Diagnose phase cancellation before claiming a finite-density result

A common lattice Dirac operator satisfies a γ5-Hermiticity relation connecting chemical potentials μ and minus its complex conjugate. At μ=0 this supports reality properties, with suitable flavor pairing and positive masses giving a nonnegative weight. For real nonzero baryon chemical potential, the determinant of one configuration need not be real or positive. The relevant dimensionless simulation parameter is aμ, while comparisons across temperature often use μ/T with chemical potential and temperature in the same energy units.

Replacing det D by its magnitude is phase quenching. For paired degenerate flavors this can correspond to opposite chemical potentials, an isospin rather than baryon ensemble. That auxiliary theory can have different phases and thresholds. It is therefore a useful sampling construction only if the omitted phase is restored or the changed physical question is explicitly intended. A positive auxiliary weight is not a proof that the target baryonic sign problem has disappeared.

Define expectations with respect to a normalized nonnegative phase-quenched ensemble. The target observable is the phase-weighted numerator divided by the average phase. A small denominator amplifies uncertainty and makes ordinary Gaussian error estimates delicate, especially when its sampling distribution has long tails or poor overlap. Numerator and denominator are correlated and must be analyzed together. A run can miss rare configurations important for the target even if the auxiliary observables appear equilibrated.

For a large-volume regime with a free-energy-density difference between target and auxiliary theories, the mean phase can fall exponentially with volume over temperature. Keeping fixed relative precision then commonly requires exponentially more independent measurements. This is a regime statement and a warning about scaling, not a claim that every theory or parameter point is equally hard. Additional autocorrelation reduces the independent count before phase cancellation is considered.

Taylor expansions about μ=0 and continuation from imaginary chemical potential can provide information within an analytically connected domain where truncation and continuation errors are tested. Nearby singularities limit that domain. Reweighting can be useful where overlap remains adequate. Special matter content or chemical-potential assignments can admit sign-free measures, while complex-Langevin or contour methods require their own correctness conditions and diagnostics. None is certified for an arbitrary baryon-density problem merely by reproducing a few small-volume points.

The probability lab illustrates binary sign cancellation only. Reset and set probability to 0.75 and trials to 4; predict mean positive-sign count 3. Reset and set probability to 0.5 and trials to 4; predict mean count 2. Assigning each positive count sign +1 and each remaining count −1 gives mean average signs 0.5 and 0. The lab displays exact binomial bars rather than samples, and has no Dirac determinant, complex phase, reweighting estimator, or finite-density gauge ensemble.