Lattice Quantum Field Theory

Transcription

1 Lattice Quantum Field Theory An introduction

2

3 1 Quarks 2 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons

4 Quarks QFT path integral: A = 1 Z = 1 Z D[U] D[ψ,ψ] e S G[U] S F [ψ,ψ,u] A[ψ,ψ, U] { } D[U] e S G[U] D[ψ,ψ] e SF[ψ,ψ,U] A[ψ,ψ, U] Bosons: usual integration (a group integral on for each link) Fermions: Grassmann variables ( anti-commuting c-numbers )

5 Quarks QFT path integral: A = 1 Z = 1 Z D[U] D[ψ,ψ] e S G[U] S F [ψ,ψ,u] A[ψ,ψ, U] { } D[U] e S G[U] D[ψ,ψ] e SF[ψ,ψ,U] A[ψ,ψ, U] Bosons: usual integration (a group integral on for each link) Fermions: Grassmann variables ( anti-commuting c-numbers )

6 Grassmann variables...are anti-commuting numbers: η i η j = η j η i for all i, j, thus η i η i = 0...have the integration rules: d N η = dη N dη N 1... dη 1, dη i dη j = dη j dη i, dη i η j = η j dη i dη i 1 = 0, dη i η i = 1, e.g. dη 2 dη 1 (1+2η 1 42η 1 η 2 ) = 42 cf.

7 These rules lead to, e.g., the Matthews-Salam formula: ( N ) Z F = dη N dη N... dη 1 dη 1 exp η i M ij η j = det[m], where M is a complex N N matrix. i,j=1 Note: This looks almost like Gaussian integration for bosons, except that there the result is det[m] 1!

8 The generating functional for fermions is then given by W [ θ,θ ] N N N N = dη i dη i exp η k M kl η l + θ k η k + η k θ k i=1 k,l=1 k=1 k=1 N ( = det[m] exp θ n M 1 ) nm θ m n,m=1 and, e.g., W [ θ,θ ] θ m θ = n θ,θ=0 N N dη i dη i η n η m exp η k M kl η l i=1 k,l=1 = det[m] ( M 1) nm

9 N-point functions can be computed by (Wick s theorem): η i1 η j1...η in η jn F = 1 N dη k dη Z k η i1 η j1...η in η jn exp F k=1 = 1... W [ θ,θ ] Z F θ j1 θ i1 θ jn θ in N η l M lm η m l,m=1 θ,θ=0 with Z F = det[m]. Quarks and anti-quarks: η i ψ (f) (x,α, a) or, e.g. u(x,α, a) η i ψ (f) (x,α, a) or, e.g. u(x,α, a) M D lattice Dirac operator = matrix

10 Some examples for propagators: Quark: u(n) u(m) F = Du 1 (n m) (Isovector) meson operator O T = d Γ u (e.g. π + ): O T (n) O T (m) F = d(n)γu(n) u(m)γd(m) F [ ] = Tr ΓDu 1 (n m)γd 1 d (m n) m (Isoscalar) meson operator O S = d Γ d (e.g. f 0 ): O S (n) O S (m) F = Tr [ ΓDu 1 (n n) ] Tr [ ΓDu 1 (m m) ] Tr [ ΓDu 1 (n m)γdu 1 (m n) ] m Connected piece of a meson correlator Disconnected piece of a meson correlator n n

11 Some examples for propagators: Quark: u(n) u(m) F = Du 1 (n m) (Isovector) meson operator O T = d Γ u (e.g. π + ): O T (n) O T (m) F = d(n)γu(n) u(m)γd(m) F [ ] = Tr ΓDu 1 (n m)γd 1 d (m n) m (Isoscalar) meson operator O S = d Γ d (e.g. f 0 ): O S (n) O S (m) F = Tr [ ΓDu 1 (n n) ] Tr [ ΓDu 1 (m m) ] Tr [ ΓDu 1 (n m)γdu 1 (m n) ] m Connected piece of a meson correlator Disconnected piece of a meson correlator n n

