From tunnels to towers: quantum scars from Lie Algebras and q-deformed Lie Algebras

We present a general symmetry-based framework for obtaining many-body Hamiltonians with scarred eigenstates that do not obey the eigenstate thermalization hypothesis. Our models are derived from parent Hamiltonians with a non-Abelian (or q-deformed) symmetry, whose eigenspectra are organized as degenerate multiplets that transform as irreducible representations of the symmetry (`tunnels'). We show that large classes of perturbations break the symmetry, but in a manner that preserves a particular low-entanglement multiplet of states -- thereby giving generic, thermal spectra with a `shadow' of the broken symmetry in the form of scars. The generators of the Lie algebra furnish operators with `spectrum generating algebras' that can be used to lift the degeneracy of the scar states and promote them to equally spaced `towers'. Our framework applies to several known models with scars, but we also introduce new models with scars that transform as irreducible representations of symmetries such as SU(3) and $q$-deformed SU(2), significantly generalizing the types of systems known to harbor this phenomenon. Additionally, we present new examples of generalized AKLT models with scar states that do not transform in an irreducible representation of the relevant symmetry. These are derived from parent Hamiltonians with enhanced symmetries, and bring AKLT-like models into our framework.


I. INTRODUCTION AND GENERAL FRAMEWORK
A central question in non-equilibrium quantum dynamics is whether reversible unitary dynamics in a closed quantum system can establish local thermal equilibrium. Much insight into quantum thermalization follows from the eigenstate thermalization hypothesis (ETH) [1][2][3][4][5], a strong version of which posits that every finite temperature eigenstate of a thermalizing system reproduces thermal expectation values locally [6]. In contrast, there are classes of interacting, typically disordered, "manybody localized" (MBL) systems that violate the ETH and never thermalize [7][8][9].
More recently, attention has focused on weak ETH violating systems with so-called 'many-body quantum scars' [10][11][12][13]. Scars are non-thermal eigenstates embedded within an otherwise thermal eigenspectrum. These typically have sub-thermal entanglement entropy ( ∼ O(log(|A|)) or ∼ O(|∂A|) for a subsystem A) and coexist at the same energy density as thermal volume-law entangled eigenstates. Scars constitute a vanishing fraction of the eigenspectrum -and hence these systems still obey a weak version of the ETH [14]; nonetheless, their presence can lead to measurable non-thermal dynamical signatures in quenches from atypical but experimentally amenable initial states [12,15]. Indeed, the recent literature on scars followed from an interesting experimental observation of non-thermal (oscillatory) quench dynamics in a Rydberg atom chain that realizes a constrained 'PXP' spin Hamiltonian.
The SM prescription [10] relies on a set of local projectors, {P i } centered around site i, that generically do not commute with one other, in addition to one or more states |ψ s that are simultaneously annihilated by all the P i and span a subspace S. Then, the |ψ s are scarred eigenstates annihilated by Hamiltonians of the form where the h i are generic local operators of finite range. The h i operators ensure that the rest of the spectrum is thermalizing and non-integrable. If, additionally, there exist special Hamiltonians H that commute with all the {P i }, then these can be added to H SM A to impart different energies to the states in S. Note that H = H SM A + H does not have explicit symmetries, but the Hilbert space nevertheless dynamically splits into disconnected 'Krylov sectors': the subspaces S and its complement do not mix because S is annihilated by H SM A . Separately, MLM in Ref. [32] furnished a complimentary framework that unified the existence of 'towers' of equally spaced in energy scar states in three different models [11,21,28]. In these models, scars {|ψ n } were generated by repeatedly acting with a particular opera-tor Q + on a particular low entanglement eigenstate of H, |ψ 0 so that |ψ n = (Q + ) n |ψ 0 . MLM showed that, in all these cases, Q + acts as a spectrum generating 'ladder' operator when restricted to the scarred subspace: [H, Q + ] − ωQ + |ψ n = 0, (2) which implies that the |ψ n are equally spaced energy eigenstates of H with E n = ωn + E 0 . Furthermore, the particular form of the Q + operator is such that the states |ψ n have low entanglement. MLM discussed various example Hamiltonians obeying Eq. (2) which had the form H = H SG + H A , such that H SG has a 'spectrum generating' algebra (SGA): and H A annihilates the scars, H A |ψ n = 0. Similar to SM, these contain a piece that annihilates the scars and one that gives them energy. While such constructions have been very useful for explicitly deriving and unifying the presence of scars in specific 'one-off' models, qualitative 'pictures' for when and how scars may arise more generally are still largely missing. For example, it is still largely unclear where in the general space of operators and states we may expect to find a set {H, Q + , |ψ 0 } such that the MLM conditions (2) leading to (weak) ETH violations are met. In contrast, we have many phenomenological notions for how (strong) ETH violation arises in MBL systems: this generally requires strong disorder and weak short-ranged interactions, and MBL systems are understood as having an extensive set of emergent local integrals of motion [38,39].
In this work, we attempt to bridge this gap by presenting a very general symmetry based framework for obtaining scar towers. We start with parent Hamiltonians, H sym , with a continuous non-Abelian symmetry G (or a q-deformed version thereof, G q ). The generators of the symmetry furnish a natural set of spectrum generating 'raising' operators Q + , that connect multiplets of degenerate eigenstates in H sym . In general, the eigenspectrum of H sym has a lot of additional structure because of constraints resulting from the symmetry; we show that there are general ways to perturb H sym that break the symmetry, but do so in manner that preserves a shadow of the symmetry in the form of scars. For example, eigenstates of H sym in superselection sectors with at most O(poly(L)) basis states will have at most O(log(L)) entanglement in a system of size L. Families of perturbations can be chosen that preserve certain such low entanglement subspaces of H sym , but generically mix between all other sectors -thereby leading to the embedding of scarred eigenstates in otherwise thermal spectra 1  Hsym has 'tunnels' of degenerate eigenstates with the same eigenvalue for the Casimir Q 2 but different eigenvalues for Q z . Each tunnel is denoted by a different color. One can move between states in a tunnel using Q ± . (b) Adding HSG ∝ Q z preserves the eigenstates, but breaks the degeneracy of the tunnels. Instead, states in each tunnel get promoted to 'towers' and acquire an evenly spaced harmonic spectrum because of the SGA [Q z , Q ± ] = ±Q ± . (c) An HA can be chosen to annihilate a specific tower of states (highlighted) but generically break all symmetries and mix between the other states so as to make the rest of the spectrum thermal. The chosen tower of states are scars in H = Hsym + HSG + HA.
perspective is reminiscent of KAM theorems that concern the fate of integrable models with extensively many symmetries to the addition of small perturbations, and specifically whether remnants of the integrability can be preserved under the action of the perturbation. Our framework makes extensive use of the generators of the Lie algebra of the symmetry group G. which furnish a natural set of spectrum generating operators (SGOs) with 'raising/lowering action'. (for example, operators {Q + , Q − , Q z } associated with an SU(2) symmetry have the SGA: [Q z , Q ± ] = ±Q ± ). We obtain scarred models via a three step process: • First, H sym contains multiplets of degenerate eigenstates -tunnels -that transform as irreducible representations (irreps) of the symmetry G. Each multiplet is labeled by its eigenvalues under the Casimir operators of G, and states within the multiplet are distinguished by their eigenvalues under the generators of G in the Cartan subalgebra. Raising operators connect between the states in a multiplet.
• Next, tunnels in H sym can be promoted to equally spaced 'towers' of non-degenerate eigenstates in the Hamiltonian H s = H sym + H SG . Here H SG is typically chosen to be a linear combination of the generators in the Cartan subalgebra, which commute with and share the eigenstates of H sym , but have a SGA with the raising operators (for example, H SG = ωQ z gives the states |ψ n energy In other words, even though the addition of H SG breaks the symmetry G, the eigenstates of H s and H sym are still the same and only their energy eigenvalues are different: in particular, degenerate tunnels of states become non-degenerate towers 2 .
We will also discuss models where the scar tower does not transform as an irrep of the symmetry and/or where H SG is not a generator of the symmetry but still has a SGA with the raising operators. This is possible when H sym has an expanded symmetry, which allows H sym to be simultaneously diagonalized with H SG and have tunnels of degenerate eigenstates that do not transform as an irrep.
• Finally, to make H s a scarred model, we introduce symmetry breaking perturbations H A that annihilate a particular low entanglement tunnel of states {|ψ n } built upon a particular low entanglement 'base state' |ψ 0 . H A can typically be chosen to be generic enough to mix states across the various symmetry sectors of H sym so as to make the rest of the spectrum generic and thermal. In all, obeys the MLM condition (2), and has towers (or pyramids) of scar states generated by raising operators of the non-Abelian symmetry G acting on a low entanglement base state |ψ 0 which is an eigenstate of each of the three terms in H.
This three-step process is schematically illustrated in Fig. 1. Our picture applies to several exactly solvable scarred models in the literature, and also furnishes a natural way to get many new scarred Hamiltonians derived from various non-Abelian and q-deformed non-Abelian symmetries. In what follows, we flesh out the ingredients for our framework in more detail in Section II. We then discuss two qualitatively distinct families of scars. In the first, discussed in Section III, the scarred eigenstates inherit the parent symmetry and transform as a single irreducible representation of G (or G q ). These represent generalizations of perturbed η-pairing models that have been discussed in the literature [34,35]. The second, discussed in Section IV, is a generalization of various AKLT like models where the scars do not inherit the symmetry G. However, as we discuss, these can be viewed as arising from parent Hamiltonians with an enhanced symmetry group larger than G. We conclude in Section V, and present various technical details in a series of appendices.

