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Numerical techniques for lattice QCD in the $\epsilon$--regime
High Energy Physics - Lattice · Submitted 6 Dec 2002
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We discuss a set of techniques (low-mode preconditioning and adapted-precision algorithms in particular) that make such computations numerically safe and more efficient by a large factor.
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In lattice QCD it is possible, in principle, to determine the parameters in the effective chiral lagrangian (including weak interaction couplings) by performing numerical simulations in the $\epsilon$--regime, i.e. at quark masses where the physical extent of the lattice is much smaller than the Compton wave length of the pion. The use of a formulation of the lattice theory that preserves chiral symmetry is attractive in this context, but the numerical implementation of any such approach requires special care in this kinematical situation due to the presence of some very low eigenvalues of the Dirac operator. We discuss a set of techniques (low-mode preconditioning and adapted-precision algorithms in particular) that make such computations numerically safe and more efficient by a large factor.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In lattice QCD it is possible, in principle, to determine the parameters in the effective chiral lagrangian (including weak interaction couplings) by performing numerical simulations in the $\epsilon$--regime, i.e. at quark masses where the physical extent of the lattice is much smaller than the Compton wave length of the pion. | background 1.00 | 0.08 | 0.10 | hedged possibility 0.96 |
| s1 | The use of a formulation of the lattice theory that preserves chiral symmetry is attractive in this context, but the numerical implementation of any such approach requires special care in this kinematical situation due to the presence of some very low eigenvalues of the Dirac operator. | background 0.94 | 0.08 | 0.05 | asserted 0.93 |
| s2 | We discuss a set of techniques (low-mode preconditioning and adapted-precision algorithms in particular) that make such computations numerically safe and more efficient by a large factor. | contribution 0.89 | 0.79 | 0.75 | asserted 1.00 |
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The paper
Original abstract
In lattice QCD it is possible, in principle, to determine the parameters in the effective chiral lagrangian (including weak interaction couplings) by performing numerical simulations in the $\epsilon$--regime, i.e. at quark masses where the physical extent of the lattice is much smaller than the Compton wave length of the pion. The use of a formulation of the lattice theory that preserves chiral symmetry is attractive in this context, but the numerical implementation of any such approach requires special care in this kinematical situation due to the presence of some very low eigenvalues of the Dirac operator. We discuss a set of techniques (low-mode preconditioning and adapted-precision algorithms in particular) that make such computations numerically safe and more efficient by a large factor.
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References
How the literature uses it
15 arXiv papers cite this in 17 sentences
Papers citing it · 7 sentences without a confident label
Uses its method or tool 7 papers
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Methodology: The overlap fermion was proposed in Ref. [ref] to construct a discretized fermion action satisfying the Ginsburg-Wilson relation $D_ovγ_5+γ_5D_ov=a/M_0D_ovγ_5D_ov$ [ref] , [equation] where $ϵ(H_w)=H_w/√(H_w^2)$ is the matrix sign function, $H_w(-M_0)=γ_5D_w(-M_0)$ and $D_w$ is the Wilson Dirac operator with a negative mass parameter such as $M_0=1.5$ to avoid the singularity in $1/D_w$. $ϵ(H)$ can be decomposed into a combination of the small and large eigenvalue regions, [equation] in which the eigenpair $(λ_H,i, v_H,i)$ satisfies the relation $H_wv_H,i=λ_H,iv_H,i$, and $d_i=0,1,2,…,N$ are the Chebyshev polynomial coefficients to approximate $ϵ(H_w)$ for all the eigenvalues larger than a given cutoff $λ^cut_H$ with a given accuracy [ref] .
Detecting the flavor content of the vacuum using the Dirac operator spectrum · Introduction · 0.86
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To this end, I use the minimax polynomial approximation [ref] , X^-1/2_ ≈∑_k=0^N c_k T_k((2X_-v_-u_)/(v_-u_)), where $T_k$ is the Chebyshev polynomial of degree $k$. $X_$ is a kernel in the time-momentum space whose size is $4(T/a-1)$ square for a fixed spatial momentum configuration $$, and the minimal and maximum eigenvalues of $X_$ are denoted by $u_$ and $v_$ respectively.
Universality check of the overlap fermions in the Schroedinger functional · Formulations of the overlap fermion in SF · 1.00
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The square-root of the remaining $(d+1)(T-1)×(d+1)(T-1)$-matrices (one for each value of $$) is computed with the min-max polynomial as explained in section 4 of [ref] .
Test of the Schr\"odinger functional with chiral fermions in the Gross-Neveu model · Free fermions · 1.00
Compares with it 1 paper
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To this end we have set $μ= 0$ and compared the relative error of the TSL approximation with the error of the minmax polynomial approximation [ref] , which works well when $H$ is a Hermitian operator.
A numerical method to compute derivatives of functions of large complex matrices and its application to the overlap Dirac operator at finite chemical potential · Numerical setup and tuning of the TSL algorithm · 0.98
Cites it as support 1 paper
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Indeed, exploratory studies of these correlation functions have already shown the great potential [ref] .
Partially quenched chiral perturbation theory in the epsilon-regime · Introduction · 0.96
Cites it as background 1 paper
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Utilizing these, precise determination of the low energy constants has been attempted through direct calculations of the low-lying Dirac eigenvalues mainly in quenched QCD [ref] .
Lattice study of meson correlators in the epsilon-regime of two-flavor QCD · Introduction · 0.90
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