# Implicit schemes for real-time lattice gauge theory

## Abstract

We develop new gauge-covariant implicit numerical schemes for classical real-time lattice gauge theory. A new semi-implicit scheme is used to cure a numerical instability encountered in three-dimensional classical Yang-Mills simulations of heavy-ion collisions by allowing for wave propagation along one lattice direction free of numerical dispersion. We show that the scheme is gauge covariant and that the Gauss constraint is conserved even for large time steps.

## 1 Introduction

Color Glass Condensate (CGC) effective theory [1] applies classical Yang-Mills theory to the area of high energy heavy-ion collisions. In the CGC description high energy nuclei can be treated as thin sheets of fast moving color charge which generate a classical gluon field. The collision of two such sheets produces the Glasma [2], which behaves classically at the earliest stages of the collision. Due to classical Yang-Mills theory being non-linear and the non-perturbative nature of the CGC, computer simulations are commonly used to investigate the time evolution of such systems [3, 4, 5, 6, 7]. Here, real-time lattice gauge theory provides a numerical treatment of classical Yang-Mills theory while retaining exact gauge invariance on the lattice. To name a few applications besides CGC and Glasma simulations, real-time lattice gauge theory is widely used for non-Abelian plasma simulations [8, 9] and hard thermal loop (HTL) simulations [10, 11, 12], classical statistical simulations of fermion production [13, 14], in studying sphalerons (in electroweak theory) [15, 16], for determining the plasmon mass scale in non-Abelian gauge theory [17, 18] or when studying perturbations on top of a non-Abelian background field [19].

The real-time lattice gauge theory approach is based on the discretization of Yang-Mills fields on a lattice in terms of so-called gauge links, i.e. Wilson lines connecting adjacent lattice sites. Using gauge link variables one can discretize the continuum Yang-Mills action in various ways, the simplest of which is the Wilson gauge action [20]. Varying this action with respect to the link variables one obtains discretized classical field equations, which are of the explicit leapfrog type. Moreover, in addition to the equations of motion, one also 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 systematically improved by adding higher order terms to the standard Wilson gauge action [21].

In previous publications [22, 23] we developed lattice Yang-Mills simulations for genuinely three-dimensional heavy-ion collisions in the CGC framework. Unlike the usual boost-invariant approach, we consider collisions of nuclei with thin, but non-vanishing support along the longitudinal direction (the beam axis) and simulate them in the laboratory frame. This has enabled us to study the effects of finite nuclear longitudinal extent (which is inversely proportional to the Lorentz gamma factor \(\gamma \)) on the rapidity profile of the produced Glasma after the collision. The numerical scheme in these simulations is based on the standard Wilson gauge action with the fields coupled to external color currents. The treatment of these color charges is closely related to the colored particle-in-cell (CPIC) method [24, 25], which is a non-Abelian extension of the particle-in-cell (PIC) method [26] commonly used in (Abelian) plasma simulations.

Unfortunately, these simulations suffer from a numerical instability that leads to an artificial increase of total energy if the lattice resolution is too coarse: even a single nucleus propagating along the beam axis, which should remain static and stable, eventually becomes unstable. Improving the resolution simply postpones the problem at the cost of much higher computational resources. We realize that this instability is due to numerical dispersion on the lattice inherent to the leapfrog scheme, which renders the dispersion relation of plane waves non-linear. As a consequence of numerical dispersion, high frequency plane waves exhibit a phase velocity that is noticeably less than the speed of light on the lattice. The shape of the pulse of color fields is lost over time. At the same time, the color current “driving” the nucleus forward will not disperse by construction: the point-like color charges making up the current are simply moved from one cell to the next as the simulation progresses. Thus the shape of the current is always kept intact. This mismatch and the resulting instability is therefore related to the numerical Cherenkov instability [27], which can occur in (Abelian) particle-in-cell simulations. Notably, simulations of laser wakefield acceleration [28], where electric charges moving at relativistic speeds are coupled to discretized electromagnetic fields, suffer from the same type of instability and many numerical schemes have been devised to cure it [29, 30, 31, 32]. A particularly simple solution to the problem is the use of semi-implicit schemes to repair the dispersion relation (i.e. making it linear) for one direction of propagation [33], which is the approach we take in this work.

