Author : Amanda M Young
Publisher :
ISBN 13 : 9781369311044
Total Pages : pages
Book Rating : 4.3/5 (11 download)
Book Synopsis Spectral Properties of Multi-dimensional Quantum Spin Systems by : Amanda M Young
Download or read book Spectral Properties of Multi-dimensional Quantum Spin Systems written by Amanda M Young and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: In this dissertation, we analyze spectral properties of frustration-free quantum spin systems, with an emphasis on multi-dimensional models. The focus of this analysis is two-fold. First, we determine the existence or non-existence of a spectral gap above the ground state for a class of multi-dimensional quantum spin systems. Second, we determine a set of criteria for which the spectral gap above the ground state energy of a frustration-free quantum spin system is stable in the presence of local perturbations. We begin by providing the mathematical formulation of quantum spin systems, including a discussion of the thermodynamic limit. We formalize the construction of infinite volume systems when we introduce Lieb-Robinson bounds. When these bounds hold, the dynamics of the finite volume quantum spin systems extend to the thermodynamic limit, and a well-defined infinite volume Hamiltonian, known as the GNS Hamiltonian, is guaranteed to exist. Determining if a quantum spin system is gapped or gapless is a question of the spectrum of the GNS Hamiltonian. It is gapped if there is an interval above the ground state energy that does not contain any points of the spectrum. Otherwise, it is gapless. We show that to prove a gap for the GNS Hamiltonian of a frustration-free system, it suffices to obtain a uniform lower bound for the spectral gap in a sequence of finite volumes. To obtain these results we apply the martingale method, which was first introduced for frustration-free quantum spin systems with open boundary conditions by Nachtergaele in 1996 [48]. We extend this method to hold for periodic boundary conditions, and apply it to the class of pure finitely correlated state models with periodic boundary conditions.We then study the the one-species, multi-dimensional PVBS models and determine the exact collection of model parameters for which the models are gapped in the thermodynamic limit. The PVBS models are a class of particle hopping models where the model parameters dictate the direction a particle prefers to move. We consider these models on two types of infinite volumes. First, we take the thermodynamic limit of the PVBS model to the d-dimensional integer lattice. Second, we consider the PVBS model on a half space of the integer lattice where the boundary is defined by a hyperplane. We find two types of closures of the spectral gap. First, there is a bulk closure due to a critical change in the set of model parameters. The second type of closure is only found in the case of the hyperplane boundary, and is due to the creation of edge states across the boundary. This is inherently different from the bulk closure, as away from the boundary the model still behaves like a gapped system. We then transition to considering the stability of gapped ground state phases. Stability results establish that if a system is gapped, it will remain gapped under the addition of small perturbations. The two main technical tools we use to prove such a result are Lieb-Robinson bounds and the spectral flow. Common to both tools is that they express locality properties for observables under a global evolution. We generalize this notion by introducing the theory of quasi-local maps. We apply our theory to prove the stability result from [47], which holds for both one-dimensional andmulti-dimensional quantum spin systems. The result from [47], which we extend to hold for a larger class of models and boundary conditions, states that a uniformly gapped family of frustration-free Hamiltonians with topologically ordered ground states will remain gapped in the presence of an exponentially decaying perturbation. We conclude with applying the stability result the finitely correlated state models with periodic boundary conditions.
