REVIEW
Lattice gauge action suppressing near-zero modes of H_W
High Energy Physics - Lattice · Submitted 16 Jul 2006 · 46 references
Pith's verdict: A solid, honest methods paper whose abstract says 'gap' where its own fit says ρ ∝ λ²; the practical case for the action survives regardless.
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Referee report
The paper
The paper claims
Adding two unphysical heavy quarks to the lattice action opens a gap at zero in the spectrum of the hermitian Wilson-Dirac operator, which is what the locality proof for overlap fermions needs and what makes their numerical sign functionche
The reading
What carries the argument
The load-bearing object is the ratio of determinants $det[H_W(m_0)^2/(H_W(m_0)^2 + mu^2)]$ generated by the extra fermion sector of equation (4): two flavors of Wilson fermion at large negative mass plus one pseudo-fermion field with a flavor-twisted mass term $i mu gamma_5 tau_3$. The numerator forces the partition function to vanish whenever an eigenvalue of $H_W(m_0)$ approaches zero, so the near-zero modes that plague the sign function are statistically excluded rather than merely projected out; the twisted mass $mu$ sets the width of the excluded window, and the pseudo-fermion is arranged so that its contribution largely cancels the heavy Wilson fermion's ultraviolet effects in the combination $[D_W + i mu gamma_5] D_W^{-1}$. The same mechanism is what freezes the spectral flow of $H_W$ through zero, and hence the topological charge.
Extended reading notes
Core claim
On a $16^3 times 32$ lattice at roughly $0.125$ fm, the collaboration generates quenched-plus-extra-fermion ensembles with the standard Wilson plaquette action, a modified plaquette action with $1/epsilon = 2/3$, and the Iwasaki (RG-improved) action, each at $mu = 0$, $0.2$ and $0.4$, with $m_0 = -1.6$ throughout. Without the extra fields the spectral density at zero is nonzero and grows with the rougher action, $rho(0) = 1.509(3) times 10^{-3}$ for the standard plaquette action, $0.494(1) times 10^{-3}$ with $1/epsilon = 2/3$, and $0.177(1) times 10^{-3}$ for the RG action. With $mu = 0.2$ or $0.4$ no eigenvalue below about $0.02$ appears at all and the density near the origin behaves as $lambda_W^2$, so the gap is present for all three gauge actions and is widest for the Iwasaki action. The paper also reports that the topological charge, defined as the index of the overlap-Dirac operator built on $H_W(m_0)$, does not change during the molecular dynamics evolution: configurations started in the trivial sector show no chiral zero mode, and configurations started at $Q = -2$ retain exactly two left-handed zero modes. Finally, a one-loop calculation gives a $\beta$ shift of $0.786(4)$ at $m_0 = -1.6$ from the extra Wilson fermions alone, falling to $0.0502(4)$ once the twisted-mass pseudo-fermions at $mu = 0.2$ are included.
Load-bearing premise
The central measurement assumes the hybrid Monte Carlo sampling is exact even though the runs show occasional large spikes in $Delta H$ and values of $langle e^{-Delta H} rangle$ that are not always consistent with one within errors; the paper states explicitly that it assumes the configurations are properly sampled.
Editorial extensions
If this is right
- The overlap-Dirac operator built on these configurations satisfies the gap condition required by the Hernandez-Jansen-Luescher locality proof, so exponential locality is no longer blocked by a nonzero density of near-zero modes.
- The numerical cost of the matrix sign function drops, since the projection of low-lying eigenmodes scales with the eigenvalue density; the paper reports that the added Wilson-fermion inversion is cheap by comparison with the overlap inversion itself.
- The reflection/refraction procedure used in dynamical overlap simulations to track sign-function jumps at $lambda_W = 0$ becomes unnecessary, because the crossing never happens for a small enough molecular dynamics step.
- Topological charge is conserved along HMC trajectories, which is an advantage for $epsilon$-regime simulations at fixed topology and a disadvantage for sampling the $theta = 0$ vacuum.
- The twisted-mass pseudo-fermions keep the beta shift small, avoiding the strong-coupling regime where the Wilson plaquette action shows a remnant of the fundamental-adjoint phase transition.
Reading between the lines
- If the gap survives at larger volumes and finer lattice spacings with the same $m_0$, the extra-fermion action becomes a generic preconditioner for any simulation that needs a well-defined sign function, not only for overlap fermions; a volume scan of $rho(0)$ at fixed physical lattice spacing would test this directly.
- The fixed-topology property is a double-edged tool: it makes $epsilon$-regime work clean, but the paper's own quenched data show a $10%$ shift in $am_{PS}$ between $|Q| = 0$ and $|Q| = 6$, so any unquenched claim of small topology dependence should be checked against the same quantity rather than assumed from the $O(1/V)$ argument.
- Because the suppression acts through a determinant ratio, its strength depends on $mu$ and on the number of extra flavors; a systematic scan of $mu$ at fixed lattice spacing would separate the desired gap from the residual beta shift more cleanly than the two-value comparison shown here.
