---
title: Group-Invariant Quantum Many-Body Scars
url: https://www.emergentmind.com/topics/group-invariant-quantum-many-body-scars
type: topic
---

# Group-Invariant Quantum Many-Body Scars

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 [2007.00845]. This perspective has been developed in several directions: exact singlet sectors in spin-$1/2$ fermion models and their conventional condensed-matter deformations [2106.10300], O$(N)$-invariant BCS-like scar towers [2411.13651], SO$(N)$ singlets in Majorana lattice systems [2212.11914], constraint-generated scars on arbitrary lattices and in disordered settings [2207.06040; 1912.13399], and higher-rank $\mathfrak{su}(3)$-invariant scar subspaces stabilized by algebraic closure rather than equal level spacing [2604.11015].

## 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
$$
H|\mathbb{S}\rangle = H_0 |\mathbb{S}\rangle.
$$
A sufficient closure condition is
$$
[H_0, C_G^2] = W \cdot C_G^2,
$$
with $C_G^2$ the quadratic Casimir. In this framework the dimension of the scar subspace is directly controlled by the choice of $G$ and can be made exponentially large [2007.00845].

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 [2007.00845].

| Setting | Invariant object | Salient consequence |
|---|---|---|
| $H=H_0+\sum_a O_aT_a$ construction | $G$-singlet sector $\mathbb{S}$ | revivals, ETH violation, ODLRO [2007.00845] |
| Spin-$1/2$ fermion lattices | $\ket{n^\eta}$, $\ket{n^\zeta}$, $\ket{n^\eta}'$ | scars in Hubbard-, Heisenberg-, and $tJU$-type models [2106.10300] |
| BCS construction | O$(N)$-invariant tower $\ket{\phi_n}$ | BCS ground state and excitations as scars [2411.13651] |
| Majorana lattices | SO$(N)$ singlets in O$(N)\times$O$(M)$ decomposition | logarithmic entanglement, non-equidistant scar spectra [2212.11914] |
| Algebraic-closure models | $\mathfrak{su}(3)$-invariant subspace $W_{\mathfrak{su}(3)}$ | multidirectional spectral lattice and multifrequency oscillations [2604.11015] |

## 2. Fermionic lattice realizations and the $H_0+OT$ paradigm

For interacting spin-$1/2$ fermions on a lattice of $N$ sites, the group-invariant construction yields explicit scar families whose algebraic content is unusually transparent. In the original formulation, one family is $U(N)$-invariant,
$$
\ket{n_U} = \frac{\zeta^n}{2^n \sqrt{\frac{N! n!}{(N-n)!}}}\,|S_1\rangle,
$$
with
$$
|S_1\rangle = \prod_a \frac{ c^\dagger_{a1} + i c^\dagger_{a2} }{ \sqrt{2} } |0\rangle,
$$
while the second family is the $\eta$-pairing tower,
$$
\ket{n_O} = \frac{\eta^n}{\sqrt{\frac{N! n!}{(N-n)!}}}|0\rangle,\qquad
\eta = \sum_{a=1}^N c_{a1}^\dagger c_{a2}^\dagger.
$$
Each family contains $N+1$ scar states, and the $\eta$ states coincide with the well-known $\eta$-pairing states [2007.00845].

The subsequent group-theoretic analysis of fermionic lattice models systematized this structure into three families, $\ket{n^\eta}$, $\ket{n^\zeta}$, and $\ket{n^\eta}'$, and showed that many standard Hamiltonians are naturally of the required form $H_0+OT$ without fine tuning. The explicitly documented examples include Hubbard, Heisenberg, extended $2$D $tJU$, 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 [2106.10300].

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 [2106.10300].

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

The BCS-based construction starts from an O$(N)$-invariant scar tower
$$
\ket{\phi_n} = \frac{(O^\dagger)^n}{P_N(n)} \ket{0_\phi}, \qquad 0 \le n \le NK,
$$
and adds a pairing potential
$$
\delta H_0 = -\gamma e^{-i\theta} O - \gamma e^{i\theta} O^\dagger.
$$
Within the scar subspace, the ground state takes the BCS-like form
$$
|\psi_0\rangle = N_\psi \prod_j e^{\frac{v}{u} O_j^\dagger} \ket{0_\phi}.
$$
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 $OT$ terms annihilate all scar states [2411.13651].

