Simulating a gauge action from the Overlap Operator ...

Simulating a gauge action from the Overlap Operator with RHMC T. Streuer

Work with A.Alexandru, I. Horváth, K.F. Liu University of Kentucky

April 21, 2007

Outline

Introduction

Overlap gauge action

RHMC

Test results

Lattice chiral symmetry

Continuum: {D, γ5 } = 0 Not possible on lattice without doublers (Nielsen-Ninomiya theorem) But: exact chiral symmetry with overlap fermions (Neuberger), Domain wall fermions (Kaplan),... Ginsparg-Wilson relation {γ5 , D} = 2Dγ5 D

Overlap fermions

Dov (m) = 1 +

  m m + 1− γ5 sgn HW 2ρ 2ρ

with Hermitian Wilson-Dirac operator HW = γ5 DW (−ρ), ρ ∈ (0, 2) Exact chiral SU(n) × SU(n) × U(1) symmetry for n massless flavours, e.g. δψ = γˆ5 τ ψ ¯ 5τ δ ψ¯ = ψγ with γˆ5 = sgn HW

Numerics

Dov = 1 + γ5 sgn HW Approximate sgn x by rational function:

r(x) sgn x

1

0.5

0

sgn(x) ≈ r (x) = x

X i

ρi 2 x + σi

such that | sgn(x) − r (x)| < ǫ for x ∈ spec HW

-0.5

-1

-3

-2

-1

Coefficients ρi , σi known analytically (Zolotarev) Compute sgn HW ψ using multi-mass solver

0

1

2

3

Cost for applying Dov determined by cond HW Can be improved by ◮

Projecting out few small eigenvalue/eigenvector pairs

◮

Using an improved gauge action (Iwasaki, Lüscher-Weisz, DBW2, ...)

◮

Link smearing (Stout, HYP, ...)

Our approach: Construct gauge action from Dov itself

Gauge action from Dov Classical continuum limit:
free trCS Dov (x, x) − Dov (x, x) = b tr Fµν (x)Fµν (x)

with known coefficient b Action: ¯ = c tr Dov S(U, ψ, ψ) with c=

1 2bg 2

Numerical value of c(g): c(g = 1) ≈ 30

Dynamical overlap fermions Action: ¯ = c tr Dov (U) + S(U, ψ, ψ)

nf X

ψ¯i Dov (U, m)ψi

i=1

Partition function: Z

=

Z

−S(U,Ψ,Ψ) ¯ DUD ΨDΨe

=

Z

DUe−c tr Dov (U) (det Dov (m, U))nf

=

Z

DU det O(U)

¯

with O(U) = e−cDov (U) Dov (m, U)nf Introducing pseudofermions: Z −1 det O(U) = DφDφ∗ e−(φ,O(U) φ)

Hybrid Monte Carlo Hamiltonian: H(U, φ, π) = S(U, φ) + 12 (π, π) Start with gauge field U ◮

Heatbath initialization: 1

P(π) ∼ e− 2 (π,π) P(η) ∼ e−(η,η) ◮

,

φ := O 1/2 η

Molecular dynamics evolution: Integrate Hamiltonian equations numerically, obtain U ′ , π ′ Can use low-precision approximation here U˙ = π π˙ = −

∂ S ∂U 

= − O

−1

 ∂O −1 φ, O φ ∂U

HMC ◮

Metropolis step: Compute δH = H(U ′ , π ′ ) − H(U, π) Accept U’ with probability   p = max 1, e−δH

Need to compute: ◮

O(U)1/2 ψ (in heatbath)

◮

O(U)−1 ψ (in molecular dynamics, Metropolis step)

where O(U) = f (M(U)) M(U) = Dov (U)† Dov (U)  nf 1 f (x) = e− 2 cx (1 − m2 )x + 4m2

