Parent Hamiltonians for stabilizer quantum many-body scars

TL;DR

This work introduces a general framework to realize stabilizer states as quantum many-body scars in locally interacting Hamiltonians. By exploiting $(2,\ell,b)$-factorizability of Pauli strings, stabilizer elements are decomposed into local, few-body operators that annihilate the stabilizer state, enabling the construction of $\\ell$-local, $b$-body parent Hamiltonians with zero-energy QMBS. The approach unifies and extends known QMBS results, producing new examples including the $D=1$ cluster state, toric code states, antipodal toric code, and volume-law entangled QMBS such as rainbow/antipodal Bell-pair and cluster states, and even identifying stabilizer QMBS in the PXP model. Exact diagonalization confirms the stabilizer states as exact zero-energy eigenstates of their parents, with entanglement properties ranging from area-law to volume-law, and the framework accommodates various generalizations to higher dimensions, qutrits, multiple scars, and Floquet settings. Overall, the paper provides a scalable, stabilizer-based route to engineer physically plausible Hamiltonians hosting rich, nonthermal QMBS structures that can be leveraged for quantum information processing and studies of ergodicity breaking.

Abstract

Quantum many-body scars (QMBS) have attracted considerable interest due to their role in weak ergodicity breaking in many-body systems. We present a general construction that embeds stabilizer states as QMBS of local Hamiltonians. The method relies on a notion of factorizability of Pauli strings on a lattice, which is used to convert stabilizer elements into local, few-body operators that annihilate the stabilizer state. This enables the systematic construction of parent Hamiltonians with zero-energy stabilizer QMBS typically near the middle of the spectrum. The method reproduces several known results in a unified framework, including recent examples of volume-law entangled QMBS, such as the ``rainbow'' QMBS and the entangled antipodal Bell pair state. We also apply the framework to construct examples of stabilizer QMBS with a more complex entanglement structure, such as the cluster state, the toric code state, and a volume-law entangled state we dub the antipodal toric code (ATC) state. Exact diagonalization confirms our results and reveal the stabilizer states as exact eigenstates of their parent Hamiltonian.

Figures (9)

• Figure 1: (a) Inset: the generators of the cluster state on the $D=1$ chain. We compute the half-chain entanglement entropy (i.e., the von Neumann entropy for the subsystem highlighted in gray) of each eigenstate of the parent Hamiltonian in Eq. \ref{['eq:H_cluster']}. The cluster state (red marker) is a zero-energy QMBS (the color of each marker denotes the overlap of the corresponding eigenstate with the cluster state). The dashed black line shows the entanglement entropy $S^{\rm cluster} = 2 \ln 2$ of the cluster state. [Parameters: $N=14$, $\vec{\theta} = (-1,-1,\hdots,-1)$ and $\omega_n$, $\omega'_n$, $J_n$, $J'_n$ are, for each $n$, generated uniformly at random from the range $[0.7,1.3]$.] (b) Inset: the vertex (blue) and plaquette (red) operators on the square lattice. We compute the half-system entanglement entropy (i.e., the von Neumann entropy of the subsystem highlighted in gray) of each eigenstate of the parent Hamiltonian in Eq. \ref{['eq:H_toric_general']}. The toric code state (red marker) is a zero-energy QMBS. Its entanglement entropy $S^{\rm toric} = N_x \ln 2$ scales with the size of the boundary of the subsystem, rather than its volume. [Parameters: $N_x = N_y = 4$, $\vec{\theta} = (-1,-1,\hdots,-1)$. The Hamiltonian parameters are given in the caption to Fig. \ref{['fig:example_toric_level_spacing']} in the End Matter.]
• Figure 2: Left: the vertex and plaquette operators for the toric code on a $N_x \times 2$ square lattice, with its coordinates $(x,y)$ mapped onto a linear chain with coordinates $n = 0, 1, \hdots, 2N_x - 1$. Right: an equivalent representation of the operators the circle, showing that they are distributed nonlocally (antipodally). For clarity, only a few vertex/plaquette operators are shown. These vertex and plaquette operators are generators for the stabilizer group of the antipodal toric code (ATC) state.
• Figure 3: The level spacing statistics for the parent Hamiltonian of the cluster state, given in Eq. \ref{['eq:H_cluster']}. All parameters are the same as for Fig. \ref{['fig:examples_cluster_toric']}(a).
• Figure 4: The level spacing statistics for the parent Hamiltonian of the toric code state, given in Eq. \ref{['eq:H_toric_general']}, in its $\hat{\Pi}^X = \hat{\Pi}^Z = +1$ parity sector. Here, the square lattice has the size $N_x = N_y = 4$, the toric code state is chosen with $\theta_{x,y} = -1$ for all $x$ and $y$, and the coupling constants $J_{x,y}^{X/Y}$ and $\tilde{J}_{x,y}^{X/Y}$ are chosen uniformly at random from the range $[0.7,1.3]$. The same parameters are also used for Fig. \ref{['fig:examples_cluster_toric']}(b), except that it also shows data in all parity sectors $\hat{\Pi}^X = \pm 1$ and $\hat{\Pi}^Z = \pm 1$.
• Figure 5: Eigenstate entanglement entropy, and level spacing statistics for two different examples of parent Hamiltonians for the product stabilizer $|1\rangle^{\otimes N}$ ($N=14$). Left column: the Hamiltonian parameters $\omega_n^{\mu,\pm}$, $J_n$, $J'_n$, $\lambda_n^\mu$ are (for each $n$) chosen at random from the interval $[0.7,1.3]$, while $\eta_{n,n'}$, $\gamma_{n,n'}$ and $\xi_{n,n'}$ are chosen at random from the interval $[0.7/N, 1.3/N]$. For this choice of parameters, (a) the mid-spectrum eigenstates follow a volume-law scaling of entanglement entropy, and (b) the Hamiltonian is nonintegrable. However, the product stabilizer QMBS is a zero-entanglement outlier [red marker in (a)]. Right column: the Hamiltonian parameters $\omega_n^{\mu,\pm}$, $J_n$, $J'_n$, $\lambda_n^\mu$ are (for each $n$) chosen at random from the interval $[0.7,1.3]$, $\xi_{n,n'}$ is chosen from the interval $[0.7/N, 1.3/N]$, while each $\eta_{n,n'}$ and $\gamma_{n,n'}$ and are chosen from the interval $[-4, 4]$. This leads to a large amount of disorder in the magnetic field $h_n$. For this choice of parameters, (c) the there are many min-spectrum eigenstates that follow a sub-volume-law scaling of entanglement entropy, and (d) the Hamiltonian is integrable. Again, the product stabilizer is a zero-entanglement eigenstate [red marker in (c)].

Theorems & Definitions (1)

• Definition 1