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arXiv:cond-mat/0011262

Low-dimensional quantum spin systems

Indrani Bose

Published Nov 15, 2000• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.cond-mat/0011262

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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.
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 is interacting many body systems for which several rigorous results are known. A large number of compounds exist which can effectively be described as low-dimensional spin systems. Frustration occurs when the interaction energies associated with all the spin pairs cannot be simultaneously minimized. The parallel spin pair may be located on any one of the bonds of the triangle. The entropy/spin has a non-zero value in the ground state and there is no ordering of the spins. The sum over permutations is in accordance with the bosonic nature of the waves, the spin waves, propagating along the chain. The energy-momentum relations suggest that the low-lying spectrum is actually a combination of two elementary excitations known as spinons. In the case of integer spin chains, the spinons axe bound and the excitation spectrum consists of spin-wave-like modes exhibiting the Haldane gap.
The investigation of the dynamics of quantum many-body systems is a concerted effort involving computational studies of mathematical models and experimental studies of material samples. Some commonalities of the two tracks of investigation are discussed in the context of the quantum spin dynamics of low-dimensional magnetic systems, in particular spin chains. The study of quantum fluctuations in such systems at equilibrium amounts to exploring the spectrum of collective excitations and the rate at which they are excited from the ground state by dynamical variables of interest. The exact results obtained via Bethe ansatz or algebraic analysis (quantum groups) for a select class of completely integrable models can be used as benchmarks for numerical studies of nonintegrable models, for which computational access to the spectrum of collective excitations is limited.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.
Introduction to integrable many-body systems I This is the first volume of a three-volume introductory course about integrable (exactly solvable) systems of interacting bodies. The aim of the course is to derive and analyze, on an elementary mathematical and physical level, the Bethe ansatz solutions, ground-state properties and the thermodynamics of integrable many-body systems in many domains of physics: Nonrelativistic one-dimensional continuum Fermi and Bose gases; One-dimensional quantum models of condensed matter physics like the Heisenberg, Hubbard and Kondo models; Relativistic models of the (1+1)-dimensional Quantum Field Theory like the Luttinger model, the sine-Gordon model and its fermionic analog the Thirring model; Two-dimensional classical models, especially the symmetric Coulomb gas. In the first part of this volume, we deal with nonrelativistic one-dimensional continuum Fermi and Bose quantum gases of spinless (identical) particles with specific types of pairwise interactions like the short-range δ-function and hard-core interactions, and the long-range 1/