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Complex Langevin and the sign problem in QCD
Felipe Attanasio
Advances in Monte Carlo Techniques for Many-Body Quantum Systems(INT-18-2b)2018-08-27
Felipe Attanasio Complex Langevin and the sign problem in QCD 1 / 28
Introduction
CLE: Motivation
Outline
Motivation(Brief) Review of lattice QCDStochastic quantisation and complex LangevinResultsSummary
Felipe Attanasio Complex Langevin and the sign problem in QCD 2 / 28
Complex Langevin Equation
CLE: Motivation
Description of QCD under different thermodynamical conditions
Reweighting, Taylor expansion
µ
T
Hadrons
Quark-GluonPlasma
Nuclear matter
ColourSuperconductor?
Critical point?
Experimental investigationsin progress (LHC, RHIC)and planned (FAIR)Perturbation theory onlyapplicable at hightemperature/density(asymptotic freedom)Full exploration requiresnon-perturbative (e.g.lattice) methods
Felipe Attanasio Complex Langevin and the sign problem in QCD 3 / 28
Complex Langevin Equation
Standard Monte Carlo methods
Monte Carlo at µ = 0
Theory is simulated on Euclidean spacetimePath integral weight eiS becomes e−SE ≥ 0Vacuum expectation values via path integrals
Configurations generated with probability proportional to e−SE
Monte Carlo at µ > 0
At µ > 0 the Euclidean action SE is no longer real (Sign Problem)For QCD, the fermion determinant acquires a complex phasee−SE cannot be interpreted as probability distributionResults from some techniques (e.g., reweighting, Taylor expansion) are notreliable when µ/T & 1
Felipe Attanasio Complex Langevin and the sign problem in QCD 4 / 28
Complex Langevin Equation
QCD phase diagram
Reweighting, Taylor expansion
µ
T
Hadrons
Quark-GluonPlasma
Nuclear matter
ColourSuperconductor?
Critical point?
Standard Monte Carlo methods cannot probe far into the phase diagram
Felipe Attanasio Complex Langevin and the sign problem in QCD 5 / 28
Complex Langevin Equation
Lattice QCD
Wilson gauge action [Wilson 1974 Phys. Rev. D 10 2445]
Gauge links Uxµ = exp (iaAxµ) ∈ SU(3) “transport” the gauge fieldsbetween x and x+ µ and have the desired gauge transformation behaviourSimplest gauge invariant objects on the lattice: Plaquettes
Ux,µν = UxµUx+µ,νU†x+ν,µU
†xν = exp
(ia2Fx,µν +O(a3)
)
S[U ] = β
3∑x
∑µ<ν
ReTr [1− Ux,µν ]
= β
3∑x
∑µ<ν
Tr[1− 1
2(Ux,µν + U†x,µν
)]
with β = 6/g2[Beane et al 2015 J. Phys. G: Nucl.Part. Phys. 42 034022]
Felipe Attanasio Complex Langevin and the sign problem in QCD 6 / 28
Complex Langevin Equation
Lattice QCD
Fermi action (naıve)
SF = ψx
[γν2(eµδν,4Uxνψx+ν − e−µδν,4Ux,−νψx−ν
)+mψx
]=∑x,y
ψxMx,yψy
Path integral
Vacuum expectation values via path integration
〈O〉 =∫DUDψDψO e−S−SF =
∫DU det(M) e−S =
∫DU e−S+ln detM
M depends on the gauge links U and chemical potential µSign problem: (detM(U, µ))∗ = detM(U,−µ∗)
Felipe Attanasio Complex Langevin and the sign problem in QCD 7 / 28
Complex Langevin Equation
Stochastic quantization
Stochastic quantization [Damgaard and Huffel, Physics Reports]
Add fictitious time dimension θ to dynamical variablesEvolve them according to a Langevin equation
∂x(θ)∂θ
= − δS
δx(θ) + η(θ) ,
where S is the action and η(θ) are white noise fields satisfying
〈η(θ)〉 = 0 , 〈η(θ)η(θ′)〉 = 2δ(θ − θ′) ,
