  We consider a family of translation-invariant quantum spin chains with
nearest-neighbor interactions and derive necessary and sufficient conditions
for these systems to be gapped in the thermodynamic limit. More precisely, let
$\psi$ be an arbitrary two-qubit state. We consider a chain of $n$ qubits with
open boundary conditions and Hamiltonian $H_n(\psi)$ which is defined as the
sum of rank-1 projectors onto $\psi$ applied to consecutive pairs of qubits. We
show that the spectral gap of $H_n(\psi)$ is upper bounded by $1/(n-1)$ if the
eigenvalues of a certain two-by-two matrix simply related to $\psi$ have equal
non-zero absolute value. Otherwise, the spectral gap is lower bounded by a
positive constant independent of $n$ (depending only on $\psi$). A key
ingredient in the proof is a new operator inequality for the ground space
projector which expresses a monotonicity under the partial trace. This
monotonicity property appears to be very general and might be interesting in
its own right. As an extension of our main result, we obtain a complete
classification of gapped and gapless phases of frustration-free
translation-invariant spin-1/2 chains with nearest-neighbor interactions.
