In lattice field theories at finite density the partition function can be written as a one-dimensional Fourier transform of the density of states ρ(s)\rho(s) of the phase ss of the fermion determinant. The Linear Logarithmic Relaxation (LLR) algorithm determines ρ(s)\rho(s) with exponential error suppression over hundreds of orders of magnitude, but the Fourier integral suffers catastrophic cancellations, and existing evaluations hinge on fitting ln⁡ρ(s)\ln\rho(s) with a low-parameter function whose success is not guaranteed. Garron and Langfeld (Eur. Phys. J. C 77, 470, 2017) proposed three alternatives — the Gaussian approximation, the telegraphic approximation, and the advanced moment expansion with universal coefficients — and left their robustness open. The open problem is to make this final integration step controlled and fit-free: establish the convergence of the advanced moment expansion for general sign-problem-afflicted theories (the exponential decay of the universal coefficients was verified numerically up to order 30 but not proven), compute the required moments directly from raw LLR data without a density fit, and determine whether any fit-free scheme works in the strong sign-problem regime where the raw-data telegraphic approximation failed.

Background

Lattice quantum chromodynamics (QCD) at finite baryon density suffers from the sign problem: at nonzero chemical potential μ\mu the fermion determinant (the determinant of the lattice Dirac operator, i.e. the discretized quark action, whose value weights each gauge-field configuration) becomes complex, so the Boltzmann weight cannot be used as a probability density for importance-sampling Monte Carlo, and no general algorithm exists (the sign problem is NP-hard in the worst case, Troyer and Wiese 2005). A standard diagnostic is the phase-quenched ensemble, in which simulations are performed with the complex phase ϕ\phi of the determinant dropped; the severity of the sign problem is then measured by the phase factor expectation value ⟨eiϕ⟩PQ=Z/ZPQ\langle e^{i\phi}\rangle_{\mathrm{PQ}} = Z/Z_{\mathrm{PQ}}, which is exponentially small in the volume. Because it is a ratio of partition functions, brute-force reweighting needs exponentially many configurations.

The density-of-states (DoS) strategy reformulates the problem: writing ss for the imaginary part of the action (the phase of the determinant), the partition function becomes Z=∫ds ρ(s) eisZ = \int ds\, \rho(s)\, e^{is}, where ρ(s)\rho(s) is the density of states of ss. The Linear Logarithmic Relaxation (LLR) method (Langfeld, Lucini, Rago, Phys. Rev. Lett. 109, 111601, 2012; refined in Eur. Phys. J. C 76, 306, 2016) is a Wang-Landau-type (flat-histogram density-of-states sampling) algorithm whose per-bin parameters are determined by Robbins-Monro stochastic approximation; it computes ln⁡ρ(s)\ln\rho(s) piecewise with exponential error suppression, making ρ(s)\rho(s) available with fixed relative accuracy over hundreds of orders of magnitude. The residual difficulty is the final Fourier integral: its integrand oscillates with cancellations as severe as the sign problem itself. In heavy-dense QCD (HDQCD — lattice QCD in the limit of static, infinitely heavy quarks, where the determinant reduces to a function of the Polyakov loops, the traced temporal Wilson lines that serve as the standard order parameters for deconfinement), Garron and Langfeld (Eur. Phys. J. C 76, 569, 2016) handled this step by fitting ln⁡ρ(s)\ln\rho(s) with a low-parameter polynomial and performing the integral semi-analytically; the same polynomial-fit route was used for dense quantum systems in Langfeld and Lucini (Phys. Rev. D 90, 094502, 2014). Garron and Langfeld (2017) call this dependence on a successful fit 'unsatisfactory', since the quest for a fit function might not always be successful.