II. INGREDIENTS OF THE FRAMEWORK
We now discuss in more detail our framework for constructing families of Hamiltonians with towers of scarred states. For specificity, we will always consider a onedimensional chain with L sites, with a spin -S degree of freedom (i.e. a (2S + 1)-state Hilbert space) on each site 3 . We denote the physical spin operators on site j as S ± j , S z j , with where S ± j are the usual spin raising and lowering operators on site j and S z j measures the z polarization of the spin. We refer to the resulting SU(2) algebra as the spin-SU(2) algebra, and to any associated symmetry in our model as a spin-SU(2) symmetry.
We will consider Hamiltonians of the form in Eq. 4, where the different pieces needing elaboration are H sym , H SG and H A . The tower of scars is built upon some low-entanglement 'base state' |ψ 0 by acting with raising/lowering 'ladder' operators Q ± associated with a non-Abelian symmetry G of H sym . Note that H sym will not generically have spin-SU(2) symmetry and the Q operators will generally be distinct from the physical spin operators in Eq. (5).

A. Lie Algebras, Raising Operators and H SG
Our first task is to characterize a suitable set of ladder operators, Q + , associated with the Lie algebra of a non-Abelian symmetry G in H sym .
We begin with symmetries G that act as a product of onsite symmetries, and raising operators that can be expressed as linear combinations of the form, where k is a momentum index, and Q ± i are raising/lowering operators associated with site i. The operators {Q ± i , Q z i } are derived from the local generators of the symmetry acting on site i. To ensure that the scar 3 It is easy to see that the general philosophy of our constructions apply mutatis mutandis to systems in higher dimensions.
tower contains only O(poly(L)) states, we also require that (Q + i ) nmax = 0, and hence on a chain with L sites, (Q † ) nmaxL = 0.
One natural choice of Q + and H SG operators is obtained by considering the generators of the spin-SU(2) symmetry so that In this case, the SGA [H SG , Q + ] = ωQ + follows from the the spin-SU(2) Lie algebra. The corresponding parent Hamiltonian H sym is spin-SU(2) symmetric. A more interesting example is furnished when the {Q ± , Q z } are distinct from the spin-SU(2) operators, for example: for spin S = 1 and k = π produces the SGO of two well-known scar models in the literature: the spin-1 AKLT model [11,27,32,33] and the spin-1 XY model [28,29,32,33]. This choice of operators Q + , Q − and Q z also obeys the Lie algebra SU (2), but the resulting Q-SU(2) symmetry is distinct from the spin-SU (2) symmetry. Thus we can consider models with Q-SU(2) symmetry that need not have spin rotation invariance.
Notice that Q ± i = (S ± i ) 2S raises (lowers) the spin −S (spin S) state to a spin S (spin −S) state, and annihilates all other states. Thus, from the (2S + 1) levels on each site, these Q i operators effectively isolate a reducible SU(2) representation wherein the 'top' and 'bottom' levels | − S and |S act as a spin-1/2 doublet, while the rest act as singlets. This is an example of an embedded SU(2)sub-algebra of SU(2S+1) on each site. For S = 1, there are three independent SU(2) sub-algebras in SU(3), and the Gell-Mann matrices provide a natural basis for these embedded sub-algebras of which the choice described in Eq. 8 represents one.
It is natural to also consider other onsite raising operators of the form Q + i = (S + i ) n for 1 ≤ n ≤ 2S. However, for n < 2S − 1 and S > 3/2, these operators do not describe an SU(2) algebra but rather form larger Lie group symmetries. These will be discussed in detail in Sec. III B, but here we review some general features about Lie groups to discuss how these naturally furnish various raising operators.
For any continuous non-Abelian symmetry group G, we can find operators Q + and H SG chosen from the associated Lie algebra that satisfy commutation relations corresponding to an SGA. The parent Hamiltonian H sym is invariant under G, meaning that it commutes with all the generators of G and also with operators analogous to the total spin, known as the Casimir operators C for G. Eigenstates of H sym necessarily come in degenerate multiplets ("tunnels") with the same eigenvalues of C. In this case we find a family of raising operators {Q α } that connect between the degenerate states, and there are several physically distinct choices for H SG leading to distinct 'pyramid' structures in the eigenspectrum.
In more detail, consider an N dimensional semi-simple Lie algebra, with generators X µ where µ = 1, · · · N [N = m 2 − 1 for SU(m)]. We denote by Q z µ with µ = 1, · · · R a maximal linearly independent set of commuting Hermitian generators that can be diagonalized simultaneously, called the Cartan subalgebra (CSA). R is known as the rank of the algebra. The N − R generators that are not in the CSA can be rearranged into pairs of raising and lowering operators of different SU(2) subalgebras. We denote these collectively as Here α is an R-component vector, known as a root. The set of raising operators can be described by choosing α to be a positive root, meaning that we include α but not −α, and that if α + β is a root, with α and β both positive roots, then α + β is also a positive root. In a chain with L sites, we may therefore choose to be a linear combination of the generators in the CSA. The raising operators (with α a positive root) then satisfy the desired SGA where ω α = µ h µ α µ . We note that for R > 1 there are multiple linearly independent choices of the coefficients h µ , which in general exhibit different spectra for the scar states.
It is convenient to note that the raising and lowering operators obey the commutation relations To summarize: a subset of the generators of a Lie algebra can always be combined to furnish one or more pairs of raising and lowering ladder operators, Q α , associated with embedded SU(2) subalgebras. The remaining generators Q z µ form the Cartan subalgebra and have spectrum-generating commutation relations with Q α (cf. Eq. 9). When H SG is chosen to be as a linear combination of Q z µ , as in Eq. (10), then H SG can be simultaneously diagonalized with H sym and the Casimir C, and has spectrum-generating commutation relations with Q α (cf. Eq. (12)). This immediately implies that specific multiplets of eigenstates of H sym with the same eigenvalue of C but different eigenvalues of H SG are degenerate. Each of these multiplets forms a "tunnel" in the spectrum of H sym that transforms as a single irreducible representation of G, and acting with Q α moves between different states in the tunnels (Fig. 1a). When H SG is added to the Hamiltonian, the degeneracies are broken and the eigenstates in the tunnels acquire energy spacings that are integer superpositions of ω α (cf. Eq. (12)), thereby getting promoted to 'towers' (or pyramids) of states ( Fig. 1(b)). The final step, discussed in the next two subsections, is to add a term H A to the Hamiltonian that annihilates a particular tunnel of low entanglement states built upon a particular 'base state' 4 ; H A generically breaks all symmetries and mixes between all other states so as to give a thermal spectrum with the chosen states embedded as low entanglement scars.
Before leaving this section, we note two further points elaborated on later. First, it is necessary in some cases, especially in our discussion of generalized AKLT models in Sec. IV, to pick H SG operators that cannot be expressed in terms of the generators of the CSA, but nevertheless have the desired 'raising action' in their commutation relations with Q α . For example, the total spin-z operator S z obeys the commutation [S z , Q + ] = 2SQ + for the 'raise by 2S' Q + operators in Eq. (8), but S z is linearly independent from Q z and does not commute with Q 2 for S > 1. In these cases, the parent Hamiltonian H sym generally has a larger symmetry, so that its eigenstates can still be simultaneously diagonalized with H SG and the picture of tunnels to towers still applies -however, the states in the tower of scars need not have a definite eigenvalue under C and are not contained within a single irreducible representation of G. Second, a different context in which operators with suitable commutation relations emerge naturally is in the context of q-deformed Lie algebras. We discuss the example of qdeformed SU(2) in detail in Sec. III C. Importantly, the q-deformation leaves the commutator between the raising operators S ± and S z unchanged. However, there is a key difference relative to the Lie algebra case: the commutation relation between the SU(2) raising and lowering operators is altered, such that Q i is not a single-site operator but has 'tails' to the right and left of i.