12 Some examples for propagators: Quark: u(n) u(m) F = Du 1 (n m) (Isovector) meson operator O T = d Γ u (e.g. π + ): O T (n) O T (m) F = d(n)γu(n) u(m)γd(m) F [ ] = Tr ΓDu 1 (n m)γd 1 d (m n) m (Isoscalar) meson operator O S = d Γ d (e.g. f 0 ): O S (n) O S (m) F = Tr [ ΓDu 1 (n n) ] Tr [ ΓDu 1 (m m) ] Tr [ ΓDu 1 (n m)γdu 1 (m n) ] m Connected piece of a meson correlator Disconnected piece of a meson correlator n n

13 Homework Compute the Grassmann integral(s): 1 dη1 dη 2 exp(η 1 +η 2 + aη 1 η 2 ). 2 dx1 dx 2 dy 1 dy 2 exp( x T A y), where A = (a ij ) is a 2 2 matrix. d(n)γu(n) u(m)γd(m) 3 F.

14 Naive fermion action: ( 4 ) S F [ψ,ψ, U] = a 4 U µ (n)ψ(n+ˆµ) U µ (n)ψ(n ˆµ) γ µ + mψ(n) 2a n Λψ(n) µ=1 = a 4 ψ(n) D(n m)ψ(m) (omitting Dirac and color indices) Free fermions (U = 1): D(n m) = 4 δ n+ˆµ,m δ n ˆµ,m γ µ 2a µ=1 + mδ n,m The lattice Dirac operator D is a matrix with n sites n color n Dirac = 12 N 3 s n t rows and columns!

15 Let us Fourier-transform the free fermion matrix: D(p q) = 1 Λ = 1 Λ n,m Λ n Λ e ip na D(n m) e iq ma e i(p q) na 4 µ=1 = δ(p q) D(p), D(p) = m1 + i 4 γ µ sin(p µ a) a µ=1 D(p) 1 = m1 ia 1 µ γ µ sin(p µ a) m 2 + a 2 µ sin(p µa) 2. γ µ e +iqµa e iqµa 2a the massless propagator D 1 has 16 poles in the Brillouin-zone to doubler fermions! + m1 (0,π/ a) (π/ a,π/ a) (0,0) (π/ a,0)

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

17 D(p) = m1 + i a 4 γ µ sin(p µ a) a µ=1 changes the denominator of D 1 : 4 ( 1 cos(pµ a) ) µ=1 sin(p µ a) 2 µ µ 4 sin(p µ a) 2 + (1 cos(p µ a)) µ= m doubler gives extra mass 2 k/a to the doublers!

18 Summing up the Wilson fermion action (N f flavors): S F [ψ,ψ, U] = D (f) (n m) = N f a 4 ψ (f) (n) D (f) (n m)ψ (f) (m), ( f=1 n,m Λ m (f) + 4 ) δ αβ δ ab δ n,m a 1 ±4 (1 γ µ ) 2a αβ U µ (n) ab δ n+ˆµ,m, µ=±1 with γ µ = γ µ, µ = 1, 2, 3, 4.

19 Why is the fermion vacuum full of loops? Let us write (removing an overall factor and for just one flavor) D = (1 κh) with the Hopping term H nm = with κ = 1/(m + 4/a). ±4 µ=±1 (1 γ µ ) U µ (n)δ n+ˆµ,m. A propagator then is D 1 nm = 1 nm +κ H nm +κ 2 (H 2 ) nm +κ 3 (H 3 ) nm... i.e., Hnm K 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!

20 Why is the fermion vacuum full of loops? Let us write (removing an overall factor and for just one flavor) D = (1 κh) with the Hopping term H nm = with κ = 1/(m + 4/a). ±4 µ=±1 (1 γ µ ) U µ (n)δ n+ˆµ,m. A propagator then is D 1 nm = 1 nm +κ H nm +κ 2 (H 2 ) nm +κ 3 (H 3 ) nm... i.e., Hnm K 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!

