Topological many-body scar states in dimensions 1, 2, and 3

We propose an exact construction for atypical excited states of a class of non-integrable quantum many-body Hamiltonians in one dimension (1D), two dimensions (2D), and three dimensins (3D) that display area law entanglement entropy. These examples of many-body `scar' states have, by design, other properties, such as topological degeneracies, usually associated with the gapped ground states of symmetry protected topological phases or topologically ordered phases of matter.

Introduction -Until recently, the study of many-body quantum systems has largely focused on ground-state properties and low-energy excitations, implicitly assuming the eigenstate thermalization hypothesis (ETH) dictating that highly excited states of generic non-integrable models are void of interesting structures [1,2]. With the discovery of quantum systems that violate the ETH, a broader interest in the physics of many-body excited states emerged. This modern development is complemented by the growing potential of quantum simulators -predominantly using ultracold atomic gases -to prepare and study quantum many-body systems that are well isolated from the environment [3,4].
Theoretical indicators for the violation of the ETH by a conserved quantum many-body Hamiltonian include (i) a sub-volume law scaling for the entanglement entropy of eigenstates, (ii) emergent local integrals of motion in a nonintegrable system [5,6], and (iii) oscillations in the expectation value of suitably chosen local observables under the unitary time-evolution [7].
Theoretical studies of such ETH-violating systems are challenging for two reasons. Analytical progress [10,15,16,19,20,22] is hard because the models in question are, by definition, non-integrable. Numerical techniques to obtain highly excited states rely on exact-diagonalization [23] and, in some cases, matrix-product state calculations [24]. These techniques are limited in that the range of available system sizes is often too small to allow an extrapolation to the thermodynamic limit. For these reasons, the majority of studies on ETH-violation have been focused on one-dimensional (1D) models.
In this work we present a generic construction that places a scar state in the spectrum of non-integrable many-body quan-tum systems in 1D, 2D, and 3D. While the construction of such states applies to many systems, our primary focus is on topological scar states. In 1D, we construct symmetryprotected topological (SPT) states [25]. In 2D, we present a non-integrable deformation of the toric-code, with 4-fold degenerate scar states on the torus. Finally, in 3D we present a deformation of the X-cube model [26,27] as an example of a system with scars that display fracton topological order [26][27][28][29][30][31].
Our construction is inspired by families of Hamiltonians that have been studied in the contexts of quantum dimer models and spin liquids [32][33][34][35][36][37][38]. In those studies, the emphasis was on the construction of parent Hamiltonians for a given ground state. Consider the Hamiltonian where s labels certain bounded regions of space, such as the elementary plaquettes of a lattice. The operators Q s (β) are Hermitian, positive-semidefinite, and local (i.e., with bounded and discrete spectra), and contain only sums of products of operators defined within the bounded region labeled by s. A family of such local operators is parametrized by the dimensionless number β, which we shall later deploy to deform solvable models and break integrability. The dimensionfull coupling constants α s ∈ R carry the units of energy. The operators Q s (β) are built so as to share a common null state |Ψ(β) , i.e., A warm-up example -We start with a simple example in 1D, which is topologically trivial, but illustrates the general ideas in a straighforward way. Consider a quantum spin-1/2 1D chain with periodic boundary conditions, i.e., a ring, with L sites. On each site i = 1, · · · , L, we denote the three Pauli operators by X i , Y i , and Z i . For any β ≥ 0, we define the local Hamiltonian with 0 < |α| < 1. The condition |α| < 1 is required to place the scar state in the middle of the spectrum; the condition α 0 is needed so as not to break the system into two independent (and integrable) transverse-field Ising chains. At β = 0, the system is equivalent to a paramagnetic spin chain in a Zeeman field, which is integrable. With β 0, all the nearest-neighbor terms no longer commute, i.e., [Q i (β), Q i±1 (β)] 0. In this case, H(β) should no longer be integrable, a fact confirmed by analysis of the energy level statistics obtained numerically as we now explain. We study the statistics of the spacings between consecutive energy levels, s n . . = E n+1 − E n , as well as the r-value defined as the average r n of the ratios r n . . = min(s n , s n−1 )/max(s n , s n−1 ). We analyze the spectrum in common eigenspaces of a maximal set of commuting symmetries of the system, namely translation, parity under inversion, and an additional Z 2 -valued parity defined by i X i = ±1. Figure 1 contains the result of this analysis for α = 0.3, β = 0.5 and L = 20. The distribution matches the distribution of eigenvalue spacings for the Gaussian Orthogonal Ensemble (GOE) of random matrices, thus supporting the claim that Hamiltonian (2) is non-integrable. The corresponding mean r-value for our distribution (averaged over the different momentum sectors) is r = 0.531, close to that of the GOE, r GOE = 0.5359, and clearly distinct from the value of the Poisson distribution, r Poisson = 0.3863.
