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Lattice Field Theory Lattice Field Theory Karl Jansen • Lattice: Non-perturbative aspects of eld theories Lattice Quantum Chromodynamics Fermion-Higgs Systems Spin Systems Chiral Gauge Theories Supersymmetry Quantum Gravity Algorithms and Machines General articles Lectures, review articles • R. Gupta Introduction to Lattice QCD, hep-lat/9807028 • C. Davies Lattice QCD, hep-ph/0205181 • M. Lüscher Advanced Lattice QCD, hep-lat/9802029 Chiral gauge theories revisited, hep-th/0102028 • A.D. Kennedy Algorithms for Dynamical Fermions, hep-lat/0607038 1 Books about Lattice Field Theory • T. DeGrand and C. DeTar Lattice methods for Quantum Chromodynamics World Scientic, 2006 • H.J. Rothe Lattice gauge theories: An Introduction World Sci.Lect.Notes Phys.74, 2005 • J. Smit Introduction to quantum elds on a lattice: A robust mate Cambridge Lect.Notes Phys.15, 2002 • I. Montvay and G. Münster Quantum elds on a lattice Cambridge, UK: Univ. Pr., 1994 • Yussuf Saad Iterative Methods for sparse linear systems Siam Press, 2003 2 The Lectures http://www-zeuthen.desy.de/∼kjansen/ The harmonic oscillator programme J. Gonzales Lopez, A. Nube, D. Renner http://www-zeuthen.desy.de/∼kjansen/ 3 Plan of Lectures • Introduction to the Lattice Quantum Mechanics on the lattice Schwinger model • Algorithms, Errors and Machines Foundations of simulation algorithms Hybrid Monte Carlo Algorithm (plus improvements Intermezzo: a demonstration Auto correlation time and error analysis A selection of supercomputers • Lattice QCD Static Potential • Lattice QCD today 4 Äîáðî ïîæàëîâàòü â òåîðèþ êàëèáðîâî÷íûõ ïîëåé íà ðåøøòêàõººº È â ìèð åø îïàñíûõ æèâîòíûõ! Dobrodosli na reˇsetku i opasne ˇzivotinje na njoj Kal¸c rjate sto plègma kai ta gria jhrÐa tou 5 ½ Quantum mechanical oscillator on a Euclidean time-lattice Feynman and Hibbs; Creutz and Freedman, Annals Phys.13,1981 We start with Feynman path integral in quantum mechanics R i S Z = Dqe~ cl • Scl classical action, e.g. quantum mechanical oscillator R 1 2 Scl = dt 2q˙ (t) − V (q(t)) • q(t) are classical paths 6 perform an analytical continuation to imaginary (in general complex) time t → −iτ f(t) → f(τ) with euclidean time τ the action then becomes " # Z 1 dq(τ) 2 S → (−i)dτ − V (q(τ)) 2 d(−iτ) Z 1 = i dτ q˙2+V (q) 2 ≡ iSE SE is the euclidean action and R −1S Z = Dqe ~ E the euclidean path integral 7 the expression for the euclidean path integral 1 R − SE R 1 2 Z = Dqe ~ ,SE = dτ 2q˙ + V (q) introduce a time lattice with N = T/a lattice points 0 { a T q(τ) → q(na) , q˙ → q˙lattice boundary condition: q(N + 1) = q(0) discretization of continuum time-derivative q˙ not unique 8 Discretizing the Quantum Mechanical Oscillator Verstegen, Phys.Lett.B122,63,1983 R 2 1 2 SE = dt q˙ (t) + 2mq (t) discr P 1 1 1 2 m 2 SE = a n a 2(1 + β)q(n + a) − βq(n) − 2(1 − β)q(n − a) + 2 q (n) β = 1 : [q(n + a) − q(n)] /a 1 0 1 00 2 = a q(n) + aq + 2q a + · · · − q(n) =q ˙ + O(a) β = 0 : [q(n + a) − q(n − a)] /2a 1 0 1 00 2 0 1 00 2 2 = 2a q(n) + aq + 2q a + · · · − q(n) + q a − 2q a =q ˙+O(a ) 9 Eects of dierent Discretizations with q(n) = √1 P eipnaqˆ(p) , p = jπ/aN , j = 0, 1, 2, ··· ,N − 1 2 p discr 1 P ∗ SE = 2a p qˆ (p)D(p)ˆq(p) 