Spectral Gap and Edge Excitations of d-Dimensional PVBS Models on Half-Spaces

Abstract

We analyze a class of quantum spin models defined on half-spaces in the d-dimensional hypercubic lattice bounded by a hyperplane with inward unit normal vector m ∈ R d . The family of models was previously introduced as the single species Product Vacua with Boundary States (PVBS) model, which is a spin-1/2 model with a XXZ-type nearest neighbor interactions depending on parameters λ j ∈ (0, ∞), one for each coordinate direction. For any given values of the parameters, we prove an upper bound for the spectral gap above the unique ground state of these models, which vanishes for exactly one direction of the normal vector m. For all other choices of m we derive a positive lower bound of the spectral gap, except for the case λ 1 = · · · = λ d = 1, which is known to have gapless excitations in the bulk.

Universality of the Hall Conductivity in Interacting Electron Systems

“…The explicit expression of P ≤M (dΨ) in terms of dΨ is 16) which motivates the appellation "Gaussian integration" that is usually given to the reference "measure" P ≤M (dΨ). Because of (5.15), P ≤M (dΨ) is also called the Gaussian integration with propagatorḡ β,L,M .…”

“…This assumption is unproven in most physically relevant cases, at least in the context of interacting fermions. As far as we know, the only cases for which it is proved are perturbations of "topologically trivial" classical reference states [24,25], or of "frustration free" Hamiltonians [16][17][18]48], that is of Hamiltonians that can be written as sums of projectors geometrically localized around the sites of the underlying lattice.…”

Gapped PVBS models for all species numbers and dimensions

“…The PVBS models can be seen as variants of the Heisenberg XXZ models in an external magnetic field. A remarkable feature of these is that, for appropriate parameter values, their gapped ground state phases can be explicitly classified in terms of edge-localized particles [2,4,6].…”

“…We expect that the methods developed here can be extended to derive a spectral gap for systems with free boundary conditions along sufficiently nice boundary shapes, like boxes; see Remark 1.3. We emphasize that the issue of boundary conditions is subtle for PVBS models, since they are known to exhibit weakly excited edge modes for certain half-plane geometries [6].…”

Quantum Hall Conductivity in the Presence of Interactions

“…This proof presumably may be extended to the other systems with the non -degenerate ground states with a gap. The latter condition remains nontrivial [20][21][22][23][24][25], and cannot be proven analytically for the majority of interacting systems. In [26] the proof has been given that the Hall conductivity is quantized for the gapped interacting system with weak shortranged interactions.…”