Instantons In A Symmetric Quartic Potential

Abstract: We extend the semi-classical analysis of the double-well potential to a quartic system featuring four degenerate minima. Utilizing the Feynman path integral in imaginary time, we identify longitudinal, transverse, and diagonal instanton configurations that mediate tunneling between minima. The zero mode is handled by transforming to a rotating frame whose origin lies on the classically determined path. By generalizing the dilute instanton gas approximation to account for these distinct pathways, we derive the coherent Rabi-type oscillations and the energy splittings of the four lowest-lying states. These semi-classical results are validated against high-precision numerical diagonalization, showing excellent agreement in the deep semi-classical limit. We further identify a critical coupling regime where the discrete $D_4$ symmetry undergoes a `melting' transition into a continuous $O(2)$ rotational symmetry, signaling a fundamental breakdown of the localized instanton description.