Ferromagnetic Scar States in Quantum Lattices

• Ferromagnetic scar states are non-thermal many-body states that violate the eigenstate thermalization hypothesis by residing in symmetric manifolds of lattice Hamiltonians.
• They are constructed via collective ladder operators acting on fully polarized states, resulting in equidistant energy towers with low entanglement.
• These states exhibit anomalous dynamics, including slow thermalization and strong revivals, making them vital for understanding quantum quench experiments and model stability.

Ferromagnetic scar states are a distinguished class of quantum many-body scar (QMBS) states that realize a striking violation of the @@@@1@@@@ (ETH) in non-integrable, interacting quantum systms. These states are non-thermal, yet coexist with an otherwise thermal spectrum, and are embedded as a highly symmetric, low-entanglement "scar manifold" in the many-body Hilbert space. Ferromagnetic scar states are characterized by their total symmetry under spatial permutations and are most naturally described in lattice systems possessing an on-site Lie algebra structure, where the scar subspace coincides with the maximally symmetric multiplet (e.g., Dicke states for spin-½). This class includes towers of exact eigenstates and magnon-like excitations in lattice models with SU($N$) symmetry and generalizes previous constructions from spin chains to broader settings (Hashimoto et al., 8 Jan 2026, Omiya, 16 Jan 2026).

1. Algebraic Structure and Definition

A ferromagnetic scar manifold consists of the totally symmetric subspace $$\mathrm{Sym}^N(\mathfrak{h}^s) \subset (\mathfrak{h}^s)^{\otimes N}$$, where $\mathfrak{h}^s$ is a "target" irreducible subspace supporting an on-site Lie algebra action. For lattice $\Lambda$ with $N$ sites, the highest-weight product state $$|\Omega\rangle \equiv \otimes_{x \in \Lambda} |\Omega\rangle_x$$ is fully polarized, with $$H^i_x |\Omega\rangle_x = \lambda^i_{\max} |\Omega\rangle_x$$, under Cartan generators $H^i_x$. The scar tower is built by action of global collective ladder operators $J^{\alpha} = \sum_{x \in \Lambda} E_x^{\alpha}$:

$$|\psi_{\vec{m}}^{\Lambda}\rangle = \frac{1}{\mathcal{N}_{\vec{m}}} \left[ \prod_{\alpha} (J^{\alpha})^{p_{\alpha}} \right] \left( \otimes_{x \in \Lambda} |\Omega\rangle_x \right)$$

where the $\vec{m}$ are global weights and $p_{\alpha}$ are chosen integers. Equivalent symmetrized-product constructions exist, and the resulting scar manifold always forms a representation whose dimension equals the number of one-row Young tableaux of length $N$. In the spin-½ case, these are the familiar Dicke states.

2. Parent Hamiltonians and the Zeeman+Annihilator Decomposition

Any local Hamiltonian $H$ that embeds the ferromagnetic scar tower as exact eigenstates admits a universal Zeeman+Annihilator decomposition:

$H = H_Z + H_A$

where $H_Z$ is a Zeeman term linear in collective Cartan generators $H_Z = \sum_{i=1}^{d_s -1} h_i H^i_{\text{tot}}$, and $H_A$ is an "annihilator" that vanishes on the scar manifold: $H_A |\psi\rangle = 0$ for all $|\psi\rangle \in \mathrm{Sym}^N(\mathfrak{h}^s)$. This structure is forced by Schur-Weyl duality and locality, and the spectrum within the scar subspace is equidistant (Omiya, 16 Jan 2026).

The annihilator $H_A$ decomposes further into a sum of strictly local projectors (on sites or bonds) that kill the scar manifold. More formally, any such $A$ can be expressed as:

$$A = \sum_{x \in \Lambda} o_{[x]}^{(1)} P_x + \sum_{\langle x, y \rangle} o_{[xy]}^{(2)} P_{xy}$$

where $P_x$ and $P_{xy}$ are projectors annihilating the symmetric sector on sites and nearest-neighbor bonds, respectively. The result is that any local Hamiltonian preserving the tower must be of generalized Shiraishi–Mori form, and the local-projector structural result is exhaustive for all ferromagnetic scar towers (Omiya, 16 Jan 2026).

3. Ferromagnetic Scar States in the SU($N$) Hubbard Model

In the one-dimensional SU($N$) Hubbard chain ($N \geq 3$), ferromagnetic scar states arise via projection onto the "doublon–holon subspace" $\mathcal{H}_P$, keeping only the holon vacuum $|0\rangle_j$ and $N-1$ doublon states per site. Exact η-pairing scar states, forming the scar tower, are

$$|S_{m_2, \dotsc, m_N}\rangle \propto \prod_{n=2}^N \left(Q_{1, n}^\dagger\right)^{m_n} | \mathrm{vac} \rangle$$

with $$Q_{1, n}^\dagger = \sum_j c_{j, \alpha_1}^\dagger c_{j, \alpha_n}^\dagger$$ and integer occupations $m_n$ such that $\sum_n m_n \leq L$.

