Topological Ising-like Fixed Points

• Topological Ising-like fixed points are universal critical points in lattice models that incorporate nontrivial topological invariants and categorical data beyond conventional symmetry-breaking.
• They are characterized by algebraic and categorical frameworks such as fusion categories and TQFT, enabling explicit construction of partition functions and dualities in enriched systems.
• Their study reveals practical insights into edge phenomena, duality transformations, and renormalization-group flows that unify conventional Ising and novel topological phases.

Topological Ising-like fixed points are universal critical points or renormalization-group fixed points in statistical and quantum lattice models that, while sharing certain structural features with the classical Ising fixed point, are distinguished by the presence of nontrivial topological invariants, protected edge phenomena, or categorical (fusion-theoretic) data. Such fixed points emerge in models with enriched symmetry, interaction range, boundary conditions, or underlying topology, and represent critical phenomena which are fundamentally beyond the reach of the pure symmetry-breaking Ginzburg–Landau–Wilson paradigm.

1. Algebraic and Categorical Frameworks Underpinning Topological Ising-like Fixed Points

The formalism of topological Ising-like fixed points is naturally articulated in the language of spherical fusion categories and topological quantum field theory (TQFT). Generic Ising-type statistical models, or their generalizations, admit categorical symmetries specified by a unitary semisimple spherical fusion category $\mathcal{C}$. Objects ${ X_i }_{i\in I}$ come equipped with tensor fusion, dualities $X_i^\vee$, fusion multiplicities $N_{ij}^k \in \mathbb{Z}_+$, associators encoded by $F$-symbols $(F^{ijk}_\ell)$, and pivotal (spherical) structure yielding quantum dimensions $d_{X_i}$ obeying $\sum_k N_{ij}^k d_k = d_i d_j$ (Delcamp et al., 2024). The categorical formalism organizes the set of possible gapped symmetric phases, algebra objects, and their module categories, and allows for explicit constructions of lattice partition functions, topological-sector projectors, and the interpretation of dualities and topological invariants.

For example, the Ising fusion category has three simple objects $\{1,\sigma,\psi\}$ with fusion rules $1 \times$ any $=$ any, $\psi\times\psi=1$, $\sigma\times\sigma=1+\psi$, $\sigma\times\psi= \psi\times\sigma = \sigma$, quantum dimensions $d_1 = d_\psi = 1$, $d_\sigma = \sqrt{2}$, and $F$-, $R$-symbols controlling associativity and braiding (Bauer et al., 2021).

2. Topological Order and Criticality in Exactly Solvable and Cluster-Ising Chains

Critical points separating symmetry-broken and symmetric or symmetry-protected topological (SPT) phases can exhibit topologically distinct Ising-like character. For the cluster-plus-transverse-field Ising chain

$H = \lambda H_{\rm TFI} + (1-\lambda) H_{\rm CI}$

with

$$H_{\rm TFI} = -\sum_{j=1}^{N-1}\sigma_j^x \sigma_{j+1}^x - h\sum_{j=1}^N \sigma_j^z,\qquad H_{\rm CI} = -\sum_{j=1}^{N-1}\sigma_j^x \sigma_{j+1}^x + h\sum_{j=1}^{N-2}\sigma_j^x \sigma_{j+1}^z \sigma_{j+2}^x$$

the model admits a global $\mathbb{Z}_2$ (spin-flip) and $\mathbb{Z}_2^T$ (time-reversal) symmetry (Yu et al., 2024). The critical manifold includes:

• Trivial Ising fixed points ($c=1/2$, $z=1$) with disorder operators $\mu(j)\sim\prod_{i=-\infty}^j \sigma_i^z$ carrying $T\mu T = +\mu$;
• Symmetry-enriched Ising$^\ast$ points with disorder operators $$\mu_*(j)\sim\prod_{i=-\infty}^j \sigma_{i-1}^x \sigma_i^z \sigma_{i+1}^x$$ and $T\mu T = -\mu$;
• A multicritical Lifshitz transition at $(h=1,\lambda=0.5)$, $z=2$, non-conformal, with quadratic dispersion and anomalous scaling for spin correlations;
• Gaussian (free boson) $c=1$ fixed points along the SPT–paramagnet boundary.

Topological invariants include string-order parameters (diagnosing SPT order), Bogoliubov winding numbers, and the time-reversal charge of disorder operators. Edge phenomena manifest as two-fold boundary degeneracy on the Ising$^\ast$ line. The overall phase diagram is organized by three distinct RG fixed points: trivial Ising, SPT-enriched Ising$^\ast$, and Gaussian, merging at the non-conformal Lifshitz point (Yu et al., 2024).

