Probing quantum scars and weak ergodicity-breaking through quantum complexity

TL;DR

This work investigates scar states and weak ergodicity breaking in the PXP model through Krylov state (spread) complexity, computed via the Lanczos (forward scattering) formalism. It shows that the Neel state exhibits an approximate SU(2) structure leading to oscillatory, revival-like complexity, whereas generic ETH-obeying states show monotonic growth; this behavior is quantified by mapping the PXP dynamics to a $q$-deformed SU(2) algebra and by analyzing a first-order perturbation that restores more exact SU(2) structure. The authors provide analytic expressions for Lanczos coefficients within the SU$_q$(2) framework and demonstrate that the PXP model lies in a broken SU$_q$(2) regime, with perturbations recapturing strong revivals and more SU(2)-like dynamics. The findings establish a concrete link between algebraic symmetry structure and the dynamical complexity of scar states, offering a diagnostic tool for weak ergodicity breaking with potential extensions to other $\mathbf{Z}$-symmetric states and deformations.

Abstract

Scar states are special many-body eigenstates that weakly violate the eigenstate thermalization hypothesis (ETH). Using the explicit formalism of the Lanczos algorithm, usually known as the forward scattering approximation in this context, we compute the Krylov state (spread) complexity of typical states generated by the time evolution of the PXP Hamiltonian, hosting such states. We show that the complexity for the Neel state revives in an approximate sense, while complexity for the generic ETH-obeying state always increases. This can be attributed to the approximate SU(2) structure of the corresponding generators of the Hamiltonian. We quantify such ''closeness'' by the q-deformed SU(2) algebra and provide an analytic expression of Lanczos coefficients for the Neel state within the approximate Krylov subspace. We intuitively explain the results in terms of a tight-binding model. We further consider a deformation of the PXP Hamiltonian and compute the corresponding Lanczos coefficients and the complexity. We find that complexity for the Neel state shows nearly perfect revival while the same does not hold for a generic ETH-obeying state.

Figures (10)

• Figure 1: Growth of $b_{n}$'s [from Eq.\ref{['cheb1']}] for $j = 1$ for various values of $q$. Although the $n$ takes a discrete value, we use a continuum $n$ for the plotting. This visualization will be useful for understanding the behavior of the Lanczos coefficients in later sections.
• Figure 2: (a) Growth of $b_{n}$'s and (b) evolution of complexity $C(t)$ for the paramagnetic Hamiltonian $H_p = \sum_{n=1}^N \sigma^x_{n}$, initialized in the $\ket{\textbf{Z}_2}$ state for a system of lattice size $N=16$. The expression for the Lanczos coefficients and the complexity is given by Eq.\ref{['bnparan']} and Eq.\ref{['comparan']}, respectively. The Lanczos coefficients exactly terminate at $n = N + 1 =17$, which implies the dimension of the Krylov subspace is $K = 17$.
• Figure 3: Growth of $b_{n}$'s (disks) for the $\ket{\mathbf{Z}_{2}}$ state versus the $q$-deformed SU(2) (thick line) result for the PXP Hamiltonian \ref{['pxp']}. The standard SU(2) result is given for comparison (dashed line). The $b_{n}$'s for a generic state ($\ket{0}$ state; without any $\mathbf{Z}$ symmetry) is also plotted (circles). Here we choose the system size $N = 16$ and $K \sim 20$ Krylov basis vectors. Around $K \sim N+1$, the state is driven out of the approximate Krylov subspace. This is in contrast with the paramagnetic Hamiltonian in Fig. \ref{['para0']}, where the Krylov subspace was exact and shown by the dashed line in this figure.
• Figure 4: $q$ values versus $1/N$ for the PXP model. Possibly due to finite-size effects, the $q$ values are different for different system sizes $N$. The asymptotic value turns out to be $q_\infty = 0.9947 \pm 0.0044$, which is close to the SU(2) value ($q = 1$). The system sizes considered in this are $N = 12, 14, \cdots, 30$. The linear regression fit has an $R^2$ value of $0.992$ and a standard error of $0.0040$.
• Figure 5: Plot of $b_{n}$'s for the $\ket{\mathbf{Z}_{2}}$ state, after adding the perturbation \ref{['pxpc']} to the PXP model for different lattice sizes. The dots indicate the numerical results while the line indicates the expression \ref{['bnlam']} ($\alpha \approx 0.7025$). Both are in excellent agreement for all system sizes considered.