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.
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.
This Thesis explores the notion of Krylov complexity as a probe of quantum chaos and as a candidate for holographic complexity. The first Part is devoted to presenting the fundamental notions required to conduct research in this area. Namely, an extensive introduction to the Lanczos algorithm, its properties and associated algebraic structures, as well as technical details related to its practical implementation, is given. Subsequently, an overview of the seminal references and the main debates regarding Krylov complexity and its relation to chaos and holography is provided. The text throughout this first Part combines review material with original analyses which either intend to contextualize, compare and criticize results in the literature, or are the fruit of the investigations leading to the publications on which this Thesis is based. These research projects are the subject of the second Part of the manuscript. In them, methods for the efficient implementation of the Lanczos algorithm in finite many-body systems were developed, allowing to compute numerically the Krylov complexity of models like SYK or the XXZ spin chain up to time scales exponentially large in system size. It was observed that the operator Krylov complexity profile in SYK, a paradigmatic low-dimensional chaotic system with a holographic dual, agrees with holographic expectations, while in the case of integrable models like XXZ complexity is affected by a novel localization effect in the so-called Krylov space which hinders its growth. Finally, an exact, analytical, correspondence between the Krylov complexity of the infinite-temperature thermofield double state in the low-energy regime of the double-scaled SYK model and bulk length in the theory of JT gravity is established.
At the heart of physics lies the human fascination with natural phenomena and the wish to understand the underlying principles. A well-known example is the butterfly effect. It describes how small changes in the initial state can cause significantly different measurement outcomes at later times. Such sensitivity to initial conditions is a key signature of chaotic systems and reflects their instability under small perturbations. The principle underlying this phenomenon is information spreading: with time, information about the perturbation propagates through the system, changing measurement results even at distant locations. In quantum systems, information not only spreads but can also scramble, meaning that the precise details about the perturbation are hidden in highly nonlocal correlations. Information scrambling has emerged as a universal feature of quantum chaos and, together with other universal features, serves as an organizing principle for quantum systems. This motivates imposing structure like locality on quantum systems in order to probe the capabilities and limits of such organizing principles. There have been significant advances in both theoretical and experimental directions in recent years. The newly gained flexibility in designing experiments and simulations also inspires the analytical study of quantum systems from the bottom up, where tractable models offer insights into the generic features of quantum mechanics. In this thesis we build upon these advances to explore the implications of adding structure to chaotic quantum systems for the behavior of such features. In particular, we consider the measures entanglement entropy and Krylov complexity as probes of structure. These quantities are closely connected to information spreading. This motivates us to study the influence of structure on information spreading in the second part. In the first part, we begin with a detailed discussion of the behavior of entanglement in chaotic systems. We impose a minimal amount of local structure by separating the system into two parts, each modeled as a single spin. We validate the predicted entanglement for typical states in such systems and show that it saturates the Page limit as the Hilbert space dimension increases. We then study how different initially unentangled product states approach the Page limit under time evolution. We consider three types of product states, each representing a distinct case. Random product states represent generic states only restricted by the absence of entanglement. Random dipole and quadrupolar states represent two distinct types of semi-classical structure. The results on the entanglement generation in the three different types of states refine our earlier conclusion, demonstrating that the semi-classical limit can suppress the generation of extensive entanglement. This suppression does not occur for random product or quadrupolar states; where the imposed structure is insufficient. We conclude the second chapter by presenting two protocols that can stabilize the dynamics at finite spin sizes, similar to the semi-classical limit. In the following chapter, we move from measuring the complexity of time-evolved states using entanglement entropy to investigating time-evolved operators through the lens of Krylov complexity. After a short introduction to Krylov complexity for Hamiltonian dynamics, we generalize the framework for discrete time evolution. We identify the linear growth of Krylov complexity as a universal feature of generic systems and analyze its behavior as we transition from chaotic to increasingly structured systems. Dual-unitary circuits converge toward this universal behavior at rates that can be as fast as exponential, consistent with their classification as maximally chaotic. We then approximately restore energy conservation by considering trotterized systems, ranging from unstructured and chaotic to highly structured and integrable ones. We find these systems to thermalize on a scale set by the trotterization step, slowly approaching the universal linear Krylov complexity growth regime. Approaching the linear growth regime is a fundamental property of trotterized systems with an unbounded spectrum, while systems with a bounded energy spectrum can evade thermalization as probed by Krylov complexity. Instead, they have a nonanalytic transition as the Trotter step size increases. Our study of Krylov complexity establishes its universal properties for discrete time evolution. The presented results deepen our understanding of the thermalizing behavior of different classes of systems in relation to their spectral properties by introducing an intuitive