Phase unwrapping and one-dimensional sign problems

Abstract

Sign problems in path integrals arise when different field configurations contribute with different signs or phases. Phase unwrapping describes a family of signal processing techniques in which phase differences between elements of a time series are integrated to construct noncompact unwrapped phase differences. By combining phase unwrapping with a cumulant expansion, path integrals with sign problems arising from phase fluctuations can be systematically approximated as linear combinations of path integrals without sign problems. This work explores phase unwrapping in zero-plus-one-dimensional complex scalar field theory. Results with improved signal-to-noise ratios for the spectrum of scalar field theory can be obtained from unwrapped phases, but the size of cumulant expansion truncation errors is found to be undesirably sensitive to the parameters of the phase unwrapping algorithm employed. It is argued that this numerical sensitivity arises from discretization artifacts that become large when phases fluctuate close to singularities of a complex logarithm in the definition of the unwrapped phase.1 Other interesting LQCD observables face distinct StN problems. For instance, isoscalar meson correlation functions are uncharged under U (1) symmetries and possess exponential StN problems but not U (1) phase fluctuations. Excited-state energies are extracted from differences of correlation functions with the same quantum numbers and face StN problems arising from the exponentially precise cancellations needed to project out ground-state contributions and leave exponentially faster decaying excited-state contributions. In large nuclei, StN problems associated with MeV excitation energies are negligible compared to the phase fluctuation StN problem associated with the multi-GeV rest mass of the nucleus. LQCD calculations of isoscalar mesons and exotic hadrons conversely face StN problems primarily from sources besides U (1) phase fluctuations, and phase unwrapping is not immediately applicable to these systems. 2 Cumulant expansions of noncompact "extensive phases" have also been applied to sign problems in QCD and other theories at nonzero chemical potential [51][52][53][54][55][56].

Lattice QCD and neutrino-nucleus scattering

“…Perhaps more realistically, significant tests of nuclear EFT frameworks beyond the fewbody sector would be enabled by lattice-QCD calculations of the spectrum and axial structure of an intermediate nucleus such as 12 C. Aspects of coherent scattering off nuclei will also be addressed by such calculations. While still challenging, a number of groups are investigating ways to perform the relevant contractions and studying improved ways to extract signals from noisy multi-baryon data through optimization methods [184] or improved estimators [43,[185][186][187][188]. For carbon targets, experimental scattering data exists and comparison of lattice-QCD calculations with this will help understand the systematics of the A dependence of nuclear EFT approaches and assess the reliability of the extrapolations to argon.…”

Systematics in nucleon matrix element calculations

“…At the physical pion mass with ∆E = 2m π (see the next subsection), the exponent is roughly −13, much larger than the −2 that is obtained when neglecting excited states. This situation could be significantly improved if multilevel methods [11,12] or other ideas [13] are able to reduce the signal-to-noise problem.…”

Unwrapping phase fluctuations in one dimension

“…access all states of the theory, see Ref. [20] for more details. The scalar boson propagator is G(t) ≡ G 1,0 (t) and in the non-interacting case V = 0 its mass is E ≡ E 0;0 = 2 arcsinh(M/2).…”

“…The analysis above suggests that avoiding sign and StN problems is equivalent to avoiding numerical sampling of circular random variables. In this simple theory, the phase can be analytically integrated out to produce a dual theory with positive-definite path integral representations for correlation functions [20,24]. As a tractable alternative for more complicated theories where all phase variables cannot be integrated out analytically, one can imagine numerically sampling the real-valued angular displacement accumulated by the phase along [0,t] including any 2π revolutions about the unit circle.…”

“…MC ensembles were also generated using the dual form of the theory in which phase fluctuations are analytically integrated out in order to obtain precise results for verifying the accuracy of phase unwrapping; see Ref. [20] for details.…”