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

Eloi Nicolau1,*, Marko Ljubotina2,3,1, and Maksym Serbyn1

  • *Contact author: eloi.nicolau@ista.ac.at
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PRX Quantum 7, 010352 – Published 13 March, 2026

DOI: https://doi.org/10.1103/sl79-1xgb

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 nonergodic 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.

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Popular Summary

Kinetically constrained models are a standard way to capture glassy dynamics: the system may relax slowly because simple local rules forbid many moves. Quantum versions of these models are now within reach in simulators, but it remains unclear which features are unique to their quantum nature, i.e., those that rely on quantum interference and not just on constrained dynamics. A special ingredient, present in many constrained models, is families of exactly degenerate eigenstates at zero energy, denoted as zero modes, but a unified understanding of these subspaces and their structure is missing.

We study a class of constrained quantum models where a degenerate zero-mode subspace is related to chiral symmetry. We identify a specific mechanism of how the interplay of chirality and constraints leads to a parametric enhancement of the number of zero modes. Within this enlarged subspace, we also build interacting many-body versions of compact localized states—states that stay confined to a small Fock space region because interference cancels every route for them to spread. We demonstrate these collective bound states in both one- and two-dimensional settings, and connect them to quantum fragmentation, where the system’s dynamics splits into many disconnected sectors that are not apparent from dynamical rules.

Our work invites further study of collective bound states, their quench responses, and their stability to deformations of the studied models. Moreover, it remains to be understood if these collective bound states can be related to ergodicity breaking, transport, and other properties of kinetically constrained models.

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PRX Quantum 7, 010352– Published 13 March, 2026
  • Received 11 June 2025
  • Revised 29 October 2025
  • Accepted 30 January 2026
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