3. Duality, Topological Sector Projectors, and Non-Abelian Kramers–Wannier Dualities

On two-dimensional graphs, partition functions $Z(\theta)$ of generalised Ising-like models can be twisted by flat $G$-connections and are naturally understood via topological sector projectors—also viewed as 't Hooft line insertions or holonomies in the TFT framework (Delcamp et al., 2024). Gauging a subsymmetry $A\subseteq G$ in lattice models yields sums over flat $A$-connections:

$$\widetilde{Z}^{\widehat{A}}(\theta) = |A|^{-1} \sum_{a \in A^E,\,\partial a = 1} Z(\theta; a)$$

Partition functions admit equivalent representations as boundary theories of 3D TFTs with input fusion category $\mathcal{C}$. The Fourier transform on group weights maps Ising models (with $\mathcal{C} = \text{Vect}_G$) to dual "Fourier–Ising" models with $\mathcal{C} = \text{Rep}\,G$ on the dual lattice. The full non-abelian Kramers–Wannier duality is realized as the composition "gauging $+$ Fourier," permuting algebra objects and topological sectors in accord with categorical Morita equivalence (Delcamp et al., 2024).

4. Topological Features in Quantum and Classical Ising Systems

In quantum Ising and related models, critical points often correspond to topological transitions in the Fermi surface of the underlying fermionized Hamiltonian (e.g., via Jordan–Wigner transformation). The one-dimensional transverse-field Ising model (TFIM) undergoes a Lifshitz transition at criticality, characterized by changes of Berry phase (Zak phase), Chern number, and $\mathbb{Z}_2$ winding number. At $h=J$ in TFIM,

$\gamma=0 \;\;(h>J), \quad \gamma=\pi \;\;(h<J)$

$(-1)^\Omega = \sgn(h)\, \sgn(h-J)$

and an emergent $SU(2)_1$ WZNW symmetry appears at criticality, connecting to the Heisenberg chain and lending the fixed point a topological character not present in the Landau paradigm (Jalal et al., 2016). The bulk–boundary correspondence is accounted for by real-space Thouless pumps and anomaly inflow, unifying aspects of topological phases, Ising criticality, and topological insulator edge effects.

In constrained Ising antiferromagnets on the kagome lattice, the devil's staircase of plateaus in defect density $\rho = n_C / n_A \in \mathbb{Z}$ is controlled by an infinite sequence of topological fixed points, each associated with a particular integer sector in the density of macroscopic strings. Each plateau is a distinct RG fixed point—an example of topological fixed points in a purely classical context (Rufino et al., 9 May 2025).

5. Renormalization Group and Classification of Topological Fixed Points

Topological Ising-like fixed points universally arise as special solutions to RG beta functions in the space of possible couplings, often corresponding to separable Frobenius algebras $\mathcal{A}\in\mathcal{C}$. In the functional RG analysis of Ising models coupled to dynamical triangulations, three classes of fixed points are distinguished (Barouki et al., 1 Apr 2025):

• Gaussian (topological gravity) fixed point: no relevant deformations; all perturbations are irrelevant; $c=0$ theory.
• Pure gravity fixed point: one relevant direction (cosmological constant).
• Ising-like fixed point: three relevant directions, matching the primary fields (identity, energy $\epsilon$, and spin $\sigma$) of the $c=1/2$ Ising minimal CFT coupled to gravity.

Renormalization group flows in topological Ising-like systems may connect families of fixed points, such as the clean Ising CFT ($c=1/2$), infinite-randomness fixed points ($c_{\rm eff} = \frac{1}{2}\ln 2$), and novel quasiperiodic fixed points with robust edge degeneracies and intermediate effective central charge $c_{\rm eff} \approx 0.63$ (Yang et al., 1 Feb 2026).

6. Chiral, Boundary, and Higher-Dimensional Extensions

Standard commuting-projector state-sum models (e.g., Turaev–Viro) cover non-chiral topological phases. To capture chiral topological Ising-like fixed points (e.g., in the 2+1D Ising UMTC, $c=1/2$), generalized "vertex-liquid" fixed-point ansatzes are required. These models assign local data (object labels, face weights, tetrahedral tensors) consistent with fusion and braiding and permit phase anomalies under Pachner moves encoded by $e^{2\pi i c/24\,\Delta P_1}$, where $c$ is the chiral central charge (Bauer et al., 2021). The Ising UMTC then admits a state-sum construction with partition function $Z(M)$ consistent with the modular $S$-matrix and fusion multiplicities.

Protected boundary phenomena are ubiquitous: e.g., robust Majorana zero modes at the edges of topological Ising chains, fractional edge excitations in stripe phases of frustrated magnets, and winding-number protected sectors in dimer models and their critical height representations (Smerald et al., 2016, Yang et al., 1 Feb 2026). Such features fundamentally distinguish topological Ising-like fixed points from conventional ones.

7. Synthesis and Implications

Topological Ising-like fixed points unify a broad landscape of critical behavior in quantum and classical lattice systems, extending and transcending the traditional Ising universality class by incorporating categorical symmetry, dualities, string order, edge modes, and topological invariants. Their study leverages algebraic, field-theoretic, and numerical frameworks and reveals the structure of universality classes in systems ranging from fracton and SPT phases, chiral spin liquids, and topological insulators to exactly solvable statistical and quantum models. Their classification and properties are deeply rooted in the categorical (fusion-theoretic) and topological properties of the underlying models, and their consequences encompass both rigorous theoretical phenomena and experimentally probed observables (Yu et al., 2024, Delcamp et al., 2024, Jalal et al., 2016, Rufino et al., 9 May 2025, Yang et al., 1 Feb 2026, Bauer et al., 2021, Smerald et al., 2016, Barouki et al., 1 Apr 2025).