Quantum expectation values are computed as averages over the Langevintime θ after the system reaches equilibrium at θ = θ0
〈O〉 ≡∫DxO e−S ≡ lim
θ′→∞
1θ′ − θ0
∫ θ′
θ0
O(θ)dθ
Felipe Attanasio Complex Langevin and the sign problem in QCD 8 / 28
Complex Langevin Equation
Complex Langevin method
Complex Langevin [Parisi, Phys. Lett. B131 (1983)][Klauder, Acta Phys. Austriaca Suppl. 25 251 (1983)]
Complexify the fields, i.e., give each component an imaginary part(x→ z = x+ iy)Rewrite action and observables in term of new fieldsIt circumvents the sign problem by deforming the path integral into thecomplex plane
Simple toy model: Anharmonic oscil-lator with complex mass
H = p2
2m + 12ωx
2 + 14λx
4
Exact partition function:
Z ∝√
4ξωeξK− 1
4(ξ) , ξ = ω2
8λ -0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
1 2 3 4 5 6 7 8 9
n
Re 〈zn〉/nIm 〈zn〉/n
Felipe Attanasio Complex Langevin and the sign problem in QCD 9 / 28
Complex Langevin Equation
CLE: Complexification
When can it be trusted? [Aarts, Seiler, Stamatescu, Phys.Rev. D81 (2010) 054508]
Sampling a complex weight ρ(x) of real x using a real weight P (x, y) ofz = x+ iy
For S and O holomorphic
〈O〉ρ = 〈O〉P∫dzO(z)ρ(z) =
∫dx dyO(x, y)P (x, y)
Sufficient conditions:Fast falloff in imaginary direction (y →∞) of P (x, y)〈LO〉 = 0 for all O, with L = [∂z − (∂zS)] ∂z
Felipe Attanasio Complex Langevin and the sign problem in QCD 10 / 28
Complex Langevin Equation
Stochastic quantization (gauge theories)
Lattice gauge theories [Damgaard and Huffel, Physics Reports]
Evolve gauge links according to the Langevin equation
Uxµ(θ + ε) = exp [Xxµ]Uxµ(θ) ,
whereXxµ = iλa(−εDa
xµS [U(θ)] +√ε ηaxµ(θ)) ,
λa are the Gell-Mann matrices, ε is the step size, ηaxµ are white noise fieldssatisfying
〈ηaxµ〉 = 0 , 〈ηaxµηbyν〉 = 2δabδxyδµν ,
S is the QCD action and Daxµ is defined as
Daxµf(U) = ∂
∂αf(eiαλ
a
Uxµ)∣∣∣∣α=0
Felipe Attanasio Complex Langevin and the sign problem in QCD 11 / 28
Complex Langevin Equation
CLE: Complexification (gauge theories)
Complexification [Aarts, Stamatescu, JHEP 09 (2008) 018]
Allow gauge links to be non-unitary: SU(3) 3 Uxµ → Uxµ ∈ SL(3,C)Use U−1
xµ instead of U†xµ asit keeps the action holomorphic;they coincide on SU(3) but on SL(3,C) it is U−1 that represents thebackwards-pointing link.
That means the plaquette is now
Ux,µν = UxµUx+µ,νU−1x+ν,µU
−1xν ,
and the Wilson action reads
S[U ] = β
3∑x
∑µ<ν
Tr[1− 1
2(Ux,µν + U−1
x,µν
)]
Felipe Attanasio Complex Langevin and the sign problem in QCD 12 / 28
Complex Langevin Equation
CLE: Complexification (gauge theories)
Gauge cooling [Seiler, Sexty, Stamatescu, Phys.Lett. B723 (2013) 213-216]
SL(3,C) is not compact ⇒ gauge links can get arbitrarily far from SU(3)During simulations monitor the distance from the unitary manifold with
d = 1N3sNτ
∑x,µ
Tr[UxµU
†xµ − 1
]2 ≥ 0
Use gauge transformations to decrease dUxµ → ΛxUxµΛ−1
x+µ
necessary, but not always sufficient
1e-35
1e-30
1e-25
1e-20
1e-15
1e-10
1e-05
1
100000
0 5 10 15 20 25 30 35 40 45
d
θ
Pure Yang-Mills, β = 6.6, 163 × 4
Without ooling
With ooling
SU( )
NSL( ,C)
N
[Aarts, Bongiovanni, Seiler, Sexty,Stamatescu, hep-lat/1303.6425]
Felipe Attanasio Complex Langevin and the sign problem in QCD 13 / 28
Complex Langevin Equation
Results
Observables considered
Polyakov loop: order parameter for confinement in pure Yang-Mills
