Emergent Order Spectrum: Unifying Collective Phenomena

• Emergent Order Spectrum is a framework that classifies complex, collective order in macroscopic, mesoscopic, and microscopic systems using entanglement spectra, topological invariants, and order-type hierarchies.
• It employs methods such as Anderson towers, chain recurrences, and excitation scaling to reveal hidden orders and continuous symmetries in models ranging from quantum spin chains to active matter dynamics.
• This unifying approach offers practical insights for experimental design and theoretical classification by bridging quantum criticality, topological phases, and dynamical order in diverse systems.

The Emergent Order Spectrum encompasses a diverse set of frameworks, physical observables, and mathematical invariants that classify how complex, collective order arises in macroscopic, mesoscopic, and microscopic dynamical systems—from quantum critical models and active matter to interacting dynamical systems and quantum spin chains. Across domains, the Emergent Order Spectrum is understood either in terms of the excitation or entanglement spectra, the organization of low-energy modes (Goldstone, topological, and collective), or in terms of topological and order-theoretic invariants. This spectrum encodes both continuous symmetry breaking (e.g., Goldstone bosons), topological order (e.g., visons, anyons), and order-type hierarchies (e.g., scattered orders), revealing the full landscape of collective phenomena accessible to a system under specific microscopic or dynamical rules.

1. Emergent Order Spectrum via Entanglement: Anderson Towers and Continuous Symmetry

In quantum many-body systems at criticality, the Emergent Order Spectrum can be uncovered by diagonalizing the reduced density matrix $\rho_A$ of a finite subsystem $A$, yielding the entanglement spectrum of an effective "entanglement Hamiltonian" $H_E = -\ln \rho_A$. This spectrum exposes the Anderson tower of states corresponding to symmetry-broken phases, without the need to identify an explicit order parameter. The characteristic scaling in the entanglement spectrum distinguishes the emergent continuous symmetry group, e.g., SO(N), and is given by

$$E_{\mathcal J}(L) - E_0(L) = \frac{c^2}{2\rho_S L^d} \mathcal J(\mathcal J + N - 2),$$

where $\mathcal{J}$ labels the Casimir eigenvalue, $L$ is the subsystem length, $d$ its dimensionality, and $N$ the effective symmetry group dimension.

Representative results include:

• O(3) Anderson tower in the columnar dimer Heisenberg model, visible as linear scaling in $S(S+1)$ in the entanglement spectrum at criticality.
• O(5) symmetry at the Néel–VBS deconfined critical point, detected by the straight-line scaling of the entanglement spectrum as $S(S+3)$.
• O(4) symmetry at the checkerboard J–Q model’s first-order transition, observed by linear scaling in $S(S+2)$.

This unbiased approach robustly identifies hidden or non-local orders and is resilient to finite-size effects, thus defining the Emergent Order Spectrum as the organization of low-lying entanglement levels into multiplets of an emergent symmetry group (Mao et al., 11 Jun 2025).

2. Emergent Order Spectra in Nonlinear Dynamics: Ω-Spectrum for Homeomorphisms

In the context of dynamical systems, the Emergent Order Spectrum $\Omega(x,y)$ is a topological invariant characterizing the order-types realized as limits of order-compatible nested $\varepsilon_n$-chains ($\varepsilon_n\to 0$) connecting points $x$ and $y$ under a transitive homeomorphism $f$ on a compact metric space. For a system of continuum cardinality, the global spectrum $\Omega_f(X^2)$ is universal at the countable scattered level—every countable scattered order-type and the rationals (dense order $\eta$) appear.

Formally, given order-compatible nested acyclic chains $\{ C_n \}$, the induced linear order on the union set $C$ allows one to extract otp$(C, \leq_\infty)$, the order-type in the spectrum. For transitive homeomorphisms, a comeagre set $\mathcal{M} \subseteq X^2$ achieves every countable infinite scattered order-type, while $\eta$ (the rationals) always appears for any $(x,y) \in X^2$. The table below summarizes main results:

Theorem	Content
Existence of finite ordinals	$K \in \Omega_f(x,y) \iff y \in O^+(x)$
Universality of $\eta$	$\eta \in \Omega_f(x, y)$ for all $(x, y)$ if $f$ is transitive and $|X| = \mathfrak{c}$
Universality at countable scattered level	Every countable scattered order-type and $\eta$ realized in comeagre subset of $X^2$

This invariant ties together the chain-recurrence theory and Cantor–Bendixson analysis, enabling classification of dynamical systems up to conjugacy by their emergent order spectra (Ciavattini et al., 14 Jan 2026).

3. Emergent Order Spectra in Topological and Quantum Systems: Anyons, Goldstone Modes, and solitons

Excitation spectra near quantum criticality are governed by emergent algebraic structures that dictate both the multiplicities and topological nature of the elementary modes:

• Affine Toda Field Theory in CoNb$_2$O$_6$: The spin excitations near the 1D QCP in CoNb$_2$O$_6$, dressed by interchain frustration, follow the $D_8^{(1)}$ affine Toda spectrum. The spectrum consists of 8 relativistic modes with masses

$m_a = 2 m_1 \sin(a\pi/28), \quad a=1,\ldots,8,$

including solitons, antisolitons, and breathers. The scaling $m_a \sim J_i^{4/7}$ and the concordance with numerical spectra highlight both integrability and topological (soliton) order (Xi et al., 2024).