B. Base state |ψ0
In order to construct our candidate scar tower, the next ingredient we need is to select a specific multiplet of degenerate tunnel states in H sym that will get promoted to form a scar tower. In order for the scars to have low entanglement, the tower should be built by acting with the raising operators Q + α on a particular low entanglement "base" state |ψ 0 . As discussed above, we will require |ψ 0 to be an eigenstate of H sym and H SG . In general, the scar space consists of a discrete set of states of the form: where α i are positive roots, 0 ≤ n i ∈ Z and k = 1 2 (N − R). It is important to note that, as for SU (2), in general on each site Q nmax αi = 0 as an operator: any state can be raised by at most a fixed amount in any direction. More generally, Eq. (13) implies that any product of positive roots must be 0 as an operator when raised to a sufficiently large power. Thus the number of states in our scar tower grows at most polynomially with the chain length L.
The base states that we consider come in two types. First, |ψ 0 can be a low-entanglement eigenstate of H sym , H SG and the relevant Casimir operators. For G=SU (2), one simple choice is the maximally spinpolarized state which is the only state in the symmetry sector labeled by (Q = Q max , Q z = −Q max ). Acting with the raising operator on this state n times generates the state in the tower that lives in the unique symmetry sector labeled by (Q = Q max , Q z = −Q max + n) For general G, the analog of the polarized state is obtained as follows. We will work in a basis {|w } of simultaneous eigenstates of all Q z µ . (This is analogous to working in the basis of σ z eigenstates in the SU(2) case.) Here w, known as the weight vector, describes the eigenvalues of the Q z µ , via The commutation relations (9) imply that i.e. acting with Q α on a state |w "raises" the eigenvalue of Q z µ by an amount α µ (which can be 0 for some choices of α, µ), while preserving the value of the Casimirs. Note that the coefficient of proportionality can be 0, in which case |w + α is not a state in our Hilbert space. There is always a unique "lowest weight" state |w min such that Q α |w min = 0 for any negative root α. (The negative roots are those root vectors that are not positive roots, and are the analog of the lowering operators Q − for the SU(2) case.) Thus more generally, we may take |ψ 0 = i |w min,i to be a product of lowest-weight states on each site in our system. By definition, this is an eigenstate of the many-body Casimirs and all Q z µ . In this case, the scar space contains all states in some irreducible representation (irrep) of the Lie algebra G. The full Hamiltonian can be viewed as a perturbation of the G-invariant Hamiltonian H sym in which the symmetry is generically broken, but preserved in a non-generic way in exactly one of the irreps. We note that for a system with L sites, the maximum number of states in any such representation grows only polynomially with L, guaranteeing that our scar subspace is sub-extensive. Base states of this form (with G=SU(2)) are relevant to the spin-1 XY -model [28,32,34,35], as well as the η-pairing states of the Hubbard model and other electronic models after appropriate mappings from spin lattices to electronic models [35].
Second, we may choose |ψ 0 to be an eigenstate of H SG , but not of the relevant Casimirs. In this case the scar pyramid is not contained within a single irreducible representation of G, and the associated parent Hamiltonian H sym must have additional degeneracies not explained by the symmetry G. This scenario arises in various AKLTlike model Hamiltonians exhibiting exact quantum scars.
We now argue that the states |ψ n1,n2,...n k are low entanglement eigenstates of (H sym + H SG ) and hence good candidate scar states once H A is added. First, they are eigenstates since where the third line follows from Eq. 12 and the fact that [H sym , Q α k ] = 0, and the last line follows from the fact that we require |ψ 0 to be an eigenstate of H SG and H sym with eigenvalue E 0 .
Second, the states |ψ n1,...n k all have entanglement that grows at most logarithmically in the subsystem size, provided that |ψ 0 has low entanglement. To see this, observe that if |ψ 0 has finite (or log(L)) entanglement, it can be approximated (up to exponentially small corrections) by a matrix product state with bond dimension d for some d that is finite (or poly(L)). In fact, the choices of |ψ 0 that we use here will all be exact matrix product states. Further, for all choices of Q † operators considered in this work -for Lie algebras and q-deformed Lie algebras -the operator (Q † ) n can be expressed as a matrix-product operator (MPO) of bond dimension n+1. We show this in Appendix A, by generalizing an argument due to Mougdalya et al in Ref. [27]. Thus the state Q n k α k ...Q n2 α2 Q n1 α1 |ψ 0 has entanglement entropy of at most S ∼ log(d) + k log(n k + 1). Since the maximum possible value of n i grows polynomially with L, we see that states in our scar tower have entanglement entropy that scales at most logarithmically, rather than linearly, with L. This is a defining characteristic of a quantum scar eigenstate.

C. Annihilation operators HA
Finally, our construction requires an operator H A that behaves like a generic, thermal Hamiltonian on the nonscarred eigenstates, but with the special property that H A |ψ n1,...n k = 0 (19) for any {n µ }-i.e. it annihilates all states in the scar tower/pyramid. In general, we will consider two types of H A operators. The first is of the Shiraishi and Mori form in Eq. (1) which requires a set of local projectors {P i } which annihilate all scar states such that for all i and any set of powers n µ , In general, we will restrict ourselves to translation- are not many-body localized. By choosing these h i operators sufficiently generically and with sufficiently large (but finite) range r, quite generally we expect that H can be chosen to be ergodic on those states that it does not annihilate.
In many cases, appropriate projectors P i can be deduced from the properties of the group, say if the scar states are chosen to have the maximum possible eigenvalue under the Casimir. For example, scars built atop a polarized state for an SU(2) spin-symmetric system will have maximum possible total spin for any pair of neighboring sites, so that bond-wise projectors onto spin states less than maximal will annihilate the scars. Letting Π max i,i+1 be the projector onto the maximal total spin state between sites i and i + 1, an appropriate set of bond-wise projectors is P i,i+1 = 1 − Π max i,i+1 . Likewise, for higher Lie groups, we exploit the fact that on each bond, the state |w min,i , w min,i+1 is symmetric under interchanging indices i and i + 1. For example, if |w min,i is a state in the fundamental representation of the group G, the corresponding many-body state is in the completely symmetric representation (a single row, in terms of Young Tableaux). We can therefore define the projector Π i,i+1 onto completely symmetric states along the bond (i, i + 1).
At this point, it is also worth commenting on the role of the momentum k in defining SU(2) generators as in Eq. (6). A priori, none of the properties discussed here depend on the choice of k. First, the commutation relations are invariant under locally re-defining: and thus under changes in k. It follows that k also does not affect the eigenvalue of Q 2 . Nevertheless, the scar models that we discuss have particular values of k, for example k = π for the spin-1 XY and AKLT models. This is because the states in the scar tower, and hence the choices of annihilating projectors P i,i+1 , will be kdependent. In general, for certain choices of the momentum k, the P i,i+1 may not have a simple, physical form in terms of the underlying spin operators. The second type of term that we include in H A are "as a sum" annihilators. These are operators of the form where O i is a local operator centered at site i which does not on its own annihilate the scar tower. In this case there is no freedom to adjust the relative coefficients β i at different sites, since only specific superpositions annihilate all scar states. Including such operators is sometimes necessary for understanding the structure of scars in a given model; for example, Ref. [32] worked out a particular H Σ A for the AKLT model. In other cases, including such terms can lead to physical and potentially experimentally realizable examples of Hamiltonian with scars, such as the one in Eq. (25) presented in Ref. [35]. Additionally, in some cases Hamiltonians of the SM form, Eq. (1), annihilate not only the desired scar tower, but also some of the states outside of the scar tower. Thus in order to ensure that the only non-ergodic states in our spectrum are the scar states, it is also useful to include "as a sum" annihilators in H A .
In order to identify the scarred models described here, we have carried out an exhaustive search for the possible contributions to H A . Specifically, we present a general algorithm which, given a particular set of 'target' states, constructs Hamiltonians for which the target states are eigenstates. This is a generalized version of 'the covariance-matrix' algorithm presented in Ref. [40], and we recapitulate some of the main points of the algorithm for completeness. (This method can also be useful for identifying H SG and H sym .) Consider any m-dimensional linear space of Hermitian operators of interest H and construct a Hermitian basis {h α } for this space. Then, given a target state |ψ , the null space of the m by m matrix (23) corresponds to the space of Hermitian operators in H for which the state |ψ is an eigenstate. That is, from any vector c in the null space, we can construct a Hermitian operator α c α h α with |ψ as an eigenstate. Because the covariance matrix has non-negative eigenvalues, the null space of a sum of covariance matrices C |ψn αβ for multiple states |ψ n corresponds to the space of Hermitian operators in H that have all the |ψ n as eigenstates. Finally, if one desires Hamiltonians that annihilate the target states, such as H A , then dropping the − ψ|h α |ψ ψ|h β |ψ piece of the covariance matrix suffices.
The dimension m of the covariance matrix depends on the size of the space of interest H, but is often quite small. For example, the space of translationally invariant sums of at-most-range-2 operators is just (2S+1) 2 ((2S+1) 2 −1) dimensional, which is independent of L. Thus, the null space of the covariance matrix can be computed very quickly. More computational effort is required to calculate the elements of the covariance matrix, and this calculation scales with the size of the eigenstates |ψ . However, in the case of translationally invariant models and states, calculations on a small size chain can capture the null space of the infinite L covariance matrix. Further, when |ψ n has a matrix product state (MPS) representation (as is the case for all of the scar states we discuss), MPS techniques are useful to calculate the elements of the covariance matrix.
A complementary algorithm for obtaining as-a-sum annihilators was discussed in reference [35], which specialized to scar towers and translationally invariant operators and relied on matrix product methods. We emphasize that the covariance-matrix algorithm above does not need such specializations, and hence can be used for a wider class of target states and Hamiltonians where matrix product methods may not be readily amenable. Relaxing the restriction on translation invariance allowed us to discover a wider class of nearest-neighbor models with the spin-1 AKLT scar states as eigenstates than had been reported previously in the literature; we discuss this example and its generalizations in Appendix B. We note that this method can also be used to directly search within different classes of operators that may be of interest to different experimental setups. For example, h α could be chosen to be a staggered field, h α = i S z i (−1) i or a particular kind of two or three body interaction. This method is also general enough to find examples of Hamiltonians that embed any specific set of target states of interest -that may or not be derived from symmetries -and hence can be used to construct special 'one-off' models with scars.
At this point, we have discussed all the ingredients that enter our framework for constructing scars from symmetries, specifically: (i) H sym with a non-Abelian symmetry G, (ii) the ladder operators Q ± derived from embedded SU(2) sub-algebras of the Lie algebra, (iii) choices for H SG that may or may not be built from the CSA of the Lie algebra, (iv) choices for the base-state |ψ 0 that may or may not be an eigenstate of the Casimirs of G and (v) choices for H A that annihilate the tower of scar states. We note that once a particular tower of states has been identified by the action of Q ± on |ψ 0 , then H A is the most important piece since it ensures that the Hamiltonian acts non-generically only on the scarred manifold but is well thermalizing on the rest of the spectrum. Indeed, in many cases, the simplest choice of H sym = 0 works. Likewise, while H SG is used to give different energies to the scar states, this is not 'required' per se and models with degenerate low entanglement are still scarred. In the next two sections, we present several examples of existing and new scarred Hamiltonians that lie within our framework.

