Implicit Schemes for Real-Time Lattice Gauge Theory

Eur. Phys. J. C (2018) 78:884 https://doi.org/10.1140/epjc/s10052-018-6323-x

Regular Article - Theoretical Physics

Implicit schemes for real-time lattice gauge theory

Andreas Ippa, David Müllerb Institut für Theoretische Physik, Technische Universität Wien, 1040 Vienna, Austria

Received: 9 May 2018 / Accepted: 10 October 2018 / Published online: 31 October 2018 © The Author(s) 2018

Abstract We develop new gauge-covariant implicit numer- so-called gauge links, i.e. Wilson lines connecting adjacent ical schemes for classical real-time lattice gauge theory. A lattice sites. Using gauge link variables one can discretize the new semi-implicit scheme is used to cure a numerical insta- continuum Yang-Mills action in various ways, the simplest bility encountered in three-dimensional classical Yang-Mills of which is the Wilson gauge action [20]. Varying this action simulations of heavy-ion collisions by allowing for wave with respect to the link variables one obtains discretized clas- propagation along one lattice direction free of numerical dis- sical field equations, which are of the explicit leapfrog type. persion. We show that the scheme is gauge covariant and that Moreover, in addition to the equations of motion, one also the Gauss constraint is conserved even for large time steps. obtains the Gauss constraint, which is exactly conserved by the leapfrog scheme, even for finite time steps. Going further, the accuracy of the numerical approximation can be 1 Introduction systematically improved by adding higher order terms to the standard Wilson gauge action [21]. Color Glass Condensate (CGC) effective theory [1] applies In previous publications [22,23] we developed lattice classical Yang-Mills theory to the area of high energy heavy- Yang-Mills simulations for genuinely three-dimensional ion collisions. In the CGC description high energy nuclei heavy-ion collisions in the CGC framework. Unlike the usual can be treated as thin sheets of fast moving color charge boost-invariant approach, we consider collisions of nuclei which generate a classical gluon field. The collision of two with thin, but non-vanishing support along the longitudinal such sheets produces the Glasma [2], which behaves classi- direction (the beam axis) and simulate them in the labora- cally at the earliest stages of the collision. Due to classical tory frame. This has enabled us to study the effects of finite Yang-Mills theory being non-linear and the non-perturbative nuclear longitudinal extent (which is inversely proportional nature of the CGC, computer simulations are commonly used to the Lorentz gamma factor γ ) on the rapidity profile of the to investigate the time evolution of such systems [3–7]. Here, produced Glasma after the collision. The numerical scheme real-time lattice gauge theory provides a numerical treat- in these simulations is based on the standard Wilson gauge ment of classical Yang-Mills theory while retaining exact action with the fields coupled to external color currents. The gauge invariance on the lattice. To name a few applications treatment of these color charges is closely related to the col- besides CGC and Glasma simulations, real-time lattice gauge ored particle-in-cell (CPIC) method [24,25], which is a non- theory is widely used for non-Abelian plasma simulations Abelian extension of the particle-in-cell (PIC) method [26] [8,9] and hard thermal loop (HTL) simulations [10–12], clas- commonly used in (Abelian) plasma simulations. sical statistical simulations of fermion production [13,14], Unfortunately, these simulations suffer from a numerical in studying sphalerons (in electroweak theory) [15,16], for instability that leads to an artificial increase of total energy determining the plasmon mass scale in non-Abelian gauge if the lattice resolution is too coarse: even a single nucleus theory [17,18] or when studying perturbations on top of a propagating along the beam axis, which should remain static non-Abelian background field [19]. and stable, eventually becomes unstable. Improving the res- The real-time lattice gauge theory approach is based on olution simply postpones the problem at the cost of much the discretization of Yang-Mills fields on a lattice in terms of higher computational resources. We realize that this instability is due to numerical dispersion on the lattice inherent to a e-mail: [email protected] the leapfrog scheme, which renders the dispersion relation of b e-mail: [email protected] plane waves non-linear. As a consequence of numerical dis- 123 884 Page 2 of 26 Eur. Phys. J. C (2018) 78 :884 persion, high frequency plane waves exhibit a phase velocity ing symplectic structure and retaining symmetries of the dis- that is noticeably less than the speed of light on the lattice. The crete action [34,35]. We will see how the exact discretiza- shape of the pulse of color fields is lost over time. At the same tion of the action affects the properties of the numerical time, the color current “driving” the nucleus forward will not scheme and in particular how numerical dispersion can be disperse by construction: the point-like color charges making eliminated. up the current are simply moved from one cell to the next as We consider a real-valued scalar field φ(x) in 2+1 with the simulation progresses. Thus the shape of the current is mostly minuses metric signature (+1, −1, −1) and set the always kept intact. This mismatch and the resulting instabil- speed of light to c = 1. The action is given by ity is therefore related to the numerical Cherenkov instability 1 μ [27], which can occur in (Abelian) particle-in-cell simula- S[φ]= ∂μφ∂ φ, (1) 2 μ tions. Notably, simulations of laser wakefield acceleration x [28], where electric charges moving at relativistic speeds are coupled to discretized electromagnetic fields, suffer from the which upon demanding that the variation of the action vanishes same type of instability and many numerical schemes have been devised to cure it [29–32]. A particularly simple solu- δS[φ] δS = δφ(x) = 0, (2) tion to the problem is the use of semi-implicit schemes to δφ(x) repair the dispersion relation (i.e. making it linear) for one x direction of propagation [33], which is the approach we take yields the equations of motion (EOM) in this work. ∂ ∂μφ( ) = ∂2φ( ) − ∂2φ( ) = . In this paper we derive an implicit and a semi-implicit μ x 0 x i x 0 (3) scheme for real-time lattice gauge theory by modifying the μ i standard Wilson gauge action. We obtain two new actions We use Latin indices i, j, k,... to denote the spatial com- that are gauge invariant (in the lattice sense), are of the same ponents, ∂2 is a shorthand for ∂ ∂ (no sum implied) and order of accuracy as the original action, but yield an implicit i i i = dx0dx1dx2. Inserting a plane-wave ansatz or semi-implicit scheme upon variation. In the case of the x semi-implicit scheme setting the lattice spacing and the time μ step to specific values can fix the dispersion relation along φ(x) = φ0 exp i kμx , (4) the longitudinal direction and thus suppress the numerical μ Cherenkov instability. We also obtain a modified version of φ ∈ R μ = (ω, 1, 2)μ the Gauss constraint that is conserved up to (in principle) with 0 and k k k into the EOM gives the arbitrary numerical precision under the discrete equations of dispersion relation motion. ω =|k|= (k1)2 + (k2)2. (5) We start with a discussion of the main ideas behind the semi-implicit scheme for the two-dimensional wave equation Obviously, the phase velocity v = ω/|k|=1 is constant, i.e. in Sect. 2 and for Abelian gauge fields on the lattice in Sect. there is no dispersion. 3. The concepts are then generalized to non-Abelian lattice Now let us consider a discretized version of this system gauge theory in Sect. 4, where we derive both a fully implicit by approximating space time as an infinite rectangular lattice and the semi-implicit scheme. Finally, we verify numerically with grid spacings aμ. We refer to a0 as the time step and ai that the Cherenkov instability can be suppressed using the as the spatial lattice spacings. The field φ(x) is replaced with new scheme and that the Gauss constraint is conserved in field values φx defined at the lattice sites x and derivatives Sect. 5. are replaced with finite difference expressions. We define the forward difference φ − φ F x+μ x 2 A toy model: the 2D wave equation ∂μ φ ≡ , (6) x aμ The basic ideas behind the numerical scheme we are after can and the backward difference be most easily explained using a simple toy model, namely φ − φ ∂ Bφ ≡ x x−μ , the two-dimensional wave equation. We start by giving a few μ x μ (7) definitions and then derive three different numerical schemes a by discretizing the action of the system in different ways and where we introduced another shorthand notation: φx±μ μ using a discrete variational principle. The schemes obtained denotes the field at a neighboring lattice site x ± a eˆμ (no through this procedure are known as variational integrators, implicit sum over μ) with the unit vector in the μ direction which exhibit useful numerical properties such as conserv- eˆμ. We also define the second order central difference 123 Eur. Phys. J. C (2018) 78 :884 Page 3 of 26 884

φ + φ − φ ∂2 φ ≡ ∂ F ∂ Bφ = x+μ x−μ 2 x . step a0 and spatial lattice spacings ai . Using the plane-wave μ x μ μ x μ (8) (a )2 ansatz (4) we find the dispersion relation The forward and backward differences are linear approxima- ωa0 a0 2 ki ai tions to the first order derivative, while the second order dif- sin2 = sin2 , (17) 2 ai 2 ference is accurate up to second order in the lattice spacing a. i Equipped with these definitions we could directly discretize which is in general non-linear and only yields real-valued the EOM (3), but this is not the approach we will take. The (stable) frequencies ω for all wave vectors k if the Courant- strategy behind variational integrators is to first discretize Friedrichs-Lewy (CFL) condition holds the action (1) and then demand that the discrete variation vanishes. a0 2 ≤ 1. (18) ai 2.1 Leapfrog scheme i The discretization errors of this finite difference scheme One possible way of discretizing the action is result in a non-linear dispersion relation, which is usually referred to as numerical dispersion, since this kind of artifi- 2 2 1 F F cial dispersive behavior of plane waves does not show up in S[φ]= V ∂ φx − ∂ φx , (9) 2 0 i 0 = 1 = 2 x i the continuum. If it were possible to set a a a (the so- called “magic time-step”) the leapfrog scheme would actu- = 0 1 2 where x is the sum over all lattice sites and V a a a ally be non-dispersive along the lattice axes, but this choice is the space-time volume of a unit cell. Introducing small of the parameters is forbidden by the CFL condition in higher δφ variations x of the discrete field at each point, the discrete dimensions than 1 + 1 and would lead to unstable modes. variation of this action reads 2.2 Implicit scheme δ = ∂ F φ ∂ F δφ − ∂ F φ ∂ F δφ S V 0 x 0 x i x i x x i Let us consider a different discretization: we define a new action 2 2 =−V ∂ φx − ∂ φx δφx (10) 0 i 1 2 x i F F F S[φ]= V ∂ φx − ∂ φx ∂ φ , (19) 2 0 i i x x i which upon setting it to zero yields the discretized EOM φ where x is the temporally averaged field ∂2φ − ∂2φ = 0, (11) 0 x i x φ + φ i φ ≡ x+0 x−0 ≈ φ + O ( 0)2 . x x a (20) ∂2 ∂2 2 where 0 and i are second order finite differences. Here, we made use of summation by parts, i.e. Note that only one of the spatial finite differences in the squared term is temporally averaged. Since this action differs ∂ F φ ∂ F δφ =− ∂ B∂ F φ δφ , 0 x 0 x 0 0 x x (12) from the leapfrog action (9) only up to an error term quadratic x x in a0, the numerical scheme derived from this action will have which is the discrete analogue of integration by parts. If the the same accuracy as the leapfrog scheme. field is known in two consecutive time slices we can explicitly Repeating the steps as before we obtain the discretized solve the EOM (11) for the field values in the next time slice: EOM 0 2 ∂2φ − ∂2φ = . a 0 x i x 0 (21) φ + = (φ + + φ − − 2φ ) − φ − + 2φ . x 0 ai x i x i x x 0 x i i (13) Footnote 1 continued 0 0 + 0 In fact, this scheme is identical to the explicit leapfrog which is defined naturally between time slices x and x a (hence the index “x + 0 ” in our notation). The EOM can then be written as scheme,1 which is accurate up to second order in the time 2 a0 π = (φ + φ − φ ) + π , 1 x+ 0 x+i x−i 2 x x− 0 (15) The connection to the leapfrog scheme becomes more apparent if we 2 ai 2 2 introduce an approximation of the conjugate momentum i π ≡ ∂ F φ , x+ 0 0 x (14) 0 2 φx+0 = φx + a π + 0 . (16) x 2

