Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations

C. J. Turner1, A. A. Michailidis2, D. A. Abanin3, M. Serbyn2, and Z. Papi´c1 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria and 3Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1211 Geneva, Switzerland (Dated: July 17, 2018) Recent realization of a kinetically-constrained chain of Rydberg atoms by Bernien et al. [Nature 551, 579 (2017)] resulted in the observation of unusual revivals in the many-body quantum dy- namics. In our previous work, Turner et al. [arXiv:1711.03528], such dynamics was attributed to the existence of “quantum scarred” eigenstates in the many-body spectrum of the experimentally realized model. Here we present a detailed study of the eigenstate properties of the same model. We find that the majority of the eigenstates exhibit anomalous thermalization: the observable expec- tation values converge to their Gibbs ensemble values, but parametrically slower compared to the predictions of the eigenstate thermalization hypothesis (ETH). Amidst the thermalizing spectrum, we identify non-ergodic eigenstates that strongly violate the ETH, whose number grows polynomi- ally with system size. Previously, the same eigenstates were identified via large overlaps with certain product states, and were used to explain the revivals observed in experiment. Here we find that these eigenstates, in addition to highly atypical expectation values of local observables, also exhibit sub-thermal entanglement entropy that scales logarithmically with the system size. Moreover, we identify an additional class of quantum scarred eigenstates, and discuss their manifestations in the dynamics starting from initial product states. We use forward scattering approximation to describe the structure and physical properties of quantum-scarred eigenstates. Finally, we discuss the sta- bility of quantum scars to various perturbations. We observe that quantum scars remain robust when the introduced perturbation is compatible with the forward scattering approximation. In con- trast, the perturbations which most efficiently destroy quantum scars also lead to the restoration of “canonical” thermalization.

I. INTRODUCTION tors, Ki, which commute with the system’s Hamiltonian, [H,Ki] = 0. For example, in many-body localized phases In recent years, significant efforts have been focused Ki correspond to deformations of simple number opera- 15 on understanding the process of quantum thermaliza- tors of Anderson localized single-particle orbitals. The tion, i.e., the approach to equilibrium of quantum sys- presence of such operators prevents the system, initial- tems which are well-isolated from any external thermal ized in a random state, from fully exploring all allowed bath. The considerable interest in this problem has come configurations in the Hilbert space, leading to strong er- hand in hand with the experimental advances in control- godicity breaking. lable, quantum-coherent systems of ultracold atoms,1,2 Despite significant progress in theoretical understand- trapped ions,3 and nitrogen-vacancy spins in diamond.4 ing of fully thermalizing16 and many-body localized sys- These systems allow one to realize highly tunable lattice tems,17 much less is known about the possibility of more models of interacting spins, bosons or fermions, and to subtle intermediate behaviors. In particular, can ergod- characterize their quantum thermalization.5 icity be broken in interacting, translationally invariant The process of quantum thermalization is believed to quantum systems? In the classical case, non-thermalizing be controlled by the properties of the system’s many- behavior without disorder is well-known in the context body eigenstates, in which physical observables have of structural glasses.18–20 The mechanism of this type of thermal expectation values. This scenario where each of behavior is the excluded volume interactions that impose the system’s eigenstates forms its own “thermal ensem- kinetic constraints on the dynamics.21,22 Similar type of ble” is known as the Eigenstate Thermalization Hypoth- physics has recently been explored in quantum systems esis (ETH).6,7 Despite the lack of a formal proof of the where a “quasi many-body localized” behavior was pro- arXiv:1806.10933v2 [cond-mat.quant-gas] 16 Jul 2018 ETH, various numerical studies of the systems of spins, posed to occur in the absence of disorder.23–35 fermions, and bosons in 1D and 2D8,9 suggest that in Recently, a striking phenomenon suggestive of a differ- many cases when the system thermalizes, all of its highly ent mechanism of weak ergodicity breaking was discov- excited eigenstates obey the ETH,10 i.e., they are typical ered experimentally.36 A Rydberg atoms platform36–38 thermal states and akin to random vectors. was used to realize a quantum model with kinetic con- However, not all quantum systems obey the ETH. straints induced by strong nearest-neighbour repulsion Indeed, in integrable systems11 and many-body local- between atoms in excited states. The experiment ob- ized phases12–14 the ETH is strongly violated due to served persistent many-body revivals after a quench from the appearance of extensively many conserved opera- a N´eel-type state. In contrast, other initial configura- 2 tions probed in the experiment exhibited fast equilibra- anomalous expectation values of local observables, low tion without any revivals. The unexpected many-body entanglement entropy, and enhanced overlap with cer- revivals are inconsistent with ergodicity and thermaliza- tain product states to be the key features distinguish- tion. Moreover, the strong dependence of relaxation dy- ing quantum scarred eigenstates. Using these signatures, namics on the initial state is unusual: for example, many- we find an additional family of quantum scarred eigen- body localized systems fail to thermalize irrespective of states. These states manifest themselves in anomalous their initial state.17 many-body revivals starting from a period-3 density wave In Ref. 39, we attributed the observed slow equilibra- initial states. tion and revivals to a special band of highly non-thermal In order to understand the properties of quantum eigenstates, and proposed an analogy to quantum scars many-body scars, in Sec.IV we formulate the forward first discovered in single-particle chaotic billiards.40 In scattering approximation (FSA), originally introduced in the semiclassical quantization of single-particle chaotic Ref. 39. After illustrating the FSA on a toy example of billiards, scars represent an enhancement of eigenfunc- a free paramagnet, we describe in detail the approximate tion density along the trajectories of classical periodic construction of scarred eigenstates, and demonstrate that orbits. Even though such classical orbits are unstable, the FSA method can be efficiently implemented in large they nevertheless leave a “scar” on the states of the cor- systems using techniques of matrix product states. This responding quantum system. The enhancement of the allows one to accurately capture even non-local proper- eigenstate probability density near a classical orbit im- ties of quantum scarred eigenstates such as the entangle- plies the breakdown of ergodicity in scarred eigenstates. ment entropy. Moreover, quantum scars are surprisingly robust to per- Finally, in Sec.V we investigate the stability of quan- turbations, and their experimental signatures have been tum scars to various perturbations to the considered detected in a variety of systems, including microwave cav- model. Using the intuition provided by the FSA, we clas- ities,41 quantum dots,42 and quantum wells.43 sify perturbations according to how effective they are in In the single-particle case quantum scars are often destroying the quantum scarred eigenstates. We show probed by preparing a particle in a Gaussian wave packet that perturbations that are most effective in destroying localized near the classical periodic trajectory. By anal- the band of scarred eigenstates are also the ones that lead ogy, in Ref. 39 an anomalous concentration of special to the fastest thermalization according to the ETH. We eigenstates in the Hilbert space was demonstrated, thus conclude with the summary of main results and a dis- providing phenomenological support to the quantum scar cussion of open questions in Sec.VI. Various technical analogy. In addition, we presented an explicit method details are delegated to the Appendices. that allowed us to construct these special eigenstates, and demonstrated the absence integrability in the stud- II. KINETICALLY CONSTRAINED PXP ied model. At the same time, many properties of these MODEL quantum scarred eigenstates remained unexplored. For example, what is the entanglement structure of special eigenstates? How many different classes of quantum scars In this Section we start with the derivation of the ef- exist? What is the relation between the presence of quan- fective Hamiltonian from the microscopic description of tum scarred eigenstates and thermalization? the Rydberg atom chain. Next, we consider the Hilbert space structure of the constrained model. Finally, we In this paper we present a detailed study of the prop- discuss the symmetries of the model. erties of scarred eigenstates which addresses the above questions. We start by introducing the model of the experimentally-realized Rydberg chain in Sec.II, and dis- A. Derivation of effective Hamiltonian cuss the structure of its Hilbert space. In Sec.III we in- vestigate properties of the eigenstates of this model from The microscopic Hamiltonian describing a chain of Ry- the point of view of the ETH and quantum entangle- 36–38 ment. We find that the many-body spectrum is distin- dberg atoms is given by guished by the presence of special eigenstates, which have L L X Ω  X atypical expectation values of local observables and thus H = X ∆Q + V Q Q , (1) 2 j − i i,j i j strongly violate the ETH. At the same time, the majority j=1 iground state ( ) or in a particular excited of local operators, and find that the spectral function of state ( ), thus the effective|◦i Hilbert space is that of L local observables has an unusual, non-monotonic form, spin-1/2|•i degrees of freedom. The Rabi term is repre- with a peak at the frequency coinciding with the energy sented by the Pauli operator Xj which flips the atom separation between special eigenstates. We identify the at site j between and states. The diagonal term |◦i |•i 3

Qj = (1 + Zj)/2 is given by the Pauli Z matrix and corresponds to the density of excitations on a given site. Note, this density is not conserved, and atoms interact with each other only when they are in the excited state. It will be convenient to introduce the projector Pj onto state at site j: |◦i 1 Z P = − j . (2) j ≡ |◦jih◦j| 2 This projector corresponds to the local density of atoms in the ground state. It is related to the density of exci- tations as Qj = 1 Pj and obeys XjPj = QjXj. We are interested− in the limit of strong nearest- Figure 1. Graph representation of the PXP model with L = 8 neighbour interactions which we denote as V = V , sites. The vertices of the graph are labelled by product state i,i+1 configurations of the atoms, where ◦ denotes an atom in the V Ω, and we set ∆ = 0, unless specified other-  ground state and • is an atom in the excited state. Edges wise. Rescaling the Hamiltonian by the (inverse) nearest- connect those configurations that map into each other under neighbour interaction strength, 1/V , and introducing the the action of the Hamiltonian. Horizontal axis shows the min- small parameter  = Ω/(2V ), we obtain the Hamiltonian imal number of excitations required to reach the N´eelstate from any given vertex, which coincides with the Hamming X X H = H0 + H1 = QjQj+1 +  Xj. (3) distance, DZ2 . j j

The dominant term H0 counts the number of adjacent Hilbert space. Restricting to the lowest-energy subspace excitations; accordingly, its eigenvalues are the non- defined by the projector in Eq. (4) amounts to ex- negative integers and they are highly degenerate. The cluding configurations withP two adjacent excitations, . •• perturbation term H1 is the trivial paramagnet; its eigen- Considering the simplest example of two sites, it is clear values are every other integer between L and +L inclu- that the lowest-energy subspace, spanned by configura- sive, where L is the number of atoms. − tions , , , cannot be obtained as a tensor product In the limit of strong interactions (small ), we derive of local{◦◦ on-site•◦ ◦•} Hilbert spaces. We note that, similar to the effective Hamiltonian via the Schrieffer-Wolff trans- other kinetically constrained models, in the lowest-energy formation.44,45 First, we introduce the low-energy sub- sector the model (4) has a “flat” potential energy land- space spanned by configurations with no adjacent excited scape. states. The projector onto this subspace can be written It is easy to see that the Hilbert space dimension of the as PXP model in Eq. (5) grows according to the Fibonacci Y sequence. Indeed, we first note that configurations of the = (1 QjQj+1). (4) constrained Hilbert space of an open system of L sites P − j can end with either or . A configuration ending with a state can be obtained• ◦ by appending to a specific Since the leading part of the Hamiltonian (3) vanishes in product• state of a system with L 2 sites.◦• On the other this subspace, we must consider the first non-trivial order hand, a configuration ending with− can be obtained by that is given by H =  H . Removing the overall ◦ SW P 1P adding to a specific configuration of a system with L 1 scale , we obtain the resulting effective “PXP” model, sites. Therefore,◦ the dimension of the Hilbert space with−

