Instantons, Tunneling, and Nonperturbative Effects

Tunneling can be too small to appear at any finite order of a power expansion and still determine the behavior we care about: a tiny level splitting, a slow oscillation between wells, or a long escape time. This chapter explains how to calculate such effects using forbidden-region WKB and stationary paths in imaginary time. The first step is to distinguish the observable: transmission, a bound-state splitting, and a decay rate are different quantities.

Helpful background. WKB and semiclassical methods introduce barrier actions; small parameters and errors explain asymptotic accuracy. Imaginary-time path integrals supply the operator meaning of Euclidean evolution. Higher-order perturbation theory prepares the later discussion of large-order series.

An effect smaller than every power

Let $\epsilon$ be a dimensionless parameter approaching zero from above. For $A>0$ and every fixed nonnegative integer $N$,

$$e^{-A/\epsilon}=o(\epsilon^N).$$

Thus adding $e^{-A/\epsilon}$ to a quantity changes none of its algebraic asymptotic coefficients. This is what beyond all orders means. A tunneling contribution proportional to $e^{-S/\hbar}$ is of this kind in a semiclassical limit with positive action $S$ held at its physical scale.

“Nonperturbative” always refers to a specified expansion. It does not mean exact, numerically computed, or necessarily strongly coupled. Instanton calculations themselves usually use an asymptotic expansion around a stationary path. Asymptotic series and nonperturbative corrections explains how useful divergent series and exponentially small terms can coexist.

Three questions about tunneling

For a one-dimensional forbidden interval, define the one-way action

$$K(E)=\int_{x_1}^{x_2}\sqrt{2m[V(x)-E]}\,dx.$$

In a thick-barrier regime, this action sets the dominant exponential dependence. What we do with the amplitude depends on the experiment or spectral problem.

Physical question	Quantity being calculated	Typical leading exponential
What fraction of incident flux emerges beyond a barrier?	Transmission probability $T$	$e^{-2K/\hbar}$
How far apart are the two lowest levels in a symmetric double well?	Energy splitting $\Delta E$	$e^{-K/\hbar}$
How rapidly does a metastable state escape?	Decay rate $\gamma$	$e^{-B/\hbar}$, with a bounce action $B$

The entries describe exponential dependence, not complete formulas. Their prefactors have different units and come from different boundary conditions. The relevant energies in $K(E)$ must also be chosen consistently. In a one-dimensional bounce, $B$ includes the outward and return portions of the imaginary-time path; this is why it differs from a one-way action.

To see why a splitting is not a transmission probability, consider an orthonormal two-state description of a symmetric double well:

$$H_{\mathrm{two}}= \begin{pmatrix}E_{\mathrm{loc}}&-J\\-J&E_{\mathrm{loc}}\end{pmatrix}, \qquad J>0.$$

Its even and odd eigenstates have energies $E_{\mathrm{loc}}\mp J$, so $\Delta E=2J$. If $J$ is proportional to $e^{-K/\hbar}$, the splitting has that exponent too. Starting in the left state instead gives a time-dependent right-well probability $\sin^2(Jt/\hbar)$. An isolated double well oscillates in this approximation; it does not undergo irreversible exponential decay.

Begin with double-well tunneling, then tunneling splittings for the matching and prefactors. For an incident scattering wave, use barrier penetration.

Why imaginary time is useful

Imaginary-time evolution uses $e^{-HT/\hbar}$, rather than the real-time unitary $e^{-iHt/\hbar}$. Its formal path integral weights a path by $e^{-S_E/\hbar}$, with

$$S_E[x]=\int d\tau\left[\frac m2\dot x^2+V(x)\right].$$

A stationary path obeys $m\ddot x=V'(x)$: the sign is that of classical motion in the inverted potential. This is a way to find the saddle, not a claim that a tunneling particle follows a hidden real-time trajectory. Euclidean time and action explains the continuation, boundary terms, and subtraction of a reference energy.

An instanton connects distinct degenerate minima in imaginary time. With the minima shifted to zero potential, its finite action supplies the leading tunneling weight. Instantons in quantum mechanics develops the saddle construction, its quartic example, and the connection to the spectral splitting.

A bounce leaves a metastable minimum and returns to it. In the standard single-coordinate escape problem, its fluctuations contain one negative mode, as well as a translation zero mode. The negative mode and the appropriate continuation distinguish a decay calculation from a bound-state splitting. Study bounce solutions together with metastable decay.

If the resonance energy is $E_R-i\hbar\gamma/2$, the corresponding pole contribution to the amplitude decays as $e^{-\gamma t/2}$ and its probability as $e^{-\gamma t}$. That describes an intermediate-time regime. Exact survival probabilities need not be exponential at very short or very long times.

Beyond the exponential

The action determines the strongest suppression when $S/\hbar\gg1$. Fluctuations around the saddle determine a prefactor and corrections. Fluctuation determinants explains why a zero mode must be replaced by an integral over the event’s center, and why a negative mode needs separate treatment. For a long-time double-well kernel, repeated instantons and anti-instantons must also be summed; one isolated event is not the whole answer.

Resurgence explores relations between large-order perturbative coefficients and other saddle contributions in suitable models. Such relations require analytic and boundary-condition information; a few perturbative coefficients do not uniquely determine every exponentially small term. The bridge to QFT instantons explains which ideas extend to fields and where gauge symmetry, topology, and renormalization add new ingredients.

For a numerical comparison, check the splitting or lifetime directly. Two individually accurate energies can lose their small difference to numerical error. Agreement of the mean double-well energy alone does not establish the accuracy of the tunneling splitting.

Exercises

Beyond all powers. Prove the limit in the first section.

Solution

Set $y=A/\epsilon$. Then $e^{-A/\epsilon}/\epsilon^N=A^{-N}y^Ne^{-y}\to0$ as $y\to\infty$, because an exponential dominates every fixed polynomial. Two functions differing by this exponential can therefore share the same algebraic asymptotic expansion.

Splitting and oscillation time. For the two-state Hamiltonian, express the first complete transfer time from left to right in terms of $\Delta E$.

Solution

The first maximum of $\sin^2(Jt/\hbar)$ occurs at $Jt/\hbar=\pi/2$. Hence $t_{L\to R}=\pi\hbar/(2J)=\pi\hbar/\Delta E$. This relation assumes the isolated two-state model and an initially localized superposition of its two eigenstates.

Action accuracy. Suppose $Q=C e^{-S/\hbar}$ and $S/\hbar=30$. Holding $C$ fixed, what happens if the computed action is one percent too high? Give a sufficient relative action tolerance for a multiplicative error below ten percent for either sign of the action error.

Solution

The ratio is $Q_{\mathrm{app}}/Q=e^{-0.3}\approx0.741$: the prediction is about 26 percent low. A sufficient two-sided condition is $|\delta S|/\hbar<\log(1.1)$, or $|\delta S|/S<\log(1.1)/30\approx0.00318$. An accurate prefactor requires controlling the action on the scale of $\hbar$, not merely relative to the much larger action itself.

References

• Coleman, S. Aspects of Symmetry. Cambridge University Press (1985), “The Uses of Instantons.” Euclidean saddles, collective coordinates, and repeated tunneling events.
• Garg, A. “Tunnel splittings for one-dimensional potential wells revisited.” American Journal of Physics 68, 430–437 (2000). doi:10.1119/1.19458. Comparison of WKB and instanton splitting formulas, including prefactors.
• Zinn-Justin, J. Path Integrals in Quantum Mechanics. Oxford University Press (2005). Imaginary-time kernels, semiclassical expansions, and large-order behavior.