Hilbert Space Fragmentation in Hardcore Bose and Fermi Hubbard Models on Generalized Lieb Lattices

TL;DR

This work addresses how Hilbert space fragmentation (HSF) arises in hardcore Bose and fermionic Hubbard models on generalized Lieb lattices. Using the restricted spectrum generating algebra (RSGA) framework, it constructs exact zero-energy eigenstates via an $\mathrm{su}(2)$-like $\eta^{+}$ operator in the large interaction limit ($V$) and demonstrates an ensuing energy tower with potential off-diagonal long-range order. Through both analytical derivations and numerical simulations, it shows that HSF is exact in the $V\to\infty$ limit and weakly manifests at finite $V$ as quasi-energy towers acting as quantum scars; it further extends the framework to fermionic Hubbard variants, where doublon-based constraints yield similar towers in effective models. The results establish a direct link between interaction-induced kinetic constraints (HSF) and RSGA-generated energy towers, broadening the scope of QMBS and providing a path to engineer non-thermal states in generalized Lieb lattices.

Abstract

We study the Hilbert space fragmentation (HSF) in hardcore Bose and Fermi Hubbard models in the framework of the restricted spectrum generating algebra (RSGA). We present a family of hardcore Bose-Hubbard models with repulsive density-density interactions on a generalized Lieb lattice. We show that this system possesses the RSGA structure in the large interaction strength limit, exhibiting quantum HSF. It allows us to construct a set of exact condensate eigenstates, possessing off diagonal long-range order. Based on numerical simulations conducted on several representative lattices, we demonstrate the existence of weak fragmentations when the constraints are not exact. As applications, we also studied the connection between HSF and RSGA in modified fermionic Hubbard models, where the η-pairing states are shown to be energy towers, acting as quantum scars.

Figures (3)

• Figure 1: Schematic illustrations for the features of the studied system. (a) A generalized Lieb lattice, consisting of three sub-lattices A, B, and C, that are denoted by red, yellow, and green-filled circles, respectively. The black lines denote the connections between sublattice A and B. Each connection corresponds to a single site of the sub-lattice C. The two nearest-neighbor hopping strengths along a connection are real and identical. (b) The conditions for the RSGA on the configurations of the boson filling specify that the doubly occupied and nearest-neighbor pair states are forbidden. (c) and (d) are two examples which represent 1D and 2D Lieb lattices.
• Figure 2: Plots of the fidelity of dynamical evolution for Hamiltonians with different structures (see Eq. (\ref{['H']})) with the initial state given by Eq. (\ref{['psi0']}). The structures corresponding to Figs. (a), (b), and (c) are labeled above each figure. (a), we take $N_{a}=3$, $N_{b}=3$, and $N_{c}=5$. (b), we take $N_{a}=3$, $N_{b}=3$, and $N_{c}=7$. (c), we take $N_{a}=4$, $N_{b}=2$, and $N_{c}=5$. In all three figures, we fix the hopping term $\kappa=1$ and the chemical potential $\mu=10$, and vary the interaction strength $V$ to plot the results. The results show that different interaction strengths $V$ indeed have an impact on the periodicity of the dynamics.
• Figure 3: Plots of the fidelity of the time evolution driven by the Fermi Hubbard Hamiltonian, given by Eq. (\ref{['H_Hubbard']}). The chain system is illustrated at the top of the figure. The orange shading indicates the on-site repulsion, and the green shading indicates the doublon-doublon interaction. The initial state is $\left\vert \phi (0)\right\rangle =2^{-2}\sum_{m=0}^{4}\sqrt{C_{4}^{m}}\left\vert \psi ^{m}\right\rangle$, where $\left\vert \psi ^{m}\right\rangle$ is given by Eq. (\ref{['FHM_IS']}). The system parameters are $\mu =5$, $N=7$, and $\kappa =1$. Three representative values of $U$ are indicated in the figure.