Group-Invariant Quantum Many-Body Scars

• Group-invariant quantum many-body scars are nonthermal subspaces defined by invariance under specific Lie group generators, resulting in reduced effective dynamics.
• They are realized in diverse models such as U(N), O(N), and SO(N)-invariant constructions, with examples in Hubbard, Heisenberg, and BCS-related systems.
• These scars offer insights into decoherence-free quantum dynamics and pave the way for experimental investigations using ultracold atoms and related platforms.

Group-invariant quantum many-body scars are nonthermal eigenstates, or finite-dimensional nonergodic subspaces, distinguished by invariance under a Lie group or related algebraic structure that need not be a symmetry of the full Hamiltonian. In the group-theoretic formulation, the scar sector is annihilated by the symmetry-breaking part of the Hamiltonian and therefore evolves under a reduced effective dynamics, leading to absence of thermalization, anomalous entanglement, and revivals (Pakrouski et al., 2020). This perspective has been developed in several directions: exact singlet sectors in spin-$1/2$ fermion models and their conventional condensed-matter deformations (Pakrouski et al., 2021), O$(N)$-invariant BCS-like scar towers (Pakrouski et al., 2024), SO$(N)$ singlets in Majorana lattice systems (Sun et al., 2022), constraint-generated scars on arbitrary lattices and in disordered settings (Tamura et al., 2022, Shibata et al., 2019), and higher-rank $\mathfrak{su}(3)$-invariant scar subspaces stabilized by algebraic closure rather than equal level spacing (Matsui, 13 Apr 2026).

1. Group-invariant scar sectors as a general construction

The basic construction considers Hamiltonians of the form

$H = H_0 + \sum_a O_a T_a,$

where $T_a$ are generators of a Lie group $G$. If $\mathbb{S}$ is the sector of $G$-invariant states, then

$$T_a |\mathbb{S}\rangle = 0 \quad \implies \quad (O_a T_a)|\mathbb{S}\rangle = 0,$$

so the full Hamiltonian acts on the sector as

$(N)$0

A sufficient closure condition is

$(N)$1

with $(N)$2 the quadratic Casimir. In this framework the dimension of the scar subspace is directly controlled by the choice of $(N)$3 and can be made exponentially large (Pakrouski et al., 2020).

This formulation is distinctive because the group need not be a symmetry of the full Hamiltonian. The non-invariant terms act trivially on the scar sector, while the rest of the spectrum can remain thermalizing. The resulting states display the characteristic scar phenomenology: ETH violation, low entanglement, and recurrent dynamics. In several fermionic realizations, the same invariant sectors also exhibit off-diagonal long-range order (ODLRO) that survives at high temperatures and is insensitive to the detailed dynamics (Pakrouski et al., 2020).

Setting	Invariant object	Salient consequence
$(N)$4 construction	$(N)$5-singlet sector $(N)$6	revivals, ETH violation, ODLRO (Pakrouski et al., 2020)
Spin-$(N)$7 fermion lattices	$(N)$8, $(N)$9, $(N)$0	scars in Hubbard-, Heisenberg-, and $(N)$1-type models (Pakrouski et al., 2021)
BCS construction	O$(N)$2-invariant tower $(N)$3	BCS ground state and excitations as scars (Pakrouski et al., 2024)
Majorana lattices	SO$(N)$4 singlets in O$(N)$5O$(N)$6 decomposition	logarithmic entanglement, non-equidistant scar spectra (Sun et al., 2022)
Algebraic-closure models	$(N)$7-invariant subspace $(N)$8	multidirectional spectral lattice and multifrequency oscillations (Matsui, 13 Apr 2026)

2. Fermionic lattice realizations and the $(N)$9 paradigm

For interacting spin-$\mathfrak{su}(3)$0 fermions on a lattice of $\mathfrak{su}(3)$1 sites, the group-invariant construction yields explicit scar families whose algebraic content is unusually transparent. In the original formulation, one family is $\mathfrak{su}(3)$2-invariant,

$\mathfrak{su}(3)$3

with

$\mathfrak{su}(3)$4

while the second family is the $\mathfrak{su}(3)$5-pairing tower,

$\mathfrak{su}(3)$6

Each family contains $\mathfrak{su}(3)$7 scar states, and the $\mathfrak{su}(3)$8 states coincide with the well-known $\mathfrak{su}(3)$9-pairing states (Pakrouski et al., 2020).

