Complex Langevin Dynamics

• Complex Langevin dynamics is a stochastic quantisation method that complexifies real degrees of freedom to simulate systems with a complex action.
• The approach uses discretized Euler–Maruyama updates, adaptive step sizing, and gauge cooling to stabilize the evolution and control singular drifts.
• Its validity depends on strict criteria such as holomorphicity, localization of the probability distribution, and the absence of problematic boundary terms.

Complex Langevin dynamics (CLD) is a stochastic-quantisation method for theories with a complex action, in which the original real degrees of freedom are complexified and evolved in a fictitious Langevin time instead of being sampled by importance sampling. In the standard formulation one replaces $x\in\mathbb{R}^n$ by $z=x+iy\in\mathbb{C}^n$, evolves $dz=K(z)\,dt+d\eta$ with complex drift $K(z)=-\partial_z S(z)$, and computes holomorphic observables as long-time averages over the induced real probability distribution on the complexified configuration space (Aarts et al., 2011, Durakovic et al., 2014). The method is intended for systems with a sign problem, but its validity depends on holomorphicity, localization of the sampled distribution, and the absence of dangerous boundary terms or singular-drift effects (Aarts et al., 2013, Aarts et al., 2017).

1. Stochastic formulation and complexification

For a lattice field theory with action $S[\phi]$, CLD introduces a fictitious Langevin time $t$ and evolves each degree of freedom according to

$$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$

with real Wiener increments satisfying $\langle dW_x(t)\rangle=0$ and $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$ (Joseph et al., 16 Sep 2025). When the action is complex, the fields must be complexified,

$\phi_x\rightarrow \phi_x^R+i\phi_x^I,$

and the evolution splits into coupled real equations for $z=x+iy\in\mathbb{C}^n$0 and $z=x+iy\in\mathbb{C}^n$1, with drift components given by the real and imaginary parts of $z=x+iy\in\mathbb{C}^n$2 evaluated on the complexified fields (Aarts et al., 2010, Joseph et al., 16 Sep 2025).

In practice one usually discretizes Langevin time by an Euler–Maruyama update. In the notation used for the three-dimensional XY model,

$z=x+iy\in\mathbb{C}^n$3

with independent Gaussian noise of zero mean and variance $z=x+iy\in\mathbb{C}^n$4 (Aarts et al., 2010). Closely related discrete updates appear in chiral random matrix theory, where each real matrix component $z=x+iy\in\mathbb{C}^n$5 is updated as

$z=x+iy\in\mathbb{C}^n$6

(Mollgaard et al., 2013).

The same structure extends to gauge theories and matrix models. For gauge links $z=x+iy\in\mathbb{C}^n$7, CLD is written directly on the complexified group manifold, while for unitary matrix models one often evolves the eigenvalue angles $z=x+iy\in\mathbb{C}^n$8 into the complex plane, so that the eigenvalues $z=x+iy\in\mathbb{C}^n$9 leave the unit circle and explore $dz=K(z)\,dt+d\eta$0 (Bongiovanni et al., 2013, Basu et al., 2018). This makes CLD fundamentally different from reweighting or dual-variable reformulations: it replaces sampling of a complex measure by sampling of a real distribution in a higher-dimensional complexified space.

2. Fokker–Planck structure and criteria for correctness

The formal basis of CLD is a real Fokker–Planck equation for the probability density $dz=K(z)\,dt+d\eta$1 of the complexified process. In the one-variable case the density satisfies

$dz=K(z)\,dt+d\eta$2

and observables are computed as

$dz=K(z)\,dt+d\eta$3

(Aarts et al., 2011). A corresponding “complex density” $dz=K(z)\,dt+d\eta$4 on the original real manifold has stationary solution $dz=K(z)\,dt+d\eta$5, and the formal argument attempts to show equality of $dz=K(z)\,dt+d\eta$6 and $dz=K(z)\,dt+d\eta$7 for holomorphic observables by transferring time evolution from the density to the observable and integrating by parts (Aarts et al., 2011, Joseph et al., 16 Sep 2025).

