Topological Critical Points Overview

• Topological critical points are defined as configurations where a system's topology changes, marked by gap closures or degeneracies.
• They govern key transitions in quantum systems, impacting phase transitions through changes in topological invariants like the Chern number.
• Experimental techniques such as band structure tuning and quantum transport measurements validate these transitions and inform theoretical models.

A topological critical point refers to a critical configuration—typically in parameter, real, or momentum space—at which the qualitative topology of a physical or mathematical system changes. These points govern topological phase transitions, topology-induced singularities in energy bands, and the structure of solutions to PDEs or thermodynamic functions. The concept has deep implications across differential geometry, condensed matter physics, topological analysis of functions, and mathematical physics, unifying approaches that involve Morse theory, invariants (e.g. Chern number, Z₂ index), and geometric/topological constraints in diverse contexts.

1. Topological Critical Points: Definitions and Frameworks

Topological critical points represent settings where the topology of solutions, energy bands, or level sets fundamentally changes, often corresponding to the vanishing of certain derivatives or the emergence of degeneracies. In the Morse-theoretic approach, these are characterized by points where the gradient of a relevant function (order parameter, Green’s function, energy dispersion, Lyapunov function) vanishes:

$\operatorname{Cr}(g) = \{x : \nabla g(x, b) = 0 \}$

A topological phase transition is generally marked by a change in the global topological invariant (e.g., Euler characteristic, Chern number, winding number) and is typically linked to degeneracies at the critical point, such as the vanishing Hessian (i.e., degenerate Morse critical points), or gap closings in quantum systems (Hu, 2023).

In the context of energy bands, a topological critical point is often a momentum $k_c$ where the electron velocity vanishes, and the band structure experiences a non-analyticity in the density of states (DOS) (Yuan et al., 2019). In thermodynamics, a topological current assigns a charge to each critical point, enabling their classification into distinct thermodynamic classes (Wei et al., 2021). For PDEs and pattern formation, critical points of solutions (e.g., stationary patterns for reaction-diffusion on manifolds) encode topological and geometric information about the domain (Enciso et al., 2010, Kamalia et al., 2020).

2. Classification: Ordinary vs. High-Order and Their Topological Indices

Critical points are broadly divided into ordinary and high-order (or singular) types:

• Ordinary Critical Points: Nondegenerate points where the Hessian is nonsingular; includes minima, maxima, and saddle points (e.g., van Hove singularities in band theory). Their topological index in 2D is $I = \pm 1$, corresponding to the sign of $\det(D_{ij})$ (Yuan et al., 2019).
• High-Order Critical Points: Occur when some or all quadratic terms vanish (degenerate Hessian), requiring higher order terms in the Taylor expansion. These points exhibit different scaling,

$$E(\lambda^a p_x, \lambda^b p_y) = \lambda E(p_x, p_y)$$

with possible non-integer (including half-integer) topological indices beyond ordinary cases. For instance, at a high-order saddle point

$E = \alpha p_x^2 + \gamma p_x p_y^2 + \kappa p_y^4$

the index becomes $I = \mathrm{sgn}(4\alpha\kappa - \gamma^2)$; such points show power-law divergent DOS and particle-hole asymmetry (Yuan et al., 2019).

The totality of topological indices for all critical points on a compact surface obeys global constraints, such as the Poincaré–Hopf theorem:

$\sum_i I_i = 0$

Other classification approaches encode the local structure at each critical point in a combinatorial object (e.g., a tree in the case of functions on 3-manifolds), with isomorphism classes serving as invariants for local topological type (Hladysh et al., 2019).

3. Topological Critical Points in Physical Systems

a. Band Theory and Topological Quantum Phase Transitions

In Bloch bands, a critical point of the dispersion $E(k)$ occurs where $\nabla_k E = 0$. Topological critical points underlie topological quantum phase transitions, for example:

• Insulator-to-Dirac-semi-metal transitions: At the critical point, bands touch (e.g., the emergence of Dirac nodes or the merging of Weyl nodes), changing the topological invariant (Chern/monopole charge) (Smith et al., 2010, Wang et al., 2018, Zhao et al., 2015). The multi-criticality of such points can be systematically categorized by their scaling properties and symmetry (Yuan et al., 2019).
• Quantum Hall and Quasicrystals: The Aubry–Andre–Harper (AAH) model transitions via a critical point into the Maryland model, which, while maintaining 1D energy gaps, features a gapless 2D spectrum signifying a change in topological class (Chern number) (1311.0882).
• Topological Superconductors: Transitions mediated by the closure and reopening of the bulk gap correspond to changes in the ground state topology, with the emergence or annihilation of Majorana zero-modes at interfaces (Tewari et al., 2011, Komijani et al., 2014, Lee et al., 2016, Takagi et al., 2018).

