Eigenvalues and Low Energy Eigenvectors of Quantum Many-Body Systems

Abstract

I first give an overview of the thesis and Matrix Product States (MPS) representation of quantum spin systems on a line with an improvement on the notation.The rest of this thesis is divided into two parts. The first part is devoted to eigenvalues of quantum many-body systems (QMBS). I introduce Isotropic Entanglement (IE), which draws from various tools in random matrix theory and free probability theory to accurately approximate the eigenvalue distribution of QMBS on a line with generic interactions. We then list some open related problems. Next, I discuss the eigenvalue distribution of one particle hopping random Schrödinger operator in one dimension from free probability theory in context of the Anderson model.There have been many great scientists who influenced my scientific life trajectory in positive ways. I am very grateful to Reinhard Nesper, Roald Hoffmann, Mehran Kardar, and Richard V. Lovelace. I also like to thank Otto E. Rössler, Jürg Fröhlich, John McGreevy, Jack Wisdom, Alexei Borodin and John Bush.I have been fortunate to have met so many wonderful people and made wonderful friendships during my PhD at MIT; too many to name here. I owe a good share of my happiness, balance in life, and the fun I had, to them. Last but certainly not least, I thank my dad, Javad Movassagh, and mom, Mahin Shalchi, for the biological existence and all they did for my sister and I to have a worthwhile future and an education. I dedicate this thesis to them.