Chiral Enough? A D Kennedy University of Edinburgh Wednesday, 22 - PowerPoint PPT Presentation

Chiral Enough? A D Kennedy University of Edinburgh Wednesday, 22 February 2012 New-Type Fermions on the Lattice

Outline We shall review algorithms for on-shell chiral lattice (approximate overlap) fermions In QCD chiral symmetry is explicitly broken by quark masses So there is no reason to require “too much”  chirality, we just need m m res q We shall show results from a preliminary dynamical study to compare the cost of using different approximations to achieve this Wednesday, 22 February 2012 A D Kennedy 2

Chiral Fermions Conventions We work in Euclidean space γ matrices are Hermitean = ⋅ γ We write D D µ µ We assume all Dirac operators are γ 5 Hermitean = γ γ † D D 5 5 Wednesday, 22 February 2012 A D Kennedy 3

On-shell chiral symmetry: I It is possible to have chiral symmetry on the lattice without doublers if we only insist that the symmetry holds on shell Such a transformation should be of the form   ( ) αγ − ζ α  − − ζ  γ ψ →   ψ ψ → ψ i 1 2 aD i 1 2 1 aD   (Lüscher) e ; e 5 5 y † y is an independent field from y y † has the same Spin(4) transformation properties as † y does not have the same chiral transformation properties y as in Euclidean space (even in the continuum) ζ is a free parameter Wednesday, 22 February 2012 A D Kennedy 4

On-shell chiral symmetry: II For it to be a symmetry the Dirac operator must ( ) α γ αγ   → − − ζ − ζ =   i i 1 2 1 aD  1 2 aD    be invariant D e De D 5 5 For an infinitesimal transformation this implies ( )   − − ζ γ + γ − ζ =   that 1 2 1 aD D D  1 2 aD  0   5 5 Which is the Ginsparg-Wilson relation γ + γ = γ D D 2 aD D 5 5 5 Wednesday, 22 February 2012 A D Kennedy 5

Neuberger’s Operator: I We can find a solution of the Ginsparg-Wilson relation as follows Let the lattice Dirac operator to be of the form [ ] = + γ γ = γ γ ⇒ γ = γ † † aD 2 1 1 ; aD aD ˆ ˆ ˆ 5 5 5 5 5 5 This satisfies the GW relation iff γ = 2 1 ˆ 5 It must also have the correct continuum limit γ   ( ) ( ) D ( ) → ∂ / ⇒ γ = γ ∂ / − + = − + 2 2 W D Z 2 aZ 1 O a  1  O a ˆ 5 5 5   M ( ) Z = → ∂ / + Where we have defined where M W D Z O a 2 aZ W W Both of these conditions are satisfied if we define − D M ( ) γ = γ =  γ −  (Neuberger) W sgn D M ˆ   5 5 ( ) ( ) 5 w † − − D M D M W W Wednesday, 22 February 2012 A D Kennedy 6

Into Five Dimensions H Neuberger hep-lat/9806025 A Boriçi hep-lat/9909057, hep-lat/9912040, hep-lat/0402035 A Boriçi, A D Kennedy, B Pendleton, U Wenger hep-lat/0110070 R Edwards & U Heller hep-lat/0005002 趙挺偉 (T-W Chiu) hep-lat/0209153, hep-lat/0211032, hep-lat/0303008 R C Brower, H Neff, K Orginos hep-lat/0409118 Hernandez, Jansen, Lüscher hep-lat/9808010 菊川芳夫 & 野口達也 (Y Kikukawa & T Noguchi) hep-lat/9902022 Wednesday, 22 February 2012 A D Kennedy 7

Neuberger’s Operator: II ( ) ( ) ( )   1 µ = + µ + − µ γ D , H 1 1 sgn H   N 5 2 0 μ 1 Is D N local? It is not ultralocal (Hernandez, Jansen, Lüscher) It is local iff D W has a gap D W has a gap if the gauge fields are smooth enough Reflection positivity? It seems reasonable that good approximations to D N will be local if D N is local and vice versa Otherwise DWF with n 5 → ∞ may not be local Wednesday, 22 February 2012 A D Kennedy 8

Neuberger’s Operator: III Four dimensional space of algorithms ( ) = γ − Kernel H D M 5 W Constraint (5D, 4D) P H ( ) ≈ ε = Approximation n sgn( H ) ( H ) n m , Q ( H ) m Representation (CF , PF , CT= DWF) Wednesday, 22 February 2012 A D Kennedy 9

Kernel Boriçi (Wilson/Whatever) kernel ( ) = γ − H D M W 5 W Shamir kernel ( ) − a D M = γ = 5 W H D ; a D ( ) + − T 5 T 5 T 2 a D M 5 W Möbius kernel ( ) ( ) + − a b c D M = γ = 5 5 5 W H D ; a D ( ) ( ) + − − M 5 M 5 M 2 a b c D M 5 5 5 W Wednesday, 22 February 2012 A D Kennedy 10

