Hidden $Z_{2}\times Z_{2}$ subspace symmetry protection for quantum scars

Abstract

We study the paradigmatic spin-1 XY chain under open boundary conditions, which hosts exact quantum many-body scars generated by an emergent Spectrum Generating Algebra (SGA). We show that the scar subspace possesses a symmetry-protected trivial (SPt) character that we attribute to a hidden $Z_{2}\times Z_{2}$ symmetry of another model, namely the commutant Hamiltonian, for which the scars are the ground states. We construct a Lieb-Schultz-Mattis (LSM) type twist operator, which, for scar states, takes the value $-1,$ and, for ergodic states, approaches zero in the thermodynamic limit. A complementary understanding of the stability of the scars under different perturbations is obtained by analyzing the Loschmidt echo and Quantum Fisher Information (QFI) of the scars. Finite-size scaling analysis of the QFI reveals that the scars are much more sensitive to perturbations as compared to the nearby thermal states. Based on the analysis of QFI and different LSM twist operators, we obtain a classification of different SGA-preserving and SGA-breaking perturbations.

Figures (19)

• Figure 1: Diagonal ETH for the extensive local observable $\hat{O}=\sum_{i} (S^{z}_{i})^{2}$ and bipartite entanglement entropy spectrum for the half-chain. The outlier data point ($\times$) near the center of the energy spectrum in each of the sub-figures indicates the single scar state in the sector. Note that with the addition of long-range terms, the diagonal ETH, and the entanglement entropy values of scars are unchanged.
• Figure 2: The plot of diagonal ETH and entropy for a fixed perturbation strength $\epsilon=0.05$ of onsite disorder. The putative scar (the point that is isolated) is found to merge with the ergodic spectrum as system size is increased, which shows this is not a true scar state despite its outlier character at smaller system sizes.
• Figure 3: The expectation value of the twist operator $U(0)$ defined in eq. \ref{['eq:twist_op']} for the middle one-third of the spectrum, with system size $L=12$, and magnetization $m=0$ sector. The first subfigure is with no perturbation, and the second figure is when the perturbation is $\epsilon V = 0.01S^{z}_{L/2}$, and the isolated point from the cluster is the scar state with no perturbation, but now it is not as it does not lie at $-1$.
• Figure 4: The expectation value of the twist operator $U(0)$ defined in eq. \ref{['eq:twist_op']} for the middle one-third of the spectrum, with system size $L=10$. The top figure is when there is no perturbation; hence, scars are exact in this regime, while the bottom figure is when the perturbation is $\epsilon V = 0.1\sum_{i}S^{x}_{i}$, which breaks the SGA, hence the scars are not exact indicated by the departure from the value of $-1$.
• Figure 5: Twist operator $\langle U(\theta)\rangle$ for whole spectrum for systems size $L=8$, and perturbation $\epsilon=0.05$. The outermost points forming a deformed circle are the putative scars that one can think of as scars numerically, see Fig. \ref{['fig:disorder_plot']}, but for larger system sizes, it mixes with the ETH prediction. Also, the leftmost points close to the unit circle are the states that were scars close to the spectral edges, and since the spectral edges do not have a high density of states, one cannot detect the deviation from the unit circle confidently.