| We consider several aspects of the XXZ quantum spin system. These aspects are: existence of a spectral gap above infinite-volume ground states for spin chains with any spin dimension; description of droplet ground states for the XXZ Hamiltonian with up-spin boundary conditions; and a constructive proof of nonexistence of spectral gap above interface ground states in dimensions two and higher.; In Chapter 1 we introduce the XXZ model, and give a complete summary of the rest of the paper. In Chapter 2, we review important results for the XXZ model, including: existence of diagonal interface ground states; the complete list of ground states in one-dimension; the SUq(2)-symmetry in one-dimension, for spin-½; and the exact formula for the spectral gap in the SUq(2)-symmetric case.; In Chapter 3, we prove that in one-dimension there is a spectral gap for all spin dimensions, not just S = 1/2. We provide numerical estimates for the gap, γ = γ(Δ, S). Based on the numerics we observe that for large enough S, γ(Δ, S) is maximized by Δ ∗ ∈ (1, ∞). We also conjecture a limit γ ∗(Δ) = limS→∞ γ(Δ, S)/S.; In Chapter 4, we introduce a model of a quantum spin droplet, which has XXZ nearest-neighbor interactions and up-spin boundary conditions. We construct droplet states by tensoring a kink state to an antikink state, and by a “cut-and-paste” argument, we prove that these droplet states are asymptotically close to the true ground states.; In Chapter 5, we prove that there is no spectral gap above the (1,…,1) interface ground states in dimensions d > 1. Previous proofs of the same fact are based on the Goldstone theorem. Our proof is similar, but constructive. We consider excitations, which are spin waves constrained to the (d−1)-dimensional interface. We derive for the first time upper bounds on the spectral gap, which decay as 1/R 2 where R is the diameter of the interface. |
