Exact 2+1 quark flavour simulations
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Exact 2+1 quark flavour simulations
International Lattice Field Theory Network Workshop
Michael Clark
(with work from Chris Maynard and Rob Tweedie)
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Edinburgh 2005
Exact 2+1 quark flavour simulations
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Motivation
Current Algorithms
Rational Hybrid Monte Carlo
2+1 ASqTad Fermions
2+1 Domain Wall Fermions
R vs RHMC
Conclusion
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Edinburgh 2005
Exact 2+1 quark flavour simulations
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Introduction and Motivation
• The starting point for numerical simulations of lattice QCD is
the path integral
1
hΩi =
[dU ]e−S(U )[det M(U )]αΩ(U )
Z
where Nf = 4α (2α) for staggered (Wilson) fermions,
M = M †M .
⇒ arbitrary number of flavours with non-integer α
• non-integer α ⇒ non-local action, conventional HMC fails
• Both ASqTad and DWF programmes want to do 2+1 physics
• Need non-local algorithms to handle 2+1 flavours
Z
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Exact 2+1 quark flavour simulations
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The R Algorithm
Rewrite the fermionic determinant:
det Mα = exp α tr ln M
• Approximate trace by noisy estimator, equivalent to pseudo
fermions
• Use integrator (O(δτ 2))
– Non-reversible
– Jacobian 6= 1
⇒ Cannot include Monte Carlo acceptance test
• R: HMD algorithm which is inexact ⇒ naı̈vely cheap, but
extrapolation to zero stepsize (strictly) required
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Edinburgh 2005
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Polynomial Hybrid Monte Carlo
Write in pseudofermion notation
det Mα =
≈
Z
Dψ̄Dψe−ψ̄M
Z
Dψ̄Dψe−ψ̄P (M)ψ ,
−α ψ
(1)
(2)
where P (M) is the MiniMax polynomial approximation over
spectral range.
• Pseudofermion heatbath easily realised since
P (M) = p†(M)p(M)
• Integrated using standard MD ⇒ can make exact
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-5-
Edinburgh 2005
Exact 2+1 quark flavour simulations
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Polynomial Hybrid Monte Carlo
• Significant Error |P (M) − M−α| > 1 ulp
– Cannot use conventional Metropolis
– Noisy acceptance test or appropriate reweighting
• Requires very high degree polynomial
– costly in memory (for non-analytic force)
– sensitive to rounding errors
– ASqTad force term very expensive
• Could also multi-boson algorithm but similar problems
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Edinburgh 2005
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Rational Hybrid Monte Carlo
• Optimal rational approximations
– generated using Remez algorithm
– roots are real and non-degenerate (poles are always +ve)
– Can easily achieve floating point precision
⇒ conventional Metropolis
– Evaluated using multimass solver
• Pseudo fermionic contribution
to the action
is written
2
P
αi
Spf = φ†r2(M)φ = φ† α0 + n
φ.
i=1 M+β
i
• Proceed as for regular HMC with r 2(M) in place of M−α
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-7-
Edinburgh 2005
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Rational Hybrid Monte Carlo
• Hybrid Molecular Dynamics Trajectory
1. Momentum refreshment heatbath using Gaussian noise
∗
(P (π) ∝ e−π π/2).
2. Pseudofermion heatbath using Gaussian noise
∗
(P (Φ) ∝ r(M)−1ξ, where P (ξ) ∝ e−ξ ξ/2).
3. MD trajectory consisting of τ /δτ steps.
– Use low degree approx r̄ ≈ r2
– Contribution to the force is
0 = φ† r̄ 0 (M)φ = −
Spf
n
X
ᾱiφ†(M − β̄i)−1M0(M − β̄i)−1φ.
i=1
• Metropolis Acceptance Test Pacc = min(1, e−δH )
• CG Computational cost per trajectory ≈ R alg,
(2 + τ /δτ ) ≈ (τ /δτ )
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Edinburgh 2005
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2+1 ASqTad Fermions
• Action has the form
Sf = ψ̄ (Mm̄)−1/2 ψ + χ̄ (Mms )−1/4 χ,
where m̄ = (mu + md)/2
• Use rational approximation for each of these contributions
2 (M)ψ + χ̄ r 2 (M)χ
Sf = ψ̄ rm̄
s
• Easy to construct approximation since M bounded explicitely
by m.
