a DCP.
  • Figure 2: Stability of DCP at the origin for (a) the Ising SSB family with the perturbation in Eq. \ref{['eq:IsingSSBPerturbation']}, and (b) the SSB family over $S^2$ with the perturbation in Eq. \ref{['eq:S2FamPerturbation']}. The DCP splits into a first order line (dashed) terminating in a pair of critical points (red 'X' marks). In (c) there are two perturbations corresponding to Eq. \ref{['eq:XZPerturbation']} which turn the DCP into a first-order ellipse.