One can verify that the state where |+ x i is the eigenstate of X i with the eigenvalue +1 and is annihilated by the operators Q i (β) for all i. Therefore |scar(β) is an eigenstate of H(β) with eigenvalue 0. That this eigenstate obeys area law entanglement entropy can be seen as follows. The operators Q i (β) are positive-semidefinite definite, owing to the identity . Therefore, |scar is the ground state of another (local) Hamiltonian, H(β) . . = i |α i |Q i (β). The spectrum of H(0) has a gap between its ground state and all excited states, a gap that remains for a finite range of values of β. Therefore, |scar(β) obeys area law entanglement entropy for a range of β [39]. Alternatively, the area-law property of |scar(β) can be argued from the form of Eq. (3) for any β, by noting that it can be represented by a quantum circuit of constant depth (independent of both β and system size), applied to a product state [40,41].
In Fig. 2, we present the entanglement entropy for the different eigenstates of H(β) for α = 0.3, β = 0.5 and L = 16. Notice that the E = 0 scar state is embedded within highly entangled states.
1D: SPT cluster model -Consider a quantum spin-1/2 ring with 2L sites. Odd and even sites are denoted by SL 1 . . = {1, 3, · · · , 2L − 1} and SL 2 . . = {2, 4, · · · , 2L}, respectively. For any β a ≥ 0 with a = 1, 2, we define the Hamiltonians Note that [H 1D 1 , H 1D 2 ] = 0 for any β 1 and β 2 . For β 1 = β 2 = 0, H 1D is exactly solvable and its ground state is a gapped SPT state [42,43]. Its topological attributes originate from symme-  [25]. Being gapped at β 1 = β 2 = 0, the SPT phase extends to nonvanishing but sufficiently small β 1 > 0 and β 2 > 0. (See Ref. [44] for another deformation of 1D SPT Hamiltonians.) The null state for β 1 = β 2 = 0 is an eigenstate of the Z i−1 X i Z i+1 operators, i = 1, · · · , 2L, with eigevalue +1. We denote this state by |+, · · · , + . For β 1 > 0 and β 2 > 0, the null state of Eq. (4a) is obtained via a similarity transformation with It remains to be shown that the Hamiltonian is nonintegrable. Since the Hamiltonian is made up of two commuting pieces H 1D 1 and H 1D 2 , one must show that each component alone is non-integrable. We shall reduce the calculation of the energy level statistics to the problem already solved for the topologically trivial warm up example of the Hamiltonian H(β) in Eq. (2), presented previously. The mapping is via a nonlocal unitary transformation FIG. 3. (Color online) Example of a lattice structure of the 2D model. Dashed sites and lines are used to represent periodic boundary conditions. (a) Starting from a (N x × N y = 2 × 4) square lattice Λ , we define the median and dual lattices Λ and Λ in such a way that sites of Λ , Λ , and Λ are represented by the symbols , , and , respectively. The red (blue) path P 1 (P 2 ) along the bonds of Λ (Λ ) goes through all sites ∈ Λ ( ∈ Λ ) without intersecting itself. (b) The toric code assigns a local spin-1/2 degree of freedom to each site of the median lattice Λ . To each site ( ) of the lattice Λ (Λ ), we assign the subset s (p) consisting of the 4 sites of Λ on the red cross (blue square) at the site ( ) and define the star (plaquette) operator A s := i∈s X i (B p := i∈p Z i ). The two orthogonal green lines are the "electric" paths l x and l y needed to define two Wilson loops W µ := i∈l µ ∩Λ Z i with µ = x, y, respectively.
