Accidental degeneracy

Accidental degeneracies¶

We have a pair of energy levels, and we ask if we can bring these levels into degeneracy by tuning Hamiltonian parameters. The energy levels (up to an overall constant) are determined by the most general 2 × 2 Hamiltonian

\(H = f_1\sigma_x + f_2\sigma_y + f_3\sigma_z\)

with an energy splitting between the levels

\(\Delta E =2\sqrt{f_1^2 + f_2^2 + f_3^2}\).

In general, in the absence of any symmetry, this cannot be accomplished by tuning just one parameter; degeneracy requires tuning all three terms to give zero simultaneously. If we focus on real Hamiltonians with time reversal symmetry, we can exclude the imaginary Pauli matrix. Then, a pair of levels can be brought into coincidence typically by tuning two parameters, since we can typically solve two equations \(\epsilon_x =0\) and \(\epsilon_z =0\) with two variables. But in the absence of any such symmetry, we need to tune three independent parameters to achieve a degeneracy.