Collective Modes and Topological Excitations in Layered Vector Phase Systems (Paper II in the Layered Phase Series)

Abstract We extend the minimal layered phase framework of Paper I 1 in two steps. First, we introduce the minimal inter-layer synchronisation potential consistent with the global U(1) symmetry of the scalar model. Spontaneous symmetry breaking produces a massless Goldstone mode and N−1 massive relative modes. Canonical normalisation of the Goldstone mode yields the exact coupling suppression geff = gN−1/2, with no adjustable parameters. In the second step, we replace the scalar target space S1 with the vector target space S2, the minimal extension that admits a non-trivial Hopf invariant QH ∈Z. We show that integrating out the massive relative modes generates a Skyrme– Faddeev quartic term in the effective Lagrangian, which evades Derrick’s theorem and stabilises three-dimensional solitons (Hopfions). The Vakulenko–Faddeev en- ergy bound E ≥C(Nf2)1/2κ1/2|QH|3/4 then implies a discrete, topologically pro- tected excitation spectrum. A variational estimate gives Emin(2)/Emin(1) ≈1.63. All results follow from the microscopic Lagrangian and established mathematical theorems without fine-tuning. Exchange statistics of Hopfions are addressed in Pa- per III