NNS Drift-Decay Test in Complex Langevin Dynamics

• The paper demonstrates that the drift-decay test ensures correctness in complex Langevin dynamics by requiring an exponential decay of the drift magnitude distribution.
• It applies a methodology where the maximal local drift from both real and imaginary components is measured and histogrammed to assess simulation convergence.
• Empirical results in the 3D XY model show that while μ² ≤ 0 produces exponential decay indicating proper convergence, μ² > 0 reveals power-law tails that signal potential errors.

Searching arXiv for the original Nagata–Nishimura–Shimasaki drift criterion papers and related complex Langevin diagnostics. The Nagata–Nishimura–Shimasaki drift-decay test is a correctness criterion for complex Langevin dynamics (CLD) that evaluates whether the probability distribution of the drift magnitude is sufficiently suppressed at large values to support the formal justification of the stochastic process. In the formulation used for the three-dimensional XY model at finite chemical potential, the test is implemented by measuring the distribution $P(u)$ of the lattice-wide maximal local drift magnitude $u$ along the Langevin history and checking whether its tail decays at least exponentially (Joseph et al., 16 Sep 2025). The criterion is motivated by the fact that, for complex actions, CLD can remain numerically stable yet still converge to an incorrect distribution if the integration-by-parts steps underlying the Fokker–Planck argument fail. The original criterion is attributed to Nagata, Nishimura, and Shimasaki, and is cited in later work as an established practical diagnostic for CLD correctness (Joseph et al., 16 Sep 2025).

1. Definition within complex Langevin dynamics

Complex Langevin dynamics evolves complexified fields in a fictitious Langevin time $\theta$. For a field $\phi_x$, the Langevin equation is written as

$$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$

with Gaussian noise $\eta_x(\theta)$ (Joseph et al., 16 Sep 2025). When the action is complex, the field is complexified as

$\phi \to \phi^{\rm R} + i \phi^{\rm I},$

and the evolution becomes a coupled stochastic process for the real and imaginary parts. In the notation reproduced in the 3D XY-model study,

$\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$

$\frac{\partial \phi_{x}^{\rm I}}{\partial \theta} = K_{x}^{\rm I} + \sqrt{N_{\rm I}} \eta^{\rm I}_x, \qquad K_{x}^{\rm I} = - \mbox{Im} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}} .$

The drift components $K_x^{\rm R}$ and $u$0 are therefore the basic quantities entering the test (Joseph et al., 16 Sep 2025).

The need for a separate reliability diagnostic arises because the standard arguments that ensure correct convergence for real actions do not automatically extend to complex actions. The formal derivation relies on holomorphicity and, crucially, on integration by parts without boundary contributions. The drift-decay test is designed to detect whether the stochastic process explores regions of the complexified configuration space where those assumptions are likely to break down (Joseph et al., 16 Sep 2025).

2. Mathematical content of the drift-decay criterion

In the implementation described for the 3D XY model, the drift magnitude variable is defined as

$u$1

This is the lattice-wide maximal local drift magnitude extracted from a given Langevin configuration (Joseph et al., 16 Sep 2025). The probability distribution $u$2 is then measured from the Langevin-time history.

The Nagata–Nishimura–Shimasaki criterion is the asymptotic condition

$u$3

Accordingly, the tail of the drift distribution must decay at least exponentially for large $u$4. A slower decay, such as a power law, is interpreted as a violation of the criterion and as a signal that the simulation may be converging to an incorrect distribution (Joseph et al., 16 Sep 2025).

In practical terms, the procedure consists of defining the drift components, forming the maximal magnitude $u$5, histogramming $u$6 over the Langevin history to obtain $u$7, and inspecting the large-$u$8 tail. This is not presented merely as an empirical rule. The criterion is explicitly linked to the validity of integration by parts in the derivation of the Fokker–Planck equation and hence to the correctness of the stochastic evolution (Joseph et al., 16 Sep 2025).

A plausible implication is that the test should be understood as a reliability condition on the sampled distribution rather than as a diagnostic of ordinary numerical instability. The later XY-model study emphasizes precisely this distinction: CLD can converge incorrectly even when no obvious instability is present (Joseph et al., 16 Sep 2025).

3. Drift structure in the 3D XY model

The 3D XY-model application provides a concrete realization of the test. The model is defined by

$u$9

on a 3D cubic lattice with temporal direction $\theta$0 and spatial directions $\theta$1 (Joseph et al., 16 Sep 2025).

