the matrix elements in terms of basis. For example, for spin 1/2 system it is common to adopt the usual representation of the Pauli matrices so that, for instance $H_0 = \vec{b}\cdot\vec{\sigma} \simeq \begin{pmatrix} b_z & b_x - i b_y \\ b_x + i b_y & -b_z \end{pmatrix}$. The numerical matrix on the rightmost side of the equal sign is written in the $|\pm z \rangle$ basis. We refer to the basis which $H_0$ as a ‘numerical matrix’ is in as the ‘laboratory frame’ with basis states $|\lambda_i\rangle$ ($\lambda$ for *l*aboratory. ) Label now the $i$th eigenstates of $H_0$ by $|\epsilon_i\rangle$. We assume that the laboratory basis is fixed, i.e. it has no dependence on any parameter in the Hamiltonian so that, for instance $\partial_{k_x} |\lambda_i\rangle = 0$. Instead of using the laboratory basis, we can also expand our states and operators in terms of the eigenbasis, and convert between the two basis through the unitary transformation $U$. Defining the $U$s so that $UH_0 U^\dagger$ is diagonal in the eigenbasis representation, we have $$U = \sum_{i,j}  |\epsilon_i \rangle \langle \epsilon_i|\lambda _j\rangle \langle j|$$, i.e. the matrix $i,j$th elements in the numerical representation of $U$ is $\langle \epsilon_i|\lambda_j\rangle$. Notice that since 