Thou shalt not tunnel: Complex instantons and tunneling suppression in deformed quantum mechanics

Abstract

The quantization of the Seiberg-Witten curve of ${\cal N}=2$ super Yang-Mills theory leads to a deformation of one-dimensional quantum mechanics with unconventional behavior. Most notably, quantum tunneling is suppressed at special points in parameter space. In this paper we examine these deformed models in the case of double-well and cubic potentials, and we find that they have a rich phase structure. In what we call the strong coupling phase, the theory behaves like conventional quantum mechanics, instantons are real, and tunneling is not suppressed. In the weak coupling phase, the instantons responsible for tunneling become complex, and tunneling suppression takes place at the so-called Toda lattice points. At the critical point between the two phases, which corresponds to a monopole point in super Yang-Mills theory, the non-perturbative amplitudes display an anomalous scaling as a function of $\hbar$. This phase structure reflects the physics of the underlying super Yang-Mills theory and can be regarded as a physical manifestation of wall-crossing behavior of the BPS spectrum, which we determine in our problem by using resurgent techniques.

Figures (18)

• Figure 1: The inverted potentials $-u_4(x)$ and $-w_4(x)$ of the double-well model, with $h=3.5$ and $E=-0.1$. In the plot of $-w_4(x)$ we also indicate the trajectory of the instanton configuration.
• Figure 2: The inverted potentials $-u_3(x)$ and $-w_3(x)$ of the cubic model, with $h=2.5$ and $E=-0.1$. In the plot of $-w_3(x)$ we also indicate the trajectory of the instanton configuration.
• Figure 3: The trajectories of the instanton $x(\tau)$ in the double well, for $\tau=x_0=0$ and $E=0$. The figure on the left is for $h=3$, while the figure on the right is for $h=3.999$. The horizontal lines denote the asymptotic values $\mp x_\star$.
• Figure 4: The trajectories of the instanton $x(\tau)$ in the cubic potential, for $\tau=0$, $x_0$ the turning point, and $E=0$. The figure on the left is for $h=2$, while the figure on the right is for $h=2.999$. The horizontal line is $x_\star$.
• Figure 5: As we change the value of the parameter $h$ in the double well and cubic potentials, the theory undergoes a phase transition at a critical value $h=h_c$. Here we show the change in the form of the potential in the case of the double well.