--- title: Geometric Criticality url: https://www.emergentmind.com/topics/geometric-criticality type: topic --- # Geometric Criticality Searching arXiv for recent papers on “geometric criticality” and closely related formulations. “Geometric criticality” denotes a family of research programs in which critical behavior is generated by geometry, encoded in geometric observables, or constrained by geometric structures. In the most literal formulation, an underlying geometry changes discontinuously or loses a degree of freedom, and the critical response follows even when the corresponding scalar law varies continuously; the $p$-adic construction on $\mathbb{Q}_p^2$ provides a particularly explicit example, where collapse of a two-scale Pontryagin filtration at $h=1$ produces jumps in coordinate diffusion constants although the radial jump law depends continuously on $h$ [2509.24234]. In other literatures, the same term refers to singular Berry phases and quantum-geometric tensors, curvature singularities of information manifolds, configuration-space statistical geometry, structural transitions in scale-invariant networks, percolative geometry in active matter, curvature-driven entropy flow on constant-energy shells, and geometric responses of black-hole observables [2505.03151], [1405.3212], [2508.00787], [2507.11348], [2506.05264], [2512.02242], [2601.17613]. The collected literature suggests that the expression is therefore best understood as a cross-disciplinary label for critical phenomena whose decisive structure is geometric rather than exclusively local or spectral. ## 1. Scope and principal meanings Current usage falls into several distinct but overlapping categories. In some works, geometry is the **mechanism** of the transition; in others it is the **diagnostic**; in still others it is the **constraint** that forbids a trivially gapped or smoothly connected phase. | Context | Geometric object | Critical signature | |---|---|---| | Ultrametric random walks | Two-scale Pontryagin filtration on $\mathbb{Q}_p^2$ | Jump in coordinate diffusion constants at filtration collapse [2509.24234] | | Quantum geometry | Berry phase, QGT, Bloch-angle manifold, geodesics | Divergent derivatives, gap closing, abrupt basis rotation [2505.03151], [2512.14302] | | Information/configuration geometry | Parameter manifold, Fisher metric, Hamming-distance manifold | Curvature singularity, universal exponents, Fisher-information peak [1405.3212], [2508.00787] | | Structural geometry | Network Laplacian RG, interaction graph, percolation clusters | Structural phase transition, fractal-dimension flow, evolving critical generations [2507.11348], [2606.20387], [2511.18462] | This multiplicity is substantive rather than terminological. The ultrametric framework of geometry-induced criticality isolates a structural mechanism in which only the geometry changes [2509.24234]. By contrast, the non-Hermitian Kitaev-chain study uses “geometric criticality” for scaling of local quantum geometry near topological transitions [2505.03151]. Information geometry defines geometric critical exponents on parameter manifolds [1405.3212], while configuration-space statistical geometry characterizes phase transitions through the distribution of pairwise distances between configurations [2508.00787]. Network and sparse-graph studies use the term for topology-driven changes of effective dimension or connectivity [2507.11348], [2606.20387]. A plausible implication is that no single invariant definition spans all usages. What is common is the claim that the decisive nonanalyticity is visible in a geometric object: a filtration, curvature, metric tensor, manifold, graph, shell, or optimal measurement basis. ## 2. Geometry as a direct generator of critical behavior The clearest direct mechanism is given by the $p$-adic random-walk construction on $\mathbb{Q}_p^2$ [2509.24234]. For $h\in(0,1]$, the weighted max-norm $$ \|(x_1,x_2)\|_h := \max(|x_1|_p,\,p^{h-1}|x_2|_p) $$ induces a duality-compatible two-scale filtration with radii $p^k$ and $p^{k+h}$. The resulting shell-uniform random walk has two characteristic scales, and its scaling limit is a càdlàg anisotropic $p$-adic Lévy process whose Fourier symbol is $M_{h,b}(y)=\alpha(y)\|y\|_h^b$ and whose generator is the anisotropic Vladimirov operator $\Delta_{h,b}$ [2509.24234]. The critical point is $h=1$: the two radii coincide, the fine filtration collapses into the coarse one, shell combinatorics must be re-merged and re-weighted uniformly, and the coordinate diffusion constants jump to the common isotropic value even though the radial convolution semigroup on shells remains continuous. The paper identifies this as the essence of “geometry-induced criticality” [2509.24234]. The same paper states that the mechanism should extend to any second countable LCAG with a compact open subgroup. Introducing a two-scale perturbation of a regular Pontryagin filtration yields hierarchical random walks with alternating shell weights, and equalizing the two scales forces a re-indexing of shells together with a jump in component diffusion constants, while the radial law remains continuous [2509.24234]. This suggests a structural route to criticality in ultrametric models that does not require tuning a local analytic parameter. A related “effective-geometry principle” is formulated by Gunning et al. for sparse long-range quantum lattice models on graphs of degree $O(\log N)$ [2606.20387]. There the ground-state phases and critical points are governed by the large-scale