• Symmetry Topological Orders (symTO): Gapless 1+1D systems, such as the Ising and Potts chains, manifest emergent non-invertible (categorical) symmetries captured by a 2+1D topological order called maximal symTO. For the Ising critical point, the maximal symTO is double-Ising topological order $\mathcal{M}_{\mathrm{dIs}}$ of dimension $D^2=9$, classifying all symmetry and dual-defect sectors. In the spin-1/2 Heisenberg chain, the spectrum exhibits the quantum double of $D_8$ with emergent SO(4) symmetry, with topological sectors and current algebras organizing all low-energy behavior (Chatterjee et al., 2022, Chen et al., 10 Sep 2025).
• Coplanar Order and Emergent Frustration: At the AFM–PSS transition in the CBJQ model, emergent O(4) symmetry on a kagome-like three-sublattice geometry creates "emergent frustration," enforcing a coplanar order and resulting in five linearly dispersing Goldstone modes—three spin-like, two plaquette-like. The entanglement entropy scaling, $$S_2(L) = aL + \frac{N_G}{2}\ln L + \gamma_{\mathrm{ord}} + \ldots$$ with $N_G = 5$, further confirms the spectrum (Deng et al., 18 Apr 2025).

4. Emergent Order Spectra in Driven and Active Systems

In nonequilibrium soft-matter and active-matter systems, the Emergent Order Spectrum can describe how order parameters and collective modes respond as control parameters (e.g., noise and interaction strength) are varied:

• Vicsek Flocking with Heterogeneous Noise: Varying the outer-region noise $\eta_b$ in a confined Vicsek model produces a U-shaped global order parameter $\phi$ as a function of $\eta_b$, with emergent rotational order $\phi_r$ peaking at high noise. The underlying phase diagram and order-parameter susceptibilities map the re-entrant and rotational order regimes. Associated escape rates and segregation are governed by mobility and virtual trapping, yielding a well-characterized dynamical spectrum (Khan, 1 Sep 2025).
• Steady-State Superradiance in 1D Atom Baths: In arrays of two-level atoms coupled to 1D electromagnetic reservoirs (ring cavity or waveguide), the Emergent Order Spectrum is revealed in synchronized phase coherence, superradiant emission scaling ($R_{\max} \sim N^2$), and linewidth narrowing of the emitted light. In the ring cavity, spontaneous chirality emerges with robust spectral narrowing as $N$ increases, whereas in the bidirectional waveguide, local chiral domains limit this effect (Cardenas-Lopez et al., 13 Nov 2025).

5. Universal Features, Classification, and Significance

Emergent Order Spectra unify discrete (topological), continuous (symmetry-breaking), and dynamical (order-type or orbital) hierarchies:

• Maximal symTO as Classification Principle: For 1+1D gapless phases and their higher-dimensional generalizations, the maximal symmetry topological order (symTO)—a modular tensor category or higher category—classifies low-energy emergent order up to holo-equivalence. This approach bridges emergent continuous symmetries and non-invertible/categorical symmetries (Chatterjee et al., 2022).
• Order-Type Hierarchies and Chain Spectra: For transitive dynamical systems, the Emergent Order Spectrum encodes all possible countable linear orders realized via limit chains, revealing a hierarchical organizing principle independent of energy spectra (Ciavattini et al., 14 Jan 2026).
• Goldstone Counting and Entropic Signatures: In quantum or statistical models with emergent (coplanar or non-collinear) order, the number of Goldstone branches and their nature can be extracted from both excitation spectra and entropic scaling laws, providing direct experimental fingerprints (Deng et al., 18 Apr 2025).
• Topological Defect Expulsion and Gauge Structures: In correlated quantum systems, transitions between confining, Higgs, and topological phases produce a spectrum of visons, photons, spinons, chargons, and electrons, each with characteristic quantum numbers and scaling laws, forming a canonical emergent order spectrum in topological matter (Sachdev, 2018).

6. Interrelations and Applications

The Emergent Order Spectrum formalism interconnects:

• Quantum criticality (entanglement/Anderson towers, symTOs, Goldstone spectra)
• Topological phases (vison/anyon structures, fusion/braiding, emergent gauge fields)
• Out-of-equilibrium phase diagrams in active/flocking systems (order parameters, bistable and re-entrant behavior)
• Order-theoretic invariants in dynamical systems (limit orders, scattered/dense types, chain structures)

Applications range from classification of numerical data (e.g., through the entanglement spectrum or symmetry twists) to experimental design (e.g., neutron scattering for Goldstone branches, spectroscopy of soliton modes), to the topological and algebraic classification of quantum phases.

7. Outlook and Open Directions

Ongoing research seeks to:

• Extend classification of Emergent Order Spectra to higher dimensions using braided fusion $d$-categories and new invariants (e.g., for non-invertible symmetries or in non-metrizable dynamics) (Chatterjee et al., 2022).
• Uncover the impact of environmental heterogeneity and noise gradients on collective and rotational order in active systems (Khan, 1 Sep 2025).
• Exploit tunable quantum materials and synthetic systems (e.g., Rydberg atom arrays, frustrated ladders) to realize new emergent integrable spectra (e.g., $D_n^{(1)}$ Toda theories) (Xi et al., 2024).
• Systematically relate entanglement-derived order spectra to experimental observables in strongly correlated and topological quantum materials (Mao et al., 11 Jun 2025, Chen et al., 10 Sep 2025).

The Emergent Order Spectrum thus constitutes a central, unifying theme across quantum, classical, dynamical, and active systems, providing a rigorous, observable-dependent, and often topologically robust means to chart the structure of collective phenomena.