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Iranian Journal of Physics Research

Introduction to quantum spin systems

A. Langari

Published Jun 15, 2008• 1 citation
OpenAlex

Abstract

This manuscript is the collection of lectures given in the summer school on strongly correlated electron systems held at Isfahan university of technology, June 2007. A short overview on quantum magnetism and spin systems is presented. The numerical exact diagonalization (Lanczos) alghorithm is explained in a pedagogical ground. This is a method to get some ground state properties on finite cluster of lattice models. Two extensions of Lanczos method to get the excited states and also finite temperature properties of quantum models are also explained. The basic notions of quantum phase transition is discussed in term of Ising model in transverse field. Its phase diagram and critical properties are explained using the quantum renormalization group approach. Most of the topics are in tutorial level with hints to recent research activities.

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This manuscript is the collection of lectures given in the summer school on strongly correlated electron systems held at Isfahan university of technology, June 2007. A short overview on quantum magnetism and spin systems is presented. The numerical exact diagonalization (Lanczos) alghorithm is explained in a pedagogical ground. This is a method to get some ground state properties on finite cluster of lattice models. Two extensions of Lanczos method to get the excited states and also finite temperature properties of quantum models are also explained. The basic notions of quantum phase transition is discussed in term of Ising model in transverse field. Its phase diagram and critical properties are explained using the quantum renormalization group approach. Most of the topics are in tutorial level with hints to recent research activities.
Aimed at introducing readers to the physics of strongly correlated many-body systems, these notes focus on numerical methods, with detailed discussions on implementing working code for exact diagonalization. A brief introduction to tensor network methods is also included. Prepared for the Summer School Quantumandu, held at Tribhuvan University (Kathmandu, Nepal) from 25 to 31 July 2024, as part of the ICTP's Physics Without Frontiers program, these notes are primarily intended for readers encountering this field for the first time.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground-state and finite-temperature properties. Symmetry-breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite-size scaling at continuous and first-order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S=1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
At a quantum critical point the ground state of a many-particle quantum system undergoes a phase transition when some control parameters such as pressure, magnetic field, or chemical composition is varied. The universal behaviour characteristic for such quantum phase transitions often affects a wide temperature range and gives rise to novel material properties. This chapter gives a short overview over five lectures held during the Les Houches summer school ‘Quantum theory from small to large scales’ in 2010. After an introductory chapter, field driven magnetic quantum phase transitions of insulators are used as an example to discuss some of the concepts underlying quantum phase transitions. Both experimentally and theoretically, quantum phase transitions in metals are much less understood compared to insulating systems. After a brief review of the standard approach to describe those systems, the importance of multiple time scales and associated multiple critical exponents z are discussed. Finally, emergent gauge theories close to critical points are investigated. As an example, it discusses why a gauge theory describes the (classical) phase transitions of a nematic, if topological defects are suppressed.
These lecture notes introduce quantum spin systems and several computational methods for studying their ground‐state and finite‐temperature properties. Symmetry‐breaking and critical phenomena are first discussed in the simpler setting of Monte Carlo studies of classical spin systems, to illustrate finite‐size scaling at continuous and first‐order phase transitions. Exact diagonalization and quantum Monte Carlo (stochastic series expansion) algorithms and their computer implementations are then discussed in detail. Applications of the methods are illustrated by results for some of the most essential models in quantum magnetism, such as the S = 1/2 Heisenberg antiferromagnet in one and two dimensions, as well as extended models useful for studying quantum phase transitions between antiferromagnetic and magnetically disordered states.
In these lecture notes, we present a pedagogical review of a number of related numerically exact approaches to quantum many‐body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson’s Numerical Renormalization Group (NRG) and White’s Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low‐dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.
We have developed a technique to replace hashing in implementing the Lancz\"os method for exact diagonalization of quantum-spin models that enables us to carry out numerical studies on substantially larger lattices than previously studied. We describe the algorithm in detail and present results for the ground-state energy, the first-excited-state energy, and the spin-spin correlations on various finite lattices for spins S=1/2, 1, 3/2, and 2. Results for an infinite system are obtained by extrapolation. We also discuss the generalization of our method to other models.