Type-I Scars: Exact Nonthermal Quantum States

• Type-I scars are exact, nonthermal eigenstates embedded in nonintegrable many-body spectra, characterized by emergent algebraic or cancellation structures.
• They manifest as equally spaced scar towers in spin chains, valence-bond solids in Heisenberg models, or effective large-spin dynamics in frustrated Rydberg systems.
• Key diagnostics include anomalous entanglement, super-extensive Quantum Fisher Information, and distinct spectral signatures that deviate from thermal eigenstate typicality.

Searching arXiv for the cited papers and closely related work on Type-I scars. Type-I scars are a class of exact quantum many-body scar states embedded in otherwise nonintegrable spectra. In the usage summarized here, the term covers several constructions that share a common nonthermal character but differ in microscopic realization: an exact equally spaced scar tower generated by an emergent Spectrum-Generating Algebra (SGA) in the spin-1 XY chain (Sharma et al., 25 Feb 2026); exact non-tower valence-bond-solid scars in the square-lattice Heisenberg model (Dai, 2024); and a graph-theoretic generalization of bipartite Néel scarring in frustrated Rydberg arrays, where locally entangled units produce an emergent large-spin structure (Desaules et al., 6 May 2026). Across these settings, Type-I scars occupy a vanishing fraction of Hilbert space, violate the expectations of thermal eigenstate typicality, and are distinguished either by exact algebraic closure, exact local cancellation mechanisms, or approximate collective precession within a specially constructed subspace.

1. Defining features

A Type-I scar tower is a family of exact eigenstates $\{|S_n\rangle\}$ of a nonintegrable Hamiltonian $H$ that all lie equally spaced in energy (Sharma et al., 25 Feb 2026). They occupy a vanishing fraction of the full Hilbert space but are connected by raising and lowering operators that form an emergent $\mathfrak{su}(2)$ algebra within the scar subspace $\mathcal W$. Concretely, one has operators

$$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$

so that the triplet $\{Q^+,Q^-,Q^z\}$ closes an $\mathfrak{su}(2)$ algebra on $\mathcal W$ even though $\mathfrak{su}(2)$ is not a symmetry of $H$ on the full Hilbert space. The resulting ladder satisfies

$H$0

(Sharma et al., 25 Feb 2026).

This algebraic definition is not exhaustive for every system called Type-I in the supplied literature. In the square-lattice Heisenberg model, the exact scar states are not part of a tower, have area-law entanglement, break translation symmetry, and exist for Heisenberg models of all spin (Dai, 2024). In frustrated Rydberg arrays, Type-I scars are described instead as a direct generalization of the bipartite Néel scar, implemented through a clique-cover construction whose quotient graph is bipartite (Desaules et al., 6 May 2026).

This suggests that “Type-I scars” functions as a family label for exact or systematically constructible nonthermal eigenstates tied either to an emergent $H$1 structure or to exact local cancellation mechanisms that preserve atypical dynamics inside a small subspace.

2. Spectrum-Generating Algebra in the spin-1 XY chain

The spin-1 XY chain under open boundary conditions provides a paradigmatic realization of exact Type-I scars (Sharma et al., 25 Feb 2026). The Hamiltonian is

$H$2

with total magnetization $H$3 fixed. Exact scars exist in every $H$4 sector (Sharma et al., 25 Feb 2026).

The ladder operators are the “bimagnon” operators

$H$5

Within the scar subspace $H$6 they obey

$H$7

so $H$8 realize $H$9 on $\mathfrak{su}(2)$0 (Sharma et al., 25 Feb 2026).

Starting from the fully polarized “vacuum”

$\mathfrak{su}(2)$1

the scar tower is

$\mathfrak{su}(2)$2

Because $\mathfrak{su}(2)$3, one has

$\mathfrak{su}(2)$4

(Sharma et al., 25 Feb 2026).

In the summary narrative accompanying this construction, the model is characterized as nonintegrable with Wigner–Dyson statistics, while the $\mathfrak{su}(2)$5 scar levels defy ETH (Sharma et al., 25 Feb 2026). A plausible implication is that this example serves as the canonical algebraic benchmark for Type-I scarring, since the tower structure, exact spacing, and subspace-restricted $\mathfrak{su}(2)$6 closure are all explicit.

3. Hidden symmetry protection and subspace topology

A central refinement of the spin-1 XY construction is the identification of a hidden $\mathfrak{su}(2)$7 subspace symmetry and an SPt characterization of the scar manifold (Sharma et al., 25 Feb 2026). The relevant auxiliary object is the commutant Hamiltonian

$\mathfrak{su}(2)$8

This highly nonlocal $\mathfrak{su}(2)$9 has the scar states as its ground-state manifold and is exactly integrable because all local terms of the original chain commute with the projectors onto $\mathcal W$0 (Sharma et al., 25 Feb 2026).