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This is an implicit scheme, which is more complicated to but k2 = 0 the propagation becomes non-dispersive, i.e. solve compared to the explicit leapfrog scheme given by Eq. ω = k1.Fork2 = 0 and k1 = 0 the propagation still exhibits (13). Here we have to find the solution to a system of linear numerical dispersion. The scheme defined by the action (23) equations, which can be accomplished using (for instance) therefore allows for non-dispersive, stable wave propagation iterative methods. along one particular direction on the lattice. This principle The dispersion relation for this scheme reads also extends to systems with more spatial dimensions, where 2 one treats a preferred direction explicitly and all other spatial a0 2 ki ai ω 0 i sin directions implicitly. 2 a = i a 2 . sin (22) 2 0 2 i i 1 + 2 a sin2 k a i ai 2 2.4 Solution method and numerical tests This relation can always be solved for real-valued frequencies ω and therefore the implicit scheme is unconditionally stable. To solve the EOM of the implicit or the semi-implicit scheme Unfortunately this does not solve the problem of numerical one has to solve a linear system of equations. This can be dispersion either, because there is no choice of lattice param- accomplished for instance by inverting a band matrix. Alter- eters that results in a linear dispersion relation. natively the equations can also be solved in an iterative man- We quickly summarize: the first action we considered ner. Taking the latter approach will be readily applicable to givenbyEq.(9) gave us the explicit leapfrog scheme, which is lattice gauge theory. One example for an iterative method is rendered non-dispersive but unstable using the “magic time- damped (or relaxed) fixed point iteration: the idea is to first step”. The second action, Eq. (19), which we obtained by rewrite the EOM (24) as a fixed point equation replacing one of the spatial finite differences with a tem- 2 φ = φ = φ − φ + 0 ∂2φ + ∂2φ , porally averaged expression, yields an implicit scheme. This x+0 F [ ] 2 x x−0 a 1 x 2 x scheme is unconditionally stable, but always dispersive. This (27) suggests that a mixture of both discretizations might solve φ(0) our problem. and then, starting with an initial guess x+0 from e.g. the explicit leapfrog evolution, use the iteration 2.3 Semi-implicit scheme φ(n+1) = αφ(n) + ( − α) φ(n) . x+0 x+0 1 F (28)

Finally, we consider the action ( + ) to obtain a new guess φ n 1 . Here the real-valued parameter 2 2 x+0 1 F F F F S[φ]= V ∂ φx − ∂ φx − ∂ φx ∂ φ , α acts as a damping coefficient. Using fixed point iteration 2 0 1 2 2 x x might induce other numerical instabilities not covered by the (23) CFL condition (26). To analyze this we make the ansatz where the derivatives w.r.t. x1 are treated like in the leapfrog φ(n) = λnϕ + φ(∞), x+0 x x+0 (29) scheme and the derivatives w.r.t. x2 involve a temporally averaged expression as in the implicit scheme. The EOM φ(∞) ϕ where x+0 isthetruesolutiontoEq.(27) and x represents now read a time-independent error term. The growth of the error is λ ∂2φ − ∂2φ − ∂2φ = 0. (24) determined by the modulus of . Employing a Fourier ansatz 0 x 1 x 2 x ϕ = i i x exp i i k x yields We call this numerical scheme semi-implicit, because the finite difference equation contains both explicitly and implic- a0 2 k2a2 λ =−2(1 − α) sin2 + α, (30) itly treated spatial derivatives. The dispersion relation asso- a2 2 ciated with Eq. (24) is given by k1 2 which is independent of the component and the corre- a0 2 ki ai 1 ω 0 i sin sponding lattice spacing a . Requiring convergence for high- 2 a i a 2 sin = , (25) k modes, i.e. |λ| < 1fork2 =±π/a2, we find 2 0 2 2 2 1 + 2 a sin2 k a a2 2 2δ − 1 <α<1, (31) which is stable if 2δ + 1

a0 2 where ≤ 1. (26) a1 2 a0 δ = . (32) The CFL condition (26) now allows us to set a0 = a1. Look- a2 ing at the dispersion relation (25) we notice that for k1 = 0, 123 Eur. Phys. J. C (2018) 78 :884 Page 5 of 26 884

1 φx tion on the lattice. As it turns out, this is just what we need 0.5 lleapfrogeapfrog to suppress the numerical Cherenkov instability we encountered in our heavy-ion collision simulations. In the next sec- 0 tion we will see how we can use the same “trick” for Abelian 1 φx gauge fields on the lattice. 0.5 iimplicitmplicit 0 3 Abelian gauge fields on the lattice 1 φx 0.5 ssemi-implicitemi-implicit Before tackling the problem of non-Abelian gauge fields x 0 on the lattice, it is instructive to see how we can derive a 0.20.40.60.8 1 dispersion-free semi-implicit scheme for discretized Abelian gauge fields. We will approach the problem as before: starting Fig. 1 This plot shows a comparison of three different numerical with a discretization of the action, we apply a discrete vari- schemes for solving the wave equation in 2+1: (top) explicit leapfrog ational principle to derive discrete equations of motion and scheme (11), (middle) the implicit scheme (21), (bottom) semi-implicit scheme (24) with “magic time-step” a0 = a1. The horizontal axis is constraints. Then we will see what modifications to the action 1 the x coordinate; the vertical axis is the field amplitude φx in arbitrary are required to obtain implicit and semi-implicit numerical units. The x2 coordinate is suppressed. The initial condition (seen on the schemes. Since we are dealing with gauge theory we will left) is a Gaussian pulse which propagates to the right under time evo- take care to retain gauge invariance also for the discretized lution. Due to numerical dispersion of the leapfrog and implicit scheme the original shape of the Gaussian is lost over time. On the other hand, system. the dispersion-free semi-implicit solver conserves the original pulse In the continuum the action for Abelian gauge fields reads shape 1 μν S[A]=− Fμν(x)F (x), (34) 4 μ,ν In d + 1 dimensions, where we treat the i = 1 direction x ≤ ≤ explicitly and all others 2 i d implicitly, the stability with the field strength tensor given by condition is given by Eq. (31) with Fμν(x) = ∂μ Aν(x) − ∂ν Aμ(x). (35) d a0 2 δ = . (33) i The field strength and the action are invariant under gauge = a i 2 transformations defined by Note that for δ<1/2 damping might not even be necessary. A similar stability condition can be derived also for the Aμ(x) → Aμ(x) + ∂μα(x), (36) implicit scheme, which by itself (just from the plane wave where α(x) is an at least twice differentiable function which analysis) is unconditionally stable. It is important to keep in defines the gauge transformation. mind that the use of fixed point iteration can introduce new Varying the action with respect to the gauge field leads to instabilities depending on the lattice spacing and the time step. μν Finally, we perform a crucial numerical test: we compare δS[A]= ∂μ F (x) δ Aν(x) = 0. (37) ν μ the propagation of a Gaussian pulse using the three different x schemes to show the effects of numerical dispersion and in The term proportional to δ A0(x) leads to the Gauss constraint particular that the semi-implicit scheme is dispersion-free. For simulations using the implicit or semi-implicit method ∂i F0i (x) = 0, (38) we solve the equations using damped fixed point iteration. i The results are shown in Fig. 1. The main insight of this section is that the specific dis- while the term proportional to the variation of spatial com- cretization of the action completely fixes the numerical ponents gives the EOM scheme of the discrete equations of motion (and the asso- ∂ F (x) = ∂ F (x). (39) ciated stability and dispersion properties) which one obtains 0 i0 j ij j from a discrete variational principle. The use of temporally averaged quantities in the action leads to implicit schemes. It is trivial to see that the EOM imply the conservation of the If we treat some derivatives explicitly and some implicitly Gauss constraint. As we will see next, it is also possible to we can end up with a semi-implicit scheme that can be non- formulate a discretization of the system where this conser- dispersive and still stable for propagation along a single direc- vation of the constraint holds exactly. 123 884 Page 6 of 26 Eur. Phys. J. C (2018) 78 :884 3.1 Leapfrog scheme 1 2 F δt F , =− Fx, i ∂ δ Ax, 2 x 0i 0 i 0 x,i x,i We consider a discretized gauge field A ,μ at the lattice sites x B = ∂ F , δ A , , (46) x. The field strength tensor Fx,μν at x is defined using forward i x 0i x 0 differences x,i = ∂ F − ∂ F , we get the constraint Fx,μν μ Ax,ν ν Ax,μ (40) B ∂ F , = 0. (47) which is antisymmetric in the Lorentz index pair μ, ν like its i x 0i i continuum analogue. Furthermore, the lattice field strength is invariant under lattice gauge transformations given by Since both the discrete EOM and the constraint follow from the same discretized, gauge-invariant action (42), the Gauss F Ax,μ → Ax,μ + ∂μ αx , (41) constraint is guaranteed to be automatically conserved under the EOM. We can show this explicitly via α where x defines the local transformation at each lattice site x. A straightforward discretization of the gauge field action ∂ B ∂ B =− ∂ B ∂ B is given by 0 i Fx,0i i 0 Fx,i0 ⎛ ⎞ i i B B =− ∂ ∂ , = , 1 ⎝ 2 1 2 ⎠ i j Fx ij 0 (48) S[A]= V F , − F , , (42) 2 x 0i 2 x ij i, j x i i, j μ which is equivalent to where V = μ a is the space-time volume of a unit cell. ∂ B = ∂ B . Due to the invariance of Fx,μν under lattice gauge transfor- i Fx,0i i Fx−0,0i (49) mations the action is also invariant. i i Performing the variation of (42) with respect to spatial This means that if the Gauss constraint is satisfied in one components Ax,i yields the equations of motion. Using the time slice then the EOM will ensure that it remains satisfied variation of the magnetic part of the action in the next time slice. We can also give a more general proof: consider an infinitesimal gauge transformation 1 1 δ 2 = δ F F , Fx,ij Fx,ij A = A ,μ + ∂ α , (50) 4 x ij 2 x,μ x μ x x,i, j x,i, j and expand the action S[A ] for small α. We then find = 1 ∂ F δ − ∂ F δ Fx,ij i Ax, j j Ax,i 2 x,i, j ∂ [ ] ∂ A [ ] [ ]+ S A x,μ α + O α2 S A S A y = ∂ B δ , ∂ A ,μ ∂α α= j Fx,ij Ax,i (43) x,y,μ x y 0 x,i, j ∂ [ ] S A F 2 = S[A]+ ∂μ δxyαy + O α ∂ A ,μ x,y,μ x and the variation of the electric part w.r.t. spatial components δ ∂ S[A] (denoted by s) = S[A]− ∂ B α + O α2 . μ ∂ x (51) ,μ Ax,μ x 1 2 F δs F , = Fx, i ∂ δ Ax,i 2 x 0i 0 0 Since S[A] is invariant for any α it must hold that x,i x,i ∂ S[A] =− ∂ B δ , ∂ B = , 0 Fx,i0 Ax,i (44) μ 0 (52) ∂ Ax,μ x,i μ or written slightly differently we find the discrete EOM ∂ S[A] ∂ S[A] ∂ B =− ∂ B . (53) B B 0 i ∂ , = ∂ , , ∂ A , ∂ A , 0 Fx i0 j Fx ij (45) x 0 i x i j If the equations of motion are satisfied in every time slice, i.e. ∂ [ ] ∂ [ ] which are of the explicit leapfrog type. We also obtain a S A = 0, then the Gauss constraint S A must be conserved ∂ Ax,i ∂ Ax,0 discretized version of the Gauss constraint by considering from one slice to the next: the variation w.r.t. temporal components A , (denoted by ∂ [ ] x 0 ∂ B S A = . δ 0 0 (54) t ). With ∂ Ax,0 123 Eur. Phys. J. C (2018) 78 :884 Page 7 of 26 884