The critical point is that this proof requires vanishing boundary terms. A necessary and sufficient condition for correctness is that all such boundary terms vanish; equivalently, at stationarity one must have

$dz=K(z)\,dt+d\eta$8

for every holomorphic observable $dz=K(z)\,dt+d\eta$9 (Aarts et al., 2011). In the related consistency-condition language, one defines

$K(z)=-\partial_z S(z)$0

and requires $K(z)=-\partial_z S(z)$1 for a sufficiently large basis of observables (Aarts et al., 2013, Giudice et al., 2013).

Localization properties of $K(z)=-\partial_z S(z)$2 are therefore central. In the quartic toy model

$K(z)=-\partial_z S(z)$3

it was shown that for real noise and $K(z)=-\partial_z S(z)$4, the equilibrium distribution has strict strip support,

$K(z)=-\partial_z S(z)$5

with an explicit $K(z)=-\partial_z S(z)$6, and in that regime correct results are expected because boundary terms vanish (Giudice et al., 2013). By contrast, when no such strip exists, power-law tails appear and the formal justification breaks down (Aarts et al., 2013, Giudice et al., 2013).

This framework also clarifies why CLD is not guaranteed merely by numerical stability. A trajectory may remain bounded and still sample a distribution whose tails or singular regions invalidate integration by parts. Conversely, a localized distribution with vanishing $K(z)=-\partial_z S(z)$7 gives a concrete a posteriori basis for trusting the method (Aarts et al., 2011, Aarts et al., 2013).

3. Singular drifts, logarithms, and mechanisms of wrong convergence

The most persistent obstructions to correctness are slow decay in imaginary directions, singular drifts generated by zeros of determinants, and multivalued logarithms. In theories with

$K(z)=-\partial_z S(z)$8

the drift contains

$K(z)=-\partial_z S(z)$9

so zeros of $S[\phi]$0 induce poles in the drift and make it meromorphic rather than holomorphic (Aarts et al., 2017). In that case the usual integration-by-parts argument acquires extra boundary terms around small neighborhoods of the poles, and correctness requires that the probability distribution be negligible near the determinant zeros (Aarts et al., 2017).

A particularly transparent example is chiral random matrix theory at nonzero chemical potential. There the action contains $S[\phi]$1, and incorrect convergence occurs for small quark masses when the determinant frequently traces out a path surrounding the origin of the complex plane during the Langevin flow (Mollgaard et al., 2013). The study identifies the failed region through scatter plots of $S[\phi]$2 and through the quantity $S[\phi]$3, the fraction of configurations with $S[\phi]$4. As soon as $S[\phi]$5, the CLD results for the condensate $S[\phi]$6 and baryon density $S[\phi]$7 deviate strongly from the exact answer, and numerically they become indistinguishable from the phase-quenched theory (Mollgaard et al., 2013). The immediate cause is an ambiguity in the drift: using $S[\phi]$8 everywhere ignores the branch cut, but if $S[\phi]$9 winds around the origin the phase changes by $t$0, and control over the drift is lost (Mollgaard et al., 2013).

Related issues appear in simpler one-link models. In the U(1) one-link model with effective action

$t$1

CL reproduces standard observables for $t$2 and moderate parameters, but for larger $t$3 or certain $t$4-windows the estimates deviate strongly; the stated culprit is the non-analyticity of the logarithm near its cut, which makes the drift ill-defined in regions explored by the process (Durakovic et al., 2014). The same paper also reports a subtler phenomenon: all moments $t$5 can be misestimated even when observables such as $t$6 are reproduced correctly by a Taylor reconstruction from those moments (Durakovic et al., 2014). This indicates that the real distribution $t$7 generated by CLD does not encode the original complex measure in a simple moment-by-moment way.

Boundary terms associated with poles have a distinct temporal behavior. In simple one-pole models, pole-induced boundary terms arise after finite Langevin time once the process reaches the pole neighborhood, but vanish again as Langevin time goes to infinity, whereas boundary terms at infinity may survive in the long-time limit (Seiler, 2021). This distinguishes transient singular-drift effects from the asymptotic delocalization that is usually associated with wrong convergence.

4. Diagnostics and empirical reliability tests

A practical CLD calculation is usually accompanied by explicit diagnostics that test localization, analyticity, and agreement with sign-problem-free limits. These diagnostics are not interchangeable: some probe the formal assumptions behind the Fokker–Planck argument, while others test thermodynamic consistency or continuity in control parameters.

