Quantum Complexity in Constrained Many-Body Models: Scars, Fragmentation, and Chaos

Abstract

Kinetic constraints in quantum many-body systems give rise to quantum states, whose behavior strongly depends on the choice of initial conditions. In recent years, these systems have drawn increasing interest because they provide insight into the mechanisms of thermalization and the situations where it can fail. In this work, we study a family of kinetically constrained models, including the celebrated Quantum Game of Life, from the perspective of quantum complexity, with a focus on entanglement, nonstabilizerness, and signatures of quantum chaos. By applying spectral diagnostics such as level statistics and spectral form factors, we demonstrate that these models show robust chaotic behavior while also supporting Hilbert space fragmentation and quantum many-body scar states. Remarkably, we find that even certain symmetry-resolved fragmented sectors can themselves host scarred eigenstates, highlighting the unexpected coexistence of chaos, scars, and fragmentation within the same family of Hamiltonians. To better understand these fragmented subspaces, we further characterize them using their quantum resource generation ability. In particular, we demonstrate that characterization of entanglement and the ability to generate nonstabilizerness can be instrumental in distinguishing different dynamically disconnected sectors.

Figures (12)

• Figure 1: Graphical representation of Hilbert space fragmentation due to kinetic constraints in the Hamiltonian $H^{\mathcal{N}(1)}$ [Eq. (\ref{['eqn:main_Ham']})], in zero momentum inversion symmetric sector. The kinetic constraint allows the quantum state at site $i$ ($s_i$) to evolve only when the combined occupation of its surrounding neighbors ($i-2, i-1, i+1, i+2$) adds up to one, as illustrated on the left. On the right, different colors denote dynamically disconnected Krylov sectors for a system of size $L=10$. Each circle represents a basis state, and an edge indicates that the state can evolve into the connected state under the action of the Hamiltonian.
• Figure 2: Krylov subspaces and their scaling in fragmented model $H^{\mathcal{N}(1)}$. In (a) and (b), we illustrate how the number of Krylov subspaces, $\mathcal{N}_L$, and the ratio $d_L/\mathcal{D}_L$ behaves with system size $L$, respectively. The maximum system size considered is $L = 22$. Both figures clearly indicate the presence of strong Hilbert space fragmentation (HSF). In contrast, $H^{\mathcal{N}(2)}$ (not shown here) displays a slower decay of $d_L/\mathcal{D}_L$ with increasing $L$.
• Figure 3: (a) Behavior of entanglement entropy for the eigenstates of kinetically constrained quantum many-body models $H^{\mathcal{N}(1)}$ and $H^{\mathcal{N}(2)}$. In (a), we present the behavior of half-chain entanglement entropy ($\mathcal{S}_{L/2}$) across the part of the spectrum of $H^{\mathcal{N}(1)}$ belonging to the zero momentum and inversion symmetric sector for $L = 18$. In contrast, for $H^{\mathcal{N}(2)}$, we obtained a similar behavior in (b) for the subspace that along with the translation symmetry ($k=0$ subspace) and inversion symmetry, has an additional spin-flip symmetry. In (c), we present the same plot as shown in (a), but highlighting the entanglement behavior of four specific Krylov subspaces using different colors. The three largest subspaces are of dimensions 749, 628, and 627, while the 50th largest subspace has a dimension of only 21. Notably, for $H^{\mathcal{N}(1)}$, the subspace with the largest dimension does not correspond to the highest entanglement entropy; instead, the highest entanglement comes from the second and third largest subspaces. Conversely, for $H^{\mathcal{N}(2)}$, as shown in (d), the largest Hilbert space, with dimension $d_L = 1003$ (marked in blue), produces the highest entanglement entropy. The next two largest fragmented subspaces have dimensions of 965 and 711, marked in red and green, respectively.
• Figure 4: Behavior of entanglement entropy for the eigenstates of the models QGL and the perturbed PPXPP with $\delta = 0.09$. In (a) and (b) we plot the behavior of half-chain entanglement entropy ($\mathcal{S}_{L/2}$) for $H^{\text{QGL}}$ and $H^{\text{Pert}}_{\text{PPXPP}}$, respectively, for the eigenstates of the zero momentum and inversion symmetric sector for L=18.
• Figure 5: The level spacing distributions along with the values of level spacing ratios $\langle r \rangle$, for (a) $H^{\text{QGL}}$ (with $L = 20$), (b) $H^{\mathcal{N}(1)}$ (with $L = 24$) and (c) $H^{\mathcal{N}(2)}$ (with $L = 22$) are analyzed. If the total number of eigenvalues in a sector is $D_{e}$, the energy list $\lbrace e_i\rbrace$ is used, where $i$ ranges from $[D_{e}/10]$ to $[D_{e}/2 - 500]$. For both $H^{\text{QGL}}$ and $H^{\mathcal{N}(1)}$, the zero momentum inversion symmetric sector is considered (for $H^{\mathcal{N}(1)}$ we further consider the largest connected subspace). In contrast, for $H^{\mathcal{N}(2)}$, the analysis focuses on the largest connected sector comprised of both zero momentum inversion symmetry as well as the spin flip symmetry. In the bottom figures, we illustrate the behavior of the Spectral Form Factor (SFF) for the same models and system sizes, (d) $H^{\text{QGL}}$ (with $L = 20$), (e) $H^{\mathcal{N}(1)}$ (with $L = 24$) and (f) $H^{\mathcal{N}(2)}$ (with $L = 22$). The structure characterized by a slope, dip, ramp, and plateau is clearly visible in each plot. We mark the slope, ramp, and plateau using red, green, and blue colors, respectively. Note that, except in the computation of the level spacing ratio, the unfolded spectra of the various symmetry-resolved sectors are considered in all cases.