Quantum chaos and phase transitions - Gupta, Nitin (Department of Mathematics and Applied Mathematics, 2025)
Quantum materials permeate the modern world - these are systems that exhibit quantum mechanical properties, like topological phases and superconductivity, at macroscopic scales. The principles governing quantum materials include entanglement and coherence -phenomena which are fundamentally quantum mechanical in nature, without classical counterparts. The semiconductors in mobile phones, computers, and solar cells; light emitting diodes (LEDs); sensors in medical equipment and other precision devices; Maglev trains and particle accelerators - all utilize quantum materials in one form or another. The trajectory of modernization over the past century has been significantly shaped by developments in quantum materials, underscoring the importance of studying their phases in detail. This thesis focuses on one aspect of the research program to improve the understanding of quantum materials: mathematical probing for the presence of quantum phases (QPs) of matter and the transitions (QPTs) among them. Specifically, it proposes that Krylov Complexity can be utilized to detect QPs and QPTs. Krylov Complexity is a quantity that has been recently proposed in the physics literature as a measure of chaotic nature of a quantum system i.e. it encodes the information transport properties of a system - exponential signatures in the Krylov Complexity typically characterize a chaotic quantum system. A priori, one may not expect Krylov Complexity to be sensitive to the presence of QPs and QPTs. This thesis gives evidence contrary to this expectation. The results demonstrate that Krylov Complexity exhibits distinctive signatures at the boundaries of QPs, such as sharp peaks or discontinuities, which correspond to the quantum critical points. Numerous techniques have been developed to study quantum materials both theoretically and experimentally: Tensor Network Methods, Renormalization Group Theory, Scanning Tunnel Microscopy, Hall Effect Measurement etc. This thesis highlights the simplicity and effectiveness of Krylov Complexity, which utilizes known information from studying Hamiltonians of many- body quantum systems with minimal additional computation. In summary, through a comprehensive review of the theoretical framework underpinning Krylov Complexity, this thesis provides compelling evidence that it is a simple yet effective tool for probing QPs and QPTs, thereby opening new avenues for understanding quantum materials, their phases, and the transitions among these phases.