L L sites, dL, satisfies the linear recurrence equation: X H = P X P , (5) j−1 j j+1 dL = dL−1 + dL−2, (6) j=1 with initial conditions d0 = 1, d1 = 2. This is the with the effective constraint that no two excitations may well-known Fibonacci recurrence, hence dL for OBC co- be adjacent. This model will be the focus of Secs.III-IV. incides with (L + 2)th Fibonacci number, dL = FL+2. Periodic boundary conditions (PBC) are imposed in the The Hilbert space for L sites with PBC can be formed usual manner by identifying atoms L + 1 and 1, while by taking the Hilbert space of the same number of sites for open boundary conditions (OBC) we add boundary with open boundary conditions and removing all con- terms X1P2 and XL−1PL. figurations which both begin and end with , hence PBC • dL = dL dL−4 = FL−1 + FL+1. Finally,− as illustrated in Fig.1, the Hilbert space B. Structure of the Hilbert space and symmetries and the Hamiltonian of the PXP model have a useful graph representation. It is instructive to first consider P The constraint on the dynamics in Eq. (5) is rather un- the free paramagnet Hamiltonian, HPM = i Xi, act- usual as it destroys the tensor product structure of the ing in the full Hilbert space of 2L product state config- 4 urations. Such a Hilbert space and the action of the III. THERMALIZATION AND Hamiltonian can be conveniently represented as a L- ENTANGLEMENT OF EIGENSTATES dimensional hypercube. All vertices of the hypercube can be uniquely labelled by product state configurations, In this Section, we investigate properties of eigenstates e.g., , , , for L = 2, and edges connect con- of the model in Eq. (5) using exact diagonalization and {◦◦ •◦ ◦• ••} figurations that differ by the state of any single atom. shift-invert algorithm.48 First, we directly test the ETH The Hamiltonian of the free paramagnet HPM is formally using the diagonal and off-diagonal matrix elements of equal to the adjacency matrix of the hypercube graph. local observables between the system’s eigenstates. We find that the majority of the eigenstates appear thermal; Next, we can consider the action of the PXP Hamil- however, the convergence of the observables to the value tonian in the full Hilbert space discussed above. Due to dictated by the Gibbs ensemble is found to be paramet- projectors dressing the X operator in Eq. (5), the state rically slower compared to the ETH predictions. In addi- of a given atom can be flipped only if its nearest neigh- tion, we find a small number of “special” eigenstates that bours are both in the state. Thus, any local two-site strongly violate the ETH. Further, we study the eigen- configuration, ...... ,◦ is frozen regardless of the state of state entanglement entropy, finding that the majority of other atoms, and•• a hypercube graph splits into a number states follow the usual scaling of entanglement entropy of disjoint components. The largest of these components with the volume of the subsystem. In contrast, the spe- contains the state ... and coincides with the sub- cial eigenstates exhibit a small, sub-thermal amount of space defined by the|◦◦◦ projector◦◦i in Eq. (4). Notably, this entanglement. implies that the Hilbert spaceP of the PXP Hamiltonian Finally, we show that the special eigenstates, identi- can be viewed as a subgraph of the L-dimensional hyper- fied as violating the ETH and having low entanglement cube. This graph is known in the mathematical and com- entropy, in fact coincide with the anomalous eigenstates 39 puter science literature under the names of Fibonacci46 which are responsible for the many-body revivals ob- and Lucas cube47 for OBC and PBC, respectively. served experimentally. The number of special eigenstates scales algebraically with the system size L, while the total number of eigenstates scales exponentially. Nevertheless, Fig.1 shows the graph representation of the con- the special eigenstates are of physical significance ow- strained Hilbert space and PXP Hamiltonian for L = 8 ing to their high overlap with the simple charge-density- sites with PBC. Vertices of the graph are classical prod- wave product states, which have been prepared in ex- uct states of atoms, which have been arranged according periment.36 While our previous paper39 focused on the to the Hamming distance from the particular product N´eel( Z2 ) initial state and anomalous states which have state, ... , where atoms on odd numbered sites are | i |•◦•◦• i a high overlap with that state, here we establish another in the excited state. This representation of the Hilbert set of special eigenstates which are distinguished by their space will play a crucial role in the forward scattering overlap with Z3 product state. These special states give approach in Sec.IV. rise to a different pattern of many-body revivals for the dynamics initialized in Z3 state. Finally, we consider symmetries of the PXP model in | i Eq. (5). Restriction to a particular symmetry sector al- lows to reach larger system sizes in exact diagonalization A. Breakdown of ETH in special eigenstates which will be used below. Moreover it is also crucial for the study of thermalization of eigenstates. The PXP Thermalization in ergodic systems is explained by the model has a discrete spatial inversion symmetry I which powerful conjecture regarding the nature of eigenstates maps site j L j + 1. With PBCs, the PXP model — the eigenstate thermalization hypothesis (ETH).6,7,9 7→ − also has translation symmetry. In, addition, the exis- The ETH states that in ergodic systems, the individ- Q tence of the operator = Zi anticommuting with the ual excited eigenstates have thermal expectation values C i Hamiltonian (5) leads to the particle-hole symmetry of of physical observables, which are identical to those ob- the many-body spectrum: each eigenstate at energy E tained using the microcanonical and Gibbs ensembles. has a partner at energy E. The expectation value of a physical observable associ- − ated with an operator O is given by the diagonal matrix Unless specified otherwise, our results below are for element Oαα = α O α , where α is an eigenstate of H, H α = E hα |. Further,| i to| describei how the sys- PBCs where translational and inversion symmetries are | i α | i explicitly taken into account. This allows us to obtain tem approaches the thermal state, Srednicki introduced an ansatz for the matrix elements of physical operators the complete set of eigenstates of large systems of up to 49,50 L = 32 sites (at this system size, the zero momentum in the basis of system’s eigenstates: inversion-symmetric sector of the Hilbert space contains −S(E)/2 Oαβ = (E)δαβ + e f(E, ω)Rαβ. (7) 0+ = 77436 states). Shift-invert algorithm allows us to O extractD a subset of eigenstates for larger systems of up to The first term describes the diagonal part of the operator L = 36 sites with = 467160. in the eigenstate basis, and (E) is a smooth function of D0+ O 5

(a) 0.3 (b) − L = 26 3 0.34 − [∆Z1] d−L 60 L = 28 ∼ 1

L = 30 Z 0.4 4 ) L = 32 |

− ln− ∆ 1

i 40 Z 1 Z ∆ | h

( 5 0.5 p − 4 5 − 20 10 10 dL

0.6 Canonical − 0 10 0 10 0.00 0.05 0.10 − E ∆Z1 | |

Z Figure 2. (a) Strong violation of the ETH revealed by the eigenstate expectation values hO i ≡ hZ1i, plotted as function of energy (color scale indicates the density of data points). While the majority of points are concentrated in the vicinity of the canonical ensemble prediction, the band of special eigenstates (indicated by crosses) is also clearly visible. For these eigenstates, hOZ i strongly deviates from the canonical prediction at the corresponding energy. The system contains L = 30 atoms in the zero momentum, inversion-symmetric sector. (b) Probability distribution for the difference in expectation value of the local observable OZ between eigenstates adjacent in energy. Inset: Mean ∆OZ decays with a power ≈ 1/3 of the Hilbert space PBC dimension dL as the system size is increased (shown up to L = 32). Averaging is performed over eigenstates in an interval between adjacent special states in the middle of the spectrum. The line shown is a linear regression to the three largest system sizes. (c) Off-diagonal matrix elements are a smooth function of the energy difference. Moreover, f 2(ω) does not depend on the system size, consistent with the ETH. At the same time, a number of features are visible in f 2(ω) at the frequency coinciding with the energy separation of special eigenstates in panel (a). The inset shows that f 2(ω), plotted as a function of energy in units of many-body level spacing ∆, does not have a well-developed plateau until ω ≤ ∆.

energy that coincides with the canonical ensemble predic- zero-momentum sector and L/2 states in π-momentum tion. The second term describes off-diagonal matrix ele- sector, resulting in the same total count. The special ments, where S(E) is the thermodynamic entropy at the eigenstates belonging to this band can be viewed as par- average energy E = (Eα+Eβ)/2, and f(E, ω) is a smooth ent states that define the ETH-breaking “towers”, visible function of E and the energy difference ω = E E . Fi- in Fig.2(a). Lower states in the towers also break the α − β nally, Rαβ is a random number with zero mean and unit ETH, though more weakly. variance. We note that the ETH ansatz (7) for the ma- In Fig.2(b) we show the distribution of differences trix elements has been verified in several low-dimensional in the expectation value of OZ between eigenstates ad- 16,51–53 Z Z Z models, while it was found to break down in many- jacent in energy, ∆Oi = Oi+1,i+1 Oii . Consistent body localized systems.54,55 with the ETH prediction, we| observe− that| this distribu- tion narrows around ∆OZ = 0 upon increasing the sys- In Fig.2(a) we test the ansatz (7) for the diagonal Z Z PL tem size. However, despite fluctuations of ∆O decaying matrix elements of the operator O = (1/L) j=1 Zj with the system size, this decay is parametrically slower in the PXP model in Eq. (5). With translation symme- compared to the standard ETH prediction. The inset try, this is equivalent to the expectation value of the Z of Fig.2(b) shows that the mean ∆OZ decays approx- operator on the first site, Z . Moreover, due to the h 1i imately as 1/ 1/3 whereas the ETH ansatz (7) would existence of the Hilbert space constraint, the operator D0+ O can be related to the nearest neighbour correlation suggest a decay which is inversely proportional to the Z square root of the density of states, 1/√ . A recent function OZZ = (1/L) PL Z Z . Note that, because 0+ j=1 j j+1 study56 of the same model with OBC alsoD reports the of the constraint, all eigenstates have negative values for scaling of diagonal matrix elements to be slower than ex- OZ , rather than OZ 0 which would be expected pected from the ETH. Note, however, that only the few inh ai generic thermalizingh i ≈ system with an unconstrained largest system sizes in Fig.2(b) appear to be in the scal- Hilbert space. Fig.2(a) shows that most of the expec- ing regime, which means that it is possible that the power tation values of OZ are close to the canonical predic- governing the decay of the diagonal matrix element con- tion, (E), which is calculated from the Gibbs states verges to 1/2 in larger systems. definedO by the density matrix ρ exp( βH). The value Finally, we test the ETH ansatz for the off-diagonal of β ( , + ) is extracted∝ by relating− the observ- matrix elements. Using Eq. (7) we define the average able expectation∈ −∞ ∞ value to the mean energy in the Gibbs matrix element at a given energy separation, ensemble. However, Fig.2 also shows that there is a num- ber of special states that clearly violate the ETH. These 2 S(E) Z 2 f (ω) = e β O α δ(Eα Eβ ω) α,β, (8) states (denoted by crosses) form a distinct band, which h| h | | i | − − i includes the ground state of the system and extends all which is rescaled by the density of states. In what follows, the way up to the middle of the spectrum. The number we refer to f 2(ω) as the infinite temperature spectral of states in this band is L+1 for OBC. For the case of sys- function, since averaging in Eq. (8) is performed over the tems with even L and PBC, there are L/2 + 1 states in middle 2/3 eigenstates in the spectrum, denoted by α, β. 6

If the off-diagonal matrix elements obey the ETH, the B. Entanglement of eigenstates function f 2(ω) ought to be smooth and independent of the system size. This is indeed confirmed by Fig.2(c), 2 which shows the collapse of f (ω) for different system Quantum entanglement is a complementary probe of sizes. With the previously chosen normalization for the Z 2 thermalization and its breakdown, which provides ad- operator O , in Fig.2(c) we have multiplied f (ω) by ditional insights compared to matrix elements of phys- L, which yields the best collapse of the curves within 16,57 2 ical observables. Equivalence of all observables to their the available system sizes. Moreover, f (ω) decays canonical values imposed by the ETH implies that the exponentially at large ω, as expected from the locality of 16,58 von Neumann entanglement entropy of a subregion A in the Hamiltonian. α α α an eigenstate α, S = trA (ρA ln ρA), is equal to the thermodynamic entropy− of A at temperature T which corresponds to the eigenstate energy E . Here, the en- Surprisingly, in the intermediate range of frequencies α tanglement of an eigenstate is defined in terms of its re- we observe non-monotonic behavior of f 2(ω). The posi- duced density matrix ρα = tr α α that is obtained by tions of the characteristic features in f 2(ω) coincide with A B tracing out the degrees of freedom| i h in| the complement of the energy separation between the ETH-breaking eigen- the spatial region A, denoted as B. Thermodynamic en- states in Fig.2(a). Such a behavior, to the best of our tropy scales proportionally to the volume of region A and knowledge, has not been reported before in the context is maximal in the middle of the band where the density of translationally invariant systems without disorder.57 of states is highest. (Note that Ref. 16 observed features in the spectral func- tion at energies O(1/L) for a system of hard-core bosons with dipolar interactions in a harmonic trap that breaks Fig.3(a) shows that entanglement entropy S for the translational invariance.) In contrast, in disordered sys- majority of eigenstates exhibits behavior that is consis- tems, the emergence of a similar peak was interpreted as tent with the predictions of the ETH. Finite-size scaling a signature of local resonances.55 In addition, the inset of of states with large entropy (S & 5) reveals volume-law 2 scaling, S L (not shown). However, in addition to the Fig.2(c) shows that f (ω) does not have a well-developed ∝ plateau until ω becomes of the order of the many-body bulk of typical highly-entangled states, we also observe outliers with much lower entropy. The outlier states with level spacing, ∆ √L/ 0+. Such a plateau is typical for thermalizing systems,∝ D and it sets the energy scale (the the lowest entanglement, labeled as 0,..., 7 in Fig.3(a), Thouless energy) below which the system essentially can span the entire bandwidth. Note that we do not label be described by a random matrix ensemble.16 states at E > 0, as they are related to states 0,..., 7 by particle-hole symmetry.