After complexification, the real and imaginary drift components are written explicitly as

$\theta$2

$\theta$3

These expressions make clear why large drifts are diagnostically important: the $\theta$4 and $\theta$5 factors are unbounded when $\theta$6. The paper explicitly states that “the drift terms become unbounded when $\theta$7” (Joseph et al., 16 Sep 2025).

This structure ties the test directly to the geometry of the complexified field space. Large imaginary excursions can generate large drifts, and a heavy-tailed $\theta$8 indicates that such excursions are not being sufficiently suppressed. That connection is central to the interpretation of the test as a proxy for whether the formal CLD justification remains plausible (Joseph et al., 16 Sep 2025).

4. Numerical implementation and operational workflow

The 3D XY-model study uses Euler-discretized Langevin evolution with adaptive step size,

$\theta$9

$\phi_x$0

with real noise satisfying

$\phi_x$1

To stabilize the evolution, the adaptive step size is

$\phi_x$2

where

$\phi_x$3

The adaptive integrator and the drift-decay test are thus closely connected, since both are organized around the magnitude of the drift (Joseph et al., 16 Sep 2025).

For the explicit XY-model study, the simulations are carried out on an $\phi_x$4 lattice with cold starts $\phi_x$5, $\phi_x$6 thermalization steps, $\phi_x$7 Langevin updates, measurements every 100 steps, and target adaptive step size $\phi_x$8. The couplings studied are

$\phi_x$9

spanning both sides of the $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$0 critical point

$$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$1

The study examines both imaginary chemical potential, $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$2, where the action is real, and real chemical potential, $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$3, where CLD is required (Joseph et al., 16 Sep 2025).

This suggests an operational characterization of the test: it is not an abstract theorem-check but a histogram-based diagnostic embedded directly in the production workflow of a CLD simulation.

5. Empirical behavior in the 3D XY model

The principal numerical result reported for the drift-decay test is sharply dichotomous. For $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$4, the drift distributions exhibit clear exponential fall-off. For $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$5, the distributions do not decay exponentially in the explored parameter sets and are described instead as having power-law tails (Joseph et al., 16 Sep 2025). The figure captions summarize this as: “The fall-off is exponential for $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$6 and power-law for $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$7” (Joseph et al., 16 Sep 2025).

The paper interprets the $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$8 behavior as consistent with correct convergence, while the $$\frac{\partial \phi_{x}(\theta)}{\partial \theta} = - \frac{\delta S[\phi; \theta]}{\delta \phi_{x}(\theta)} + \eta_x(\theta),$$9 behavior is taken to signal violation of the drift criterion and possible convergence to incorrect distributions despite the absence of obvious numerical instabilities (Joseph et al., 16 Sep 2025). In this application, the test is therefore highly discriminating between the real-action and complex-action regimes.

At the same time, observable-based diagnostics yield a more differentiated picture. The real part of the action density $\eta_x(\theta)$0, plotted as a function of $\eta_x(\theta)$1, is smooth across $\eta_x(\theta)$2 at large $\eta_x(\theta)$3, but develops a discontinuity across $\eta_x(\theta)$4 at smaller $\eta_x(\theta)$5. This is summarized as CLD succeeding in the ordered phase at large $\eta_x(\theta)$6 and failing systematically for $\eta_x(\theta)$7 in the disordered phase (Joseph et al., 16 Sep 2025). By contrast, the drift-decay criterion appears to flag all explored $\eta_x(\theta)$8 cases as suspect.

A plausible implication is that, in this study, the drift-decay test behaves as a stricter or more conservative diagnostic than the action-density continuity check. The paper does not formally label it “conservative,” but that interpretation follows from the juxtaposition of the two diagnostics (Joseph et al., 16 Sep 2025).

6. Relation to other diagnostics and interpretive limits

A major contribution of the 2025 XY-model study is its comparison of the Nagata–Nishimura–Shimasaki test with a configurational temperature, or configurational coupling, estimator $\eta_x(\theta)$9. The paper reports that for $\phi \to \phi^{\rm R} + i \phi^{\rm I},$0, $\phi \to \phi^{\rm R} + i \phi^{\rm I},$1 closely tracks the input $\phi \to \phi^{\rm R} + i \phi^{\rm I},$2, whereas for $\phi \to \phi^{\rm R} + i \phi^{\rm I},$3, $\phi \to \phi^{\rm R} + i \phi^{\rm I},$4 deviates from $\phi \to \phi^{\rm R} + i \phi^{\rm I},$5, indicating thermodynamic inconsistency and suggesting convergence to an incorrect distribution (Joseph et al., 16 Sep 2025). No excursion problems were observed, so the mismatch is interpreted as wrong-ensemble sampling rather than runaway instability.