Diagnostic	Criterion	Representative source
Continuity in $t$8	Values from $t$9 and $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$0 should coincide	(Aarts et al., 2010)
Hot/cold start at $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$1	Complexified and purely real starts should converge to the same result	(Aarts et al., 2010)
Drift-decay test	For large $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$2, $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$3	(Joseph et al., 16 Sep 2025)
Configurational temperature	$$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$4 should match the input $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$5	(Joseph et al., 16 Sep 2025)
Consistency conditions	$$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$6 for a basis of holomorphic $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$7	(Aarts et al., 2013)
Determinant winding	Monitor $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$8 and require $$d\phi_x(t)=-\frac{\partial S[\phi]}{\partial \phi_x}\,dt+dW_x(t),$$9	(Mollgaard et al., 2013)

In the three-dimensional XY model, continuity around $\langle dW_x(t)\rangle=0$0, comparison with the world-line formalism, hot/cold starts at $\langle dW_x(t)\rangle=0$1, the width

$\langle dW_x(t)\rangle=0$2

and the distribution of the maximal drift $\langle dW_x(t)\rangle=0$3 were all used to diagnose failure (Aarts et al., 2010). These tests show that CLD works in the ordered phase at larger $\langle dW_x(t)\rangle=0$4, but fails in the disordered phase at smaller $\langle dW_x(t)\rangle=0$5, and that the failure is not driven by sign-problem severity; it is already visible at $\langle dW_x(t)\rangle=0$6, where no sign problem exists (Aarts et al., 2010, Aarts et al., 2010).

The configurational-temperature diagnostic adds a thermodynamic consistency test. For the 3D XY model one defines

$\langle dW_x(t)\rangle=0$7

and correct sampling requires $\langle dW_x(t)\rangle=0$8 (Joseph et al., 16 Sep 2025). In the ordered phase, $\langle dW_x(t)\rangle=0$9 matches the input $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$0, the action density is continuous across $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$1, and the drift histograms fall exponentially. In the disordered phase at real $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$2, $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$3 deviates strongly from $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$4, the action density violates analytic continuation, and $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$5 develops power-law tails (Joseph et al., 16 Sep 2025).

Taken together, these results establish a recurrent empirical pattern: correct CLD is associated with localization, continuity, and exponentially decaying drift histograms, whereas wrong convergence is associated with broad complex excursions, singular regions, or winding of logarithmic arguments (Aarts et al., 2010, Mollgaard et al., 2013, Joseph et al., 16 Sep 2025).

5. Stabilisation, control, and algorithmic modifications

The most basic stabilization device is adaptive step sizing. In the XY-model implementation one monitors

$\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$6

and chooses

$\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$7

which suppresses runaway excursions caused by unbounded drifts in the complexified theory (Aarts et al., 2010, Aarts et al., 2010). Because the steps are unequal, observables are then averaged with Langevin-time weights $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$8 (Aarts et al., 2010). This cures the numerical problem of runaway solutions in the XY model, but it does not by itself guarantee correctness (Aarts et al., 2010).

For non-Abelian gauge theories, gauge cooling is the standard control mechanism. Since CLD evolves $\langle dW_x(t)\,dW_y(t)\rangle=2\delta_{xy}dt$9 links into $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$0, one introduces complexified gauge transformations chosen to minimize a distance from unitarity, such as

$\phi_x\rightarrow \phi_x^R+i\phi_x^I,$1

(Bongiovanni et al., 2013). Cooling steps alternate with Langevin updates, and adaptive choices of the cooling parameter,

$\phi_x\rightarrow \phi_x^R+i\phi_x^I,$2

accelerate the reduction of the unitarity norm and suppress excursions into non-compact directions (Bongiovanni et al., 2013). In toy models a force-based adaptive choice $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$3 yields a faster decay $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$4, while in full gauge theory adaptive cooling keeps $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$5 orders of magnitude smaller than without cooling (Bongiovanni et al., 2013).