From the absence of saturation in the matrix elements For even system size L, there are L/2+1 special eigen- at small energies, we expect the level statistics to show states in the zero momentum sector, and L/2 1 such − deviations from the Wigner-Dyson form. Indeed, previ- states in π-momentum sector. Thus in total, we observe ously it was demonstrated39 that for small system sizes L+1 special states. These states coincide with the states L 28 the level statistics is approximately described that maximally violate the ETH, depicted by crosses in by≤ the Semi-Poisson distribution.59 This is consistent Fig.2(a). Furthermore, as shown in Ref. 39, and as we with the approximately critical form of f 2(ω) for ω ∆ discuss in more detail in the following Section, these spe- in Fig.2(c). 55,60 In addition, we also expect the level≥ cial states can also be identified as ones that have highest compressibility to be enhanced compared to the Wigner- overlap with Z2 product state defined in Eq. (9) below, | i Dyson ensemble. However, the slow development of the as illustrated in Fig.3(b). plateau for L 30 suggests that both the level statistics and compressibility≥ approach the Wigner-Dyson ensem- In Sec.IV we present an approach based on forward ble for larger system sizes. scattering, which accurately captures the highly excited eigenstates with low-entropy labeled in Fig.3. (A brief account of this approach was presented in Ref. 39.) The absence of a Thouless plateau in the off-diagonal Within the forward scattering approximation, we will matrix elements, along with the slow decay of fluctua- be able to demonstrate that these special eigenstates tions in diagonal matrix elements, ∆OZ , and deviations are highly atypical from the entanglement point of view: from purely Wigner-Dyson level statistics, suggests that their entanglement entropy scales with the logarithm of thermalization of the bulk of eigenstates in the PXP system size, i.e., S ln L. This type of behavior, which model may not fully follow the ETH. We return to the is very different from∝ the ETH prediction, is commonly discussion of thermalization in Sec.V. There we will show encountered in ground states of critical systems61 and that full thermalization is restored, and the system fol- systems with Fermi surfaces.62,63 Similar phenomenology lows the canonical ETH predictions, once the PXP model is found in recent work,64,65 where exact expressions for is perturbed in a way that fully destroys the special bands special excited eigenstates in the non-integrable AKLT of eigenstates. model were found. 7

(a) 140 states 6 120 Zk = ...... , (9) 100 | i | |•◦{z ◦} • i 4 80 k S 60 where the atoms in the excited state are separated by 2 5 6 7 40 2 3 4 k 1 atoms in the ground state. In this Section we show 1 20 that− the simplest CDW states, the period-2 ( or N´eel) 0 Z2 0 state and the period-3 (Z3) state, allow one to identify a 20 10 0 10 20 dominant subset of special states in the PXP model. − − E Fig.3(b) shows the squared overlap between all the (b) 40 0 eigenstates of the PXP model and Z2 product state on 2 6 7 35 | i

| 5 4 30 the logarithmic scale. From this plot, we see that there ψ 3 | 3 25 exists a set of eigenstates with anomalously large over- 2 − 2 lap, which form regular tower structures. The states at Z 20

| 1 6 15 the top of towers coincide with the special eigenstates 10 − 0 10 identified via the breakdown of the ETH in Fig.2(a) and log 9 5 entanglement entropy in Fig.3(a). We also see that for − each of the special states labeled 0,... 7, there are further 20 10 0 10 20 eigenstates belonging to the same tower (i.e., with sim- − − E ilar eigenenergy), which have much larger overlap with the N´eelstate compared to the majority of thermalizing Figure 3. (a) Bipartite entanglement entropy of eigenstates, states. S, as a function of energy E. Region A is chosen as one half Interestingly, the L/2 + 1 special eigenstates from the of the chain. The bulk of the states have large volume-law zero-momentum sector, half of which are highlighted in entropy (S 5), however some outliers with anomalously low & Fig.3, are nearly equidistant in energy. Near the cen- entropy (S . 2) are also visible. These states are labelled by 0,..., 7, and they span the entire energy range between the ter of the many-body band, they are separated in en- ground state (state 0) and the middle of the band (state 7). ergy by ∆E 2.66. In addition, the L/2 1 special (b) Density plot showing the joint distribution of energy and states from the≈ π-momentum inversion-antisymmetric− overlap with |Z2i product state among the energy eigenstates. sector have energies exactly between the special states The states with largest overlap are identified with the low from the zero-momentum sector. Thus, combining both entropy states from the top panel. Data shown is for L = 30 sectors, the energy separation between special states be- sites in the zero-momentum and inversion symmetric sector.

1.34

1.32 C. Overlap of special eigenstates with product states E 1.30 ∆

1.28 ∆FSAE1 ∆E2 We have demonstrated that the PXP model breaks ∆FSAE2 ∆E3 ∆E the ETH because of the existence of a relatively small 1.26 1 (algebraic in the system size) number of highly atypi- cal, non-thermal eigenstates. These states are distin- 0.00 0.02 0.04 0.06 0.08 0.10 guished by anomalous matrix elements of local observ- 1/L ables, Fig.2(a), as well as by sub-thermal entanglement entropy, Fig.3(a). However, there exist only L + 1 such Figure 4. Finite size scaling of the energy gaps between states embedded among an exponentially many (slowly) special states closest to the middle of the spectrum shows thermalizing eigenstates. Hence, naively one may expect the convergence of all gaps to the same value in the ther- that these states do not have direct physical relevance, as modynamic limit. Note that adjacent special states belong to different momentum sectors. The gaps accurately follow they might be hidden by the contribution of a much larger quadratic dependence on 1/L in the range of available system number of typical eigenstates. Below we show that this is sizes L = 10,..., 36. The prediction of the forward scatter- not the case because special eigenstates have anomalously ing approximation, discussed in Sec.IV for systems with up high overlaps with certain product states. This implies to L = 48 sites, is shown by blue/orange points, correspond- that superpositions of special eigenstates can be experi- ing to the two states closest to the middle of the spectrum. mentally prepared and probed using a global quench. For Within this approximation, the energies appear to follow lin- example, a class of product states which was studied in ear dependence in 1/L (dashed line). recent experiments36 are the charge density wave (CDW) 8

(a) (b) 1.0 d = 2 2 d = 3 i| 2.5 ψ − 2 2 0.8 d = 4 | − i | ) 3

5.0 t Z (

d 0.6

|h −

3 Z | 10 − 7.5 d 0.4 log − Z

4 | h 10.0 − 0.2 − 20 10 0 10 20 2.5 0.0 2.5 − − E − E 0.0 0 10 20 30 Figure 5. (a) The overlap between eigenstates of the PXP t model and |Z3i state as a function of energy E reveals another special band of eigenstates. (b) Same plot but zoomed in Figure 6. Quantum fidelity shows periodic in time revivals around energy = 0. Black dots mark individual eigenvalues, E for |Z2i and |Z3i initial product states. In contrast, |Z4i initial while the blue curve is a Gaussian convolution of the overlap state shows a complete absence of revivals. Data is for system 2 probability |hZ3|ψ(E)i| viewed as a function of energy. This with L = 24 sites with periodic boundary conditions. reveals a number of peaks subdividing the interval between the highest overlap states. Both plots are for L = 30 in the zero-momentum and inversion symmetric sector together with the ±2π/3 momentum sectors. D. Dynamical signatures of special eigenstates

Anomalously high overlaps of special eigenstates with product states like Z2 or Z3 make them amenable to a simple experimental| i probe| –i global quench. In partic- comes ∆E 1.33 in the middle of the spectrum. Fig.4 ular, the quench from state was studied experimen- ≈ Z2 shows the finite-size scaling of the energy gaps between tally in Ref. 36 and in| numericali simulations on small the four special eigenstates closest to the energy E = 0. systems.66–68 We initialize the system at time t = 0 in All the gaps accurately follow the finite size scaling the state ψ(0) = Zk , and then follow the evolution 2 | i | i ∆Ei = 1.337+ci/L , where a linear term is absent. Con- of this initial state with the PXP Hamiltonian, Eq. (5), stants ci depend on the chosen pair of eigenstates, with ψ(t) = exp( iHt) ψ(0) . This evolution is determined | i − | i c1 = 0.582 corresponding to the gap between the special by how ψ(0) is decomposed in terms of the system’s state at E = 0 and the closest one with non-zero energy. eigenstates.| i In contrast, the distance between special eigenstates at Figs.3(b) and5(a) demonstrate that there are a few the edge of the spectrum, e.g., the ground state (0th eigenstates with high overlaps and constant energy sep- special eigenstate), which always belongs to the zero- aration in the middle of the band where the overlaps are momentum sector, and the first special eigenstate that largest (see Fig.4). Therefore, we expect that quantum lives in π-momentum sector is ∆E0 0.96. This behav- quench from 2 or 3 product state will give rise to co- ≈ 64 Z Z ior should be contrasted with the AKLT model, where herent oscillations,| i with| i a frequency determined by the the special excited states are equidistant in energy. energy separation between the towers of special states in Fig.3 or Fig.5. These oscillations in the dynam- Finally, we note that in addition to the special states ics can be observed by measuring the expectation values 36,39 identified via the overlap with Z2 state, there are fur- of certain local observables, or more generally, using ther states that also violate the| ETHi but more weakly. the quantum fidelity (or return probability) defined as 2 To identify some of them, in Fig.5(a) we plot the over- Zk exp( iHt) Zk . | h | − | i | lap of PXP eigenstates with Z3 product state. Here we Fully consistent with the expectations described above, | i can also observe the existence of a band of states with fidelity for quenches from Z2 , Z3 initial states shown in | i | i anomalously high overlap. In contrast to the Z2 case, Fig.6 reveals pronounced periodic revivals. The period of this band is less clearly separated from the bulk of the these revivals is given by T = 2π/∆E∞, where ∆E∞ Z2 ≈ spectrum. A natural question is whether the set of spe- 1.33 is the energy separation between the Z2 special | i cial states revealed by Z2 intersects with that of Z3 . states. We note that revivals of a local observable – the Cross comparison of overlaps| i (not shown) reveals| thati density of domain walls – were found in Ref. 36 for the these two sets of special states are different from each Z2 case. The frequency of these revivals is identical other. Zooming in on the overlap plot around energy to| thei frequency found here using quantum fidelity. By E = 0, shown in Fig.5(b), we can observe several “mini- contrast, for Z4 initial state, we do not observe any towers” between the highest overlap states. This feature revivals in the| fidelity.i This is in agreement with the will give rise to more complicated dynamics in the Z3 absence of anomalously high overlaps between eigenstates case, which is discussed in the following Section. and Z4 product state. | i 9

