The exploration of topological phases of matter has predominantly focused on equilibrium systems. However, recent advances in experimental techniques allow for precise manipulation and probing of quantum systems far from equilibrium, opening new avenues for discovering and engineering novel topological states. Current theoretical tools, largely categorized into variational and perturbative methods, often struggle to capture the full complexity of these non-equilibrium phenomena. Variational approaches offer broad applicability but can lack microscopic detail, while perturbative methods provide precise insights within specific regimes but are limited by their applicability range. This paper introduces a synergistic variational-perturbative framework designed to overcome these limitations. By integrating the strengths of both methodologies, our framework aims to provide a robust and accurate description of non-equilibrium topological matter, spanning both weakly and strongly driven regimes, and accommodating many-body interactions. We elaborate on the conceptual underpinnings, methodological details, and potential applications of this combined approach, demonstrating its capacity to predict and characterize dynamic topological invariants, non-equilibrium phase transitions, and emergent phenomena in driven quantum systems. This framework promises to be a powerful tool for advancing our understanding and control of topological properties in dynamical settings.
The exploration of topological phases of matter has predominantly focused on equilibrium systems. However, recent advances in experimental techniques allow for precise manipulation and probing of quantum systems far from equilibrium, opening new avenues for discovering and engineering novel topological states. Current theoretical tools, largely categorized into variational and perturbative methods, often struggle to capture the full complexity of these non-equilibrium phenomena. Variational approaches offer broad applicability but can lack microscopic detail, while perturbative methods provide precise insights within specific regimes but are limited by their applicability range. This paper introduces a synergistic variational-perturbative framework designed to overcome these limitations. By integrating the strengths of both methodologies, our framework aims to provide a robust and accurate description of non-equilibrium topological matter, spanning both weakly and strongly driven regimes, and accommodating many-body interactions. We elaborate on the conceptual underpinnings, methodological details, and potential applications of this combined approach, demonstrating its capacity to predict and characterize dynamic topological invariants, non-equilibrium phase transitions, and emergent phenomena in driven quantum systems. This framework promises to be a powerful tool for advancing our understanding and control of topological properties in dynamical settings.
Non-equilibrium quantum phase transitions (NEQPTs) represent a frontier in condensed matter physics and quantum statistical mechanics, characterized by abrupt changes in the steady-state properties of open quantum systems or by critical dynamics in closed systems far from thermal equilibrium. Unlike their equilibrium counterparts, NEQPTs are profoundly influenced by driving forces, dissipation, and the specific dynamics of quantum quenches. Understanding these complex phenomena requires theoretical frameworks that can accurately capture both strong correlations and real-time evolution in the presence of environmental interactions. Traditional theoretical tools, such as purely variational methods or purely perturbative expansions, often fall short when dealing with the intricate interplay of coherent and incoherent processes characteristic of NEQPTs. Variational approaches, while adept at capturing ground state properties and short-time dynamics within a chosen ansatz, struggle with long-time thermalization and the inclusion of subtle quantum and thermal fluctuations beyond the variational manifold. Conversely, perturbative methods, particularly those based on the Keldysh formalism, can systematically include interactions and environmental effects but typically rely on a small expansion parameter, limiting their applicability to weakly interacting or weakly driven regimes. This paper proposes a novel "Variational-Perturbative Unification" framework designed to overcome these limitations. The core idea is to combine the strengths of both approaches: using a variational ansatz to capture the dominant non-equilibrium many-body state or dynamics, and then systematically incorporating quantum and thermal fluctuations, as well as dissipation, through a perturbative expansion around this optimized variational state. This hybrid methodology aims to provide a robust and accurate description of NEQPTs, offering insights into critical exponents, universality classes, and the dynamic pathways to steady states. We explore the theoretical underpinnings of this unified approach, discuss its potential applications to driven-dissipative systems and quantum quenches, and highlight how it can reveal novel non-equilibrium phenomena.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
