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.

A minimal example makes the sector geometry explicit. Take $\epsilon>0$, $A>0$, $r>0$, and

$$\Delta S=A r e^{i\theta}.$$

The candidate Stokes rays are

$$\sin\theta=0 \quad\Longrightarrow\quad \theta=0,\pi,$$

provided a connecting flow actually exists. Equal exponential magnitude instead occurs on

$$\cos\theta=0 \quad\Longrightarrow\quad \theta=\pm\frac{\pi}{2}.$$

Because $\Delta S=S_\tau-S_\sigma$, the $\tau$ sector is more suppressed and $\sigma$ dominates when $\cos\theta>0$; $\tau$ dominates when $\cos\theta<0$. Thus phase-alignment rays and dominance-sector boundaries are rotated by $\pi/2$ in this example. The action equations locate candidate rays, but only the global flow and contour data decide whether a thimble jump occurs there.

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.

Berry smoothing does not turn the integer thimble coordinates into continuous functions: the relative-homology basis and its integer coordinates still jump across the Stokes ray. The smooth object is the effective Stokes multiplier of a subdominant exponential after the dominant series has been optimally truncated and exponentially improved. If $\mathfrak F=\Delta S/\epsilon$ is the singulant with $\operatorname{Re}\mathfrak F>0$, its local transition has the schematic form

$$\mathfrak s_{\rm eff} \simeq \mathfrak s_- +\frac{\mathfrak s_+-\mathfrak s_-}{2} \operatorname{erfc}\!\left( -s\,\frac{\operatorname{Im}\mathfrak F} {\sqrt{2\operatorname{Re}\mathfrak F}} \right), \qquad s=\pm1,$$

where $s$ records the crossing orientation. The angular boundary layer has width $O(|\mathfrak F|^{-1/2})$; outside it the effective multiplier approaches the usual step-function values. This is a statement about an exponentially improved asymptotic representation, not about smoothing the underlying homological jump. Berry 1989, pp. 7–21 derives this universal error-function behavior 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. At $h=0$, uniform WKB and instanton quantization determine the tunneling splitting for degenerate minima and organize its multi-instanton corrections; see Dunne and Ünsal 2014, §§II–III, pp. 2–18. That result supplies the symmetric-well origin and transseries structure of $\kappa$. It does not by itself derive the biased avoided crossing: once the $h\sigma_z$ term is assumed, the displayed $h$ dependence follows directly from diagonalizing the stated two-state Hamiltonian.

Complexifying $h$ exposes the associated branch points at

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

These are exceptional points of the complexified two-state Hamiltonian: both eigenvalues and eigenvectors coalesce there. They set the radius of convergence of the Taylor series about $h=0$ and approach the origin as $g\to0$. They are not, by themselves, Picard–Lefschetz Stokes walls of the underlying path integral. Establishing such a wall requires the complex classical solutions, their action differences, a phase-alignment condition, and an actual connecting flow. The exceptional points diagnose the analyticity of the effective two-state model; the saddle-flow analysis is a separate calculation.

From finite-volume mixing to field-theory branches

The same algebra gives a useful bridge to a QFT with two phase-like states in a finite spatial volume $V$. Let $m>0$ be the magnitude of an order-parameter density, let $h$ be its source, and suppose a barrier-crossing saddle produces an off-diagonal mixing energy $\delta(V)$. The two-state Hamiltonian is

$$H_{\rm eff}(V,h) =V e_0\mathbf1-\delta(V)\sigma_x+Vmh\,\sigma_z.$$

Its lower energy density is

$$e_-(V,h) :=\frac{E_-(V,h)}{V} =e_0-\sqrt{(mh)^2+\left[\frac{\delta(V)}{V}\right]^2}.$$

If

$$\delta(V)\sim e^{-\sigma V}$$

up to a subexponential prefactor, every finite-$V$ energy is analytic for real $h$, but the mixing disappears in the thermodynamic limit. The source and volume limits then do not commute:

$$\lim_{V\to\infty}\lim_{h\to0} \frac{\partial e_-}{\partial h}=0, \qquad \lim_{h\to0^\pm}\lim_{V\to\infty} \frac{\partial e_-}{\partial h}=\mp m.$$

The limiting density $e_0-|mh|$ is a branch envelope, whereas the finite-volume ground state remains the smooth mixed state. This exponential two-state model applies only after the theory’s boundary conditions and a mixing saddle with action proportional to $V$ have been established; conserved sectors or a continuous-symmetry tower of states require different finite-size scaling. Tunneling, Superselection, and the Infinite-Volume Limit develops the corresponding state, clustering, and order-of-limits questions.

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.

Calling every complex exceptional point a Stokes wall. An exceptional point limits analytic continuation of an effective spectrum. A Picard–Lefschetz wall additionally requires phase-aligned saddle actions and a connecting gradient flow for the declared integration problem.

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.

4. Evidence analysis. A paper claims: “The cusp in the limiting vacuum-energy density proves that the finite-volume path integral undergoes a Picard–Lefschetz Stokes jump.” Build an assumption graph that separates the finite-volume spectral statement, the thermodynamic-limit statement, and the thimble statement. Identify which implication is invalid and assign an evidence or source requirement to every nontrivial edge.

Solution

One minimal assumption graph has the following nodes:

• $A$: a regulated finite-volume theory has a fixed contour or boundary prescription;
• $B$: its two lowest states are described by the displayed two-state Hamiltonian with $\delta(V)>0$;
• $C$: a model-specific calculation establishes $\delta(V)\sim e^{-\sigma V}$ up to a subexponential prefactor;
• $D$: the thermodynamic limit is taken before the source is removed;
• $E$: the limiting energy density is $e_0-|mh|$ and has a cusp at $h=0$;
• $F$: two complex saddles have phase-aligned actions and a connecting gradient flow for the declared contour;
• $G$: the oriented thimble basis and its integer coordinates undergo the corresponding Picard–Lefschetz transformation.

The valid spectral chain is

$$A+B\longrightarrow \text{finite-volume analyticity}, \qquad B+C+D\longrightarrow E.$$

The first arrow is exact two-by-two diagonalization conditional on the effective description. The second also needs a model-specific saddle or finite-volume calculation for $C$ and an explicit declaration of the limit order $D$; the superselection page develops the state and clustering interpretation.

The thimble chain is instead

$$F\longrightarrow G,$$

and requires an action-difference calculation, oriented flow analysis, and intersection numbers, as in the Witten reference and the preceding thimble page. The invalid edge is

$$E\not\Longrightarrow F \quad\text{and therefore}\quad E\not\Longrightarrow G.$$

A cusp can arise from a nonuniform infinite-volume limit of an analytic finite-volume spectrum without any thimble jump at finite volume. Conversely, a thimble basis can jump while the exact observable remains analytic and has no thermodynamic cusp. Berry smoothing supports only the exponentially improved asymptotic multiplier near an already identified Stokes ray; it does not supply the missing flow evidence. The research claim is therefore conditional on two independent evidence branches, spectral/volume and saddle/contour, which must not be merged without an additional model-specific argument.

Continue from a Stokes diagram to an observable

• To combine the contributing sectors with renormalized insertions and a requested accuracy, continue to Renormalized Saddle Contributions and Validity Tests.
• To reconstruct lateral sums and match their ambiguities to correlated saddles, continue to Resurgence, Transseries, and Large-Order Structure.
• To study branch competition in a field-theoretic vacuum problem, return to Theta Dependence, CP, and Model-Dependent Branches, keeping a finite-system avoided crossing distinct from an infinite-volume phase transition.

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.