notion of a bath with chaotic characteristics. We link this thermalization behavior to information spreading, which we explore in the second part of this thesis. Information scrambling, along with the associated spreading, is a key phenomenon in chaotic systems with local structure. In the fourth chapter, we investigate information spreading in proximity to an integrable system exhibiting spontaneous symmetry breaking, the transverse field Ising model (TFIM). As expected, integrability prevents the scrambling of information. At the same time, information spreading probed by quenched out-of-time ordered correlators (OTOCs) exhibits features reminiscent of the quantum phase transition present in the TFIM. We unravel the intricate behavior of these OTOCs and identify the underlying mechanism that stabilizes the long-time values of OTOCs: dynamical relocalization. We provide two variations of the TFIM in which the relocalization phenomenon is not directly related to the presence of a quantum phase transition. This shows that no direct, universal relation exists between the two. Finally, we discuss the effects of breaking the integrable structure of the TFIM. Our results show that for short times, chaotic systems in proximity of integrability can retain signatures of integrability. In contrast, we observe significant changes at large times. This suggests that information scrambling, as a universal feature of chaotic systems, is restored at large times—even in systems near integrability that exhibit order. Following the analysis of information spreading close to integrable regimes, we study the stability of information spreading in more generic systems. First, we consider charge-conserving random circuits. In these systems, charge-charge correlators exhibit large fluctuations across different circuit instances with a slow polynomial decay in time. At the same time, they remain stable against fine-tuned perturbations, which can be interpreted as dephasing noise in the eigenbasis of the charge density operators. Equivalently, charge-charge correlators are mostly insensitive to decoherence, with interference effects contributing only subleading corrections. In this sense, the charge information spreading probed by charge-charge correlators is effectively classical. We perform a detailed analysis and quantify the dephasing insensitivity of these correlators. This suggests that circuit-to-circuit fluctuations can be captured by considering small regions where “quantum” effects are included, embedded within otherwise “classical” dynamics. This observation motivates the development of an algorithm to compute charge-charge correlators for individual circuits on classical computers with high accuracy. We quantitatively relate the algorithm’s performance to the dephasing analysis. We demonstrate the algorithm’s accuracy numerically for charge-charge correlators. Then, we argue that in generic circuits without conservation laws, the information spreading, as probed by OTOCs, follows similar principles. We conclude by comparing OTOC values measured on quantum devices with up to 95 superconducting qubits and 1000 gates to predictions from the approximate algorithm. This cross-validates experimental measurements and theoretical predictions. In this thesis, we explore the implications of adding structure to chaotic quantum systems, contributing to a deeper understanding of the organizing principles governing quantum chaos. We establish that entanglement can be suppressed in the semiclassical limit and that Krylov complexity behaves differently in thermalizing or integrable time-dependent systems. We show that the phenomenon information scrambling is also present in chaotic quantum systems even in the vicinity of a quantum phase transition. Conversely, when charge conservation is added, information spreading is stabilized and becomes reminiscent of that in classical systems. This work opens up further research directions towards a better understanding of the quantum nature of chaotic quantum systems and the development of probes of chaos in time dependent systems with different types of structure.
Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are a prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an order parameter for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum-mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs while introducing an operator-independent diagnostic for quantum chaos, offering more universal insights and a deeper understanding of the general properties of quantum chaotic systems.
At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. \nThe first approach in this thesis is a technique to measure entanglement in finite size \nlattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm \nis developed, implemented, and used to explore anomalous entanglement scaling terms in \nthe spin-1/2 Heisenberg antiferromagnet. \nIn the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. \nEntanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
The study of complex quantum systems is a fundamental aspect of modern physics, providing insights into the behavior of matter at microscopic scales. Mathematical physics plays a crucial role in developing the theoretical frameworks necessary for understanding these systems, yet challenges remain in applying these concepts to real-world scenarios. This research aims to investigate the application of mathematical techniques in analyzing complex quantum systems. The focus is on identifying effective mathematical models and methods that can enhance our understanding of quantum phenomena. A comprehensive literature review was conducted, analyzing various mathematical approaches utilized in quantum mechanics, including perturbation theory, group theory, and numerical simulations. Case studies were examined to illustrate the successful application of these methods in real-world quantum systems. Findings indicate that advanced mathematical techniques significantly improve the modeling and analysis of complex quantum systems. The application of perturbation theory and numerical simulations provided deeper insights into system behaviors, while group theory facilitated a better understanding of symmetry properties. This research highlights the indispensable role of mathematical physics in the study of complex quantum systems. By emphasizing the integration of mathematical techniques, the study contributes to the advancement of theoretical physics and offers pathways for future research in quantum mechanics.