Lattice QCD for Pedestrians Quarks Chiral Symmetry and Fermions Species

Quarks Chiral symmetry and fermions species

Lattice Quantum Field Theory

An introduction

Christian B. Lang

Christian B. Lang Lattice QCD for Pedestrians Quarks Chiral symmetry and fermions species

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

1 Quarks Fermion integration Fermion doubling Wilson fermions Monte-Carlo simulation of fermions 2 Chiral symmetry and fermions species Ginsparg-Wilson condition Fermion species Eigenmodes and instantons

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Quarks Fermion integration

QFT path integral:

1 A = [U] [ψ, ψ] e−SG[U]−SF [ψ,ψ,U] A[ψ, ψ, U] h i Z D D Z 1 = [U] e−SG[U] [ψ, ψ] e−SF [ψ,ψ,U] A[ψ, ψ, U] Z D D Z n o Bosons: usual integration (a group integral on for each link)

Fermions: Grassmann variables (“anti-commuting c-numbers”)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Quarks Fermion integration

QFT path integral:

1 A = [U] [ψ, ψ] e−SG[U]−SF [ψ,ψ,U] A[ψ, ψ, U] h i Z D D Z 1 = [U] e−SG[U] [ψ, ψ] e−SF [ψ,ψ,U] A[ψ, ψ, U] Z D D Z n o Bosons: usual integration (a group integral on for each link)

Fermions: Grassmann variables (“anti-commuting c-numbers”)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Grassmann variables

...are anti-commuting numbers:

ηi ηj = ηj ηi −

for all i, j, thus ηi ηi = 0 ...have the integration rules:

N d η = dηN dηN−1 ... dη1 , dηi dηj = dηj dηi , dηi ηj = ηj dηi − −

dηi 1 = 0 , dηi ηi = 1 , Z Z e.g.

dη2 dη1 (1 + 2 η1 42 η1η2)= 42 − − Z

cf. http://en.wikipedia.org/wiki/42_(number)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

These rules lead to, e.g., the Matthews-Salam formula:

ZF = dηN dηN ... dη1dη1 exp ηi Mij ηj = det[M], ! Z iX,j=1 where M is a complex N N matrix. × Note: This looks almost like Gaussian integration for bosons, except that there the result is det[M]−1!

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

The generating functional for fermions is then given by

N N N N W θ, θ = dη dη exp η M η + θ η + η θ i i  k kl l k k k k  Z i=1 k,l=1 k=1 k=1 Y X X X   N −1 = det[M] exp θn M θm − nm  n,m=1 X   and, e.g.,

N N ∂ ∂ W θ, θ = dηi dηi ηn ηm exp ηk Mkl ηl ∂θm ∂θ   n θ,θ=0 Z i=1 k,l=1 Y X  

= det[M] M−1 − nm

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

N-point functions can be computed by (Wick’s theorem):

1 N N d d exp M ηi1 ηj1 ...ηin ηjn F = ηk ηk ηi1 ηj1 ...ηin ηjn ηl lmηm h i ZF   Z kY=1 l,Xm=1 1 ∂ ∂ ∂ ∂   = ... W θ, θ Z ∂θ ∂θ F j1 ∂θi1 jn ∂θin θ,θ=0

with ZF = det[M]. Quarks and anti-quarks:

(f ) ηi ψ (x, α, a) or, e.g. u(x, α, a) → (f ) η ψ (x, α, a) or, e.g. u(x, α, a) i → M D lattice Dirac operator = matrix →

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions Some examples for propagators: Quark: u(n) u(m) = D−1(n m) h iF u |

m n + (Isovector) meson operator OT = d Γ u (e.g. π ):

Connected piece of a meson correlator OT (n) OT (m) = d(n)Γu(n) u(m)Γd(m) F F D E = Tr ΓD−1(n m)ΓD−1(m n) − u | d | h i m n

(Isoscalar) meson operator OS = d Γ d (e.g. f0): Disconnected piece of a meson correlator

−1 −1 OS(n) OS(m) = Tr ΓDu (n n) Tr ΓDu (m m) F | | D E TrΓD−1(n m)Γ D−1(m n) − u | u | Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions Some examples for propagators: Quark: u(n) u(m) = D−1(n m) h iF u |

m n + (Isovector) meson operator OT = d Γ u (e.g. π ):