Original abstract
We propose a lattice action including unphysical Wilson fermions with a negative mass m_0 of the order of the inverse lattice spacing. With this action, the exact zero mode of the hermitian Wilson-Dirac operator H_W(m_0) cannot appear and near-zero modes are strongly suppressed. By measuring the spectral density rho(lambda_W), we find a gap near lambda_W=0 on the configurations generated with the standard and improved gauge actions. This gap provides a necessary condition for the proof of the exponential locality of the overlap-Dirac operator by Hernandez, Jansen, and Luescher. Since the number of near-zero modes is small, the numerical cost to calculate the matrix sign function of H_W(m_0) is significantly reduced, and the simulation including dynamical overlap fermions becomes feasible. We also introduce a pair of twisted mass pseudo-fermions to cancel the unwanted higher mode effects of the Wilson fermions. The gauge coupling renormalization due to the additional fields is then minimized. The topological charge measured through the index of the overlap-Dirac operator is conserved during continuous evolutions of gauge field variables.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We propose a lattice action including unphysical Wilson fermions with a negative mass m_0 of the order of the inverse lattice spacing. With this action, the exact zero mode of the hermitian Wilson-Dirac operator H_W(m_0) cannot appear and near-zero modes are strongly suppressed. By measuring the spectral density rho(lambda_W), we find a gap near lambda_W=0 on the configurations generated with the standard and improved gauge actions. This gap provides a necessary condition for the proof of the exponential locality of the overlap-Dirac operator by Hernandez, Jansen, and Luescher. Since the number of near-zero modes is small, the numerical cost to calculate the matrix sign function of H_W(m_0) is significantly reduced, and the simulation including dynamical overlap fermions becomes feasible. We also introduce a pair of twisted mass pseudo-fermions to cancel the unwanted higher mode effects of the Wilson fermions. The gauge coupling renormalization due to the additional fields is then minimized. The topological charge measured through the index of the overlap-Dirac operator is conserved during continuous evolutions of gauge field variables.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a lattice action including unphysical Wilson fermions with a negative mass m_0 of the order of the inverse lattice spacing. | contribution 0.78 | 0.82 | 0.06 | asserted 0.98 |
| s1 | With this action, the exact zero mode of the hermitian Wilson-Dirac operator H_W(m_0) cannot appear and near-zero modes are strongly suppressed. | result 0.73 | 0.62 | 0.42 | asserted 1.00 |
| s2 | By measuring the spectral density rho(lambda_W), we find a gap near lambda_W=0 on the configurations generated with the standard and improved gauge actions. | result 1.00 | 0.95 | 0.15 | asserted 1.00 |
| s3 | This gap provides a necessary condition for the proof of the exponential locality of the overlap-Dirac operator by Hernandez, Jansen, and Luescher. | interpretation 0.97 | 0.27 | 0.16 | asserted 0.98 |
| s4 | Since the number of near-zero modes is small, the numerical cost to calculate the matrix sign function of H_W(m_0) is significantly reduced, and the simulation including dynamical overlap fermions becomes feasible. | interpretation 0.74 | 0.67 | 0.27 | asserted 0.99 |
| s5 | We also introduce a pair of twisted mass pseudo-fermions to cancel the unwanted higher mode effects of the Wilson fermions. | method 0.81 | 0.91 | 0.04 | asserted 1.00 |
| s6 | The gauge coupling renormalization due to the additional fields is then minimized. | result 0.43 | 0.67 | 0.05 | asserted 0.99 |
| s7 | The topological charge measured through the index of the overlap-Dirac operator is conserved during continuous evolutions of gauge field variables. | result 0.99 | 0.83 | 0.09 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
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- references and citing sentences
- relations to other papers
- P abstract
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Figures
Showing 4 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-lat/0607020 by the authors.
References
How the literature uses it
43 arXiv papers cite this in 55 sentences
Papers citing it · 23 sentences without a confident label
Uses its method or tool 25 papers
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In this work, we avoid this problem by introducing a topology fixing term to a gauge field action [ref] , with which we never encounter the discontinuity of the sign function.
B_K with two flavors of dynamical overlap fermions · Introduction · 0.94
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To study the chiral magnetic effect in 2+1 flavor QCD at non-zero temperature and topological charge, we use DWF on a lattice of size $16^3×8$, the RHMC algorithm of Clark and Kennedy, and an auxiliary determinant (to fix the topological charge) [ref] .
Chiral magnetic effect in 2+1 flavor QCD+QED · Simulation details · 0.99
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For the gauge sector we use the Iwasaki gauge action together with extra Wilson fermions and ghosts producing a factor [equation] in the partition function such that the near-zero modes of $H_W(-m_0)$ is naturally suppressed [ref] .
Lattice simulation of 2+1 flavors of overlap light quarks · Dynamical overlap fermion · 0.92
Cites it as support 1 paper
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As was shown in Refs. [ref] , this term successfully suppresses the near-zero modes of $H_W$.
Exploring the chiral regime with dynamical overlap fermions · Topology fixing term · 0.99
Cites it as background 1 paper
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Several strategies to allow for both small $m_res$ and $L_5$ exist in the literature [ref] , and can be combined with the Möbius Domain Wall Fermion (MDWF) prescription [ref] .
$F_K/F_\pi$ as a precision test of a new four flavor Domain Wall Fermion action · Introduction · 0.83
Discussion
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