Distinct Types of Parent Hamiltonians for Quantum States: Insights from the $W$ State as a Quantum Many-Body Scar

TL;DR

This work introduces a threefold locality-based classification of parent Hamiltonians that share a given set of eigenstates, extending QMBS concepts beyond ground-state frustration-freeness. Focusing on the simple yet rich $W$ state, the authors derive the most general finite-range Hermitian-local Hamiltonians for which $|W\rangle$ is an eigenstate, and show a complete decomposition into $H=\Omega I+\omega \hat{N}_{\rm tot}+t H_{\rm ImHop}+\sum_{|X|\le R_{\max}}h_X$, with $H_{\rm ImHop}$ non-Hermitian and $h_X|W\rangle=0$, while $|\bar{0}\rangle$ is also an eigenstate. This leads to a unique Type II equivalence class represented by $H_{\rm ImHop}$, and a Type III class represented by $\hat{N}_{\rm tot}$, with a separate Type I subset formed by strictly local Hermitian annihilators. The paper then connects these structural results to dynamical signatures (e.g., droplet motion and boundary melting) and to asymptotic QMBS ($|W_q\rangle$, $|W^2\rangle$, $|W^p\rangle$), discusses a boundary-action theorem that ties Hamiltonian type to patch truncations, and extends the framework to short-range-entangled states and MPS symmetry generators. Collectively, the results provide a robust, locality-aware platform for classifying and understanding the dynamical and spectral properties of QMBS-rich Hamiltonians, with broader implications for MPS, symmetry, and potential experimental realizations.

Abstract

The construction of parent Hamiltonians that possess a given state as their ground state is a well-studied problem. In this work, we generalize this notion by considering simple quantum states and examining the local Hamiltonians that have these states as exact eigenstates. These states often correspond to Quantum Many-Body Scars (QMBS) of their respective parent Hamiltonians. Motivated by earlier works on Hamiltonians with QMBS, in this work we formalize the differences between three distinct types of parent Hamiltonians, which differ in their decompositions into strictly local terms with the same eigenstates. We illustrate this classification using the $W$ state as the primary example, for which we rigorously derive the complete set of local parent Hamiltonians, which also allows us to establish general results such as the existence of asymptotic QMBS, and distinct dynamical signatures associated with the different parent Hamiltonian types. Finally, we derive more general results on the parent Hamiltonian types that allow us to obtain some immediate results for simple quantum states such as product states, where only a single type exists, and for short-range-entangled states, for which we identify constraints on the admissible types. Altogether, our work opens the door to classifying the rich structures and dynamical properties of parent Hamiltonians that arise from the interplay between locality and QMBS.

Figures (6)

• Figure 1: Dispersion of a type I Hamiltonian $H_{\rm ReHop}$ [Eq. (\ref{['eq:HI']})] versus a type II Hamiltonian $H_{\rm ImHop}$ [Eq. (\ref{['eq:HII']})]. Here, we have diagonalized the respective Hamiltonians for the single-particle states $\ket{W_q}$. We see that the variational state $\ket{W_q}$ with $q\ll1$ has energy $O(q^2)$ for $H_{\rm ReHop}$, while it is linear for $H_{\rm ImHop}$, exemplifying a key difference between the two types of Hamiltonians. While the figure is an illustration for free-particle Hamiltonians, the qualitative difference between "dispersions" $q^2$ vs $q$ holds for trial energies of generic type I vs type II Hamiltonians, see text for details.
• Figure 2: Relative energy locations of exact (black) and asymptotic (green) QMBS for Hamiltonians with $\ket{W}$ as an exact eigenstate (i.e., QMBS). The energy separation between $\ket{\bar{0}}$ and $\ket{W}$ is $\omega$ defined in Eq. (\ref{['eq:HW']}), with asymptotic $\ket{W_q}$ QMBS (for $q \ll 1$) around $\omega$ (with lifetime $\gtrsim 1/q \sim N$) and $\ket{W^2}$ around $2\omega$ (with lifetime $\gtrsim \sqrt{N}$). We also prove that there exist asymptotic scars $\ket{W^p}$ at energies $p\omega$ for $p\in\{3,4,...\}$, where $\ket{W^p}$ is given by Eq. \ref{['eq:Whigher']}.
• Figure 3: Evolution of the $W$ droplet under three Hamiltonians: (a) $H_{\rm ReHop}$; (b) $H_{\rm ImHop}$; and (c) $H_{\rm CHop}$ (with $\alpha = \beta = 0.5$). The system is $[-100,\dots,100]$ ($N=201$) with PBC and the droplet starts on $[-25,\dots,25]$ ($M=51$). The colors indicate different values of $n_j(t) := |\!\braket{j}{\phi(t)}\!|^2$. Note that there is a clear directionality in the evolution in the ImHop and CHop cases, which indicates a type II component.
• Figure 4: Here we depict three possibilities when we apply a truncated Hamiltonian $H_\Lambda$ (on a contiguous region $\Lambda$) to a state $\ket{\psi}$, with $H\ket{\psi}=\epsilon\ket{\psi}$. (a) Upon truncation, the $H_\Lambda\ket{\psi}$ can be written as boundary terms actions $(A_\ell+B_r)\ket{\psi}$, where $A_\ell$ and $B_r$ are both Hermitian. (b) $H_\Lambda\ket{\psi}$ can also result in an action on a state with boundary terms that are non-Hermitian $A_\ell$ and $B_r$, with no Hermitian choices existing. We refer to these Hamiltonians as type II. (c) Type III Hamiltonians are all other cases such as when the truncation cannot be decomposed as any boundary term action. Note that although the depictions resemble a quantum circuit, the pieces are additive instead of multiplicative since we are dealing with a Hamiltonian, not a unitary.
• Figure 5: Evolution of the orbitals $\ket{\phi_j(t)}$ visualized using $n_j(t)$ under the Hamiltonians (a) $H_{\rm ReHop}$, (b) $H_{\rm ImHop}$, and (c) $H_{\rm CHop}$ (with $\alpha = \beta = 0.5$). Data shown for the initial droplet of size $M=51$ in the PBC chain of length $N = 201$ (same as in Fig. \ref{['fig:Wdropletnumerics']}), with uniform time steps from $t = 0$ (darkest) to $t = 200$ (lightest) in steps of $20$. Note the shift in the $x$ axes in the ImHop and CHop panels that compensates for the ballistic propagation of the droplet seen in Fig. \ref{['fig:Wdropletnumerics']}. The insets show the "leakage" of the particle number from the initial domain in the ReHop case and the appropriately shifted domains in the ImHop and CHop cases as a function of time [i.e., $\sum_{j,\, |j - G(t)| > M/2} n_j(t)$ with appropriate $G(t)=0$, $wt$, and $\beta wt$ for the three cases respectively]. The dashed guidelines in the insets of (a) and (c) denote $\sim \sqrt{t}$, and the dotted guideline in the inset of (b) denotes $\sim \sqrt[3]{t}$, which is consistent with the analysis in the text.

Theorems & Definitions (45)

• Theorem 1
• Definition 1
• Corollary 1
• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Corollary 2
• proof
• Corollary 3