P = 1N3s
∑~x
〈P~x〉 , P~x =∏τ
U4(~x, τ)
Spatial plaquette:
13N3
sNτ
∑x
∑1<µ<ν<3
〈Ux,µν〉 , Ux,µν = UxµUx+µ,νU−1x+ν,µU
−1xν
Chiral condensate: order parameter for confinement for massless quarks
〈ψψ〉 = ∂
∂mlnZ
Felipe Attanasio Complex Langevin and the sign problem in QCD 14 / 28
Complex Langevin Equation
Heavy-dense QCD
Heavy-dense approximation [Bender et al, Nucl. Phys. Proc. Suppl. 46 (1992) 323]
Heavy quarks → quarks evolve only in Euclidean time direction:
detM(U, µ) =∏~x
{det[1 + (2κeµ)Nτ P~x
]2det[1 +
(2κe−µ
)Nτ P−1~x
]2}
Polyakov loopP~x =
∏τ
U4(~x, τ)
Exhibits the sign problem: [detM(U, µ)]∗ = detM(U,−µ∗)Siver-blaze problem at T = 0
Felipe Attanasio Complex Langevin and the sign problem in QCD 15 / 28
Complex Langevin Equation
Phase boundary of HDQCD [Aarts, Attanasio, Jager, Sexty, JHEP 09 (2016) 087]
T[M
eV]
µ /µ0c
63
83
103
0
50
100
150
200
250
300
350
0 0.2 0.4 0.6 0.8 1
Felipe Attanasio Complex Langevin and the sign problem in QCD 16 / 28
Complex Langevin Equation
CLE: Instabilities
Gauge cooling (mild sign problem)
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
50 100 150 200 250 300 350 400 450 500
Pol
yako
vlo
op
θ
Gauge coolingReweighting
0
0.2
0.4
0.6
0.8
1
1.2
50 100 150 200 250 300 350 400 450 500U
nita
rity
norm
θ
Gauge cooling
Left: Langevin time history of Polyakov loopRight: Langevin time history of unitarity norm (“large” but under control)
Felipe Attanasio Complex Langevin and the sign problem in QCD 17 / 28
Complex Langevin Equation
CLE: Dynamic stabilisation
Dynamic stabilisation [Attanasio, Jager, hep-lat/1808.04400]
New term in the drift to reduce the non-unitarity of Ux,ν
Xxν = iλa(−εDa
x,νS − εαDSMax +√ε ηax,ν
).
with αDS being a control coefficient
Max : constructed to be irrelevant in the continuum limit
Max only depends on Ux,νU†x,ν (non-unitary part)
Felipe Attanasio Complex Langevin and the sign problem in QCD 18 / 28
Complex Langevin Equation
CLE: Dynamic stabilisation
Dynamic stabilisation (mild sign problem)
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
50 100 150 200 250 300 350 400 450 500
Pol
yako
vlo
op
θ
Gauge coolingDyn. stab.
Reweighting
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
1e+01
50 100 150 200 250 300 350 400 450 500U
nita
rity
norm
θ
Gauge coolingDyn. stab.
Left: Langevin time history of Polyakov loopRight: Langevin time history of unitarity norm (notice log scale!)
Felipe Attanasio Complex Langevin and the sign problem in QCD 19 / 28
Complex Langevin Equation
Deconfinement in HDQCD
Good agreement with reweighting – even when GC converges to the wrong limit
0.46
0.48
0.5
0.52
0.54
0.56
0.58
0.6
0.62
5.4 5.6 5.8 6.0 6.2
Spat
ialp
laqu
ette
β
HDQCD 64, κ = 0.12, µ/µ0c = 0.6
ReweightingDyn. stab. αDS = 1000Gauge cooling, d < 0.03
Gauge cooling
Spatial plaquette as a function of the inverse coupling β for HDQCD
Felipe Attanasio Complex Langevin and the sign problem in QCD 20 / 28
Staggered quarks
Dynamical fermions
Staggered quarks
The Langevin drift for Nf flavours of staggered quarks
Dax,νSF ≡ Da
x,ν ln detM(U, µ)
= NF4 Tr
[M−1(U, µ)Da
x,νM(U, µ)]
Inversion is done with conjugate gradient method
Trace is evaluated by bilinear noise scheme – introduces imaginarycomponent even for µ = 0!