Two symmetries of $\mathcal W$1 are identified. The first is flipping of $\mathcal W$2 by $\mathcal W$3, with

$\mathcal W$4

which interchanges $\mathcal W$5 and thus $\mathcal W$6. The second is a sublattice $\mathcal W$7-phase operation that multiplies odd sites by $\mathcal W$8, under which $\mathcal W$9 are invariant. Together these form $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$0 (Sharma et al., 25 Feb 2026). Because the scar manifold is the ground space of $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$1, it is a symmetry-protected trivial phase under these two $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$2s (Sharma et al., 25 Feb 2026).

The corresponding diagnostic is a Lieb-Schultz-Mattis type twist operator,

$$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$3

For the scar states,

$$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$4

so for $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$5 they all sit at $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$6 on the unit circle. For generic ergodic states in the middle of the spectrum, numerical diagonalization shows $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$7 with fluctuations vanishing as $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$8 (Sharma et al., 25 Feb 2026). Thus $$Q^+,\;Q^-,\;Q^z \quad\text{such that}\quad [H,Q^\pm]\,\mathcal W = \pm\Delta\,Q^\pm\,\mathcal W, \quad [Q^z,Q^\pm]\,\mathcal W=\pm Q^\pm\,\mathcal W,$$9 distinguishes scars, with value $\{Q^+,Q^-,Q^z\}$0, from thermal states, with value $\{Q^+,Q^-,Q^z\}$1 in the thermodynamic limit (Sharma et al., 25 Feb 2026).

If one mixes scar levels among themselves but stays in $\{Q^+,Q^-,Q^z\}$2, then $\{Q^+,Q^-,Q^z\}$3 remains $\{Q^+,Q^-,Q^z\}$4. To detect mixing within $\{Q^+,Q^-,Q^z\}$5, one must turn on $\{Q^+,Q^-,Q^z\}$6; the phases $\{Q^+,Q^-,Q^z\}$7 then dephase, and $\{Q^+,Q^-,Q^z\}$8 (Sharma et al., 25 Feb 2026). This establishes that the twist operator diagnoses not only scar-versus-ergodic separation but also the internal coherence structure of the scar subspace.

4. Stability diagnostics: Loschmidt echo, QFI, and perturbation classes

The stability of the spin-1 XY scar tower under perturbations is analyzed through the Loschmidt echo and the Quantum Fisher Information (QFI) (Sharma et al., 25 Feb 2026). For a reference eigenstate $\{Q^+,Q^-,Q^z\}$9 of $\mathfrak{su}(2)$0 and a small static perturbation $\mathfrak{su}(2)$1,

$\mathfrak{su}(2)$2

At short times,

$\mathfrak{su}(2)$3

where $\mathfrak{su}(2)$4 is the pure-state Quantum Fisher Information (Sharma et al., 25 Feb 2026).

For scar states with superextensive QFI, $\mathfrak{su}(2)$5, the Loschmidt echo decays on a time scale $\mathfrak{su}(2)$6. For ergodic thermal states, QFI is $\mathfrak{su}(2)$7 or constant, so $\mathfrak{su}(2)$8 is $\mathfrak{su}(2)$9 (Sharma et al., 25 Feb 2026). The QFI for a pure state $\mathcal W$0 and generator $\mathcal W$1 is

$\mathcal W$2

If $\mathcal W$3 preserves the scar subspace, it can be rewritten in the SGA basis $\mathcal W$4. Since $\mathcal W$5 forms the spin-$\mathcal W$6 multiplet with $\mathcal W$7, one finds for

$\mathcal W$8

that

$\mathcal W$9

which at mid-tower scales as $\mathfrak{su}(2)$0. Hence the QFI density satisfies $\mathfrak{su}(2)$1 (Sharma et al., 25 Feb 2026).

Finite-size scaling numerically confirms the distinction:

• scar states satisfy $\mathfrak{su}(2)$2;
• coherent spin-$\mathfrak{su}(2)$3 states or random ergodic states satisfy $\mathfrak{su}(2)$4 or constant;
• “asymptotic” scars also yield $\mathfrak{su}(2)$5 (Sharma et al., 25 Feb 2026).

The perturbations are classified by whether they preserve or break the SGA. Local one-site and two-site operators $\mathfrak{su}(2)$6 are cataloged according to overlap with $\mathfrak{su}(2)$7 (Sharma et al., 25 Feb 2026). In the rotated basis that removes $\mathfrak{su}(2)$8 phases, the effective building blocks

$\mathfrak{su}(2)$9

act only on $H$0 and preserve the scar subspace, whereas

$H$1

create or destroy local $H$2 and break the subspace (Sharma et al., 25 Feb 2026).