The return probability in Fig.6 shows that Z3 initial that the FSA is exact. After explaining the basics of the state also exhibits many-body revivals. The| periodi of method on this simple model, we consider the more inter- these revivals is approximately given by TZ3 (3/4)TZ2 . esting case of the PXP Hamiltonian (5). This model dif- In addition to the revivals, the dynamics displays≈ a beat- fers from a free paramagnet by the projection imposed on ing pattern modulating the amplitude of the revivals. the Hilbert space, which makes the FSA scheme approxi- These modulations can be attributed to additional tow- mate. We formulate the FSA scheme for the PXP model ers of special states, which are illustrated by the blue line and benchmark it on a number of different properties in Fig.5(b). These additional towers are situated be- (more detailed analysis of errors introduced by the FSA tween the highest overlap states. This secondary band of scheme can be found in AppendixB). We demonstrate special eigenstates generally has enhanced overlaps with that the FSA can be efficiently implemented in large sys- product states containing a domain wall between two dif- tems using matrix product state methods. Finally, in ferent Z3 patterns, each spanning one half of the system, the last part of this Section we discuss the notion of a e.g., ...... The existence of such a state trajectory which allows us to relate special eigenstates in a finite◦◦• system◦◦•|◦•◦ requires◦•◦ L to be divisible by 6, L = 6`. to quantum scars in the many-body case. In addition, These “secondary” special states introduce an additional we discuss the implications of the FSA for the stabil- frequency that is ` = L/6 times smaller compared to the ity of special eigenstates to various perturbations of the energy difference between the adjacent Z3 special states Hamiltonian. from all momentum sectors. Consequently,| i the beating pattern also appears for system sizes divisible by 6, and in Fig.6 for L = 24 we observe an enhancement of every ` = 4 revival. A. Forward scattering on the hypercube Finally, we mention that the PXP model, in addition to special eigenstates, also exhibits an exponentially large number of states with energy E = 0. These states can We start with the FSA on the L-dimensional hyper- be understood as arising from the intricate interplay be- cube graph corresponding to the free paramagnet Hamil- PL tween the bipartite structure of the graph which describes tonian, HPM = i=1 Xi. Hence there is no constraint the Hilbert space and Hamiltonian (see Fig.1) and inver- imposed on the Hilbert space throughout this subsection. sion symmetry present in the problem. In AppendixA we In this case the FSA method is exact, and it results in discuss these zero-energy states in greater detail and ob- a Hamiltonian whose non-zero matrix elements are those tain the lower bound on their number that was reported of the spin operator 2Sx for a spin of size L/2. Although in Ref. 39. this result can be obtained via other means, the approach In the following Section, we introduce a forward scat- outlined here allows us to introduce the basic ingredients tering method that allows us to construct accurate ap- that will be needed for the non-trivial case of the PXP proximations of special eigenstates in the PXP model. model. Moreover, this method allows to build special eigenstates The FSA method is a version of the Lanczos recur- starting from Z2 and Z3 product states, explaining 69 | i | i rence. Lanczos recurrence is used to construct the the anomalously enhanced overlaps. In addition, forward Krylov subspace and obtain an approximation to the scattering will provide an insight into the different dy- given Hamiltonian by its projection onto this subspace. namical behavior of the PXP model depending on the The usual Lanczos iteration starts with a given vector in initial state. the Hilbert space, v0, usually chosen to be random. The orthonormal basis is constructed by recursive application of the Hermitian matrix H (i.e., the Hamiltonian) to the IV. FORWARD-SCATTERING starting vector. The basis vector vj+1 is obtained from APPROXIMATION vj by applying H and orthogonalizing against vj−1:

So far we have studied spectral properties of the PXP βj+1vj+1 = Hvj αjvj βjvj−1, (10) model using exact diagonalization and identified a set of − − special eigenstates. Here we explicitly construct a sub- where α = Hv v and β > 0 are chosen such that set of those eigenstates which are related to Z2 product j j j | i v = 1. Hereh we| i observe that the action of H re- state, and whose number scales linearly with the system k jk size. The basic idea behind the construction of these spe- sults in the next vector vj+1 (“forward propagation”), cial states is a modification of the Lanczos iteration.69 but also gives some weight on the previous basis vector, Below we start with applying this modification, dubbed vj−1 (“backward propagation”). “forward scattering approximation” (FSA), to the solv- In the case of the free paramagnet Hamiltonian, the able example of a free paramagnet. In this toy exam- above scheme can be simplified. Let us choose the spe- ple, the Hilbert space and the Hamiltonian can be repre- cific initial vector as the N´eelbasis state v0 = Z2 = | i sented as a hypercube and its adjacency matrix, respec- ... . Moreover, we split the Hamiltonian HPM = P|•◦•◦ i tively (see Sec.IIB). The advantage of this toy model is i Xi = H+ + H− into the forward and backward scat- 10 tering operators, tri-diagonal matrix form of HFSA in the basis of vj:   X − X + 0 β1 H+ = σj + σj , (11a)  β1 0 β2  j∈ odd j∈ even   X X  ..  + − HFSA =  β2 0 .  . (16) H− = σj + σj . (11b)   j∈ odd j∈ even  .. ..   . . βL  βL 0 For the free paramagnet considered in this section, it can be seen that H+ and H− obey the standard algebra of Taking into account the expression for βj, we see that spin raising and lowering operators. This can be used to this matrix coincides with the 2Sx operator for a spin of immediately write down the Hamiltonian matrix. Nev- size L/2, resulting in a set of L + 1 equidistant energy ertheless, we show how the same result can be obtained levels. Likewise, the wave functions in the basis of vj can via a more general procedure, which can be directly gen- be obtained from the Wigner rotation matrix. eralized to the PXP model. Let us consider the first step of the recurrence (10) in this case. Operator H− annihilates the state ... , B. Forward scattering for PXP model |•◦•◦ i and we obtain the vector β1v1 = H+ Z2 , which is an | i equal superposition of all states with a single spin flip on Above we demonstrated how the FSA allows to find top of Z2 , | i a subset of eigenstates in the case of a free paramag- net. Now we return to the problem of the constrained β1v1 = ... + ... + ... + .... PXP model that is defined on the subgraph of the L- |◦◦•◦•◦ i |•••◦•◦ i |•◦◦◦•◦ i (12) dimensional hypercube, where the FSA method is no Hence we see that H+ ensures forward propagation in longer exact. To see this, we again start the FSA from this case, and the action of H− has vanished. The vector v0 = Z2 state, and split the Hamiltonian Eq. (5) | i v1 is automatically orthogonal to v0, thus we set α0 = 0, into the forward and backward propagating parts, H = √ and β1 = L by normalization. H+ + H− with In the second step of the recurrence, we can observe X ± X ∓ that the action of H+ on v1 will produce a state contain- H± = Pj−1σj Pj+1 + Pj−1σj Pj+1. (17) ing a pair of defects atop the N´eelstate, which is thus j∈ even j∈ odd orthogonal to both v1, and v0. On the other hand, the action of the backward-scattering part gives us the orig- Similar to the case of free paramagnet, in such a decom- inal state v0, H−v1 = β1v0, where we explicitly used the position H+ always increases the Hamming distance from value of β1. In the case of a free paramagnet, one can the N´eelstate and H− always decreases it. In the Hilbert show that space graph in Fig.1, H+ always corresponds to moving from left to right. Hence, the FSA recurrence closes after

H−vj = βjvj−1 (13) L+1 steps once forward propagation reaches the opposite edge of the graph, Z0 . | 2i holds more generally at every step of the iteration. This Now, we observe that the key property that enabled the FSA recurrence, Eq. (13), holds only approximately. allows to cancel H−vj with the last term in Eq. (10), yielding the FSA recurrence: More specifically, if one starts from the N´eel state, Eq. (13) is exact for j = 1, 2, but at the third step of the β v = H v , (14) recurrence this property does not hold any more. Never- j+1 j+1 + j theless, we can still apply the FSA recurrence as defined in Eq. (14). The error is quantified by the vector where we also omitted the αjvj term since all αj = 0. This follows from the fact that H± operators change the δwj = H−vj−1 βj−1vj−2. (18) Hamming distance from Z2 state by 1. Hence, the − | i ± new state vj+1 is always orthogonal to vj. Moreover, The error per individual step of the FSA iteration can be by the same argument, the FSA recurrence closes after shown to depend on the commutator [H ,H−], and will 0 + L+1 steps as it reaches the vector vL = Z = ... , | 2i |◦•◦• i be discussed in more detail in AppendixB. Generally, this which is the translated N´eelstate that vanishes under the error is smaller for states that are closest to the middle action of H+. of the spectrum. This is because, as shown in Fig.7(a), Finally, using induction one can demonstrate that the special eigenstates closest to E = 0 have their wave function concentrated near the edges of the graph. As the p βj = j(L j + 1), (15) first few steps of the FSA approximation near the edges − of the graph are exact, we expect it to better capture which, as anticipated, is the well-known matrix element those states that are close to zero energy. In contrast, the of a spin ladder operator. This reesults in the effective ground state and other low-lying special eigenstates live 11

(a) (b) 1.0 C. Extracting physical properties of special 0.8 eigenstates within FSA 0.6

0.4 Diagonalizing the tridiagonal matrix with βj deter- 0.2 mined either directly from Eq. (14) or via linear recur- 0 rence method explained in AppendixC, we obtain a set of approximate eigenvectors and their energies. Pre- Figure 7. Two eigenstates of the PXP model represented viously, in Ref. 39 we demonstrated that eigenenergies on the Hilbert space graph for L = 10 sites and PBC. The color of the vertices reflects the weight of the corresponding agree within a few percent with exact diagonalization product state in the eigenvector, where the largest weight is data for the largest available system of L = 32 atoms. normalized to one. Similar to Fig.1, the N´eelstates are the Here we perform a more detailed study of scaling of the left/right most vertices of the graph, while the fully polarized FSA results. The finite size scaling in Fig.4 reveals that state is located in the center of the graph. (a) Wave function the energy spacing between special eigenstates within the of the special eigenstate closest to zero energy is concentrated FSA approximation saturates to a value that differs by in the vicinity of the N´eelstates. In contrast, the wave func- 2.6% from the one extracted from exact diagonaliza- tion of the ground state (b) is concentrated in the center of tion.≈ Moreover, finite-size corrections to the FSA energy the graph. are linear in 1/L, while exact results appear to follow 1/L2 corrections. The origin of this discrepancy remains to be understood. Moreover, earlier we reported a good agreement be- primarily in the center of the graph, i.e., in the vicinity tween the FSA eigenvectors and the projection of exact of the fully polarized state, ... , as seen in Fig.7(b). eigenvectors onto the FSA basis.39 The FSA also cor- |◦◦◦ i rectly reproduces the expectation values of local observ- The resulting vectors vj obtained from the FSA re- ables. In particular, crosses in Fig.2(a) represent the currence, Eq. (14), starting from v0 = Z2 , form an or- | i expectation values of local observables within the FSA thonormal subspace because each is in a different Ham- for a chain with L = 30 sites. They agree very well ming distance sector and the recurrence closes after L+1 with the exact diagonalization data. Given the ability of steps. At present we do not have closed analytical expres- the FSA to capture the values of local observables, it is sions for βj coefficients, however they can be obtained by natural to ask if it also describes non-local properties of a number of efficient means for systems on the order of special eigenstates, such as entanglement. L . 100 sites, as we discuss in AppendixC. Fig.8 shows the scaling of the bipartite entanglement Diagonalizing the tridiagonal matrix of size (L + 1) entropy in special eigenstates extracted using a MPS im- × plementation of the FSA. Despite entropy being a non- (L+1) with βj determined either directly from Eq. (14) or via linear recurrence method explained in AppendixC, local quantity, we again find good agreement between the one can obtain a set of approximate eigenenergies and FSA and exact diagonalization results for system sizes eigenvectors in the FSA basis. However, rotating the up to L = 30. The logarithmic growth of entanglement eigenvectors to the physical basis requires one to store at entropy with system size L suggests that special eigen- least L + 1 FSA basis vectors, each of the dimension of states cannot be efficiently represented by MPS in the the full Hilbert space, i.e., exponentially large in L. Ear- thermodynamic limit. We note that jumps in the entropy lier we demonstrated in Fig.3(a) that special eigenstates growth of special eigenstates obtained via exact diagonal- have considerably lower entropy than other eigenstates ization, visible in Fig.8, can be understood as accidental at the same energy density. This suggests that matrix hybridization with eigenstates in the bulk. Because the product state (MPS)70 representation of the FSA basis majority of eigenstates carry an extensive amount of en- and special eigenstates should be highly efficient in the tropy (volume-law) in the middle of the spectrum, such present case. jumps can be attributed to two-eigenstate resonances. Notably, the FSA overestimates the entanglement en- In order to formulate the FSA recurrence in the MPS tropy for L 30. This trend is even more pronounced basis, we use the matrix product operator representation in the inset≤ of Fig.8, which shows the scaling of the of H+ from Eq. (17) and construct the basis by applying ground state entanglement entropy obtained with the the MPS operator to the N´eelstate. The only difference FSA. From exact diagonalization it is known that the sys- with respect to the exact FSA is that a compression sim- tem is gapped, and the bipartite entanglement entropy is ilar to DMRG algorithms70 is performed every time an expected to saturate at the value S 0.346 in the ther- operator is applied to a state or two states are summed. modynamic limit. The observed slow≈ linear growth is an That is, we truncate the state for all bipartitions, so that indication of the error of the FSA and we expect it to re- for each reduced density matrix the truncated probabil- side within all eigenstates. However, since the prefactor ity is < 10−8, and then renormalize the state. Below of the observed linear growth is very small, for the sys- we discuss the physical properties of special eigenstates tem sizes considered, it is not visible in the logarithmic obtained within the FSA. entropy growth of the highly excited states. 12