III. SYMMETRIC SCARS
In this section, we discuss several examples of models in which the scarred subspace transforms as an irreducible representation of G (or a q-deformed version thereof), even though the Hamiltonian as a whole is not invariant under the symmetry. In all such models, the scar tower is obtained by acting with raising operators derived from the generators of G on a 'base' low entanglement state that has a definite eigenvalue under all the Casimir operators of G as well as the generators of the Cartan subalgebra of G. Section III A presents various examples, several of which have been presented in the literature previously, where the scars are derived from an SU (2) or "η-pairing SU(2)" symmetry. Section III B generalizes to higher Lie groups, while Section III C considers models with q-deformed non-Abelian symmetries. For most of our examples, the base state will be a state with maximum eigenvalue under the Casimir, such as a spin-polarized state for SU(2) or its analog for general G, but we also present new examples of scar towers built on non-polarized states in Sec. III A 3.

Spin-SU(2) symmetry
We start with a particularly simple example where the symmetry group G is the spin-SU(2) symmetry represented by spin-1/2 operators on each site. Here Q ± = S ± , the only generator in the Cartan subalgebra is Q z = S z , and the Casimir is Any Hamiltonian H sym with spin-SU(2) symmetry has a tunnel of (L + 1) degenerate states built upon the polarized base state |ψ 0 = | ↓↓↓ · · · ↓ by acting with S + . Each of these has maximal S 2 eigenvalue but different S z eigenvalues, and take the form Each |ψ n is the unique eigenstate in a particular symmetry sector characterized by (Q 2 = Q max (Q max + 1), Q z ) eigenvalues. As discussed above, the form of Q + ensures that that these states have at most logarithmic entanglement. The degeneracy of these states can be lifted by adding a term H SG = ΩS z to H sym , which promotes the tunnels to towers. Finally, we can consider a Shiraishi-Mori type H A , as in Eq. 1, with projectors onto two-site singlets on neighboring sites P i,i+1 = (1/4 − S i · S i+1 ). Because the |ψ n have maximal total spin, they are annihilated by each of these singlet projectors. Indeed, Ref. [18] constructed a model of 'perfect scars' of exactly this form: where V i,j = µ,ν J µ,ν i,j S µ i S ν j is an arbitrary operator that is used to break the spin-SU(2) symmetry. In this "perfect scar" model, . Note that even though H sym = 0, the action of H A makes the model well thermalizing outside the scarred subspace, and the scarred states still inherit the SU(2) algebra.
A different example with the same maximal spin scar states is given in reference [35]: Unlike the previous model, Eq. 24, this model has a non-trivial H sym = J 1 S i · S i+1 + J 2 S i · S i+2 with spin-SU(2) symmetry, but has H SG = 0 so that all the scars are degenerate (one could, of course, equally well add a term of the form H SG = S z ). The final term, (2) symmetry and annihilates the scars, but it is not of the SM form since it only annihilates the scar states as a complete sum, whereas previously each local projector individually annihilated the scar states.

Q-SU(2) symmetry
Next, we consider a model where the operators {Q ± , Q z } satisfy SU(2) commutation relations, but are distinct from the spin-SU(2) operators. In particular, we can choose Q ± , Q z according to Eq. 8 with k = π and spin S = 1. As before, we use the operators Q + to construct scar states built upon a base state that is an eigenstate of both Q z and Q 2 so that all of the scars share the same eigenvalue of Q 2 , but are distinguished by their eigenvalues under Q z .
A particular example of this kind in furnished in the spin-1 XY model : The scars are built by the action of Q + on the fully polarized down state |ψ 0 = | − − − · · · − . Note that the first term ∝ J breaks Q-SU(2) symmetry and annihilates the scars, the term hS z acts as H SG and gives energy to the scars, while the term ∝ D commutes with Q z and Q + . The third neighbor term is added to further break a non-local SU(2) symmetry that is present in 1D -this nonlocal symmetry has a ladder operator that is the same as Q + except that it replaces (−1) i → e iπ j =1 i S z j . This ladder operator also generates the same scar tower starting from the same base state, so the scar states would be alone in their symmetry sectors unless this non-local SU(2) and its Casimir are broken.
Similar physics is also at play in the η-pairing states of the Hubbard model on bipartite lattices [41]. The Hubbard model has both a spin SU(2) symmetry and an independent "η-pairing" SU(2) symmetry (which plays the role of the Q-SU(2) symmetry). The η-pairing states have low-entanglement [42], and are the unique states in the symmetry sector of maximal "η-pairing" total spin (i.e. states with maximal eigenvalues under Q 2 ). Analogous to the examples above, the Hubbard model can be perturbed by a suitable H A to break the η-pairing SU(2) symmetry while preserving the η-pairing states as scarred eigenstates in the perturbed model [34,35]; the Hirsch model furnishes a notable example [35]. Strikingly, there exists a simple mapping from spin-1 models above to electronic models that allows for translation between the scar states of the spin-1 XY model and the eta-pairing scars of the Hirsch model and some related electronic models [35].

Scar towers from base states of non-maximal spin
The above examples, drawn from previous literature, contain scar towers generated from a fully polarized state for the base state. In each case, this meant that the scar tower transformed in an irreducible representation of Q-SU(2) with maximal spin. We emphasize, however, that maximal spin (or, more generally, extremal Casimir eigenvalues) are not necessary for scar states, though they are useful for enumerating the bond-wise annihilators.
To demonstrate this, we offer a simple example. Consider a spin-1 chain described by the Hamiltonian H = H sym + H SG + H A with: This model has a scar tower, generated by acting with which is of the form in Eq. 8 with k = 0 and S = 1 on the base state |ψ 0 = 1 As promised, |ψ 0 is an eigenstate of the Casimir Q 2 with eigenvalue L/4(L/4+1) which is less than the maximal Casimir L/2(L/2 + 1).
To see that the terms act as labeled, observe that the terms with proportionality constants B 1 , B 2 , and J z all annihilate all states in this scar tower bond-wise, because every state in the scar tower has |0 on every other site. Similarly, J 1 is equal to the identity plus three times the projector onto the singlet state and is hence also a bondwise annihilator on subtracting out the identity. J 2 is another bond-wise annihilator up to a factor of the identity, as it is equal to a linear combination of the identity and a projector onto the antisymmetric spin-1 states.
We emphasize that the J 2 term is sensitive to the momentum of Q + ; further, without this term, any states of the form |a 1 0a 2 0a 3 0... where a i = ±1 would be eigenstates. Finally, the terms B 1 and B 2 help us to break symmetries and make the model thermal. Collectively, the terms in H A are sufficient to render all but a few of the states outside the scar space thermal, as seen in Figure 2; the fully polarized up and down states remain as eigenstates despite S z being broken, so those states are also scars. We also note that Eq. (27) contains only a subset of the operators that could be added to H A to make the model thermal, others can be found using the covariance-matrix algorithm.
More generally, base states with other eigenvalues under Q 2 and Q z , such as those with eigenvalue (Q max − p)(Q max − p + 1) under Q 2 , and (−Q max + p) under Q z for some finite p, will also have low entanglement and can be used to build scar tunnels.

B. Higher Lie group symmetric scars
Another new class of examples that our symmetrybased perspective on scars makes natural is scar states associated with continuous symmetry groups G other than SU (2). As discussed in Section II, these differ from the SU(2) case in a few important ways. First, in general there are multiple choices of raising operators. Second, there are multiple choices of H SG , which in general satisfy commutation relations of the form (12). Depending on the choice of H SG , we can therefore engineer scar states with multiple distinct frequencies, or with exact degeneracies in their spectra that reflect the more complex Lie group symmetry.
The general idea of the construction closely parallels the SU(2) case. Choosing |ψ 0 = i |w min,i to be a product of the lowest weight state at each site, we have Q α,i |w min,i = 0 for any negative root α. This is the analog of the polarized state, and has maximal eigenvalue under the Casimir. The scar tower then consists of all states of the form (15), where α i are positive roots.
As a simple example, we consider a spin-1 chain. The three states a given site can be viewed as transforming in the 3-dimensional fundamental representation of SU(3), which has N = 8 generators and rank R = 2. The six generators that are not in the CSA furnish three raising and three lowering operators. In the S z i basis |+ i = (1, 0, 0) T , |0 i = (0, 1, 0) T , |− i = (0, 0, 1) T , the three raising operators of SU (3) are Using the following basis for the 2 generators of the Cartan subalgebra on site i: the roots are Note that Q α3,i = [Q α2,i , Q α1,i ] = Q α2,i Q α1,i ; hence Q α1,i and Q α2,i generate the complete set of states on site i from the lowest weight state |− i . We now consider the global raising operators Q αj = L i=1 Q αj ,i . With a base state |ψ 0 = i |− i , the scar space is spanned by the 1 2 (L + 1)(L + 2) states i.e. by all states in the 1 2 (L + 1)(L + 2)-dimensional irreducible representation of SU(3) containing the lowest weight state |ψ 0 . These states will have at most log(m + 1)+log(n + 1) entanglement entropy, since both Q n α1 and Q n α2 have MPO representations with bonddimension n + 1; see Appendix A.
There are two natural physical operators for H SG : S z = i S z i and i (S z i ) 2 . These can be expressed as Using Eqs. (30), (10), (12), both Q α1 and Q α2 raise S z i by one, while Q α1 (Q α2 ) decreases (increases) the eigenvalue of (S z i ) 2 by 1. Correspondingly, the eigenvalue of S z on the state in Eq.(31) is −L + m + n, while the eigenvalue of i (S z i ) 2 is L − n + m. We now turn to H A . At the two-site level, the states in the tower will only contain the six symmetric states | − − , | − 0 + |0− , |00 , | + − + | − + , | + 0 + |0+ and | + + . The three antisymmetric states do not appear, and so we can use projectors onto these states as Shiraishi-Moritype projectors. Correspondingly, there are 9 bond-wise annihilators on a given pair of sites (i, i + 1) that we can use in H A : We also found eight "as-a-sum" annihilators through the covariance-matrix algorithm discussed above, but we will not discuss these annihilators further here. The terms in Eq. (34) break SU(3) symmetry and keep the Hamiltonian from commuting with the two SU(3) Casimirs; hence taking H A to be a linear combination of these bond-wise annihilators at each site is sufficient to eliminate the symmetry, and indeed can lead to a spectrum that is ergodic in the non-scarred Hilbert space. This is seen in Figure 3, which shows the entanglement entropy in the k = 0 and inversion-symmetric sector of the Hamiltonian We've colored the scar states according to their values of m from equation 31; we have an evenly spaced tower of states for each value of m, with n ranging from m to L. As we increase m by one, the resulting tower has one fewer state than the previous. For the parameters chosen here, increasing m corresponds to increasing the energy by 2.3, while increasing n corresponds to decreasing the energy by .3.

