Constraints and unconventional symmetries in quantum many-body systems can lead to fascinating phases of matter. In this thesis, we study transport properties and low-energy physics of constrained quantum matter with particular focus on systems that conserve the dipole moment. We demonstrate the slow emergence of hydrodynamic transport in the presence of vastly different kinetic energy scales and find exotic ground state phases of bosonic lattice models with dipole conservation.
Constraints and unconventional symmetries in quantum many-body systems can lead to fascinating phases of matter. In this thesis, we study transport properties and low-energy physics of constrained quantum matter with particular focus on systems that conserve the dipole moment. We demonstrate the slow emergence of hydrodynamic transport in the presence of vastly different kinetic energy scales and find exotic ground state phases of bosonic lattice models with dipole conservation.
Quantum phenomena at macroscopic scales are fascinating, given that quantum mechanical effects usually manifest only at atomic scales due to their fragile nature. Enriched with symmetry, topology, and entanglement, quantum matters exhibit exotic emergent properties that defy our classical intuitions. This dissertation is devoted to the study of quantum matters, consisting of three distinct topics. The first part studies quantum phases of matter and their phase transitions, with an emphasis on the role of symmetry and their exotic natures. Employing both analytical and numerical methods, we study the deconfinement physics in quantum Hall systems and quantum magnets. The second part focuses on the physics of Moir\'e systems in two-dimensional materials that exhibit interesting single-particle as well as correlated many-body physics. Using a continuum model description, we illuminate several recent experimental findings in twisted double bilayer graphenes. The last part touches on a more unconventional topic, non-Hermitian systems. We provide mathematical foundations to classify their topological structure as well as the potential connection to anomalous or driven systems.
Phases of matter have always attracted great interest within the scientific community, as they represent emergent phenomena arising from the collective behavior of a large number of interacting degrees of freedom. Their macroscopic properties, often impossible to predict directly from the microscopic scale, can be effectively captured in terms of quasiparticles, \textit{i.e.} particle-like low-energy excitations that emerge from interactions among the underlying microscopic constituents and may exhibit properties entirely different from those of any fundamental particle, reflecting the rich structure of the many-body system. In this Thesis, we investigate two classes of quasiparticles that arise in distinct exotic phases of matter. The first class appears in antiferromagnetic quantum spin chains with discrete symmetries and frustrated boundary conditions, namely periodic boundary conditions with an odd number of sites. In this setting, the competition between global and local orders produces exotic gapless, non-relativistic excitations above the ground state, which can dramatically modify the physics even in the thermodynamic limit. In particular, we focus on how this specific choice of boundary conditions profoundly modifies the phase diagram of one of the most famous integrable models, the XY chain. The second class consists of subdimensional quasiparticles, excitations whose mobility is restricted to lower-dimensional subspaces and which characterize the low-energy sector of emerging fracton phases of matter. Quasiparticles with restricted mobility first appeared in exactly solvable quantum spin models, and it was later realized that they can also emerge from higher-rank gauge theories. In this part of the Thesis, we develop Lorentz-covariant gauge field theories for fractons in 2+1 dimensions, starting from the first principles of Quantum Field Theory (fields and symmetries), with the aim of embedding the features observed in the condensed matter literature into a more formal field theoretical framework.
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.
This doctoral thesis studies low dimensional quantum systems that can be realized in recent cold atom experiments. From the viewpoint of quantum statistical mechanics, the main emphasis is on the detailed study of the different quantum and thermal phases and their transitions using numerical methods, such as quantum Monte Carlo and the Tensor Network Renormalization Group. The first part of this work deals with a lattice Boson model subject to strong three-body losses. In a quantum-Zeno like manner, a dynamical three-body constraint arises that restricts the evolution of the quantum system to the subspace with at most two particles per site i.e., three-body losses will be suppressed. Its low-energy description by a Feshbach model leads to the appearance of a pair condensate phase with intriguing quantum critical properties controlled by the Coleman-Weinberg mechanism. A comprehensive numerical study, presented in Chapter 3, identifies the pair phase and provides evidence for an unconventional Berezinksii-Kosterlitz-Thouless transition due to the unbinding of half-vortices. The study of the pair phase with worm-like algorithms turns out to be challenging and an extension of the directed loop algorithm is presented that overcomes the sampling limitations posed by the appearance of fat-tail distributions. In the following chapters, attention is drawn to lattice systems of polar molecules in optical lattices. Their intriguing dipolar interaction and sensitivity towards external fields make them ideal candidates to realize new states of matter. Particularly, it has been shown that one can drive these systems into a regime where the interaction is dominated by competing two- and three-body potentials. First, the triangular lattice supersolid is considered in Chapter 4. An extensive numerical study combined with a detailed symmetry analysis answers the question of the nature of the supersolid quantum nucleation transition. Following this discussion, Chapter 5 investigates whether the characteristic features of this model persist in the presence of interactions for a realistic system with polar molecules. The three-body interaction regime in a honeycomb Bose-Hubbard model turns out to give rise to a set of highly complex phases in terms of valence bond solids and leads to frustration. Chapter 6 studies the phase diagram and shows how the competition between two- and three-body interactions exhibiting a cascade of phases in the regime of intermediate couplings. In addition, it is demonstrated how the Tensor Network Renormalization Group can be applied to systems with multibody interactions and generic algorithmic limitations are described. Finally, preliminary results concerning the effect of quantum disorder on the triangular supersolid and recent calculations for the entropy of one-dimensional trapped SU(N) Hubbard models are discussed in Chapter 7.
The physics that emerges when large numbers of particles interact can be complex and exotic. The collective behaviour may not reflect the underlying constituents, for example fermionic quasiparticles can emerge from models of interacting bosons. Due to this emergent complexity, manybody phenomena can be very challenging to study, but also very useful. A theoretical understanding of such systems is important for robust quantum information storage and processing. The emergent, macroscopic physics can be classified using the idea of a quantum phase. All models within a given phase exhibit similar low-energy emergent physics, which is distinct from that displayed by models in different phases. In this thesis, we utilise tensor networks to study many-body systems in a range of quantum phases. These include topologically ordered phases, gapless symmetry-protected phases, and symmetry-enriched topological phases.