2 1 2 4 1 2 D(p) = 4 sin (2pa) + 4(β − 1) sin (2pa) + a m • dierent β : obtain continuum p2 with dierent rate • β = 0 : obtain continumm p2 behaviour for p → 0 and p → π 10 Lessons from discretizing the Oscillator • dierent choices of lattice derivatives can lead to O(a), O(a2), O(a3), ··· lattice artefacts → very dierent rates to approach continuum limit • there are wrong choices of lattice derivatives → lead to wrong continuum theory 11 The Transfer Matrix Restricting ourselves to the choice qn+1−qn (T=N ) q˙ → a a “q −qn ”2 ff −a P 1 n+1 +V (q) Q R ~ n 2 a Z = n dqne we obtain he continuum expression by sending N → ∞ while keeping T xed ⇒ a → 0 12 relation to canonical quantization: 2 1 2 1 q0−q a2q˙n + V (qn) = a2 a + V (q)a introduce coordinate and momentum operators: pˆ|pi = p|pi;q ˆ|qi = q|qi pˆ = qˆ˙ = (qˆ0 − qˆ)/a 2 2 „ 0 « „ ˆ0 « a1 q −q +V (q)a a1 q −qˆ +V (ˆq)a e 2 a = hq0|e 2 a |qi Z 2 = dphq0|e−pˆ a|pihp|e−V (ˆq)a|qi Z 2 = dp hq0|pihp|qi e−p ae−V (q)a | {z 0 } eip(q−q ) (1) 13 using the Baker-Hausdor formula 1 eAeB = eA+B+2[A,B]+... we obtain for an innitesimal change from q(t) → q0(t + a) 2 hq0|e−pˆ ae−V (ˆq)a|qi + O(a2) = hq0|e−aH|qi with Hamilton operator H 1 2 H = 2pˆ + V (ˆq) 14 pathintegral and operator picture classical fields operator picture picture Z 1 1 − SE − HT Dqe ~ = hqN|e ~ |q0i Z Y −1Ha = dqihqN|e ~ |qN−1i · ... i −1Ha ·hqi+1|e ~ |qii · ... −1Ha ·hq1|e ~ |q0i 15 −1Ha the operator T = e ~ is called the transfer matrix and describes innitesimal time steps of the system the matrix elements of the transfer operator are −1Ha hqi+1|e ~ |qii with qN = q0 ≡ q Z = TrTN → partition function in statistical mechanics 16 inserting an energy eigenbasis −1Ha X −1Ha hqi+1|e ~ |qii = hqi+1|e ~ |EihE|qii E X −1Ea = hqi+1|EihE|qiie ~ E → euclidean Hamilton operator ~ H = −a ln T 17 ⇒ transfer matrix has to be positive denite • positivity of T is sucient to guarantee that all n-point functions can be rotated back to Minkowski space (reconstruction theorem) • positivity of T ⇔ Osterwalder-Schrader reection positivity • Schrödinger equation in euclidean time 1 ∂ 2∆Ψ = ∂τ Ψ becomes analoguous to a diusion equation (with diusion constant 1 ) D = 2 18 Schwinger model: 2-dimensional Quantum Electrodynamics Schwinger 1962 the free Dirac equation (~ = c = 1) (iσµ∂µ − m) Ψ = 0 can be derived from the Lagrange density Ψ(¯ x)(iσµ∂µ − m) Ψ(x) σµ :-Pauli matrices (will also use γµ) Ψ 2-component Grassmann vector → represent fermionic elds (electron and anti-electron) 19 Principle of Local Gauge Invariance interaction of elds mediated by gauge degrees of freedom: photon → replace derivative with the covariant derivative ∂µΨ(x) → (∂µ − ig0Aµ(x))Ψ(x) ≡ DµΨ(x) with Aµ gauge potential, g0 bare coupling R 2 Sferm = d xΨ(¯ x)[Dµ + m] Ψ(x) is locally gauge invariant under a gauge transformation Λ(x): iΛ(x) Aµ(x) → Aµ(x) + ∂µΛ(x) , Ψ(x) → e Ψ(x) Gauge Field Self-Interaction R 2 Sgauge = d xFµνFµν ,Fµν(x) = ∂µAν(x) − ∂νAµ(x) equations of motion: obtain classical Maxwell equations 20 Schwinger Model Quantization via Feynman path integral R −Sgauge−S Z = DAµDΨDΨ¯ e ferm • existence of bound states (mass gap) • asymptotic free (g0 → 0 for distance between charges going to