Rational HMC RHMC algorithm(Clark, Kennedy): Approximate f (x)−1 , f (x)1/2 by rational functions: X αi f (x)−1 ≈ ri (x) = βi + x i

f (x)1/2 ≈ rs (x) =

X i

αi′ βi′ + x

Coefficients αi , βi , αi′ , βi′ : Remez algorithm Compute r (M)ψ using multi-shift conjugate gradient Coefficients βi not always real → can use 3-term CG

RHMC Problem: condition number of O huge 2e+21

Coefficients |αi |: (c = 25, d = 11)

1e+21 0 0

2

4

6

8

10

f (0) = O(1) → enormous loss of precision Possible solution (Clark, Kennedy): n 1 det(O n ) Z Z 1 P − = dΦ1 . . . dΦn e i (φi ,O n φi )

det O =



2e+07

Coefficients |αi |: (c = 25, d = 11, n = 6)

1e+07 0 0

2

4

6

8

10

RHMC Pseudofermion heatbath: φi =

X j

αj′ M + βj′

ηi

Molecular dynamics: X ∂S =− αi ∂U i,j



∗ −1

(M + β )

∂M φj , (M + β)−1 φj ∂U

Hamiltonian: X H= φi , i,j

αj φi M + βj



+ 12 (π, π)



Overlap Derivative “Inner” derivative: Need to compute   ′ ∂ χ, Dov χ ∂U   ∂ = χ′ , γ5 sgn HW χ ∂U Derivative of rational approximation(Fodor et al, Cundy): X ρi sgn x ≈ r (x) = x x 2 + σi i



∂HW ∂r (HW ) X 2 = ρi (HW +σi )−1 σi − HW ∂U ∂U i



∂HW ∂U





2 HW (HW +σi )−1

Eigenvalue zero crossings

Dov (U) discontinuous w.r.t. U Different possibilities to treat discontinuity: ◮

Ignore it Correct algorithm, but no acceptance → inefficient

◮

Reflection/refraction step (Fodor et al.): Treat derivative exactly

◮

Use smooth approximation of sign function

If gauge action suppresses rough gauge fields, there should be no crossings anyway → topological charge remains fixed Ergodicity?

Eigenvalue zero crossings Our choice: Approximate sign function smoothly during molecular dynamics: sgn x

1

0.5

0

0

-0.5

-0.5

-1 -0.001

r(x)

1

0.5

-1 -0.0005

0

0.0005

0.001

-0.001

-0.0005

0

0.0005

0.001

Heatbath/Metropolis Molecular dynamics Does not give satisfactory acceptance unless τ small enough: 370000

370000

360000

360000

0

0.05

0.1

0.15

0.2

0.012

0.012

0.006

0.006

0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

0

0

0

0.05

0.1

τ = 0.01

0.15

0.2

τ = 0.001 for t > 0.17

Test results

Simulation parameters:

V

= 84

ρ = 1.8 am = 0.5 τMD = 0.01 c

∈ {25, 35, 45}

a = ?

Energy conservation violation

δH = H(U ′ , π ′ ) − H(U, π) c = 35, 45: ◮

δH acceptable for τ = 0.01

c=45

1

0

-1

◮

δH ∼ τ 2

0

20

40

60

80

100

120

140

160

180

200

120

140

160

180

200

c=35 2

c = 25: ◮

◮

δH huge (O(105 )) whenever eigenvalue crosses zero δH ≈ 1 would require very small τ

1

0

-1 0

20

40

60

80

100 c=25

200000

100000

0 0

5

10

15

20

25

30

35

40

45

Spectral density

250 200

ρ(HW ) spectral density of HW

150 100 50 0 0

0.05

0.1

0.15

0.2

0

0.05

0.1

0.15

0.2

350 300

c = 35, 45: ◮

Gap around zero

c = 25: ◮

ρ(0) > 0

250 200 150 100 50 0

250 200 150 100 50 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Summary

◮

Simulation of overlap gauge action possible with RHMC...

◮

... but expensive

◮

Need to find c corresponding to a ≈ 0.1fm

◮

Need better algorithm to treat zero crossings