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Lattice QCD with open boundary conditions and twisted-mass reweighting

Martin Lüscher, Stefan Schaefer

arXiv:1206.2809v1hep-lat

TL;DR

Lattice QCD near physical quark masses and fine lattice spacings is challenged by topological trapping and near-zero-mode instabilities. This paper combines open temporal boundaries with twisted-mass determinant reweighting and algorithmic improvements in 2+1-flavour O(a)-improved Wilson QCD. The study finds the approach stable and applicable to full QCD with open boundaries, while identifying boundaries in correlation-function analysis and stochastic reweighting accuracy.

  • Problem

    Fine-lattice simulations near physical quark masses can be trapped in fixed topological-charge sectors or destabilized by accidental near-zero Dirac modes.

  • Method

    The paper combines open temporal boundary conditions, twisted-mass determinant reweighting, and frequency-splitting algorithmic improvements for 2+1-flavour O(a)-improved Wilson QCD.

  • Results

    Twisted-mass reweighting works as expected with open boundaries, while observed HMC energy deficits are well behaved and large |∆H| values are rare.

  • Takeaways & Limitations

    Open boundaries and twisted-mass reweighting provide a stable, efficient formulation that addresses topological ergodicity problems and near-zero-mode instabilities in the studied setting.

  • Takeaways & Limitations

    Reweighting-factor estimates may require greater accuracy for low-mode-sensitive observables when exceptionally small eigenvalues occur, and open boundaries complicate correlation-function analysis near temporal boundaries.

Abstract

from arXiv · show

Lattice QCD simulations at small lattice spacings and quark masses close to their physical values are technically challenging. In particular, the simulations can get trapped in the topological charge sectors of field space or may run into instabilities triggered by accidental near-zero modes of the lattice Dirac operator. As already noted in ref. [1], the first problem is bypassed if open boundary conditions are imposed in the time direction, while the second can potentially be overcome through twisted-mass determinant reweighting [2]. In this paper, we show that twisted-mass reweighting works out as expected in QCD with open boundary conditions and 2+1 flavours of O(a) improved Wilson quarks. Further algorithmic improvements are tested as well and a few physical quantities are computed for illustration.

1. Introduction

Physical-volume lattice QCD simulations face topological-sector trapping and near-zero-mode instabilities. The paper studies open temporal boundaries and twisted-mass determinant reweighting in 2+1-flavour improved Wilson QCD.

  • Simulations require lattice spacings below 0.05 fm, spatial extents of at least 4 fm, and ideally physical quark masses.
  • Open temporal boundary conditions bypass topological-charge trapping by allowing charge to flow through the lattice boundaries.
  • O(a)-improved Wilson quarks retain accidental near-zero Dirac modes, which can destabilize HMC and potentially compromise simulation correctness.
  • Twisted-mass determinant reweighting addresses near-zero modes through intermediate infrared regularization of the quark determinant.
  • The paper presents a complete study of full QCD with open boundaries, 2+1 flavours, nearly physical quark masses, and additional algorithmic improvements.

2. Twisted-mass determinant reweighting

The formulation combines open temporal boundaries and twisted-mass regularization with determinant reweighting for O(a)-improved Wilson quarks. The regularized ensembles are reweighted to recover correct expectation values, with the second regularization often fluctuating less.

  • The simulations use O(a)-improved Wilson quarks with mass-degenerate light quarks and one included strange quark.
  • Open boundary conditions are imposed in time, while the gauge action includes tree-level Symanzik-improved and Iwasaki forms.
  • Twisted-mass regularization replaces the light-quark determinant using a positive regulator mass µ, usually comparable to the light-quark mass.
  • Regularized gauge-field ensembles require determinant reweighting to obtain correct expectation values, with reweighting factors estimated stochastically.
  • The second determinant regularization is favored because its reweighting factor tends to fluctuate less than the first, though behavior depends on µ.
  • Even-odd preconditioning replaces D with a preconditioned operator and adds the twisted-mass term only on even lattice sites, making the resulting regularization different.
  • Typically 12 to 48 random fields estimate the reweighting factor, but more accurate determinations may be needed for low-mode-sensitive observables with exceptionally small eigenvalues.

3. Frequency splitting of the quark determinant

The paper factorizes light- and strange-quark determinants to split force frequencies and improve HMC integration. Log-scale twisted-mass spacing performed well empirically, although mass selection lacks firm theoretical guidance.

