Quantum and classical Hilbert space fragmentation

Thesis

Hilbert space fragmentation has provided fruitful grounds for the study of ergodicity violations and unusual symmetry structures in a variety of quantum systems. In this thesis, I use analytic and numerical approaches to examine these features in two contrasting Hilbert-space-fragmented models. One of these models exhibits "quantum fragmentation" (where the fragmentation can only be described in an entangled basis), the other displays "classical fragmentation" (where the fragmentation can be described in a product-state basis).

The quantum-fragmented system is an SU(M)-symmetric disordered bipartite spin model which, for M≥3, has a large non-trivial nullspace whose dimension grows exponentially with system size. This exponential growth leads to a rare example of Hilbert space fragmentation in a system with long-range interactions. I characterise the nullspace and the resulting fragmentation in detail, and show that the symmetry algebra responsible for the large degeneracy is a non-trivial subalgebra of the Read-Saleur commutant algebra. I also discuss perturbations of the model, including one that transforms certain states in the nullspace into quantum many-body scars.

The classically fragmented system is a family of quantum chains with local range-k interactions subject to the conservation of a total charge and its dipole moment. Such models exhibit a continuous "freezing" phase transition between weakly fragmented (ergodic) and strongly fragmented (non-ergodic) phases as the charge density ν is varied. I use a variety of innovative approaches to analyse these models, and show that the transition occurs at a critical charge density of ν꜀=(k-2)^(-1) independently of the onsite Hilbert space dimension. I also obtain numerous results characterising the properties of the different phases and their impact on the models' dynamical evolution. Together, the work in this thesis presents new insights into areas of fundamental importance to the exploration of Hilbert space fragmentation and ergodicity breaking.

Authors: Classen-Howes, J

• DOI: 10.5287/ora-e9z8medpv
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: quantum phases of matter; Hilbert space fragmentation; symmetry structures; eigenstate thermalisation hypothesis
• Subjects: Condensed matter theory