The two diagnostics are described as “complementary,” and the discussion states that both the drift-decay test and the configurational coupling estimator “flag the same regions of failure” (Joseph et al., 16 Sep 2025). The difference lies in interpretive frame. The drift-decay test is mathematically motivated through the tail behavior of the drift and the validity of integration by parts, whereas the configurational estimator is “directly tied to physical observables” and rooted in “thermodynamic consistency” (Joseph et al., 16 Sep 2025).

The paper also attributes additional scope to the configurational estimator, stating that it can detect mis-scaled noise, step-size artifacts, and thermalization issues, and citing earlier work for the claim that it can identify algorithmic issues early in regions where the drift criterion may fail to signal problems (Joseph et al., 16 Sep 2025). Within the XY-model paper itself, however, the concrete comparison is mainly that the NNS test offers a mathematically crisp tail classification while $\phi \to \phi^{\rm R} + i \phi^{\rm I},$6 provides a more physically transparent account of how the sampled distribution is wrong.

The interpretive limits of the drift-decay test are stated with some care. The paper treats the exponential-decay condition as necessary for the standard formal justification of CLD because it supports the required integration-by-parts step. It does not, however, claim a full theorem of sufficiency for correct convergence in all practical settings. It also explicitly presents the test as incomplete on its own, which is one of the main motivations for introducing an additional physics-based diagnostic (Joseph et al., 16 Sep 2025).

7. Context, scope, and common confusions

The original NNS works cited in the 3D XY-model study are Nagata, Nishimura, and Shimasaki, Phys. Rev. D 94 (2016) 114515 and JHEP 05 (2018) 004, summarized there as establishing the drift-tail criterion that the distribution of the drift magnitude must decay at least exponentially (Joseph et al., 16 Sep 2025). The later study does not revisit the original derivations in detail; instead, it adopts the criterion as an established practical tool.

Within the XY-model application, the test also helps separate CLD failure from sign-problem severity. For small $\phi \to \phi^{\rm R} + i \phi^{\rm I},$7, the strong-coupling expansion gives

$\phi \to \phi^{\rm R} + i \phi^{\rm I},$8

with numerical values

$\phi \to \phi^{\rm R} + i \phi^{\rm I},$9

$\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$0

For the phase factor at $\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$1, $\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$2, the study estimates

$\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$3

These values are used to argue that the sign problem is only mild in that regime, even though CLD still fails in the disordered phase (Joseph et al., 16 Sep 2025). The drift-decay test is therefore interpreted as diagnosing a problem in the CLD sampling process itself rather than merely reflecting a severe sign problem.

A common confusion arises from the word “drift.” In the present context, “drift” refers to the deterministic part of the complex Langevin evolution generated by the complexified action derivative. It is unrelated to the $\frac{\partial \phi_{x}^{\rm R}}{\partial \theta} = K_{x}^{\rm R} + \sqrt{N_{\rm R}} \eta^{\rm R}_{x}, \qquad K_{x}^{\rm R} = - \mbox{Re} \frac{\delta S}{\delta \phi_{x}} \Big|_{\phi \to \phi^{\rm R} + i \phi^{\rm I}},$4 drift effect used in charged-particle momentum spectroscopy, such as in NoMoS, which concerns guiding-center motion in curved magnetic fields and does not discuss any procedure called the Nagata–Nishimura–Shimasaki drift-decay test (Moser et al., 2019). The two uses of the term belong to different technical domains.

Taken in its modern usage, the Nagata–Nishimura–Shimasaki drift-decay test is best understood as a mathematically motivated reliability criterion for CLD that probes whether large-drift excursions are sufficiently suppressed. In the 3D XY model, it robustly distinguishes real-action from complex-action regimes, aligns broadly with a thermodynamic consistency diagnostic, and supports the broader conclusion that CLD can fail even when the sign problem is mild and no overt numerical instability is visible (Joseph et al., 16 Sep 2025).