SU(2) Lattice Gauge Theory with a Topological Action
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1 SU(2) Lattice Gauge Theory with a Topological Action Lorinc Szikszai in collaboration with Zoltan Varga Supervisor Daniel Nogradi November 08, 2017
2 Outline Gauge Theory Lattice Gauge Theory Universality Topological action Our work
3 Pure SU(N) Gauge Theory Action density Field strength L = 1 2g 2 TrF µνf µν F µν = µ A ν ν A µ + [A µ, A ν ] A µ is the gluon filed g is the bare gauge coupling constant Quantization with the Euclidean path integral formalism O = DAµ O(A µ )e S(Aµ) DAµ e S(Aµ).
4 Lattice Gauge Theory Defining the theory on hypercubic space-time lattice V = a 3 N 3 σ T = 1 N τ a Non-perturbative gauge invariant UV regularization A µ (x) U µ (x) = exp(ag 0 A µ (x)) DA µ DU µ = x R,µ=1...4 du µ (x) Both high and low energy observables can be computed O = duo(u)e S Lattice (U). due S Lattice (U) Finite integral, but what is the S Lattice
5 (a) U µ(x) = exp(ag 0A µ(x)) (b) U P (x; µν) Figure: Ingredients of lattice gauge theory: link and plaquette
6 Universality Locality is vital for the viability of universality Universality classes are determined by the space-time dimension the symmetries all action good, if its continuum S = β 4N a 4 TrF µν (x)f µν (x) + O(a 5 ) x β = 2N g 2 0 if we demand the reproduction of the classical action in the classical continuum we still have infinite number of choices for the action
7 Wilson plaquette Action One good choice the Wilson plaquette action which is in the case of SU(2) gauge theory S = x,µ<ν S plaquette where S plaquette is the contribution of squares with sides a S plaquette = β(1 1 2 TrU P (x; µν)) U P (x; µν) is the ordered product of U µ (x) in the plaquette U P (x; µν) U (x+aˆν)( ν) U (x+aˆµ+aˆν)( µ) U (x+aˆµ)ν U xµ lot of other lattice action but all looks in the continuum S = β a 4 TrF µν (x)f µν (x) + O(a 5 ) 8 x
8 Topological Lattice Action Is the classical continuum limit really important? for the topological action as before S = But, with S plaquette = x,µ<ν S plaquette { 0 if (1 1 2 TrU P (x; µν)) < δ if (1 1 2 TrU P (x; µν)) > δ
9 The topological action is invariant against small deformations of the fields (a) Wilson Figure: Continuum limit (b) topological
10 Topological Lattice Action So e S = { 1 for all alowed state 0 otherwise All allowed state has the same weight in the path integral the action does not vary do not have classical continuum limit Cannot be treated using perturbation theory Our aim: to show they yield the correct quantum continuum limit which shows the robustness of universality
11 Former results with topological action 1d O(2),O(3) and 2d O(2), O(3) model Bietenholz, Wolfgang, et al. Topological lattice actions. Journal of High Energy Physics (2010): 20. Bietenholz, W., et al. Topological lattice actions for the 2d XY model. Journal of High Energy Physics (2013): 141. They show that this type of action has the correct continuum limit 4d U(1) model Akerlund, Oscar, and Philippe de Forcrand. Aspects of topological actions on the lattice. (2015). No continuum limit in the U(1)
12 Our aim is decide whether the topological action works well so we determine dimensionless quantity with the standard Wilson action and the topological action, than compare them in the continuum We measure Zero temperature (topological susceptibility) High temperature (Deconfinement phase transition) Small volume (running coupling)
13 Topological susceptibility The topological charge in the continuum Q(x) = dx π 2 ɛ µνϱσtrf µν F ϱσ On lattice we determine the Q 2 with Wilson flow at t = t 0 The topological susceptibility χ = Q2 (t 0 ) V where V is the space-time volume of the lattice χ a 4 so we use the t 0 a 2 to make it dimensionless
14 The t 0 scale Flow equation A µ = δs = D ν F µν δa µ The gauge filed at t > 0 is a renormalized filed We can define a dimension full quantity (t 0 ) E is the action density t 2 E t0 = 0.3 E = 1 4 F a µνf a µν t 2 E is a dimensionless observable applications of t 0 : determines the relative lattice spacings
15 Dimensionless quantity We can measure dimensionless quantity with our simulations So we get from out simulation χ a 4 and t 0 a 2 Our dimensionless observable is χ t 2 0 we can extrapolate to the continuum with a 2 t 0
16 The continuum extrapolation of χt topological χ 2 /dof = 3.4 wilson χ 2 /dof = χt a 2 /t 0 Figure: χt 2 0 continuum
17 Deconfinement phase transition Second order phase transition Same University class as 3d Ising model We investigate the Binder Cumulant L 4 g r = L Where L is the order parameter, so called Polyakov loop g r is an universal quantity at the critical coopling g r = 1.41 in the 3d Ising model The reduced temperature can be approximated by x = β β c β c
18 The ν critical exponent The Binder Cumulant in the vicinity of the critical temperature We expand the Q function g r (N σ, x) = Q(xN 1/ν σ ). Q(xNσ 1/ν ) = Q(0) + Q (0) xnσ 1/ν + O(x 2 ). }{{} g r(x=0) g r (x = 0) is an universal quantity The slop S(b N σ ) of the g r (β/δ) will be S(b N σ ) = S(N σ )b 1/ν This way we can estimate the ν exponent ( ) log S(b Nσ) S(N σ) 1/ν =. log(b) Do not need for the critical couplings or the explicit form of the scaling function.