In this paper we derive an implicit and a semi-implicit scheme for real-time lattice gauge theory by modifying the standard Wilson gauge action. We obtain two new actions that are gauge invariant (in the lattice sense), are of the same order of accuracy as the original action, but yield an implicit or semi-implicit scheme upon variation. In the case of the semi-implicit scheme setting the lattice spacing and the time step to specific values can fix the dispersion relation along the longitudinal direction and thus suppress the numerical Cherenkov instability. We also obtain a modified version of the Gauss constraint that is conserved up to (in principle) arbitrary numerical precision under the discrete equations of motion.

We start with a discussion of the main ideas behind the semi-implicit scheme for the two-dimensional wave equation in Sect. 2 and for Abelian gauge fields on the lattice in Sect. 3. The concepts are then generalized to non-Abelian lattice gauge theory in Sect. 4, where we derive both a fully implicit and the semi-implicit scheme. Finally, we verify numerically that the Cherenkov instability can be suppressed using the new scheme and that the Gauss constraint is conserved in Sect. 5.

## 2 A toy model: the 2D wave equation

The basic ideas behind the numerical scheme we are after can be most easily explained using a simple toy model, namely the two-dimensional wave equation. We start by giving a few definitions and then derive three different numerical schemes by discretizing the action of the system in different ways and using a discrete variational principle. The schemes obtained through this procedure are known as variational integrators, which exhibit useful numerical properties such as conserving symplectic structure and retaining symmetries of the discrete action [34, 35]. We will see how the exact discretization of the action affects the properties of the numerical scheme and in particular how numerical dispersion can be eliminated.

*x*and derivatives are replaced with finite difference expressions. We define the forward difference

*a*. Equipped with these definitions we could directly discretize the EOM (3), but this is not the approach we will take. The strategy behind variational integrators is to first discretize the action (1) and then demand that the discrete variation vanishes.

### 2.1 Leapfrog scheme

^{1}which is accurate up to second order in the time step \(a^0\) and spatial lattice spacings \(a^i\). Using the plane-wave ansatz (4) we find the dispersion relation

*k*if the Courant-Friedrichs-Lewy (CFL) condition holds

### 2.2 Implicit scheme

We quickly summarize: the first action we considered given by Eq. (9) gave us the explicit leapfrog scheme, which is rendered non-dispersive but unstable using the “magic time-step”. The second action, Eq. (19), which we obtained by replacing one of the spatial finite differences with a temporally averaged expression, yields an implicit scheme. This scheme is unconditionally stable, but always dispersive. This suggests that a mixture of both discretizations might solve our problem.

### 2.3 Semi-implicit scheme

### 2.4 Solution method and numerical tests

*k*modes, i.e. \({|}\lambda {|} < 1\) for \(k^2=\pm \pi / a^2\), we find

Finally, we perform a crucial numerical test: we compare the propagation of a Gaussian pulse using the three different schemes to show the effects of numerical dispersion and in particular that the semi-implicit scheme is dispersion-free. For simulations using the implicit or semi-implicit method we solve the equations using damped fixed point iteration. The results are shown in Fig. 1.

The main insight of this section is that the specific discretization of the action completely fixes the numerical scheme of the discrete equations of motion (and the associated stability and dispersion properties) which one obtains from a discrete variational principle. The use of temporally averaged quantities in the action leads to implicit schemes. If we treat some derivatives explicitly and some implicitly we can end up with a semi-implicit scheme that can be non-dispersive and still stable for propagation along a single direction on the lattice. As it turns out, this is just what we need to suppress the numerical Cherenkov instability we encountered in our heavy-ion collision simulations. In the next section we will see how we can use the same “trick” for Abelian gauge fields on the lattice.

## 3 Abelian gauge fields on the lattice

Before tackling the problem of non-Abelian gauge fields on the lattice, it is instructive to see how we can derive a dispersion-free semi-implicit scheme for discretized Abelian gauge fields. We will approach the problem as before: starting with a discretization of the action, we apply a discrete variational principle to derive discrete equations of motion and constraints. Then we will see what modifications to the action are required to obtain implicit and semi-implicit numerical schemes. Since we are dealing with gauge theory we will take care to retain gauge invariance also for the discretized system.

### 3.1 Leapfrog scheme

*x*. The field strength tensor \(F_{x,\mu \nu }\) at

*x*is defined using forward differences

*x*. A straightforward discretization of the gauge field action is given by

*S*[

*A*] is invariant for any \(\alpha \) it must hold that

*S*[

*A*]. Consequently, it does not matter what kind of discretization of the action we use. As long as

*S*[

*A*] retains lattice gauge invariance in the sense of Eq. (41), we are guaranteed to find that the discrete Gauss constraint is conserved under the discrete equations of motion.

