Fragmentation, Zero Modes, and Collective Bound States in Constrained Models

Abstract

Kinetically constrained models were originally introduced to capture slow relaxation in glassy systems, where dynamics are hindered by local constraints instead of energy barriers. Their quantum counterparts have recently drawn attention for exhibiting highly degenerate eigenstates at zero energy -- known as zero modes -- stemming from chiral symmetry. Yet, the structure and implications of these zero modes remain poorly understood. In this work, we focus on the properties of the zero mode subspace in quantum kinetically constrained models with a $U(1)$ particle-conservation symmetry. We use the $U(1)$ East, which lacks inversion symmetry, and the inversion-symmetric $U(1)$ East-West models to illustrate our two main results. First, we observe that the simultaneous presence of constraints and chiral symmetry generally leads to a parametric increase in the number of zero modes due to the fragmentation of the many-body Hilbert space into disconnected sectors. Second, we generalize the concept of compact localized states from single particle physics and introduce the notion of collective bound states, a special kind of non-ergodic eigenstates that are robust to enlarging the system size. We formulate sufficient criteria for their existence, arguing that the degenerate zero mode subspace plays a central role, and demonstrate bound states in both example models and in a two-dimensional model, the $U(1)$ North-East, and in the pair-flip model, a system without particle conservation. Our results motivate a systematic study of bound states and their relation to ergodicity breaking, transport, and other properties of quantum kinetically constrained models.

Figures (12)

• Figure 1: Illustration of the kinetic constraints of the $U(1)$ East and East-West models. Allowed hoppings for a kinetic constraint (a) of arbitrary range in the $U(1)$ East model and (b) of range $r=2$ in both models.
• Figure 2: Adjacency graphs for the $U(1)$ East model with $N=4$ particles on $L=10$ sites before [Eq. \ref{['EqHamiltonianEast']} without the constraint operator $\hat{\mathcal{K}}_{i,r}$] and after fragmentation. The mismatch $\mathcal{M}$ between the number of blue vertices (with an even number of particles in the odd sublattice) and orange vertices (with an odd number) is indicated below. Fragmentation enhances the mismatch considerably, and thus it increases the lower bound on the number of ZM. The increase is determined by those sectors where the dominant set of vertices is the opposite of the one before fragmentation.
• Figure 3: Number of ZMs in the $U(1)$ East and East-West models for $r=2$ and $L=3N-2$ as function of the number of particles $N$, and the analytical lower bound given by the mismatch $\mathcal{M}$, Eq. \ref{['EqMismatch']}, and the asymptotic expansion of Eq. \ref{['EqMismatchFit']}. Due to chiral symmetry, the number of ZMs increases exponentially, while the fragmentation in the $U(1)$ East model yields a faster exponential growth given by $\mathcal{M}_{\rm frag}$, which tightly bounds the number of ZMs from below.
• Figure 4: Illustration of the procedure to construct a bound state within the ZM subspace of the $U(1)$ East model restricted to the largest classical fragmentation sector. (i) Consider a set of ZM $\{|\psi\rangle\}$ on a graph $G_n^\ell$ of a lattice with $\ell$ sites and $n$ particles. (ii) Rotate the subspace to construct a ZM localized on the set of vertices $V_1$ (blue vertices), $|\psi_{BS}\rangle$. (iii) Add one site to the right boundary of the lattice, which yields the graph $G_n^{\ell+1}$. Note that $G_n^{\ell+1}$ contains $G_n^\ell$ as an induced subgraph and also an extra set of vertices of the form $|\cdots\bullet\rangle$ (red vertices). As the red vertices are only connected to the vertices in the set $V_2$, the state $|\psi_{BS}\rangle\otimes|\circ\rangle$ remains an eigenstate of the model, and thus, $|\psi_{BS}\rangle$ is a right bound eigenstate. For the $U(1)$ East model, adding one site to the left also yields the same graph, $G_n^{\ell+1}=G_n^{\ell+2}$, as no particle can reach that site, and thus, $|\psi_{BS}\rangle$ is not only right bound but is a full bound state.
• Figure 5: Ratio of factorizable ZMs to the total number of ZMs as a function of the number of particles $N$ for the largest sector of the $U(1)$ East model and $r=2,3$. The proportion of factorizable ZMs increases with $N$, reaching a high saturation value for $r=2$ and exhibiting a modulated behavior for $r=3$. As the ZM subspace grows exponentially with $N$, any non-decreasing ratio $N_{FS}/N_{ZM}$ suggests that the number of factorizable ZMs also increases exponentially with $N$.