Semiclassical partition function for the double-well potential

We compute the partition function and specific heat for a quantum-mechanical particle under the influence of a quartic double-well potential nonperturbatively, using the semiclassical method. Near the region of bounded motion in the inverted potential, the usual quadratic approximation fails due to the existence of multiple classical solutions and caustics. Using the tools of catastrophe theory, we identify the relevant classical solutions, showing that at most two have to be considered. This corresponds to the first step towards the study of spontaneous symmetry breaking and thermal phase transitions in the nonperturbative framework of the boundary effective theory.

Figure 1

Single-well inverted potential.

Figure 2

Double-well inverted potential.

Figure 3

Plots of the time of flight $\mathrm{\Theta }$ vs. the turning point $q_t$ for $q_0=0$ (upper) and $q_0=0.3$ (lower). In the upper panel, the trivial solution $q_0=q_t=0=q(t)\forall t$, although valid for all $\mathrm{\Theta }$, is not shown.

Figure 4

The plane $(q_0,\mathrm{\Theta })$ is divided into regions with different number of classical solutions. The frontiers between those regions, named caustics, are shown above. The smooth curves indicate the coalescence of strictly periodic solutions, i.e, those that begin and end at the same position and at the same velocity.

Figure 5

Three-dimensional plot of $q_t\times q_0\times \mathrm{\Theta }$ (upper panel) and its projections onto the planes $q_t\times \mathrm{\Theta }$ (bottom left) and $q_0\times \mathrm{\Theta }$ (bottom right).

Figure 6

Cuspid $27v^2+4u^3=0$ in the control parameter space. See text and Figs. 7 and 8 for explanation of the arrows.

Figure 7

Behavior of the action in functional space when the cusp is crossed at the vertex. See arrow 1 in Fig. 6.

Figure 8

Behavior of the action in functional space when the cusp is crossed at a point that is not the vertex. See arrow 2 in Fig. 6.

Figure 9

Level urves of the imaginary part of the action (13), where $a\equiv \mathcal{R}(z)$ (horizontal axis), $b\equiv \mathcal{I}(z)$ (vertical axis). We adopted $v=1$ and $u=-1$, $-2$, $-3$, for the first, second and third panels, respectively, i.e, following arrow 2 in Fig. 6.

Figure 10

Plots of the classical solutions $q(\theta )$ for $q_0=0.3$ and $\mathrm{\Theta }=3$ (upper), before the first caustic, and for $\mathrm{\Theta }=6$ (lower), after the first caustic. In the latter, the middle curve corresponds to a (local) maximum of the action and, thus, will not be taken into consideration in the subsequent calculation of the partition function.

Figure 11

Specific heat (20) vs. $1/\mathrm{\Theta }$ for $g=0.1$. Dashed/green: Classical result; solid/red: current approach.

Figure 12

Same as previous plot—specific heat (20) vs. $1/\mathrm{\Theta }$ for $g=0.1$—zooming in around the temperature where the first catastrophe takes place ($\mathrm{\Theta }=\pi$). Dotted/magenta line: Only the global minimum is taken into account. Dashed/blue line: Both global and local minima are taken into account. Note that the local minimum (and therefore also this curve) exists only at low temperatures, on the left-hand side of the plot. Solid/red line: Current approach.