Chiral Fermions on the Lattice:

Chiral Fermions on the Lattice: A Flatlander s Ascent into Five Dimensions (ETH Zürich) with R. Edwards (JLab), B. Joó (JLab), A. D. Kennedy (Edinburgh), K. Orginos (JLab) LHP06, Jefferson Lab, Newport News (VA), 1 August 2006

1 Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations 2 Schur complement Continued fractions Partial fractions Cayley transform 3 Panorama view Chiral symmetry breaking Numerical studies 4 Conclusions

QCD on the Lattice Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Quantumchromodynamics is formally described by the Lagrange density: L QCD = ψ(i/d m q )ψ 1 4 G µνg µν Non-perturbative, gauge invariant regularisation Lattice regularization: discretize Euclidean space-time Continuum limit a 0 Poincaré symmetries are restored automatically Naive discretisation of Dirac operator introduces doublers restoration of chiral symmetry requires fine tuning

On-shell chiral symmetry Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations 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 ψ e iαγ 5(1 ad) ψ; ψ ψe iα(1 ad)γ 5 and the Dirac operator must be invariant: D e iα(1 ad)γ 5 De iαγ 5(1 ad) = D For an infinitesimal transformation this implies that (1 ad)γ 5 D + Dγ 5 (1 ad) = 0 which is the Ginsparg-Wilson relation γ 5 D + Dγ 5 = 2aDγ 5 D

Overlap Dirac operator I Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations We can find a solution D GW of the Ginsparg-Wilson relation as follows: Let the lattice Dirac operator be of the form ad GW = 1 2 (1 + γ 5ˆγ 5 ); ˆγ 5 = ˆγ 5; ad GW = γ 5aD GW γ 5 This satisfies the GW relation if ˆγ 2 5 = 1 And it must have the correct continuum limit D GW / ˆγ 5 = γ 5 (2a/ 1) + O(a 2 ) Both conditions are satisfied if we define D ρ ˆγ 5 = γ 5 (D ρ) (D ρ) = sgn [γ 5(D ρ)]

Overlap Dirac operator II Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations The resulting overlap Dirac operator: D(H) = 1 2 (1 + γ 5sgn [H( ρ)]) has exact zero modes with exact chirality index theorem no additive mass renormalisation, no mixing Three different variations: Choice of kernel Choice of approximation: polynomial approximations, e.g. Chebyshev rational approximations sgn(h) R n,m(h) = Pn(H) Q m(h) Choice of representation: continued fraction, partial fraction, Cayley transform

Kernel choices Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Simplest choice is the Wilson kernel H W = γ 5 D W ( ρ) Domain wall fermion kernel is H T = γ 5 D T ; D T = D W( ρ) 2 + ad W ( ρ) The generic Moebius kernel interpolates between the two: H M = γ 5 D M ; D M = (b + c)d W( ρ) 2 + (b c)ad W ( ρ) Use UV-filtered covariant derivative: overlap operator becomes more local no tuning of ρ, better scaling, cheaper,...

Tanh Approximation Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Use a tanh expressed as a rational function sgn(x) R 2n 1,2n (x): ( ) tanh 2n tanh 1 (x) = (1 + x) 2n (1 x) 2n (1 + x) 2n + (1 x) 2n Properties: f (x) x=0 = 0 lim x = 0 f (x) = f (1/x)

Zolotarev s Approximation I Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations By means of Zolotarev s theorem we have: c r = sn( u sn(u, k), λ) = M M sn2 ( rk n,k 2 ) 1 sn 2 ( rk n,k 2 ) [ n 2 ] r=1 1 + sn2 (u,k) c 2r 1 + sn2 (u,k) c 2r 1 ξ = sn(u, k) is the Jacobian elliptic function defined by the elliptic integral u = ξ 0 dt (1 t 2 )(1 k 2 t 2 ), 0 < k < 1. Setting x = k sn(u, k) we obtain the best uniform rational approximation on [ 1, k] [k, 1]:

Zolotarev s Approximation II Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations sgn(x) R n+1,n (x) = (1 l) x kd [ n 2 ] r=1 x 2 + k 2 c 2r x 2 + k 2 c 2r 1

Cayley Transform Representation Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Represent the rational function as a Euclidean Cayley transform: R(x) = 1 T (x) 1 + T (x) It is an involutive automorphism, T (x) = 1 R(x) 1 + R(x), and the oddness of R(x) translates into the logarithmic oddness of T (x) and vice versa, R( x) = R(x) T ( x) = T 1 (x) How do you evaluate this?

Continued Fraction Representation Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Continued fraction is obtained by applying Euclid s division algorithm: sgn(x) R 2n+1,2n (x) = k 0 x + k 1 x + + 1 1 k 2n 1 x + 1 k 2n x where the k i s are determined by the approximation. How do you evaluate this?

Partial Fraction Representation Lattice formulation of QCD On-shell chiral symmetry Kernels, Approximations and Representations Partial fraction decomposition is obtained by matching poles and residues:! nx c sgn(x) R 2n+1,2n (x) = x c 0 + k x 2 + q k=1 k use a multi-shift linear system solver Physics requires inverse of D(µ) (propagators, HMC force) leads to a two level nested linear system solution How can this be avoided? introduce auxiliary fields extra dimension five-dimensional representation of the sgn function nested Krylov space problem reduces to single 5d Krylov space solution

Schur Complement Schur complement Continued fractions Partial fractions Cayley transform ( ) A B Consider the block matrix C D It may be block diagonalised by a LDU decomposition (Gaussian elimination) ( ) ( ) ( ) 1 0 A 0 1 A CA 1 1 0 D CA 1 1 B B 0 1 The bottom right block is the Schur complement In particular we have ( ) A B det = det(a) det(d CA 1 B) C D

