complex langevin dynamics for nonabelian gauge...
TRANSCRIPT
QCD phase diagram
QCD partition function
Z =
∫
DUDψDψ e−SYM−SF =
∫
DU detD e−SYM
at nonzero quark chemical potential
[detD(µ)]∗ = detD(−µ∗)
fermion determinant is complex
straightforward importance sampling not possible
sign problem
⇒ phase diagram has not yet been determinednon-perturbatively
XQCD 14, June 2014 – p. 2
Outline
complex Langevin: exploring a complexified field space
gauge theories: from SU(N ) to SL(N,C)
gauge cooling
status report:
SU(3) + θ-term
heavy dense QCD
full QCD
summary
XQCD 14, June 2014 – p. 3
Complex Langevin dynamics
various scattered results since mid 1980s
here: recent results obtained with
Erhard Seiler, Dénes Sexty, Nucu Stamatescu
Benjamin Jaeger
Lorenzo Bongiovanni, Felipe Attanasio
. . .
some refs:0807.1597 [GA, IOS]
0912.3360 [GA, ES, IOS], 0912.0617, 1101.3270 [GA, FJ, ES, I OS]
1211.3709 [ES, DS, IOS], 1307.7748 [DS]
1306.3075 [GA, PG, ES], 1308.4811 [GA]
1311.1056 [GA, LB, IOS, ES, DS]
reviews: 1302.3028 [GA], 1303.6425 [GA, LB, IOS, ES, DS]
XQCD 14, June 2014 – p. 4
Complexified field space
partition function Z =∫
dx ρ(x) with complex weight ρ(x)
dominant configurations in the path integral?
x
Re
ρ(x)
⇒
y
x
real and positive distribution P (x, y): how to obtain it?
⇒ solution of stochastic process
complex Langevin dynamicsParisi 83, Klauder 83
XQCD 14, June 2014 – p. 5
Complex Langevin dynamics
does it work?
for real actions: stochastic quantization Parisi & Wu 81
equivalent to path integral quantization
Damgaard & Huffel, Phys Rep 87
for complex actions: formal proof was notably absent
recent progress:
theoretical foundation given
practical criteria for correctness formulated
severe sign and Silver Blaze problems explicitly solved
first results for gauge theories and even full QCD
XQCD 14, June 2014 – p. 6
Localised distributions
if
the action is holomorphic
[log dets are an additional worry, see Kim Splittorff ]
and
the distribution is localised, i.e.
P (x, y) = 0 for |y| > ymax [or P (x, y) → 0 fast enough]
then
the correct result is obtained GA, ES, IOS 09 + FJ 11
applications to gauge theories ...
XQCD 14, June 2014 – p. 7
Gauge theories
SU(N ) gauge theory: complexification to SL(N,C)
links U ∈ SU(N ): CL update
U(n+1) = R(n)U(n) R = exp[
iλa(
ǫKa +√ǫηa
)]
Gell-mann matrices λa (a = 1, . . . N2 − 1)
drift: Ka = −Da(SB + SF ) SF = − ln detM
XQCD 14, June 2014 – p. 8
Gauge theories
SU(N ) gauge theory: complexification to SL(N,C)
links U ∈ SU(N ): CL update
U(n+1) = R(n)U(n) R = exp[
iλa(
ǫKa +√ǫηa
)]
Gell-mann matrices λa (a = 1, . . . N2 − 1)
drift: Ka = −Da(SB + SF ) SF = − ln detM
complex action: K† 6= K ⇔ U ∈ SL(N,C)
deviation from SU(N ): unitarity norms
1
NTr
(
UU † − 11)
≥ 01
NTr
(
UU † − 11)2 ≥ 0
XQCD 14, June 2014 – p. 8
Gauge theories
deviation from SU(3): unitarity norm GA & IOS 08
1
3TrUU † ≥ 1
heavy dense QCD, 44 lattice with β = 5.6, κ = 0.12, Nf = 3XQCD 14, June 2014 – p. 9
Gauge theories
controlled evolution: stay close to SU(N ) submanifold when
small chemical potential µ
small non-unitary initial conditions
in presence of roundoff errors
XQCD 14, June 2014 – p. 10
Gauge theories
controlled evolution: stay close to SU(N ) submanifold when
small chemical potential µ
small non-unitary initial conditions
in presence of roundoff errors
in practice this is not the case
⇒ unitary submanifold is unstable!
process will not stay close to SU(N )
wrong results in practice, e.g. jumps when µ2 crosses 0
also seen in abelian XY model
XQCD 14, June 2014 – p. 10
Gauge cooling
what is the origin? can this be fixed?