Higher Lie symmetries from spin operators
A priori, it is not obvious under what conditions higher Lie group symmetries would arise in real solid-state systems, such as spin chains. In fact, however, we are naturally lead to these if we consider raising operators of the form for 1 ≤ n ≤ 2S, where N is a normalization constant. As noted above, for n = 1 or n = 2S, with a suitable choice of N the operators Q + i , Q − i , and Q z i form an SU(2) algebra. For n < 2S − 1 and S > 3/2, however, in general Eq. (36) does not describe an SU(2) algebra. Indeed, the set Q + i , Q − i , Q z i is not closed under commutation. Closing these operators under commutation leads to a set of raising operators associated with a larger Lie group symmetry.
For general spin S and n = 2, one can show the following. For half-integer S, the relevant Lie algebra is SU(S + 1/2), with the 2S + 1 states on each site dividing into a copy of the (S + 1/2)-dimensional fundamental representation, and a copy of its conjugate. For integer S, the algebra can be divided into two sets of operators, which act only on even and odd integer spins, respectively. This leads to a Lie algebra structure SO(S +1) × Sp(S) for even S, and SO(S) × Sp(S + 1) for odd S. For even S, the Hilbert space at each site corresponds to a copy of the S + 1-dimensional vector representation of SO(S + 1), containing the even-integer spins, and a copy the S dimensional fundamental representation of Sp(S), containing the odd-integer spins. For odd S the S + 1 odd-integer spins transform in the (S + 1) dimensional fundamental representation of Sp(S + 1), while the S even-integer spins transform in the S-dimensional vector representation of SO(S).
For these examples, though the Cartan generators Q z µ,i all commute with S z i , it is not in general the case that S z i can be expressed as a linear combination of the Q z µ,i , since it is not necessarily traceless when acting on each irreducible representation of the relevant Lie group in the Hilbert space. Thus a natural alternative to an H SG of the form (10) is to take H SG = S z = h i S z i , which satisfies an SGA commutation relation of the form (12) for all raising operators Q αj . In this case the value of ω is fixed by how much Q αj raises S z . Thus all frequencies are integer multiple of the elementary frequency 2nh, and in general multiple Q + i operators will be associated with the same frequency.
With this choice, we find degeneracies in the scar tower characteristic of the underlying larger Lie group symmetry. For example, consider the spin-5/2 system described above, with an SU(3) symmetric scar tower. The two operators Q α1 , Q α2 both raise S z by 2, while Q α3 = Q α2 Q α1 raises S z by 4. Taking |ψ 0 = | − 5/2, −5/2, −5/2, ... to have energy 0, we see that (Q α1 ) 2 |ψ 0 and Q α2 Q α1 |ψ 0 are linearly independent states with the same energy of 4h. In contrast, in the SU(2) case, all states in the scar tower have distinct energies, since each power of Q + applied to |ψ 0 necessarily raised the eigenvalue of S z by the same amount.

C. q-deformed towers
In the preceding sections, we considered scar states that transformed in a single irreducible representation of some group. However, we can also consider scar states transforming in representations of "q-deformed groups". q-deformed groups have found many applications, including solving the quantum Yang-Baxter equation [43], describing anyons [44], and phenomenologically describing perturbations to otherwise symmetric models [45]. For the purposes of this work, we restrict our attention to SU q (2), though we expect that our key results generalize to other q-deformed groups.
The characteristic feature of q-deformed groups is a parameter q that modifies the generator algebra. For example, SU q (2) has the following algebra: where The deformation is such that q → 1 returns the algebra to the usual SU(2) algebra. For real, positive q, the representations of q-deformed SU(2) that satisfy the algebra share many similarities with the usual representations. The irreducible representations are 2S + 1-dimensional with S z independent of q and with TheS ± operators are the same as S ± for spin S < 1. The Casimir operator that commutes with the generators and labels the multiplets isS 2 =S −S+ + [S z ] q [S z + 1] q with eigenvalues [S] q [S + 1] q ; that such an operator commutes with the generators can be checked by explicit computation. We will also definẽ for use below.
However, because of the deformation, some of the usual properties of representations of Lie algebras no longer hold. In the regular SU(2) algebra, if we had a representation {S + , S − , S z } we could form a direct product representation, (44) which would also satisfy the algebra. This is how we would describe the action of SU(2) on, say, two spin-S particles. Such a set of would-be generators generally fail to satisfy the q-deformed algebra -instead, for SU q (2), we have that the operators satisfy the deformed algebra if {S + ,S − , S z } do. Similarly,S ± acting on a chain of length L picks up 'tails' of diagonal operators to the left and right for each site: whileS z is the same as S z . For generating scar towers, we'll consider a single-site representation of SU q (2): From the single-site representation, we can construct a chain-wide representation through for arbitrary phase-factor φ i . The freedom to choose an arbitrary phase factor while maintaining the commutation relations may seem surprising, asQ ± is a sum of tailed operators. Nevertheless, commute for i = m, and the phase factors cancel for i = m, so the phases don't affect the commutation relations.
For φ i = kr i , powers of these operators, (Q ± ) n , have a translationally-invariant MPO representation with bonddimension n + 1; see Appendix A. It is striking that the MPO representation is linear in n rather than exponential in n. This means that (Q ± ) n will only increase entanglement entropy by a factor of at most O(log(n)), rather than by n. Thus,Q + with an associated q-deformed symmetry and symmetric base states are good candidates for scar states with additional q-deformed symmetry relative to the Hamiltonian.
To illustrate some of these ideas, consider the following q-deformations of previously discovered models. The simplest is a q-deformation of the model with SU(2) scars in Eq. (24): whereP i,i+1 is a projector onto the two-site q-deformed spin-singlet. Explicitly, The new scar states are those simultaneous eigenstates of S z andS 2 generated byS + on the S z = −L/2 state.
We can also modify more complicated models and extract q-deformed scar states. We introduce here a deformed version of the spin-1 XY model [28,32,34,35] in Eq. 26, Here,S x andS y are deformed using the value of the deformation parameter given in the superscript ofh. This model has scar states generated by a deformed version of the raising operator in Eq. (8) with k = π, S = 1 and with a deformation parameter of 1 q 2 : i acts as a reducible 'doublet' representation of SU (2) on the |± states and a singlet on the |0 state, and hence its single-site q-deformed version is unchanged (cf. Eq. (42)). One can check directly that thisQ + is a ladder operator for SU 1/q 2 (2). There's a separate ladder operator for which (−1) i → e iπ i j=1 S z j that also generates these same scar states. This second operator is a ladder operator for a separate SU 1/q 2 (2) symmetry, so we must be careful to break the two different SU 1/q 2 (2) Casimirs associated with the different ladder operators. The tower of scar states is annihilated by the J and J 3 terms in Eq. (51), and these terms keep the Casimirs from commuting with the Hamiltonian: the nearest neighbor term J is sufficient to violate conservation of the Casimir of the first ladder operator, while the J 3 term violates conservation of the Casimir corresponding to the first and second ladder operators. The similarities to the discussion of the original spin-1 XY model in Eqn. 26 should be clear.
In figure 4, we plot the entanglement entropy in the S z = −2 sector of the q-deformed XY Hamiltonian in equation 51, for q = 1.

