Quantum many-body scarring from Kramers-Wannier duality

TL;DR

The paper addresses how dualities, specifically Kramers-Wannier (KW) duality, influence the stability of quantum many-body scars (QMBS) in a nonintegrable spin chain. It uses stochastic matrix form (SMF) Hamiltonians and sequential quantum circuits (SQC) to realize and track KW duality across symmetry and twist sectors, revealing that some scars remain nonthermal while others map to thermal-like states under duality. First-order perturbation theory aligns well with exact diagonalization for finite systems, demonstrating a quantitative handle on fidelity loss and the robustness of certain scars, and highlighting the role of twist sectors in governing ergodicity breaking. Collectively, the work casts duality as a general diagnostic and constructive tool for weak ergodicity breaking and paves the way for discovering new QMBS in more complex settings and near-term quantum devices.

Abstract

Kramers-Wannier duality, a hallmark of the Ising model, has recently gained renewed interest through its reinterpretation as a non-invertible symmetry with a state-level action. Using sequential quantum circuits (SQC), we argue that this duality governs the stability of quantum many-body scar (QMBS) states in a nonintegrable model, depending on whether the dual preserves the embedding conditions for scarring. This is supported by good agreement between first-order perturbation theory and numerics, which capture scar dynamics despite chaotic spectra. Our results establish non-invertible dualities as both a generative mechanism and a diagnostic tool for quantum many- body scarring, offering a generalized symmetry-based route to weak ergodicity breaking.

Figures (7)

• Figure 1: (a) Entanglement entropy versus energy obtained from an exact diagonalization procedure for a chain of size $L=16$ for the $H_{\text{ND}}$ Hamiltonian. The scar state correspond to the isolated state at zero energy at low entropy in comparison to the rest of the spectrum. (b) Level statistics of paramagnetic Hamiltonian with $\alpha=0.3$ and $\beta=0.5$. Obtained from exact diagonalization for a chain of size $L=20$ sites. The plot is obtained by using the middle $\sim70\%$ of the spectrum in each momentum sector excluding $k=0,\,\pi$.
• Figure 2: (a) Entanglement entropy versus energy obtained from an exact diagonalization procedure for a chain of size $L=16$ for the $H_\text{D}$ Hamiltonian. The two degenerate scar states sit at exactly zero energy and zero entropy, due to the product state form of the wavefunction. (b) Level statistics of nondegenerate Hamiltonian with $\alpha=0.3$ and $\beta=0.5$. Obtained from exact diagonalization for a chain of size $L=20$ sites. The plot is obtained by using the middle $\sim70\%$ of the spectrum in each momentum sector excluding $k=0,\,\pi$.
• Figure 3: Profile of entanglement entropy versus energy for (a) the twisted Hamiltonian $H_{\text{ND}}$ and (b) the twisted Hamiltonian $H_{\text{D}}$. Both plots are obtained from exact diagonalization of the corresponding Hamiltonians for a chain of size $L=16$. It is clear from the plot that there are no signs of low-entangled states at the $E=0$ line. Giving further evidence that the twisted sectors does not contain QMBS states at zero energy, for which the antisymmetric $\ket{\mathcal{S}_-}$ can be dual to.
• Figure 4: Matrix elements distributions of the corresponding perturbations $H_{\text{ND}}$ or $H_\text{D}$. The data was obtained from exact diagonalization for a chain of size $L=16$. The distribution for thermal states are represented by inverted triangles, meanwhile, the ones for scar states are represented by circles. We show the distributions for (a) the anti-symmetric scar $\ket{\mathcal{S}_-}$ in comparison to a thermal state. (b) the symmetric scar $\ket{\mathcal{S}_+}$ in comparison to a thermal state and (c) the symmetric scar $\ket{\bar{\mathcal{S}}}$ in comparison to a thermal state. Overall, the symmetric ones have a clear distinct behavior, with a suppression near $\Delta E\approx 0$, meanwhile the anti-symmetric scar have a thermal-like distribution.
• Figure 5: Distribution of the matrix elements for both (a) $\ket{\mathcal{S}_+}$ and (b) $\ket{\bar{\mathcal{S}}}$ QMBS states in an energy window around the $\Delta E=0$ point, for sizes $L=12$ (cyan) and $L=16$. The suppression for the matrix elements involving the scar states, does not seem to decrease significantly with system size.