This paper presents an overview of the properties of magnetic insulators, with emphasis on quantum effects. In particular, the consequences of strong quantum fluctuations on the low-energy properties of a number of systems are reviewed: the occurrence of a spin gap in spin-1 chains and other low-dimensional magnets; the presence of low-lying singlet excitations in several frustrated magnets; the interplay of orbital and spin fluctuations in orbitally degenerate Mott insulators. This review is intended as a pedagogical introduction to the field, and the formalism has been kept at a minimal level.
Quantum magnetism in low dimensions has been one of the central areas of theoretical research for many decades now. One of the key reasons for the long standing interest in this field has been the existence of simplified models, which serve as paradigms for understanding the role of strong interactions in many-electron systems. Although simple, these models quite often can not be solved exactly. In this review, we discuss a variety of analytical and numerical methods, which treat the system in a systematic and controlled manner. The central method employed in all the studies is the density matrix renormalization group (DMRG) method. This is supported by small scale numerical exact calculations, and by analytical methods using field theoretic techniques. We have considered a number of magnetic systems, from magnetic clusters to extended lattices, and have found some novel quantum ground states and low-energy elementary excitations. In some cases, we have also employed the finite-temperature DMRG method to accurately compute the low-temperature thermodynamic properties such as specific heat and magnetic susceptibility.
Abstract Quantum integrability provides a unique and powerful framework for accurately understanding quantum magnetism. In this review, we focus specifically on several quantum integrable low-dimensional quantum Ising models. We begin with the transverse field Ising chain (TFIC) at quantum critical point and examine how it evolves under perturbations, such as an applied longitudinal field or weak coupling to another quantum critical TFIC. These perturbations reveal a wealth of emergent quantum integrable field theories with exotic many-body excitations, elegantly characterized by conformal invariance and the E 8 and D 8 ( 1 ) Lie algebras, respectively. In exploring these models, we also delve into the framework of exact scattering matrices, which is related to determining spin dynamics within these systems. Finally, we show how the emergent phenomena in these integrable quantum Ising models find experimental realization in Co-based quasi-one-dimensional quantum magnetic materials. The substantial theoretical and experimental advancements in these systems highlight the profound connections between quantum integrable field theory, statistical field theory, and condensed matter physics.
In this topical review, we explore low-dimensional spin-
21 systems, focusing on key models such as the Majumdar-Ghosh (MG) model on a spin-
21 chain and the Heisenberg spin-
21 model applied to various ladder geometries, including normal, zigzag, two-leg honeycomb, skewed, and diamond. We discuss the key physical features and experimental realizations of these models in real materials, providing a detailed overview. The review also includes the Kitaev-Heisenberg model on a two-leg ladder, a recent development in quantum magnetism. It discusses emerging quantum phases, such as spin liquids and magnetization plateaus, and highlights open research areas, including the investigation of unexplored exotic ground states. Additionally, we briefly discuss how quantum simulators based on ion traps, superconducting circuits, optical lattices, etc., are being used as powerful tools to explore the ground-state phases of spin-
21 systems. Furthermore, the review emphasizes the potential for new material design, offering an exciting avenue for future experimental and theoretical advancements in low-dimensional quantum systems.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In these lectures, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.
Low-dimensional quantum spin systems are interacting many body systems for which several rigorous results are known. Powerful techniques like the Bethe Ansatz provide exact knowledge of the ground state energy and low-lying excitation spectrum. A large number of compounds exists which can effectively be described as low-dimensional spin systems. Thus many of the rigorous results are also of experimental relevance. In this Lecture Note, an introduction is given to quantum spin systems and the rigorous results known for such systems. Some general theorems and exactly-solvable models are discussed. An elementary introduction to the Bethe Ansatz technique is given.