Connected piece of a meson correlator OT (n) OT (m) = d(n)Γu(n) u(m)Γd(m) F F D E = Tr ΓD−1(n m)ΓD−1(m n) − u | d | h i m n

(Isoscalar) meson operator OS = d Γ d (e.g. f0): Disconnected piece of a meson correlator

m n + (Isovector) meson operator OT = d Γ u (e.g. π ):

Connected piece of a meson correlator OT (n) OT (m) = d(n)Γu(n) u(m)Γd(m) F F D E = Tr ΓD−1(n m)ΓD−1(m n) − u | d | h i m n

(Isoscalar) meson operator OS = d Γ d (e.g. f0): Disconnected piece of a meson correlator

Homework Compute the Grassmann integral(s):

1 dη1dη2 exp (η1 + η2 + a η1 η2). T 2 dx1dx2dy1dy2 exp (~x A ~y) , where A = (aij ) is a 2 2 matrix. R · · × 3 Rd(n)Γu(n) u(m)Γd(m) F .

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Fermion doubling

Naive fermion action:

4 U (n)ψ(n+ˆµ) U− (n)ψ(n µˆ) S [ψ, ψ, U] = a4 ψ(n) γ µ − µ − + m ψ(n) F µ 2a n∈ ! XΛ Xµ=1 = a4 ψ(n) D(n m) ψ(m) | (omitting Dirac and color indices)

Free fermions (U = 1):

4 δn+ˆµ,m δn−µ,ˆ m D(n m)= γ − + m δn m | µ 2a , 1 µX= The lattice Dirac operator D is a matrix with 3 nsites ncolor nDirac = 12 N nt rows and columns! × × s ×

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Let us Fourier-transform the free fermion matrix: 1 D(p q) = e−ip·na D(n m) eiq·ma | Λ | n m∈ | | X, Λ 4 e 1 e+iqµa e−iqµa = e−i(p−q)·na γ − + m1 Λ  µ 2a  n∈ 1 | | XΛ Xµ= = δ(p q) D(p) ,   − 4 (0,π/a ) (π/ ,π/ ) i a a D(p) = m1 + e γ sin(p a) a µ µ µ=1 1 −X1 e m ia γµ sin(pµa) (0,0) D(p)−1 = − µ . (π/ a ,0) m2 a−2 sin p a 2 + P µ ( µ ) e the massless propagatorPD−1 has 16 poles in the→ Brillouin-zone to “doubler fermions”!

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Homework Compute the naive continuum limit (ﬁxed p) of the lattice Dirac operator D(p) for free (naive) fermions.

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Wilson fermions

i 4 1 4 D(p) = m1 + γ sin(p a) + 1 1 cos(p a) a µ µ a − µ µ=1 µ=1 X X e changes the denominator of D−1: → 2 4 sin(p a)2 sin(p a)2 + (1 cos(p a)) µ → µ  − µ  µ µ X X Xµ=1 4.0  4.0  3.5 3.5

3.0 3.0

2.5 2.5 2.0 2.0 mdoubler 1.5 1.5

1.0 1.0

0.5 0.5

-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 ... gives extra mass 2 k/a to the doublers!

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Summing up the Wilson fermion action (Nf ﬂavors):

Nf 4 (f ) (f ) (f ) SF [ψ, ψ, U] = a ψ (n) D (n m) ψ (m) , | n m∈ Xf =1 X, Λ (f ) (f ) 4 D (n m) = m + δ δab δn m | a αβ , 1 ±4 (1 γ ) U (n)ab δn m, −2a − µ αβ µ +ˆµ, µX=±1 with γ− = γ , µ = 1, 2, 3, 4 . µ − µ

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Why is the fermion vacuum full of loops?

Let us write (removing an overall factor and for just one ﬂavor)

±4 D = (1 κH) with the “Hopping term” Hnm = (1 γ ) U (n) δn m . − − µ µ +ˆµ, µX=±1 with κ = 1/(m + 4/a).

A propagator then is

−1 2 2 3 3 Dnm = 1nm + κ Hnm + κ (H )nm + κ (H )nm ... K i.e., Hnm is sum of paths of length K from site n to site m. This is called the “Hopping expansion”. Theoretically intriguing. Practically unfeasible due to convergence problems for smaller quark masses! Sorry!