IV. GENERALIZED AKLT SCARS
Thus far, we have constructed scarred Hamiltonians in which the scarred eigenstates transform as a single irreducible representation of a (possibly q-deformed) symmetry group G, and have a unique eigenvalue for the Casimir operator(s) C. Here, we describe a qualitatively different family, in which the scarred states are not eigenstates of C and do not transform as an irrep of the symmetry.
For specificity, we focus on one-dimensional spin-S generalized AKLT chains for which the Hamiltonians can be written as a sum of projectors. In Sec. IV A we present two new models, the q-deformed and SO(2S + 1) generalizations of the spin-S AKLT model, and show that they have towers of scarred eigenstates generated by the action of the ladder operator: on their respective ground states. While Q + AKLT is the same as the raising operator associated with Q-SU(2) discussed earlier, Eq. (8) with k = π, the projector onto this asymmetric scarred manifold does not commute with the Q-SU(2) symmetry. Previous work showed that the spin-S AKLT model also has an asymmetric tower of scarred eigenstates generated by the same Q + AKLT [11,32].
To understand how this fits with our broader symmetry based picture, we note that Refs. [32,33] showed that the AKLT model can be deformed to an H sym with Q-SU(2) symmetry while preserving the scars. Now, we generally expect H sym to have degenerate eigenstates labeled by the same value of the Casimir Q 2 , but multiplets in different irreps with different values of C will generically not be degenerate. However, the fact that the AKLT ground state is an eigenstate of H sym -despite not being an eigenstate of Q 2 -implies that tunnels of states with different values of C are degenerate in H sym so that they can be superposed to give an eigenstate with indefinite S 2 . This points to an expanded symmetry in H sym leading to a much larger set of degeneracies. By taking advantage of these, one can prepare base states that are eigenstates of H sym and H SG , even if they are not eigenstates of Q 2 and Q z . Because of such considerations, it is important to systematically study perturbations of the scarred models to connect them to high-symmetry points, and Section IV B considers families of such perturbations with continuously varying scarred states.
The scars in the AKLT model and its generalizations are a consequence of the following general structure, first discussed by Refs. [32,33]. Consider a Hamiltonian H = j h j,j+1 , a base state |ψ 0 with zero energy, and a ladder operator Q + = j e ikj O j . We assume periodic boundary conditions with kL being a multiple of 2π; we discuss the generalization to open boundary conditions briefly in the next subsection and in more detail in Appendix F. We also assume that we can group the two-site Hilbert space into the three disjoint subspaces, G, R, and M . The subspace G (not to be confused with the non-Abelian symmetry group G of H sym ) contains all 2-site configurations (which we will refer to as bonds) that are present in the base state. The subspace R contains the image of all bonds in G under the action of q + j,j+1 = O j + e ik O j+1 , while the subspace M is the complement of G ∪ R. If h j,j+1 and q + j,j+1 have the following general forms, then the model has a scar tower generated by Q + with energy spacing 2ω. The above result follows from explicitly calculating the commutator of H and Q + , as shown in Eq. (61). That is, if G and R are disjoint, and (q + j,j+1 ) 2 = 0 when acting on G, then the Hamiltonian takes the form above, and the model is scarred. We show that the generalized AKLT models in Sec. IV A satisfy these conditions (with ω = 1), and their ground states are not eigenstates of Q 2 . Hence, they each have asymmetric scar towers.
Remarkably, we can also construct a large, continuously connected class of matrix product states (MPS) such that G and R are disjoint for Q + = Q + AKLT (see Sec. IV B). Each such state can function as a base state, and we can enumerate the states in G, R, M to construct new scarred Hamiltonians with these base states as ground states. Because the MPS are continuously connected, we can give continuous deformations between scarred Hamiltonians along which the asymmetric scarred states persist and are continuously deformed.
To demonstrate the power of this large class of states, we revisit Ref. [33]'s deformation of the spin-1 AKLT model to an integrable point along a path with (continuously varying) scarred eigenstates. In particular, we show that there are many such deformations between the spin-S AKLT model and corresponding high-symmetry integrable points.
A. The generalized AKLT models Affleck, Kennedy, Lieb and Tasaki (AKLT) introduced the spin-1 AKLT model to analytically describe the Haldane gap in integer spin chains [46]. Subsequent work discovered that the AKLT chain has fractionalized edge spins in open chains, and is a symmetryprotected topological phase with non-local string order [47,48]. The spin-1 AKLT chain's interesting properties prompted many generalizations, including generalizations to spin-S, q-deformed spin-S [49][50][51][52][53], and other symmetry groups like SO(2S+1) [54]. These generalizations are all examples of the Haldane phase with exactly known ground states. We demonstrate that these models have a second curious property in common, not directly related to the Haldane phase: they all have scar towers generated by Q + AKLT on appropriate ground states.
of a generalized AKLT model of type α can be expressed as a sum of projectors. For the spin-S, q-deformed spin-S, and the SO(2S + 1) AKLT models, we have: Here, the two-site operators P j,j+1 project onto total spin t and q-deformed total spin t respectively. We give their explicit forms in terms of spin operators and q-deformed spin operators in Appendix C. The projectors P j,j+1 is SU q (2)symmetric for all j except for j = L, implying that H Sq is SU q (2)-symmetric with open boundary conditions but not periodic boundary conditions. Notice that H S and H SO(2S+1) agree for S = 1. We also emphasize here that although H Sq has q-deformed SU(2) symmetry, its scar tower is generated by the "usual" raising operator Q + AKLT defined in Eq. 55, and not by q-deformed raising operators of the form Eq. (48).
The three models have exactly known matrix product ground states. With periodic boundary conditions, the ground states are frustration-free and unique. With open boundary conditions, the regular and q-deformed spin-S AKLT models have (S + 1) 2 frustration-free ground states, and the SO(2S + 1) AKLT Hamiltonian has 4 S frustration-free ground states.
We briefly comment on the difference between the scar towers in periodic and open chains (see Appendix F for more details). Assume that we have disjoint G and R with h and q as given in Eq. (56) and (57). In periodic chains, we have where Computing the operator A j,j+1 using Eqs. (56) and (57), we see that A j,j+1 annihilates all the 2-site configurations on sites (j, j + 1) that appear in the states of the scar tower. The form in Eq. (61) matches the MLM condition in Eq. (2). In open chains, the commutator is modified to be: The critical change is the presence of O 1 and O L acting on the physical edge spins, which restricts which of the ground states in open boundary conditions will be a good base state for the tower of states. We argue in Appendix F that O 1 and O L must individually annihilate the physical edge spins in these models, which we show occurs for S 2 out of (S + 1) 2 ground states of the regular and q-deformed spin-S AKLT models and 4 S−1 out of 4 S ground states in the SO(2S + 1) AKLT model. This discussion of open boundary conditions is especially important for the q-deformed AKLT models, as the models lose their interesting SU q (2) symmetry with periodic boundary conditions.
In Appendices D and E, we prove that the G and R subspaces are disjoint, and that Eqs. (56) and (57) hold with ω = 1 and h M M = I for all three models. This furnishes the proof that the spin-S, q-deformed spin-S, and the SO(2S + 1) AKLT models have asymmetric scar towers generated by Q + AKLT on their respective ground states with periodic boundary conditions, while the discussion of open boundary conditions follows additionally from Appendix F. Fig. 5 shows the eigenstate entanglement entropy of the open q-deformed spin-1 AKLT model in a fixed S z sector vs the energy density. In every S z = 1 + 2m sector with integer m ≥ 0, we expect a unique scar state at energy density 2m/L generated by the action of (Q + AKLT ) m on the S z = 1 ground state 6 . The circled state at E/L = 0.5 is thus the predicted scar state in the S z = 7 sector.

B. A large class of Matrix Product States with disjoint G and R subspaces
We now consider consider a broader class of MPSs that can serve as base states for towers generated by Q + AKLT . These will also allow us to study deformations of the generalized AKLT models to Q-SU(2) symmetric H sym with enhanced symmetries. In Appendix D, we show that any spin-S, bond-dimension S + 1, translationally invariant Matrix Product State (MPS) of the form: with m i = 1, 2, · · · 2S + 1 and A [m] being nonzero only on the mth diagonal has disjoint G and R under Q + AKLT . As Q + AKLT raises the z-magnetization S z by 2S, all but one of the bonds in G are mapped out of G under the action of Q + AKLT . The π momentum carried by the Q + AKLT operator furthermore ensures that the remaining bond is also mapped out of G. This result was useful in proving that the q-deformed AKLT models have scars.
More broadly, we can use the above result to construct a large family of scarred Hamiltonians with an MPS of the form given by Eq. (63) functioning as the base state and Q + AKLT functioning as the ladder operator. The Hamiltonians have the general form given by Eq. (56). Different choices of ω and h M M provide different Hamiltonians with the same scarred manifold. The choice of ω tunes the energy spacing of the scar tower, while changing h M M corresponds to adding or subtracting bond-wise annihilators of the scar manifold. One simple choice of Hamiltonian is h M M the identity with ω = 1; this choice leads to a frustration-free Hamiltonian with the chosen base MPS as the zero energy ground state.
We can also use the result in Appendix D to construct paths in the parameter space of the Hamiltonian along which the scar manifold varies continuously. For a given spin-S, we may obtain different paths in parameter space by varying the coefficients on the mth diagonal of A [m] for each m. There are thus (S + 1) 2 complex numbers we can vary continuously for spin S, although the number of free parameters will be smaller on taking into account redundancies in MPS descriptions.
As the MPS along the path functions as a base state for the scar tower generated by Q + AKLT , we can generalize Ref. [33]'s deformation of the spin-1 AKLT model to an integrable point. Ref. [33] described a deformation of the matrices in the spin-1 AKLT ground state to that of an eigenstate of the integrable pure-biquadratic model. That is, they considered for varying c ±,0 . The spin-1 AKLT ground state has co- corresponds to an eigenstate of the integrable pure-biquadratic model. The authors used "numerical brute force" to verify that G and R are disjoint, and that Eq. 57 holds for every choice of the c ±,0 coefficients. They thus constructed a family of Hamiltonians with the form in Eq. 56 that connected the spin-1 AKLT model to the pure-biquadratic model. However, we see that numerical brute force is not needed; the conditions on G, R and q + follow as an immediate corollary of our results on MPS for which A [m] is nonzero only on the mth diagonal, as the matrices in Eq. 64 are only nonzero on the correct diagonals.
We can generalize Ref. [33]'s deformation to spin-S. We note that the Hamiltonian of the spin-1 integrable pure biquadratic point is equivalent to a sum of projectors onto two-site spin-singlets. A spin-S chain with Hamiltonian given by a sum of projectors onto spinsinglets (the singlet-projector model) is similarly integrable 7 [55,56]. Furthermore, we note that there is a simple matrix product eigenstate of H SP , 7 It is Temperley-Lieb equivalent to the Bethe-ansatz solvable XXZ model [55] which can be written in terms of the matrices A SP is only nonzero on the mth diagonal, and hence serves as a nice endpoint for a deformation between the spin-S AKLT model and the integrable spin-S singletprojector model. This construction reduces to the form of the MPS in Ref. [33] for S = 1.
We note that the spin-S singlet-projector model is spin-SU(2) and Q-SU(2) symmetric 8 . The Q-SU(2) invariance arises because the spin singlet is annihilated by Q + and Q − , as mentioned in Appendix E. Because the model is Q-SU(2) invariant, the scar states all have the same energy. The model in fact corresponds to ω = 0, and h M M zero except for the projector onto the spinsinglet. However, at the cost of breaking the spin-SU(2) and Q-SU(2) invariance, we can assign energies to the scar states by setting ω > 0. We can furthermore make the model thermalizing outside the scar manifold by introducing generic h M M , e.g. a sum of projectors with unit coefficients.
Between the endpoints of the spin-S AKLT model and singlet-projector model are many different paths along which the scar states deform continuously. For example, one could take the path