This holds regardless of the exact form of the gauge invariant The factor of 1/2in(63) is introduced for convenience. For [ ] χ 2 = action S A . Consequently, it does not matter what kind of differences with respect to the time coordinate we find 0 discretization of the action we use. As long as S[A] retains sin2 ωa0/2 . Using these definitions the Gauss constraint lattice gauge invariance in the sense of Eq. (41), we are guar- (47) for the plane wave ansatz can be reduced to anteed to find that the discrete Gauss constraint is conserved χ B = . under the discrete equations of motion. i Ai 0 (65) The EOM (45) alone are not enough to uniquely determine i the time evolution of the field Ax,μ: we must specify a gauge The discrete EOM (56) can be written as condition. Here we use temporal gauge χ 2 A =− χ Bχ F A + χ 2 A . (66) = , 0 i j i j j i Ax,0 0 (55) j which we will also use in the case of non-Abelian lattice χ Bχ F One can eliminate the mixed terms j i A j using the Gauss gauge fields. The EOM (45) then read constraint and find the dispersion relation 2 B F 2 − ∂ Ax,i = ∂ ∂ Ax, j − ∂ Ax,i . (56) χ 2 = χ 2, 0 j i j 0 j (67) j j Using a plane wave ansatz which is equivalent to the dispersion relation of the wave i ωx0− ki xi equation Eq. (17), i.e. A , = A e i , (57) x i i ωa0 a0 2 k j a j with amplitude Ai , we find the same non-trivial dispersion sin2 = sin2 . (68) 2 a j 2 relation and CFL stability condition as in the case of the j leapfrog scheme for the wave equation in 2+1 [see Eq. (17) and Eq. (18)]. To show this explicitly we first introduce some 3.2 Implicit scheme notation. Taking either a forward (backward) finite difference of the plane wave ansatz yields An implicit scheme analogous to the one we derived for the

i i wave equation [see Eq. (19)] can be found using the action −ik a − F e 1 ⎛ ⎞ ∂ , = , , i Ax j i Ax j (58) a 1 1 + i i [ ]= ⎝ 2 − ⎠ , − ik a S A V Fx,0i Fx,ijMx,ij (69) B 1 e 2 2 ∂ A , = A , , (59) x i i, j i x j ai x j which suggests the definition of the forward (backward) lat- where we introduce the temporally averaged field-strength tice momentum 1 Mx,ij = F x,ij = Fx+0,ij + Fx−0,ij . (70) −iki ai − 2 κ F = e 1, i i (60) ia Note that Mx ,ijdiffers from Fx,ij only by an error term pro- +iki ai 0 2 1 − e portional to a . Replacing one of the field strengths Fx,ij κ B = . (61) i iai in the quadratic term in the action with its temporally aver- † aged expression Mx,ij is analogous to replacing the wave It holds that κ F = κ B. We define the squared lattice φ φ = 1 (φ + φ ) i i amplitude x with x 2 x+0 x−0 in the action (19). momentum as The averaged field strength M , is also invariant under lat- x ij 2 i i tice gauge transformations: κ2 ≡ κ F κ B = 2 2 k a . i i i sin (62) ai 2 → + ∂ F ∂ F α − ∂ F ∂ F α = . Mx,ij Mx,ij i j x j i x Mx,ij (71) Furthermore, we can render the lattice momenta dimension- Varying the action as we did for the leapfrog scheme we find less by multiplying with a0. We define the dimensionless the EOM lattice momentum as ∂ B = ∂ B , / a0 / 0 Fx,i0 j Mx,ij (72) χ F B = κ F B, (63) i 2 i j and and employing temporal gauge we have a0 2 ki ai − ∂2 = ∂ B∂ F − ∂2 , χ 2 = χ F χ B = 2 . 0 Ax,i j i Ax, j j Ax,i (73) i i i sin (64) ai 2 j 123 884 Page 8 of 26 Eur. Phys. J. C (2018) 78 :884

2 where the fields Ax,i have been replaced with temporally = ∂ F ∂ B + ∂ F (α − α ) . 1 0 Ax,0 1 x x (79) = 1 + 0 2 averaged expressions Ax,i 2 Ax+0,i Ax−0,i on the a right-hand side. The Gauss constraint that arises from varying w.r.t. temporal components is simply Eq. (47), because Here we made use of the exact relation we did not modify the term involving Fx,0i or introduce new 2 1 0 2 dependencies on A , . The discrete EOM (72) still conserve a ∂ αx = αx − αx . (80) x 0 2 0 the Gauss constraint due to lattice gauge invariance. Perform- Therefore, the transformation property of the properly aver- ing the same steps as in the previous section, we can show that ˜ this implicit scheme is unconditionally stable and exhibits the aged gauge field Ax,1 is the same as Ax,1: same non-trivial dispersion relation as the implicit scheme for ˜ ˜ F Ax,1 → Ax,1 + ∂ αx . (81) the wave equation, see Eq. (22). 1 It still holds that in the continuum limit the properly averaged ˜ 3.3 Semi-implicit scheme gauge field Ax,1 is the same as Ax,1 up to an error term quadratic in a0. While the definition (78) seems a bit arbitrary In this section we want to develop a semi-implicit scheme at first sight, this way of averaging is more natural using the for Abelian gauge fields. Specifically we need an action that language of lattice gauge theory: in Sect. 4.3 we will find an allows for dispersion-free propagation of waves in the direc- intuitive picture in terms of Wilson lines that reduces to (78) tion of the x1 coordinate (which we refer to as the longitudinal in the Abelian limit for small lattice spacings. direction). Wecall x2 and x3 the transverse coordinates. From The properly semi-averaged gauge-invariant field strength now on Latin indices i, j, k,...refer to transverse indices and is then given by the longitudinal index will always be explicit. Our goal is to F ˜ F W , ≡ ∂ A , − ∂ A , . (82) define the action in such a way that we end up with equations x i1 i x 1 1 x i of motion that include explicit differences in the x1 direction, In order to keep the field strength explicitly antisymmetric 2 3 but temporally averaged finite differences in the x and x we define Wx,1i =−Wx,i1. Using these definitions we can 2 direction. This means that we have to modify the Fx,i1 term guess the action of the leapfrog action (42) such that it results in terms like ∂ F A , ∂ F A , i x 1 1 x 1. A first guess for a semi-averaged version of 1 2 2 1 S[A]= V F , + F , − Fx,ijMx,ij field strength Fx,i1 that could accomplish this is 2 x 01 x 0i 2 x i i, j F F ∂ A , − ∂ A , , (74) i x 1 1 x i − , Fx,1i Wx,1i (83) = 1 + i with Ax,1 2 Ax+0,1 Ax−0,1 . However, it turns out that such a term is not invariant under lattice gauge transforma- , tions (41). The problem is that Ax,1 transforms differently where the indices i j denote transverse components. Since than Ax,i .Wehave we have built the new action from gauge invariant expressions it is also invariant under lattice gauge transformations. Note → + ∂ F α , ˜ Ax,1 Ax,1 1 x (75) that the use of Ax,i in Wx,1i introduces new terms in the action → + ∂ F α , (83) dependent on the temporal component of the gauge field. Ax,i Ax,i i x (76) Although these terms disappear in temporal gauge (our pre- which yields ferred choice), they still have an effect on the scheme since we have to perform the variation before choosing a gauge. ∂ F − ∂ F → ∂ F − ∂ F i Ax,1 1 Ax,i i Ax,1 1 Ax,i Therefore we will obtain a modified Gauss constraint com- + ∂ F ∂ F (α − α ) , patible with the equations of motion derived from the action i 1 x x (77) (83) even after setting Ax,0 = 0. At this point one might ask if the “proper” averaging pro- where the last term ∂ F ∂ F (α − α ) breaks gauge invariance. i 1 x x cedure has any effect on the implicit scheme of the previ- To fix this we introduce the “properly” averaged field strength ˜ ous section as well. It turns out that if one defines the aver- Ax,1 given by aged field-strength Mx,ij of (70) using the properly averaged 2 ˜ , ˜ 1 0 F B gauge field Ax i , the action of the implicit scheme (and by Ax, ≡ Ax, − a ∂ ∂ Ax, , (78) 1 1 2 1 0 0 extension the EOM and the constraint) remains unchanged. This is due to the fact that the terms proportional to A , in where the last term transforms as x 0 (78) cancel: ∂ F ∂ B → ∂ F ∂ B + ∂ F ∂2α ∂ F ˜ − ∂ F ˜ = ∂ F − ∂ F . 1 0 Ax,0 1 0 Ax,0 1 0 x i Ax, j j Ax,i i Ax, j j Ax,i (84) 123 Eur. Phys. J. C (2018) 78 :884 Page 9 of 26 884

Therefore, no such modification is required in the implicit If we consider the special case of a plane wave with a purely scheme. transverse amplitude and which propagates in the x1 direc- Varying this action w.r.t temporal components Ax,0 yields tion (i.e. setting the transverse momenta χ2 = χ3 = 0) we the modified Gauss constraint find that both dispersion relations agree: 2 d a0 ω 0 = ω 0 = − χ 2 . ∂ B + ∂ B∂ B∂ F = . 1a 2a arccos 1 2 1 (94) i Fx,0i i 1 0 Fx,1i 0 (85) = 2 i 1 i This dispersion becomes linear if we set a0 = a1. This explic- Since Wx,1i explicitly depends on Ax,0, we obtain a correc- itly shows that the semi-implicit scheme for Abelian gauge tion term to the standard leapfrog Gauss constraint (47). The fields allows for dispersion-free, stable propagation along discrete EOM read the longitudinal direction if we use the “magic time-step”. ∂ B = 1 ∂ B + , The dispersion relations also agree if we set the longitudinal 0 Fx,10 i Wx,1i Mx,1i (86) 2 momentum χ1 to zero for arbitrary transverse momenta. In i general however, wave propagation in this scheme is bifrac- B B 1 B ∂ Fx,i = ∂ Mx,ij + ∂ Fx,i + Wx,i . (87) tive. It would be interesting to see if there are alternative 0 0 j 2 1 1 1 j=i discretizations of the action, which allow for dispersion-free We have separate EOM for the longitudinal and transverse propagation without being bifractive. components of the gauge field. Note that by replacing the The main result of this section is the action (83) which averaged expressions M and W with F the EOM reduce to gives rise to the semi-implicit scheme. Here we used a com- the leapfrog equations as expected. bination of differently averaged field strengths, Mx,ij and The propagation of waves in the semi-implicit scheme Wx,1i , in the action. Our next goal is to generalize these turns out to be more complicated compared to the leapfrog expressions to non-Abelian gauge fields. or implicit scheme: given a wave vector k and a field amplitude Ai (such that the Gauss constraint (85) is satisfied) the dispersion relation becomes polarization dependent, i.e. the 4 Non-Abelian gauge fields on the lattice scheme exhibits birefringence. The continuum action for non-Abelian Yang-Mills fields is The amplitude Ai of an arbitrary plane wave given by i ωx0− ki xi Ax,i = Ai e i , (88) 1 μν S[A]=− tr Fμν(x)F (x) , (95) has to be split into a longitudinal and two momentum- 2 μ,ν dependent transverse components x ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ where the field strength is ⎨ 1 0 0 ⎬ , , , , = ⎝ ⎠ , ⎝ −χ B ⎠ , ⎝ χ F ⎠ , Fμν(x) = ∂μ Aν(x) − ∂ν Aμ(x) + ig Aμ(x), Aν(x) . (96) AL AT 1 AT 2 ⎩ 0 3 2 ⎭ (89) 0 χ B χ F 2 3 The constant g is the Yang-Mills coupling constant and / a a a χ F B Aμ(x) = Aμt is a non-Abelian gauge field, where t where i are dimensionless lattice momenta given by Eq. a (63). In Appendix A we find that the components AL and are the generators of the gauge group. In the following we use the normalization tr tatb = 1 δab. Through variation of AT,2 have the dispersion relation 2 the action we obtain the Gauss constraint and the equations 2 2 2 1 − χ 2 + χ + χ of motion: ω a0 = arccos 1 2 3 , (90) 1 + χ 2 − χ 2 + χ 2 − χ 2 1 2 2 1 3 2 1 i0 Di F (x) = 0, (97) and the component AT,1 has a second different dispersion i relation 0i ( ) =− ij( ), D0 F x D j F x (98) 1 − 2χ 2 j ω a0 = arccos 1 . (91) 2 + χ 2 + χ 2 1 2 2 2 3 where the gauge-covariant derivative acting on an algebra element χ is given by Analyzing the stability of the scheme using the two dispersion relations yields that it is stable if Dμχ(x) ≡ ∂μχ(x) + ig Aμ(x), χ(x) . (99) χ 2 ≤ , 1 1 (92) In real-time lattice gauge theory, instead of gauge fields Aμ(x), we use gauge links (or link variables) U ,μ which which, when requiring stability for all modes, reduces to x are unitary matrices and interpreted as the Wilson lines con- 0 1 a ≤ a . (93) necting nearest neighbors on the lattice. Ux,μ is the shortest 123 884 Page 10 of 26 Eur. Phys. J. C (2018) 78 :884 possible Wilson line on the lattice starting at the site x and where i denotes a sum over all spatial components. Using ending at x + μ. This is also reflected in the gauge transfor- Eq. (108) it is clear that the action (109) is a discretization mations of the continuum Yang-Mills action ⎛ ⎞ → † , Ux,μ Vx Ux,μVx+μ (100) [ ]=1 ⎝ a ( ) a ( )− 1 a ( ) a ( )⎠ , S A F0i x F0i x Fij x Fij x where Vx is a gauge transformation defined at the lattice site 2 , 2 , , x. Gauge links with negative directions are identified as x a i a i j (110) = † . Ux,−μ Ux−μ,μ (101) where we made the split into temporal and spatial compo- In the continuum limit the gauge links can be approximated nents explicit. Since it is built from gauge-invariant expres- by sions, the Wilson gauge action (109) is invariant under lattice μ Ux,μ exp iga Ax,μ , (102) gauge transformations (100). At this point we remark that the continuum Yang-Mills = a a where Ax,μ a Ax,μt . We use “lattice units” for the μ action (110) and its discretization (109) look very different: gauge fields, i.e. we absorb a factor of ga in the definition while the Yang-Mills action is given in terms of squares of of the gauge field. The gauge links then read the field strength tensor, the Wilson gauge action is linear in ˆ plaquette variables. In terms of plaquettes it is not immedi- Ux,μ exp i Ax,μ , (103) ately clear how we might generalize our approach from Sect. ˆ μ where Ax,μ ≡ ga Ax,μ. For the rest of this paper we drop 3. Fortunately, the action (109) can be written differently so the hat symbol and just remember to restore factors of gaμ that its functional form is more similar to (110). We define whenever necessary. (see for instance p. 94 of [36]) The smallest possible Wilson loops that can be constructed C ,μν ≡ U ,μU +μ,ν − U ,νU +ν,μ, (111) on the lattice are the so-called “plaquettes” x x x x x which transforms non-locally Ux,μν ≡ Ux,μUx+μ,νUx+μ+ν,−μUx+ν,−ν, (104) → † . which can also be written as Cx,μν Vx Cx,μν Vx+μ+ν (112) = † † . Ux,μν Ux,μUx+μ,νUx+ν,μUx,ν (105) For comparison to the plaquette Ux,ij, the path traced by C , is shown in Fig. 2 on the right. In the continuum limit The path traced by the plaquette is shown in Fig. 2 on the x ij C ,μν can be (up to constant factors) identified with the field left. In the continuum limit the plaquettes can be identified x strength Fμν(x): expanding for small lattice spacing we find with the field strength tensor μ ν C ,μν iga a Fμν(x). (113) Ux,μν exp iFx,μν . (106) x μ ν Here, Fx,μν contains a factor of ga a . Plaquettes represent Most noteworthy is the exact relation a closed Wilson loop and therefore transform locally at the † = − − † , starting (and end) point x: Cx,μνCx,μν 2 Ux,μν Ux,μν (114) → †. with which we can identically rewrite the action as Ux,μν Vx Ux,μν Vx (107)