3.5 which allows to systematically construct a manifold of FSA E 1.33 0.40 ≈ low-entangled states that furnish an effective “semiclas- FSA E 2.66

≈ S 0.37 sical” description of many-body dynamics. In particular, 3.0 Exact E 1.33 ≈ Ref. 36 captured the revivals using bond dimension 2 Exact E 2.66 ≈ 0.34 variational ansatz for the collective Rabi oscillations of 2.5 S 25 50 75 atoms between two different configurations of L the unit•◦ cell. ↔ ◦• These oscillations can be viewed as a tra- 2.0 jectory connecting Z2 = ... product state and its | 0i |•◦•◦ i 72 translated version, Z2 = ... . In recent work, 1.5 the TDVP approach| wasi extended|◦•◦• i to a wider class of spin models, thus providing a general framework to ex- 16 24 32 40 48 64 80 plore quantum scarring in the dynamics of many-body L systems by an analogy with the single-particle case. While the TDVP approach allows one to extract some Figure 8. Logarithmic scaling of entropy for two adjacent characteristics of special eigenstates (for example, the os- FSA eigenstates in the middle of the spectrum. Black trian- cillation frequency approximately agrees with the energy gles correspond to the state at energy ≈ 1 33 and blue E1 . separation between adjacent special eigenstates), it is not crosses to E2 ≈ 2.66 (the two eigenstates have approximately the same entanglement entropy with difference ∆S ∼ 0.1%). clear if such an approach can be used for describing the The fit gives S ∝ 0.48 log(L). Green curve corresponds to properties of individual scarred eigenstates, such as the the entropy of the exact special eigenstate at E1 ≈ 1.33. The entanglement structure and expectation values of local non-monotonic behavior of entropy in this case is attributed observables, and for understanding the finite-size behav- to weak hybridization with volume-law entangled states at ior. In this respect the FSA approach provides a descrip- nearby energies. The inset displays the entropy of the FSA tion of the nearly periodic Hilbert-space trajectory that is ground state. The weak growth of entropy with L is an arte- complementary to TDVP. Above we demonstrated that fact of the approximation, since the exact ground state is the FSA constructs a basis of L + 1 states directly in gapped and obeys area law for entropy. the many-body Hilbert space of a finite-size system. The special property of this basis is that it effectively cap- −iHt tures the unitary evolution e Z2 that connects the | i 0 N´eelstate and its translated version, Z . Indeed, the We demonstrated that the FSA allows one to extract | 2i eigenenergies and other characteristics of special eigen- dynamics in the many-body Hilbert space starting from v0 = Z2 = ... proceeds via an increasing number states. Overall, we find good agreement of these results | i |•◦•◦ i with exact diagonalization. The fact that one can cap- of flips that are generated by the forward-propagation ture many-body eigenstates in the Hilbert space (that part of the Hamiltonian, H+. In particular, at the first scales exponentially in L) with a basis of L + 1 vectors is step the dynamics generates one delocalized defect within Z2 state. This state coincides with the second basis vec- unexpected. As we discuss below, this reflects the rela- | i tion between special eigenstates and unstable periodic or- tor in the FSA basis, v1, see Eq. (12). Similarly, the v2 vector from the FSA, with two defects on top of Z2 bits. The FSA provides a basis in the many-body Hilbert | i space that approximately captures the dynamics associ- initial state, corresponds to the second step of the tra- ated with the periodic orbit. jectory. Hence, we conclude that the FSA captures the dominant subspace of the Hilbert space where the dy- 0 namics connecting Z2 and Z states occurs. This is | i | 2i D. FSA subspace as a basis for quantum scarred further supported by Fig. 12 in Appendix. eigenstates Finally, let us discuss other families of quantum scarred eigenstates within the language of TDVP and FSA. In ad- Until now we have discussed the phenomenology of dition to special eigenstates with enhanced overlap with special eigenstates. Several properties of these special the Z2 product state, we also observed Z3 -generated eigenstates suggest their similarity to quantum scarred band| ofi special states in Fig.5. This shows| i that the eigenstates in single-particle systems. In particular, spe- PXP model has more than one periodic trajectory that cial eigenstates are concentrated in parts of the Hilbert leads to quantum scars. In particular, the Z3 -band of space,39 have approximately equal energy spacing, and special eigenstates is related to oscillations| betweeni the are easily accessible by preparing the system in certain three-site configuration, , and configurations , , product states. However, in order to put the relation be- obtained from it by translations.•◦◦ These oscillations◦•◦ ◦◦• can tween special eigenstates and quantum scars on a firm also be described within TDVP.72,73 We note that it is basis, one needs to generalize the notion of a classical also possible to describe the corresponding scarred eigen- trajectory to the many-body quantum case. states using the FSA scheme starting from Z3 product One promising route for defining an analogue of a state. Moreover, the first step of the FSA| recurrencei classical trajectory in the many-body case is provided still remains exact. However, in this case the FSA re- by the time dependent variational principle (TDVP),71 currence is frustrated: starting from state, forward •◦◦ 13

2
propagation brings one into either of the translated con-  H1 (1) + (1/2) (2) H1 , where (1,2) are projec- figurations, or . This fact may potentially explain torsP ontoP subspaces withP oneP or two adjacentP excitations. ◦•◦ ◦◦• the observation that the Z3 -band of special eigenstates The last perturbation, δHnnn, physically corresponds to is less separated from the| continuumi of other eigenstates correlated next-nearest neighbour flips. We note that in Fig.5(a). In other words, the trajectory starting from these perturbations lift the zero-mode degeneracy as they Z3 product state is more unstable, leading to weaker commute with both particle-hole symmetry and inversion quantum| i many-body scars. Nevertheless, one still ob- operators. Thus, all perturbations in Eq. (19) effectively serves distinct periodic revivals of the many-body fidelity remove the bipartite structure of the Hilbert space graph starting from Z3 state, see Fig.6. that is responsible for the appearance of zero modes, see | i The observation of Z2 and Z3 trajectories and un- AppendixA. | i | i derlying sets of scarred eigenstates naively suggests that As we discussed in SectionIVD, the FSA allows to density wave states with larger periods will also give rise quantify the structure of the Hilbert-space orbit under- to scars. Clearly, Fig.6 shows that this is not the case as lying quantum-scarred eigenstates. Hence, we use the already Z4 product state features a complete absence intuition provided by the FSA to qualitatively under- | i of revivals. We attribute this to the fact that the FSA stand sensitivity to different perturbations in Eq. (19). approximation ceases to be exact at the first step for Zn In the case when the PXP model is perturbed by the uni- | i product state with n 4. This signals that the underly- form chemical potential, Eq. (19a), the FSA recurrence ≥ ing trajectories become too unstable to produce quantum remains exact at the first and second steps. However, scars. On the other hand, product states that contain do- δH0 will introduce on-site energies in the FSA, mak- main walls between different Z2 and Z3 patterns can ing the diagonal of the tridiagonal matrix in Eq. (16) | i | i potentially lead to another set of scarred eigenstates. We non-zero. Hence, we expect that δH0 will change the leave a detailed investigation of this issue to future work. frequency of oscillations and also contribute to their de- phasing by removing the periodic energy spacing between special eigenstates. For the weak perturbation g0 = 0.2, V. STABILITY AGAINST PERTURBATIONS we demonstrated almost no change in oscillations, see the Supplementary Material of Ref. 39. Moreover, in Fig.9 Our discussion so far has demonstrated that the FSA we compare the structure of the spectral function in the is helpful for developing intuition about the structure of PXP model to that in the perturbed model with g0 = 1. various families of quantum scarred eigenstates at high We observe that the peak in off-diagonal matrix elements 2 energy densities. Here we investigate the stability of Z2 f (ω) at ω 2.66 shifts to slightly larger frequencies, but special eigenstates with respect to various perturbations| i still remains≈ strongly pronounced. of the Hamiltonian. We will rely on the FSA to develop Next, we consider the case of the nearest neighbour intuition why special eigenstates are robust with respect hopping perturbation. This perturbation leaves the first to certain perturbations, or which kind of perturbations step of the FSA exact, but introduces an error already at are most efficient in removing the periodic orbits. Fur- the second step. Yet, Fig.9 shows that the peak in the thermore, we discuss several deformations that bring the spectral function associated with the separation between PXP model to exactly solvable points. Some of these special eigenstates shifts while keeping the same magni- deformations were found in Refs. 74 and 75. Below we tude when we add nearest neighbor hopping, gnn = 0.5. demonstrate that these perturbations are strong and re- Note that the magnitude of the perturbation is chosen in move the special eigenstates that are found in the PXP such a way that it has comparable operator norm to the model. previously considered chemical potential with g0 = 1. Finally, we considered next nearest neighbor correlated flips, Eq. (19c), as a perturbation that introduces error A. Physical perturbations even at the first step of the FSA approximation. Hence, we expect such a term to have the strongest effect of all We consider the following perturbations of the PXP three terms considered in Eq. (19). Fully consistent with Hamiltonian, these expectations, we observe that the perturbation of X magnitude g = 0.25, which has operator norm com- δH = g Q , (19a) nnn 0 0 j parable to earlier perturbations, suffices to significantly j broaden the peak in the spectral function. In addition, X + − − + δHnn = gnn Pj−1(σj σj+1 + σj σj+1)Pj+2, (19b) we observe in Fig. 10 that this perturbation is the most j efficient one in damping the oscillations of the local two- X site entanglement after about three periods. We note δHnnn = gnnn Pj−1XjPj+1Xj+2Pj+3. (19c) that while the perturbations gnnn = 0.25 and g0 = 1 j both lead to the strongly enhanced growth of bipartite The uniform chemical potential, δH0, and constrained entanglement with very similar slopes (not shown), the nearest neighbour hopping, δHnn, result from the former is more efficient in damping the local oscillations. (2) second order Schrieffer-Wolf transformation, HSW = Above we observed that perturbations δH0 and δHnn 14

