Hilbert Space Fragmentation and Commutant Algebras

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 fragmentation, discuss the connection to previously-studied "Statistically Localized Integrals of Motion" (SLIOMs), and analytically obtain new or improved Mazur bounds for autocorrelation functions of local operators that explain previous numerical results. In addition, we show how Quantum Many-Body Scars, a related form of weak ergodicity breaking, can be captured within a similar framework of commutant algebras.