Non-equilibrium quantum many-body systems like periodically driven (Floquet) systems exhibit much richer dynamics than their equilibrium counterparts. A striking example is a time crystal, a novel non-equilibrium phase of matter which is forbidden in equilibrium. This thesis concentrates on studying Floquet many-body systems, in particularly, higher-order Floquet topology, light-matter interactions in the Floquet systems, and interesting physics of Floquet many-spin systems with a view towards quantum simulation, e.g. with trapped ions. The model Hamiltonians studied in this thesis are of two types: Floquet-Bloch Hamiltonian and Floquet many-spin Hamiltonian. We report the theoretical discovery and characterization of higher-order Floquet topological phases dynamically generated in a periodically driven system with mirror symmetries. We demonstrate numerically and analytically that these phases support lower-dimensional Floquet bound states, such as corner Floquet bound states at the intersection of edges of a two-dimensional system, protected by the nonequilibrium higher-order topology induced by the periodic drive. We characterize higher-order Floquet topologies of the bulk Floquet Hamiltonian using mirror-graded Floquet winding number. We also show that bulk vortex structures can be dynamically generated by a spatially inhomogeneous drive and can host multiple Floquet bound states. Inspired by the effectiveness of the optical probes in condensed matter systems, we study the light-matter interactions in Floquet systems in strong and weak optical field regimes. In the weak field limit, we formulate the linear response theory of a periodically driven system. We illustrate the applications of this formalism by giving general expressions for optical conductivity of Floquet systems, including its homodyne and heterodyne components and beyond. We obtain the Floquet optical conductivity of specific driven models, including two-dimensional Dirac material such as the surface of a topological insulator, graphene, and the Haldane model irradiated with circularly or linearly polarized laser, as well as a semiconductor quantum well driven by an ac potential. In the strong field limit, we formulate a theory of bulk optical current for a periodically driven system, which accounts for the mixing of external drive and laser field frequencies and, therefore, the broadening of the harmonic spectrum compared to the undriven system. We illustrate the application of this theory by studying high harmonic generation in the periodically driven Su Schrieffer-Heeger model. We find significant enhancement in higher harmonics when the system is driven, even for low field amplitudes and obtain harmonics forbidden in the undriven model. Motivated by applications on a trapped-ion quantum simulator, we study many-spin systems with periodically driven two-body interactions and external fields in the high-frequency limit. We show that a hierarchy of multi-spin interactions can be generated in powers of inverse-frequency, such as Dzyaloshinskii-Moriya interaction (DMI), three- and four-spin interactions, and additional Zeeman terms. We introduce the concept of a Floquet gauge pump where we exploit the dynamically generated interactions of a Floquet Hamiltonian to detect the topology of the ground state and edge states in interacting systems. We demonstrate this concept in 1D XY model with periodically driven couplings and a transverse field and discuss the requirements to realize the Floquet gauge pump with trapped ions.

This book is a self-contained advanced textbook on the mathematical-physical aspects of quantum many-body systems, which begins with a pedagogical presentation of the necessary background information before moving on to subjects of active research, including topological phases of matter. The book explores in detail selected topics in quantum spin systems and lattice electron systems, namely, long-range order and spontaneous symmetry breaking in the antiferromagnetic Heisenberg model in two or higher dimensions (Part I), Haldane phenomena in antiferromagnetic quantum spin chains and related topics in topological phases of quantum matter (Part II), and the origin of magnetism in various versions of the Hubbard model (Part III). Each of these topics represents certain nontrivial phenomena or features that are invariably encountered in a variety of quantum many-body systems, including quantum field theory, condensed matter systems, cold atoms, and artificial quantum systems designed for future quantum computers. The book’s main focus is on universal properties of quantum many-body systems. The book includes roughly 50 problems with detailed solutions. The reader only requires elementary linear algebra and calculus to comprehend the material and work through the problems. Given its scope and format, the book is suitable both for self-study and as a textbook for graduate or advanced undergraduate classes.

This book studies the fundamental aspects of many-body physics in quantum systems open to an external world. Recent remarkable developments in the observation and manipulation of quantum matter at the single-quantum level point to a new research area of open many-body systems, where interactions with an external observer and the environment play a major role. The first part of the book elucidates the influence of measurement backaction from an external observer, revealing new types of quantum critical phenomena and out-of-equilibrium dynamics beyond the conventional paradigm of closed systems. In turn, the second part develops a powerful theoretical approach to study the in- and out-of-equilibrium physics of an open quantum system strongly correlated with an external environment, where the entanglement between the system and the environment plays an essential role. The results obtained here offer essential theoretical results for understanding the many-body physics of quantum systems open to an external world, and can be applied to experimental systems in atomic, molecular and optical physics, quantum information science and condensed matter physics.

The book explains concepts and ideas of mathematics and physics that are relevant for advanced students and researchers of condensed matter physics. With this aim, a brief intuitive introduction to many-body theory is given as a powerful qualitative tool for understanding complex systems. The important emergent concept of a quasiparticle is then introduced as a way to reduce a many-body problem to a single particle quantum problem. Examples of quasiparticles in graphene, superconductors, superfluids and in a topological insulator on a superconductor are discussed.The mathematical idea of self-adjoint extension, which allows short distance information to be included in an effective long distance theory through boundary conditions, is introduced through simple examples and then applied extensively to analyse and predict new physical consequences for graphene.The mathematical discipline of topology is introduced in an intuitive way and is then combined with the methods of differential geometry to show how the emergence of gapless states can be understood. Practical ways of carrying out topological calculations are described.

Intended for graduates in physics and related fields, this is a self-contained treatment of the physics of many-body systems from the point of view of condensed matter. The approach, quite traditionally, covers all the important diagram techniques for normal and superconducting systems, including the zero-temperature perturbation theory, and the Matsubara, Keldysh, and Nambu-Gorov formalisms. The aim is not to be exhaustive, but to present just enough detail to enable students to follow the current research literature or to apply the techniques to new problems. Many of the examples are drawn from mesoscopic physics, which deals with systems small enough that quantum coherence is maintained throughout the volume, and which therefore provides an ideal testing ground for many-body theories. '

Quantum annealing is a new-generation tool of information technology, which helps in solving combinatorial optimization problems with high precision, based on the concepts of quantum statistical physics. Detailed discussion on quantum spin glasses and its application in solving combinatorial optimization problems is required for better understanding of quantum annealing concepts. Fulfilling this requirement, the book highlights recent development in quantum spin glasses including Nishimori line, replica method and quantum annealing methods along with the essential principles. Separate chapters on simulated annealing, quantum dynamics and classical spin models are provided for enhanced learning. Important topics including adiabatic quantum computers and quenching dynamics are discussed in detail. This text will be useful for students of quantum computation, quantum information, statistical physics and computer science.

"Unabridged republication of the second edition of the work, originally published in the Pure and applied physics series by Academic Press, Inc., New York, in 1972"--Title page verso.