By taking the trace of a plaquette one obtains a gauge invari- [ ]= V 1 † ant expression. In particular it holds that S U tr Cx,0i Cx,0i g2 a0ai 2 x i † 2 1 1 tr 2 − Ux,μν − U ,μν tr F ,μν † x x − tr Cx,ijC , . (115) 2 i j 2 x ij i, j a a 1 μ ν a 2 ga a Fμν(x) . (108) 2 a This functional form of the rewritten action (115)isnowvir- This leads to the standard Wilson gauge action [20] tually the same as the continuum case (110). We will exploit this when generalizing the implicit (69) and semi-implicit V 1 † schemes (83) to non-Abelian gauge ﬁelds. S[U]= tr 2 − Ux,0i − U , g2 0 i 2 x 0i Performing the variation of (109)or(115) is a bit more x i a a involved compared to the wave equation or Abelian gauge 1 1 † − tr 2 − Ux,ij − U , , (109) ﬁelds on the lattice. The degrees of freedom, the gauge links 2 i j 2 x ij i, j a a Ux,μ, are not just completely general complex matrices but 123 Eur. Phys. J. C (2018) 78 :884 Page 11 of 26 884

x + i x + i

x x + i + j x x + i + j

x + j x + j

Fig. 2 Left: the Wilson line associated with the plaquette Ux,ij. Right: and ends at the same lattice site x (red dot), the lattice ﬁeld-strength Cx,ij the Wilson lines associated with the lattice ﬁeld strength Cx,ij. Spatial starts at x and ends at x + i + j (blue dot) link variables are drawn as solid black arrows. While the plaquette starts

1 a † † elements of SU(N), i.e. unitary matrices with determinant 1: =− P Ux,i Ux+i, j C , + C − , Ux− j, j , i j 2 x ij x j ji j a a † = 1, Ux,μUx,μ (116) (121) det Ux,μ = 1. (117) which upon using the definition of Cx,ij can be written in the This means that one has to perform a constrained variation more familiar form of the degrees of freedom, which can either be done rather tediously by including the constraints explicitly in the action 1 a 1 a P Ux,i0 + Ux,i−0 = P Ux,ij + Ux,i− j . using Lagrangian multipliers or with the help of a variation 0 i 2 i j 2 a a j a a that conserves the unitary and determinant constraint. An (122) appropriate variation is given by As before, time evolution under the EOM conserves the δUx,μ = iδ Ax,μUx,μ, (118) Gauss constraint exactly (see Appendix D). In order to actu- δ = δ a a ally solve the equations we specify the temporal gauge where Ax,μ a Ax,μt is an element of the Lie algebra su(N), i.e. hermitian and traceless. An extended discussion U , = 1, (123) can be found in the Appendix B. x 0 which enables us to compute the spatial link variables of the 4.1 Leapfrog scheme next time-slice using past links and the temporal plaquette: = . Following the same procedure as in the case of Abelian gauge Ux+0,i Ux,0i Ux,i (124) fields, we vary w.r.t. spatial components Ux,i to obtain the The temporal plaquette Ux,0i has to be determined from Eq. discrete EOM and w.r.t. temporal components U , to find x 0 (122). For SU(2) this can be done explicitly [17,18]: We use the Gauss constraint. the parametrization Starting with the constraint we get (see Appendix C.1 for details) 0 a a Ux,i0 = u 1 + i u σ , (125) a 1 a P Ux,0i + Ux,0−i = 0. (119) 0 i 2 where σ a, a ∈{1, 2, 3} are the Pauli matrices. The real- i a a valued parameters u0, ua fulfill the constraint Here we introduced the shorthand Pa (...) to denote (u0)2 + uaua = 1. (126) a a P (U) = 2Im tr t U . (120) a

Varying the spatial link variables we obtain the EOM (see The EOM (122) can then be written as

Appendix C.2 for more details) 0 2 a a a a 2u =−P U , − + P U , + U , − . x i 0 a j x ij x i j j 1 a P Ux,i0 + Ux,i−0 a0ai 2 (127) 123 884 Page 12 of 26 Eur. Phys. J. C (2018) 78 :884

To obtain u0 we take the positive branch of Eq. (126) Condensing our notation even further we write ! (+0) C = U , C + , U + + ,− , (136) u0 = 1 − uaua, (128) x,ij x 0 x 0 ij x i j 0 (− ) a 0 = , Cx,ij Ux,−0Cx−0,ijUx+i+ j−0,0 (137) assuming that the changes from one time slice to the next are and “small”, i.e. the temporal plaquette Ux,0i will be “closer” to 1 (+0) (−0) 1 than to −1. Mx,ij = C + C . (138) 2 x,ij x,ij 4.2 Implicit scheme It still holds that 2 0 M , C , + O a , (139) Guided by what we learned from the Abelian case in Sect. x ij x ij 3, we would like to replace one of the Cx,ij expressions in so the use of M , instead of C , does not change the the Wilson action (115) with a time-averaged equivalent. At x ij x ij accuracy of the scheme. the same time we need to retain the gauge invariance of the Note that temporal gauge renders all temporal link vari- action. Simply using the temporally averaged expression ables trivial and Eq. (133) reduces to the simple time average. 1 The inclusion of temporal links in Mx,ij breaks a symme- Cx+0,ij + Cx−0,ij , (129) 2 try on the lattice: the diagonal from x to x + i + j is now is not enough because Cx+0,ij and Cx−0,ij transform differ- a preferred direction, which can be seen in Fig. 3.Theloss ently. A solution is to include temporal gauge links in order of symmetry can be mitigated by also including terms like , − to “pull back” Cx+0,ij and Cx−0,ij to the lattice site x.This Mx i j in the action. Therefore we propose the action leads us to the definition of the “properly” averaged field strength V 1 † S[U]= C , C 2 2 tr x 0i x,0i g a0ai x i 1 Mx,ij ≡ Ux,0Cx+0,ijUx+i+ j,−0 1 1 † 2 − tr Cx,ijM , , (140) 4 i j 2 x ij i,| j| a a + Ux,−0Cx−0,ijUx+i+ j−0,0 , (130) where we explicitly include terms with negative spatial indices using the sum | | over positive and negative com- which transforms like Cx,ij, i.e. j ponents j to keep the action as symmetric as possible. The → † . Mx,ij Vx Mx,ijVx+i+ j (131) action is also invariant under time reversal, real-valued (see Appendix E for a proof) and gauge invariant. While we have This gauge-covariant averaging procedure can be general- not made any changes to the terms involving temporal pla- X ized: consider an object x,y that transforms like quettes the spatial plaquette terms now include temporal links X → X †. and therefore we will also obtain a modified Gauss constraint x,y Vx x,y Vy (132) like in the semi-implicit scheme for Abelian fields. As an example, Xx,y could be a Wilson line connecting Performing the variation the same way we did for points x and y along some arbitrary path. A time-averaged the leapfrog scheme, we obtain the Gauss constraint (see version of Xx,y is given by Appendix E.1)

1 1 a X x,y = Ux,0Xx+0,y+0Uy+0,−0 P Ux,0i + Ux,0−i 2 0 i 2 i a a 1 1 (+ ) + Ux,−0Xx−0,y−0Uy−0,0 , (133) a 0 † =− P C , C , , (141) 8 i j 2 x ij x ij |i|,| j| a a where Xx±0,y±0 is simply Xx,y shifted up (or down) by one time step. It still transforms like Eq. (132), i.e. where |i| denotes the sum over positive and negative indices i. The left hand side (LHS) of (141)isthesame X → X †. x,y Vx x,y Vy (134) as in the leapfrog scheme Eq. (119), but now there is also a new term on the right hand side (RHS) from varying the Using this we can write spatial part of the action. Performing the continuum limit for 0 Mx,ij = C x,ij. (135) the Gauss constraint (after multiplying both sides with a ), 123 Eur. Phys. J. C (2018) 78 :884 Page 13 of 26 884

x +0+i x +0+1

x +0 x +0+i + j x +0 x +0+1+i

x x + i + j x x +1+i

x − 0 x − 0+i + j x − 0 x − 0+1+i

x − 0+j x − 0+i

Fig. 3 Left: the Wilson lines associated with the properly time- temporal link connections only at the start and end points. Therefore the averaged field strength Mx,ij used in the implicit scheme. Right: the diagonal from x to x + i + j is a preferred direction. This asymmetry Wilson lines associated with the partially averaged field-strength Wx,1i can be repaired by including Mx,i− j terms in the action (140). We can used in the semi-implicit scheme. Spatial link variables are drawn as also see that in the continuous time limit (the vertically stacked time solid black arrows; temporal links as dashed arrows. The shaded planes slices would merge into one) the paths traced by Mx,ij and Wx,1i would represent (equal) time slices, i.e. the spatial lattice (only two dimensions become identical to Cx,ij shown) in subsequent time steps. The averaged field-strength Mx,ij has

( ) 0 2 0 the RHS term vanishes as O a . This shows that the ing from an initial guess for the future link variable Ux+0,i , RHS is not a physical contribution, but rather an artifact of for instance by performing a single evolution step using the the implicit scheme. Note that the correct continuum limit of leapfrog scheme, we iterate the constraint (and the EOM) is already guaranteed by the 1 a a action. U + P Ux,i−0 a0ai 2 In a similar fashion as before we perform the variation 1 1 a (n)† (n)† w.r.t. spatial link variables to get the discrete EOM. We ﬁnd =− P Ux,i Ux+i, j M , + M − , Ux− j, j , 2 i j 2 x ij x j ji (see Appendix E.2) | j| a a (143) 1 a + a (n+1) a (n) a P Ux,i0 Ux,i−0 P U , = αP U , + (1 − α) U , (144) a0ai 2 x i0 x i0

