Exploring Complex-Langevin Methods for Finite-Density QCD
aa r X i v : . [ h e p - l a t ] O c t Exploring Complex-Langevin Methods forFinite-Density QCD
D. K. Sinclair ∗ † HEP Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439,USAE-mail: [email protected]
J. B. Kogut
Department of Energy, Division of High Energy Physics, Washington, DC 20585, USAandDepartment of Physics – TQHN, University of Maryland, 82 Regents Drive, College Park, MD20742, USAE-mail: [email protected]
QCD at non-zero chemical potential ( m ) for quark number has a complex fermion determinantand thus standard simulation methods for lattice QCD cannot be applied. We therefore simulatethis theory using the Complex-Langevin algorithm with Gauge Cooling in addition to adaptivemethods, to prevent runaway behaviour. Simulations are performed at zero temperature on a12 lattice with 2 quarks which are light enough that m N / m p / m < m N /
3, beyond which it increases, eventually reaching its saturation value of 3 for m suffi-ciently large. The chiral condensate decreases as m is increased approaching zero at saturation,while the plaquette increases towards its quenched value. We have yet to observe the transitionto nuclear matter at m ≈ m N /
3, presumably because the runs for m between m N / The 33rd International Symposium on Lattice Field Theory14 -18 July 2015Kobe International Conference Center, Kobe, Japan ∗ Speaker. † This research was supported in part by US Department of Energy contract DE-AC02-06CH11357 c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). http://pos.sissa.it/ xploring Complex-Langevin Methods for Finite-Density QCD
D. K. Sinclair
1. Introduction
QCD at a finite chemical potential m for quark number has a complex action which prevents thedirect application of simulation methods based on importance sampling. The Langevin equation isa stochastic differential equation for the evolution of the classical fields in a fictitious time, whichdoes not rely on importance sampling. It is, in fact, a special case of the hybrid molecular-dynamicsalgorithm, where each trajectory consists of a single update.The Langevin equation can be extended to complex actions by complexifying the fields [1, 2,3, 4]. In the case of QCD this means promoting the gauge fields from SU ( ) to SL ( , C ) . Unfor-tunately, there is no proof that the long-time evolution of the fields under this Complex Langevinequation (CLE) provides a limiting value for observables. Even when this process does converge,the values it provides for observables are not guaranteed to be correct.After successfully applying the CLE to spin models (see for example [5]), people were en-couraged to apply it to lattice QCD at finite m . Early attempts at applying the CLE to QCD werestymied by runaway behaviour, which was not corrected by adaptive methods. Recently it has beennoted that at least some of this undesirable behaviour is due to the production of unbounded gaugetransformations of compact gauge fields. Such behaviour can be controlled by gauge transformingto a gauge which minimizes the magnitudes of the gauge fields and hence their distance from the SU ( ) manifold[6]. This is called Gauge Cooling.This has revived interest in the CLE for QCD at finite m . These methods have been tested onsimple models and for quark masses large enough that hopping-parameter methods can be applied[7, 8, 9, 10, 11, 12]. In addition, studies have been made of the conditions under which the CLEconverges to the correct results [13, 14, 15, 16, 17, 18, 19]. QCD at finite m and small masses hasbeen simulated and the results compared with the heavy quark methods, for larger masses [20].Very recently this has been extended to larger lattices at finite temperatures, where the transitionfrom hadron/nuclear matter to a quark-gluon plasma is observed and results are compared withthose from reweighting methods [21].We are simulating QCD at zero temperature and finite m for light quarks using the CLE, totest directly if it converges and produces believable results. We present preliminary results of ourexplorations.
2. Complex Langevin for finite density Lattice QCD If S ( U ) is the gauge action after integrating out the quark fields, the Langevin equation for theevolution of the gauge fields U in Langevin time t is: − i (cid:18) ddt U l (cid:19) U − l = − i dd U l S ( U ) + h l (2.1)where l labels the links of the lattice, and h l = h al l a . Here l a are the Gell-Mann matrices for SU ( ) . h al ( t ) are Gaussian-distributed random numbers normalized so that: h h al ( t ) h bl ′ ( t ′ ) i = d ab d ll ′ d ( t − t ′ ) (2.2)2 xploring Complex-Langevin Methods for Finite-Density QCD D. K. Sinclair
The complex-Langevin equation has the same form except that the U s are now in SL ( , C ) . S ,now S ( U , m ) is S ( U , m ) = b (cid:229) (cid:3)(cid:3)(cid:3) (cid:26) −
16 Tr [ UUUU + (
UUUU ) − ] (cid:27) − N f { ln [ M ( U , m )] } (2.3)where M ( U , m ) is the staggered Dirac operator. Note: backward links are represented by U − not U † . Note also that we have chosen to keep the noise-vector h real. h is gauge-covariantunder SU ( ) , but not under SL ( , C ) . This means that gauge-cooling is non-trivial. Reference [18]indicates why this is not expected to change the physics. After taking − i d S ( U , m ) / d U l , the cyclicproperties of the trace are used to rearrange the fermion term so that it remains real for m = F ( U ) = V (cid:229) l Tr h U † l U l + ( U † l U l ) − − i ≥ , (2.4)where V is the space-time volume of the lattice.
3. Zero-temperature simulations m = m = SU ( ) manifold. For b = . m = .
05 on an 8 lattice we observe runaway solutions, even after Gauge Cooling!For b = . m = .
025 on a 12 lattice without gauge cooling, we observe runaway solutions.With gauge cooling, the trajectory moves slowly off the SU ( ) manifold. We perform 100,000updates with input dt = .
01. Adaptively rescaling to keep the drift(force) term under control dt is reduced to dt adaptive ≈ . ≈
108 Langevin time units, at the end ofwhich the unitarity norm ≈ . × − . This we can probably tolerate, especially since we expectit to improve with weaker couplings and larger lattices. Figure 1 shows the time evolution of theunitarity norm with and without gauge-cooling.For this run plaquette = . ( ) , whereas the RHMC algorithm gives 0 . ( ) , and h ¯ yy i = . ( ) , RHMC 0 . ( ) . This is reasonable agreement for a short run with an in-exact algorithm. m = lattice at b = . m = .
025 with m >
0. Potentially important m valuesinclude m p / ≈ .
21 and m N / ≈ .
33 (masses from HEMCGC collaboration [25, 26, 27]). Thefirst is the position of the expected transition for the phase-quenched approximation. The secondis the approximate position of the expected transition to nuclear matter. We start with a limitednumber of m values to probe the various regions of the zero-temperature phase diagram. The3 xploring Complex-Langevin Methods for Finite-Density QCD D. K. Sinclair
Figure 1:
Unitarity norms for runs on a 12 lattice.Red curve is for run without gauge cooling. Bluecurve is for run with 10-step gauge cooling. Figure 2:
Unitarity norms for run at m = . lattice with 5-step gauge-cooling. values we choose are 0 .
1, 0 .
2, 0 .
25, 0 .
35, 0 .
5, 0 . .
5. In each case we start the simulationfrom an ordered start and use 5-step gauge-cooling.The first thing we look for, is evidence that the trajectories for a given set of parameters arerestricted to a compact region of the SL ( , C ) manifold. Without this it is (almost) impossiblefor these simulations to produce meaningful results. If the simulations do converge to a limitingdistribution, one must then address the question as to whether this is the correct limit.At m = . SL ( , C ) V The evolution of this norm over the trajectory is shown in figure 2. The quarknumber density j = . ( ) . Hence the system has reached saturation where j =
3, as expectedfor large m . This is where each site is occupied with 3 quarks in a colour singlet state (nucleon).The chiral condensate h ¯ yy i = . ( . ) × − – small and consistent with zero as expected. Theplaquette P = . ( ) , consistent with the idea that, at saturation, the quarks are frozen out andthe system approximates quenched QCD. The quenched plaquette at b = . P = . ( ) . Thetotal trajectory length ≈
46 time-units. In fact, after equilibration, dt adaptive ≈ . m values we are simulating, m = . m = . m = .
25 appears to be close to equilibrating. m = . m = . m = . m = . m with the understanding that the points at m = .
35, 0 . . m = . SU ( ) manifold, then as the system equilibrates, thegauge links move away from this SU ( ) manifold. Because not all m s are equilibrated, we havenot included error-bars in these figures. 4 xploring Complex-Langevin Methods for Finite-Density QCD D. K. Sinclair
Figure 3:
Quark number density, normalized to onestaggered quark (4-flavours), as a function of m . Er-rors not known. Figure 4:
Chiral condensate, normalized to one stag-gered quark (4-flavours), as a function of m . Errorsnot known. These preliminary results (each point represents 90,000 – 500,000 sweeps/updates of the lat-tice) agree qualitatively with our expectations. The quark-number density remains close to zero for m < m N /
3. For larger m s it becomes non-zero, increasing towards its saturation value of 3 as m isincreased. The chiral condensate decreases monotonically from its m = m is increased,approaching zero at saturation. We will need to wait until each point has equilibrated to where it isclear that the gauge fields are varying over a compact region in the SL ( , C ) manifold, before wecan get truly quantitative results. This takes longer for m > m N / m increases, until close to saturation, and dt adaptive is smaller. Sinceeach run is starting from the SU ( ) manifold, we expect metastability for m > m N / m ≈ m N /
3. This will also slow down the approachto equilibrium just above the transition.Because the runs for m just above the transition have yet to equilibrate, we have been unableto observe this transition to nuclear matter.
4. Summary, discussion and outlook
We apply Complex-Langevin simulations with gauge cooling to lattice QCD at finite quark-number chemical-potential ( m ) at zero temperature. Our current simulations are on a 12 latticewith N f = b = . m = . SU ( ) gauge transformations –it is possible that further gauge fixing might improve the situation. Fixing to Landau gauge in the SU ( ) subgroup suggests itself. 5 xploring Complex-Langevin Methods for Finite-Density QCD D. K. Sinclair
We need answers to the following. Do these simulations converge and converge to the correctlimit? Do we observe a phase transition to nuclear matter at m ≈ m N /
3? Is there a spurious transi-tion at m ≈ m p /
2? Do these simulations produce the expected 2-flavour colour-superconductor atlarge m ( m > m N / m = . m at the transition to nuclear matter ( m c ) would yield thebinding energy/nucleon ( e b ) in the absence of electromagnetic interactions, since m c = ( m N − e b ) / e b < m N , this will be a formidable task. More accessible nuclear physics willbe to study the propagation of hadrons in the nuclear-matter medium. If our 2-flavour simulationsare successful, we will also simulate N f = ( + ) -flavour QCD with independentchemical potentials for the ( u , d ) and s quarks. Acknowledgements
These simulations are being performed on PCs belonging to Argonne’s HEP Division, Edisonand Carver at NERSC, and Blues at LCRC, Argonne.
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