Quantum scars as embeddings of weakly broken Lie algebra representations

We present an interpretation of scar states and quantum revivals as weakly “broken” representations of Lie algebras spanned by a subset of eigenstates of a many-body quantum system. We show that the PXP model, describing strongly interacting Rydberg atoms, supports a “loose” embedding of multiple $\mathrm{su}\left(2\right)$ Lie algebras corresponding to distinct families of scarred eigenstates. Moreover, we demonstrate that these embeddings can be made progressively more accurate via an iterative process which results in optimal perturbations that stabilize revivals from arbitrary charge density wave product states, $|{\mathbb{Z}}_n\rangle$, including ones that show no revivals in the unperturbed PXP model. We discuss the relation between the loose embeddings of Lie algebras present in the PXP model and recent exact constructions of scarred states in related models.

Figure 1

An example of an allowed (a) and forbidden (b) transition under the Hamiltonian in Eq. (1).

Figure 2

Summary of various mechanisms for embedding scarred eigenstates in a many-body system. (a) An exactly embedded Krylov subspace (purple tridiagonal matrix, with red lines symbolizing the nonzero elements). Such a scenario can emerge in models exhibiting the phenomenology of fractonic systems [40], where if the Krylov subspace is exponentially large this effect is coined “Krylov-restricted thermalization” [56]. Lifting the restriction the embedded subspace be tridiagonal, models of type (a) can also be generically realized by the “projector embedding method” (Sec. 3a). (b) Exact scars featuring perfect revivals due to a dynamical symmetry of certain terms in the Hamiltonian generated by $Q^{+}$ (see Sec. 3b). Type (b) scars have being realized in a variety of spin models such as spin-1 XY model [51, 52, 53, 54]. (c) PXP-like scarring [49, 50], where a Krylov subspace which approximately acts as an $\mathrm{su}\left(2\right)$ representation is sparsely coupled to the thermal bulk, such that the subspace has a low subspace variance (which is equivalent to the Frobenius norm of the block labeled couplings). By fixing various broken Lie algebra representations in models of type (c) we can also realize the scarred subspace of approximate type (a), where the nearly exactly embedded subspace forms a representation of the Lie algebra (as will be discussed in Secs. 6b and 7a).

Figure 3

Schematic illustration of our iterative scheme which identifies corrections to broken Lie algebras, specifically an $\mathrm{su}\left(2\right)$ Lie algebra in this case. The optimization of ${\lambda }_n$ is with respect to the error measures described in the text, such as maximizing the first fidelity peak $|\langle H^z,LW|e^{-iHt}|H^z,LW{\rangle |}^2$ or minimizing the subspace variance of $H$ with respect to the $\mathrm{su}\left(2\right)$ basis defined in Eq. (23).

Figure 4

${\mathbb{Z}}_2$ revival in PXP model. (a) Eigenstate overlap with the Néel $|{\mathbb{Z}}_2\rangle$ state. (b) Eigenstate overlap after including the first order $\mathrm{su}\left(2\right)$ correction [Eq. (34)]. (c) Eigenstate overlap after including the second order $\mathrm{su}\left(2\right)$ correction [Eqs. (37, 38, 39, 40, 41, 42, 43, 44)]. (d) Quantum fidelity in ${\mathbb{Z}}_2$ quench, with and without perturbations. Perturbation coefficients are those that maximize the first fidelity revival peak. (e) Bipartite entropy, Eq. (36), of the eigenstates of PXP model after including second order ${\mathbb{Z}}_2\mathrm{su}\left(2\right)$ corrections. The states labeled “Exact Scars” are exact diagonalization results identified from the top band of states in (c). Red crosses in (a)–(c), and (e) indicate approximate scar states obtained by projecting the Hamiltonian to the broken $\mathrm{su}\left(2\right)$ basis and diagonalizing. Color scale in (a)–(c), and (e) indicates the density of data points, with lighter regions being more dense.