1 1 a † † ( ) ( ) =− P Ux,i Ux+i, j M , + M − , Ux− j, j , n n 2 i j 2 x ij x j ji using U + , in M , from the last iteration step to determine | j| a a x 0 i x ij (n+1) a( ) (142) Ux,i0 . In the above equations we replaced P Ux,i0 by an unknown variable U a.WefirstsolveEq.( 143)forU a and which is formally similar to the leapfrog scheme (121) with a (n+1) α then update P Ux,i0 using Eq. (144). The parameter is Mx,ij in place of Cx,ij, a sum over positive and negative com- used as a damping coefficient to mitigate numerical instabil- ponents j (instead of just positive indices) and an additional ities induced by fixed point iteration. For SU(2) we construct / factor of 1 2 to avoid overcounting. As a simple check one (n+1) a (n+1) the temporal plaquette U , from P U , using Eqs. can replace Mx,ij with Cx,ij in Eq. (142) (only introducing an x i0 x i0 irrelevant error term quadratic in a0) and recover Eq. (121). (127) and (128) and using temporal gauge we update the link Compared to the leapfrog scheme, solving Eq. (142)is variables via more complicated: it is not possible to explicitly solve for (n+1) = (n+1) . Ux+0,i Ux,0i Ux,i (145) the temporal plaquette Ux,i0 anymore because Mx,ij on the RHS involves contributions from both “past” and “future” Then we can repeat the iteration and keep iterating until con- link variables. Completely analogous to the case of Abelian vergence. gauge fields on a lattice, we obtain an implicit scheme by This iteration scheme can be used to solve the EOM (142) introducing time-averaged field strength terms in the action. until the Gauss constraint (141) is satisfied up to the desired Similar to what we did in Sect. 2.3, we propose to solve numerical accuracy. Conversely, this means that unlike the Eq. (142) iteratively using damped fixed point iteration. Start- leapfrog scheme, where the Gauss constraint (119)isalways 123 884 Page 14 of 26 Eur. Phys. J. C (2018) 78 :884 satisfied up to machine precision in a single evolution step, 2 1 1 0 F B the implicit scheme, solved via an iterative scheme, only U x, 1 + iga Ax, − a ∂ ∂ Ax, 1 1 2 1 0 0 approximately conserves the Gauss constraint (141). How- 2 ever, in Sect. 5 we will show that using a high number of + O a1 . (152) iterations the constraint can be indeed fulfilled to arbitrary accuracy. In practice however we find that a lower number of iteration is sufficient for stable and acceptably accurate We find that the linear term of the gauge-covariant aver- simulations at the cost of small violations of the constraint. age (150) agrees with the expression we constructed in the It is also immediately obvious that solving the implicit Abelian semi-implicit scheme (78). While we had to include scheme requires higher computational effort compared to an “arbitrary” correction term in the Abelian scheme to fix the leapfrog scheme. Considering that one has to use the gauge invariance, the link formalism of lattice gauge the- leapfrog scheme for a single evolution step once (as an ini- ory forces us to only consider closed paths constructed from tial guess) and then use fixed point iteration, where every step gauge links in the action and thus naturally leads us to the is at least as computationally demanding as single leapfrog “proper” averaging procedure. step, it becomes clear that the use of an implicit scheme is The Wilson line path traced by Wx,1i , as compared to only viable if increased stability allows one to use coarser Mx,ij, is shown in Fig. 3 on the right. As before, we keep lattices while maintaining accurate results. the action as symmetric as possible by also including terms with negative transverse directions, i.e. Wx,1−i . Inspired by 4.3 Semi-implicit scheme the Abelian semi-implicit case (83), we define the new action as Using our knowledge from Sects. 3 and 4.2 we can now generalize the semi-implicit scheme to real-time lattice gauge V 1 † S[U]= tr Cx,01C , g2 0 1 2 x 01 theory. An appropriate generalization of the semi-averaged x a a field strength (82) is given by + 1 † tr Cx,0i Cx,0i a0ai 2 1 (+0) (−0) i Wx,1i = U , + U , Ux+1,i 2 x 1 x 1 1 1 † − tr Cx,ijM , i j 2 x ij 1 (+0) (−0) 4 ,| | a a − Ux,i U + , + U + , , (146) i j 2 x i 1 x i 1 1 1 † − tr Cx,1 j W , + h.c. , (153) 4 1 j 2 x 1 j where | j| a a

(+0) = , Ux,μ Ux,0Ux+0,μUx+μ+0,−0 (147) where the sum over i and j only run over transverse coordi- (−0) = . nates and x1 is the longitudinal coordinate. Ux,μ Ux,−0Ux−0,μUx+μ−0,0 (148) The purely transverse part of the action uses the same terms as the implicit scheme, see Eq. (140). The longitudinal- We also deﬁne Wx,i1 ≡−Wx,1i . Using the time-averaging notation [see Eq. (133)] this can be written more compactly transverse part is now given in terms of Cx,1 j and Wx,1 j anal- as ogous to Eq. (69). We have to explicitly include the hermitian conjugate in order to keep the action real-valued. Wx,1i = U x,1Ux+1,i − Ux,i U x+i,1, (149) Varying with respect to temporal components yields the Gauss constraint (see Appendix F.1) where 1 (+ ) (− ) 3 ≡ 0 + 0 . 1 U x,1 Ux,1 Ux,1 (150) a 2 P Ux,0i + Ux,0−i 0 i 2 i=1 a a Note that =− 1 1 a (+0) † 2 P Cx,ij Cx,ij 0 i j 2 U , U , + O a , (151) | |,| | 8 a a x 1 x 1 i j 1 1 (+ ) (+ ) − a 0 † + 0 † which shows that the semi-averaged ﬁeld strength W , only 2 P Ux,1 Tx,1 Tx,1 Ux,1 x 1i 8 a1 differs from Cx,1i by an irrelevant error term. Taking the Abelian limit (i.e. neglecting commutator terms) of Eq. (150) + (+0) † + (+0) † , Ux−1,1Tx−1,1 Tx−1,1Ux−1,1 (154) and expanding for small lattice spacing yields 123 Eur. Phys. J. C (2018) 78 :884 Page 15 of 26 884 where we use the shorthand and the transverse components are given by

1 ≡ − 1 a a Tx,1 Cx,1 j Ux+1+ j,− j Ux,− j Cx− j,1 j , + , − = , , [ , ] j 2 P Ux i0 Ux i 0 P Ux i Kx ij U M | | a 0 i 2 j a a | j| 1 1 = 2 − Ux, j1 − Ux,− j1 Ux,1. (155) + [ , ]+ [ , ] . j 2 Kx,i1 U W Kx,i1 U C (160) | j| a 2 |1|

Varying w.r.t. spatial links the discrete semi-implicit EOM These equations can be solved numerically using damped read (see Appendix F.2) fixed point iteration completely analogously to Sect. 4.2. (0) First, one obtains an initial guess Ux+0,i for “future” link 1 a variables from a single leapfrog evolution step using Eqs. P U , + U , − 2 x 10 x 1 0 = a0a1 (122) and (124). Then iterate from n 1 until convergence: 1 1 a † † =− P U , U + , W + W U − , 2 x 1 x 1 j x,1 j x− j, j1 x j j 1. Compute the next iteration using damped fixed point iter- 4 a1a j | j| a → U a ation: in Eqs. (156) and (157) replace P Ux,10 1 + † + † , a , → U a U Ux+1, j Cx,1 j Cx− j, j1Ux− j, j (156) and P Ux i0 i , solve for the unknown ’s and update the temporal plaquettes using and for transverse components a (n) = α a (n−1) + ( − α) U a P Ux,10 P Ux,10 1 1 (161) 1 a P U , + U , − ( ) 2 x i0 x i 0 n U a α a0ai and analogously for Ux,i0 and i . is the damping coefficient. =−1 1 a † + † P Ux,i Ux+i, j Mx,ij Mx− j, jiUx− j, j 2 ai a j 2 2. For SU(2) we can reconstruct the full temporal plaquette | j| a from its components P (U) with the identity 1 1 a † † − P Ux,i Ux+i,1W , + W − , Ux−1,1 4 i 1 2 x i1 x 1 1i ! a a | | 1 1 i U = 1 − Pa (U)21 + σ a Pa (U) , (162) + † + † , U x+i,1Cx,i1 Cx−1,1i U x−1,1 (157) 4 a 2 a

= (n) = (n) where |1| simply means summing over the terms with pos- for U Ux,10 and U Ux,i0. itive and negative longitudinal directions. We now have two (n) (n) (n) 3. Using Ux,10 and Ux,i0, compute the spatial links Ux+0,1 sets of equations: one for longitudinal components, and two ( ) and U n via for transverse components. x+0,i The equations of motion can be written more compactly (n) = (n) . by introducing the symbol Ux+0,i Ux,0i Ux,i (163)

4. Repeat with n → n + 1. 1 1 † † Kx,ij[U, C]=− Ux+i, j C , − C − , Ux− j, j , 2 i j 2 x ij x j ij a a As with the implicit scheme, our approach to solving the (158) equations in the semi-implicit scheme is an iterative one. The Gauss constraint (154) is only approximately satisﬁed, where C can be exchanged for corresponding expressions depending on the degree of convergence.2 with M or W and U can be exchanged for its temporally averaged version U. The longitudinal component of the EOM 4.4 Coupling to external color currents then reads 1 a Up until now we have only considered pure Yang-Millsﬁelds. P U , + U , − 2 x 10 x 1 0 In the continuum we can include external color currents by a0a1

1 a 2 = P Ux, Kx, i [U, W]+K x, i [U, C] , This is also true for the Abelian semi-implicit scheme. If the equations 2 1 1 1 |i| of motion are solved only approximately using an iterative method, then the conservation of the Gauss constraint is also only approximate (159) depending on the degree of convergence. 123 884 Page 16 of 26 Eur. Phys. J. C (2018) 78 :884 adding a J · A term to the action. and (156), (157)). Due to conservation of the Gauss constraint without external charges, the continuity equation for 1 μν the implicit and semi-implicit scheme is simply Eq. (171)as S[A]=SYM + SJ =− tr Fμν(x)F (x) 2 μ,ν well. This implies that our treatment of the external currents x in terms of parallel transport as detailed in our previous pub- ν − 2 tr J (x)Aν(x) . (164) lication [14] does not require any modiﬁcations when using ν x the newly derived schemes.

The equations of motion then read μν ν 5 Numerical tests Dμ F (x) = J (x), (165) μ In this last section we test the semi-implicit scheme on the and due to gauge-covariant conservation of charge we have propagation of a single nucleus in the CGC framework. For an observer at rest in the laboratory frame, the nucleus moves μ( ) = , Dμ J x 0 (166) at the speed of light and consequently exhibits large time μ dilation. As the nucleus propagates, the interactions inside which is the non-Abelian continuity equation. On the lattice appear to be frozen and the field configuration is essentially we can simply add a discrete J · A term to the action as well: static. On the lattice we would like to reproduce this behavior as well, but depending on the lattice resolution we run into the 3 V g b b g b b numerical Cherenkov instability, which leads to an artificial SJ = − ρ A , + j , A , , (167) g2 a0 x x 0 ai x i x i increase of the total field energy of the system. x,b i=1 As previously stated the root cause of the instability is a μ where Ax,μ includes a factor of ga (“lattice units”). We also numerical dispersion: in the CGC framework a nucleus con- a( ) ρa made the split into 3+1 dimensions explicit using J0 x x sists of both propagating field modes and a longitudinal cur- a( ) a and Ji x jx,i . The variation of SJ simply reads rent generating the field around it. It is essentially a non- Abelian generalization of the field of a highly relativistic 3 V g b b g b b electric charge. In our simulation the color current is mod- δSJ = − ρ δ A , + j , δ A , , (168) g2 a0 x x 0 ai x i x i eled as an ensemble of colored point-like particles moving at x,b i=1 the speed of light along the beam axis. The current is unaf- which gives the appropriate contributions to the Gauss con- fected by dispersion, i.e. it retains its shape perfectly as it straint and the EOM. In the leapfrog scheme the constraint propagates. The field modes suffer from numerical disper- now reads sion, which over time leads to a deformation of the original 1 a + = g ρa longitudinal profile. The mismatch between the color current P Ux,0i Ux,0−i x (169) 0 i 2 a0 and the field leads to creation of spurious field modes, which i a a interact with the color current non-linearly through parallel and the EOM read transport (color rotation) of the current. This increases the mismatch further and more spurious fields are created. As the 1 a , + , − simulation progresses this eventually leads to a large artifi- 2 P Ux i0 Ux i 0 a0ai cial increase of total field energy. The effects of the instability = 1 a + − g a . can be quite dramatic as seen in Fig. 4. The main difference P Ux,ij Ux,i− j jx,i (170) i j 2 ai to the numerical Cherenkov instability in Abelian PIC sim- j a a ulations is that in electromagnetic simulations the spurious The constraint taken together with the EOM imply the local field modes interact with the particles through the Lorentz conservation of charge (see Appendix D) force [27]. In our simulations we do not consider any acceleration of the particles (i.e. their trajectories are fixed), but † ρ − ρ − jx,i − U − , jx−i,i Ux−i,i interaction is still possible due to non-Abelian charge con- x x 0 = x i i , (171) a0 ai servation (171), which requires rotating the color charge of i the color current. Therefore our type of numerical Cherenkov which is the discrete version of the continuity equation. For instability is due to non-Abelian effects. the implicit and semi-implicit schemes the procedure is the We now demonstrate that the instability is cured (or at least same: including the SJ term simply leads to the appearance highly suppressed for all practical purposes) by using the of ρ on the RHS of the Gauss constraint (see Eqs. (141) semi-implicit scheme. We test the schemes ability to improve and (154)) and jx,i on the RHS of the EOM (see Eqs. (142) energy conservation the following way: we place a single 123 Eur. Phys. J. C (2018) 78 :884 Page 17 of 26 884