Figure 9. Peak in the energy dependence of matrix ele- Figure 10. The dynamics of two-site entanglement in the ments of the unperturbed PXP model at ω ≈ 2.66 softens quantum quench from |Z2i initial state is strongly influenced and shifts upon the addition of perturbations. Moreover, the by perturbations to the PXP model. We note that nearest- inset shows that the plateau in f 2(ω) only slightly increases neighbor hopping is the least effective in damping the oscilla- its size when the added perturbation is chemical potential tions. In contrast, for the perturbation gnnn = 0.25, when the or correlated hopping. In contrast, upon adding δHnnn, the bulk of eigenstates becomes fully thermal, the oscillations in plateau increases to values of ω/∆ ≤ 200, fully consistent entanglement and local observables are strongly damped. The with the restoration of conventional thermalization. entanglement is normalized by the maximal possible value for two sites, which is equal to ln 3 due to the presence of a constraint. The data is obtained with iTEBD, the maximal evolution time is limited by the bond dimension χ = 1200. are less effective in destroying the special eigenstates and the corresponding oscillations in the PXP model. The correlated flips δHnnn is most effective in destroying the oscillations. At the same time, we observe that the latter bonacci anyonic model.76 In addition, there exist a one- perturbation is most effective at removing the traces of parameter family of frustration-free Hamiltonians that “slow” thermalization in the bulk of other eigenstates. In includes the PXP term.75 particular, Fig.2(b) demonstrated that while the fluctu- In particular, by adding the operator ations in local observables decay exponentially with the X  −1
 system size, this decay is slower than expected from the δH = vQ − Q + v v Q (20) v − P j 1 j+1 − j P ETH. We checked that for gnnn = 0.25 the fluctuations j of local observables decay as a square root of the Hilbert Z p space dimension, ∆O 1/ 0+, fully consistent with with one free real parameter v to the Hamiltonian of the ∝ D the ETH expectations. In addition, the inset of Fig.9 PXP model, we obtain a Bethe-ansatz solvable model.74 shows that δHnnn perturbation corresponds to the best- For v = 1 this perturbation amounts to the constant developed plateau in the spectral function at small energy next-nearest neighbor interaction of Rydberg atoms. An- separations. other special point is v = 2−1/4, when the total norm of Thus we conclude that the existence of well-defined the operators in Eq. (20) takes a minimal value. More- special eigenstates on the one side, and anomalies in ther- over, this integrable line has a quantum critical/tricritical malization of the bulk of eigenstates on the other hand, point at v = ((√5 + 1)/2)5/2, respectively.74,77 3,2 ∓ are related to each other. In other words, the existence In the integrable models all eigenstates violate conven- of strongly scarred quantum many-body eigenstates and tional ETH and can be described only via the generalized their “protection” from the bulk of other eigenstates is in- Gibbs ensemble that incorporates additional conserved tertwined with slower thermalization of other eigenstates. quantities. At the same time, we explicitly checked for all these cases that the special eigenstates, found in unper- turbed PXP model via overlap with Z2 product state, are B. Integrable deformations of PXP model either strongly perturbed or completely destroyed. More- over, we did not observe any low-entanglement eigen- After illustrating that perturbations that are effective states at energy E close to zero, unlike for the unper- in restoring “canonical ETH” thermalization also destroy turbed PXP model, see Fig.3. the bands of special eigenstates, we discuss deformations Another family of perturbations with a free parame- of the PXP model that make it exactly solvable. PXP ter z, Hamiltonian can be deformed to become integrable by a 74 X −1 one-parameter family of deformations. The set of inte- δHz = (zPj−1PjPj+1 + z Pj−1QjPj+1), (21) grable models includes the so-called “golden chain” Fi- j 15 brings the PXP model into a frustration-free Hamilto- the case when the system is initialized in Z3 product | i nian, which allows for the exact solution of its ground state, with some additional features compared to the Z2 state75 for any real value of z. When z 0 (z ), case, and we identified the corresponding family of| spe-i the first (second) term dominates, and this→ perturbation→ ∞ cial eigenstates. always has finite magnitude O(1) for any values of z. The phenomenology of the special eigenstates de- Analytically minimizing the energy of the PXP model scribed above allowed us to draw parallels with the ubiq- Eq. (5) using a frustration-free MPS ansatz of Ref. 75 uitous phenomenon of quantum scarring, thus lending 1/2 results in the value zPXP = (√5 + 1) /(2√2) 0.636. support to the term quantum many-body scars. In the Such an approximation reproduces local observables≈ of case of a single particle in a chaotic billiard, a single un- the exact ground state, such as energy density, with high stable periodic classical orbit leads to a set of scarred 40 precision. Perturbing the PXP model using δHz, we find eigenstates. These eigenstates have their wave func- that fidelity oscillations always decay faster compared to tions localized in the vicinity of their parent trajectory, the unperturbed PXP model. For z z we find the and can be efficiently prepared by initializing the wave ∼ PXP slowest decay for all z so that the perturbation δHzPXP packet near the classical orbit. Moreover, in chaotic bil- has weakest effect. Damping of oscillations increases with liards one usually finds more than one periodic trajectory increasing z zPXP . that gives rise to quantum scars. The classical trajecto- In summary,| − we| observed that deforming the PXP ries which are less unstable give rise to more localized model to nearby solvable points does not improve the ro- wave functions, corresponding to stronger scarring. Fi- bustness of quantum scarred eigenstates. Instead, such nally, one also expects some degree of stability of quan- deformations either strongly perturb these states or lead tum scars to perturbing the system, unless the perturba- to their complete disappearance. This result suggests tion destroys the periodic trajectory. that it is the unperturbed PXP model that should be Similarly, in the PXP model, we observed a set of L+1 viewed as a parent model for the quantum-scarred eigen- special eigenstates that are well described within the FSA states, and despite proximity of several integrable points basis, whose dimension scales linearly with the system to this model, Bethe-ansatz integrability cannot be used size. This suggests that special eigenstates are concen- to explain this behavior. trated in a small part of the Hilbert space, analogous to the case of a chaotic billiard. The system effectively ac- cesses these eigenstates when prepared in the initial Z2 0 | i VI. DISCUSSION AND OUTLOOK state or its translated partner, Z2 . Additionally, in this work we reported a second family| i of scarred eigenstates arising from Z3 density wave product state. This second In this paper we studied the eigenstate and dynamical | i properties of the PXP model, which describes a chain family of eigenstates has larger entanglement, suggesting of Rydberg atoms realized in recent experiments.36 We that the underlying orbit is less stable. The enhanced sta- bility of Z2 special eigenstates compared to their Z3 found that the majority of the eigenstates of this model | i | i thermalize more slowly compared to other microscopic counterparts can also be understood within the FSA. models that are usually used to test the ETH.16 On the Finally, we demonstrated the stability of the many- one hand, the origin of such behavior may be related body revivals with respect to perturbations of the PXP to the constraints in the PXP model, which make the model. We confirmed that perturbations which do not Hamiltonian sparse in the Hilbert space. On the other introduce any immediate errors in the FSA approach are hand, conventional ETH was shown to hold for other less effective in destroying the bands of special eigen- kinematically constrained models.78 Hence, we speculate states and the revivals. We also identified a perturbation that anomalies in the thermalization in the PXP model that quickly removes the non-ergodic scarred eigenstates, may be related to the existence of quantum scarred eigen- restoring the ETH for all states. In addition, we also con- states. sidered several deformations of the PXP model that bring These quantum scarred eigenstates, identified in it to solvable points. Although the PXP model can be de- Ref. 39, strongly violate the ETH. In particular, these formed into Bethe-ansatz integrable models which do not special eigenstates stand out due to their anomalous ex- follow the ETH, the characteristic many-body revivals pectation values of local observables, as well as their from simple product states do not persist in these inte- much smaller entanglement entropy compared to thermal grable models. Thus, the proximity of those integrable eigenstates with similar energies. Besides the ETH viola- lines is likely unrelated to the weak ergodicity breaking tion, special eigenstates are characterized by their large in the PXP model. overlaps with charge density wave states Z2 and Z3 . While our study sheds new light onto the structure and The energies of these special eigenstates| arei (approxi-| i stability of quantum many-body scars, many interesting mately) multiples of the same fundamental frequency. questions remain open. We used the FSA approximation Consequently, as discussed in Ref. 39 and in this paper, throughout this paper and provided a simple estimate of these eigenstates play a key role in the experimentally the incurred errors in AppendixB. However, quantify- observed many-body revivals in the quantum quench ing the final error in the FSA remains an open problem. setup.36 We also predicted the existence of revivals for Furthermore, it would be highly desirable to identify a 16 parameter that governs the stability of quantum scars AKLT model with logarithmic scaling of the entangle- in the generic case. Better understanding of the errors ment entropy. As discussed in Ref. 65, the existence of in the FSA would allow to obtain more rigorous under- such eigenstates is suggestive of the presence of quantum standing of quantum-scarred eigenstates in the thermo- scars in the AKLT model. dynamic limit. While the FSA suggests their persistence, Furthermore, during the completion of this numerical studies have revealed an onset of accidental hy- manuscript, we became aware of two related works bridizations between scarred eigenstates and the thermal- on the PXP model.56,72 In Ref. 72, a generalization izing bulk of eigenstates. These hybridizations resulted of the TDVP approach for various spin models and a in irregular behavior of entanglement entropy for larger connection with periodic orbits has been developed. In system sizes L 34, despite the energies of special eigen- a different direction, Ref. 56 has argued that special states still following≥ accurate finite-size scaling. Gener- properties of the PXP model result from a “proximate ally, one may expect special eigenstates to get “dissolved” integrable point”, to which the model can be driven by in the bulk in the thermodynamic limit. Nevertheless, applying a particular perturbation. there will be signatures remaining in the properties of local operators and dynamics at short and intermediate time scales. In particular, the structure of the spectral ACKNOWLEDGMENTS function reported in this work is expected to be robust in the thermodynamic limit. For instance, the unusual We thank Paul Fendley, Misha Lukin, Hannes Pichler, peaks in the off-diagonal matrix elements at the energy Marcos Rigol, Harry Levine, and Wen Wei Ho for illumi- difference of order one, shown in Fig.2(c) and Fig.9, are nating discussions. C.J.T. and Z.P. acknowledge support converged with the system size. Revealing and identify- by EPSRC grants EP/P009409/1 and EP/M50807X/1, ing other experimentally observable signatures remains and the Royal Society Research Grant RG160635. D.A. an interesting problem. acknowledges support by the Swiss National Science More broadly, it would be desirable to understand Foundation. A.M. and M.S. acknowledge support pro- whether there are wider classes of models that display vided by J. Kiss and A. Schl¨oglfrom the HPC Scientific quantum scars. On the one hand, the FSA suggests that Service Unit of IST Austria. Statement of compliance the constraint present in the PXP model plays a crucial with EPSRC policy framework on research data: This role in protecting and enabling such behavior. There- publication is theoretical work that does not require sup- fore, it would be natural to search for other types of porting research data. (constrained) Hilbert spaces and models with similar be- havior. For example, we note that the model in Eq. (5) is related to a class of models that represent interactions Appendix A: Zero-energy states between fundamental excitations in topological phases of matter in two dimensions.33,76,78–83 A wide class of In the main text it was mentioned that one of the spe- such phases are the fractional quantum Hall states, in cial features of the PXP model is the existence of an which electrons fractionalize into Abelian or non-Abelian exponentially large number of states which are annihi- anyons. In particular, in a ν = 12/5 fractional quan- lated by the PXP Hamiltonian in Eq. (5). In Ref. 39 tum Hall state, the fundamental excitation is a Fibonacci (see also Ref. 85) it was shown that the degeneracy of anyon τ.84 The rules of anyon fusion place a formally this zero-energy subspace, , grows with system size similar constraint to the allowed number of anyons as L according to a Fibonacci numberZ F . More precisely, for our constraint on the allowed excitations in the Rydberg open boundaries, depending on whether the system size atom chain. Thus, it would be interesting to explore L is even or odd, we have analogous models (in the context of cold atomic gases or trapped ions) for different types of anyon models, and in- = F , = F . (A1) vestigate the occurrence and stability of quantum scars Z2n n+1 Z2n+1 n in them. For periodic boundaries in the zero-momentum sector Finally, the issues discussed above naturally connect to questions about practical uses of quantum many-body (0) (0) = Fn−1, = Fn−1, (A2) scars and their dynamical signatures. Preparing the sys- Z2n Z2n+1 tem in a superposition of quantum-scarred eigenstates whilst in the π-momentum sector instead effectively shields it from thermal relaxation for much longer times. Hence, a better understanding of funda- (π) 2n = Fn−2. (A3) mental properties of quantum many-body scars, their Z stability and tunability may be of potential use in experi- We note that there are zero energy levels in other sym- ments studying dynamics of non-equilibrium many-body metry sectors, but they are fewer in number and they quantum systems. will not be explicitly considered here. Note added.— Very recently, Ref. 65 has analytically In this Appendix, we formally derive the above count- constructed a set of non-thermalizing eigenstates in the ing for both OBC and PBC (in the zero momentum sec- 17 tor). The key to this is the particle-hole symmetry, gen- subspaces spanned by even invariant elements and Ke erated by the operator odd invariant elements o. Each two-element orbits con- tains one inversion-evenK irreducible representation and Y = Zi (A4) one inversion-odd irreducible representation. We denote C ± i these as , where + means reflection even and Mo/e − means reflection odd. In what follows we will use Latin which anticommutes with the PXP Hamiltonian, H = letters for the dimension of the vector spaces labelled by H . Each eigenstate ψ with energy E = 0 thereforeC the corresponding script letter. −hasC a partner ψ with| energyi E. The6 graph has a bipartite structureC| i with vertex subsets− that are even and In each of the I-sectors there is a lower bound on the odd in the number of excitations, which are measured number of zero energy states given by the difference be- by . It is well known that the difference in dimensions tween the dimensions of the subspaces of even and odd of theseC two subspaces lower bounds the number of zero numbers of excitations. Putting this together, energy states.86,87 However, applying this idea directly, + + − − L e o + gives us the difference between sectors with an even and Z ≥ |Me ⊕ K | − |Mo ⊕ K | |Me | − |Mo | = M + K M K + M M odd number of excitations to be at most one, which is | e e − o − o| | e − o| not a useful lower bound. Missing from this analysis is Ke Ko , (A8) ≥ | − | consideration of the inversion symmetry I, where we have used the triangle inequality. All that re- I : j L j + 1, (A5) mains is to calculate the vector space dimensions K and 7→ − e Ko. which commutes with and hence in its symmetry sec- C Before deriving general expressions for Ke and Ko, we tors the bipartite structure is preserved. The combined present a simple example to illustrate the above. For a action of these two symmetries will be shown to provide chain of size L = 4 with OBC, the Hilbert space contains a tight bound for the number of zero energy states. We 8 states in total, four of which are even in the number of note that the exponentially-large number of zero-energy excitations, states is an interesting feature of the PXP model because energy E = 0 corresponds to the middle of the many- , , , , (A9) body spectrum. By contrast, in 2D models endowed with •◦•◦ •◦◦• ◦•◦• ◦◦◦◦ supersymmetry, exponentially many zero-energy states and four odd ones, can occur in the ground state manifold.88 Curiously, the zero-mode degeneracy is robust for , , , . (A10) even L to perturbation by the staggered magnetic field •◦◦◦ ◦•◦◦ ◦◦•◦ ◦◦◦• P j j( 1) Zj, yielding for open boundary conditions Two of these ( , ) are invariant under I, while the − rest can be organized•◦◦• ◦◦◦◦ into two-dimensional orbits. Thus, 2n = Fn+1, 2n+1 = 0. (A6) Z Z + n o e = + , For periodic boundaries, the staggered field explicitly M •◦•◦ ◦•◦• n o breaks translation symmetry to a subgroup, thereby + ZL/2 o = + , + , combining the zero- and π-momentum sectors M •◦◦◦ ◦◦◦• ◦•◦◦ ◦◦•◦ n o = , , (0,π) e = F − + F − . (A7) K •◦◦• ◦◦◦◦ Z2n n 1 n 2 no o = , From the analysis below, it follows that zero modes are K − n o generally present if the Hamiltonian anticommutes with e = , the product of particle-hole symmetry and inversion I. M •◦•◦ − ◦•◦• C − n o Staggered field is a special case of this as it commutes o = , . with particle-hole symmetry and anticommutes with M •◦◦◦ − ◦◦◦• ◦•◦◦ − ◦◦•◦ C inversion I. Plugging into Eq. (A8), we find 3 2 + 2 1 = ZL=4 ≥ | − | | − | 2, which indeed agrees with the exact result L=4 = 2. Next, we consider the case of general L. ForZ any con- 1. Open chain figuration An−1 on the open chain of length n 1, there is a corresponding invariant element on the length− L = 2n Our Hilbert space = C[V ], where V is the vertex chain (with = +1) given by set, decomposes intoH subspaces containing states with C T even and odd numbers of excitations. Those are mea- An−1 An−1 e, (A11) sured by , and will be denoted by subscripts e and o. ◦◦ ∈ K C T Each of these subspaces further decomposes into orbits where An is the spatially reversed pattern of An. Ev- under the action of the inversion operator I. The or- ery element of e is of this form because the central two bits of I are either one- or two-dimensional. Denote the sites cannot containK excitations as they would then be 18 adjacent. These configurations are in one-to-one corre- which is tight, i.e., matches the exact number of zero- spondence and therefore Ke = Fn+1 and Ko = 0, giving energy states in the presence of staggered field. 2n = Fn+1 for open chains with even length. Interestingly, for odd L the situation is completely dif- Z Similarly, for L = 2n + 1 odd, there is again a one- ferent. Now the S-blocks go along the diagonal, which to-one correspondence between invariant configurations removes the bipartite structure. Thus, we do not expect of fixed excitation parity and configurations of smaller any zero-energy states in this case, which is indeed con- open chains. In particular, the invariant configurations firmed by exact calculation. can be constructed for the two sectors as follows Beyond this example we can see that the bipartite block-structure between ↑ and ↓ is undisturbed by = +1 : A AT , (A12) H H C n◦ n ∈ Ke the addition of terms to the Hamiltonian which anticom- T = 1 : A − A . (A13) mute with I and do not violate the adjacency constraint. C − n 1◦•◦ n−1 ∈ Ko C This is because ↑ and ↓ are the subspace of I = +1 This reveals that Ke = Fn+2 and Ko = Fn+1 for odd and 1 respectivelyH sinceH measures the numberC parity length open chains, which altogether gives = F . − C Z2n+1 n of excitations.

