The decay of a discrete 3PB state |𝑎⟩ for n0 = 1 reveals a continuum without discrete 3PB states whereas for n0 ≥ 2 the G(ω) shows discrete states as a point-like gap that slows down the decay (Fig. ). In the latter, discrete states at frequency Ω exist provided 𝜌(Ω)|𝑣(Ω)| 2 = 0 and Ω -𝜔 𝑎 = 𝜆 2 Δ(Ω) are simultaneously satisfied in a tight-binding Hamiltonian . Assuming t is normalized to 1/κ0, the coupling of |𝑎⟩ scales as ~ (κa/κ0) and λ ≡ κa/κ0 is a measure of interaction strength where κa is the electron hopping amplitude between |𝑎⟩ and |𝑛 0 ⟩ (Fig. ). Discrete states + ΔGbulk > 0 where ΔGsurface is positive and proportional to surface area and ΔGbulk is negative and proportional to volume. Assuming negligible co-existent phase equilibria of species, the thermodynamic length between species along path γ is given by an equation of state and explicitly depends on the path taken: