# Dynamical stabilisation of complex Langevin simulations of QCD

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## Abstract

The ability to describe strongly interacting matter at finite temperature and baryon density provides the means to determine, for instance, the equation of state of QCD at non-zero baryon chemical potential. From a theoretical point of view, direct lattice simulations are hindered by the numerical sign problem, which prevents the use of traditional methods based on importance sampling. Despite recent successes, simulations using the complex Langevin method have been shown to exhibit instabilities, which cause convergence to wrong results. We introduce and discuss the method of dynamic stabilisation (DS), a modification of the complex Langevin process aimed at solving these instabilities. We present results of DS being applied to the heavy-dense approximation of QCD, as well as QCD with staggered fermions at zero chemical potential and finite chemical potential at high temperature. Our findings show that DS can successfully deal with the aforementioned instabilities, opening the way for further progress.

## 1 Introduction

Strongly interacting matter at finite baryon number density and temperature has been, and remains, an active research subject to understand QCD under extreme conditions. Features of QCD are typically studied in thermodynamic equilibrium, where the theory has two external parameters: the temperature *T* and and baryon chemical potential \(\mu _B\). Varying those allows the exploration of the QCD phase diagram in the *T*–\(\mu _B\) plane. Known phases include ordinary nuclear matter and the quark-gluon plasma (QGP), with a colour superconducting phase expected at large \(\mu _B\). Of great appeal are also the boundaries that mark the transition between these phases. This phase diagram has a fascinating structure, which is of significance for the study of hot and/or dense systems, such as the early universe and heavy-ion collisions.

Heavy-ion collisions have been successfully used to investigate the high temperature behaviour of QCD at the relativistic heavy ion collider (RHIC) and the large hadron collider (LHC). These facilities, together with future ones, namely the Facility for Antiproton and Ion Research (FAIR) and the nuclotron-based ion collider facility (NICA), will further explore the phase diagram of QCD. They will allow the study of hadronic interactions under extreme conditions, such as higher baryonic density or very high temperatures.

From a theoretical perspective, some insight, at high temperature or density, can be gained from perturbation theory. A full picture of the phase diagram, however, requires non-perturbative methods. Recent lattice results at non-zero temperature include [1, 2]. Typically, lattice QCD simulations at finite baryon/quark density are carried out using the grand canonical ensemble, with the chemical potential introduced as conjugate variable to the appropriate number density (quark, baryon, etc). At finite quark chemical potential, the simulations have to overcome the infamous *sign problem*—a complex weight in the Euclidean path integral. This imposes severe limitations on the applicability of standard numerical methods [3, 4]. Many approaches to deal with the sign problem have been proposed, including the complex Langevin method [5, 6, 7, 8], strong coupling expansions [9, 10, 11], Lefschetz thimbles [12, 13, 14, 15, 16], holomorphic gradient flow [17], density of states [18, 19, 20, 21] and sign-optimized manifolds [22].

The complex Langevin (CL) method is an extension of the stochastic quantisation technique [23] to a complexified configuration space, without requiring a positive weight [5, 7, 8]. The complex nature of the method allows the circumvention of the sign problem, even when it is severe [24, 25, 26]. However, convergence to wrong limits has been observed both at Euclidean time [27, 28, 29, 30, 31], and real time [32, 33]. These cases of incorrect convergence can be identified a posteriori, based on the theoretical justification of the method [34, 35, 36, 37]. Further discussions on the criteria for correct convergence of complex Langevin can be found in Ref. [38, 39]. Moreover, gauge cooling (GC) [40] has improved the convergence of complex Langevin simulations for gauge theories. The effects of gauge cooling on the complex Langevin method have been studied analytically in [41]. Investigations of gauge cooling in random matrix theories has been performed in [42, 43].

Complex Langevin simulations, combined with gauge cooling, have successfully been used in QCD with a hopping expansion to all orders [44], with fully dynamical staggered fermions [45] and to map the phase diagram of QCD in the heavy-dense limit (HDQCD) [46]. In that work, we noticed that, despite the use of gauge cooling, instabilities might appear during the simulations. Here, we introduce and elaborate on our method of dynamic stabilisation (DS), which has been constructed to deal with these instabilities.

This paper is organised as follows: in Sect. 2 we review the complex Langevin method. Section 3 motivates and introduces the method of DS. Tests of this procedure, applied to QCD in the limit of heavy-dense quarks (HDQCD) [47, 48] are discussed in Sects. 4 and 5. Section 6 shows the outcome of applying dynamic stabilisation to simulations with staggered quarks at zero chemical potential and at finite chemical potential and high temperatures. We summarise our findings in Sect. 7. Appendices A and B review the HDQCD approximation and the staggered formulation of lattice quarks, which have been used in our investigations.

Preliminary results on DS have already appeared in Refs. [49, 50, 51].

## 2 Complex Langevin

*U*represents the gauge links, \(S_{\mathrm {YM}}\) is the Yang–Mills action, \(S_{\mathrm {F}}=\int d^4x \overline{\psi }M\psi \) is the fermion action with

*M*being the fermion matrix, which depends on the gauge links and the chemical potential, and \(S = S_{\mathrm {YM}} - \ln \det M\).

*S*,

Poles may appear in the drift in the presence of quarks, when \(\det M = 0\) and \(M^{-1}\) does not exist. In some situations this has a negative impact on the results [30, 53, 54], but, as far as understood, this is not the case in HDQCD [44, 55]. For further reference, we refer the reader to the extended discussion on the issues arising from the branch cuts of the logarithm of the determinant [53, 54, 55, 56]. In [57] it was clarified that it is the drift’s behaviour around the poles, rather than the branch cuts, that affects the reliability of the complex Langevin method. It is also necessary to employ adaptive algorithms to change the Langevin step size \(\varepsilon \), in order to avoid runaways due to large values of the drift [58].

*d*with respect to a gauge transformation is negative semi-definite. The coefficient \(\alpha \) can be changed adaptively to optimise the cooling procedure [59]. A variable number of gauge cooling steps, depending on the rate of change of the unitarity norm, can be applied between successive Langevin steps [60].

*V*is the spatial volume. The average Polyakov loop is an order parameter for Yang-Mills theories, as it is related to the free energy of a single quark by \(\langle P \rangle \sim e^{-F_q / T}\). In the presence of dynamical quarks, it is no longer an order parameter. However, it still provides information on whether quarks are free or confined within hadrons. Another useful observable is the average phase of the quark determinant, measured in a phase quenched ensemble,

## 3 Dynamic stabilisation

Figure 1 shows the Langevin time evolution of the Polyakov loop and of the unitarity norm. This situation has a very mild sign problem, with average phase \(\langle e^{2i\phi } \rangle = 0.9978(2) - 0.0003(57)i\), and thus results from reweighting are reliable. We observe two distinct regions: one is characterised by a sufficiently small unitarity norm and agreement between gauge cooling and reweighting results. At a larger Langevin time, i.e. \(\theta \gtrsim 50\), the agreement disappears as the unitarity norm becomes too large. It has been concluded in Ref. [46] that a large unitarity norm is an indicator of these instabilities, with 0.03 being a conservative threshold, after which results become unreliable.

*d*and to be directed towards the SU(3) manifold. One possible implementation is given by the substitution:

We point out that \(M^a_x\) is not invariant under general SL(\(3, \mathbb {C}\)) gauge transformations, but it is with respect to SU(3) transformations. Moreover, it is not holomorphic, since it is constructed to be a function of only the non-unitary part of the gauge links, i.e., of the combination \(UU^\dagger \). This is necessary to make \(M^a_x\) scale with the unitarity norm, such that explorations of the non-unitary directions can be controlled. Therefore, it cannot be obtained from a derivative of the action. This invalidates the standard justification for the validity of complex Langevin [34, 35], which require a holomorphic Langevin drift. Nevertheless, numerical evidence of the convergence to the correct limit of CL simulations with dynamic stabilisation will be shown in Sects. 5 and 6.

^{1}and DS is shown in Fig. 2, where a comparison of results from DS, gauge cooling

^{2}and reweighting is shown. We have used the same parameters of Fig. 1 and found agreement with reweighting for the entire length of the simulation. Figure 2 also demonstrates that it is possible to stabilise complex Langevin simulations in a way that gauge cooling alone is not able to, allowing for longer simulation times and thus smaller statistical errors.

## 4 Dependence of observables on \(\alpha _{\mathrm {DS}}\)

The complexity of gauge theories makes it difficult to predict the effect of the control parameter \(\alpha _{\mathrm {DS}}\) on the Langevin dynamics. However, two limiting cases can be expected: for small \(\alpha _{\mathrm {DS}}\) the DS drift becomes very small, essentially not affecting the dynamics. For large values of \(\alpha _{\mathrm {DS}}\), the DS force heavily suppresses excursions into the non-unitary directions of SL(\(3,\mathbb {C}\)), which can be interpreted as a gradual reunitarisation of the gauge links. We illustrate the effect of different \(\alpha _{\mathrm {DS}}\) on complex Langevin simulations of HDQCD, see appendix A, in two cases. The first scenario corresponds to an average phase of the quark determinant close to unity, i.e., when the sign problem is mild and comparisons with reweighting are possible. In the second case, the average phase is very small, indicating a severe sign problem. Both scenarios have been simulated with inverse coupling \(\beta =5.8\) and hopping parameter \(\kappa =0.04\). Additionally, one gauge cooling step has been applied between consecutive Langevin updates.

*O*(0.1). We use the region before the unitarity norm rises as a reference point to test dynamic stabilisation. Due to this small sampling region, the statistical uncertainties of the gauge cooling simulations are comparatively large. The results are compatible for a wide region of \(\alpha _{\mathrm {DS}}\), as shown in Fig. 5

^{3}, where the results from gauge cooling are indicated by green bands. We find disagreement when \(\alpha _{\mathrm {DS}}\) is outside a certain window. This can be understood as follows: for \(\alpha _{\mathrm {DS}}\) very small the DS drift is too small to be effective; on the other hand, large values of the control parameter cause a heavy suppression of the exploration of the non-unitary directions.

^{4}. In other words, the region of least sensitivity to \(\alpha _{\mathrm {DS}}\) seems to provide the best estimate. However, a cross-check with another method is necessary to verify that these values provide the correct result.

For \(\alpha _{\mathrm {DS}}=10^{0}\) we observe larger values of the product \(\alpha _{\mathrm {DS}} \varepsilon M^a_x\), due to the larger unitarity norm. We remind the reader that \(M^a_x\) is a function of the combination \(UU^\dagger \) (see Eq. 14), similar to the unitarity norm. As \(\alpha _{\mathrm {DS}}\) increases, the unitarity norm decreases and then plateaus (seen in Fig. 6), and so does \(M^a_x\). Intuitively, for very large \(\alpha _{\mathrm {DS}}\) the DS drift overshadows the Langevin drift coming from the physical action.

Figures 7 and 8 show histograms with compact distributions, with no skirts or “tails”. This indicates an absence of boundary terms, which would spoil the proof of convergence [63]. A more in-depth analysis, however, is necessary to verify whether the lack of exponential fall-off in this situation breaks the proof of convergence.

## 5 Continuum behaviour of dynamic stabilisation

The deconfinement transition of the heavy dense approximation of QCD was studied in Refs. [40, 59]. The gauge coupling was varied in the interval \(5.4 \le \beta \le 6.2\). These simulations have a lattice of volume \(6^3 \times 6\), chemical potential of \(\mu =0.85\) and hopping parameter of \(\kappa =0.12\). It was found that the average plaquette from complex Langevin with just gauge cooling disagrees with reweighting for \(\beta \lesssim 5.5\). We have investigated whether DS can remedy this discrepancy. As in our previous studies, we added one step of gauge cooling between consecutive Langevin updates.

\(\beta \) | RW | DS | GC |
---|---|---|---|

5.4 | \(0.47164\, (33)\) | \(0.472007\, (86)\) | \(0.504292\, (75)\) |

5.5 | \(0.49687\, (38)\) | \(0.49708\, (11)\) | \(0.516607 \,(56)\) |

5.6 | \(0.52461\, (47)\) | \(0.52441\, (12)\) | \(0.530817\, (72)\) |

5.7 | \(0.55086\, (63)\) | \(0.55064\, (19)\) | \(0.547050 \,(97)\) |

5.8 | \(0.57097\, (58)\) | \(0.570849\, (69)\) | \(0.56547\, (22)\) |

5.9 | \(0.58417\, (47)\) | \(0.584086\, (37)\) | \(0.58220 \,(16)\) |

6.0 | \(0.59533 \,(42)\) | \(0.595220 \,(28)\) | \(0.594490\, (67)\) |

6.1 | \(0.60533\, (38)\) | \(0.605332\, (24)\) | \(0.604713\, (50)\) |

6.2 | \(0.61460 \,(36)\) | \(0.614567\, (22)\) | \(0.614275\, (33)\) |

## 6 Staggered quarks

### 6.1 Staggered quarks at \(\mu = 0\)

In order to evaluate the fermionic contribution to the Langevin drift of Eq. (4) we employ a bilinear noise scheme and the conjugate gradient method to calculate the trace and inverse, respectively. The fermion matrix we use is that of staggered quarks, reviewed in Appendix B. One characteristic of the bilinear noise scheme is that, at \(\mu =0\), the drift is real only on average [45]. Therefore, a non-zero unitarity norm is expected even for vanishing chemical potential. This can cause simulations to diverge, and has been addressed in Ref. [31] where the necessary elements of the inverse were calculated exactly. It has been pointed out that solutions in the confined phase are wrong, showing that stability does not imply correctness.

We have investigated whether DS is able to successfully keep the unitarity norm under control, by comparing complex Langevin and hybrid Monte-Carlo (HMC) simulations^{5}. We have used four different lattice volumes, \(6^4\), \(8^4\), \(10^4\) and \(12^4\).

