Hilbert Space Fragmentation and Commutant Algebras

Abstract

We study the phenomenon of Hilbert space fragmentation in isolated Hamiltonian and Floquet quantum systems using the language of commutant algebras, the algebra of all operators that commute with each term of the Hamiltonian or each gate of the circuit. We provide a precise definition of Hilbert space fragmentation in this formalism as the case where the dimension of the commutant algebra grows exponentially with the system size. Fragmentation can hence be distinguished from systems with conventional symmetries such as $U(1)$ or $SU(2)$, where the dimension of the commutant algebra grows polynomially with the system size. Further, the commutant algebra language also helps distinguish between "classical" and "quantum" Hilbert space fragmentation, where the former refers to fragmentation in the product state basis. We explicitly construct the commutant algebra in several systems exhibiting classical fragmentation, including the $t-J_z$ model and the spin-1 dipole-conserving model, and we illustrate the connection to previously-studied "Statistically Localized Integrals of Motion" (SLIOMs). We also revisit the Temperley-Lieb spin chains, including the spin-1 biquadratic chain widely studied in the literature, and show that they exhibit quantum Hilbert space fragmentation. Finally, we study the contribution of the full commutant algebra to the Mazur bounds in various cases. In fragmented systems, we use expressions for the commutant to analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that agree with previous numerical results. In addition, we are able to rigorously show the localization of the on-site spin operator in the spin-1 dipole-conserving model.

23+15 pages, 4 figures, v2: Added references, minor clarifications, table of contents

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• Hilbert space fragmentation in a 2D quantum spin system with subsystem symmetries
• Emergence of Hilbert Space Fragmentation in Ising Models with a Weak Transverse Field
• Unified theory of local quantum many-body dynamics: Eigenoperator thermalization theorems
• Fractonic Luttinger Liquids and Supersolids in a Constrained Bose-Hubbard Model
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• Pauli topological subsystem codes from Abelian anyon theories
• Dipole condensates in tilted Bose-Hubbard chains
• Hilbert space shattering and dynamical freezing in the quantum Ising model
• Exact solution for the filling-induced thermalization transition in a 1D fracton system
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• Exhaustive Characterization of Quantum Many-Body Scars using Commutant Algebras
• Ergodicity breaking provably robust to arbitrary perturbations
• Holographic Quantum Scars
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• Height-conserving quantum dimer models
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• Measurement catastrophe and ballistic spread of charge density with vanishing current
• Ergodicity Breaking Under Confinement in Cold-Atom Quantum Simulators
• Multipole groups and fracton phenomena on arbitrary crystalline lattices
• The frustration-free fully packed loop model
• Local Hilbert space fragmentation and weak thermalization in Bose-Hubbard diamond necklaces
• Exact weak bosonic zero modes in a spin/fermion chain
• Coherence generation, symmetry algebras and Hilbert space fragmentation
• Circuit complexity and functionality: a thermodynamic perspective
• Real space thermalization of locally driven quantum magnets
• Quantum local random networks and the statistical robustness of quantum scars
• Linear response and exact hydrodynamic projections in Lindblad equations with decoupled Bogoliubov hierarchies