gauge freedom: link at site k
Uk → ΩkUkΩ−1k+1 Ωk = eiω
k
aλa → e−αfk
aλa
in SU(N ): ωka ∈ R ⇒ in SL(N,C): ωk
a ∈ C
choose ωka purely imaginary, orthogonal to SU(N )
control unitarity norm D =1
NTr
(
UU † − 11)
≥ 0
gauge cooling ES, DS & IOS 12
after one gauge cooling step at site k, D → D′, linearise
D′ − D = − α
N(fka )
2 +O(α2) ≤ 0XQCD 14, June 2014 – p. 11
Gauge cooling
what is fka? Ωk = e−αfk
aλa D′ − D = −α/N(fka )2 + . . .
choose fka as the gradient of the unitarity norm
fka = 2Trλa
(
UkU†k − U †
k−1Uk−1
)
if U ∈ SU(N ): fka = 0, D = 0, no effect
cooling brings the links asclose as possible to SU(N )
SU( )
NSL( ,C)
N
XQCD 14, June 2014 – p. 12
Gauge cooling
simple example: one-link model GA, LB, ES, DS & IOS 13
S =1
NTrU U → ΩUΩ−1
D =1
NTr
(
UU † − 11)
fa = 2Trλa(
UU † − U †U)
note: c = TrU/N, c∗ = TrU †/N invariant under cooling
cooling dynamics:
D′ − D ≡ ˙D = − α
Nf2a = −16α
NTrUU †[U,U †]
in SU(2)/SL(2,C):
˙D = −8α(D2 + 2
(
1− |c|2)D+ c2 + c∗2 − 2|c|2
)
XQCD 14, June 2014 – p. 13
Gauge cooling
SU(2)/SL(2,C) one-link model
˙D = −8α(D2 + 2
(
1− |c|2)D+ c2 + c∗2 − 2|c|2
)
c = 12TrU, c∗ = 1
2TrU† invariant under cooling
if c = c∗: U gauge equivalent to SU(2) matrix
˙D = 8α(D+ 2− 2c2)D D(t) ∼ e−16α(1−c2)t → 0
if c 6= c∗: U not gauge equivalent to SU(2) matrixD(t) → D0 = |c|2 − 1 +√
1− c2 − c∗2 + |c|4 > 0
minimal distance from SU(2)reached exponentially fast
XQCD 14, June 2014 – p. 14
Langevin with gauge cooling
complex Langevin dynamics with gauge cooling:
alternate CL updates with gauge cooling updates
monitor unitarity norm
stay fairly close to SU(N )
under investigation:
SU(3) with a θ-term Lorenzo Bongiovanni, GA, ES, DS & IOS
13 + in prep
heavy dense QCD GA & IOS 08, ES, DS & IOS 12
Benjamin Jaeger, Felipe Attanasio, GA, ES, DS & IOS in prep
full QCD Denes Sexty 13 + in prep
XQCD 14, June 2014 – p. 15
SU(3) with aθ-term
pure SU(3) Yang-Mills theory (no fermions)
S = SYM − iθQ Q =g2
64π2
∫
d4xF aµνF
aµν
on the lattice:
S = SW − iθL∑
x
qL(x) qL(x) = discretised lattice version
θL bare parameter, requires renormalisation
lattice QL =∑
x qL is not topological (top. cooling)
complex action for real θL, real action for imaginary θL
imaginary θL: real Langevin and hybrid Monte Carlo (HMC)
real θL: use complex LangevinXQCD 14, June 2014 – p. 17
Analytical continuation
Z(θL) =
∫
DU e−SYM+iθLQ = e−Ωf(θL)
partition function and free energy even functions of θL
real θL:
−i〈Q〉θL =∂ lnZ
∂θL= ΩχLθL
(
1 + 2b2θ2L + 3b4θ
4L + . . .
)
imaginary θL = iθI :
〈Q〉θI = −∂ lnZ∂θI
= ΩχLθI(
1− 2b2θ2I + 3b4θ
4I + . . .
)
determine topological charge susceptibility χL andhigher-order coefficients bk
XQCD 14, June 2014 – p. 18
Analytical continuation
−i〈Q〉θL = −∂ lnZ∂θL
= ΩχLθL(
1 + 2b2θ2L + 3b4θ
4L + . . .