The spectrum of H 1D a can be related to that of H by noticing that the operators X i with i ∈ SL 2 that appear in the exponentials in Eq. (7) have no dynamics within H 1D 1 , and vice versa, the X i with i ∈ SL 1 have no dynamics within H 1D 2 . For the purpose of obtaining the eigenvalues of H 1D 1 , one can freeze the X i , i ∈ SL 2 ; there are only two gauge inequivalent choices depending on the Z 2 sector selected, i.e., the choice of i∈SL 2 X i = ±1. (This symmetry is one of the two Z 2 's in the Z 2 × Z 2 .) The spectrum of H 1D 1 in the + sector (equivalent to fixing X i = +1, i ∈ SL 2 ) reduces to that of H that we studied previously. We thus conclude that the 1D SPT scar from Eq. (5a) is an exceptional state in the spectrum of a nonintegrable Hamiltonian H 1D 1 +H 1D 2 . Example in 2D: Toric code -In 2D we study a lattice model derived from the toric code [45]. The Hamiltonian H 2D . . = H 2D 1 + H 2D 2 is defined by the pair of commuting operators where s labels a star and p a plaquette (see Fig. 3), A s = i∈s X i and B p = i∈p Z i . (Notice that β 1,2 = 0 yields the usual toric code up to an additive constant.) We define α s . . = α + (−1) ρ s [α p . . = α + (−1) ρ p ] such that ρ s (ρ p ) is equal to 0 on one sublattice and 1 on the other sublattice of the lattice Λ (Λ ). Here, Λ is the lattice formed by the centers of all the stars, and Λ is the lattice formed by the centers of all the plaquettes. Our deformation of the toric code for β 1,2 0 uses the paths P 1 and P 2 , on Λ and Λ , respectively. These paths are connected, non-intersecting, and chosen such that all the spins are on either of the two paths. (An example of such paths P 1,2 is presented in Fig. 3, and in the Supplemental Material we give further examples.) These conditions on P 1,2 guarantee that (a) [H 2D 1 , H 2D 2 ] = 0, (b) there is no further integral of motion besides H 2D 1 or H 2D 2 as well as space group symmetries, and (c) the spectrum of H 2D 1 alone is equal to that of H 1D 1 for a path P 1 of length L (up to exact degeneracies due to a different number of integrals of motion in 1D and 2D). To obtain (c), one notes that Z i for spins not in P 2 are integrals of motion of H 2D 2 . Replacing them by their eigenvalue ±1 reduces H 2D 2 to the form of H 1D 2 for an appropriate choice of its integrals of motion X j for j ∈ SL 2 in Eq. (4b), upon labeling the spins along P 2 in the order of the 1D chain. We conclude that the level statistics of H 2D 1 and H 1D 1 are identical up to exact degeneracies. Hence the numerical evidence for the non-integrability of H 1D 1 directly carries over to H 2D 1 . In our model, the extensive symmetries at β 1 = β 2 = 0 arising from [A s , B p ] = 0 are lifted when β 1,2 0 (in which case H 2D 1,2 are no longer sums of commuting projectors).
