Stokes Jumps, Saddle Dominance, and Contour Dependence

An exact integral can be analytic while the most useful asymptotic saddle representation changes. Across a Stokes ray, the thimble basis and its integer coefficients jump in compensating ways. Across an equal-magnitude curve, the exponentially dominant saddle changes. Sharp phase-like behavior appears when a singular limit is taken before the exponentially small mixing that smooths the crossover.

Required background. Complex saddles and Lefschetz thimbles supplies the cycle decomposition and intersection numbers; multi-saddle sums and dilute ensembles supplies exponentially small level mixing; asymptotic scales, remainders, and uniformity supplies the meaning of a parameter-dependent asymptotic expansion.

Helpful background. Theta dependence, CP, and branches supplies a field-theoretic setting in which competing branches and limit order are physically consequential.

Stokes conditions and equal magnitude

For two saddle sectors,

$$F(\epsilon,\lambda) \sim n_\sigma(\lambda) e^{-S_\sigma(\lambda)/\epsilon}\Phi_\sigma(\epsilon,\lambda) +n_\tau(\lambda) e^{-S_\tau(\lambda)/\epsilon}\Phi_\tau(\epsilon,\lambda),$$

define $\Delta S=S_\tau-S_\sigma$. Two distinct conditions occur:

$$\begin{aligned} \operatorname{Im}\frac{\Delta S}{\epsilon}=0 &\quad &&\text{phase alignment},\\ \operatorname{Re}\frac{\Delta S}{\epsilon}=0 &\quad &&\text{equal exponential magnitude}. \end{aligned}$$

This chapter uses Stokes ray for a phase-alignment locus on which a connecting flow exists and the thimble basis jumps. It uses anti-Stokes curve for an equal-magnitude locus on which dominance can exchange. Some references reverse the two names, so a calculation should always state the defining equation.

On a Stokes ray, suppose

$$\mathcal J_\tau^{+} =\mathcal J_\tau^{-}+m\mathcal J_\sigma^{-}.$$

Then the coefficient of $\mathcal J_\sigma$ changes by $-mn_\tau$. The exact cycle and exact integral remain the same. What jumps is the decomposition into asymptotic sectors. Witten 2011, §3, preprint pp. 14–23 gives this relative-homology mechanism.

Shared calculation. The quartic thimble and Stokes map displays the critical points, downward and dual cycles, intersection numbers, and compensating basis change. Shared comparison. The canonical saddle comparison records the same contour requirement alongside the other saddle-control data.

At finite asymptotic parameter, optimally truncated expansions do not develop a literal discontinuity. The multiplier of a subdominant exponential changes rapidly but smoothly over a boundary layer; the step-function jump is the strict asymptotic limit. Berry 1989, pp. 7–21 derives the universal error-function smoothing for ordinary isolated saddles.

Dominance is not contribution

A saddle can contribute with nonzero $n_\sigma$ and still be exponentially subleading. It can also have the smallest real action but have $n_\sigma=0$. These are separate questions:

1. Contribution: does the original cycle contain the saddle’s thimble?
2. Dominance: among contributing sectors, which has the smallest $\operatorname{Re}(S/\epsilon)$?
3. Accuracy: is a subdominant sector larger than the stated remainder?

Near an anti-Stokes curve, two retained sectors must be compared uniformly. If

$$\operatorname{Re}\frac{\Delta S}{\epsilon}=O(1),$$

neither exponential is uniformly negligible. Near a saddle coalescence, even a two-exponential sum can fail because the separate Gaussian expansions diverge; a canonical Airy-, Pearcey-, or higher uniform approximation is then required.

Tilted double well and a smoothed crossing

The dilute-instanton calculation reduces the two lowest double-well states to

$$H_{\rm eff} =E_{\rm pert}\mathbf1-\kappa\sigma_x+h\sigma_z,$$

where $2h$ is the perturbative energy bias between the localized wells and $\kappa>0$ is the instanton mixing amplitude. The exact eigenvalues of this two-state problem are

$$E_\pm(h) =E_{\rm pert}\pm\sqrt{h^2+\kappa^2}.$$

For every fixed $\kappa>0$, the levels are analytic on the real $h$ axis and exhibit an avoided crossing with minimum gap $2\kappa$. The ground-state polarization is

$$\langle\sigma_z\rangle_- =-\frac{h}{\sqrt{h^2+\kappa^2}},$$

which changes smoothly over $|h|\sim\kappa$.