Continued fractions I Schur complement Continued fractions Partial fractions Cayley transform Consider a five-dimensional matrix of the form 0 B @ A 0 1 0 0 1 A 1 1 0 0 1 A 2 1 0 0 1 A 3 and its LDU decomposition where S 0 = A 0 ; S n + 1 = A S n n 1 0 1 0 1 0 0 0 B S 1 0 1 0 0 @ 0 S 1 C B 1 1 0 A @ 0 0 S 1 2 1 1 C A 1 0 S 0 0 0 0 1 S 1 0 0 0 0 S 1 0 0 C B 0 1 S 1 1 0 0 0 S 2 0 A @ 0 0 1 S 1 2 0 0 0 S 3 0 0 0 1 The Schur complement of the matrix is the continued fraction S3 = A 3 1 1 1 = A 3 S 2 A 2 1 S 1 = A 3 A 2 1 A 1 1 A 0 1 C A

Continued fractions II Schur complement Continued fractions Partial fractions Cayley transform We may use this representation to linearise our continued fraction approximation to the sign function: sgn n 1,n (H) = k 0 H + k 1 H + 1 1 k 2 H +... + 1 k nh as the Schur complement of the five-dimensional matrix 0 B @ k 0 H 1 1 k 1 H 1 1 k 2 H... 1 1 k nh 1 C A

Continued fractions II Schur complement Continued fractions Partial fractions Cayley transform We may use this representation to linearise our continued fraction approximation to the sign function: sgn n 1,n (H) = k 0 H + c 1 k 1 H + c 1 c 1 c 2 c 2 k 2 H +... + c n 1 c n c nk nh as the Schur complement of the five-dimensional matrix 0 k 0 H c 1 c 1 c1 2k 1H c 1 c 2 c 1 c 2 c2 2k 2H B @... cn 1 c n c n 1 c n cnk 2 nh 1 C A Class of operators related through equivalence transformations parametrised by c i s

Partial fractions Schur complement Continued fractions Partial fractions Cayley transform Consider a five-dimensional matrix of the form: 0 B @ A 1 1 0 0 1 1 B 1 0 0 0 0 0 A 2 1 1 0 0 1 B 2 0 1 0 1 0 R 1 C A where A i = x p i, B i = p i x q i Compute its LDU decomposition and find its Schur complement R + p 1x x 2 + q 1 + p 2x x 2 + q 2 So we can use this representation to linearise the partial fraction approximation to the sgn-function: nx p j sgn n 1,n (H) = H H 2 + q j=1 j

Partial fractions Schur complement Continued fractions Partial fractions Cayley transform Consider a five-dimensional matrix of the form: 0 B @ A 1 1 0 0 1 1 B 1 0 0 0 0 0 A 2 1 1 0 0 1 B 2 0 1 0 1 0 R 1 C A where A i = x p i, B i = p i x q i Compute its LDU decomposition and find its Schur complement R + c 1 p 1 x c 1 (x 2 + q 1 ) + c 2 p 2 x c 2 (x 2 + q 2 ) So we can use this representation to linearise the partial fraction approximation to the sgn-function: nx p j sgn n 1,n (H) = H H 2 + q j=1 j

Cayley Transform Schur complement Continued fractions Partial fractions Cayley transform Consider a five-dimensional matrix of the form (transfer matrix form): 0 B @ 1 T 1 0 0 0 1 T 2 0 0 0 1 T 3 T 0 0 0 1 1 C A with its Schur complement 1 T 0 T 1 T 2 T 3 So we can use this representation to linearise the Cayley transform of the approximation to the sgn-function: sgn n 1,n (H) = 1 Q n j=1 T j (H) 1 + Q n j=1 T j (H) This is the standard Domain Wall Fermion formulation

What do we see... Panorama view Chiral symmetry breaking Numerical studies...each representation of the rational function leads to a different five-dimensional Dirac operator...they all have the same four-dimensional, effective lattice fermion operator the overlap Dirac operator...each five-dimensional operator has different symmetry properties different calculational behaviour

What do we see... Panorama view Chiral symmetry breaking Numerical studies...each five-dimensional operator can be even-odd preconditioned...lowest modes of the kernel can be projected out...lowest modes of the kernel can be suppressed: R (x) 1 R(x)

Chiral symmetry breaking Panorama view Chiral symmetry breaking Numerical studies Ginsparg-Wilson defect γ 5 D + Dγ 5 2aDγ 5 D = γ 5 : it measures chiral symmetry breaking for the approximate overlap operator ad = 1 2 (1 + γ 5R n (H)) it is a n = 1 2 (1 R n(h) 2 ) The residual quark mass is m res = G ng G G G is the π propagator it can be calculated directly in four and five dimensions m res is just the first moment of n higher moments might be important for other physical quantities

Setup Panorama view Chiral symmetry breaking Numerical studies We use 15 gauge field backgrounds from dynamical DWF dataset: V = 16 3 32, L s = 8, 12, 16, N f = 2, µ = 0.02 Matched π mass for all representations All operators are even-odd preconditioned, no projection

Comparison of Representations Panorama view Chiral symmetry breaking Numerical studies

m res per configuration Panorama view Chiral symmetry breaking Numerical studies

Cost versus m res Panorama view Chiral symmetry breaking Numerical studies

Conclusions Conclusions We have a thorough understanding of various five dimensional formulations of chiral fermions More freedom and possibilities in 5 dimensions Physically they are all the same From a computational point of view there are better alternatives than the commonly used Domain Wall Fermions Hybrid Monte Carlo simulations: 5 versus 4 dimensional dynamics?