Aspects of quantum states of matter - Slayen, Ruach (Department of Mathematics and Applied Mathematics, 2025)
In this thesis we explore two aspects of the spectra of low-dimensional quantum systems with potential relevance for modern condensed matter and holography. We begin with a study of two-dimensional systems in magnetic fields whose spectra exhibit Landau level structure. We then study disordered quantum field theories whose spectra exhibit the correlations char- acteristic of quantum chaotic/integrable systems. In Part I, we review established results concerning the eigenstates and spectra of spin-0 and spin-1/2 quantum fields confined to two-dimensional planes and spheres, in homoge-neous magnetic field configurations. We then study a novel variation of Haldane's spherical monopole system called the spherical dipole system. We review and expand on the results for the single-particle Hilbert space and spectra for the spin-0 case, then extend these to the spin-1/2 case. The latter is relevant for the study of experimentally realisable systems such as C60 fullerine. We find that in the strong-field limit, the spectrum exhibits a Landau level structure, which is explained by the tendency of a strong dipole field to localise the particles at the poles of the sphere. The spin-1/2 system features a new (approximately) zero-energy lowest Landau level for certain values of the angular momentum quantum number relative to the dipole strength. In Part II, we give an overview of random matrix theory and its relation to the study of quantum chaos via spectral statistics, with a focus on the spectral form factor (SFF) as a diagnostic of chaos. After reviewing a 0 + 1 dimensional, disordered quantum field theory called the Sachdev-Ye-Kitaev (SYK) model and its chaos properties, we study a novel variation of the model: the gauged complex SYK 2 model. This model describes N complex fermions with a disordered quadratic interaction term (the SYK2 model) coupled to a one-dimensional external gauge field, where the introduction of the external gauge field is equivalent to a twisting of the boundary conditions of the fermions. We probe the large N chaos properties of this model from the perspective of the SFF. We find that the gauge field does not affect the integrability of the original SYK2 model, but nonetheless gives rise to notable effects on the slope-dip-ramp structure of the SFF. Namely, by tuning the gauge field, one may control both the decay of the early time slope as well as the explicit timescale needed for the appearance of zero modes. These zero modes are responsible for an exponential ramp of the SFF, which is conjectured to be a feature of all non-interacting, disordered systems. While the timescale governing their appearance takes a fixed finite value in the ungauged model, in our model it may be made arbitrarily small.