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arXiv:2503.03564

Beginner's Lecture Notes on Quantum Spin Chains, Exact Diagonalization and Tensor Networks

Guglielmo Lami, Mario Collura, Nishan Ranabhat

Published Mar 5, 2025• 0 citations
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Abstract

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.

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<!--HTML-->In recent years, tensor networks have become a viable alternative to Monte Carlo calculations and exact diagonalization for the simulation of many-body systems. As they represent a formulation of quantum mechanical wavefunctions with polynomially many parameters, they make calculations of large systems feasible. They have already found wide application in condensed matter physics and start to be an interesting tool for high energy physics as well. In this lecture series, I will introduce the basic concepts of tensor networks. We will start with an introduction of the necessary basics of quantum mechanics and linear algebra and focus on the algorithmic side of tensor networks in the second lecture.
These notes accompany lectures given in June 2025 at the summer school \emph{Fundamental Problems in Statistical Physics XVI}. They offer a short introduction to random quantum circuits as simple models for generic many-body quantum systems. They give an outline of the motivation for introducing these models, starting from ideas of random matrix theory. They also provide a sketch of calculations of some of the quantities of most physical interest, based on an average over an ensemble of systems. These quantities give insights into operator spreading, entanglement dynamics and spectral correlations.
In these notes, we present a rigorous and self-contained introduction to the fundamental concepts and methods of quantum many-body theory. The text is designed to provide a solid theoretical foundation for the study of interacting quantum systems, combining clarity with mathematical precision. Core topics are developed systematically, with detailed derivations and comprehensive proofs that aim to make the material accessible to graduate students and beginning PhD students. Special attention is given to formal consistency and pedagogical structure, so as to guide the reader through both the conceptual and technical aspects of the subject. This work is intended as a reliable starting point for further exploration and research in modern quantum many-body physics.