Given two Hamiltonians H 1 and H 2 , it may be possible to continuously deform H 1 into H 2 while maintaining the symmetry constraint and gap (that is, there exists continuous function H ( t , k → → ) such that for all 0 ≤ ≤ t ≤ ≤ 1 the Hamiltonian has no zero eigenvalue and symmetry condition is maintained, and H ( 0 , k → → ) = H 1 ( k → → ) and H ( 1 , k → → ) = H 2 ( k → → ) ). Then we say that H 1 and H 2 are equivalent.