V. CONCLUSIONS AND OUTLOOK
In this work, we have presented a general framework for understanding how quantum scars emerge from parent Hamiltonians with non-Abelian (and possibly qdeformed) symmetries. Generators of the symmetry furnish a natural set of operators with spectrum generating commutation relations, and the parent Hamiltonians have rich structure in their eigenspectrum as a consequence of the symmetry. In particular, the spectrum of H sym is organized as degenerate multiplets ('tunnels') that transform as irreps of the symmetry. Scars emerge when perturbations generically destroy the symmetry and give a thermal spectrum, but do so in a manner that preserves a shadow of the symmetry so that a particular multiplet of low entanglement states fails to mix with the rest of the Hilbert space. This furnishes one qualitative 'picture' for how and when one might expect scars to arise generally, something that has thus far been largely missing in the literature.
Our framework applies to several known models with scars in the literature, but it has allowed us to also introduce several new models with exact quantum scars, significantly generalizing the types of systems known to harbor this phenomenon. These models fall into two broad classes. In the first class, the scar states transform in a single irreducible representation of the symmetry group. Our examples in this class include models where the symmetry is a q-deformation of SU(2), as well where the relevant symmetry is SU(3) rather than SU (2). In the second class, the scar states do not belong to a single representation of the relevant symmetry group, which requires the parent Hamiltonian to have an enhanced symmetry. We have presented examples of this type not previously known in the literature, including generalizations of the AKLT model and families of scarred Hamiltonians that can be smoothly deformed into each other. It is interesting to note that prior studies have tried to explain scars in the PXP model via deformations towards integrable models [17] and those with approximate SU(2) symmetry [18,26]. Hence this symmetry based framework may prove key to eventually fully understanding scars in the PXP model, too.
Our framework leaves open many important questions about the qualitative features that distinguish Hamiltonians with quantum scars from their ETH-satisfying peers. For example, what distinguishes models in which the symmetry is broken in a generic way from those in which a scarred subspace persists? A key general question concerns the stability of scars to perturbations, and whether scars can survive these perturbations either in an asymptotic or 'prethermal' sense [17,57]. Indeed, to classify scars as a new kind of dynamical phase of matter with 'intermediate' thermalization properties -neither fully ergodic nor fully localized -requires scarred models to display some degree of stability in phase space. One important consequence of our picture is that it furnishes a family of scarred models emanating from symmetric parent Hamiltonians, and thus shows that scarred models can have at least some degree of stability to certain classes of perturbations.
Acknowledgments-The authors are grateful to Daniel Renard, Cheng-Ju Lin and Lesik Motrunich, for helpful discussions. A.C. is supported by the Sloan Foundation through a Sloan Research Fellowship. This work was supported by the National Science Foundation through the awards NSF DMR-1752759 (A.C.) and DMR-1928166 (F.J.B.) Note Added-While we were preparing our manuscript, we became aware of two related works, Refs. [58] and [59], which also apply group theoretic considerations to scarred Hamiltonians. Our results agree where they overlap, although the scope of our work is broader than Ref. [58] which only considers Casimir singlets, and our results on q-deformed and asymmetric scar towers also lie outside the constructions of Ref. [59].
Appendix A: MPO representations for (Q + ) n In the text, we discuss creating scars by repeated action of some operators on a base state. We show in this appendix that for certain operators Q + , (Q + ) n has a simple MPO representation with a bond dimension of only n + 1. This implies that acting (Q + ) P oly(L) on some base state can only increase the entanglement entropy of the base state by at most O(log(L)). Our approach is inspired by the examples given in [27]. We extend the results there and take a different style of proof.
We will consider Q + as some momentum k sum of tailed-operators: We will additionally require that at the single-site level, for some arbitrary numbers r and l.
As an example, we could have O + = S + , L = R = I, and k = 0; the commutation relations are satisfied with r, l = 0. This choice corresponds to Q + = S + . As another example, consider O + =S + , L = q −S z , R = q S z and k = 0; this has r = q and l = 1 q . This choice corresponds to the ladder operator for SU q (2).
We'll prove the case of k = 0 first. Define the n + 1 by n + 1 matrix (A3) for β ≥ α and 0 otherwise. Here, we're using the notation, for m and n positive integers, Additionally define the vectors (v r ) i = δ i,1 and (v l ) i = δ i,n+1 , and the matrix Note that we've made the length-of-the-chain Ldependence of Q + explicit.
We'll prove A6 by way of a stronger result: This stronger result implies equation A6 by the definition A3 through We will prove equation A7 by induction on L. The base case of L = 1 follows by inspection.
Assume the form in equation A7 holds for L − 1. Then We can simplify the sum on γ by noting that for two matrices A and B such that BA = xAB, This fact may be found in the "q-Calculus" chapter of reference [60]. Then it follows This concludes the proof. For k = 0, note that L j = e ik L j , O + j = e ik O + j , R j = R j also satisfy the commutation relations in equation A2. Thus, the proof above holds for these primed single-site operators. This implies that for ( i.e. the correct form for a momentum-k sum.

Appendix B: A new class of as-a-sum annihilators
We note in this appendix that our covariance-matrix algorithm for finding Hamiltonians with sets of states as eigenstates has found a previously unknown and rather physical as-a-sum annihilator of the tower of states in the spin-1 AKLT model. This as-a-sum annihilator is the alternating spin-1 AKLT model: H A = j (−1) j P (2) j,j+1 . This means that the alternating spin-1 AKLT model has the scar states all at zero-energy, and further that the staggered AKLT model H A = j c j P (2) j,j+1 with c j = c j+2 has scar states separated by energy spacing c 1 + c 2 . These results are novel and add to the classes of Hamiltonians with the spin-1 AKLT tower of states found in previous papers [33,35].
We can greatly generalize this as-a-sum annihilator to a large class of models with a quick proof. We will assume that the models have the form discussed in Section IV: namely, they satisfy the "disjoint G and R" condition and equations 56 and 57.
As a reminder of the notation, H = j h j,j+1 , Q + = j e ikj O j , q + j,j+1 = O j + e ik O j+1 , and the base state has zero energy. The form discussed in section IV ensures that A j,j+1 = [h j,j+1 , q + j,j+1 ] − ωq + j,j+1 annihilates every state in the scar tower. Then our claim is that annihilates the scar states for all these models. As noted in Section IV, this large class of models includes the spin-S AKLT chain, the q-deformed spin-S AKLT chain, and the SO(2S+1) AKLT chain.
In periodic boundary conditions, with chain length even and kL an integer multiple of 2π, the above assumptions imply that:, Thus, [H A , Q + ] is of the form of equation 2 with ω = 0, while H A |ψ 0 = 0. This completes the proof that H A is an annihilator in periodic boundary conditions with L even and kL a multiple of 2π. It is certainly an as-a-sum annihilator, as without the alternating sign it would give energy to the scar states.
In open boundary conditions, one will get edge terms O 1 and O L , These edge terms could potentially spoil this as-a-sum annihilator in open boundary condtions. However, see F for a discussion of choosing the right ground states in the AKLT models to use as the base for a tower of exact eigenstates; these states are exactly the ones for which the edge terms will annihilate the tower of states. Thus H A in open boundary conditions satisfies the right commutation relation and has the scar states all at the same energy of 0 for arbitrary L.