LF 256 LF 256 1 LF 1024 0.4 LF 512 LF 1024 SI 256 (0) LF 2048 E

(0) SI 256

− 0.2 ) ) [a.u.] E t 1 (

x 0.5 ( E ε 0 00.511.52 0 t [fm/c] 2.32.35 2.42.45 2.52.55 2.62.65 2.7 1 x [fm] Fig. 5 The relative increase of the total field energy E(t) as a function of time t for the propagation of a single nucleus in a box of volume Fig. 4 The energy density of a right-moving nucleus averaged over V = (3fm) × (6fm)2 with a longitudinal length of 3 fm and a trans- 1 2 2 the transverse plane as a function of the longitudinal coordinate x verse area of (6fm) . The lattice size is NL × N with the transverse = / T after t 2fm c. The results were obtained from the same simulations lattice fixed at NT = 256. Starting with the same initial condition, we as in Fig. 5. We compare the performance of the leapfrog (LF) scheme evolve forward in time using the leapfrog (LF) and the semi-implicit (with NL = 256 and NL = 1024) to the semi-implicit (SI) scheme with (SI) method. In the case of the leapfrog scheme we vary the resolu- = NL 256. In the most extreme example (LF 256) the nucleus becomes tion along the beam axis using the number of longitudinal cells NL of completely unstable due to the numerical Cherenkov instability. By the lattice. For NL = 256 the numerical Cherenkov instability leads eliminating numerical dispersion (SI 256) the nucleus retains its original to catastrophic failure, increasing the energy to many times its original shape almost exactly value. The effect is suppressed when increasing the longitudinal resolution, but the instability is still present. In the case of the semi-implicit scheme it is possible to set NL = 256 and still obtain (approximate) gold nucleus described by the McLerran-Venugopalanmodel energy conservation. After t = 2fm/c the energy increase for the semi- . in a simulation box of volume V = (3fm) × (6fm)2 and set implicit scheme is roughly 0 02%, compared to 1% for the leapfrog with N = 2048. For the simulation using the semi-implicit scheme we used the longitudinal extent of the nucleus to roughly correspond L Ni = 10 iterations and a damping coefficient of α = 0.45. The time to a boosted nucleus with Lorentz factor γ = 100. These step is set to the longitudinal lattice spacing. In the case of the leapfrog parameters correspond to a similar setting as in our previous simulation we used a0 = a1/4 work [23]. We refer to [22] for a detailed description of the initial conditions. After setting up the initial condition we let the nucleus freely propagate along the longitudinal axis. As the simulation runs we record the total field energy scheme. However, using the new scheme might not always 1 E(t) = d3x Ea (t, x)2 + Ba (t, x)2 , (172) be economical: finer lattices suppress the instability as well 2 i i V i,a and since the leapfrog scheme is computationally cheaper than the semi-implicit scheme, the leapfrog can be favorable where Ea(x) and Ba(x) are the color-electric and -magnetic i i in practice. On our test system (a single 256 GB node on fields at each time step. On the lattice the electric and mag- the VSC 3 cluster) the simulation using the semi-implicit netic fields are approximated using plaquettes: scheme (SI 256) takes ∼ 4 hours to finish, while the same simulation using the leapfrog with NL = 1024 (LF 1024) a 1 a E (x) P U , , ∼ . = i 0 i x 0i (173) takes roughly 2 5 hours and with NL 2048 (LF 2048) ga a ∼ 10 hours. Even though energy conservation is not as good a 1 a B (x) − εijk P Ux, jk . (174) as SI 256, the longitudinal resolution is much better in com- i 2ga j ak j,k parison enabling us to extract observables with higher accuracy. It should be noted however that our implementation of We compute the relative change (E(t) − E(0)) /E(0), which the leapfrog scheme is already highly optimized, while the we plot as a function of time t. In the continuum we would implementation of the semi-implicit scheme is very basic and have E(t) = E(0), but due to numerical artifacts and the should be considered as a proof of concept. Further optimiza- Cherenkov instability this is not the case in our simulations. tions and simplifications of the semi-implicit scheme might The numerical results are shown in Fig. 5. We see that make it the better choice in most cases. the leapfrog scheme leads to an exponential increase of the As a second test we look at the violation of the Gauss con- total energy over time, which can be suppressed using finer straint. The leapfrog scheme (122) conserves its associated lattices. On the other hand, the semi-implicit scheme leads Gauss constraint (119) identically, even for finite time-steps to better energy conservation even on a rather coarse lattice. a0. In numerical simulations this conservation is not exact Therefore, the resolution that is usually required to obtain due to floating point number errors, but the violation is zero accurate, stable results is lowered by using the semi-implicit up to machine precision. On the other hand, the semi-implicit 123 884 Page 18 of 26 Eur. Phys. J. C (2018) 78 :884

−9 6 Conclusions and outlook

δg 10 In this paper we derived new numerical schemes for real- −13 10 time lattice gauge theory. We started our discussion based on two simpler models, namely the two-dimensional wave e violation −17 10 equation and Abelian gauge fields on the lattice. It turns out that using a discrete variational principle to derive numer- relativ 10−21 ical schemes for equations of motion is a very powerful 10 20 30 40 50 tool: the use of time-averaged expressions in the discrete Ni action yields implicit and semi-implicit schemes depending on how exactly the time-averaging is performed and what Fig. 6 The relative Gauss constraint violation for the semi-implicit scheme as a function of the number of iterations Ni of the damped terms are replaced by their averages. We extended this con- fixed point iteration. For this plot we used a simulation box of volume cept to real-time lattice gauge theory, allowing us to make = ( . ) × ( )2 = V 1 5fm 6fm on a lattice with NT 256 points in the trans- modifications to the standard Wilson gauge action that yield = verse directions and NL 128 points in the longitudinal direction. new numerical schemes, which have the same accuracy as the Otherwise, we use the same initial conditions as in Figs. 4 and 5.Atthe beginning of the simulation at t = 0fm/c the constraint is conserved up leapfrog scheme and, most importantly, are gauge-covariant to machine precision by construction. We then let the nucleus propagate and conserve the Gauss constraint. Finally, we demonstrated until t = 0.5fm/c and compute the violation of the Gauss constraint a peculiar property of the semi-implicit scheme: it allows for (black dots). We also compare to the constraint violation after only a dispersion-free propagation along one direction on the lat- single evolution step (red crosses), which does not differ much from the violation after a larger number of time steps. It is evident that the vio- tice, thus curing a numerical instability that has plagued our lation systematically converges towards zero (up to machine precision) simulations of three-dimensional heavy-ion collisions. as we increase the number of iterations Ni Although all numerical tests in this work have been performed for the propagation of a single nucleus, we expect that the new semi-implicit scheme will improve simulations scheme has to be solved iteratively and therefore the results of nucleus-nucleus collisions in multiple ways. Primarily, using the new scheme we can be sure that the color fields depend on the number of iterations Ni used in the fixed point iteration method. We define the relative violation of the Gauss of incoming nuclei have not been altered up until the colli- constraint as the ratio of the absolute (squared) Gauss con- sion event and all changes to the fields afterwards are solely straint violation to the total (squared) charge on the lattice. due to interaction between the colliding nuclei during the In the case of the leapfrog scheme this reads collision event itself. This also helps to improve numerical accuracy in the forward and backward rapidity region: at later simulation times, when the now outgoing nuclei are well sep-

2 arated, all field modes (i.e. “gluons”) with momenta at high 1 a g a , | | P U , − ρ x a i ( 0 i )2 x 0i a0 x rapidity must have been created in the collision and con- δ ( ) = a a , g t (175) tributions from artificial modes emitted by the nucleus due g 2 ρa x ,a a0 x to numerical Cherenkov radiation are strongly suppressed. Furthermore, from the dispersion relations (90) and (91)we can infer that these gluons with almost purely longitudinal where the sum x runs over the spatial lattice of a single momentum kL (and small transverse momentum kT ) exhibit time slice at t. The numerator depends on the Gauss con- a phase velocity approximately the speed of light (up to an 0 straint of the scheme and has to be adjusted according to the error term quadratic in a kT ). This means any interactions implicit and semi-implicit method (either Eq. (141)or(154) between high rapidity gluons produced in the collision and including the charge density on the RHS as discussed in Sect. the finite-thickness color fields of the nuclei directly after the 4.4). In Fig. 6 we show how the Gauss constraint violation collision event can be considered physical and are not tainted converges systematically towards zero as we increase the by numerical dispersion as in our previous simulations. Con- number of iterations. Therefore, even though we can not use sequently, it should be possible to extract space-time rapidity an arbitrarily high number of iterations due to limited com- profiles of the local rest frame energy density of the Glasma putational resources, the semi-implicit scheme conserves the as in [23] valid for larger ranges of rapidity as previously Gauss constraint in principle. The same holds for the purely considered. implicit scheme. In practice it is not necessary to satisfy the In conclusion, we hope that this new treatment of solving constraint up to high precision, as observables such as the the Yang-Mills equations on the lattice allows us to perform energy density seem to converge much faster up to satisfying better, more accurate simulations using more complex mod- accuracy. els of nuclei. 123 Eur. Phys. J. C (2018) 78 :884 Page 19 of 26 884

Acknowledgements The authors thank A. Dragomir, C. Ecker, T. we will use the Gauss constraint to first reduce the number of Lappi, J. Peuron and A. Polaczek for helpful discussions and com- degrees of freedom and then compute the dispersion relation ments. This work has been supported by the Austrian Science Fund ω(k). Inserting the ansatz into the Gauss constraint we find FWF, Project No. P26582-N27 and Doctoral program No. W1252-N27. The computational results have been achieved using the Vienna Scien- tific Cluster. + χ 2 χ B + − χ 2 χ B = , 1 i 1 A1 1 1 i Ai 0 (A.8) i i Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecomm where we use the dimensionless lattice momenta (63). The ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, constraint equation can alternatively be written as and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative χ B =−β χ B , 1 A1 i Ai (A.9) Commons license, and indicate if changes were made. i Funded by SCOAP3. where β is a momentum-dependent factor given by 1 − χ 2 β = 1 . (A.10) Appendix A: stability analysis of the semi-implicit scheme + χ 2 1 i i for Abelian gauge fields The temporal average Ax,i reduces to a multiplication with In this section of the appendix we prove stability for the semi- a frequency dependent factor: implicit scheme for Abelian fields derived in Sect. 3.3.Using 0 Ax,i = cos ωa Ax,i = cAx,i , (A.11) temporal gauge, A , = 0, the equations of motion (86) and x 0 (87) read where we used the shorthand c = cos ωa0 . Inserting the plane wave ansatz into the EOM yields 2 1 B −∂ Ax,1 = ∂ Wx,1i + Mx,1i , (A.1) 0 2 i 1 i 2 2 B F χ A = c χ A − (1 + c) χ χ Ai , (A.12) 0 1 i 1 2 i 1 −∂2 = ∂ B + 1∂ B + , i i 0 Ax,i j Mx,ij 1 Wx,i1 Fx,i1 (A.2) 2 χ 2 = χ 2 + χ 2 − χ Bχ F j=i 0 Ai c j Ai 1 Ai c j i A j j=i j=i where W , reduces to x 1i − 1 ( + ) χ Bχ F . 1 c 1 i A1 (A.13) = ∂ F − ∂ F ¯ . 2 Wx,1i 1 Ax,i i Ax,1 (A.3) Note that both χ and c depend on ω. After making use of The Gauss constraint (85) reads 0 the Gauss constraint the longitudinal EOM reads d 0 2 2 2 2 1 −1 2 B F a B B F F F χ A = c χ + χ A + (1 + c) β χ A , (A.14) ∂ ∂ Ax,i + ∂ ∂ ∂ ∂ Ax,i − ∂ Ax, = 0. 0 1 2 3 1 1 1 i 0 2 1 i 0 1 i 1 2 i=1 i (A.4) and the two transverse equations read