### 3.2 Implicit scheme

### 3.3 Semi-implicit scheme

*i*,

*j*denote transverse components. Since we have built the new action from gauge invariant expressions it is also invariant under lattice gauge transformations. Note that the use of \({\tilde{A}}_{x,i}\) in \(W_{x,1i}\) introduces new terms in the action (83) dependent on the temporal component of the gauge field. Although these terms disappear in temporal gauge (our preferred choice), they still have an effect on the scheme since we have to perform the variation before choosing a gauge. Therefore we will obtain a modified Gauss constraint compatible with the equations of motion derived from the action (83) even after setting \(A_{x,0} = 0\).

*M*and

*W*with

*F*the EOM reduce to the leapfrog equations as expected.

The propagation of waves in the semi-implicit scheme turns out to be more complicated compared to the leapfrog or implicit scheme: given a wave vector *k* and a field amplitude \(A_{i}\) (such that the Gauss constraint (85) is satisfied) the dispersion relation becomes polarization dependent, i.e. the scheme exhibits birefringence.

The main result of this section is the action (83) which gives rise to the semi-implicit scheme. Here we used a combination of differently averaged field strengths, \(M_{x,ij}\) and \(W_{x,1i}\), in the action. Our next goal is to generalize these expressions to non-Abelian gauge fields.

## 4 Non-Abelian gauge fields on the lattice

*g*is the Yang-Mills coupling constant and \(A_\mu (x) = \sum _a A^a_\mu t^a\) is a non-Abelian gauge field, where \(t^a\) are the generators of the gauge group. In the following we use the normalization \({{\mathrm{tr}}}\left( t^a t^b \right) = \frac{1}{2} \delta ^{ab}\). Through variation of the action we obtain the Gauss constraint and the equations of motion:

*x*and ending at \(x+\mu \). This is also reflected in the gauge transformations

*x*. Gauge links with negative directions are identified as

*x*:

### 4.1 Leapfrog scheme

Following the same procedure as in the case of Abelian gauge fields, we vary w.r.t. spatial components \(U_{x,i}\) to obtain the discrete EOM and w.r.t. temporal components \(U_{x,0}\) to find the Gauss constraint.

### 4.2 Implicit scheme

*x*. This leads us to the definition of the “properly” averaged field strength

*x*and

*y*along some arbitrary path. A time-averaged version of \({\mathscr {X}}_{x,y}\) is given by

*x*to \(x+i+j\) is now a preferred direction, which can be seen in Fig. 3. The loss of symmetry can be mitigated by also including terms like \(M_{x,i-j}\) in the action. Therefore we propose the action

*j*to keep the action as symmetric as possible. The action is also invariant under time reversal, real-valued (see Appendix E for a proof) and gauge invariant. While we have not made any changes to the terms involving temporal plaquettes the spatial plaquette terms now include temporal links and therefore we will also obtain a modified Gauss constraint like in the semi-implicit scheme for Abelian fields.

*i*. The left hand side (LHS) of (141) is the same as in the leapfrog scheme Eq. (119), but now there is also a new term on the right hand side (RHS) from varying the spatial part of the action. Performing the continuum limit for the Gauss constraint (after multiplying both sides with \(a^0\)), the RHS term vanishes as \({\mathscr {O}} \left( \left( a^0 \right) ^2 \right) \). This shows that the RHS is not a physical contribution, but rather an artifact of the implicit scheme. Note that the correct continuum limit of the constraint (and the EOM) is already guaranteed by the action.

*j*(instead of just positive indices) and an additional factor of 1 / 2 to avoid overcounting. As a simple check one can replace \(M_{x,ij}\) with \(C_{x,ij}\) in Eq. (142) (only introducing an irrelevant error term quadratic in \(a^0\)) and recover Eq. (121).

Compared to the leapfrog scheme, solving Eq. (142) is more complicated: it is not possible to explicitly solve for the temporal plaquette \(U_{x,i0}\) anymore because \(M_{x,ij}\) on the RHS involves contributions from both “past” and “future” link variables. Completely analogous to the case of Abelian gauge fields on a lattice, we obtain an implicit scheme by introducing time-averaged field strength terms in the action.

This iteration scheme can be used to solve the EOM (142) until the Gauss constraint (141) is satisfied up to the desired numerical accuracy. Conversely, this means that unlike the leapfrog scheme, where the Gauss constraint (119) is always satisfied up to machine precision in a single evolution step, the implicit scheme, solved via an iterative scheme, only approximately conserves the Gauss constraint (141). However, in Sect. 5 we will show that using a high number of 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 simulations at the cost of small violations of the constraint.

