Spectral Gap and Edge Excitations of $d$-dimensional PVBS models on half-spaces

TL;DR

This paper investigates spectral gaps and edge excitations in a class of quantum spin models on half-spaces, revealing conditions under which the spectral gap is positive or vanishes, depending on model parameters and boundary orientations.

Contribution

It provides bounds on the spectral gap for PVBS models on half-spaces, identifying cases with gapless edge excitations and those with a positive spectral gap.

Findings

Spectral gap upper bound vanishes for one boundary direction.

Positive lower bound for spectral gap in other boundary directions.

Gapless excitations occur when parameters are all equal to one.

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 $m\in\mathbb{R}^d$. 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 $\lambda_j\in (0,\infty)$, 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 $\lambda_1 =\cdots =\lambda_d=1$, which is known to have gapless excitations in the bulk.