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Why is the fermion vacuum full of loops?

Let us write (removing an overall factor and for just one ﬂavor)

±4 D = (1 κH) with the “Hopping term” Hnm = (1 γ ) U (n) δn m . − − µ µ +ˆµ, µX=±1 with κ = 1/(m + 4/a).

A propagator then is

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Why is the fermion vacuum full of loops?

Let us write (removing an overall factor and for just one ﬂavor)

±4 D = (1 κH) with the “Hopping term” Hnm = (1 γ ) U (n) δn m . − − µ µ +ˆµ, µX=±1 with κ = 1/(m + 4/a).

A propagator then is

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Why is the fermion vacuum full of loops?

Let us write (removing an overall factor and for just one ﬂavor)

±4 D = (1 κH) with the “Hopping term” Hnm = (1 γ ) U (n) δn m . − − µ µ +ˆµ, µX=±1 with κ = 1/(m + 4/a).

A propagator then is

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

The determinant

∞ 1 det[D]= det[1 κH]= exp Trln 1 κH = exp κj Tr Hj − − − j  j=1 X   is a sum of closed loops (due to the trace).

“When fermions, the most antisocial type of quantum particle, do get together, they pair up in a wondrous dance...” from: ScienceDaily (Dec. 23, 2005)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Monte-Carlo simulation of fermions

“Full QCD”:

C(t) [U] [ψ, ψ] e−SG[U]−ψ D[U] ψ N(t)N¯ (0) ∝ D D −S [U] = R [U] e G (det Du det Dd ...) D −1 −1 R Du D ... + ... × d h i Set det D 1 (no dynamical fermion vacuum, i.e.≡ no sea quarks) Gauge ﬁeld vacuum is fully dynamical (Monte Carlo) Consider only the valence quarks Hadron correlation functions are built from the quark propagators

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Monte-Carlo simulation of fermions

Quenched approximation:

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Dynamical fermions:

−1 −N −φ†Aφ Bosons: det[A] = π [φ] [φI ] e D D Z Fermions: det[A] = [ψ] [ψ] e−ψAψ D D Z Replace fermions by pseudofermions:

† † −1 −ψ Dψu −ψ Dψd −N −φ (DD ) φ [ψ] [ψ] e u d = π [φR] [φI ] e . D D D D Z Z Doubling is necessary in order to ensure positivity: det[D] det[D]= det[D] det[D†]= det[DD†] 0 ≥

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Simulation with Hybrid Monte Carlo (HMC) algorithm:

[U] exp( S[U]) O[U] O = D − h iQ [U] exp( S[U]) R D − 1 2 R[U] [P] exp( 2 P S[U]) O[U] = D D − − = O P,U . [U] [P] exp( 1 P2 S[U]) h i R D D − 2 − R For the dynamical fermion simulation S[U] and [U] include the pseudofermion terms. D

Microcanonical ensemble canonical ensemble →

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Hybrid Monte Carlo (HMC) algorithm Given a configuration U generate random set of conjugate momenta P with probability distribution exp( P2/2). − Evolve configuration (P, U)with the P Hamiltonian (P2/2 + S[U]): 100 small steps (molecular dynamics trajectory, leapfrog U algorithm) Correct at the end of that trajectory with a Monte Carlo accept/reject step, accepting the new configuration with probability

exp( H[P′, U′] min 1, − exp( H[P, U] − Repeat many times.

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Custom computers

Abakus/Suanpan/Soroban (0.1 FLOPS?)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Custom computers

Staatl. Kunstsammlungen Dresden, Mathematisch-Physikalischer Salon; Computer, 1650, (0.1 FLOPS estimated)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Custom computers PC Clusters "Self-made" computers (apeNEXT, QCD-on-a-chip "QCDOC", QPACE (QCD PArallel computing on CEll)) GPUs (graphics processing units) and GPU farms SuperMUC at LRZ Munich (3.2 PetaFLOPS)

Juqueen at Jülich (5.9 PetaFLOPS) QPACE/Jülich/Wuppertal/Regensburg (200 TFLOPS)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

“Berlin Wall”

(from C. Urbach, LATTICE 2006)

Christian B. Lang Lattice QCD for Pedestrians Fermion integration Quarks Fermion doubling Chiral symmetry and fermions species Wilson fermions Monte-Carlo simulation of fermions

Key points In the path integral fermions are Grassmann variables. The lattice Dirac operator is a huge matrix. Quark propagators are entry of the inverse matrix, hadron propagators are built from quark propagators. In the naive action there are 16-fold too many fermions. Wilson’s action moves these doublers to higher masses. The hopping expansion visualizes the quark paths. Dynamical fermions are “simulated” by pseudofermions (=bosons) in the Hybrid Monte Carlo algorithm.