2. Open chain with alternating field 3. Periodic chain

In this subsection we generalize the lower bound on For a periodic chain, in addition to and I, we also the zero-energy degeneracy to the case of open chains in need to consider the cyclic translation generatorC σ, the presence of a staggered field S P ( 1)jZ . First, ≡ j − j notice that σ : j (j + 1) mod L. (A17) ( 7→ IS, if L even, Every translation orbit of a periodic chain of length L SI = − (A14) +IS, if L odd, that is left invariant by I contains at least one element invariant under I or Iσ, i.e., under either a site or a bond i.e., [S, I] = 0 if L is odd and S, I = 0 if L is even, inversion. { } where I is the inversion symmetry in Eq. (A5). First, take L = 2n + 1 odd, and consider the orbits Assume L even and let X be our unperturbed Hamilto- which are excitation odd; the inversion invariant orbits nian. We can partition the Hilbert space in the following must contain at least one element of the form manner, + − + − • o e e e o A◦ ◦ T M M M K M +  A  XXS  o M− ↓ ◦◦ (A18) SX e H X + S  M . (A15) ' XS  + )  Me This diagram depicts a configuration wrapped around a X e ↑ K − H ring. As before, A denotes an arbitrary pattern (which SX o connects to and on its two ends), while AT is the M ◦◦ ◦•◦ In this block diagram, an X or S denotes a non-zero block spatially reversed pattern of A. Suppose then that this of X+S according to the partitioning of the Hilbert space configuration is non-unique with in terms of orbits of and I that was discussed in the C previous section. Note that o does not appear because ◦•◦ T j ◦•◦ T it’s trivial for L even. K A A = σ B B (A19) The block structure arises from symmetry considera- ◦◦ ◦◦ tions. Since X anticommutes with and commutes with C for some j and B = A. Take these two mirror planes I, its non-zero matrix elements couple sectors which have and generate the full6 set of mirror planes. The invariant different excitation parities and the same inversion pari- element then takes the form ties. Opposite to this, S commutes with and anticom- C mutes with I. Accordingly, its non-zero matrix elements Y Y T couple sectors with the same excitation parities but dif- ◦•◦ ◦ ◦ ferent inversion parities. Y◦T ◦Y The blocks in Eq. (A15) have been judiciously arranged ◦ ◦ • T• to reveal a bipartite structure between subspaces ↑ and ◦Y Y ◦ H ◦◦ (A20) ↓. This occurs because X + S anticommutes with I. H C From this bipartite structure we get a bound on the zero- from which it follows that both A and B must have energy degeneracy, the form Y Y T Y , thus they are equal. This demonstrates◦ ◦◦ that the◦•◦ inversion-invariant element in each H↑ H↓ = M + K + M M M Z2n ≥ | − | | e e o − o − e| inversion-invariant translation orbit is unique. This one- = Ke = Fn+1, (A16) to-one correspondence with the allowed configurations of 19

(0) open chains of length n 2 provides Ko = Fn. If the 0.3 orbit is instead excitation− even, the diagram is instead

◦ L A AT 0.2 √ k ◦◦ (A21) j δv (0) k 0.1 and the same reasoning can be applied to find Ke = L = 24 L = 50 Fn+1. L = 32 L = 80 Now take L = 2n even and consider the excitation-odd L = 40 inversion-invariant translation orbits; these must contain 0.0 an inversion-invariant element of the form 0.0 0.2 0.4 (j 3)/(L 6) ◦•◦ − − A AT . (A22) √Figure 11. Normalized errors kδvj k rescaled by a factor of ◦ L for PBC in in the parity symmetric sector. With this The previous reasoning can again be applied to find rescaling, the errors for different L are well collapsed, sug- (0) 2 2 3 Ko = Fn. gesting kδvj k = O(j /L ). The final case is that of L even and excitation-even orbits. An elementary method like the previous is more involved for this case because the invariant elements are Appendix B: Errors in forward scattering no longer unique. We instead observe that our lower approximation bound for is K 2Ko where K = Ke + Ko is the number ofZ invariant− elements irrespective of excitation Here we present a more detailed analysis of the er- parity. The number K can be found as K = 2(M + ror made in individual steps of the forward-scattering K) (2M + K), where M = Me + Mo is the number of approximation. The vector δwj introduced in Eq. (18) two-element− orbits irrespective of excitation parity. Note corresponds to the error made in a single FSA iteration. 2M +K is the number of translation orbits and M +K is The squared-norm of the error vector can be brought to the number of orbits of the combined dihedral symmetry the following form of translation and inversion symmetry. These are integer 2 2 2 δw = v − [H ,H−] v − + β β . (B1) sequences of system size and can be found in the OEIS k jk h j 1| + | j 1i j − j−1 as A000358 and A129526 respectively.89,90 The ordinary generating functions of these sequences are known to be From here it is clear that this error is governed by the commutator [H+,H−] between forward and backward X φ(k) 1 propagation terms in the Hamiltonian. Explicit evalu- 2M + K = [xN ] ln (A23) k 1 xk(1 + xk) ation of this commutator gives: k≥1 −  N 1 X φ(k) 1 X j M + K = [x ] ln (A24) [H+,H−] = ( ) Pj−1ZjPj+1 2 k 1 xk(1 + xk) − − k≥1 − j 1 (1 + x)(1 + x2) L X j . = Dˆ ( ) P − P P , (B2) −2 x4 + x2 1 Z2 − 2 − − j 1 j j+1 − j where φ(k) is the Euler totient function, i.e., the num- ˆ P P ber of positive integers up to k that are relatively prime where the operator DZ2 = j∈ odd Pj + j∈ even Qj is to k. Taking the appropriate linear combination of the diagonal in the basis of product states with eigenvalues generating functions giving the Hamming distance of a given product state from Z2 state. The final term in Eq. (B2) measures (1 + x)(1 + x2) | i 2(M + K) (2M + K) = [xN ] , the imbalance of “forward-holes” and “backward-holes”. − − x4 + x2 1 Here we define forward-hole as a pattern centered on − ◦◦◦ (A25) a site with odd j, where H+ could introduce an excita- we recognize on the right hand side the generating func- tion in the middle. Using the fact that for j = 1, 2 no tion of the Fibonacci sequence, hence forward/backward holes exist in the system, and using explicit values of β1,2 from Eq. (15) one can check that 2 2(M + K) (2M + K) = F . (A26) Eq. (B1) indeed gives δwj = 0. − bN/2c+2 k k Fig. 11 shows the normalized errors δvj , obtained (0) k k From here we arrive at the desired result Ke = Fn+1, numerically from the definition in Eq. (18), where we which completes the derivation of the zero energy de- have introduced the error vector δvj, defined as βjδvj = generacy for even L in the zero momentum sector of a δwj. This represents the relative error in the Lanczos periodic chain. vector v . Moreover, in Fig. 11 we have rescaled δv j k jk 20

duced here: 1.0 R = u u j | j  j | R = 2 2 0.8 |Z Z | βjvj = H+vj−1. (C1) Similarly, we must have 0.6

