Constrained models in aperiodic systems

Singh, Shobhna 2025. Constrained models in aperiodic systems. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

Constrained models play a pivotal role in condensed matter physics by capturing essential features of strongly interacting many-body systems. These models impose local or global restrictions that mirror real physical constraints, enabling the study of complex emergent behaviors. For instance, hard-core dimer models reflect physical systems with strong local constraints due to repulsive interactions. Similarly, loop models, vertex models, and height representations provide constrained frameworks that reveal rich phase structures and critical phenomena. In quantum systems, constraints often emerge naturally from microscopic Hamiltonian and lead to phenomena such as fractionalization and quantum spin liquids. Constrained models act as fertile grounds for theoretical insight, enabling exact solutions, mapping to field theories, and exploring universal behaviors. They also bridge condensed matter with disciplines like statistical mechanics, combinatorics, and computational complexity, offering a unified language to study order, disorder, and transitions in systems ranging from magnets to artificial spin ices and cold atomic gases. This thesis investigates critical phenomena, strongly correlated models, and optimization problems on aperiodic lattices, with a particular focus on NP-hard problems, dimer models, and their classical and quantum variants. Moving beyond periodic lattices like square and honeycomb structures, this research delves into the less understood area of aperiodic systems, such as quasicrystals, which exhibit long-range order without translational symmetry. These structures challenge traditional paradigms of condensed matter physics and open avenues for discovering new critical behaviors. I present original contributions that combine analytical methods and numerical algorithms to study constrained systems on quasicrystalline tilings. These include the solution of NP-hard optimization problems like the Hamiltonian cycle on Ammann-Beenker tilings, an energy-based worm algorithm for simulating classical dimers on random graphs and modified Penrose tilings, and the analytical solution of the dimer partition function on the newly discovered aperiodic Spectre tiling. I further extend the study to quantum systems by analytically investigating the quantum dimer model (QDM) on the Spectre tiling and numerically analyzing it on the Ammann-Beenker tiling, revealing distinctive ground-state properties and quantum phase transitions that arise due to aperiodicity and local tiling constraints. This work builds interdisciplinary bridges between condensed matter physics, graph theory, and computational complexity. It also sheds light on the feasibility and limitations of transferring universal results from periodic systems to aperiodic ones.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Physics and Astronomy
Subjects:	Q Science > QC Physics
Uncontrolled Keywords:	Aperiodic tiling, Quasicrystals, NP-hard problems, Dimer Models, Constrained Models, DMRG, Monotiling
Funders:	EPSRC
Date of First Compliant Deposit:	6 October 2025
Last Modified:	07 Oct 2025 15:22
URI:	https://orca.cardiff.ac.uk/id/eprint/181483

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