The scar states are built as follows. Because A s and B p square to unity and satisfy s A s = p B p = 1 1, we can build a vector λ ∈ {−, +} 2N x N y −2 out of the distinct eigenvalues of (N x N y − 1) independent A s 's and (N x N y − 1) independent B p 's to label an orthogonal basis |λ of a 2 2N x N y −2 -dimensional subspace of the 2 2N x N y -dimensional Hilbert space on which H 2D acts. To complete the basis of the Hilbert space, we use the eigenstates |ω with the eigenvalues ω ≡ (ω x = ±, ω y = ±) of the pair of Wilson-loop operators W µ with µ = x, y defined in Fig. 3. The following four scar states (one in each of the 4 topological sectors) are eigenstates of H 2D with the eigenvalues E = 0: 3D Example: X-cube model -Our 2D construction can be extended in a straightforward way to 3D toric code-type Hamiltonians [46]. Here, we derive scar states for the slightly more exotic fracton topological order, which only arises in three or more dimensions [26-28, 30, 31]. Fracton phases carry excitations which are (at least partially) immobile in that they cannot be moved infinitesimally by applying local operators. In addition, they can support topological ground state degeneracies that scale exponentially in the system size. Here, we introduce a Hamiltonian based on the X-cube model [27], which supports fracton topological order in its ground state, to construct a set of 3D scar states with the same exponential degeneracy. The Hamiltonian H 3D . . = H 3D 1 + H 3D 2 is, once again, The center of a cross + + + joining its 4 nearest-neighbor sites from Λ defines a site from Λ and the subset s ⊂ Λ . There are three oriented crosses for any site from Λ . They are in one-toone correspondence with the three oriented planes in the Cartesian coordinates of R 3 . For any such oriented cross, we define A s by taking the product of all four Pauli matrices X i with i ∈ s.

defined by the pair of commuting operators
where s labels a star and c a cube (see Fig. 4), A s = i∈s X i and B c = i∈c Z i . (Notice that β 1,2 = 0 yields the usual Xcube model up to a constant.) We define α s . . = α + (−1) ρ s (α c . . = α + (−1) ρ c ) analogously to that in the 2D model, such that ρ s (ρ c ) is equal to 0 on one sublattice and 1 on the other sublattice of the lattice Λ (Λ ). The paths P 1 and P 2 are defined on Λ and Λ , respectively, and they obey the same conditions as in the 2D construction. These conditions guarantee that [H 3D 1 , H 3D 2 ] = 0 for any β 1,2 , while lifting the extensive symmetries at β 1 = β 2 = 0 arising from A s , B c = 0 because H 3D 1,2 are no longer sums of commuting projectors. The Hilbert space for a cubic lattice of linear size L is 2 3L 3dimensional (there are L 3 sites in Λ and 3L 3 in Λ ). The counting of independent stars and cubes delivers the vector λ ∈ {−, +} 3L 3 −6L+3 of eigenvalues. These quantum numbers are complemented by the sub-extensive vector ζ ∈ {−, +} 6L−3 of topological quantum numbers. The number of scar states that are eigenstates of H 3D with the eigenenergy E = 0 thus grows sub-extensively with the linear size L of Λ , and are written as Conclusions -We proposed a scheme to analytically construct highly excited states of non-integrable local Hamiltonians with sub-volume-law entanglement entropy scaling that are embedded in a dense spectrum of volume-law scaling states. We gave further examples of constructions of scar states using stochastic matrix form Hamiltonians [35,37,38] with a notion of SPT or topological orders. This allowed us to construct sets of degenerate scar states. Whether these degeneracies are topological in that they carry a sense of protection against small generic local perturbations is left as a problem for future work.

Construction of Hamiltonians containing null states
Here we demonstrate the construction of Hamiltonians hosting null eigenstates starting from a solvable model.

Consider first operators A s satisfying
where the s label bounded regions in space, for instance any finite subset of sites from a lattice. The notion of locality is tied to the fact that the region on which the operators act nontrivially is bounded. More precisely, for two sites i, j ∈ s, the distance between the sites is bounded, |i − j| < d s , where d s is the finite "diameter" of the region s. Notice that the operators 1 1 − A s are commuting projectors. Second, we define where the operators O i need not just act at one site i, but on a bounded subset of sites centered around i. The operators O i are chosen to be Hermitian and to commute, as well as such that (Notice that if O i contains exclusively operators at site i, that A s , M s = 0 follows trivially from the fact that no common site belongs to s and its complement.) Third, we define and Notice that Q s is Hermitian, while F s is not. They are related by In addition to being Hermitian, Q s is local, because A s is local and the exponential of the local operator M s is also local; and it is positive-semidefinite, as can be inferred by squaring it, and observing that cosh(β M s ) is positive-definite.
