Topological Multicritical Points

• Topological multicritical points are loci in parameter space where several topological phase boundaries intersect, revealing overlapping critical and unstable fixed-point curves.
• They emerge in one-dimensional chiral systems and beyond through simultaneous gap closings, anomalous curvature-function scaling, and transitions between distinct winding-number phases.
• Analytical frameworks like curvature renormalization group and invariant zero analysis elucidate their role in mediating transitions in both gapped and gapless regimes.

Searching arXiv for the specified paper and closely related work on topological multicritical points. Topological multicritical points are loci in control-parameter space where several topological phase boundaries intersect and where the organization of criticality is constrained by topological data rather than solely by conventional symmetry-breaking criteria. In the literature summarized here, the term covers several related but distinct settings: intersections of curvature-function critical lines and unstable fixed-point curves in one-dimensional chiral systems (Abdulla et al., 2020); multicritical points between gapless critical phases with different topological character (Kumar et al., 2020, Kumar et al., 2022, Kumar et al., 2021); multicritical points that themselves remain topologically nontrivial and host localized zero modes despite a gapless bulk (Kumar et al., 15 Jul 2025); and topologically enforced Lifshitz multicriticality in which topology of neighboring critical lines forces a higher-order band touching (Chou et al., 5 Jun 2026). Across these settings, recurring signatures include simultaneous gap closings at different momenta, changes in winding-number data or related invariants, anomalous curvature-function scaling, and a renormalization-group structure in which critical manifolds, fixed manifolds, and multicritical junctions are geometrically intertwined.

1. Curvature-function formulation and the basic multicritical geometry

A central framework for topological multicriticality is the curvature renormalization group (CRG), in which a topological invariant is written as an integral over a curvature function $F(\mathbf{k},\mathbf{M})$, with $\mathbf{M}$ denoting couplings or masses (Abdulla et al., 2020). In one dimension, for the chiral models emphasized there, the invariant is a winding number,

$W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$

and the curvature function is the derivative of a momentum-dependent phase or, equivalently, a one-dimensional Berry connection in a chiral gauge (Abdulla et al., 2020).

Within this framework, a topological phase transition is identified by two simultaneous conditions: the bulk gap closes at some momentum $k_0$, and the curvature function diverges at that momentum. Such points form critical curves in parameter space. The CRG flow is defined so as to preserve the topological invariant while smoothing the curvature peak near a chosen $k_0$; the flow diverges at critical points and vanishes on fixed-point curves where the curvature becomes flat around the expansion momentum (Abdulla et al., 2020).

The distinctive multicritical result is that in the one-dimensional BDI and AIII examples studied in "Curvature function renormalisation, topological phase transitions and multicriticality" (Abdulla et al., 2020), unstable segments of fixed-point lines overlap with critical lines associated with other momenta. The paper states that "the unstable part of the fixed point line overlaps with a critical line, thereby showing that unstable fixed point lines also denote topological phase transitions," and that "at the crossing point where an unstable fixed point line meets or intersects a critical curve, three phases coexist and hence the crossing point turns out to be a multi-critical point" (Abdulla et al., 2020). In this language, a topological multicritical point is therefore a point where a critical curve and an unstable fixed-point curve intersect and where three or more topological phases coexist.

This geometry is not merely pictorial. It implies that local curvature data near a chosen momentum can encode nonlocal information about other transitions in the phase diagram. A plausible implication is that CRG can reconstruct a larger portion of the phase-boundary network than the local derivation of the flow equations might initially suggest.

2. One-dimensional chiral chains: BDI and AIII realizations

The two canonical realizations in (Abdulla et al., 2020) are an extended Kitaev chain in class BDI and an extended SSH chain in class AIII. In both cases the Hamiltonian is $2\times2$, chiral, and characterized by an integer winding number.

