Hilbert space fragmentation in quantum Ising systems induced by side coupling

Abstract

We study Hilbert space fragmentation and quantum scars in quantum spin systems with Ising interactions. The system consists of two sets of quantum spins, A and B. As the parent system, set A is an Ising model on arbitrary lattices with a transverse field, while set B comprises free spins that are coupled to set A. We show that the Hilbert space is fragmented into exponentially many decoupled sectors when the transverse field and the side coupling strength are at resonance. As examples, several typical systems with quantum scars are studied analytically. Numerical simulations of probability distribution of entanglement entropy for finite-size chains, square and triangular lattices are performed using the Monte Carlo method. The results show that Hilbert space fragmentation and the corresponding quantum scars become pronounced when the system approaches resonance.

Figures (3)

• Figure 1: Schematic illustration of the main idea of this work. (a) We consider an Ising model with a transverse field on an arbitrary lattice. When the local transverse field at a site vanishes, the spin state at this site is pinned. When a set of such pinned spins (red dots) forms the boundary of a region (green dots), the kinetic constraint for the domain walls results in Hilbert-space fragmentation. (b) The structure of a system with side couplings. It consists of two sets of quantum spins, an upper set and a lower set. The red shading represents the Ising interaction within the upper set of spins, where each spin also couples to its counterpart in the lower set.
• Figure 2: The entanglement entropy $S$ for the eigenstates of the Hamiltonian in Eq. (\ref{['H']}) for three geometries of lattice A shown in the insets [(a) an open chain, (b) a square lattice, and (c) a triangular lattice] and the corresponding probability density distribution $P(S)$. $S$ denotes the entanglement entropy between the odd sites and even sites of an eigenstate with eigenenergy $\varepsilon$. The parameters $\kappa _{j}$ in Eq. (\ref{['H']}) and $g_{j}$ in Eq. (\ref{['H0']}) are uniform, and $\kappa _{j}=3, J=1$ are fixed. (a1), (a2), and (a3) correspond to $g_{j}=1$, $g_{j}=3$, and $g_{j}=5$, respectively. (b) and (c) have the same parameter settings as (a). The size of the system is taken as $N=12$, and the numerical simulations are performed by utilizing the Monte Carlo method. Specifically, we randomly select 1000 eigenstates of lattice B in the basis of $\sigma ^{x}$, with each constructing an invariant subspace of the complete Hamiltonian. These results indicate that the proportion of zero entropy significantly increases for the resonant case where $g=\kappa$ and that the distribution of entropy is affected by the geometry of the lattice.
• Figure 3: Panels (a1) and (b1) display the dynamic fidelity defined in Eq. (\ref{['fidelity']}) for two structures of lattice A with periodic boundary conditions, illustrated in the insets of (a) and (b). Panels (a2) and (b2) present the corresponding bipartite entanglement entropy between odd sites and even sites. The Hamiltonian is given by Eq. (\ref{['HA']}), with lattice B prepared in the state specified by Eq. (\ref{['psiB']}). Other parameters are set to $J=1$, $\kappa=3$, and the total number of sites is 12. For the resonant case where $g=\kappa$, the fidelity demonstrates perfect periodic oscillations while the entanglement remains at zero---both hallmark signatures of quantum many-body scars.