zero) • super-renormalizable • exactly solvable for zero fermion mass (Coleman) ⇒ valuable test laboratory for QCD (simulations can be done on your desk top) 21 Going to the Lattice introduce a 2-dimensional lattice with lattice spacing a elds Ψ(x), Ψ(¯ x) on the lattice sites x = (t, x) integers we then dene discrete derivatives 1 ∗ 1 ∇µΨ(x) = a [Ψ(x + aµˆ) − Ψ(x)] , ∇µΨ(x) = a [Ψ(x) − Ψ(x − aµˆ)] with µˆ a unit vector in direction µ replace (symmetric derivative): 1 ∗ ∂µ → 2 ∇µ + ∇µ 22 Lattice Partition Function 2 P ¯ 1 ∗ Z = R DΨ¯ DΨe−a x Ψ(x)[2σµ(∇µ+∇µ)+m]Ψ(x) R Q R DΨ = x dΨ(x) with R dΨ(x) a usual Grassmann integral version of naive lattice Schwinger model involves only classical elds Mathews-Salam formulae: for a n × n matrix D Z Y ¯ dΨ(¯ x)dΨ(x)e−ΨxDx,yΨy = detD x Z ¯ Y ¯ ¯ −ΨxDx,yΨy −1 dΨ(x)dΨ(x)ΨiΨje = D ij detD x 23 a rst look at the continuum limit we introduce external sources η, η¯ and write the action as S = Ψ¯ KΨ Z(η, η¯) = R DΨ¯ DΨe−ψD¯ Ψ+ψη¯ +¯ηΨ the 2-point function (Green's function) G is obatined by 1 ∂ ∂ G = Z(0,0) ∂η ∂η¯Z(η, η¯) η=¯η=0 we have −1 −1 −1 −1 Z(η, η¯) = R DΨ¯ DΨe−(Ψ¯ −ηD¯ )D(Ψ−D η)+¯ηD η = det(D)eηD¯ η and obtain −1 1 ∗ −1 G = D = 2γµ ∇µ + ∇µ + m 24 we evaluate the 2-point function in momentum space impose periodic boundary conditions Ψ(x + Lµ) = Ψ(x) This means that • the momenta at restricted −π/a < pµ ≤ π/a (the rst Brillouin zone) • the momenta are quantized (⇐ eip(x+L) = eipx) pµ = 2nπ/Lµ, n = 0, 1,...,Lµ − 1 and we have 1 P ˜ ipx Ψ(x) = V p Ψpe 2 with V = L the lattice volume assuming Lµ ≡ L for each µ 25 h i 1 ∗ ˜ ipx 1 ip(x+aµ) ip(x−aµ) ˜ ˜ ipx 2γµ ∇µ + ∇µ + m Ψpe = 2aγµ e − e Ψp + mΨpe | {z } eipx(eipaµ−e−iapµ) −1 i ˜ 1 i −1 −aγµ sin pµa+m G = a (iγµ sin pµa) + m ≡ a (ˆpµγµ + m) = pˆ2+m2 continuum limit a → 0 i −aγµ sin pµa → γµpµ and we obtain ˜ −ipµγµ+m lima→0 G = p2+m2 this is exactly the free continuum propagator! 26 but now comes this little devil, asking what if pµ ≈ π ? with pµ = kµ + π/a : sin pµa = sin(kµ + π) = − sin kµ which means that for a → 0 we obtain ˜ −ikµγµ+m G → − k2+m2 ⇒ again we obtain the continuum 2-point funktion! the opposite sign can be interpreted such that this continuum fermion at p ≈ π/a has the opposite chirality from the one at p ≈ 0 → nd proliferation of fermions (0, 0) (π, 0) Brillouin zones 4 fermions (0, π) (π, π) 27 The Wilson-Dirac Operator can we repair this? Wilson suggested to add a second derivative term 2 P ¯ 2 S → a x Ψ[γµ∂µ − r ∂µ +m] Ψ(x) |{z}∗ ∇µ∇µ h i−1 ˜ 1 r P → G = a (iγµ sin pµa) + a µ(1 − cos pµa) + m now, for a → 0, pµ π/a ˜ −ipµγµ+m G → p2+m2 we play the same devils trick pµ = kµ + π/a, kµ 1 r r r a(1 − cos [kµ + π/a] a) = a(1 + cos kµa) → a ˜ r G → a(1 + cos kµa) ⇒ at the other corners of the Brillouin zone the fermions become innitely heavy and decouple in the continuum limit 28 how to implement covariant derivative? matter elds live on lattice points separated by the lattice spacing a geometrical interpretation of the gauge elds if we have a curve zµ(s) parametrized by s, 0 ≤ s ≤ 1 a phase v at y (s = 0) would be transported along this curve to x (s = 1) £¡£ ¤¡¤ £¡£ ¤¡¤ £¡£ ¤¡¤ y ¡ ¢¡¢ ¡ ¢¡¢ ¡ ¢¡¢ x mathematically this is described
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