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Chiral symmetry

Continuum QCD Massless u/d quarks (N = 2) chiral symmetry f → SU(2)R SU(2)L U(1)V U(1)A. × × ×

U(1)A broken by anomaly (non-invariance of fermion integration measure)

SU(2)R SU(2)L is spontaneously broken by QCD: × SU(2)V -multiplets + Goldstone bosons (pions)

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

The symmetry is manifest by

D γ5 + γ5 D = 0

Left-handed or right-handed fermions are “zero modes” and eigenstates of the Dirac operator:

γ5 ψ± = ψ± ± D ψ± = 0

Massless fermions have deﬁnite chirality. Atiyah-Singer Index Theorem: topological charge of gauge ﬁeld ν = n− n . − + Topological charge is a concept for differentiable manifolds, i.e. continuum; lattice implementation? The condensate ψψ is the order parameter of chiral symmetry breaking h i

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Lattice QCD Chiral symmetry is a problem for LQCD! The formulation should allow explicite chiral symmetry, such that it can be broken spontaneously! No-go theorem (Nielsen, Ninomiya, 1982): Lattice theories do not allow simultaneously chiral invariance, locality, and correct continuum behavior of quark propagators. Finally excavated (Hasenfratz):

1 D γ + γ D = aD γ D 5 5 2 5 Ginsparg-Wilson condition (1982!) for chiral lattice fermions. Consequences for the spectrum of D: zero modes, Banks-Casher! “Lattice chiral symmetry” transformation (Lüscher). The GWC is violated for simple Dirac operators (simple fermion actions)!

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Fermion species

Non-GW type: Wilson improved Staggered Twisted mass

Approximate GW type: Domain Wall Fixed Point Chirally Improved

Exact GW type: Overlap

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

“Clash of discretizations”

Staggered fermions

Reduction from 16 down to 4 Dirac fermions (i.e. 16 Grassmann variables), distributed over the 16 sites of the hypercube Remnant chiral symmetry Asqtad: a2 and tadpole improved 4th root trick: (det D)1/4??? tastes → Simple implementation , harder interpretation Squared pion mass as a (taste splitting) → HISQ - Highly Improved function of the light quark Staggered Quarks masses (MILC,PoS LAT2006:163)

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Wilson improved

λ Symanzik improvement program: (a2) Wilson improvement by clover leaf term O

5 i S + a cSW ψ(x) σ Fˆ ψ(x) W x 4 µν µν m ?

0.2 A1 (Sheikoleslami/Wohlert),P 0.1 0

0.2 coefﬁcient tuned A2 C1 non-perturbatively 0.1 0

0.2 Simple implementation A3 0.1 Doubler modes 0 0.2 A4 Problem with small quark masses: Spurious 0.1 0 0 20 40 60 80 low-lying eigenmodes of the Dirac operator µ[MeV] (DelDebbio et al., JHEP 0602(2006) Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Phase diagram for Wilson fermions:

κ = 1/4 κ (β) ch

κ =1/8

κ = 0 κ = 0 β = 0 β = ∞

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Twisted mass: tmQCD (Frezotti, Grassi, Sint, Weisz)

Wilson + "twisted mass" i µ ψγ5τ3 ψ

m = mcrit, µ> 0 “maximal twist” is (a) improved O No spurious zero modes λ tm

m ? breaks parity and ﬂavor

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Domain Wall

Kaplan, Furman, Shamir: Introduce extra 5th dimension N5 Left-handed and right-handed part of the fermion is bound to the 4-dimensional interface walls.

They decouple for N5 → ∞

5th dimension In the limit one gets exact GW type: the Overlap operator (Neuberger)

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Overlap Dirac operator(Neuberger) Can be constructed explicitly (a = 1 for simplicity):

D(m = 0) = (1 + γ5 sign(H)) with H = γ5 DW (m < 0)

(γ5DW is hermitian.) Sign function of an hermitian matrix?

