Systematic approximation for QCD at non-zero density
Gert Aartsa, Benjamin Jägera, Erhard Seilerb, Dénes Sextyc and Ion-Olimpiu Stamatescuc
a
Phys. Dept., Swansea University, UK, b M.P.I. f. Physik, Munich, Germany,
c
Inst. Theor. Physik, Univ. Heidelberg, Germany
The simulation methods.
The HD-QCD approximation
We consider QCD with Wilson fermions at nonzero chemical potential. The QCD grand canonical
partition function for nf = 1 flavour is:
Z
(1)
Z =
DU ρ, ρ = e −S = e −SYM det M
M = 1 − 2κ
3 X
The CLE method: set up
The CLE method: convergence
The Complex Langevin Equation (CLE) has the
potential to simulate lattice models for which usual
importance sampling fails. In many cases, especially for QCD at large chemical potential the CLE
in principle provides the (only) model independent
procedure. To develop it to a reliable method is
both rewarding and tough.
The LE is a stochastic process in which the updating of the variables is achieved by addition of a drift
term (or ”force”) and a suitably normalized random
noise. For a complex action the drift is also complex
and this implies setting up the problem in the complexification of the original manifold Rn −→ C n or
SU (n) −→ SL(n, C) . The CLE then amounts to
two related, real LE with independent noise terms
- here for just one variable x → z = x + i y and
with K = −∂z S(z):
One can formally prove for the observables O(z)
Z
Z
O(x + iy)P (x, y; t)dxdy = O(x)ρ(x; t)dx.
The higher orders (Nq LO)
The HD -limit (LO)
†
Γ+iUx,iTi + Γ−iUx,iT−i
(2)
i=1
†
−2 κγ e µΓ+4Ux,4T4 + e −µΓ−4Ux,4T−4
1
Γ±µ = (1 ± γµ), Γ2±µ = Γ±µ, Γ+µΓ−µ = 0
2
We can go beyond the static limit (leading order, LO) with successive Nq LO to approximate full
QCD using the analytic hopping expansion.
For q = 1 the NLO can be defined directly in
the loop expansion by using decorated Polyakov
loops, for explicit formulae see [2]. The determinant still factorizes, but the quarks have
some mobility. Higher Nq LO become increasingly
cumbersome, however, because of the combinatorics. The effective expansion parameter is Ntκ.
with T : lattice translations, κ (hopping parameter) ∼ 1/M , γ: anisotropy parameter. The temperature is introduced as aT = Nγ (γ = 1 below).
τ
The HD-QCD [1] relies on the double limit
κ → 0, µ → ∞, ζ = κ e µ : fixed .
N
(κ e −µ) τ
τ
σ
Nτ
κ2 (κ e µ)
(3)
δx(t) = Re K(z) δt + η(t)
δy(t) = Im K(z) δt
hηi = 0 , hη 2i = 2 δt
N
(κ e µ ) τ
Then only the Polyakov loops Px survive in the
loop expansion for the fermionic deteminant which
becomes a product of local terms
2
det M 0(µ) = detc (1 + CPx)2 detc 1 + C ′Px−1
where we also introduced the non-dominant factors from the inverse Polyakov loops Px−1 to explicitly account for the det M (µ) = [det M (−µ)]∗
symmetry. We obtain ”quenched QCD at non-zero
density” describing gluonic interactions in a static
charged background.
κn
T = 1 / Nτ a τ
Alternatively we can expand to higher orders ”algebraically” (explicit formulae in a forthcoming paper). The drifts for CLE in Nq LO can be systematically derived and used for simulations with the full
YM action to any desired order q, see plots below.
The probability distribution P (x, y; t) realized in
the process evolves according to a real FPE:
HD-QCD has been used as a model in LO and NLO
to study the phase diagram and other properties
both with the full Y-M action [2], [3], and in strong
coupling also to higher orders [4].
One can also define a complex distribution ρ(x, t)
∂tP (x, y, t) = LT P (x, y, t) ,
L = (∂x + ReK(z))∂x + ImK(z)∂y
∂tρ(x, t) = LT0 ρ(x, t) ,
L0 = (∂x + K(x))∂x
with the asymptotic solution ρ(x) ≃ exp(−S(x)).
Preliminary CLE results for Nq LO
Such as for the any stochastic process the proof
of convergence depend on some conditions to be
fulfilled. For CLE these include:
- rapid decay of P (x, y, t) in y and
- holomorphy of the drift and of the observables.
There are, moreover, a number of numerical problems, due to the amplification of numerical imprecisons by unstable modes in the drift dynamics.
For gauge theories we set up a method (”gauge cooling”) to obtain a narrow y−distribution. Together
with using adaptive step size this also eliminates divergences triggered by numerical imprecisions [5].
Zeroes in the original measure (the complex distribution ρ(z)) lead to poles in the drift which occasionally lead to wrong convergence, see also [6].
In the cases of physical interest the effects due to
poles do not appear quantitatively relevant, however a systematic understanding is still missing.
For QCD at finite chemical potential the method
has been defined in [7] and applied, e.g., in [3], [8].
The reweighting (RW) method
As usual we split the Boltzmann factor in (1) and
calculate the expectation values by reweighting [2]
e −SYM det M = H w, H > 0, hOiρ = hO wiH/hwiH.
′
′
−S
+C
tr
P+C
tr
P
YM
H=e
allows a fast updating.
