Ferromagnetic Scarred States: Theory & Models

• Ferromagnetic scarred states are nonthermal magnetic configurations defined by collective magnon excitations on a ferromagnetic background.
• They are generated via precise Hamiltonian architectures that combine local projector annihilators with Zeeman splitting to protect the scar manifold.
• These states emerge in both quantum many-body models and classical antidot arrays, offering insights into controlled dynamical revivals and topologically constrained textures.

Ferromagnetic scarred states are nonthermal, highly structured magnetic states that remain embedded in spectra or phase-space regions that are otherwise expected to be thermal or ergodic. In the quantum many-body setting, the term most directly denotes exact scar eigenstates that can be understood as collective magnon excitations on top of a ferromagnetic background, or as symmetric ferromagnetic multiplets protected by a specific Hamiltonian architecture. In the long-range transverse-field Ising model, it also denotes parity-resolved energy subspaces that retain nonzero magnetization in spectral regions where thermal equilibrium should be paramagnetic. A related but distinct classical usage arises in sub-micron ferromagnetic periodic antidot arrays, where the literature does not use the word “scar,” but the metastable vortex–antivortex textures can be interpreted as scar-like because the perforated geometry constrains defect localization and magnetization flow along structure-selected paths and sites (Omiya, 16 Jan 2026, Corps et al., 22 Jul 2025, Lee et al., 2020, Chertkov et al., 2021, Bogatyrëv et al., 2018).

1. Terminological scope and representative realizations

Taken together, the literature uses “ferromagnetic scarred states” in several technically precise but nonidentical senses. In all cases, the common element is the coexistence of a sparse, atypical, low-entanglement or topologically constrained state manifold with a much larger surrounding set of thermal or generic states.

Context	Scarred object	Defining structure
Ferromagnetic quantum many-body scars	Totally symmetric scar manifold	Collective fixed-momentum magnon excitations on a ferromagnetic background
Long-range TFIM	Two-dimensional parity subspaces $\mathcal E_n$ with nonzero $\lambda_n$	Ferromagnetic-like subspaces embedded among paramagnetic states
Staggered and motif magnetic models	Ferromagnetic multiplets or motif-compatible aligned states	Exact eigenstates protected by symmetry, staggering, or motif constraints
Periodic antidot arrays	Metastable vortex–antivortex textures	Geometry-imprinted, topologically constrained defect patterns

A common misconception is that ferromagnetic scarred states are simply conventional ferromagnets in unusual parameter regimes. The cited works show a narrower and more structured phenomenon. In some cases the scar manifold is a maximal-spin symmetric multiplet; in others it is a set of symmetry-breaking parity doublets; in motif constructions it can be a ferromagnetic or spiral-colored exact eigenstate family; and in antidot arrays the “scarred” characterization is interpretive rather than terminological (Omiya, 16 Jan 2026, Corps et al., 22 Jul 2025, Lee et al., 2020, Chertkov et al., 2021, Bogatyrëv et al., 2018).

2. Ferromagnetic quantum many-body scars as symmetric magnon towers

The most explicit definition is given by the theorem-level analysis of ferromagnetic quantum many-body scars. These states live in a totally symmetric subspace of a reduced on-site Hilbert space $\mathcal h^s \subset \mathcal h$, and can be generated by repeatedly applying a collective magnon-creation operator of fixed momentum to a ferromagnetic or product reference state. After a local unitary transformation that removes the momentum twist, they become fully symmetric Dicke-like states in $\mathrm{Sym}^N(\mathcal h^s)$. In the spin-1 XY example, the tower is written as

$$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$

and, after a gauge transformation $\hat U$,

$$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$

Within the reduced two-level subspace, these states span the maximal-spin symmetric multiplet, $$\{\ket{S_n}\}_{n=0}^L \cong \mathrm{Sym}^L(\mathcal h^s)$$. The paper therefore calls them “ferromagnetic” because they reproduce the same algebraic structure as a ferromagnet’s spin-$L/2$ multiplet, while remaining exact embedded scar states rather than ordinary low-energy ferromagnetic excitations.

The relation to magnons is structural rather than quasiparticle-theoretic in the conventional low-energy sense. The relevant procedure is: start from a polarized background state, apply a collective creation operator with fixed momentum, and obtain exact eigenstates that lie far outside the low-energy sector. The states are therefore “magnons” in the sense of collective algebraic construction, but not conventional low-energy quasiparticles. This distinction is essential for understanding why these states violate the generic eigenstate thermalization picture while remaining compatible with an otherwise thermal spectrum (Omiya, 16 Jan 2026).

