= Solution
A <Bloch state> on the loop has $\psi(x)=u(x)e^{i\phi x}$, with $u(x+a)=u(x)$. Its circumference is $L=Na$, so $u(x+Na)=u(x)$. Single-valuedness after one complete circuit requires the <periodic boundary condition>
$$
\psi(x+Na)=\psi(x),\qquad e^{i\phi Na}=1.
$$
Therefore
$$
\boxed{\phi=\frac{2\pi n}{Na},\qquad n\in\mathbb Z.}
$$
The allowed Bloch <wavevectors> are spaced by $2\pi/(Na)$. Values of $\phi$ differing by $2\pi/a$ label the same <crystal momentum> modulo a <reciprocal lattice> vector, so there are $N$ inequivalent choices in one <Brillouin zone>. In this notation $\phi$ is the Bloch <wavenumber>, not a dimensionless phase; $\phi a$ is dimensionless. Spin gives additional states but does not alter this condition.
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