H = λ λ λ f (H)= f (λ) λ λ | ih | → | ih | Xλ Xλ Thus sign(H)= sign(λ) λ λ | ih | Xλ is expensive (hardly possible).

Alternative: H sign(H)= √H2 approximated by Chebyshef polynomial series or rationals.

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Exact Ginsparg-Wilson type Nice circular spectrum: λ overlap

D γ5 + γ5 D = D γ5D → † † γ5 D γ5 + D = γ5 D γ5 D D + D = D D → → λ∗ + λ = λ∗λ (λ 1)(λ∗ 1)= 1 → − − → λ 1 = 1 m | − | - exact zero modes - spectral density related to condensate via Banks-Casher (and RMT studies): ψψ lim lim ρ(Imλ, V ) h i∝ Imλ→0 V →∞

Allows to implement nf = 1. 50-100 times more expensive than Wilson’s operator. Problems with sector tunneling in HMC implementations.

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Homework Prove that the Wilson Dirac operator (like most others, except the tm- operator) is γ5-hermitian, i.e.,

† γ5 D = (γ5 D) .

Show that a Ginsparg-Wilson operator is “normal”, i.e., it commutes with its hermitian conjugate: [D, D†] = 0. What does this imply for the eigenvalues and eigenvectors?

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Approximate GW-operators:

Fixed Point action

A perfect action would follow a renormalized trajectory and have no corrections to scaling. The ﬁxed point action (Hasenfratz, Niedermayer) may deviate from the renormalized trajectory renormalized trajectory

FP fixed point action

Parameterized form of the action with tuned parameter based on blockspin transformations and operator matching. Studied in the BGR-collaboration

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Chirally Improved Dirac operator

Wilson

s1 +s2 +s3 s +4 .... General ansatz for fermion action: − + γ −+ − + 16 + µ v1 +v2 +v3 .... α − + Dmn = Γα cp Ul δn,m+p p∈Pα αX=1 Xm,n Yl∈p + − γ γ − + .... γ γ γ γ + µ ν t1 +µ ν ρ a1 .... + 5 p1 .... + − − + (Gattringer, PRD63(2001)114501) Insert the ansatz in the GW-equation, truncate the length of the contributions (to,e.g., 4) and compare the coefﬁcients!

Leads to a set of (e.g. 50) algebraic equations, which can be solved (norm minimization).

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Eigenmodes and instantons

Exact zero modes are chiral and “deﬁne” the topological sector. Non-GW-operators: real eigenmodes play this role.

∗ Iso-surfaces of eigenvector density p0(x)= v (c, α, x) v(c, α, x) : c,α X

Wilson Chirally improved Overlap (λ = 0.14) (λ = 0.016) (λ = 0)

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Near zero modes (small real part, small imaginary part) deﬁne the density ρ(m, V , λ) related (via Banks-Casher) to the condensate: σ = lim lim lim ρ(m, V , λ). − m→0 Im(λ)→0 V →∞ ChPT and RMT predict the shape of the distributions in universality classes.

Kieburg/Verbaarschot/Zafeiropoulos; arXiv:1307.7251 (2013) Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Phase diagram for fermions (idealized):

continuum chiral limit limit m = 0

constant physicsconstant a

m = ∞ β = 0 ( g = ∞) quenched limit β = ∞ ( g = 0)

(Lines of constant physics: e.g., ﬁxed fK /mK and mπ/mK ,or...)

Christian B. Lang Lattice QCD for Pedestrians Ginsparg-Wilson condition Quarks Fermion species Chiral symmetry and fermions species Eigenmodes and instantons

Key points The Ginsparg-Wilson condition ensures lattice chiral symmetry. Simple (computationally less expensive) lattice Dirac operators violate the GWC. The overlap operator obeys the GWC but is very costly and numerically demanding. Today most “full” LQCD calculations on large lattices are for staggered or Wilson-improved actions. (Then: Twisted mass, domain wall, ...) The eigenvalues of the Dirac operator provide information on instantons, the condensate and the mechanism of spontaneous chiral symmetry breaking.

Christian B. Lang Lattice QCD for Pedestrians