connectivity of the interaction graph. For even-$p$ power-of-$p$ graphs, the classical critical point is $$ s_c(p)=-1-\log_p 2, $$ marking the change in dominant-shell geometry; for Fibonacci graphs, a single classical transition at $s_c=0$ appears for odd $N$, associated with a reordering that exchanges short- and long-bond geometry under $s\to -s$ [2606.20387]. The criticality is therefore attached to graph geometry rather than to a local symmetry-breaking term. Two microcanonical formulations push this idea further. In mean-field rotor Hamiltonians, the trace of the Weingarten operator on the constant-energy shell admits a universal collective expansion whose quadratic form has eigenvalues $\lambda_i(E)$; geometric criticality is the vanishing of one of these curvature coefficients, $\lambda_i(E_c)=0$, selecting the critical channel directly from shell geometry [2603.29074]. In “Phase Transitions as Emergent Geometric Phenomena,” entropy obeys the deterministic equation $$ \frac{d^2S}{dE^2}+\Bigl(\frac{dS}{dE}\Bigr)^2=\Upsilon_g^{(2)}(E), $$ with $\Upsilon_g^{(2)}$ built from curvature invariants of the constant-energy manifold, so thermodynamic nonanalyticities arise from singularities or sign changes of a geometric source term [2512.02242]. ## 3. Quantum-state geometry, Berry phases, and geometric observables A major line of work uses geometric phases and quantum geometry as critical probes. In the anisotropic XY chain under linear quench, the Berry phase per mode is $$ \gamma_k=\pi[1-\cos\theta_k(t)], $$ and the total ground-state Berry phase develops a nonanalytic cusp at the critical field; in the XX limit $\gamma\to0$, the mode phases become a sequence of non-contractible geometric phases, which the paper identifies as the geometric signature of the extended XX critical region [1108.3406]. Closely related results were obtained for coupled optical cavity arrays modeled by an anisotropic Heisenberg spin-$1/2$ lattice under linear quench, where the Berry-phase susceptibility develops a non-analytic cusp as $\alpha\to0$, and summing over $k$ yields the XX critical exponent $\alpha'=1/2$ [1310.0003]. In the non-Hermitian long-range Kitaev chain, the derivative of the geometric phase and the quantum geometric tensor distinguish several universality classes [2505.03151]. For $\alpha>2$ and for $1<\alpha<2$ at $k_0=\pi$, one obtains $\nu\approx1$ and $$ \frac{dG}{d\mu}\sim |\mu-\mu_c|^{-1}, $$ whereas for $1<\alpha<2$ at $k_0=0$ the peak in $dG/d\mu$ does not diverge and the Wannier-state correlation function develops a non-decaying plateau $C(R)\simeq\mathrm{const}$, which the paper interprets as an anomalous universality class [2505.03151]. Near exceptional points, the metric component $g_{\lambda\lambda}=\mathrm{Re}\,T_{\lambda\lambda}$ diverges with $\beta=1$ on the Hermitian-Hermitian boundary and with $\beta=-1$ on the topological-coalescing boundary, defining a new EP universality class [2505.03151]. The driven Jaynes–Cummings model provides an eigenstate-level version of the same theme [2602.07795]. With drive amplitude $\eta$ and phase $\phi$ as control parameters, the quantum metric and Berry curvature of each eigenstate diverge near the photon-blockade-breakdown critical point. Because quasienergy gaps scale as $(1-\eta^2)^{3/4}$, the leading singularity is $$ g_{\mu\nu}^{(n)},\,F_{\mu\nu}^{(n)}\sim (1-\eta^2)^{-3/2}, $$ and the divergence is stronger for bright states than for the unique dark state [2602.07795]. Quantum geometry also appears through geodesics and geometric defects. In the many-body analysis of precursors of criticality, finite-$N$ diabolical points are isolated geometric singularities whose local geometry is that of a Bloch-sphere puncture with $g_{rr}=0$ and angular divergence; they act as “seeds of irregular geodesics,” and chains of such defects can condense along a first-order separatrix in the thermodynamic limit [2411.03967]. In a different direction, the study of optimal Bell operators distinguishes “geometric criticality” from “geometric locking”: in the former, the optimal Bloch angles $(\theta^*,\phi^*)$ jump discontinuously with simultaneous cusp in $|\lambda_1|$, divergence of $\partial_\mu|\lambda_1|$, and nonlocal-gap closing; in the latter, a strong anisotropy pins one angle so that spectral indicators remain critical while the optimal basis geometry stays fixed [2512.14302]. ## 4. Information geometry, configuration-space geometry, and bounded domains Information geometry supplies one of the most systematic uses of the term. The parameter manifold with coordinates $\lambda=(\lambda^1,\lambda^2)$ carries metric $$ g_{ij}(\lambda)=\partial_i\partial_j\Phi(\lambda), $$ and near a curvature singularity of a two-dimensional manifold the Ricci scalar and affine parameter scale as $$ R\sim x^{-2},\qquad s\sim x^{-1}, $$ defining geometric critical exponents $\alpha_g=2$ and $\nu_g=1$ [1405.3212]. The same exponents were found for the Van der Waals gas, Curie–Weiss ferromagnet, one-dimensional Ising chain, transverse XY chain, and RN–AdS black hole, which the paper presents as evidence for universality across classical and quantum models [1405.3212]. Configuration-space statistical geometry shifts attention from parameter space to the ensemble of sampled configurations [2508.00787]. For Ising variables, the normalized Hamming distance $$ r_H(s,s')=\sum_{i=1}^N \frac{1-s_is_i'}{2N} $$ has variance determined by magnetization and two-point correlators. At a $Z_2$ symmetry-breaking critical point with $m_\gamma=0$ and $4\beta/\nu