3. Hamiltonian structure: Zeeman term, annihilator, and generalized Shiraishi–Mori architecture

For this ferromagnetic scar class, the Hamiltonian structure is sharply constrained. If a local Hamiltonian has the full set of totally symmetric weight-basis states as exact eigenstates, then it must decompose as

$\hat H=\hat H_A+\hat H_Z,$

where the annihilator satisfies

$\lambda_n$0

and the Zeeman term acts diagonally on the scar states. In the local formulation,

$\lambda_n$1

with the first two sums forming the annihilator sector and the final term $\lambda_n$2 the Zeeman term, expressed as a linear combination of on-site Cartan generators (Omiya, 16 Jan 2026).

A central lemma establishes that any operator annihilating the full symmetric scar manifold must contain strictly local projector factors. The proof uses Young symmetrizer decompositions, antisymmetrizing components of nontrivial Young tableaux, and a telescoping decomposition of non-nearest-neighbor transpositions into nearest-neighbor projector terms. The result is that the annihilation mechanism is always attributable to local forbidden components detected by projectors, even when the prefactors are more general.

This yields a converse-type statement for ferromagnetic scars: the generalized Shiraishi–Mori construction is not merely sufficient but “essentially exhaustive” for this class. The allowed non-annihilating dynamics within the scar manifold reduces to a simple collective Zeeman splitting. A practical implication is that coherent revival dynamics can be interpreted as collective large-spin precession within an exactly protected manifold rather than as an accidental finite-size phenomenon. The same Zeeman-splitting logic appears in explicit magnetic constructions, where evenly spaced levels generate periodic Loschmidt revivals from simple product states (Omiya, 16 Jan 2026).

4. Lattice and motif constructions: ferromagnetic multiplets, staggered interactions, and motif magnetism

Independent constructions realize ferromagnetic scarred states through local magnetic motifs and staggered sign structures. In the frustrated kagome XXZ setting at the special point $\lambda_n$3, valid three-colorings form exact product-state ground states. By introducing a staggered sign structure,

$\lambda_n$4

the same coloring states become exact zero modes embedded in the many-body spectrum. Beyond this frustrated route, the paper gives a directly ferromagnetic construction based on isotropic Heisenberg interactions on lattices with staggered motifs. There, SU(2) symmetry guarantees that the fully polarized ferromagnetic state $\lambda_n$5 and all members of its multiplet $\lambda_n$6 are exact eigenstates with zero energy in the absence of a magnetic field. A Zeeman term splits the multiplet and produces revivals with period $\lambda_n$7 in the Loschmidt echo for the fully $\lambda_n$8-polarized initial state $\lambda_n$9 (Lee et al., 2020).

The motif-magnetism generalization extends this logic from triangles to $\mathcal h^s \subset \mathcal h$0-spin polygons and polyhedra. The local motif Hamiltonians support exact spiral-colored product eigenstates, and their $\mathcal h^s \subset \mathcal h$1-projected versions remain exact eigenstates under a Zeeman field. In frustration-free assemblies these states are ground states; in sign-alternating frustrated assemblies they become exact excited states, often in the middle of the spectrum, with subthermal entanglement scaling $\mathcal h^s \subset \mathcal h$2. The ferromagnetic limit appears already for the $\mathcal h^s \subset \mathcal h$3 two-spin bond, whose motif Hamiltonian is

$\mathcal h^s \subset \mathcal h$4

The associated aligned $\mathcal h^s \subset \mathcal h$5-plane state is the simplest ferromagnetic scar-related motif. The paper explicitly warns that this is not ordinary $\mathcal h^s \subset \mathcal h$6-polarized ferromagnetism, but a coherent aligned or spiral configuration compatible with the motif constraints. Under Zeeman splitting, the projected exact eigenstates yield periodic dynamics with revival period $\mathcal h^s \subset \mathcal h$7, because $\mathcal h^s \subset \mathcal h$8 connects sectors differing by $\mathcal h^s \subset \mathcal h$9 (Chertkov et al., 2021).

These constructions establish two complementary routes. The first uses frustration and local coloring constraints; the second uses symmetry and staggered cancellation. In both, the ferromagnetic sector functions as an exact scar manifold, and simple product states with large overlap onto that manifold display revival dynamics that sharply distinguish them from nearby thermal states (Lee et al., 2020, Chertkov et al., 2021).

5. Scarred ferromagnetism in the long-range transverse-field Ising model

A different use of the term arises in the one-dimensional long-range transverse-field Ising model with power-law interactions. The Hamiltonian has $\mathrm{Sym}^N(\mathcal h^s)$0 parity symmetry, and the analysis is organized in parity-paired energy subspaces

$\mathrm{Sym}^N(\mathcal h^s)$1

Within each $\mathrm{Sym}^N(\mathcal h^s)$2, the magnetization operator has projected eigenvalues $\mathrm{Sym}^N(\mathcal h^s)$3. Using the generalized ETH framework for discrete symmetry breaking, the criterion for symmetry-breaking equilibrium states is that $\mathrm{Sym}^N(\mathcal h^s)$4. The paper then identifies ferromagnetic scarred states as those subspaces for which $\mathrm{Sym}^N(\mathcal h^s)$5 remains significantly nonzero even in parameter regimes where thermal equilibrium should be paramagnetic (Corps et al., 22 Jul 2025).