We shall now construct a common null state to all the Q s operators. First, notice that the state is annihilated by (1 1 − A s ), for all s, for where we used the fact that A 2 s = 1 1. The state |Ω is arbitrary, as long as it is not annihilated by the projectors (1 1 + A s ). Second, let It follows that, for any s, and consequently Therefore, the state |Ψ β is a common null state of all the local operators Q s , and also of any local Hamiltonian written as a weighted sum of the Q s , say for any weights α s ∈ R. In Eq.
(2), we chose, in place of A s and M s , X i and −β (Z i−1 Z i + Z i Z i+1 ), respectively.

Symmetries in 1D
One finds the commutation relations Therefore, H 1D 1 , H 1D 2 , and H 1D can be diagonalized simultaneously.
Hence, H 1D 1 , H 1D 2 , and H 1D can be simultaneously diagonalized with the Hermitian generator of the unitary operators representing the transformations (23), i.e., the momentum operator associated to the sublattice SL 1 , say.
Inversion symmetry: For any site j ∈ SL 1 , H 1D 1 is invariant under the inversion For any site j ∈ SL 2 , H 1D 2 is invariant under the inversion Hence, H 1D has the Z 2 × Z 2 symmetry that is generated by the two independent involutive unitary transformations (24) and (25). This is to say that H 1D 1 , H 1D 2 , and H 1D are invariant under any inversion of the ring that leaves one site of the ring unchanged. Two independent involutive symmetries: Hamiltonian H 1D 1 is invariant under the involutive unitary transformation that acts trivially on the sites of the ring. Hamiltonian H 1D 2 is invariant under the involutive unitary transformation that acts trivially on the sites of the ring. Hence, H 1D has the Z 2 × Z 2 symmetry that is generated by the two independent involutive unitary transformations (26a) and (26b). With this choice for the paths P 1 and P 2 , the condition β 1 , β 2 > 0 is sufficient to guarantee that the sum over s in H 2D 1 (the sum over p in H 2D 2 ) can never be arranged into the sum of two non-vanishing Hermitian operators that commute pairwise and commute with H 2D 2 (H 2D 1 ). (e) The choice made for the path P 1 colored in red and the path P 2 colored in blue fails to guarantee that the sum in H 2D a can be arranged into the sum of two non-vanishing Hermitian operators that commute pairwise and with H 2D a when β a , β¯a > 0. Indeed, of all Hermitian operators B p entering H 2D 2 , those sites from the dual lattice Λ that are identified by the symbol are not traversed by P 2 . They give a set of operators {B }, whereby B commutes with both H 2D 1 and H 2D 2 .
Relation to the construction for scar states from Ref. [17] In this section, we show that there exists a unitary transformation that brings Hamiltonian (1a) with the property (1b) into the form of the family of Hamiltonians defined in Eqs. (1) and (2) from Ref. [17]. However, we emphasize that Hamiltonian (1a) stems from the stochastic matrix form Hamiltonians introduced in Refs. [37], wherein the property (1b) was proven.
We present the local Hermitian operator Q s in Eq. (1a) (the β dependence is implicit) as Q s = a(s) λ a(s) |ψ a(s) ψ a(s) | , where a(s) labels the orthogonal eigenstates |ψ a(s) with the real-valued eigenvalues λ a(s) of Q s . The consequence of the locality of Q s , in this paper, is that its spectrum is bounded and discrete. Moreover, by construction, Q s has zero eigenvalues. We denote by T (s) the kernel of Q s , i.e., the subspace spanned by the eigenvectors with vanishing eigenvalues λ a (s) = 0. [From here, we use primed label a (s) for a (s) ∈ T (s) and unprimed label a(s) for a(s) T (s).] We shall define the local projector The projector defined by Eq. (36b) and H fulfill all the conditions of their counterparts in Eqs. (1) and (2) from Ref. [17], respectively. Since U ∈ R is allowed to take the value 0, in which case Q s = Q s , H = H, and [ H , P s ] = 0, our Hamiltonian H in Eq. (1a) belongs to the family of Hamiltonians defined by Ref. [17].