Mathematical Physics Seminar

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.