This is especially significant for $\mathrm{Sym}^N(\mathcal h^s)$6, where the equilibrium ferromagnetic phase is absent at finite temperature. For $\mathrm{Sym}^N(\mathcal h^s)$7, the spectrum does not simply become uniformly paramagnetic. Instead, it breaks into bands containing both subspaces with $\mathrm{Sym}^N(\mathcal h^s)$8 and others with clearly nonzero $\mathrm{Sym}^N(\mathcal h^s)$9, extending to high excitation energies. In the numerical counting used for one diagnostic, “scarred states” are subspaces with $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$0, and for $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$1 their number grows approximately exponentially with $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$2. For $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$3, the growth remains exponential but slower, while for $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$4 exponential growth is not evident.

The dynamical consequence is a nonequilibrium “scarred ferromagnetic phase.” At $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$5, $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$6, $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$7, and $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$8, two initial states with similar average energy but different domain structures behave differently. The state $$\ket{S_n^\pi}\coloneqq \frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (-1)^x (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x,$$9, composed of three very small magnetic domains, has local density of states concentrated on nonzero-magnetization subspaces and satisfies

$\hat U$0

whereas $\hat U$1, composed of two relatively large magnetic domains, has weight near $\hat U$2 and satisfies

$\hat U$3

Both states populate mainly the central $\hat U$4 of the spectrum, corresponding to $\hat U$5. The point is not merely slow thermalization but selective equilibration: small-domain initial states relax toward ferromagnetic equilibrium states because they preferentially populate ferromagnetic scarred subspaces, whereas larger-domain or structureless states relax to the expected paramagnetic equilibrium. This gives the phrase “scarred ferromagnetic phase” a dynamical rather than thermodynamic meaning (Corps et al., 22 Jul 2025).

6. Classical ferromagnetic scar-like patterns in periodic antidot arrays

The classical micromagnetic counterpart is provided by sub-micron ferromagnetic periodic antidot arrays. The system is a thin ferromagnetic film in the $\hat U$6-$\hat U$7 plane, perforated by a doubly periodic lattice of cylindrical holes with axes parallel to $\hat U$8. The periodicity is described by

$\hat U$9

and the ferromagnetic region is

$$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$0

where $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$1 is the shape of a single hole in one unit cell. The decisive feature is infinite connectivity: the periodic perforated topology imposes stronger constraints than those of simply connected films or ordinary multiply connected islands (Bogatyrëv et al., 2018).

The magnetization textures are treated as a “soup of 2-d topological solitons,” comprising vortices, antivortices, and skyrmion-like objects. The authors construct an approximate analytical family of metastable states by sequential energy minimization: first minimizing exchange energy, then enforcing boundary conditions that suppress magnetostatic surface charges. The soliton-meron join is written as

$$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$2

with free constants $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$3. For periodic antidot arrays, $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$4 is constrained by the geometry and can be written through real meromorphic differentials on the doubly periodic perforated surface as

$$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$5

where $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$6 and $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$7 are rational functions with real coefficients.

A central result is conservation of topological charge under the boundary condition that the magnetization is tangent to the antidot boundary, eliminating normal surface poles. The proof relies on the “image” structure implied by the real-coefficient rational functions: real roots lie on the boundary line or appear in symmetric pairs, and vortices or antivortices inside the medium are mirrored by “imaginary” partners outside. The conclusion is explicit: “no vortex or antivortex can ever leave or enter the medium.” Different choices of the degrees and coefficients of $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$8 and $$\ket{S_n}=\frac{1}{\mathcal N_n} \left(\sum_{x=1}^L (\hat S_x^+)^2\right)^n \bigotimes_{x=1}^L \ket{-}_x.$$9 produce different metastable states; the square-array ground state is one example, and many more metastable states exist in the same family.

The paper does not call these configurations scarred states. A precise reading is that they are topologically constrained metastable magnetization textures in a medium of infinite connectivity. The scar-like interpretation is therefore inferential: the antidot lattice imprints preferred channels and defect placements on the texture, so the geometry acts as a global organizer of localized topological structure. This suggests a classical analog of ferromagnetic scarred behavior, but not a strict quantum many-body scar in the sense used for nonthermal eigenstates (Bogatyrëv et al., 2018).