21 Why is the fermion vacuum full of loops? Let us write (removing an overall factor and for just one flavor) D = (1 κh) with the Hopping term H nm = with κ = 1/(m + 4/a). ±4 µ=±1 (1 γ µ ) U µ (n)δ n+ˆµ,m. A propagator then is D 1 nm = 1 nm +κ H nm +κ 2 (H 2 ) nm +κ 3 (H 3 ) nm... i.e., Hnm K 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!

22 Why is the fermion vacuum full of loops? Let us write (removing an overall factor and for just one flavor) D = (1 κh) with the Hopping term H nm = with κ = 1/(m + 4/a). ±4 µ=±1 (1 γ µ ) U µ (n)δ n+ˆµ,m. A propagator then is D 1 nm = 1 nm +κ H nm +κ 2 (H 2 ) nm +κ 3 (H 3 ) nm... i.e., Hnm K 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!

23 The determinant ( det[d] = det[1 κh] = exp Tr ln ( 1 κh )) 1 = exp j is a sum of closed loops (due to the trace). j=1 κ j Tr [ H j] When fermions, the most antisocial type of quantum particle, do get together, they pair up in a wondrous dance... from: ScienceDaily (Dec. 23, 2005)

24 Full QCD : C(t) D[U] D[ψ,ψ] e S G[U] ψ D[U]ψ N(t) N(0) = D[U] e SG[U] (det D u det D d...) [ ] Du 1 D 1 d 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

25 Quenched approximation: C(t) D[U] D[ψ,ψ] e S G[U] ψ D[U]ψ N(t) N(0) = D[U] e SG[U] (det D u det D d...) [ ] Du 1 D 1 d 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

26 Dynamical fermions: Bosons: det[a] 1 = π N D[φ]D[φ I ] e φ Aφ Fermions: det[a] = D[ψ]D[ψ] e ψaψ Replace fermions by pseudofermions: D[ψ]D[ψ] e ψ u Dψu ψ d Dψ d = π N D[φ R ]D[φ I ] e φ (D D ) 1φ. Doubling is necessary in order to ensure positivity: det[d] det[d] = det[d] det[d ] = det[d D ] 0

27 Simulation with Hybrid Monte Carlo (HMC) algorithm: O Q = = D[U] exp( S[U]) O[U] D[U] exp( S[U]) D[U]D[P] exp( 1 2 P2 S[U]) O[U] = O D[U]D[P] exp( 1 P,U. 2 P2 S[U]) For the dynamical fermion simulation S[U] and D[U] include the pseudofermion terms. Microcanonical ensemble canonical ensemble

28 Hybrid Monte Carlo (HMC) algorithm Given a configuration U generate random set of conjugate momenta P with probability distribution exp( P 2 /2). Evolve configuration (P, U)with the Hamiltonian (P 2 /2+S[U]): 100 small steps (molecular dynamics trajectory, leapfrog algorithm) Correct at the end of that trajectory with a Monte Carlo accept/reject step, accepting the new configuration with probability ( min 1, exp( H[P, U ) ] exp( H[P, U] P U Repeat many times.

29 Custom computers Abakus/Suanpan/Soroban (0.1 FLOPS?)

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

31 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)

32 Berlin Wall (from C. Urbach, LATTICE 2006)

33 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.

34 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons 1 Quarks 2 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons

35 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Chiral symmetry Continuum QCD Massless u/d quarks (N f = 2) chiral symmetry 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)

36 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons The symmetry is manifest by Dγ 5 +γ 5 D = 0 Left-handed or right-handed fermions are zero modes and eigenstates of the Dirac operator: γ 5 ψ ± = ±ψ ± Dψ ± = 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

37 Ginsparg-Wilson condition Fermion 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): Dγ 5 +γ 5 D = 1 2 a Dγ 5 D 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)!

38 Ginsparg-Wilson condition Fermion 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): Dγ 5 +γ 5 D = 1 2 a Dγ 5 D 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)!

39 Ginsparg-Wilson condition Fermion 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): Dγ 5 +γ 5 D = 1 2 a Dγ 5 D 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)!