For the BDI model, the Bogoliubov–de Gennes Hamiltonian is

$H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$

with

$$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$

and the curvature function

$$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$

where $\tan\theta_k=d_3/d_2$ (Abdulla et al., 2020). Gap closure requires $\mathbf{M}$0, giving high-symmetry critical lines $\mathbf{M}$1 at $\mathbf{M}$2, together with a non-high-symmetry critical line

$\mathbf{M}$3

with $\mathbf{M}$4 (Abdulla et al., 2020). The phase diagram contains regions with $\mathbf{M}$5 and three multicritical points labeled $\mathbf{M}$6 (Abdulla et al., 2020).

For the AIII model, the momentum-space Hamiltonian is

$\mathbf{M}$7

with

$\mathbf{M}$8

and curvature function

$\mathbf{M}$9

(Abdulla et al., 2020). For fixed $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$0, gap closures occur along three critical lines in the $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$1-$W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$2 plane, including high-symmetry lines at $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$3 and a non-high-symmetry line at $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$4 for $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$5, or at $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$6 for $W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$7 (Abdulla et al., 2020).

In both models, the CRG derived around a high-symmetry gap-closing momentum produces not only the corresponding high-symmetry critical line but also a fixed line whose unstable segment coincides with another critical line. This is the specific mechanism by which multicriticality appears as an overlap between critical and unstable fixed structures (Abdulla et al., 2020).

3. Critical lines, non-high-symmetry criticality, and transitions between critical phases

A major extension beyond gapped-to-gapped topological transitions is the existence of topological transitions between gapless phases. The transverse-field Ising chain with three-spin interaction studied in "Multi-critical topological transition at quantum criticality" (Kumar et al., 2020) provides a representative example. In Jordan–Wigner fermion form, it becomes an extended Kitaev chain with Bloch Hamiltonian

$W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$8

where

$W=\frac{1}{2\pi}\int dk\,F(k,\mathbf{M}),$9

The gap closes along three lines in $k_0$0 space: $k_0$1 at $k_0$2, $k_0$3 at $k_0$4, and $k_0$5 at incommensurate $k_0$6 (Kumar et al., 2020).

The line $k_0$7 splits into two distinct gapless phases, CP-1 and CP-2, separating $k_0$8 and $k_0$9 gapped phases, respectively (Kumar et al., 2020). These gapless phases have different universality classes: CP-1 has $k_0$0, whereas CP-2 has $k_0$1, and the point $k_0$2 is a topologically active multicritical point in the Lifshitz universality class (Kumar et al., 2020). The paper explicitly identifies two multicritical points: one topologically trivial at $k_0$3, and one topologically active at $k_0$4, where the transition between gapless phases occurs (Kumar et al., 2020).

The companion development in "Topological phase transition between non-high symmetry critical phases and curvature function renormalization group" (Kumar et al., 2022) generalizes this structure to non-high-symmetry critical phases. There, a one-dimensional two-band Bloch Hamiltonian

$k_0$5

with up to third-neighbor terms supports high-symmetry critical lines at $k_0$6 and non-high-symmetry critical lines at paired momenta $k_0$7 (Kumar et al., 2022). Along the non-high-symmetry critical lines, the system is gapless yet can be either topologically trivial or topologically non-trivial, distinguished by the decay length of edge modes and a winding number defined at criticality (Kumar et al., 2022).

The linear multicritical points $k_0$8 and $k_0$9, located at $2\times2$0 for fixed $2\times2$1, separate trivial and topological non-high-symmetry critical phases (Kumar et al., 2022). Across these points the winding number at criticality jumps by an integer, and the edge-mode decay length diverges (Kumar et al., 2022). This is a direct realization of a topological transition between critical phases rather than between gapped phases.

"Signatures of topological phase transition on a quantum critical line" (Kumar et al., 2021) analyzes closely related physics in a generic extended SSH/Kitaev-type model with $2\times2$2. With $2\times2$3, the multicritical points $2\times2$4 at $2\times2$5 and $2\times2$6 at $2\times2$7 separate topological and non-topological critical phases along high-symmetry critical lines (Kumar et al., 2021). The critical phases are gapless on both sides; the difference is the presence or absence of localized edge modes. The decay length of the Dirac bound-state solution diverges at the multicritical point, and the generalized winding number on the critical line changes accordingly (Kumar et al., 2021).

