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 $m\in {\mathbb{R}}^d$. 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.

References in corpus (5)

• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Valence Bond Solids for Quantum Computation
• On topological phases of spin chains
• A class of asymmetric gapped Hamiltonians on quantum spin chains and its characterization I
• $C^1$-classification of gapped parent Hamiltonians of quantum spin chains

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• Divide and conquer method for proving gaps of frustration free Hamiltonians
• Random translation-invariant Hamiltonians and their spectral gaps
• Gapped PVBS models for all species numbers and dimensions
• Quasi-Locality Bounds for Quantum Lattice Systems. Part II. Perturbations of Frustration-Free Spin Models with Gapped Ground States
• Topologically ordered states in infinite quantum spin systems
• Knabe's spectral gap method for open boundary conditions
• Quantum Hall conductivity in the presence of interactions