)
β = 6.1
Ω = 104
lines are fits:
y = b0θL(
1± 2b2θ2L
)
with b0 = 0.026for both andb2 ∼ 1 · 10−5
no renormalisation of θL yet!XQCD 14, June 2014 – p. 19
Analytical continuation
−i〈Q〉θL = −∂ lnZ∂θL
= ΩχLθL(
1 + 2b2θ2L + . . .
)
expected β dependence: drop in χL
no renormalisation of θL yet!XQCD 14, June 2014 – p. 20
Heavy dense QCD
consider static quarks: fermion determinant simplifies
detM =∏
x
det(
1 + heµ/TPx
)2det
(
1 + he−µ/TP−1x
)2
with h = (2κ)Nτ and P(−1) (conjugate) Polyakov loops
full Wilson gauge action is included
nontrivial phase diagram:
thermal deconfinement transition (as in pure glue)µ-driven transition at µc ∼ − ln(2κ)
test case for full QCD
XQCD 14, June 2014 – p. 22
Heavy dense QCD
consider static quarks: fermion determinant simplifies
detM =∏
x
det(
1 + heµ/TPx
)2det
(
1 + he−µ/TP−1x
)2
with h = (2κ)Nτ and P(−1) (conjugate) Polyakov loops
preliminary results for Polyakov loop and density for
β = 5.8 (a ∼ 0.15 fm)
κ = 0.12 (µc ∼ − ln(2κ) = 1.43)
volume 83 ×Nτ
Nτ = 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24 26 28
Benjamin Jaeger
XQCD 14, June 2014 – p. 23
Heavy dense QCD (in progress)
density µc = − ln(2κ) = 1.43 nsat = 12
first order transition at T = 0 (expected)see poster of Ben for more results
XQCD 14, June 2014 – p. 24
Beyond heavy dense QCD
include higher-order κ corrections DS, IOS et al, in prep
closer to real physicse.g. mB/3 6= mπ/2, no immediate saturation, . . .
0
0.05
0.1
0.15
0.2
14 14.5 15 15.5 16 16.5 17 17.5
0103.16, nf=2, NLO, beta=5.8, kappa=0.12 vs mu/T; denstot
dens0dens2
〈n〉/T 3 vs µ at NLO (including O(κ2) corrections)see poster of Nucu
XQCD 14, June 2014 – p. 25
Beyond heavy dense QCD
include higher-order κ corrections DS, IOS et al, in prep
closer to real physicse.g. mB/3 6= mπ/2, no immediate saturation, . . .
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 5 10 15 20 25 30
dens
ity
order of NLO corrections
43*4β=5.9κ=0.12 NF=2µ=0.9
fullqcdκ expansion
convergence in systematic expansion in κ2 (DS)XQCD 14, June 2014 – p. 25
Full QCD
first application to full QCD: Denes Sexty arXiv:1307.7748
fermion determinant: additional drift term in CLE
requires inversion of fermion matrix
stochastic inversion using conjugate gradient
staggered fermions with 4 flavours(Wilson fermions in progress)
monitor unitarity norm, log det, distributions, . . .
compare with HDQCD for heavy quarks andreweighting for light quarks
XQCD 14, June 2014 – p. 27
Full QCD
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
n/n s
at
µ/T
83*6 latticeNf=4β=5.8
HQCD ma=1staggered ma=1
HQCD ma=4staggered ma=4
density / densitysat vs µ/T
for heavier quarks and lighter quarks
comparison with HDQCD Sexty 13
XQCD 14, June 2014 – p. 28
Full QCD
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 5 10 15 20 25
Pol
yako
v lo
op
µ/T
83*6 latticeNf=4β=5.8
HQCD ma=1staggered ma=1
HQCD ma=4staggered ma=4
Polyakov loop vs µ/T
for heavier quarks and lighter quarks
comparison with HDQCD Sexty 13
XQCD 14, June 2014 – p. 28
Full QCD
comparison with reweighting
0.27
0.28
0.29
0.3
0.31
0.32
0.33
0.34
0.35
0.36
0.37
0 0.5 1 1.5 2 2.5 3 3.5 4
µ/T
83*4 latticeβ=5.4mass=0.05NF=4
Polyakov loop CLEinverse Polyakov CLEPolyakov reweighting
inv. Polyakov reweighting
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
Re
<ex
p(i 2
φ)>
µ/T
83*4 latticeβ=5.4mass=0.05NF=4
rewCLE
Polyakov loop vs µ/T average sign
for light quarks
agreement until reweighting breaks down
Fodor, Katz & Sexty in prep
XQCD 14, June 2014 – p. 28