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.

Figures (9)

• Figure 1: Schematic of the four well potential with parameters $b_p=2,b_q=1,c=1/2$ (see Eq. \ref{['Pot']}.) Saddle points are clearly visible. An instanton can travel from any one well to any other well by a variety of different paths and thus back to its starting point as well.
• Figure 2: Decomposition of the fluctuation matrix: In the frame comoving with the instanton, the action is unchanged for motion along the longitudinal direction whereas for transverse fluctuations a restoring force pushes it back on the trajectory.
• Figure 3: P, Q, R instantons:$P$ flips the $p$ value by 2 units leaving $q$ unchanged, $Q$ flips only the $q$ value, while the diagonal instanton $R$ flips both $p,q$.
• Figure 4: Auxiliary functions: Plots of $q_1(\tau)$ and $p_2(\tau)$ in the perturbative expansion Eq. \ref{['pqex']} where $q_1$ is the source function for $p_2$.
• Figure 5: $\chi_T(\mu)$, which is the ratio of gamma functions in Eq. \ref{['CHI']} plotted against the coupling constant $\mu$.