Perturbation class	Representative form	Reported effect
SGA-preserving (class I)	$H$3	QFI $H$4, scars remain exact
Extensive but SGA-breaking (class II)	$H$5	QFI becomes $H$6 or constant; scars wash out in the thermodynamic limit
Intensive and SGA-breaking (class III)	single-site impurity $H$7	QFI $H$8; scars are destroyed for large $H$9

Within the summary narrative, scar-preserving perturbations are also described as showing super-extensive QFI $H$00, rapid dephasing, and robust fidelity revivals, whereas broken SGA yields only linear or constant QFI and thermalization (Sharma et al., 25 Feb 2026). The coexistence of rapid dephasing and robust fidelity revivals reflects the fact that the dephasing is controlled within a constrained algebraic sector rather than by generic thermal mixing.

5. Exact non-tower Type-I scars in the square-lattice Heisenberg model

The square-lattice Heisenberg model furnishes a distinct realization in which Type-I scars are exact valence-bond solids rather than an $H$01 tower (Dai, 2024). The Hamiltonian is

$H$02

and the nonintegrable cases of interest are two-leg ladders of size $H$03 with $H$04 even, and full two-dimensional lattices of size $H$05 with $H$06 even and each at least $H$07 (Dai, 2024).

On any two sites $H$08, the unique spin-0 singlet is

$H$09

which satisfies $H$10 on that bond (Dai, 2024). For ladders, the scar wavefunction is the diagonal VBS

$H$11

with translation partner

$H$12

For the two-dimensional even-by-even lattice, the 2×2 supercell VBS is

$H$13

with three symmetry-shifted partners $H$14, where $H$15 (Dai, 2024).

The proof that these are exact eigenstates relies on angular-momentum algebra and factorization of $H$16 into dot products of spin sums that annihilate the valence bonds. In ladders,

$H$17

and the bond singlets obey

$H$18

so each term annihilates $H$19 and therefore $H$20 (Dai, 2024). In two dimensions, a corresponding plaquette factorization gives $H$21 (Dai, 2024).

The construction generalizes to all $H$22 because the singlet $H$23 exists for all spin and the proofs use only SU(2) commutators (Dai, 2024). Even-length ladders also host two families of daughter scars:

• the one-magnon state

$H$24

with $H$25 and $H$26;

• the two-magnon bound state

$H$27

with $H$28 and $H$29 (Dai, 2024).

In this setting, the Type-I label is tied to several properties stated explicitly in the source: the states are exact mid-spectrum eigenstates, they obey area-law entanglement, they are not generated by an SU(2) tower, and they break translation symmetry (Dai, 2024). Numerical exact diagonalization further reports that level statistics in a generic symmetry sector follow GOE, that only the predicted VBS states and few-magnon towers near saturation appear as exact low-$H$30 scars in the examined ladders, and that in a $H$31 $H$32 system the four zero-modes $H$33 appear in the mid-spectrum with no other $H$34, $H$35 eigenstates found (Dai, 2024).

This example clarifies that Type-I scarring need not imply a tower structure. A plausible implication is that the term encompasses exact nonthermal states stabilized either by algebraic raising/lowering or by exact frustration-free cancellation on an atypical manifold.

6. Frustrated Rydberg arrays and graph-theoretic Type-I constructions

In Rydberg-blockaded lattices, Type-I scars are introduced as a systematic extension of bipartite Néel-state scarring to frustrated geometries (Desaules et al., 6 May 2026). In the strong blockade limit the dynamics is described by the PXP model

$H$36

where the projector forbids flipping atom $H$37 if any nearest neighbor is excited (Desaules et al., 6 May 2026).

A Type-I scar construction consists of two graph-theoretic ingredients (Desaules et al., 6 May 2026):

1. a clique cover $H$38 of the blockade graph $H$39, where each $H$40 induces a complete subgraph and the $H$41 are disjoint and cover $H$42;
2. a quotient graph $H$43 that is bipartite, with nodes given by the subsets $H$44 and edges indicating inter-clique blockade.

Each clique $H$45 can hold at most one Rydberg excitation; restricting to its symmetric subspace of the all-down state and the $H$46-state realizes an effective spin-$H$47 (Desaules et al., 6 May 2026). For a clique of size $H$48,

$H$49

In this two-dimensional subspace one defines Pauli-like operators $H$50, etc. Then

$H$51

where $H$52 is the bipartition of $H$53. One also constructs

$H$54

satisfying $H$55 approximately, exactly in the subspace spanned by the maximal-spin multiplet (Desaules et al., 6 May 2026).