  • Frequency splitting is introduced because mass shifts or domain decomposition can stabilize molecular-dynamics integration and permit larger step sizes.
  • Light-quark determinant: The light-quark determinant uses twisted masses µ0=µ<µ1<...<µn and a factorized representation with n+2 independent pseudo-fermion fields.
  • Light-quark determinant: The factorization assigns determinant factors mainly to spectral intervals bounded by the twisted masses, producing an approximate frequency split.
  • Light-quark determinant: The choice of twisted masses lacks solid theoretical guidance and may require fine-tuning, although the reported studies found good results without fine-tuning.
  • Light-quark determinant: The log-scale rule µk≈0.1×µk+1 splits the Dirac-operator spectral range into equal logarithmic segments and implicitly fixes the number of twisted masses.
  • Strange-quark determinant: RHMC provides a natural frequency splitting for the strange-quark determinant through rational-function factorization and pseudo-fermion representations.
  • Strange-quark determinant: Zolotarev rational-approximation parameters and twisted masses are uniquely determined by the approximation degree and chosen spectral range, with masses roughly log-spaced.
  • Force integration: Strongly ordered pseudo-fermion forces allow different integration step sizes, including roughly tenfold force hierarchies under log-scale splitting.

4. Integration of the molecular-dynamics equations

The simulations integrate molecular-dynamics equations with leapfrog and OMF schemes, using hierarchical force splitting and solver choices tailored to quark masses and pseudo-fermion forces. Solver tolerances must account for condition-number effects because loosening them can compromise integration accuracy and stability.

  • Integrator construction: The simulations use leapfrog, second-order OMF, and fourth-order OMF integrators built from elementary gauge-field and momentum updates.The fourth-order OMF scheme has five update steps and no tunable parameters.
  • Hierarchical integration: Hierarchical integration recursively assigns different step sizes to forces, starting with the smallest forces at the top level and replacing gauge-field updates at lower levels.For two forces, the average step sizes are τ/(10n0n1) for F0 and τ/(2n1) for F1.
  • Force computation: Frequency splitting produces many pseudo-fermion forces, requiring repeated twisted-mass Dirac solves during molecular-dynamics evolution.This motivates solver acceleration and force-specific integration strategies.
  • Solver acceleration: Multi-shift CG handles rational-function actions and large quark masses, while GCR with local deflation and SAP preconditioning is faster at small and intermediate masses.Local-deflation overhead is rapidly amortized along trajectories when several pseudo-fermion forces are present.
  • Solver tolerances: Relaxing solver tolerances for smaller forces can be unsafe because computed-field errors depend on the Dirac-operator condition number and may scale as δ/µk and δ/µk^2.The resulting loss of accuracy can compromise the stability of molecular-dynamics integration.

5. Algorithm stability and performance

The simulations combine twisted-mass reweighting with algorithmic choices designed to maintain stability and efficiency on challenging lattices. Energy deficits and reweighting factors remain well behaved, while reweighting adds little computational effort.

  • The study tests whether twisted-mass determinant reweighting works on large lattices near physical quark masses and also evaluates the simulation algorithm.
  • The simulated ensembles use light-quark and 2+1-flavour QCD runs with O(a)-improved Wilson quarks and open-boundary lattice setups.
  • The algorithm combines twisted-mass determinant factorization, hierarchical molecular-dynamics integration, and locally deflated SAP-preconditioned GCR solvers.
  • 5.3 Integration instabilities: |∆H| significantly larger than 2 is rare, occurring with an estimated probability of at most a few permille across the test runs.The authors attribute the stability primarily to low-mode regularization, with possible contributions from determinant frequency splitting and the integrator.
  • 5.4 Reweighting efficiency: In the most critical run I2, the normalized reweighting factor stays below 1.5, and only 15% of configurations have weight below 0.5.The authors conclude that reweighting does not compromise simulation efficiency on the studied lattices and regulator masses.
  • 5.6 Simulation cost: Twisted-mass reweighting adds very little computational effort, while estimated exponential autocorrelation times are about 32, 20, and 15 molecular-dynamics time units in runs D6, E8, and I1.These autocorrelation estimates may require correction because they are based on relatively short data series.

6. Computation of physical quantities

With open temporal boundaries, the paper measures the Wilson-flow reference scale and pseudoscalar observables while accounting for boundary effects. The measured flow-energy profile develops a central plateau, and leading-order chiral fits describe the pion and kaon propagators reasonably well.

