[Paper Review] Stochastic calculation of the QCD Dirac operator spectrum with Mobius domain-wall fermion

Guido Cossu, Hidenori Fukaya+3 more|arXiv (Cornell University)|Jan 5, 2016

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper presents a stochastic method using Chebyshev filtering and eigenvalue counting to compute the full spectrum of the QCD Dirac operator with Möbius domain-wall fermions. The approach enables precise, single-measurement spectral analysis across the entire eigenvalue range, yielding a chiral condensate of $[\Sigma(2~\text{GeV})]^{1/3} = 260.0(1.7)~\text{MeV}$ via one-loop chiral perturbation theory fits.

ABSTRACT

We calculate the spectral function of the QCD Dirac operator using the four-dimensional effective operator constructed from the Mobius domain-wall implementation. We utilize the eigenvalue filtering technique combined with the stochastic estimate of the mode number. The spectrum in the entire eigenvalue range is obtained with a single set of measurements. Results on 2+1-flavor ensembles with Mobius domain-wall sea quarks at lattice spacing ~ 0.08 fm are shown.

Motivation & Objective

• To develop a computationally efficient method for calculating the full spectrum of the QCD Dirac operator in lattice QCD.
• To overcome the limitations of traditional eigenvalue solvers in large-volume simulations by enabling stochastic, flexible spectral estimation.
• To extract the chiral condensate non-perturbatively using the Banks-Casher relation and chiral perturbation theory.
• To validate the method on 2+1-flavor Möbius domain-wall fermion ensembles at $a \sim 0.08$ fm with varying sea quark masses.

Proposed method

• Utilizes Chebyshev polynomial filtering to stochastically estimate the number of eigenvalues in a given interval $[a,b]$ of the Dirac operator $D^\dagger D$.
• Employs a stochastic estimator with $N_v$ Gaussian random vectors to compute $\bar{n}[a,b] = \frac{1}{N_v} \sum_{k=1}^{N_v} \xi_k^\dagger h(A) \xi_k$, where $h(A)$ acts as a spectral filter.
• Applies the Chebyshev expansion $h(x) = \sum_{j=0}^{p} g_j^p \gamma_j T_j(x)$ with Jackson damping to suppress oscillations and improve convergence.
• Leverages the recurrence $T_j(x) = 2xT_{j-1}(x) - T_{j-2}(x)$ for efficient polynomial evaluation and reduces computational cost.
• Uses the 4D effective Dirac operator from Möbius domain-wall fermions, whose eigenvalues of $D^{(4)\dagger}D^{(4)}$ are confined to $[0,1]$, enabling high-precision filtering at moderate polynomial order.
• Performs chiral extrapolation using one-loop $N_f=2$ chiral perturbation theory to extract $\Sigma$ from the spectral density near zero.

Experimental results

Research questions

• RQ1 Can Chebyshev filtering combined with stochastic estimation provide an efficient and accurate method for computing the full Dirac spectrum in lattice QCD?
• RQ2 How does the spectral density $\rho(\lambda)$ depend on the sea quark mass $m_{ud}$ in the $p$-regime?
• RQ3 To what extent can one-loop chiral perturbation theory describe the lattice data for $\rho(\lambda)$ near zero?
• RQ4 What is the value of the chiral condensate $\Sigma$ extracted from the spectral density using non-perturbative lattice data?

Key findings

• The method successfully computes the entire Dirac spectrum with a single set of measurements using stochastic Chebyshev filtering.
• The spectral function $\rho(\lambda)$ shows a clear dependence on $m_{ud}$, with a peak near $\lambda=0$ for heavier sea quarks due to reduced suppression of near-zero modes.
• The chiral condensate is extracted as $[\Sigma(2~\text{GeV})]^{1/3} = 260.0(1.7)~\text{MeV}$, consistent with the Banks-Casher relation.
• One-loop chiral perturbation theory describes the data well near $\lambda=0$ for $\delta=0.01$, with $\chi^2/\text{dof} = 1.13$.
• The systematic error from Chebyshev approximation is below statistical error, with 0.8% and 1.5% errors for $\delta=0.01$ and $\delta=0.005$, respectively.
• The 2+1-flavor chiral perturbation theory fit fails to reproduce the data, likely due to the strange quark being too heavy for the one-loop approximation.