• Proceed as for regular conventional RHMC with two fields
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-9-
Edinburgh 2005
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2+1 ASqTad Fermions
• ASqTad simulations involve costly computation of force ≈ CG
cost
⇒ Naı̈ve RHMC has factor n more force computation than R
n
X
n
X
†
†
U...U XiXi U...U = U...U
XiXi U...U
i=1
i=1
• Calculate link products independently of outer product sum
⇒ at least 3×overhead, BUT
– For 2+1 can perform force calculation simultaneously
– No heatbath overhead at each MD step
• No difference in naive cost
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-10-
Edinburgh 2005
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2+1 ASqTad Fermions
• First run shall be V = 24364, β = 6.76, m̄ = 0.0078,
ms = 0.039.
• Large volume light masses ideal for examining acceptance rate
behaviour
• Better compararison of algorithmic cost
• MILC δτ = 2
3 m̄ ≈ 0.005
• Use Hasenbusch type tricks to improve acceptance rates
• Can we do any better?
• No results yet.............
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-11-
Edinburgh 2005
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2+1 Domain Wall Fermions
• Action now takes the form of
!1/2
M1
M1
ψ + χ̄
χ
Mm̄
Mms
Use rational approximation for second contribution
BUT mf not diagonal in the fermion matrix
⇒ cannot write rational approximation as a shifted matrix
Forced to write action as
M
Sf = ψ̄ 1 ψ + φ̄ (M1)1/2 φ + χ̄ (Mms )−1/2 χ
Mm̄
⇒ must use two fermion fields to simulate 1 flavour
contribution
CG inversion required for 1 flavour bosonic Pauli-Villars field
Cost of including φ and χ fields small since mf << ms
Sf = ψ̄
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Edinburgh 2005
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2+1 Domain Wall Fermions
• Additional complication - lower bound of DWF matrix?
• Use speculative lower bound, and monitor
• Need to verify RHMC is both correct and efficient
• Compare observables against the R algorithm results
• VERY PRELIMINARY
• V = 16332 × 8, β = 0.72, DBW2, ms = 0.04, m̄ = 0.02, 0.04
• R algorithm δτ = 0.01
1, 1)
• RHMC δτ = 0.02, multiple pseudo-fermion fields ( 1
,
2 2 2
• hA2+1i = 68%, hA3i = 72%.
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Plaquette, m̄ = 0.04, ms = 0.04
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Plaquette, m̄ = 0.02, ms = 0.04
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Lowest Eigen-value Behaviour
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Highest Eigen-value Behaviour
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Effective mass, m̄ = 0.02, ms = 0.04
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Pseudo-Scalar2 vs valence, m̄ = 0.04, ms = 0.04
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Pseudo-Scalar2 vs dynamical, ms = 0.04
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Edinburgh 2005
Exact 2+1 quark flavour simulations
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Pseudo-Scalar Correlator (t=14), m̄ = 0.04, ms = 0.04
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Vector vs dynamical, ms = 0.04
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Histogram of ρ correlator, (t = 1), m̄ = 0.04, ms = 0.04
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Edinburgh 2005
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mres vs dynamical, ms = 0.04
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Edinburgh 2005
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Conclusions and outlook
• Fast exact algorithm for non-local quantum field theories
• 2+1 ASqTad
– Naive cost R ≈ cost RHMC
– Full cost comparison required
– Full physics comparison required
– Awaiting stability.......
• 2+1 DWF
– RHMC verified for DWF
– Need *more* statistics
– Naive cost R ≈ cost RHMC
– Parameter tuning required
– Future work shall use RHMC exclusively
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Edinburgh 2005