Preliminary RW results for the baryon density in NLO
Simulations
Convergence of the Nq LO series
On-set of the baryonic density
Outlook
As pointed out in [3] there appears to be a reliability
lower threshold for the present CLE simulations at
β ≃ 5.7. Fortunately the threshold does not appear
to depend on either the lattice size or on µ and low
enough temperatures can be attained increasing the
former. With β → ∞ we reach scaling.
RW can go to lower β but the signal/noise ratio
decreases, moreover RW cannot go beyond µ ≃ 1.
We use β = 5.8 and 5.9 and nf = 2 degenerate
flavours with κ = 0.12 for which the κ expansion
is expected to converge. We use Nt = 8, 10, 12, 16
except for the comparison with full QCD where we
use a lattice of 44. The full QCD data are obtained
by CLE, the HD-QCD data by CLE and RW [3],[8].
Although we do not aim at physical results at
this stage, we can get a rough idea of the physical parameters from the scale estimate obtained
by gradient flow for HD-QCD in LO. This indicates quark masses in the region 300-500 Mev,
that is rather heavy, and temperatures between
90 and 180 MeV for the lattices considered.
Since HD-QCD is significantly easier to simulate
we want to estimate how well we approximate full
QCD in the Nq LO of HD-QCD. The results gathered in the Figures below are obtained with CLE.
Here we use Ns = 10, Nt = 8 and 16 at β = 5.8.
Taking the scale from the first figure this means
temperatures of ≃ 180 and 90 MeV. The low µ
region can be scanned by RW in NLO. The small
Aspect ratio introduces finite spatial size effects and
a ”residual temperature”. Here we show only the
baryonic density, the Polyakov loop plots are similar. We see silver-blaze and an on-set earlier than
the hadron-quark transition as seen in LO. We show
separately the contributions from straight and decorated Polyakov loops, but notice that this is a
NLO calculation, therefore also the former are not
the same as in a LO calculation (for a comprehensive CLE calculation in LO cf. the poster/talk of
Benjamin Jäger at XQCD/LATTICE 2014).
We here present a program to extend the HD-QCD
as an expansion to higher orders, approaching in
this way QCD within a controllable approximation,
permiting comprehensive calculations with the full
YM action. This appears feasible for not too small
(bare) quark masses. Forthcoming data (presently
in production) pertain the region of low temperatures (below 120 Mev) on larger lattices, with RW
(NLO) and with CLE to higher orders.
Support from DFG is thankfully acknowledged.
HD kappa=0.12 w0 2444
pure w0 24
pure w0 biglat 3244
kappa=0.45 w0 24
Sommer hep-lat/0108008
0.24
0.22
0.2
6.35
fullqcd
κ expansion
6.25
chiral condensate
chiral condensate
fullqcd
κ expansion
6.3
5.6
43*4
β=5.9
κ=0.12 NF=2
µ=0.9
5.55
5.5
5.45
43*4
β=5.9
κ=0.14 NF=2
µ=0.9
6.2
6.15
6.1
6.05
6
5.95
5.4
5.9
5.35
5.85
0
5
10
15
20
order of NLO corrections
0.596
0.595
30
0
5
0.601
10
15
20
order of NLO corrections
25
30
fullqcd
κ expansion
0.6
0.599
0.594
spatial plaquettes
spatial plaquettes
25
fullqcd
κ expansion
0.593
43*4
β=5.9
κ=0.12 NF=2
µ=0.9
0.592
0.591
0.598
0.597
43*4
β=5.9
κ=0.14 NF=2
µ=0.9
0.596
0.595
0.594
0.589
3
10 .8, nf=2, beta=5.8, kappa=0.12 vs mu/T; denstot
dens0
dens2
0.593
0
5
10
15
20
order of NLO corrections
1
25
30
0
2
fullqcd
κ expansion
0.95
5
25
30
0.15
fullqcd
κ expansion
1.9
0.9
10
15
20
order of NLO corrections
[4] M. Fromm, J. Langelage, S. Lottini, and O.
Philipsen, JHEP 1201 (2012) 042 (2010) 054508
[arXiv 0912.3360]
1.8
density
density
0.85
0.8
43*4
β=5.9
κ=0.12 NF=2
µ=0.9
0.7
0.05
1.6
43*4
β=5.9
κ=0.14 NF=2
µ=0.9
1.5
1.4
0.65
0.6
0
4
5
10
15
20
order of NLO corrections
25
30
0
5
10
15
20
order of NLO corrections
25
30
0.14
0.12
0.1
0.08
0.06
0.04
5.5
5.6
5.7
5.8
5.9
6
beta
6.1
6.2
6.3
6.4
6.5
5
5.5
6
6.5
[5] G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty,
I.-O. Stamatescu, Eur.Phys.J. A 49 (2013) 89
0.2
0
3
10 .16, nf=2, NLO, beta=5.8, kappa=0.12 vs mu/T; denstot
dens0
dens2
0.18
0.16
4.5
1.3
0
From the figures we see that for κ = 0.12 the approximation becomes reasonable for q ≥ 7 , which
is still much easier to simulate than full QCD. For
κ = 0.14, however, some quantities need larger q
or do not converge at all. We conclude that at
κ = 0.12 we can obtain a good approximation.
[1] I. Bender, T. Hashimoto, F. Karsch, V. Linke,
A. Nakamura, M. Plewnia, I.-O. Stamatescu,
W. Wetzel, N. Ph. Proc. Suppl. 26 (1992) 323
[3] E.Seiler, D. Sexty and I.-O. Stamatescu, Phys.
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0.1
1.7
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14
14.5
15
15.5
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16.5
17
17.5