= Berry connection
{c}
{title2=$A_j=i\langle\psi|\partial_{R_j}\psi\rangle$}
For a normalized, smoothly chosen instantaneous <energy eigenstate> $|\psi(\mathbf R)\rangle$, the <Berry connection> has components $A_j=i\langle\psi|\partial_{R_j}\psi\rangle$. Its <line integral> around a closed parameter path $C$ gives the <Berry phase> in a periodic choice of state vector:
$$
\gamma=\oint_C\sum_j A_j\,dR_j\pmod{2\pi}.
$$
Normalization makes the <inner product> $\langle\psi|\partial_{R_j}\psi\rangle$ purely imaginary, so the <Berry connection> is real. Changing the representative to $e^{i\chi(\mathbf R)}|\psi\rangle$ changes the <Berry connection> to $A_j-\partial_{R_j}\chi$. A single-valued phase factor along the closed path changes the <line integral> by an integer multiple of $2\pi$, leaving $e^{i\gamma}$ unchanged. If the chosen vector is not periodic, include the endpoint phase:
$$
\gamma=\arg\langle\psi(0)|\psi(T)\rangle+i\int_0^T\langle\psi(t)|\dot\psi(t)\rangle\,dt\pmod{2\pi}.
$$
Here $\psi$ denotes the instantaneous <energy eigenstate> representative, with its <dynamical phase> removed. This is freedom to choose a representative of a state ray. https://davidtong.org/pdfs/teaching/quantum-hall-effect/qhe1.pdf[David Tong, The Quantum Hall Effect, sections 1.5.1–1.5.2: Berry connection and a spin example; the spin convention below is stated explicitly]{ref}.
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