  • Reference flow time: Open boundaries break time-translation invariance, but boundary effects decrease exponentially away from the temporal boundaries.The central region can therefore approximate infinite-volume behavior when the Wilson-flow smoothing range is much smaller than the lattice dimensions.
  • Reference flow time: The measured ⟨E(x)⟩ profile is consistent with a broad central plateau within 1–2% statistical fluctuations.The plateau is assessed in the central region, where boundary contamination is reduced.
  • Reference flow time: t0 = 2.792(10) is obtained for run I1 from central-region measurements of ⟨E(x)⟩.The associated Wilson-flow smoothing range is 4.7 lattice spacings, corresponding to 0.43 fm.
  • Reference flow time: The dimensionless combination t2⟨E(x)⟩ rises nearly linearly beyond a smoothing range of about 0.2 fm.Its behavior over the plotted flow-time range is practically the same as in the pure gauge theory.
  • Pseudo-scalar meson masses: The pion and kaon propagators decrease roughly exponentially from a source at y0 = 1 toward the opposite temporal boundary.The kaon propagator falls more rapidly but otherwise behaves essentially like the pion propagator.
  • Pseudo-scalar meson masses: The fitted lattice-unit meson masses are mπ = 0.0925(19) and mK = 0.2373(10).Effective masses are constant within errors in the central region; the corresponding physical values are 203(4) and 520(2) MeV.
  • Pseudo-scalar meson masses: Near the temporal boundaries, pseudoscalar propagators deviate from a single-exponential form, while small x0 also receives higher-energy-state contributions.Near the chiral limit, the stated boundary-distance condition is not smaller than about 0.5 fm.
  • Pseudo-scalar meson masses: Leading-order chiral perturbation theory formulae fit the measured propagators quite well.The results support, while not conclusively establishing, the conjecture that the chiral limit with open boundaries is described by standard chiral effective theory with Dirichlet boundary conditions.

7. Concluding remarks

The concluding remarks find open boundaries and twisted-mass determinant reweighting useful for stable and efficient numerical lattice QCD. The combined approach addresses topology-related ergodicity problems and near-zero-mode instabilities, while the tested algorithm achieved strong performance with little tuning.

  • Concluding remarks: Open temporal boundaries bypass ergodicity problems associated with topological charge sectors without changing the physical states or Hamiltonian.They slightly complicate the analysis of correlation functions.
  • Concluding remarks: Twisted-mass determinant reweighting ensures the absence of instabilities and sampling inefficiencies caused by accidental near-zero Dirac-operator modes in the Wilson formulation.The conclusion attributes this benefit specifically to the reweighting approach used with Wilson quarks.
  • Concluding remarks: The simulation algorithm combines twisted-mass reweighting, log-scale Hasenbusch factorization, and a hierarchical Omelyan-type integrator.The runs achieved excellent stability and performance with very little parameter tuning.
  • Concluding remarks: The authors expect the stability situation to remain essentially unchanged on larger and finer lattices.They also report that reweighting efficiency should depend only weakly on lattice parameters for an appropriate regulator mass in the second reweighting variant.

Appendix A. Gauge action

The gauge actions are built from oriented plaquette and rectangular loops with boundary weights, using improvement coefficients to control lattice artifacts. The coefficient cG is required for O(a) improvement of boundary-sensitive correlation functions.

  • Gauge action: The gauge action is formed from oriented 1 × 1 plaquette and 1 × 2 rectangular loops whose time coordinates lie between 0 and T.Each loop contributes through an ordered product of link variables and a specified weight factor.
  • Gauge action: The Wilson, tree-level Symanzik-improved, and Iwasaki actions correspond to c1 = 0, c1 = −1/12, and c1 = −0.331, respectively.The inverse bare coupling uses the convention β = 6/g0^2.
  • Gauge action: Boundary weights equal 1 except for space-like loops on the lattice boundaries at times 0 and T.These exceptional weights encode the temporal boundary treatment in the gauge action.
  • Gauge action: Setting cG = 1 ensures on-shell tree-level improvement for correlation functions involving fields close to or at the boundaries.The coefficient is required for O(a) improvement of boundary-sensitive correlation functions.

Appendix B. Wilson flow

The Wilson flow evolves lattice gauge fields from their initial configuration using an action-driven flow time. With open temporal boundaries, the flow is modified and improved to avoid O(a) effects in gauge-invariant observables.

  • Periodic boundary conditions: On periodic lattices, the Wilson flow V_t(x, μ) is defined by differential equations with the initial field U(x, μ).The parameter t ≥ 0 is called the flow time.
  • Open boundary conditions: For open boundary conditions, the Wilson-flow equation takes a slightly different form.The modified formulation applies across the temporal interval 0 ≤ x_0 ≤ T for spatial directions k = 1, 2, 3.
  • Open boundary conditions: The open-boundary construction uses the Wilson action with c_G = 1 and includes a weight factor.The weight factor is part of the formulation used for the flow with open boundaries.
  • O(a) improvement: The weight factor ensures the absence of O(a) lattice effects in expectation values of gauge-invariant quantities at positive flow time.This condition applies at flow time t > 0.
  • O(a) improvement: O(a) improvement is guaranteed when the flow is improved at tree level, and the same equations can also be derived through an orbifold construction.In the orbifold formulation, symmetry excludes O(a) lattice effects.
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