19 N τ b N σ 1/ν topological 1/ν Wilson 4 5/ (12) 1.54(7) 6 5/ (16) 1.67(24) 8 5/ (25) 1.46(23) 10 4/ (25) 1.74(26) Table: The ν exponent value for the topological and Wilson action(in the case of the 3d Ising model 1/ν = 1.59). Here we use our data to estimate the 1/ν W ilson.
20 g r -1 N σ =20 N σ = g r -0.4 N σ =40 N σ = /ν x(n σ1 =20,N σ2 =12)N σ /ν x(n σ1 =40,N σ2 =30)N σ (a) N τ = 4 (b) N τ = 10 Figure: The scaling of the Binder Cumulant with the topological action.
21 Critical couplings The finite-volume critical couplings (β c (N σ, b)) is determed from the intersection points of the Binder cumulant N σ is the spatial size of the smaller lattice b is the ratio N σ N σ To extrapolate infinite volume we use β c (N σ, b) = β c, (1 aɛ) (1) where ɛ = N y 1 1/ν σ 1 b y 1 b 1/ν 1 y 1 = 1 (2)
22 N σ =20 N σ =16 N σ =12 N σ =8-0.6 N σ =20 N σ =16 N -0.8 σ =12 N σ = g r g r β (a) N τ = 4, Wilson δ (b) N τ = 4, topological Figure: The intersection of the Binder cumulant. We investigate N τ = 4, 6, 8, with N σ = 2, 3, 4, 5 N τ and N τ = 10, with N σ = 2, 3, 4 N τ.
23 topological, χ 2 /dof = wilson, χ 2 /dof = β c ε ε (a) N τ = 4, topological (b) N τ = 4, Wilson Figure: Extrapolation of the critical couplings to infinite volume in the case of N τ = 10.
24 Extrapolation to the continuum We know N τ = 1 T c a(β c ) We need dimensionless quantity, so we measure t 0 a(β c ) 2 N τ β c t 0 /a 2 δ c t 0 /a ( 6) (51) (5) (166) (30) 3.368(65) (5) (261) (40) 5.84(15) (10) (994) (24) 8.86(13) (21) (3129) Table: Values of t 0 /a 2 at the critical couplings.
25 Our dimensionless quantity is ( 8 t ) a 2 Nτ = 8t 0 T c topological, χ 2 /dof = 0.04 Wilson, χ 2 /dof = (8t 0 ) 0.5 T c a 2 2 T c Figure: Continuum extrapolation of the dimensionless ratio 8t 0 T c
26 Summary Our results agree with the two different action We investigate low and high temperature observables. in the continuum with finite lattice spacing In progress: we will check another quantity which connected to the running coupling
27 Foresight Fermion lattice action Nielsen-Ninomiya No-Go Theorem If we require Locality one fermion flavor correct classical continuum limit Then we do not have chiral symmetry but the correct classical continuum limit is really important?
arxiv: v1 [hep-lat] 31 Oct 2014
arxiv:4.88v [hep-lat] 3 Oct 24 Zoltán Fodor University of Wuppertal, Department of Physics, Wuppertal D-4297, Germany Jülich Supercomputing Center, Forschungszentrum Jülich, Jülich D-52425, Germany Eötvös
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