Understanding strongly coupled quantum systems is one of the most profound challenges in modern theoretical physics, encompassing phenomena from quark confinement in Quantum Chromodynamics to emergent properties in strongly correlated materials. Traditional perturbative methods, reliant on small coupling constants, inherently fail in these regimes. This paper proposes a novel framework for "Variational-Asymptotic Unification," aiming to combine the non-perturbative strengths of variational principles with the analytical power of asymptotic expansions. Variational methods offer a direct route to approximating ground states and low-lying excitations, providing robust bounds and insights into the non-perturbative structure of a theory. Asymptotic expansions, on the other hand, offer systematic approximations in certain limits, such as large coupling, low energy, or large-N limits. We outline a methodology where variational ansatzes are constructed to incorporate leading asymptotic behaviors, and the variational parameters are subsequently determined by matching to known asymptotic limits or by extremizing an effective action derived from such an ansatz. This unified approach is posited to bridge the gap between perturbative and non-perturbative regimes, offering a more comprehensive and accurate description of strong-coupling physics. Potential applications range from calculating spectral properties of hadrons and glueballs in lattice gauge theories to elucidating phase transitions in condensed matter systems described by models like the Hubbard model. The paper discusses the theoretical foundations, outlines concrete methodological steps, and explores the advantages and challenges of this interdisciplinary approach, suggesting a pathway towards robust analytical tools for previously intractable problems.
This paper explores the application of real-time Feynman diagrammatics as a powerful theoretical framework for understanding and predicting emergent topological phenomena in quantum materials under non-equilibrium conditions. Traditional approaches, primarily focused on equilibrium systems, often fall short in describing the rich dynamics and transient phases induced by external drives or sudden quenches. We delve into the intricacies of the Keldysh formalism, a crucial tool for non-equilibrium quantum field theory, and detail how it can be combined with diagrammatic expansions to capture time-dependent correlations and interactions. The methodology presented allows for the analysis of phenomena such as Floquet topological insulators, dynamically induced topological phase transitions, and the evolution of topological invariants in time-dependent settings. Through this framework, we aim to uncover novel topological states that emerge far from equilibrium, offering pathways for their experimental realization and manipulation. The discussion highlights the unique insights provided by a real-time perspective, contrasting it with static descriptions and emphasizing its potential to drive advancements in both fundamental condensed matter physics and the engineering of quantum devices.
This paper explores topological thermodynamics in driven dissipative quantum materials, a field at the intersection of non-equilibrium statistical mechanics, quantum topology, and condensed matter physics. We review theoretical frameworks and experimental advancements, highlighting how driving forces and dissipative processes influence the formation, stability, and characterization of topological phases. A special focus is placed on Floquet engineering for open quantum systems, where periodic driving and environmental coupling lead to novel non-equilibrium steady states with exotic topological properties. We discuss challenges in defining thermodynamic quantities like temperature and entropy in these far-from-equilibrium settings and examine how topological invariants manifest in dissipative dynamics and transport. The paper uniquely synthesizes advancements in periodically driven, open quantum systems by critically examining how dissipation, often seen as detrimental, can be precisely engineered to reveal or even stabilize topological order. This offers a distinct perspective on designing next-generation quantum technologies. Key theoretical tools, including generalized Floquet theory, Keldysh formalism, and non-Hermitian descriptions, are discussed alongside experimental platforms like photonic quantum gases and cold atomic systems.
The emergence of topological entanglement, a fundamental phenomenon in condensed matter physics, is often overlooked in the context of complex, non-equilibrium systems. This research proposes a novel framework, "Phase-Steering," to quantify and control the dynamics of topological entanglement within such systems. We develop a continuous, dynamically evolving algorithm that analyzes real-time feedback to actively manipulate the topological structure of entanglement, offering a fundamentally new approach to understanding and potentially engineering topological phases. The core claim is that this method can facilitate the transition between distinct topological phases and provides a pathway for designing systems exhibiting enhanced robustness and adaptability. The work presented here details the algorithm's implementation, provides preliminary results demonstrating its effectiveness in simulating a specific non-equilibrium system, and discusses potential applications in materials science and quantum computing.