F βjvj−1 = H−vj. (C2) 0.4 Calculating the off-diagonal elements of the Hamiltonian 0.2 matrix directly from these formulas is, however, ineffi- cient due to an exponential growth of the dimension of 0.0 the connected subspace. For this reason, in this Ap- 0 10 20 30 pendix we develop a more efficient method. First, in t Eq. (C5), it will be shown that β coefficients are deter- mined by the number of loops reaching a given distance Figure 12. Red curve shows the time dependence of the from the initial state. Recursive expressions for these probability to be found within the forward-scattering sub- loop countings will then be derived for OBC, yielding as space, measured by Eq. (B3). Blue curve shows the fidelity main results Eqs. (C9), (C10) and (C11). Analogous ex- for the N´eelinitial state. Data is for system size = 32. L pressions for PBC are given in Eqs. (C17), (C18) and (C20). These results allow for an efficient and high- precision implementation of the FSA approximation in by a factor of √L. Fig. 11 shows that the error for fixed large systems on the order of L 100 sites. j decreases with L, which is promising for applications . of the FSA method to larger systems. However, a more complete error analysis requires an analytical description 1. Off-diagonal matrix elements from loop counting of how the errors in the individual steps compound to produce final errors in the physical quantities of interest. This is a more challenging question that requires further By repeated application of Eqs. (C1)-(C2) we obtain investigation. 2 β = v − H−H v − In Fig. 12 we assess the quality of the forward- j h j 1| +| j 1i j j scattering approximation for the dynamics. The red v (H−) (H ) v = h 0| + | 0i curve tracks the probability that the system remains Qj−1 β2 within the forward-scattering subspace over time, start- k=1 k j j ing out in the N´eelstate. This is quantified by calculating v0 (H−) (H+) v0 = h | j−1 j−| 1 i . (C3) the generalization of quantum fidelity defined as v (H−) (H ) v h 0| + | 0i iHt −iHt = Z2 e Re Z2 , (B3) From this, we recognize the amplitude F h | | i P j j where the operator R = j uj uj projects onto the W = v (H−) (H ) v (C4) | ih | N,j h 0| + | 0i forward-scattering subspace spanned by vectors uj . In the same figure, we also show the actual fidelity| (returni as the number of shortest closed paths reaching a dis- probability) for the N´eelstate, since Eq. (B3) reduces to tance j from the initial state. Subscript N indicates the −iHt 2 = Z2 e Z2 , when R = Z2 Z2 . The fact dependence of Wj’s on the system size, which we denote thatF the| h generalized| | i fidelity | changes| overih time| indicates by N in this section. In terms of W ’s, the off-diagonal that the weight of the wave function contained within matrix elements are the FSA subspace is not invariant under unitary evolu- s tion generated by H. Equivalently, this implies that the WN,j βj = . (C5) operator R is only an approximate integral of motion. WN,j−1 The gap between the revival probability and the sub- space probability shows the effect of dephasing within Our goal here is to derive a linear recurrence system for the forward-scattering subspace. Thus, we observe that calculating WN,j and our strategy will be to count loops the leakage of the many-body wave function outside the recursively in terms of loops on smaller subsystems. We FSA subspace is the main cause of fidelity decay in a will first present results for OBC and then generalize to quantum quench. PBC.

Appendix C: Linear recurrence method 2. Loop counting for open chains

The forward-scattering approximation in the main text Consider a loop of valid spin flips on an open system of was defined by the recurrence in Eq. (14), which is repro- N sites. This loop can be projected into two subsystems 21 where each flip is assigned to the subsystem in which the These equations can be made explicit by introducing spin in flips is located. We will choose the two subsystems the counting sequences for the classes. Let fN,j be the to comprise of the two leftmost spins and the remaining number of shortest loops on the open chain of N sites N 2 sites of the system. These subsystem loops are valid reaching a Hamming distance of j from the N´eelstate − a,b loops on the corresponding open systems of 2 and N 2 where the leftmost spin is invariant, and let l count − N,j sites. The original loop is one of the ways in which the those loops where the leftmost spin is flipped. These are spin flips of the subsystem loops can be interlaced such the counting sequences for and , respectively. The that the constraints are never violated. Given a pair previous equations (C7)-(C8F) then becomeL of loops on the two subsystems, we only need to know when the left-most spin of the right subsystem and the X a0,b0 fN,j = fN−2,j + lN−2,j, (C9) right-most spin of the left subsystem are flipped, if at all, a0,b0 in order to count ways in which they can be interlaced. 0 0 a,b X a,a0 a ,b b,b0 Because the loops discussed are properly shortest loops, lN,j = fN,j−1 + T lN−2,j−2T , (C10) these boundary spins are either flipped once moving away a0,b0 from the N´eelstate and once again on the return journey, 0 0 or not at all. where we have introduced T a,a = min(a, a + 1). The most general shortest loop looks like a word Eq. (C9) captures the idea that we may glue two addi- tional sites and the loop of doing nothing onto any loop c d on the reduced system. Eq. (C10) captures the idea that z }| { z }| { A AL A AR A A A A LA ARA A the loop that flips only the leftmost spin may be glued ··· | ··· {z ··· } | | ···{z } ··· ··· a b to any loop on the reduced system, but when the interior | {z } | {z } j j site is excited in the left-subsystem then the leftmost site right-subsystem must first have its excitation removed. (C6) Finally, the class of all loops in + can be counted by F L where the symbol A represents any spin flip (and each instance is different) of a bulk spin, L represents flipping WN,j = fN+2,j, (C11) the leftmost spin and R the rightmost spin. The verti- cal line separates the forward and backward steps in the because for every loop on N sites we may glue the trivial loop. The order in which the L and R flips appear, if at loop on two sites onto its left boundary to get a distinct all, in the forward and backward half-words is not fixed, loop on N + 2 sites, and for every loop on N + 2 sites we Eq. (C6) represents only one possible ordering. may cut off the leftmost two sites to get a distinct loop We will start by considering only the case of open on N sites. boundaries. Let be the combinatorial class of forward- Let us illustrate how the recurrence works on an exam- scattering loops onF a system with open boundaries where ple with N = 4 site open chain. Directly from Eq. (C4), the left-most spin remains fixed throughout the process. it is easy to show that the number of loops for different This class is graded by the size of the system N and the j sectors is given by W4,1 = 2, W4,2 = 5, W4,3 = 13, number of forward and backward transitions j. Similarly, and W4,4 = 25. Using the recurrence, we can obtain let be the class where the left-most spin is flipped at these values starting from a smaller N = 2 site chain. In L that case, the admissible j are given by 0, 1 and 2, and some point of the process and is additionally graded by 0,0 1,1 a and b. The index a specifies the number of forward the only non-zero coefficients are f2,0 = l2,1 = l2,2 = 0. steps which follow the flip of the leftmost spin, b is the Then, applying Eqs. (C11), (C9), and (C10), we have number of backward steps preceding the return flip of the X a,b leftmost spin. These classes are defined recursively from W4,1 = f6,1 = f4,1 + l4,1, the following equations, a,b=0 X a0,b0 f4,1 = l2,1 = 1, = ( + ) (C7) a0,b0 F •◦ ∗ F L •◦ a,b l4,1 = f4,0 = 1, (C12) •◦ •◦ ◦◦ = ◦◦ ( + ) + ◦• (C8) thus we get W4,1 = 1+1 = 2. Analogous calculation gives L ◦◦ ∗ F L ◦• ∗ L •◦ ◦◦ W4,2 = 5. For W4,3 we need to include the T tensors. We •◦ have where the operation glues a two-site system onto the 2 ∗ X left, and for each pair of elements in the classes appearing W = f = f + la,b, on the left and right hand sides produces all the interlac- 4,3 6,3 4,3 4,3 a,b=0 ings that satisfy the Fibonacci constraint. The vertical direction in the loop diagrams are successive steps in the f4,3 = 0, a,b X a,a0 b,b0 a0,b0 a,0 b,0 loop and the horizontal line separates the forward steps l4,3 = f4,2 + T T l2,1 = 1 + T T . (C13) from the backward steps. a0,b0 22

3. Loop counting on periodic chains direct efficient For periodic boundaries we must keep track of what 5 happens at the subsystem right boundary, to retain the information required to ensure the constraints are not vi-

β olated when the left and right sides are glued together. 4 We reinterpret and with the additional requirement that for the loopsF in thesesL classes the rightmost site is at no point excited. This gives us two new classes, and R 3 , which are the analogs of and , except that the rightmostM site is now excitedF at someL point of the pro- 5 10 15 cess. These classes are graded by indices c and d which j mark when the the right boundary site is excited and unexcited. The new classes satisfy analogous equations Figure 13. A demonstration of exact agreement between to the previous classes direct computation of β coefficients from Eq. (C1) and com- putation from the efficient recurrence, Eq. (C11). Data is for = ( + ) (C15) N = 16 site chain with open boundary conditions. R •◦ ∗ R M •◦ •◦ •◦ ◦◦ = ◦◦ ( + ) + ◦• . (C16) M ∗ R M ◦• ∗ M ◦◦ ◦◦ •◦ •◦ Therefore, we obtain This is because the gluing process never changes whether the rightmost site is excited or left invariant. Keeping track of how all these indices are changed as loops are interlaced during gluing results in the following 2 set of equations: X W4,3 = (1 + min(a, 1) min(b, 1)) = 9+4 = 13, (C14) c,d c,d X a0,b0,c,d a,b=0 rN,j = rN−2,j + mN−2,j , (C17) a0,b0 a,b,c,d c−δ(c≥a), d−δ(d≥b) mN,j = rN,j−1 X c,c0,a,k a0,b0,c0,d0 d,d0,b,m + T mN−2,j−2T , (C18) as anticipated. Repeating this procedure and using a0,b0,c0,d0 Eq. (C5), we can obtain the set of N + 1 coefficients βj that form the tridiagonal matrix in the forward- where scattering approximation. min(a0,a−1) a,c,a0,c0 X T = δ δ δ 0 . (C19) The β calculated by the linear recurrence system, a=6 c k=6 c c , c−δ(c≥a)−δ(c≥k) k=0 Eqs. (C9) and (C10), are compared against direct com- putation with Eq. (C1) in Fig. 13. The two methods Finally, the total number of loops is found by demanding indeed agree to machine precision for the given system compatibility between the left and right boundaries size, but the recurrence method provides much higher accuracy in larger system sizes. Computation time for X a,b X a,b,c,d WN,j = fN,j + lN,j + mN,j . (C20) calculating βN,N/2 for an open chain of N sites is found a,b a>c, b>d to scale roughly as N 5. ∝

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