Felipe Attanasio Complex Langevin and the sign problem in QCD 21 / 28
Staggered quarks
Staggered quarks (β = 5.6, m = 0.025, NF = 4)
Comparison between CLE + DS runs and HMC (results by P. de Forcrand)
0.1190
0.1195
0.1200
0.1205
0.1210
0.1215
10−4 10−2 100 102 104 106 108
〈ψψ〉 µ
=0
αDS
HMCDS
Green band represents results from HMC simulationsαDS scan of the chiral condensate for a volume of 64
Felipe Attanasio Complex Langevin and the sign problem in QCD 22 / 28
Staggered quarks
Staggered quarks (β = 5.6, m = 0.025, NF = 4)
Comparison between CLE + DS runs and HMC (results by P. de Forcrand)
0.58100.58110.58120.58130.58140.58150.58160.58170.58180.58190.58200.5821
0 20 40 60 80 100 120 140
Pla
quet
te
〈ǫ〉/10−6
HMCDS
Linear fit
Grey band represents results from HMC simulationsLangevin step size extrapolation of the plaquette for a volume of 124
Felipe Attanasio Complex Langevin and the sign problem in QCD 23 / 28
Staggered quarks
Staggered quarks (β = 5.6, m = 0.025, NF = 4)
Comparison between CLE + DS runs and HMC (results by P. de Forcrand)
Plaquette ψψVolume HMC Langevin HMC Langevin
64 0.58246(8) 0.582452(4) 0.1203(3) 0.1204(2)84 0.58219(4) 0.582196(1) 0.1316(3) 0.1319(2)104 0.58200(5) 0.58201(4) 0.1372(3) 0.1370(6)124 0.58196(6) 0.58195(2) 0.1414(4) 0.1409(3)
Expectation values for the plaquette and chiral condensate for full QCDLangevin results have been obtained after extrapolation to zero step size
Felipe Attanasio Complex Langevin and the sign problem in QCD 24 / 28
Staggered quarks
Staggered quarks (β = 5.6, m = 0.025, NF = 2)
Preliminary results at µ > 0 (not extrapolated to zero step size)
−2
0
2
4
6
8
10
12
14
0 0.5 1 1.5 2
〈n〉/T
3
µ/T
Nτ = 2Nτ = 4Nτ = 6Nτ = 8
−2
0
2
4
6
8
10
12
0 0.5 1 1.5 2
∆p/T
4
µ/T
Nτ = 2Nτ = 4Nτ = 6Nτ = 8
VeryPreli
minary
VeryPreli
minary
Vertical lines indicate position of critical chemical potential for each temperatureLeft: Density as a function of chemical potentialRight: Pressure as a function of chemical potential
Felipe Attanasio Complex Langevin and the sign problem in QCD 25 / 28
Staggered quarks
Pure Yang-Mills at θ2 > 0
Euclidean Yang-Mills lagrangian with a topological term
LYM = 14Tr[FµνFµν ]− iθ g2
64π2 εµνρσTr[FµνFρσ]
Sign problem for real θ
5.38 5.4 5.42 5.44 5.46 5.48 5.5 5.52 5.54 5.56 5.58 5.6 5.62
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
β
〈|P|2 〉
θL = 0
θL = 10
θL = 30
5.54 5.56 5.58 5.6 5.62 5.64 5.66 5.68 5.7 5.72 5.74 5.76
0.00
0.01
0.01
0.02
0.02
0.03
0.03
0.04
β
〈|P|2 〉
θL = 0
θL = 10
θL = 30
Very Preliminary
Very Preliminary
Volumes of 123 × 3 (left) and 163 × 4 (right)
Felipe Attanasio Complex Langevin and the sign problem in QCD 26 / 28
Staggered quarks
Non-relativistic fermions in 1D [Drut, Loheac, Phys.Rev. D95 (2017) 094502]
Sign problem for polarised systemsStabilisation method similar to Dynamic Stabilisation
0.8
0.9
1
1.1
1.2
1.3
0.1 1 10 100 1000
n/n
0
t
ξ0.0
0.005
0.05
0.1
0.2
0.4
-0.1
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
logρ
sin
θ
logρ cosθ
ξ = 0.0
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
logρ
sin
θ
logρ cosθ
ξ = 0.1
-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60
log
ρ s
inθ
logρ cosθ
ξ = 0.005
-800
-400
0
400
800
-800 -400 0 400 800
log
ρ s
inθ
logρ cosθ
ξ = -0.1
Left: Running average of the density for different ξRight: Plot of e−S = ρeiθ. Each point is one snapshot of the Langevin evolution
Felipe Attanasio Complex Langevin and the sign problem in QCD 27 / 28
Staggered quarks
Summary and outlook
Conclusions
Complex Langevin provides a way of circumventing the sign problemResults in QCD and non-relativistic fermions possible with modified process
Future plans
Map the phase diagram of QCD with light quarksFurther applications in condensed matter
Felipe Attanasio Complex Langevin and the sign problem in QCD 28 / 28