The subsequent group-theoretic analysis of fermionic lattice models systematized this structure into three families, $H = H_0 + \sum_a O_a T_a,$0, $H = H_0 + \sum_a O_a T_a,$1, and $H = H_0 + \sum_a O_a T_a,$2, and showed that many standard Hamiltonians are naturally of the required form $H = H_0 + \sum_a O_a T_a,$3 without fine tuning. The explicitly documented examples include Hubbard, Heisenberg, extended $H = H_0 + \sum_a O_a T_a,$4D $H = H_0 + \sum_a O_a T_a,$5, and Haldane-Hubbard models, as well as Hamiltonians containing spin-orbit coupled hopping and superconducting pairing terms. The same framework extends to non-Hermitian open systems, where the scar subspace continues to undergo coherent time evolution and exhibit revivals (Pakrouski et al., 2021).

A central consequence is that the invariant subspace functions as a decoherence-free sector. The fermionic group-invariant scars are described as insensitive to electromagnetic noise, and the low-energy sector can be engineered so that it is comprised solely of scars. In this setting the scar phenomenon is not tied to translational invariance or to integrability, but to the annihilation of selected states by an explicitly identifiable set of group generators (Pakrouski et al., 2021).

3. BCS, Majorana, and higher-rank invariant scar structures

The BCS-based construction starts from an O$H = H_0 + \sum_a O_a T_a,$6-invariant scar tower

$H = H_0 + \sum_a O_a T_a,$7

and adds a pairing potential

$H = H_0 + \sum_a O_a T_a,$8

Within the scar subspace, the ground state takes the BCS-like form

$H = H_0 + \sum_a O_a T_a,$9

For single-flavour spin-full fermions this is a special case of the BCS wavefunction written in real space and invariant under any site index relabelling; for multi-orbital fermions it includes higher-order terms corresponding to “pairing” of more than two fermions. The dynamics inside the scar subspace are governed exactly by the BCS mean-field Hamiltonian, while the $T_a$0 terms annihilate all scar states (Pakrouski et al., 2024).

The Majorana construction reorganizes the Hilbert space of a lattice with $T_a$1 Majorana fermions per site under O$T_a$2O$T_a$3. The scars are the SO$T_a$4 singlets. For any even $T_a$5 there are two families: the $T_a$6 states, symmetric under O$T_a$7, and the $T_a$8 states, with SO$T_a$9 invariance. For $G$0 these reduce to the $G$1 $G$2-pairing states and the $G$3 states of maximum spin; for $G$4 explicit formulae permit an analytic calculation of bipartite entanglement entropy, which grows logarithmically with subsystem size at large $G$5 (Sun et al., 2022).

Higher-rank generalization replaces the familiar one-dimensional $G$6 tower by a multidirectional scar lattice. In the $G$7 construction, the invariant basis is

$G$8

and the scar energies are

$G$9

The spectrum in the subspace is therefore not equally spaced, but forms a two-dimensional lattice. The resulting dynamics show multifrequency oscillations governed by integer linear combinations of distinct energy scales, and the invariant subspace survives algebra-preserving perturbations even when individual eigenstates become analytically intractable (Matsui, 13 Apr 2026).

4. Constraints, disorder, gauge structure, and chiral symmetry

A separate but closely related line of work realizes exact scars through local constraints that effectively define a symmetry-respecting sector. In spinless fermion models with density-assisted hopping, exact scarred eigenstates are constructed on any bipartite lattice in any dimension, with translation invariance unnecessary and site-dependent interactions allowed. The exact tower is

$\mathbb{S}$0

with $\mathbb{S}$1, and these states are exact zero-energy eigenstates for arbitrary choices of hopping matrix and site-dependent interactions. The same work constructs a positive-semidefinite parent Hamiltonian for which the scarred states are the unique zero-energy ground states under mild conditions (Tamura et al., 2022).

Disorder need not destroy the mechanism. In disordered spin chains with Onsager symmetry, arbitrary disorder is compatible with exact scar states for arbitrary spin quantum number $\mathbb{S}$2. The two stated classes are coherent states associated to an Onsager-algebra element and one-magnon scar states. Both are highly excited, have area-law entanglement, admit matrix-product-state representations, and generate perfectly periodic nonthermal dynamics, even though the bulk spectrum is non-integrable and thermalizing (Shibata et al., 2019).

Gauge theories provide another symmetry-based stabilization mechanism. In $\mathbb{S}$3 and $\mathbb{S}$4 lattice gauge theories, robustness against gauge-breaking errors is achieved by adding a term linear in the gauge-symmetry generator or a simplified pseudogenerator,

$\mathbb{S}$5

thereby confining the dynamics to the physical gauge-invariant sector. The reported explanation is quantum Zeno dynamics, and the method is presented as experimentally feasible in large-scale ultracold-atom and Rydberg-atom platforms (Halimeh et al., 2022).