First, we have identified a suitable value for the control parameter \(\alpha _{\mathrm {DS}}\) following the procedure in Sect. 4, i.e. using Eq. (18). After finding the optimal values for \(\alpha _{\mathrm {DS}}\) for each lattice size, we extrapolated the results to zero Langevin step size. We have performed studies with four degenerate quark flavours of mass \(m = 0.025\) and inverse coupling \(\beta = 5.6\). We have analysed the average values of the plaquette and (unrenormalised) chiral condensate. For these parameters, the chiral symmetry is broken, as indicated by \(\langle \overline{\psi } \psi \rangle \ne 0\).

Average values for the plaquette and chiral condensate from simulations of four flavours of naïve staggered fermions at \(\beta =5.6\), \(m=0.025\) and \(\mu =0\). The Langevin results have been obtained after extrapolation to zero step size

Volume | Plaquette | \(\overline{\psi } \psi \) | ||
---|---|---|---|---|

HMC | Langevin | HMC | Langevin | |

\(6^4\) | \(0.58246\, (8)\) | \(0.582452\,(4)\) | \(0.1203\, (3)\) | \(0.1204\, (2)\) |

\(8^4\) | \(0.58219 \,(4)\) | \(0.582196\, (1)\) | \(0.1316\, (3)\) | \(0.1319\, (2)\) |

\(10^4\) | \(0.58200\, (5)\) | \(0.58201\, (4)\) | \(0.1372\, (3)\) | \(0.1370\, (6)\) |

\(12^4\) | \(0.58196\, (6)\) | \(0.58195\, (2)\) | \(0.1414\, (4)\) | \(0.1409\, (3)\) |

Recent works on complex Langevin and gauge cooling applied to staggered fermions include [66, 67]. There, a discrepancy between the CLE and exact results is reported for \(V=12^4\) at the same inverse coupling and quark mass used here, but with two flavours of staggered fermions. This tension could be removed by using dynamic stabilization and careful extrapolation to zero step size.

### 6.2 Staggered quarks at \(\mu \ne 0\)

## 7 Summary and outlook

Dynamic stabilisation was introduced to deal with instabilities found in complex Langevin simulations, especially when the inverse coupling is small (typically \(\beta \lesssim 5.5\)) or the unitarity norm rises steadily. The method is based on adding a non-holomorphic drift to the complex Langevin dynamics to keep simulations in the vicinity of the SU(3) manifold. We have studied the dependence of the observables on the control parameter \(\alpha _{\mathrm {DS}}\) and have presented a criterion to tune it appropriately. We also found numerical evidence that the DS drift decreases when the lattice spacing is reduced and has a localised distribution. DS improved results on the deconfinement transition for HDQCD, previously shown in [40], where a discrepancy between reweighting and complex Langevin was observed. We find good agreement with reweighting for all gauge couplings in both confined and deconfined phases.

We presented a study of complex Langevin simulations of QCD with naïve staggered fermions at vanishing chemical potential. After extrapolating the Langevin results to zero step size, we found excellent agreement between complex Langevin and hybrid Monte-Carlo simulations for the plaquette and chiral condensate for four different lattice volumes, despite DS adding a non-holomorphic drift. Our findings rectify the discrepancy found in earlier studies in [66, 67]. For \(\mu > 0\), we were able to observe changes in the chiral condensate as the chemical potential increases at high temperatures. In those cases, DS kept the unitarity norm under control and allowed for long simulations. However, the extent of these studies were limited, as simulations at lower temperatures showed a numerical difficulty arising from the inversion of the fermion matrix.

More analytical work on the justification of DS is desirable, as DS formally violates the proof of convergence of CL. Nevertheless, numerical evidence clearly shows no difference between HMC and CL, even at a sub-permille level. This needs to be confirmed at non-zero \(\mu \), by comparing with other approaches, such as those mentioned in Sect. 1.

## Footnotes

- 1.
We have checked that multiple gauge cooling steps lead to a negligible improvement. At least one gauge cooling step is required, since DS does not affect the real part of the drift, which can develop large fluctuations [59].

- 2.
We have verified that HDQCD simulations, using gauge cooling, with different initial conditions exhibit the same qualitative behaviour.

- 3.
- 4.
We thank Gert Aarts for suggesting this.

- 5.
We thank Philippe de Forcrand for providing the results from hybrid Monte-Carlo simulations.

## Notes

### Acknowledgements

We would like to thank Gert Aarts, Dénes Sexty, Erhard Seiler and Ion-Olimpiu Stamatescu for invaluable discussions and collaboration. We are indebted to Philippe de Forcrand for providing us with the HMC results for staggered fermions. We are grateful for the computing resources made available by HPC Wales. This work was facilitated though the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington. The work of FA was supported by US DOE Grant no. DE-FG02-97ER-41014.

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