The key advantage afforded by the generalized formulation lies in its computational ease: Bargmann invariants lend themselves directly to numerical computation, and thus, crucially, to scenarios where parts of the circuit under consideration represent non-integrable Hamiltonians, as is the case in the present work. An obvious potential drawback is that the discretization procedure may lack the simple interpretational underpinnings of the Berry phase in terms of physical adiabatic processes.

The Spin Model {#section:SpinModel}
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Consider a spin chain with nearest-neighbour $XX$ interactions in the presence of an external magnetic field, which is aligned along an arbitrary direction $\overrightarrow{n}$. The system is described by the following Hamiltonian: $$\label{eq:generalham}
    H_{\overrightarrow{n}} = J \sum_{k=1}^{N}\sigma^{x}_{k}\sigma^{x}_{k+1}
            + B \sum_{k=1}^{N}\sigma^{\overrightarrow{n}}_{k},
            \quad N \geqslant 2,$$ where, $\overrightarrow{n} = (n_x,n_y,n_z)$ denotes a vector of unit length, so that $\sigma^{\overrightarrow{n}} = \sum_{\alpha}
n_{\alpha}\sigma^{\alpha}$ with $\sum_{\alpha} n_{\alpha}^2 = 1$, $\alpha \in \{x,y,z\}$. Note that we impose periodic boundary conditions, so that $\sigma^{x}_{N+1} \equiv \sigma^{x}_1$.

As already outlined in the previous two sections, our aim is now to analyze the Bargmann invariant induced by the ground states of a family of Hamiltonians (\[eq:generalham\]) that is characterized by a set of $\mathcal{N}$ unit vectors $\{\overrightarrow{n}\}$. Typically, we will imagine the magnetic field 