b. Thermodynamics and Black Hole Physics

In black hole thermodynamics, critical points in the $T-S$ plane are assigned topological charges via $\phi$-mapping theory, classifying them as "conventional" ($Q = -1$) or "novel" ($Q = +1$). Only the former are associated with extendable first-order phase transitions (Wei et al., 2021).

c. PDEs, Pattern Formation, and Manifold Geometry

Critical points of Green’s functions and pattern solutions to reaction-diffusion equations (e.g., on tori) connect the topology of the domain (e.g., Betti number of a surface) to the maximal number of allowable critical points (Enciso et al., 2010, Kamalia et al., 2020). In three-manifolds, the trees constructed from neighborhoods of isolated critical points provide a complete set of local invariants; their global arrangement yields a topological classification of the function (Hladysh et al., 2019).

4. Methodologies: Morse Theory, Invariants, and Renormalization Procedures

The Morse-theoretic framework equates the number and ordering of Morse critical points (MCPs) to the underlying topology:

$$\text{Critical point:}\ \nabla g(x, b) = 0;\quad \text{Degeneracy:}\ |\nabla^2 g(x, b_c)|_{x = x_c} = 0$$

A topological phase transition ensues when the MCP count changes, triggered by the emergence or annihilation of degenerate MCPs (interpreted as particle–antiparticle pairs) (Hu, 2023). The local criterion is connected globally via the Euler characteristic,

$\chi(b) = \sum_i (-1)^i H_i(b)$

where $H_i(b)$ counts MCPs with $i$ negative Hessian eigenvalues.

In renormalization group (RG)-based approaches, one tracks divergences in the "curvature function" (e.g., Berry curvature), with RG flows revealing phase boundaries and multicriticalities in the parameter space (Abdulla et al., 2020, Kumar et al., 2021).

In computational topology, invariants such as the sequential or parametrized topological complexity (TC), closely tied to the Lusternik–Schnirelmann category, yield lower bounds on the number of critical points in smooth functions (Mescher et al., 4 Nov 2024).

5. Physical Consequences: DOS Divergences, Stability, and Critical Dynamics

A key signature of a topological critical point in electronic systems is a non-analyticity in the DOS:

$$\rho(\varepsilon) = \int \frac{d^2p}{(2\pi)^2} \delta(\varepsilon - E(p))$$

For ordinary van Hove singularities, this is a logarithmic divergence; for high-order points, it is a power-law divergence, potentially with particle-hole asymmetry (Yuan et al., 2019). These singularities impact physical properties, driving anomalies in magnetoresistance, Hall responses, and superconductivity instabilities (Zhao et al., 2015, Wang et al., 2018). In fracton critical points, the breakdown of the area law for entanglement entropy ($S \sim L \ln L$) marks a fundamentally new regime of quantum criticality (You et al., 2020).

Topological phase transitions are often induced by external parameters: twist angle (moiré systems), strain, pressure, exchange field, or electric fields (Yuan et al., 2019, Duong et al., 2016).

6. Realization, Measurement, and Experimental Probes

Realizing and probing topological critical points involves:

• Band Structure Tuning: Twist angle, strain, pressure, field-tuning to access criticality in engineered materials (twisted bilayer graphene, photonic crystals) (Yuan et al., 2019, Zhao et al., 2015).
• Quantum Transport and Optical Probes: Detection via quantized conductance jumps, fractional Josephson effects, or band-edge shift current sign reversals (optical nonlinear responses), which provide precise markers for criticality (Tewari et al., 2011, Yan, 2018).
• Topology in Thermodynamics: Topological currents and winding numbers computed from thermodynamic gradients to distinguish classes of black hole critical points (Wei et al., 2021).
• Topological Robotics: Sequential and parametrized TC measurable through motion planning algorithms, with complexity bounds reflecting the minimal number of critical regions (rules) in navigation tasks (Mescher et al., 4 Nov 2024).

7. Mathematical and Conceptual Impact

The concept of the topological critical point unifies analytic, geometric, and algebraic methods. It links Morse theory and homotopy invariants to concrete bounds on critical points, enables local-to-global transitions in understanding phase changes, and enables new classifications of functions and quantum phases. Recent results demonstrate that topology-driven criticality extends traditional paradigms: area law violations, edge-bulk correspondence beyond gapped systems, and critical phases with coexisting edge modes and gapless bulk (You et al., 2020, Kumar et al., 2021). The framework also fosters advances in classifying and engineering complex models in mathematics and physics, including reaction–diffusion, black hole thermodynamics, and quantum materials.

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This synthesis reviews foundational and contemporary developments in the theory, classification, realization, and application of topological critical points, drawing on geometric analysis, condensed matter theory, topological field theory, and mathematical physics (Enciso et al., 2010, Yuan et al., 2019, Wei et al., 2021, Hu, 2023, Abdulla et al., 2020, Mescher et al., 4 Nov 2024).