Schur Complement Consider the block matrix Equivalently a matrix over a skew field = division ring It may be block diagonalised by an LDU factorisation (Gaussian elimination) −              1 1   0 0 A A   B 0  1 A B 1 0 0 1 1 A B = = = ⋅ ⋅ ⋅ ⋅ ⋅                   − − − −           1 1 1 1        − −  C D 1 0 0 1           CA CA 1 1 0 0 D D CA CA B B 0 1 The bottom right block is the Schur complement   = ( ) A B − − 1 In particular det   det AD ACA B   C D Wednesday, 22 February 2012 A D Kennedy 11

Constraint: I So, what can we do with the Neuberger operator represented as a Schur complement? Consider the five-dimensional system of linear equations       φ φ φ       0 0 1  1      1 φ φ  φ      0 0     2 2 2     =        − = − = = 1   1   L D L DU D L     φ     φ φ 5 0 0     − − − n 2       n 2 n 2 φ   φ φ 0 0           − n 1     − − χ χ ψ n 1 n 1       ψ ψ     ψ = ψ = χ The bottom four-dimensional component is D D n n , N Wednesday, 22 February 2012 A D Kennedy 12

Constraint: II Alternatively, introduce a five-dimensional ( ) Φ = φ φ φ ψ  pseudofermion field − 1 n 1 2 Then the pseudofermion functional integral is n = ∏ ∫ − −Φ Φ Φ Φ † 1 ∝ = = † D d d e det D det LDU det D det D 5 5 j j , = j 1 So we also introduce n-1 Pauli-Villars fields − 1   − − n 1 n 1 ∏ ∏ ∫ − ξ ξ † ξ ξ ∝  D †  j j j , j det d d e D , j j j j   = = j 1 j 1 and we are left with just det D n,n = det D N Wednesday, 22 February 2012 A D Kennedy 13

Approximation: tanh Pandey, Kenney, & Laub; Higham; Neuberger (KLN) ω j For even n (analogous formulæ for odd n ) n −   − 1 x 1 ( )   ( ) − ε = =  +  1 1 x x tanh n tanh x − n 1, n n   + − 1 x 1    +  1 x − n 1 ( ) ∏ 2 π k + 2 2 x tan n 2 x ∑ n 2 1 = = = k 1 xn π π     ( ) ( ) ( ) n n   − + − 2 2 2 1 1 x cos k sin k ∏ = 2 + 1 π     k 1 k  2   2  n n   2 + 2 2 x tan   n = k 1 Wednesday, 22 February 2012 A D Kennedy 14

Approximation: Золотарев ( ) 2 sn z k ; − 1 ( ) ( ) ′ 2 λ     n / 2 sn z / M ; sn 2 iK m / n k ; 1 ∏ = ( ) ( ) 2 sn z k ; M sn z k ; = m 1 − 1 ( ) ( ) 2 ′ − sn 2 iK m 1 / n k ; 2 sn( z/M, λ ) sn( z,k ) ω j Wednesday, 22 February 2012 A D Kennedy 15

Approximation: Errors 0.01 ε(x) – sgn( x ) The fermion sgn problem Approximation over 10 -2 < | x | < 1 0.005 Rational functions of degree (7,8) log 10 x -2 1.5 -1 -0.5 0 0.5 -0.005 Золотарев tanh(8 tanh -1 x ) -0.01 Wednesday, 22 February 2012 A D Kennedy 18

Approximation: Errors II Dependence on degree Wednesday, 22 February 2012 A D Kennedy 19

Approximation: Errors III Error over approximation interval as function of degree and interval Wednesday, 22 February 2012 A D Kennedy 20

Approximation: Slope at x =0 Derivative of Золотарев approximation at origin (maximum MD force)  0 49 0 025 . . N       e  e  0 0 ; N e 0 87 Z N , KLN . Wednesday, 22 February 2012 A D Kennedy 21

Approximation: Δ H HMC energy change (32 3 × 64 lattice) Nigel Cundy, A.D. Kennedy, Andreas Schäfer, Nucl.Phys. B845, 30—47 (2011) [arXiv:1010.5629] Wednesday, 22 February 2012 A D Kennedy 22

Approximation: m Res N dependence of (32 3 × 64 lattice) Residual mass Wednesday, 22 February 2012 A D Kennedy 23

Approximation: Low Mode m Res Ratio of “bulk” to “low mode” contributions to residual mass (32 3 × 64 lattice) Wednesday, 22 February 2012 A D Kennedy 24