It is also immediately obvious that solving the implicit scheme requires higher computational effort compared to the leapfrog scheme. Considering that one has to use the leapfrog scheme for a single evolution step once (as an initial guess) and then use fixed point iteration, where every step is at least as computationally demanding as single leapfrog step, it becomes clear that the use of an implicit scheme is only viable if increased stability allows one to use coarser lattices while maintaining accurate results.

### 4.3 Semi-implicit scheme

*i*and

*j*only run over transverse coordinates and \(x^1\) is the longitudinal coordinate.

The purely transverse part of the action uses the same terms as the implicit scheme, see Eq. (140). The longitudinal-transverse part is now given in terms of \(C_{x,1j}\) and \(W_{x,1j}\) analogous to Eq. (69). We have to explicitly include the hermitian conjugate in order to keep the action real-valued.

*C*can be exchanged for corresponding expressions with

*M*or

*W*and

*U*can be exchanged for its temporally averaged version \({\overline{U}}\). The longitudinal component of the EOM then reads

- 1.Compute the next iteration using damped fixed point iteration: in Eqs. (156) and (157) replace \(P^a \left( U_{x,10}\right) \rightarrow {\mathscr {U}}^a_1\) and \(P^a \left( U_{x,i0} \right) \rightarrow {\mathscr {U}}^a_i\), solve for the unknown \({\mathscr {U}}\)’s and update the temporal plaquettes usingand analogously for \(U^{(n)}_{x,i0}\) and \({\mathscr {U}}^{a}_i\). \(\alpha \) is the damping coefficient.$$\begin{aligned} P^a \left( U^{(n)}_{x,10} \right) = \alpha P^a \left( U^{(n-1)}_{x,10} \right) + \left( 1 - \alpha \right) {\mathscr {U}}^{a}_1 \end{aligned}$$(161)
- 2.For SU(2) we can reconstruct the full temporal plaquette from its components \(P^a \left( U \right) \) with the identityfor \(U = U^{(n)}_{x,10}\) and \(U = U^{(n)}_{x,i0}\).$$\begin{aligned} U = \sqrt{1- \frac{1}{4} \sum _a P^a \left( U \right) ^2} \mathbb {1}+ \frac{i}{2} \sum _a \sigma ^a P^a \left( U \right) , \end{aligned}$$(162)
- 3.Using \(U^{(n)}_{x,10}\) and \(U^{(n)}_{x,i0}\), compute the spatial links \(U^{(n)}_{x+0,1}\) and \(U^{(n)}_{x+0,i}\) via$$\begin{aligned} U^{(n)}_{x+0,i} = U^{(n)}_{x,0i} U_{x,i}. \end{aligned}$$(163)
- 4.
Repeat with \(n\rightarrow n+1\).

^{2}

### 4.4 Coupling to external color currents

## 5 Numerical tests

In this last section we test the semi-implicit scheme on the propagation of a single nucleus in the CGC framework. For an observer at rest in the laboratory frame, the nucleus moves at the speed of light and consequently exhibits large time dilation. As the nucleus propagates, the interactions inside appear to be frozen and the field configuration is essentially static. On the lattice we would like to reproduce this behavior as well, but depending on the lattice resolution we run into the numerical Cherenkov instability, which leads to an artificial increase of the total field energy of the system.

As previously stated the root cause of the instability is numerical dispersion: in the CGC framework a nucleus consists of both propagating field modes and a longitudinal current generating the field around it. It is essentially a non-Abelian generalization of the field of a highly relativistic electric charge. In our simulation the color current is modeled as an ensemble of colored point-like particles moving at the speed of light along the beam axis. The current is unaffected by dispersion, i.e. it retains its shape perfectly as it propagates. The field modes suffer from numerical dispersion, which over time leads to a deformation of the original longitudinal profile. The mismatch between the color current and the field leads to creation of spurious field modes, which interact with the color current non-linearly through parallel transport (color rotation) of the current. This increases the mismatch further and more spurious fields are created. As the simulation progresses this eventually leads to a large artificial increase of total field energy. The effects of the instability can be quite dramatic as seen in Fig. 4. The main difference to the numerical Cherenkov instability in Abelian PIC simulations is that in electromagnetic simulations the spurious field modes interact with the particles through the Lorentz force [27]. In our simulations we do not consider any acceleration of the particles (i.e. their trajectories are fixed), but interaction is still possible due to non-Abelian charge conservation (171), which requires rotating the color charge of the color current. Therefore our type of numerical Cherenkov instability is due to non-Abelian effects.