Garron and Langfeld (Eur. Phys. J. C 77, 470, 2017) therefore tested three fit-avoiding extraction schemes on HDQCD (an 848^4 lattice, β=5.8\beta=5.8, κ=0.12\kappa=0.12), building on their PoS LATTICE2016 moment-expansion proceedings (arXiv:1611.01378). (i) The Gaussian approximation models the extensive density of states ρE(s)\rho_E(s) by a normal distribution, which yields ⟨eiϕ⟩PQ\langle e^{i\phi}\rangle_{\mathrm{PQ}} analytically but is not systematically improvable. (ii) The telegraphic approximation reduces the Fourier integral to an alternating difference of neighboring bin values of ρE\rho_E; its relative systematic error is of the order of the phase factor itself, which makes it excellent exactly in the strong sign-problem regime, but when implemented directly on raw histogrammed data (bin size δs=π/5\delta_s = \pi/5) no signal could be extracted at μ=1.2921\mu = 1.2921, where the fit-based answer is ⟨eiϕ⟩≈2.19×10−6\langle e^{i\phi}\rangle \approx 2.19\times 10^{-6}. (iii) The advanced moment expansion assumes that ρ(s)\rho(s) on [−π,π][-\pi,\pi] is well represented by a polynomial in s2s^2 and rewrites ⟨eiϕ⟩PQ\langle e^{i\phi}\rangle_{\mathrm{PQ}} as ∑nα2n+2M2n+2\sum_n \alpha_{2n+2} M_{2n+2}, where the advanced moments M2kM_{2k} are linear combinations of the elementary moments ⟨s2i⟩\langle s^{2i}\rangle with universal coefficients, and the coefficients α2n+2\alpha_{2n+2} are likewise universal (independent of the theory) and are obtained from solving a linear system — the only dependence on the theory under investigation enters via the moments. The authors solved the associated Cholesky recursion analytically up to order N0=30N_0 = 30 and found numerical evidence that α2N0\alpha_{2N_0} decays exponentially, guaranteeing fast convergence when the moments stay bounded — but no analytic proof, and the elementary moments were still computed from the fitted density rather than from raw LLR data.

Later work within the LLR program has not closed these gaps. For the finite-density Bose gas, Francesconi, Holzmann, Lucini, and Rago (Phys. Rev. D 101, 014504, 2020; see also PoS LATTICE2018, 051) showed that piecewise (raw-data) reconstruction of the phase factor fails in the strong sign-problem regime — fluctuations of order 45 magnitudes on a 10410^4 lattice — while the polynomial fit works down to O(10−480)\mathcal{O}(10^{-480}), and that alternative interpolations (L2L^2 expansions, local fits, Padé) all perform worse; they note it is unclear whether the polynomial's success generalizes. Körner, Langfeld, Smith, and von Smekal (Phys. Rev. D 102, 054502, 2020) applied LLR to the repulsive Hubbard model on the honeycomb lattice using different, Fourier-mode-based reconstruction schemes, and explicitly list the Gaussian and telegraphic approximations and the advanced moments approach of the 2017 paper as unapplied future options. Buividovich and Ostmeyer (Phys. Rev. B 107, 024302) stress-tested the DoS approach on real-time quantum spin chains and confirmed that the one-dimensional oscillatory integral over the density of states remains the bottleneck, with smooth approximations to ρ\rho being the ingredient that can make or break the method. The fit-free, provably convergent extraction of the phase factor from LLR data is thus the recognized open step of this route to finite-density field theory.

Problem Statement

The overarching objective is to make the final, fit-free extraction of the phase factor expectation value ⟨eiϕ⟩PQ\langle e^{i\phi}\rangle_{\mathrm{PQ}} from LLR density-of-states data a controlled procedure — one whose accuracy and convergence are established rather than contingent on the success of an intermediate fit of ln⁡ρ(s)\ln\rho(s) — for sign-problem-afflicted finite-density quantum field theories, with heavy-dense QCD (where the fit-based answers are known) as the reference benchmark. This objective decomposes into the four linked questions below, as left open by Garron and Langfeld (2017): all four are required components of the objective, and each is independently resolvable, so that resolving one of them advances the objective without by itself completing it. (1) Convergence: establish the convergence of the advanced moment expansion for general sign-problem-afflicted theories — prove or bound the exponential decay of the universal coefficients α2N0\alpha_{2N_0} (so far verified numerically only up to N0=30N_0 = 30), or determine sharp conditions on the density ρ(s)\rho(s) (equivalently on the boundedness of the advanced moments) under which the expansion converges, including theories of a different nature than heavy-dense QCD and Z3\mathbb{Z}_3-type models (effective three-state theories of the Polyakov loop). (2) Fit-free moments: demonstrate — or rule out — that the elementary moments ⟨s2n⟩\langle s^{2n}\rangle entering the expansion can be computed directly from raw LLR simulation data with sufficient precision for the advanced moment expansion to converge, without fitting the density, including the associated systematic study of the truncation order N0N_0 and bin size δs\delta_s. (3) Strong sign-problem regime: determine whether a fit-free scheme (e.g., the telegraphic approximation with finer binning or modified LLR parameter choices near s=0s=0 and s=πs=\pi) can extract a signal where the raw-data version failed (heavy-dense QCD at μ=1.2921\mu = 1.2921). (4) Reach: establish how these schemes perform at larger volumes, for different lattice fermion actions, and for other finite-density theories where the sign problem differs in nature (e.g., the Hubbard model or finite-isospin QCD), benchmarking against fit-based or otherwise known results. In questions (1) and (4), the enumerated volumes, lattice actions, and theories delimit the class of systems these schemes are meant to cover; they are illustrative of that scope rather than a conjunctive list of cases each required separately.

References

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