These works collectively correct a common misconception: gaplessness does not by itself erase topological distinctions. Gapless critical phases can remain topologically distinct, and multicriticality can mediate transitions between them (Kumar et al., 2020, Kumar et al., 2022, Kumar et al., 2021).

4. Topologically nontrivial and topologically enforced multicritical points

A more stringent notion appears in "Topologically nontrivial multicritical points" (Kumar et al., 15 Jul 2025). In a third-neighbor chiral chain described by

$2\times2$8

with

$2\times2$9

the authors fix $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$0 and $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$1 and study the $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$2 plane (Kumar et al., 15 Jul 2025). In the regime $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$3, only nontrivial gapped phases with $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$4 occur, separated by topological critical lines whose invariants $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$5 remain nonzero (Kumar et al., 15 Jul 2025).

The topology of both gapped and critical phases is formulated in terms of the zeros of the complex function

$H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$6

where the topological invariant is the number of zeros inside the unit circle, $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$7, because there are no poles (Kumar et al., 15 Jul 2025). The multicritical points $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$8 and $H(k)=\sum_{i=1}^3 d_i(k)\sigma_i,$9, at $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$0 and $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$1, are intersections of high-symmetry and non-high-symmetry critical lines (Kumar et al., 15 Jul 2025). They remain gapless, have quadratic dispersion $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$2 with $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$3, and nevertheless host one localized zero mode per edge: $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$4 (Kumar et al., 15 Jul 2025).

This notion is sharper than the earlier CRG-based definition because the multicritical point itself is topologically nontrivial. It is characterized by one zero strictly inside the unit circle and two degenerate zeros on the unit circle, the latter generating the quadratic band touching (Kumar et al., 15 Jul 2025). The paper further shows that the discriminant of the cubic polynomial,

$$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$5

vanishes at the multicritical point and changes sign across it along a high-symmetry critical line. In that model, $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$6 is necessary but not sufficient for multicriticality; the sign change across the point distinguishes a topological multicritical point from a trivial one (Kumar et al., 15 Jul 2025).

A related but conceptually distinct proposal appears in "Topologically Enforced Lifshitz Multicriticality in One Dimension" (Chou et al., 5 Jun 2026). There, one-dimensional chiral chains labeled by an integer $$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$7 define distinct topological critical lines,

$$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$8

with Bloch function

$$d_1(k)=0,\quad d_2(k)=-2\lambda_1\sin k-2\lambda_2\sin 2k,\quad d_3(k)=2g-2t_1\cos k-2t_2\cos 2k,$$9

(Chou et al., 5 Jun 2026). Different $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$0-critical lines share the same local critical data, $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$1, $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$2, and $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$3, but are topologically distinct as critical phases because they carry different edge-mode and entanglement-spectrum structures (Chou et al., 5 Jun 2026).

Interpolating between two such critical lines with indices $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$4 and $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$5 while forcing all topology-changing zeros of the auxiliary complex function to collide at a single momentum yields a topologically enforced Lifshitz multicritical point with

$$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$6

and dynamical exponent

$$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$7

(Chou et al., 5 Jun 2026). The multicriticality is therefore driven by topology of neighboring critical lines rather than by a change of central charge or standard universality class. A plausible implication is that multicriticality can be symmetry-enforced in the space of critical theories themselves, not only at boundaries between gapped phases.

5. Scaling theory, CRG flows, and correlation functions near multicriticality

Across the one-dimensional examples, the curvature function typically acquires a Lorentzian or generalized Lorentzian form near the relevant critical momentum. In (Abdulla et al., 2020), near a high-symmetry critical point one has

$$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$8

with $$F(k,\mathbf{M})=\frac{d\theta_k}{dk} =\frac{d_2(k)\partial_k d_3(k)-d_3(k)\partial_k d_2(k)}{d_2(k)^2+d_3(k)^2},$$9 and $\tan\theta_k=d_3/d_2$0. For the BDI and AIII models discussed there, $\tan\theta_k=d_3/d_2$1 along the high-symmetry critical lines (Abdulla et al., 2020).