Appendix C: Explicit forms for AKLT Projectors
In discussing the AKLT Hamiltonians, we noted that all of them could be written as sums of two-site projectors onto manifolds with various total spin or total q-deformed spin eigenvalues. We did not need the explicit forms of the projectors for our discussion. In this appendix, we give forms for the AKLT projectors in terms of spin operators, which are useful for generating the AKLT Hamiltonians for exact diagonalization.
For total spin-S, we can project into the total-spin t manifold by projecting out everything else: will vanish when acting on a two-site state of total spin s = t and will reduce to 1 acting on a state of total spin t.
Similarly, for the q-deformed models, we can write In this appendix, we verify that G and R are disjoint under the action of Q + = 1 in the qdeformed AKLT model. We also show that the form for q + matches that of equation 57. As noted in the text, these two facts will complete our proof of scars in the qdeformed AKLT model in periodic boundary conditions. We'll prove these facts for a much wider set of base states than just the q-deformed spin-S ground states. Namely, we will show that for towers built with Q + on top of a translationally-invariant bond-dimension χ = S + 1 matrix product state for which A [m] has all its non-zero entries on the m-th diagonal alone that G and R are disjoint and q + has the stated form. The ground state of the q-deformed spin-S AKLT model is indeed a matrix product state of this form [53].
The space of two-site bonds of some translationallyinvariant matrix product state will be contained within the span of the bond-dimension-squared number of states kj |m 1 m 2 . Now, note that the product of a matrix with non-zero elements only on the kth diagonal and a matrix with non-zero elements only on the k th diagonal will be a matrix with non-zero elements only on the k + k th diagonal. Thus |AA ij has contributions only from kets with m 1 + m 2 = j − i. Then we see we have −S ≤ m 1 + m 2 ≤ S for all the states in G, and there are S +1−|m 1 +m 2 | states with magnetization m 1 + m 2 .
To see that G and R are disjoint, note that the action of q + increases the z-magnetization of m 1 + m 2 by 2S. Thus, for the all-but-one states in G with m 1 +m 2 > −S, q + takes the states to those with total z-magnetization > S, which is outside of G. There's only one state within G with magnetization m 1 + m 2 = −S , |AA S+1,1 , and there's only one state in G with magnetization S, |AA 1,S+1 . In order to complete the proof of disjoint G and R, we have to verify that despite having the same z-magnetization as |AA 1,S+1 , the state q + |AA S+1,1 is orthogonal to |AA 1,S+1 .
Thus, looking at the explicit form of |AA ij , we see q + |AA S+1,1 ∝ A 11 |0S + "terms with different m1, m2 ", so q + |AA S+1,1 and |AA 1,S+1 are orthogonal. That is, |AA S+11 is mapped to a state outside of G, and hence we've verified that all states in G are mapped to states outside of G under the action of q + . We've thus shown that R and G are disjoint for towers built with Q + on top of the matrix product states described above.
Further, noting that (q + ) 2 raises the total zmagnetization by 4S and hence annihilates all the states in G, we see that q + annihilates the states in R. Putting all the information together, we see that the form for q + (namely, the blocks of zeros) is indeed the one given above.
Thus, the lemma about the forms for q + and h is proven for the spin-S q-deformed AKLT models, completing the proof that the tower of states are indeed eigenstates for these models. We emphasize that disjoint G and R, and the form for q + , were all satisfied for the large class of matrix product states in equation D1 j,j+1 , which projects onto two-site total spin even and greater than zero. Since the ground state is frustration-free, the bonds of the ground state necessarily live within either total spin odd, or zero total spin. That is, G is contained within the span of total spin odd states and the zero spin state.
Under the action of q + , the total spin odd states in G are mapped to total spin even. This is because (S + 1 ) 2S − (S + 2 ) 2S is odd under exchange of 1 and 2, and states with total spin even (odd) are even (odd) under exchange of 1 and 2. Then the action of (S + 1 ) 2S − (S + 2 ) 2S acting on a total spin odd (even) state yields a total spin even (odd) state. This useful property of q + was noted in reference [32].
Under the action of q + , the total spin zero state, |t = 0, m = 0 , is annihilated. Further, no states are mapped via q + to the total spin zero state. That's because q + acting on a total spin-z zero state needs to return a state with total spin-z 2S, but the only such state, |t = 2S, m = 2S , has the same total spin parity (even) as |00 . Similarly, the the only state satisfying the total spin-z constraint that could be mapped to |t = 0, m = 0 would be |t = 2S, m = −2S , but that state has the same total spin parity as |t = 0, m = 0 .
Putting this together, this means that the total spin odd states in G are mapped to total spin even and greater than zero, i.e. states outside of G, while the total spin zero state in G is just annihilated, so R and G are disjoint.
This proves the lemma for the AKLT-like point of the SO(2S+1) AKLT model and completes the proof that the tower of states built off the ground state are all eigenstates of the Hamiltonian.

Appendix F: Scar towers in generalized AKLT models with OBC
Our discussion about the generalized AKLT models in the main text was limited to periodic boundary conditions (PBC). However, there are several key differences between PBC and open boundary conditions (OBC): in OBC, the ground state is no longer unique and instead the models have growing numbers of ground states with S. Correspondingly, the models have growing numbers of linearly independent scar towers using different ground states as base states. However, not every ground state will be a good base state for a scar tower. In particular, we will show that, in OBC, the q-deformed and regular spin S AKLT models have S 2 out of (S + 1) 2 ground states forming the base of linearly independent scar towers, while the SO(2S+1) models have 4 S−1 out of 4 S ground states forming the base of linearly independent scar towers. In past work on the subject, reference [11] described one out of the S 2 towers in the regular spin-S AKLT model in OBC.
In order to count the ground states, note that the ground states of the PBC models were frustration free, unique, and could be represented as matrix product states Such states are ground states in OBC because they are in the kernel of the projectors in the Hamiltonian: They are composed of the same bonds as in the PBC ground state save for the bond between the edge spins at 1 and L. Since the bond dimension of A is S +1 for the q-deformed and regular AKLT models, while the bond-dimension of A is 2 S for the SO(2S+1) models, there are (S + 1) 2 ground states of the q-deformed and regular AKLT models, while there are 4 S ground states of the SO(2S+1) models. A small subtlety is that the SO(2S+1) models need to have long enough chain lengths L for all the ground states found this way to be linearly independent; we will assume that is the case. The ground states of the OBC Hamiltonian, given in equation F2, and towers of states built on top of them by Q + will be annihilated by each A j,j+1 in L−1 j e ikj A j,j+1 . This follows because the set of bonds in the states and the towers built on top of the states are the same as in P BC; i.e. G, R, and M are independent of boundary conditions, the towers contain only bonds in G and R, and A j,j+1 annihilates G and R. (Of course, these ground states generically have bonds between L and 1 that are not in G or R, but A L,1 is not in the sum For the OBC ground states, there's a simple sufficient condition for the whole tower to be annihilated by O 1 and O L : each individual spin-z-basis product state within our base state must have edge spins that are annihilated by O 1 and O L . It's clear that this condition ensures the base state is annihilated by O 1 and O L . Furthermore, the condition guarantees that the action of Q + on a given product state within the base state won't be able to change a given product-state's edge spins, so each state in the tower generated by Q + will be composed of product states whose edge spins are still annihilated by O 1 and O L . Thus the whole tower of states will be annihilated by O 1 and O L if we satisfy the sufficient condition that the edge spins of all the product states within the base state are annihilated by O 1 and O L . In our cases, O 1 and O L are proportional to (S + ) 2S , so satisfying the above condition simply means that the edge spins in the product states comprising the base state can't be | − S .
For the q-deformed AKLT models, S 2 out of the (S + 1) 2 OBC ground states satisfy this condition and hence host towers of eigenstates. This follows from explicit form of the ground states in equation F2 for the q-deformed and regular spin-S AKLT models: A [m] is an S + 1 by S + 1 dimensional matrix that lives only on the mth diagonal, which means that (A ..m L , which contain some product states with | − S at the right and left edges respectively. This means that the PBC ground state, despite also being one of the ground states of the OBC Hamiltonian, does not satisfy the above sufficient condition of O 1 and O L annihilating all the edge states. While we've been careful to identify the condition as sufficient but not necessarily necessary, it is indeed necessary for these models. ) ij can have left spin between 1 − i and S + 1 − i and the right spin between −S − 1 + j and j − 1, the states that don't satisfy the sufficient condition, i.e. do NOT satisfy i < S + 1 or j > 1, will generically contain for L > 2 some product states of the form with 0 as the left spin and −S as the right spin, or vice versa.
Moreover, we cannot eliminate these poorly behaved product states by taking superpositions of different OBC ground states failing to satisfy i < S + 1 or j > 1. That's because we could only bring about such a cancellation between ground states with the same eigenvalue under S ztot , and, by inspection, there are at most two such ground states with the same eigenvalue under S ztot . When there are two such ground states, one will generically have some product states of the form with 0 as the left spin and −S as the right spin and will necessarily not have any product states of the form with −S as the left spin and 0 as the right spin, while the other ground state with the same eigenvalue under S ztot satisfies the opposite. Thus the above sufficient condition is necessary for these models.
We can make a similar set of arguments for the spin-S SO(2S+1) AKLT model to show that 4 S−1 of the 4 S OBC ground states host towers of exact eigenstates generated by Q + . For these models, A [0] = −⊗ S i=1 σ z i and for m > 0, A [±m] = (±1) m √ 2(⊗ S−m i=1 σ z i )⊗σ ± S+1−m ⊗(⊗ S j=S+2−m σ 0 j ). This form of the MPS, though not quite explicitly given in Ref. [54] for general S, follows from that reference up to similarity transformations of the A. This means Similarly to the discussion for the q-deformed models of whether O 1 and O L annihilating all the edge spins is necessary and not simply sufficient, this sufficient condition again appears to be necessary for this model. We will omit the proof of necessity -it follows similarly that each of the ground states in equation F2 that fail to satisfy i ≤ 2 S−1 or j > 2 S−1 will contain poorly behaved product states for large enough L, but it is more challenging to prove that it is impossible that a superposition of these ground states could cancel out the poorly behaved product states.