*t*. In the continuum we would have \(E(t)=E(0)\), but due to numerical artifacts and the Cherenkov instability this is not the case in our simulations.

*t*. The numerator depends on the Gauss constraint of the scheme and has to be adjusted according to the implicit and semi-implicit method (either Eq. (141) or (154) including the charge density on the RHS as discussed in Sect. 4.4). In Fig. 6 we show how the Gauss constraint violation converges systematically towards zero as we increase the number of iterations. Therefore, even though we can not use an arbitrarily high number of iterations due to limited computational resources, the semi-implicit scheme conserves the Gauss constraint in principle. The same holds for the purely implicit scheme. In practice it is not necessary to satisfy the constraint up to high precision, as observables such as the energy density seem to converge much faster up to satisfying accuracy.

## 6 Conclusions and outlook

In this paper we derived new numerical schemes for real-time lattice gauge theory. We started our discussion based on two simpler models, namely the two-dimensional wave equation and Abelian gauge fields on the lattice. It turns out that using a discrete variational principle to derive numerical schemes for equations of motion is a very powerful tool: the use of time-averaged expressions in the discrete action yields implicit and semi-implicit schemes depending on how exactly the time-averaging is performed and what terms are replaced by their averages. We extended this concept to real-time lattice gauge theory, allowing us to make modifications to the standard Wilson gauge action that yield new numerical schemes, which have the same accuracy as the leapfrog scheme and, most importantly, are gauge-covariant and conserve the Gauss constraint. Finally, we demonstrated a peculiar property of the semi-implicit scheme: it allows for dispersion-free propagation along one direction on the lattice, thus curing a numerical instability that has plagued our simulations of three-dimensional heavy-ion collisions.

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 of nucleus-nucleus collisions in multiple ways. Primarily, using the new scheme we can be sure that the color fields of incoming nuclei have not been altered up until the collision event and all changes to the fields afterwards are solely due to interaction between the colliding nuclei during the 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 separated, all field modes (i.e. “gluons”) with momenta at high rapidity must have been created in the collision and contributions from artificial modes emitted by the nucleus due 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 momentum \(k_L\) (and small transverse momentum \(k_T\)) exhibit a phase velocity approximately the speed of light (up to an error term quadratic in \(a^0 k_T\)). This means any interactions between high rapidity gluons produced in the collision and the finite-thickness color fields of the nuclei directly after the collision event can be considered physical and are not tainted by numerical dispersion as in our previous simulations. Consequently, it should be possible to extract space-time rapidity profiles of the local rest frame energy density of the Glasma as in [23] valid for larger ranges of rapidity as previously considered.

In conclusion, we hope that this new treatment of solving the Yang-Mills equations on the lattice allows us to perform better, more accurate simulations using more complex models of nuclei.

## Footnotes

- 1.The connection to the leapfrog scheme becomes more apparent if we introduce an approximation of the conjugate momentumwhich is defined naturally between time slices \(x^0\) and \(x^0+a^0\) (hence the index “\(x+\frac{0}{2}\)” in our notation). The EOM can then be written as$$\begin{aligned} \pi _{x+\frac{0}{2}} \equiv \partial ^F_0 \phi _x, \end{aligned}$$(14)$$\begin{aligned} \pi _{x+\frac{0}{2}} = \sum _i \frac{a^0}{\left( a^i \right) ^2} \left( \phi _{x+i} + \phi _{x-i} - 2\phi _x \right) + \pi _{x-\frac{0}{2}}, \end{aligned}$$(17)$$\begin{aligned} \phi _{x+0} = \phi _x + a^0 \pi _{x+\frac{0}{2}}. \end{aligned}$$(18)
- 2.
This is also true for the Abelian semi-implicit scheme. If the equations of motion are solved only approximately using an iterative method, then the conservation of the Gauss constraint is also only approximate depending on the degree of convergence.

## Notes

### Acknowledgements

The authors thank A. Dragomir, C. Ecker, T. Lappi, J. Peuron and A. Polaczek for helpful discussions and comments. This work has been supported by the Austrian Science Fund FWF, Project No. P26582-N27 and Doctoral program No. W1252-N27. The computational results have been achieved using the Vienna Scientific Cluster.

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