Abstract

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 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 the Hamiltonian 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. This is joint work with Sergey Bravyi.