  Energy gap, the difference between the energy of the ground state of a given
Hamiltonian and the energy of its first excited state, is a parameter of a
critical importance in analysis of phase transitions and adiabatic quantum
computation. We present a concrete technique to determine the upper bound for
the energy gap of a Hamiltonian $H_0$ based on properties of the set of
expectation values of $H_0$ and an additional auxiliary Hamiltonian $V$. This
formalism can be applied to obtain an effective criterion of gaplessness, which
we illustrate with a concrete example of the XY model -- a physical system with
vanishing energy gap.

Quantum Physics