χ 2 = χ 2 +χ 2 − χ Bχ F − 1 ( + ) χ Bχ F Splitting the equations of motion into Laplacian terms and 0 A2 c 3 A2 1 A2 c 3 2 A3 1 c 1 2 A1 mixed derivative terms we find 2 1 = cχ 2 A + χ 2 A + (1 + c) βχ2 A 3 2 1 2 2 2 2 −∂2 =− ∂2 + 1 ∂ B∂ F + , 0 Ax,1 i Ax,1 i 1 Ax,i Ax,i 1 2 + (1 + c) β − c χ F χ B A , (A.15) i i 2 2 3 3 (A.5) χ 2 = χ 2 +χ 2 − χ B χ F − 1 ( + ) χ Bχ F −∂2 =− ∂2 − ∂2 0 A3 c 2 A3 1 A3 c 2 3 A2 1 c 1 3 A1 0 Ax,i j Ax,i 1 Ax,i 2 = j i = χ 2 + χ 2 + 1 ( + ) βχ2 c 2 A3 1 A3 1 c 3 A3 + ∂ B∂ F + 1∂ B∂ F + . 2 j i Ax, j 1 i Ax,1 Ax,1 2 1 B F j=i + (1 + c) β − c χ χ A . (A.16) 2 2 3 2 (A.6) This system of equations can be written in matrix notation Using a plane wave ansatz as an eigenvalue problem ω 0− i i = i x i k x , = χ 2 , Ax,i Ai e (A.7) MA 0 A (A.17) 123 884 Page 20 of 26 Eur. Phys. J. C (2018) 78 :884 where the vector A is simply (A.23) are restricted to [−1, 1]. Both dispersion relations ⎛ ⎞ remain stable if the CFL condition A1 A = ⎝ A ⎠ . (A.18) χ 2 ≤ , 2 1 1 (A.25) A3 holds. Using The eigenvectors of M are ⎧⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎫ a0 k1a1 ⎨ ⎬ χ 2 = 2 , 1 0 0 1 sin (A.26) , , = ⎝ ⎠ , ⎝ −χ B ⎠ , ⎝ χ F ⎠ , a1 2 AL AT,1 AT,2 ⎩ 0 3 2 ⎭ χ B χ F 0 2 3 and requiring stability for all values of k1 yields the constraint (A.19) a0 ≤ a1. (A.27) where we find two transverse, momentum dependent eigenvectors AT,1, AT,2 and the longitudinal unit vector AL .The This concludes the proof that the semi-implicit scheme, even two transverse vectors AT,1 and AT,2 can be interpreted as though exhibiting peculiar wave propagation phenomena, is transverse polarization modes and are orthogonal in the sense stable. † of AT,1 · AT,2 = 0. The three eigenvectors yield three different equations for the eigenvalue problem, namely Appendix B: variation of gauge links = λ ( ,χ ) = χ 2 , MAL L c i AL 0 AL (A.20) = λ ( ,χ ) = χ 2 , ∈ , , We introduce the infinitesimal variation of a gauge link vari- MAT,l T,l c i AT,l 0 AT,l l 1 2 (A.21) able where λL (c,χi ) and λT,k(c,χi ) are expressions which δUx,μ = iδ Ax,μUx,μ, (B.28) depend on the momenta χi and the frequency ω via c = ω 0 λ ( ,χ ) = χ 2 ω cos a . Solving the first equation L c i 0 for where the variation of the gauge field δ Ax,μ is traceless yields the dispersion relation of the longitudinal component: and hermitian. In the continuum limit δ Ax,μ becomes the ( ) − χ 2 + χ 2 + χ 2 infinitesimal variation of the gauge field Aμ x . The infinites- 0 1 1 2 2 3 δ ωL a = arccos . imal variation Ux,μ preserves the unitary of gauge links. Let 1 + χ 2 2 − χ 2 + χ 2 2 − χ 2 = + δ 2 1 3 1 Ux,μ Ux,μ Ux,μ then we find (A.22) λ ( ,χ ) = χ 2 † = + δ † + δ † Solving the two transverse equations T,l c i 0 yields Ux,μUx,μ Ux,μ Ux,μ Ux,μ Ux,μ − χ 2 1 + δ † + δ † + O δ 2 0 1 2 1 Ux,μUx,μ Ux,μ Ux,μ A ω , a = arccos , (A.23) T 1 + χ 2 + χ 2 1 2 2 2 3 2 1 + iδ Ax,μ − iδ Ax μ + O δ A and 1 + O δ A2 . (B.29) − χ 2 + χ 2 + χ 2 0 1 1 2 2 3 ω , a = arccos T 2 + χ 2 − χ 2 + χ 2 − χ 2 The determinant is also unaffected for infinitesimal varia- 1 2 2 1 3 2 1 0 tions. = ωL a . (A.24) = + δ ω det Ux,μ det Ux,μ Ux,μ It turns out that the expressions for associated with the 2 different eigenvectors are not the same, although AL and 1 + tr adj Ux,μ δUx,μ + O δ A AT,2 share the same dispersion relation. We interpret this as + † δ + O δ 2 numerical (or artificial) birefringence. Given a momentum k, 1 tr Ux,μ Ux,μ A the amplitude of an arbitrary wave has to be split into two + O δ 2 . components: a part which is projected into the plane spanned 1 A (B.30) by AL and AT,2 which oscillates with ωL = ωT,2 and a part parallel to AT,1 which oscillates with frequency ωT,1. The variation δUx,μ therefore preserves the constraints and We require the propagation of a wave to be stable, i.e. we allows us to vary the action without Lagrange multipliers, require the frequencies ω to be real-valued. This is guaranteed which dramatically simplifies the derivation of equations of if the arguments of the arccos expressions in Eqs. (A.22) and motion. 123 Eur. Phys. J. C (2018) 78 :884 Page 21 of 26 884

Appendix C: variation of the leapfrog action and subsequently δ [ ]= V 1 a + δ a . In this section of the appendix we give a derivation of the dis- t SE U P Ux,0i Ux,0−i Ax,0 g2 0 i 2 crete equations of motion obtained from the standard Wilson x,i,a a a action (109) using the constraint preserving variation of link (C.38) variables of the previous section. To make the calculation δ [ ]= more organized, we first split the action into two parts: a part We require that the variation vanishes, i.e. S U 0. Since all gauge links can be varied independently we find containing temporal plaquettes SE [U] (“E” for electric) and a part containing spatial plaquettes S [U] (“B” for magnetic). B 1 a P Ux,0i + Ux,0−i = 0. (C.39) We write 0 i 2 i a a S[U]=SE [U]−SB[U], (C.31) Appendix C.2: equations of motion where V 1 † First we consider the variation of SE [U] w.r.t spatial links. SE [U]= tr Cx,0i C , , (C.32) g2 0 i 2 x 0i We find a result that is similar to the expression for the Gauss x,i a a constraint V 1 1 † SB[U]= tr Cx,ijC , . (C.33) g2 2 i j 2 x ij x,i, j a a δ † ∼ δ † − † s Cx,0i Cx,0i i Ax,i Ux,i Cx−0,0i Ux−0,0 Ux+i,0Cx,0i Appendix C.1: Gauss constraint + h.c. (C.40)

For the Gauss constraint in the leapfrog scheme we only have and to consider the variation δt SE [U] as SB[U] does not contain any temporal links. In the following sections we make use of δ S [U] s E the “∼” symbol, denoting equality under the sum over lattice =−V 1 δ a a † − † sites x and under the trace. We then have Ax,i P Ux,i Cx− , i Ux−0,0 Ux+i,0Cx, i g2 0 i 2 0 0 0 x,i,a a a = V 1 δ a a + . Ax,i P Ux,i0 Ux,i−0 (C.41) δ † = δ − δ † + g2 0 i 2 t Cx,0i Cx, i Ux,0Ux+0,i Ux,i Ux+i,0 Cx, i h.c. x,i,a a a 0 0 ∼ δ † − † + Ux,0 Ux+0,i Cx,0i Cx−i,0i Ux−i,i h.c. [ ] For the variation of SB U we use = δ † − † + i Ax,0 Ux,0 Ux+0,i Cx,0i Cx−i,0i Ux−i,i h.c. δ † (C.34) s Cx,ijCx,ij i, j

To go from the first to the second line, we applied a shift ∼ δU , U + , + U , δU + , x → x − i in the right term of the first line and made use x i x i j x i x i j i, j of the cyclicity of the trace. In the third line we simply used − δ − δ † + the definition of the variation of gauge links. The variation Ux, j Ux+ j,i Ux, j Ux+ j,i Cx,ij h.c. of the action therefore reads † † ∼ i δ A , U , U + , C + C U − , + 2 x i x i x i j x,ij x− j, ji x j j h.c. V 1 a † i, j δt SE [U]=− P Ux,0 Ux+0,i C , 2 0 i 2 x 0i g , , a a (C.42) x i a − † δ a , Cx−i,0i Ux−i,i Ax,0 (C.35) The variation then reads δ [ ]=−V 1 δ a s SB U Ax,i whereweused 2 i j 2 g , , , a a x i j a Pa (U) ≡ 2Im tr taU =−i tr ta U − U † . (C.36) × a † + † . P Ux,i Ux+i, j Cx,ij Cx− j, jiUx− j, j Replacing the Cx,ij terms with link variables we find (C.43) † † U , U + , C − C U − , = − U , − U , − , x 0 x 0 i x,0i x−i,0i x i i 2 x 0i x 0 i We set δS = 0 and after canceling some constants we find (C.37) the discrete EOM 123 884 Page 22 of 26 Eur. Phys. J. C (2018) 78 :884

1 a Doing the same at x − i and parallel transporting from x − i P Ux,i0 + Ux,i−0 a0ai 2 to x yields =− 1 a † + † , P Ux,i Ux+i, j C , C − , Ux− j, j 1 i j 2 x ij x j ji + j a a Ux,0−i Ux,−0−i 0 i 2 ah (C.44) i a a 1 g † = Ux, j−i − U − , jx−i,i Ux−i,i . which can also be written as i j 2 ah ai x i i i, j a a i 1 a 1 a (D.52) P Ux,i0 + Ux,i−0 = P Ux,ij + Ux,i− j . 0 i 2 i j 2 a a j a a (C.45) Subtracting the above two equations gives