In (Kumar et al., 2022), the same Ornstein–Zernike form is adapted to non-high-symmetry criticality and to multicriticality between critical phases. The critical exponents at both non-high-symmetry critical points and the linear multicritical points satisfy

$\tan\theta_k=d_3/d_2$2

(Kumar et al., 2022). In (Kumar et al., 2020), by contrast, the Lifshitz-type multicritical point between gapless phases has $\tan\theta_k=d_3/d_2$3, and the scaling law becomes $\tan\theta_k=d_3/d_2$4 when the dominant curvature form is quartic in momentum (Kumar et al., 2020).

In "Multi-critical Behavior in Topological Phase Transitions" (Rufo et al., 2019), multicriticality in generalized SSH models is characterized in terms of the penetration depth of edge modes, identified with the correlation length, together with Berry-connection scaling. The synthetic-potential model exhibits a line $\tan\theta_k=d_3/d_2$5 of multicritical points with $\tan\theta_k=d_3/d_2$6, and the Berry connection behaves differently depending on the path of approach: it may remain finite at the gap-closing momentum while its width shrinks, or it may diverge directly, even though the same multicritical point is approached (Rufo et al., 2019). This path dependence is itself a hallmark of multicritical structure.

The Wannier-state correlation function provides a complementary bulk diagnostic. In (Kumar et al., 2020), the Fourier transform of the curvature function defines

$\tan\theta_k=d_3/d_2$7

which acts as a Wannier-state correlation function and decays with a correlation length $\tan\theta_k=d_3/d_2$8 that diverges at the transition (Kumar et al., 2020). In (Kumar et al., 2022), the analogous $\tan\theta_k=d_3/d_2$9 at criticality near multicritical points exhibits the same diverging length scale and oscillatory structure reflecting non-high-symmetry gap closings (Kumar et al., 2022). In (Malard et al., 2020), a basis-independent Wannier-state skew correlation function is derived for a one-dimensional class CII topological insulator; its decay length diverges at both ordinary critical lines and the multicritical point, even when the two phases on either side share the same winding number (Malard et al., 2020).

The CRG perspective remains especially useful because it encodes multicriticality geometrically. In (Abdulla et al., 2020), the intersection of a critical curve and an unstable fixed curve defines the multicritical point. In (Kumar et al., 2022), generalized CRG equations based on non-high-symmetry peak tracking identify conventional topological transitions through non-high-symmetry critical points and also the transitions between critical phases through multicritical points (Kumar et al., 2022). In (Kumar et al., 2021), the multicritical points appear as the loci where one line is critical and another is an unstable fixed line, depending on which expansion momentum is chosen (Kumar et al., 2021). These constructions suggest that multicriticality in topological systems is often encoded in the topology of RG flow itself.

6. Entanglement, disorder, and broader extensions

Entanglement spectra provide additional resolution when conventional bulk invariants are ill-defined. In (Kumar et al., 2021), the two multicritical points $\mathbf{M}$00 and $\mathbf{M}$01 are sharply distinguished by entanglement entropy. At $\mathbf{M}$02, the entropy is maximal along the critical line and scales as

$\mathbf{M}$03

with $\mathbf{M}$04, consistent with a conformal critical point (Kumar et al., 2021). At $\mathbf{M}$05, by contrast, the entropy shows a minimum, reflecting the fact that this point is simultaneously a multicritical point and a fixed point from the viewpoint of the gapped-phase RG flow (Kumar et al., 2021). This demonstrates that topological multicritical points need not all be conformal in the same way, even within a single model.

The topologically nontrivial multicritical points of (Kumar et al., 15 Jul 2025) are robust to weak disorder. When randomness is introduced in $\mathbf{M}$06 and $\mathbf{M}$07, the multicritical points survive although they shift in parameter space, and their zero-energy edge modes persist (Kumar et al., 15 Jul 2025). At strong disorder, the clean multicritical points are washed out and replaced by a gapless Anderson-localized phase with zero-energy edge modes, described there as a topologically nontrivial gapless Anderson-localized phase (Kumar et al., 15 Jul 2025). This indicates that disorder can both preserve and qualitatively transform topological multicritical structures.