The parent Hamiltonian projected to $\mathcal{H}_P$ is the SU($N$) ferromagnetic Heisenberg model:

$$H_p = 2J^2 \sum_{j=1}^L \left( I_{j, j+1} - \sum_{\mu, \nu=0}^{N-1} F_j^{\mu\nu} F_{j+1}^{\nu\mu} \right)$$

where $F_j^{\mu\nu}$ are local matrix-unit operators. The scar tower forms the exact zero-energy manifold of $H_p$. Magnon excitations (with explicit analytic dispersions) are asymptotic QMBS states orthogonal to the tower (Hashimoto et al., 8 Jan 2026).

4. Spectral Properties, Entanglement, and Orthogonality

One-magnon excitations above the fully polarized vacuum are constructed as Fourier modes,

$$F_{\mathrm{PBC}}^{\mu 0}(k) = \frac{1}{\sqrt{L}}\sum_{j=1}^L e^{ikj} F_j^{\mu 0}$$

yielding exact eigenstates of $H_p$ with energies $E(k) = 4J^2 (1 - \cos k)$ for periodic boundary conditions, displaying quadratic gaplessness near $k = 0$. For open boundaries, standing-wave modes $F_{\mathrm{OBC}}^{\mu 0}(p)$ yield $E(p) = 4J^2 [1 - \cos(\pi p / L)]$.

These magnon states are strictly orthogonal to the scar tower since the former lie outside the ground-state manifold of $H_p$. In the thermodynamic limit, the energy variance $\Delta E$ for such excitations vanishes relative to their excitation energy: $\Delta E / E \to 0$ as $L \to \infty$. The entanglement entropy for scar states, as measured by the half-chain von Neumann entropy $S_{\mathrm{vN}}$, is strictly subvolume: $$S_{\mathrm{vN}} \le \ln 2 + \sum_\mu \ln(m_\mu+1) = O((N-1) \ln L)$$, with exact matrix product operator representations for both the scar states and single-magnon excitations (Hashimoto et al., 8 Jan 2026).

5. Relation to the Shiraishi–Mori Construction and Exhaustiveness

The general framework for constructing Hamiltonians with ferromagnetic scar manifolds was pioneered by Shiraishi and Mori, involving local projectors $P_a$ whose common kernel is the scar manifold, plus an intra-kernel term $H_{\mathrm{intra}}$ that commutes with all $P_a$. The central structural theorem establishes that all local Hamiltonians embedding the totally symmetric (ferromagnetic) scar subspace must be of this Shiraishi–Mori form—there exists no alternative architecture for embedding such towers within strictly local quantum Hamiltonians. The Zeeman+Annihilator decomposition is thus exhaustive for the ferromagnetic scar case, with the annihilator reducible to sums of local projectors and the intra-kernel Zeeman term generating the ladder structure (Omiya, 16 Jan 2026).

6. Physical Implications, Model Design, and Stability

Ferromagnetic scar states possess unique dynamical signatures: they exhibit anomalously slow thermalization and strong revivals in quantum quench experiments, stemming from their symmetry, low entanglement, and orthogonality to the typical thermal spectrum. The projector structure of the parent Hamiltonian underwrites the stability of these eigenstates to local perturbations, though deviations that weakly fail to annihilate the scar manifold only produce approximate scars with long but finite lifetimes—enabling perturbation-theoretic analysis of their robustness.

Model construction is algorithmic: one specifies any set of strictly local projectors with a symmetric kernel equal to $\mathrm{Sym}^N(\mathfrak{h}^s)$, and adds a Zeeman ladder within the kernel. This produces Hamiltonians with perfect scars and equidistant ladders, generalizing across onsite representations (spin-1, higher SU($N$), etc.). The result unifies all known ferromagnetic QMBS models and provides a classification of their possible forms (Omiya, 16 Jan 2026).

7. Examples in One Dimension

Representative 1D Hamiltonians exhibiting ferromagnetic scars include:

Model	Hilbert structure	Projectors (Annihilator)	Zeeman term
Spin-1 XY chain	$h_x^s = \text{span} \{|+\rangle,|-\rangle\}$	$P_{x,x+1}^{(1)}, P_{x,x+1}^{(2)}$	$hS_x^3 + D(S_x^3)^2$
Spin-½ Heisenberg DM	$$h_x^s = \text{span} \{|\uparrow\rangle, |\downarrow\rangle\}$$	$(1-\mathrm{Swap}_{x,x+1})/2$ (singlet projector)	$h_z S^3_x$
SU($N$) Hubbard chain	$\mathcal{H}_P$: holon/doublon	$F_j^{\mu \nu} F_{j+1}^{\nu \mu}$	none

These constructions anchor the generalized scar framework and illustrate the principal features of ferromagnetic scar states and their associated Hamiltonians (Hashimoto et al., 8 Jan 2026, Omiya, 16 Jan 2026).