1 , + , − − , − − ,− − Ux i0 Ux i 0 Ux 0 i Ux 0 i ah Appendix D: conservation of the Gauss constraint in the a0ai 2 i leapfrog scheme g † =− j , − U j − , U − , . (D.53) ai x i x−i,i x i i x i i We now explicitly show that the leapfrog EOM preserve the i associated Gauss constraint. We use the identity for the fun- Using antisymmetry we have damental representation of SU(N) U , =− U , (D.54) 1 1 x i0 ah x 0i ah ta Pa (X) = X − X † − tr X − X † 1, i iN a 2 2 and (D.46) ,− − =− ,− − . Ux 0 i ah Ux i 0 ah (D.55) which can be shown using the Fierz identity for the generators Moreover, in temporal gauge we have U , − = U − , and of su(N) x i 0 x 0 0i Ux,−i−0 = Ux−0,0−i , which leads to 1 1 ta ta = δ δ − δ δ , (D.47) ij kl il jk N ij kl a 2 1 Ux,i0 + Ux,i−0 − Ux,0−i − Ux,−0−i 0 i 2 ah where i, j, k, l are fundamental representation matrix indices. i a a Using a shorthand we can write = 1 + − + Ux−0,0i Ux−0,0−i Ux,0i Ux,0−i ah a0ai 2 a a ( ) = , i t P X [X]ah (D.48) g =− ρx − ρx−0 , (D.56) a a0 where “ah” denotes the anti-hermitian traceless part of X. The constraint and the equations of motion then read (includ- where we used the Gauss constraint in the last line to replace ing external charges) the temporal plaquette terms with charge densities. This yields the gauge-covariant continuity equation 1 g U , + U , − = ρ , 1 1 † 2 x 0i x 0 i ah 0 x (D.49) (ρ − ρ ) = − . 0 i a x x−0 jx,i Ux−i,i jx−i,i Ux−i,i i a a a0 ai i 1 , + , − (D.57) Ux i0 Ux i 0 ah a0ai 2 If there are no external charges, we simply have the conser- 1 g = Ux,ij + Ux,i− j − jx,i . (D.50) vation of the Gauss constraint: assume that the constraint in i j 2 ah ai j a a the previous time slice holds, i.e. 1 We take (D.50) and sum over i.Dueto Ux,ij being anti- Ux−0,0i + Ux−0,0−i = 0, (D.58) ah 0 i 2 ah symmetric in the index pair i, j we ﬁnd i a a 1 then the EOM guarantee that it will also hold in the next one, , + , − Ux i0 Ux i 0 ah a0ai 2 i.e. i 1 1 g U , + U , − = 0. (D.59) = Ux,i− j − jx,i . (D.51) 2 x 0i x 0 i ah i j 2 ah ai a0ai i, j a a i i 123 Eur. Phys. J. C (2018) 78 :884 Page 23 of 26 884

Appendix E: variation of the implicit action This is a useful property that we will need in the following derivation. We now consider the action

S[U]=SE [U]−SB[U], (E.60) Appendix E.1: Gauss constraint where We perform the variation of SE [U] and SB[U] w.r.t. temporal links to derive the Gauss constraint in the implicit scheme. [ ]= V 1 † , SE U tr Cx,0i C , (E.61) Since SE [U] is the same for all schemes, we do not have to g2 0 i 2 x 0i x,i a a repeat it. On the other hand, the variation of the magnetic part V 1 1 † † involves terms like δt Cx,ijM , , which we now discuss SB[U]= tr Cx,ijM , . (E.62) x ij g2 4 i j 2 x ij x,i,| j| a a explicitly. First, we make use of δ † ∼ δ † The electric part is the same as in the leapfrog scheme. How- t Cx,ijMx,ij t Mx,ijCx,ij ever, the magnetic part SB[U] now contains temporal gauge 1 (+0) (−0) † links and gives a new contribution to the Gauss constraint. = δt C , + δt C , C , . (E.70) 2 x ij x ij x ij Before we vary this action, we must verify that SB[U] is indeed real-valued. While it is easy to see that the original Then, after some algebra we ﬁnd leapfrog action is real-valued because of the obvious her- † † micity of Cx,ijC , ,thetermCx,ijM , is not hermitian in † x ij x ij δ (+0) † ∼ δ (+0) † − (+0) general. Still, we can show that the action is real: we have t Cx,ij Cx,ij i Ax,0 Cx,ij Cx,ij Cx,−i− j Cx,−i− j † 1 (+0) (−0) † (E.71) Cx,ijM = Cx,ij C + C , (E.63) x,ij 2 x,ij x,ij where we can rewrite and (+0)† † † Cx,ijC , = Cx,ij Ux,0Cx+0,ijU + + , x ij x i j 0 (−0) † (+0) † (+0) † δ C C ∼ iδ A , −C , C +C C = † † t x,ij x,ij x 0 x ij x,ij x,−i− j x,−i− j Cx,ijUx+i+ j,0Cx+0,ijUx,0 (+ ) † ∼ † † = δ C 0 C† , (E.72) Ux,0Cx,ijUx+i+ j,0Cx+0,ij t x,ij x,ij (−0) † = C + , C + , . (E.64) x 0 ij x 0 ij which yields

Here we used the cyclicity of the trace. Then using a shift x → x − 0wehave δ † ∼ i δ (+0) † − t Mx,ijCx,ij Ax,0 Cx,ij Cx,ij h.c. (+0)† ∼ (−0) † . 2 Cx,ijCx,ij Cx,ij Cx,ij (E.65) (+ ) + C 0 C† − h.c. . (E.73) Likewise we have x,−i− j x,−i− j (−0)† (+0) † Cx,ijC , ∼ C , C , , (E.66) x ij x ij x ij The variation of the magnetic part therefore reads which leads to V 1 1 † † δ [ ]=− δ a C , M ∼ M , C . t SB U Ax,0 x ij x,ij x ij x,ij (E.67) g2 8 i j 2 x,a,i,| j| a a Incidentally, the RHS term is exactly the hermitian conjugate a (+0) † (+0) † of the LHS term. In other words ×P C , C , + C ,− − C ,− − x ij x ij x i j x i j † † = † ∼ † . V 1 1 (+ ) Cx,ijMx,ij Mx,ijCx,ij Cx,ijMx,ij (E.68) =− δ a a 0 † . Ax,0 P Cx,ij Cx,ij g2 8 ai a j 2 This means that under the sum over x and the trace, the x,a,|i|,| j| † (E.74) expression Cx,ijMx,ij is indeed real-valued and by extension SB[U] is real-valued as well. We have also shown that the In the last line we consolidated terms with index i and −i time-average in Mx,ij can be “shifted” to the other term Cx,ij under the sum and trace, i.e. into a single term using the sum |i|. Taking the result for δt SE [U] from the leapfrog scheme, we ﬁnd the Gauss con- † ∼ † . Cx,ijC x,ij C x,ijCx,ij (E.69) straint 123 884 Page 24 of 26 Eur. Phys. J. C (2018) 78 :884

Appendix F: variation of the semi-implicit action 1 a P Ux,0i + Ux,0−i 0 i 2 i a a In the semi-implicit scheme the action reads 1 1 a (+0) † =− P C , C , . (E.75) [ ]= [ ]− [ ], 8 i j 2 x ij x ij S U SE U SB U (F.82) |i|,| j| a a where SE [U] is the same as before. The magnetic part com- Appendix E.2: equations of motion prises of SB[U]=SB,M [U]+SB,W [U], where SB,M [U] [ ] is the same as SB U from the implicit scheme except that For the EOM we vary S[U] w.r.t spatial links. Again, we i| j| only runs through transverse components: already have the result for δs SE [U] from the leapfrog scheme δ [ ] V 1 1 † and only need to calculate s SB U . In particular we consider SB,M [U]= tr Cx,ijM , . (F.83) g2 4 i j 2 x ij the term x,i,| j| a a δ † = δ † + δ † . Therefore we can take the results from the previous section s Cx,ijMx,ij sCx,ijMx,ij Cx,ij s Mx,ij (E.76) for δSB,M [U],Eqs.(E.74) and (E.78). The new part is given Since the variation only acts on spatial links we can shift the by † time-average of the right term from M , to Cx,ij. This gives x ij V 1 1 † SB,W [U]= tr Cx,1 j W , + h.c. . g2 4 1 j 2 x 1 j x,| j| a a δ † ∼ δ † + δ † s Cx,ijMx,ij Cx,ijMx,ij Mx,ij Cx,ij (F.84) = δ † + Cx,ijMx,ij h.c. (E.77) Appendix F.1: Gauss constraint The variation then proceeds analogously to the derivation of δ [ ] δ [ ] the leapfrog scheme. We find We already know t SE U and t SB,M U from previous sections, so we only have to compute δt SB,W [U]. The relevant terms are δ [ ]=−V 1 1 δ a s SB U Ax,i g2 2 i j 2 x,i,| j| a a 1 † δt Wx,1 j C , j 2 x 1 j a † † | | a × P Ux,i Ux+i, j M , + M − , Ux− j, j . j x ij x j ji 1 † (E.78) = δt U x,1Ux+1, j − Ux, j δt U x+ j,1 C , j 2 x 1 j | j| a δ [ ] With the result for s SE U we obtain the discrete EOM 1 † † ∼ δt U x,1 Ux+1, j C , − C − , Ux− j, j j 2 x 1 j x j 1 j | | a j 1 a + P Ux,i0 Ux,i−0 = δ † , a0ai 2 t U x,1Tx,1 (F.85) 1 1 a † † =− P U , U + , M + M U − , . 2 x i x i j x,ij x− j, ji x j j where we defined 2 ai a j | j| (E.79) † 1 † † T , = Ux+1, j C , − C − , Ux− j, j . (F.86) x 1 j 2 x 1 j x j 1 j | j| a Introducing the shorthand Using the same techniques as before we find 1 1 † † K , [U, M]=− U + , M − M U − , , x ij 2 x i j x,ij x− j,ij x j j † i (+0) † (+0)† 2 ai a j δt U x, T ∼ δ Ax, U T − Ux, T 1 x,1 2 0 x,1 x,1 1 x,1 (E.80) − † (+0) + (+0)† , Tx−1,1Ux−1,1 Tx−1,1 Ux−1,1 (F.87) allows us to write the EOM rather compactly as

and the variation of SB,W [U] reads 1 a a P Ux,i0 + Ux,i−0 = P Ux,i Kx,ij[U, M] . 0 i 2 a a | j| δ [ ]=−V 1 δ a × t SB,W U Ax,0 (E.81) g2 1 2 x,a 8 a 123 Eur. Phys. J. C (2018) 78 :884 Page 25 of 26 884 (+ ) (+ ) (+ ) (+ ) † † × a 0 † + 0 † + 0 † + 0 † , δ , + =δ , − + P Ux,1 Tx,1 Tx,1 Ux,1 Ux−1,1Tx−1,1 Tx−1,1 Ux−1,1 j Wx 1 j Cx,1 j h.c. j Wx 1 j Cx,1− j h.c. − j j (F.88) ∼ δ † + δ † + . j Wx,−1 j Cx,−1 j j Cx,−1 j Wx,−1 j h.c. which (with the previous results taken into account) yields j the Gauss constraint in the semi-implicit scheme, see Eq. (F.93) (154). This allows us to write Appendix F.2: equations of motion δ † ∼ δ j Wx,1 j Cx,1 j i Ax, j Ux, j | | | | For the variation w.r.t. spatial links we have to distinguish j 1 two cases: the longitudinal and the transverse links. Starting † † × U x+ j,1C , − C − , U x−1,1 , with the variation of longitudinal links we ﬁnd x j1 x 1 j1 (F.94) † † δ , + δ ∼ δ 1 Cx 1 j Wx,1 j h.c. j Cx,1 j Wx,1 j i Ax, j Ux, j | | | | = δ † + δ † + j 1 1Cx,1 j Wx,1 j Cx,1 j 1Wx,1 j h.c. † † × Ux+ j,1W , − W − , Ux−1,1 , ∼ δ † + † x j1 x 1 j1 i Ax,1Ux,1 Ux+1, j Wx,1 j Wx− j, j1Ux− j, j (F.95) † † + Ux+1, j C + C Ux− j, j + h.c. (F.89) x,1 j x− j, j1 where |1| stands for summing over terms with component indices 1 and −1. The variation of SB,W [U] then reads Again we used the fact that the time-average can be shifted to other terms by exploiting the sum over x and the cyclicity V 1 1 † δ j SB,W [U]= δ j tr Cx,1 j W , + h.c. [ ] g2 4 1 j 2 x 1 j of the trace. The variation of SB,W U then reads x,| j| a a V =− 1 1 δ a V 1 1 Ax, j δ [ ]=− δ a 2 1 j 2 1 SB,W U Ax,1 g 4 a a 2 1 j 2 x, j,a g , , 4 a a x a j × a † − † a † † P Ux, j U x+ j,1Cx, j1 Cx−1, j1U x−1,1 × P Ux,1 Ux+1, j W , + W − , Ux− j, j x 1 j x j j1 |1| † † + U + , C + C U − , . (F.90) + † − † . x 1 j x,1 j x− j, j1 x j j Ux+ j,1Wx, j1 Wx−1, j1Ux−1,1 (F.96)

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