A distinct disorder-driven realization occurs in two dimensions in "Multicriticality of Two-dimensional Class D Disordered Topological Superconductors" (Wang et al., 2021). There the tricritical point TCP1 is where three phases meet: diffusive thermal metal, thermal quantum Hall phase with $\mathbf{M}$08, and thermal quantum Hall phase with $\mathbf{M}$09 (Wang et al., 2021). It is repulsive under RG and has its own critical exponents,

$\mathbf{M}$10

distinct from both the metal–insulator transitions and the plateau transitions (Wang et al., 2021). This is a topological multicritical point in a broader sense: it is where topology-changing plateau transitions terminate against a metallic phase.

Surface topological quantum criticality in interacting systems offers another extension. "Surface topological quantum criticality: Conformal manifolds and Discrete Strong Coupling Fixed Points" (Vijayan et al., 2024) studies surfaces of three-dimensional topological insulators with $\mathbf{M}$11 half-Dirac cones and attractive pairing interactions. For $\mathbf{M}$12, the one-loop RG features a ring conformal manifold of fixed points; for $\mathbf{M}$13, an $\mathbf{M}$14 conformal manifold appears (Vijayan et al., 2024). In both cases, strongly interacting isolated fixed points with many relevant directions occur as endpoints or special loci in the phase-boundary manifold and are interpreted there as multicritical surface topological quantum critical phenomena rather than generic surface criticality (Vijayan et al., 2024). This suggests that topological multicriticality can naturally generalize from isolated points to higher-dimensional fixed manifolds.

7. Conceptual synthesis and recurring distinctions

Several distinctions recur across the literature.

First, there is a distinction between a multicritical point that merely sits at the intersection of topological phase boundaries and one that is itself topologically nontrivial. The CRG-based multicritical points of (Abdulla et al., 2020) are identified geometrically through intersecting critical and unstable fixed structures. The multicritical points of (Kumar et al., 15 Jul 2025) go further: they themselves carry localized zero modes and a nonzero invariant $\mathbf{M}$15.

Second, there is a distinction between multicriticality driven by changing universality class and multicriticality driven by topology of critical lines. The gapless-to-gapless transition in (Kumar et al., 2020) changes from Ising-like to Lifshitz-like behavior, so topology and universality change together. In (Chou et al., 5 Jun 2026), by contrast, neighboring critical lines have the same $\mathbf{M}$16, $\mathbf{M}$17, and $\mathbf{M}$18, and the Lifshitz multicriticality is enforced solely by their topological inequivalence.

Third, there is a distinction between ordinary bulk–boundary correspondence and its failure at multicriticality. The works (Kumar et al., 2021, Kumar et al., 2022), and (Kumar et al., 15 Jul 2025) all use edge modes to characterize critical or multicritical topology. But (Chou et al., 5 Jun 2026) shows that entanglement-spectrum topology can survive at a Lifshitz multicritical point even when physical zero modes do not, because the higher-order derivative structure spreads the real-space coupling over several sites and prevents dangling boundary degrees of freedom.

Finally, there is a distinction between control-parameter scaling and gap scaling. In the class CII Aubry–André–Harper model of (Malard et al., 2020), ordinary critical lines and the multicritical point differ in the order of nonanalyticity of the ground-state energy and in how the spectral gap closes with the control parameter. Yet the critical exponents defined with respect to the gap remain the same in both cases (Malard et al., 2020). This suggests that what appears as higher-order topological criticality in parameter space can still belong to the same gap-based scaling structure.

Taken together, these results establish that topological multicritical points are not a single phenomenon but a family of related structures. They may be defined geometrically in RG flow (Abdulla et al., 2020), spectrally via simultaneous gap closings at multiple momenta (Kumar et al., 2022, Kumar et al., 2021), algebraically through zeros and discriminants of complex functions (Kumar et al., 15 Jul 2025), or entanglement-theoretically through protected midgap degeneracies (Chou et al., 5 Jun 2026). What unifies them is that multicriticality is organized by topological data of phases or critical lines, not solely by the conventional Landau pattern of relevant directions and symmetry breaking.