40 Ginsparg-Wilson condition Fermion 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

41 Ginsparg-Wilson condition Fermion 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: a 2 and tadpole improved 4th root trick: (det D) 1/4??? tastes Simple implementation, harder interpretation (taste splitting) HISQ - Highly Improved Staggered Quarks Squared pion mass as a function of the light quark masses (MILC,PoS LAT2006:163)

42 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Wilson improved Symanzik improvement program: O(a 2 ) improvement by clover leaf term λ Wilson S W + a 5 c SW x ψ(x) i 4 σ µν ˆF µν ψ(x) m? (Sheikoleslami/Wohlert), coefficient tuned non-perturbatively Simple implementation Doubler modes Problem with small quark masses: Spurious low-lying eigenmodes of the Dirac operator 0.2 A A 2 C A A µ[mev] (DelDebbio et al., JHEP 0602(2006)

43 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Phase diagram for : κ = 1/4 κ ch (β) κ =1/8 κ = 0 β = 0 κ = 0 β =

44 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Twisted mass: tmqcd (Frezotti, Grassi, Sint, Weisz) Wilson + "twisted mass" iµψγ 5 τ 3 ψ m = m crit, µ > 0 maximal twist is O(a) improved No spurious zero modes λ tm m? breaks parity and flavor

45 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Domain Wall Kaplan, Furman, Shamir: Introduce extra 5th dimension N 5 Left-handed and right-handed part of the fermion is bound to the 4-dimensional interface walls. They decouple for N 5 5th dimension In the limit one gets exact GW type: the Overlap operator (Neuberger)

46 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Overlap Dirac operator(neuberger) Can be constructed explicitly (a = 1 for simplicity): D(m = 0) = (1+γ 5 sign(h)) with H = γ 5 D W (m < 0) (γ 5 D W is hermitian.) Sign function of an hermitian matrix? H = λ λ λ λ f(h) = λ f(λ) λ λ Thus sign(h) = λ sign(λ) λ λ is expensive (hardly possible). Alternative: sign(h) = H H 2 approximated by Chebyshef polynomial series or rationals.

47 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Exact Ginsparg-Wilson type Nice circular spectrum: Dγ 5 +γ 5 D = Dγ 5 D γ 5 Dγ 5 + D = γ 5 Dγ 5 D D + D = D D λ +λ = λ λ (λ 1)(λ 1) = 1 λ overlap λ 1 = 1 m - exact zero modes - spectral density related to condensate via Banks-Casher (and RMT studies): ψψ lim lim ρ(imλ, V) Imλ 0 V Allows to implement n f = times more expensive than Wilson s operator. Problems with sector tunneling in HMC implementations.

48 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Homework Prove that the Wilson Dirac operator (like most others, except the tmoperator) is γ 5 -hermitian, i.e., γ 5 D = (γ 5 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?

49 Ginsparg-Wilson condition Fermion 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 fixed point action FP Parameterized form of the action with tuned parameter based on blockspin transformations and operator matching. Studied in the BGR-collaboration

50 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Chirally Improved Dirac operator General ansatz for fermion action: 16 D mn = Γ α α=1 p P α m,n c α p U l δ n,m+p l p Wilson s + s 2 + s 3 s γ v 1 + v 2 + v 3... µ + + γ µ γ ν t γ µ γ ν γ ρ a 1 γ 5 p 1 (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). + +

51 Ginsparg-Wilson condition Fermion 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 p 0 (x) = v (c,α, x) v(c,α, x) : c,α (λ = 0.14) Wilson Chirally improved (λ = 0.016) (λ = 0) Overlap

52 Ginsparg-Wilson condition Fermion 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 m 0 Im(λ) 0 V ρ(m, V,λ). ChPT and RMT predict the shape of the distributions in universality classes. Kieburg/Verbaarschot/Zafeiropoulos; arxiv: (2013)

53 Ginsparg-Wilson condition Fermion species Eigenmodes and instantons Phase diagram for fermions (idealized): m = 0 chiral limit constant physics constant a continuum limit m = β = 0 ( g= ) quenched limit β = ( g= 0) (Lines of constant physics: e.g., fixed f K /m K and m π/m K, or... )

54 Ginsparg-Wilson condition Fermion 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.