The broader control strategy emphasized in reviews is to combine adaptive step size, cooling, localization diagnostics, and consistency checks. Gauge cooling is effective because it localizes the sampled distribution in the complexified gauge manifold, removes broad “skirts,” and correlates with stable observables in Polyakov-chain models and heavy-quark QCD (Aarts et al., 2013). However, gauge cooling and adaptive stepping do not remove ambiguities associated with a multi-valued logarithm; the chiral-random-matrix study states explicitly that one still needs a prescription for the logarithmic branch structure (Mollgaard et al., 2013).

Other stabilizing modifications are available. The analysis of effective models shows that kernels or coordinate transformations can generate additional restoring drift and mimic the stabilizing role of a nontrivial Haar measure (Aarts et al., 2012). In real-time CLD, modern implicit solvers replace explicit Euler updates by $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$6-schemes. For $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$7, these schemes are unconditionally stable; backward Euler–Maruyama is L-stable, and Crank–Nicolson–Maruyama is A-stable and preserves free-theory norms exactly (Alvestad et al., 2021). The same work interprets implicitness as an intrinsic regularization of the real-time path integral and shows that comparatively large Langevin time steps can be used without runaway behavior (Alvestad et al., 2021).

6. Model systems, related frameworks, and scope

The three-dimensional XY model remains a canonical test bed for CLD. Two independent studies found that CLD is reliable at larger $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$8 and fails at smaller $\phi_x\rightarrow \phi_x^R+i\phi_x^I,$9, with the boundary between good and bad behavior tracking the ordered-to-disordered transition rather than the severity of the sign problem (Aarts et al., 2010, Aarts et al., 2010). The later configurational-temperature study reached the same phase-structure conclusion and added a physics-driven reliability test that complements the Nagata–Nishimura–Shimasaki drift-decay criterion (Joseph et al., 16 Sep 2025).

The three-dimensional SU(3) spin model provides an important contrast. A criteria-for-correctness analysis found strong indications that CLD yields correct results in this theory (Aarts et al., 2011). The subsequent stability analysis argues that the difference relative to the XY model is due to the nontrivial Haar measure, which exerts a stabilizing effect on the complexified dynamics (Aarts et al., 2012). In this sense, the group manifold is not a passive kinematic background; it directly shapes the localization properties on which CLD correctness depends.

Chiral random matrix theory at nonzero chemical potential isolates the effect of logarithmic determinants and determinant winding in a setting close in spirit to finite-density QCD (Mollgaard et al., 2013). Large-$z=x+iy\in\mathbb{C}^n$00 unitary matrix models show that CLD can also reproduce analytically known phase structures, including a series of Gross-Witten-Wadia transitions, Polyakov lines, and quark number density, with excellent agreement at large $z=x+iy\in\mathbb{C}^n$01 and $z=x+iy\in\mathbb{C}^n$02 (Basu et al., 2018). In supersymmetric quantum mechanics with complex actions, CLD together with drift-distribution and Langevin-operator diagnostics suggests that the method can reliably predict the absence or presence of dynamical supersymmetry breaking when those tests are satisfied (Joseph et al., 2020).

CLD also has a close but non-identical relationship to Lefschetz-thimble methods. In quartic, U(1), and SU(2) examples, the sampled CL distribution is typically related to the contributing thimble or thimbles, but it remains a genuine two-dimensional probability distribution and may avoid repulsive saddle directions or terminate differently near singular points (Aarts et al., 2014). A thimble-motivated modification strategy goes further by altering a model so that the modified integral has a single relevant thimble, after which CLD can reproduce the exact result in cases where the naive implementation fails (Tsutsui et al., 2015). This suggests that thimble geometry can be used not only for interpretation but also for algorithm design.

Across these settings, the central lesson is consistent. CLD is neither a purely formal replacement for importance sampling nor a universally reliable black box. It is a stochastic method whose success depends on the geometry of the complexified manifold, the analytic structure of the drift, and the measured localization properties of the sampled distribution (Aarts et al., 2011, Aarts et al., 2013). Where distributions are localized, boundary terms vanish, and diagnostics are satisfied, CLD can reproduce nontrivial finite-density and real-time observables. Where trajectories probe poles, logarithmic branch cuts, or delocalized regions, the method can converge to stable but incorrect limits (Mollgaard et al., 2013, Aarts et al., 2017).