Chiral symmetry organizes several additional exact-scar settings. In the density-difference-dependent Hamiltonian,

$\mathbb{S}$6

the transformation $\mathbb{S}$7 implies $\mathbb{S}$8, placing the model in the BDI symmetry class after Fock-basis reordering. The model hosts two classes of scars: a charge density wave scar and an edge-mode scar, both diagnosed by low entanglement and robust thermalization breaking time dynamics (Faugno et al., 7 Mar 2025). In the spin-$\mathbb{S}$9 $G$0 chain, the interplay of $G$1 magnetization conservation and chiral symmetries yields extensive zero-energy manifolds, interference-protected Fock-space cage states, a tower of volume-entangled states, and mirror-dimer states. The same paper shows that these nonthermal states can be organized as simultaneous eigenstates of non-commuting local operators within a commutant algebra framework (Mohapatra et al., 18 Nov 2025).

5. Entanglement structure, correlation functions, and nonthermal dynamics

Across constructions, the most persistent diagnostic is anomalously small entanglement relative to thermal eigenstates. Group-invariant scars are argued in general to have entanglement entropy parametrically smaller than that of typical states, and explicit examples include area-law, logarithmic, and cut-dependent subthermal scaling (Sun et al., 2022). In the original group-invariant framework, the scars are area-law entangled and often display ODLRO; in particular, correlators such as $G$2 are independent of spatial separation throughout the scar sector (Pakrouski et al., 2020).

The same qualitative pattern appears in exact lattice realizations. In the density-assisted hopping model, the scarred states are identified as outliers by entanglement entropy and correlation functions, and quench dynamics from initial states with large scar overlap exhibit nonthermalizing evolution with slow entanglement growth and persistent fidelity (Tamura et al., 2022). In the BCS construction, the O$G$3-invariant product structure implies

$G$4

so the pairing correlations do not decay with distance and realize ODLRO within the scar subspace (Pakrouski et al., 2024).

A two-dimensional translationally invariant example is provided by quantum dimer models on the kagome lattice. There the exact scar state

$G$5

is an equal-amplitude superposition over all dimer coverings in a topological sector. For a specific bipartition of the $G$6-site lattice into two $G$7-site ribbons, the scar has exact $G$8, far below the surrounding states, while the non-scar eigenstates exhibit GOE or GUE level statistics within the appropriate symmetry sectors. Fidelity dynamics reveal strong revivals for special initial configurations and rapid decay for generic ones (Wildeboer et al., 2020).

Dynamically, equally spaced towers produce periodic revivals, but equal spacing is not necessary. Onsager scars are described as trapped in a perfectly periodic orbit in Hilbert space (Shibata et al., 2019), Fock-space-cage scars form equally spaced ladders in a transverse field whose coherent superpositions display long-lived fidelity oscillations (Mohapatra et al., 18 Nov 2025), and $G$9-invariant scars generate multifrequency oscillations rather than single-frequency revivals because their spectrum forms a multidirectional lattice (Matsui, 13 Apr 2026).

6. Relation to quantum chaos and conceptual boundaries

The phrase “group-invariant quantum many-body scar” does not denote a single universally agreed mechanism. One line of work studies scarring in the quantum-chaotic sense: in a broad family of spin chains including Ising, XX, XXZ, and Heisenberg models, scarred eigenstates are identified by enhanced Husimi weight along unstable periodic orbits, especially on translationally invariant and interaction-suppressing manifolds. In that setting, symmetry plays the role of generating manifolds of unstable periodic orbits through translation and mirror operations; it does not by itself guarantee scarring, and the authors explicitly distinguish this phenomenon from non-thermal eigenstates generated by constraints or emergent symmetries (Pizzi et al., 2024).

A further conceptual complication comes from semiclassical analysis. A large-$$T_a |\mathbb{S}\rangle = 0 \quad \implies \quad (O_a T_a)|\mathbb{S}\rangle = 0,$$0 bosonic model constructed as a counterexample to the conjecture that QMBS correspond to weakly unstable periodic orbits shows that scar-associated dynamics can be classically stable rather than chaotic. The reported out-of-time-ordered correlators behave as

$$T_a |\mathbb{S}\rangle = 0 \quad \implies \quad (O_a T_a)|\mathbb{S}\rangle = 0,$$1

for the scarred sector, in contrast with the conventional chaotic form

$$T_a |\mathbb{S}\rangle = 0 \quad \implies \quad (O_a T_a)|\mathbb{S}\rangle = 0,$$2

for generic non-scarred orbits. The robustness is attributed to the Shiraishi-Mori projector structure rather than to a conventional global symmetry (Omiya, 2024).

This suggests that group invariance enters the scar literature in at least two technically distinct senses. In one, it denotes exact invariant sectors or singlet subspaces defined by Lie-group generators, commutant algebras, or algebraic closure. In the other, it denotes symmetry-related manifolds of periodic orbits that enhance scarring in an otherwise chaotic phase space. The two viewpoints overlap in their emphasis on nonthermal subspaces and weak ergodicity breaking, but they do not impose the same microscopic mechanism or the same relation to classical chaos.