The scarred subspace is then the spin-$H$56 irreducible representation of this $H$57, with extremal states

$H$58

$H$59

and the entire $H$60-dimensional “scar manifold”

$H$61

with $H$62 (Desaules et al., 6 May 2026). Under $H$63 these states precess collectively as an $H$64 level large spin, producing periodic revivals when quenching from $H$65 (Desaules et al., 6 May 2026).

Two examples are detailed. On the Shastry–Sutherland lattice with $H$66 sites, a unique clique cover by $H$67 disjoint nearest-neighbor dimers exists; the quotient graph is a bipartite square lattice with $H$68 sites, and quenches from the “dimer Néel” state show large-amplitude fidelity revivals, with overlap spectrum clustered in a ladder of $H$69 states equally spaced by $H$70 (Desaules et al., 6 May 2026). On the honeycomb lattice with $H$71 hexagons on a torus, there is one stripe dimer cover plus $H$72 distinct zigzag covers, each giving a valid bipartite quotient and a corresponding Néel-like initial state; quenches from any of these states show clear revivals, yielding an exponential family of Type-I scars (Desaules et al., 6 May 2026).

The geometric conditions are explicit: the blockade radius $H$73 must be large enough that each $H$74 is a clique and that no two same-color cliques block each other; the direct van-der-Waals tail should not spoil the projector structure; and the graph-theoretic requirement is the existence of a clique cover whose quotient is bipartite (Desaules et al., 6 May 2026). Mild frustration only weakly breaks the $H$75 algebra, leading to slowly decaying revivals (Desaules et al., 6 May 2026). The only source of decay is the non-closure of $H$76 within the maximal-spin subspace, arising from edges that connect sites in distinct $H$77 but lie within the same $H$78 or $H$79 block of the quotient graph (Desaules et al., 6 May 2026).

7. Comparative interpretation and recurring themes

The supplied literature identifies several recurring signatures of Type-I scars, but it also shows that the label spans more than one microscopic mechanism.

First, exactness or controlled subspace closure is central. In the spin-1 XY chain, exactness follows from an emergent SGA and yields an equally spaced tower of $H$80 eigenstates (Sharma et al., 25 Feb 2026). In the square-lattice Heisenberg model, exactness follows from angular-momentum identities and local factorization, giving only $H$81 exact states rather than an $H$82 tower (Dai, 2024). In Rydberg arrays, exact or approximate closure arises from reducing frustrated blockade graphs to effective bipartite large-spin dynamics on clique covers (Desaules et al., 6 May 2026).

Second, atypical entanglement and atypical dynamics recur. The Heisenberg valence-bond scars have area-law entanglement, with ladder cuts giving entanglement rank $H$83 and $H$84, while in two dimensions $H$85 (Dai, 2024). The spin-1 XY scars are distinguished dynamically by $H$86 and associated Loschmidt-echo scaling, in contrast to $H$87 or constant behavior for thermal states (Sharma et al., 25 Feb 2026). The Rydberg constructions are distinguished by long-lived or clear revivals from specially prepared initial states $H$88 or related Néel-like states (Desaules et al., 6 May 2026).

Third, the relation to symmetry differs by platform. In the XY chain, the scar manifold is assigned symmetry-protected trivial character via a hidden $H$89 symmetry of the commutant Hamiltonian (Sharma et al., 25 Feb 2026). In the Heisenberg construction, the emphasis is instead on SU(2) commutators and translation-symmetry breaking by the VBS pattern (Dai, 2024). In the Rydberg setting, the central symmetry input is bipartiteness of the quotient graph, which guarantees an approximate $H$90 acting on effective spins (Desaules et al., 6 May 2026).

A common misconception would be to identify Type-I scars exclusively with equally spaced towers. The square-lattice Heisenberg example explicitly states the opposite: the scars are not part of a tower, yet are still designated Type-I (Dai, 2024). Another possible misconception would be that scarring requires an unfrustrated lattice. The frustrated Rydberg construction directly addresses this by showing that locally entangled states can overcome mild frustration through the clique-cover mechanism (Desaules et al., 6 May 2026).

Taken together, these works present Type-I scars as a category of exact or systematically constructible nonthermal many-body eigenstates that can arise from emergent $H$91 algebras, hidden subspace symmetries, or exact local cancellation structures. The category therefore unifies tower and non-tower realizations, one-dimensional and higher-dimensional settings, and both spin and Rydberg platforms, while preserving a common emphasis on low-dimensional atypical subspaces embedded within nonintegrable many-body spectra (Sharma et al., 25 Feb 2026, Dai, 2024, Desaules et al., 6 May 2026).