The Majorana construction reorganizes the Hilbert space of a lattice with $M$ Majorana fermions per site under O$(N)\times$O$(M)$. The scars are the SO$(N)$ singlets. For any even $M$ there are two families: the $\eta$ states, symmetric under O$(N)$, and the $\zeta$ states, with SO$(N)$ invariance. For $M=4$ these reduce to the $N+1$ $\eta$-pairing states and the $N+1$ states of maximum spin; for $M=6$ explicit formulae permit an analytic calculation of bipartite entanglement entropy, which grows logarithmically with subsystem size at large $N$ [2212.11914].

Higher-rank generalization replaces the familiar one-dimensional $\mathfrak{su}(2)$ tower by a multidirectional scar lattice. In the $\mathfrak{su}(3)$ construction, the invariant basis is
$$
|\Psi_{n_1,n_2}\rangle =
\left(\Delta^{(L)}(f_2)\right)^{n_2}
\left(\Delta^{(L)}(f_1)\right)^{n_1}
|\Psi_{0,0}\rangle,
$$
and the scar energies are
$$
E_{n_1,n_2} = \lambda_1 (L - 2 n_1 + n_2) + \lambda_2 (n_1 - 2 n_2).
$$
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 [2604.11015].

## 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
$$
\left| \Psi_k \right\rangle = Q^k \left| \overline{\mathrm{vac}} \right\rangle,
$$
with $Q=\sum_{x,y\in\Lambda} q_{x,y} c_x c_y$, 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 [2207.06040].

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 $S$. 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 [1912.13399].

Gauge theories provide another symmetry-based stabilization mechanism. In $\mathrm{U}(1)$ and $\mathbb{Z}_2$ lattice gauge theories, robustness against gauge-breaking errors is achieved by adding a term linear in the gauge-symmetry generator or a simplified pseudogenerator,
$$
\hat{H} = \hat{H}_0 + \lambda \hat{H}_1 + V \sum_j c_j \hat{G}_j,
$$
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 [2203.08828].

Chiral symmetry organizes several additional exact-scar settings. In the density-difference-dependent Hamiltonian,
$$
H = \sum_j a^\dagger_{j+1}\left[-J+\gamma(n_{j+1}-n_j)\right]a_j + \text{h.c.},
$$
the transformation $a_j^{(\dagger)} \rightarrow (-1)^j a_j^{(\dagger)}$ implies $H\rightarrow -H$, 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 [2503.05252]. In the spin-$1$ $XY$ chain, the interplay of $U(1)$ 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 [2511.14878].

## 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 [2212.11914]. In the original group-invariant framework, the scars are area-law entangled and often display ODLRO; in particular, correlators such as $c_{i1}^\dagger c_{i2} c_{j2}^\dagger c_{j1}$ are independent of spatial separation throughout the scar sector [2007.00845].

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 [2207.06040]. In the BCS construction, the O$(N)$-invariant product structure implies
$$
\langle O_i^\dagger O_j \rangle = \langle O_i^\dagger \rangle \langle O_j \rangle \quad \forall\, i,j,
$$
so the pairing correlations do not decay with distance and realize ODLRO within the scar subspace [2411.13651].

A two-dimensional translationally invariant example is provided by quantum dimer models on the kagome lattice. There the exact scar state
$$
|\Psi\rangle = \sum_D |D\rangle
$$
is an equal-amplitude superposition over all dimer coverings in a topological sector. For a specific bipartition of the $48$-site lattice into two $24$-site ribbons, the scar has exact $S^{\text{vN}}=7$, 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 [2009.00022].

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 [1912.13399], Fock-space-cage scars form equally spaced ladders in a transverse field whose coherent superpositions display long-lived fidelity oscillations [2511.14878], and $\mathfrak{su}(3)$-invariant scars generate multifrequency oscillations rather than single-frequency revivals because their spectrum forms a multidirectional lattice [2604.11015].

## 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 [2408.10301].

A further conceptual complication comes from semiclassical analysis. A large-$N$ 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
$$
\mathrm{OTOC}(t) \sim \frac{1}{N} e^{\lambda t/N}
$$
for the scarred sector, in contrast with the conventional chaotic form
$$
\mathrm{OTOC}(t) \sim \frac{1}{N} e^{2\lambda_L t}
$$
for generic non-scarred orbits. The robustness is attributed to the Shiraishi-Mori projector structure rather than to a conventional global symmetry [2410.16916].

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.

Source: https://www.emergentmind.com/topics/group-invariant-quantum-many-body-scars