Figure 5

Improving the ${\mathbb{Z}}_3$ revival in the PXP model. (a) Eigenstate overlap with $|{\mathbb{Z}}_3\rangle$ state for PXP model. (b) Eigenstate overlap after including first order correction in Eq. (52, 53, 54). (c) Eigenstate overlap after including second order perturbations listed in Appendix pp2. (d) Quantum fidelity when the system is quenched from $|{\mathbb{Z}}_3\rangle$ state at various perturbation orders. The perturbation coefficients are those which maximize the first fidelity revival peak. (e) Bipartite entropy [Eq. (36)] of eigenstates of the PXP model after including second order ${\mathbb{Z}}_3\mathrm{su}\left(2\right)$ corrections. Points labeled “Exact Scars” are exact diagonalization results identified from the top band of states in (c). Red crosses in (a)–(c), and (e) indicate approximations to the scar states obtained by projecting the Hamiltonian to the broken representation basis and diagonalizing. Color scale in (a)–(c), and (e) indicates the density of data points, with lighter regions being more dense.

Figure 6

${\mathbb{Z}}_4$ revival in PXP model. (a) Eigenstate overlap with $|{\mathbb{Z}}_4\rangle$ state for PXP model. (b) Eigenstate overlap with $|{\mathbb{Z}}_4\rangle$ state after including first order $\mathrm{su}\left(2\right)$ corrections, Eqs. (64), (65). (c) Eigenstate overlap after including second order $\mathrm{su}\left(2\right)$ corrections (Appendix pp3). (d) ${\mathbb{Z}}_4$ quench fidelity. $|{\mathbb{Z}}_4\rangle$ state does not revive in pure PXP model, but it does revive in the new model obtained by correcting the $\mathrm{su}\left(2\right)$ algebra. (e) Bipartite entropy [Eq. (36)] of eigenstates of the PXP model after including second order ${\mathbb{Z}}_4\mathrm{su}\left(2\right)$ corrections. Points labeled “Exact Scars” are exact diagonalization results identified from the top band of states in (c). Red crosses in (a)–(c), and (e) indicate approximate scar states obtained by projecting the Hamiltonian to the broken representation basis and diagonalizing. Color scale in (a)–(c), and (e) indicates the density of data points, with lighter regions being more dense.

Figure 7

Local autocorrelation function $\langle {\sigma }_{2i}^z\left(t\right){\sigma }_{2i}^z\left(0\right)\rangle$ of the model given by Eq. (70), for various initial states given in the legend. Results are for $N=20$. We consider sites $2i$ as the translation symmetry of Eq. (70) is broken to a subgroup corresponding to translations by two units. Generic initial states such as the polarized state $|000\cdots \rangle$ equilibrate, whereas the autocorrelation function exhibits non stationary behavior for all times when the system is initialized in the $|{\mathbb{Z}}_4\rangle =|10001000\cdots \rangle$ state.

Figure 8

$|{\mathbb{Z}}_2\rangle$ revival in PXP spin-1 model. (a) Eigenstate overlap with $|{\mathbb{Z}}_2\rangle$ for pure PXP model. (b) Eigenstate overlap including the $PPXP+PXPP$ perturbation inspired by spin-$\frac{1}{2}$ PXP model. (c) Eigenstate overlap including first order $\mathrm{su}\left(2\right)$ correction, $PP{X}_{01}P+P{X}_{01}PP$. (d) $|{\mathbb{Z}}_2\rangle$ quench fidelity with the various perturbations. (e). Bipartite entropy of PXP spin-1 after including the first order ${\mathbb{Z}}_2\mathrm{su}\left(2\right)$ correction $PP{X}_{01}P+P{X}_{01}PP$. Points labeled “Exact Scars” are exact diagonalization results identified from the top band of states in (c). Red crosses in (a)–(c), and (e) indicate approximate scar states obtained by projecting the Hamiltonian to the broken representation basis and diagonalizing. Color scale in (a)–(c), and (e) indicates the density of data points, with lighter regions being more dense.