If one drops the exponentially small instanton sector first, $\kappa\to0$, then

$$E_-(h)\longrightarrow E_{\rm pert}-|h|.$$

The resulting cusp is the lower envelope of two competing localized saddles. It is not a singularity of the finite-$g$ quantum-mechanical spectrum. The limits are nonuniform:

$$\lim_{\kappa\to0^+}\lim_{h\to0} \frac{\partial E_-}{\partial h}=0, \qquad \lim_{h\to0^\pm}\lim_{\kappa\to0^+} \frac{\partial E_-}{\partial h}=\mp1.$$

Since $\kappa\sim e^{-\mathcal S_I/g}$, resolving the crossover requires retaining an effect beyond every finite perturbative order. Uniform WKB and instanton quantization reproduce this avoided crossing and organize its multi-instanton corrections; see Dunne and Ünsal 2014, §§II–III, pp. 2–18.

Complexifying $h$ exposes the associated branch points at

$$h=\pm i\kappa.$$

They approach the origin as $g\to0$. In the limiting asymptotic description, Stokes curves emanating from these nearby singularities control how the subdominant saddle and its fluctuations turn on. This explains how a smooth finite-$g$ observable can approach a piecewise saddle envelope.

Contour dependence and lateral answers

If the original integral is convergent throughout a parameter domain, analytic continuation of that same cycle gives one exact function there. Its thimble coordinates may be piecewise constant and jump at Stokes rays.

If instead the parameter crosses into a region where the original real contour ceases to converge, the upper and lower lateral deformations can define distinct analytic continuations:

$$F_+(\epsilon)-F_-(\epsilon) \sim C\,e^{-A/\epsilon}.$$

This difference is not a contradiction; $F_+$ and $F_-$ are different prescriptions. In a complete transseries, a corresponding change in the nonperturbative parameter can cancel the lateral ambiguity in a real observable. One must state which contour or lateral prescription is being used.

A uniform decision procedure

When parameters vary, use the following order:

1. continue the exact contour or boundary prescription;
2. track all critical points and singularities without yet discarding sectors;
3. test phase alignment and the existence of connecting flows;
4. update the thimble basis and intersection numbers together;
5. compare real actions among contributing sectors;
6. retain every sector above the requested error threshold;
7. replace separate saddle series by a uniform approximation near coalescence.

Failure at step 1 cannot be repaired by local saddle algebra. Failure at step 7 often appears as a divergent prefactor, a vanishing Hessian eigenvalue, or rapid loss of numerical accuracy.

Common pitfalls

Calling a dominance exchange a Stokes jump. Equal magnitude concerns $\operatorname{Re}\Delta S$; a thimble jump concerns phase alignment, a connecting flow, and intersection data. They can occur on different curves.

Interpreting a limiting cusp as an exact finite-parameter singularity. The tilted double well is smooth for $\kappa>0$. The cusp appears only after exponentially small mixing is discarded.

Changing coefficients without changing the thimble basis. This produces a spurious discontinuity in the exact integral. Basis and coordinates transform inversely.

Exercises

1. Determine the width of the avoided-crossing region in the tilted double well.

Solution

Mixing is important when the diagonal bias and off-diagonal matrix element are comparable:

$$|h|\lesssim\kappa.$$

Equivalently, the polarization $|h|/\sqrt{h^2+\kappa^2}$ is not close to one in this region. Since $\kappa$ is exponentially small in $g$, the crossover is invisible on any fixed algebraic scale in $g$.

2. Verify the two noncommuting derivative limits displayed above.

Solution

For $\kappa>0$,

$$\frac{\partial E_-}{\partial h} =-\frac{h}{\sqrt{h^2+\kappa^2}},$$

so setting $h=0$ first gives zero for every $\kappa$. If $\kappa\to0$ first at fixed nonzero $h$, the derivative becomes $-\operatorname{sgn}h$, whose one-sided limits are $\mp1$.

3. Suppose $\operatorname{Re}(\Delta S/\epsilon)=5$. Estimate the exponential ratio and explain why this alone does not decide whether the second saddle is present.

Solution

The magnitude ratio is $e^{-5}\simeq6.7\times10^{-3}$, before prefactors. This determines relative suppression only if both intersection numbers are nonzero. A vanishing intersection number removes the second saddle from that contour; an anomalously large prefactor or a requested accuracy below $10^{-2}$ can require retaining it.

References

• Berry, Michael V. “Uniform Asymptotic Smoothing of Stokes’s Discontinuities.” Proceedings of the Royal Society A 422 (1989): 7–21. DOI.
• Dunne, Gerald V., and Mithat Ünsal. “Uniform WKB, Multi-Instantons, and Resurgent Trans-Series.” Physical Review D 89 (2014): 105009. DOI.
• Witten, Edward. “Analytic Continuation of Chern–Simons Theory.” In Chern–Simons Gauge Theory: 20 Years After, AMS/IP Studies in Advanced Mathematics 50 (2011): 347–446. arXiv:1001.2933.