On Spectral Gap,U(1) Symmetry and Split Property in Quantum Spin Chains

Understand

We consider a class of ground states for quantum spin chains on an integer lattice.

• First we show that presence of the spectral gap between the ground state energy and the rest of spectrum implies the split property of certain subsystems.As a corollary, we show that gapless excitation exists for spinless Fermion if the pure gauge invariant ground state is non-trivial and translationally invariant.