Eigenvalues and Low Energy Eigenvectors of Quantum Many-Body Systems

TL;DR

This thesis explores eigenvalues, eigenvectors, and ground state properties of quantum many-body systems using matrix product states, random matrix theory, and free probability, providing new bounds, examples, and techniques for understanding quantum spin chains.

Contribution

Introduces new methods and results on eigenvalue distributions, ground state entanglement bounds, and specific highly entangled frustration-free Hamiltonians with unique properties.

Findings

Eigenvalue distribution approximation for QMBS using Isotropic Entanglement.

New lower bounds for ground state entanglement.

Construction of a highly entangled, translation-invariant spin-1 chain with critical behavior.

Abstract

I first give an overview of the thesis and Matrix Product States (MPS) representation of quantum spin chains with an improvement on the conventional 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, which draws from various tools in random matrix theory and free probability theory (FPT) to accurately approximate the eigenvalue distribution of QMBS on a line with generic interactions. Next, I discuss the energy distribution of one particle hopping random Schr\"odinger operator in 1D from FPT in context of the Anderson model. The second part is devoted to ground states and gap of QMBS. I first give the necessary background on frustration free (FF) Hamiltonians, real and imaginary time evolution within MPS representation and a numerical implementation. I then prove the degeneracy and FF condition for quantum spin chains with generic local interactions, including corrections to our earlier assertions. I then summarize my efforts in proving lower bounds for the entanglement of the ground states, which includes some new results, with the hope that they inspire future work resulting in solving the conjecture given therein. Next I discuss two interesting measure zero examples where FF Hamiltonians are carefully constructed to give unique ground states with high entanglement. One of the examples (i.e., $d=4$) has not appeared elsewhere. In particular, we calculate the Schmidt numbers exactly, entanglement entropies and introduce a novel technique for calculating the gap which may be of independent interest. The last chapter elaborates on one of the measure zero examples (i.e., $d=3$) which is the first example of a FF translation-invariant spin-1 chain that has a unique highly entangled ground state and exhibits signatures of a critical behavior.