\rangle$}})$ is manifestly gauge-invariant and may therefore potentially represent a physically relevant, measurable quantity. This observation features strongly in the early works of Pancharatnam [@Pancharatnam] and it also appears in the form of a remark in the celebrated proof of Wigner’s theorem by Bargmann [@Bargmann].

Introducing some notation, we define the *Bargmann invariant* with respect to a general $\mathcal{N}$-vertex circuit in parameter space by $$\label{eq:phasedefn2}
  \mathcal{C} =
  \prod_{s=0}^{\mathcal{N}-1}{\mbox{$\langle \psi_s  | \psi_{s+1} \rangle$}} =
  {\operatorname{tr}}{\prod_{s=0}^{\mathcal{N}-1}{\mbox{$|\psi_s\rangle \!\langle \psi_s |$}}},
  \quad
  \psi_{\mathcal{N}} = \psi_{0}.$$ The associated *Bargmann phase* is denoted by $\varphi =
\text{arg}(\mathcal{C})$. Modulo $2\pi$, the global Bargmann phase is just the sum of the individual phases, i.e. $\varphi =
\sum_{s=0}^{\mathcal{N}-1}\text{arg}{\mbox{$\langle \psi_s  | \psi_{s+1} \rangle$}}$. Note that we require $\mathcal{N} \geqslant 3$ for non-trivial Bargmann phases.

Suppose now, for argument’s sake, that we are dealing with state vectors ${\mbox{$| \psi(\underline{\alpha}_s) \rangle$}}$ that represent the instantaneous ground states of a family of parameter-dependent Hamiltonians $H(\{\underline{\alpha}\})$. Then, in the continuum limit where the sum above is replaced by a contour integral over an infinitesimal phase along a (smooth) closed circuit in parameter space, the Bargmann invariant reduces to the usual Berry phase. In this sense Bargmann invariants may be regarded as generalized Berry phases. 