First Summit resultsLattice QCD at non-zero baryon density
April 26, 2019 Patrick Steinbrecher, HotQCD
Quantum Chromodynamics (QCD)
Quantum Chromodynamics
Proton
Quarks
Gluons
Color Charge
Strong Force
Confinement
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 1
1% of proton mass
= quark mass
99% of proton mass
= kinetic energy ofgluons & quarks
What happens if you make things hotter and hotter?
What happens if you keep squeezing and squeezing?
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 2
Phases of Water
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 3
The QCD phase diagram
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 4
Relativistic Heavy Ion Collider
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 5
Beam Energy Scan Theory
HotQCD is part of BEST and its mission is to provide critical
input for the Beam Energy Scan II which is performed now
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 6
Simulating Quantum Chromodynamicsfrom first principles
discretize Dirac equation
(iγµ∂µ − m)Ψ = 0
calculate path integral using
monte carlo methods
〈O〉 = 1Z
∫
DU O exp (−S)
Lattice QCD
✉ ✉ Q
✉ Q
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✉
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 7
e.g. 4-dimensional Lattice N3σ×Nτ
V = (aNσ)3
T = 1aNτ
The Lattice QCD Kernel
TrM -1 = limN→∞
1
N
N∑
k=1
η†kM -1ηk
evaluate traces using N random noise vectors η
solve M-1ηk using Conjugate Gradient
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 8
condition:
limN→∞
1
N
N∑
k=1
η∗kiηkj = δij
up to 2000 η per gauge field configuration
99% of runtime
Conjugate Gradient
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 9
1. calculate stencil operator (66% of runtime)
2. call multiple STREAM kernels (33% of runtime)
3. repeat
no optimizations possible for 2.
performance determined by memory bandwidth
Stencil Operator
wn =
4∑
µ=0
[(
Un,µvn+µ − U†n−µ,µvn−µ
)
+(
Nn,µvn+3µ − N†n−3µ,µvn−3µ
)]
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 9
complex 3-dim vector
complex 3×3 matrix U(3) matrix→֒ reconstruct from 14 floats
ν
µmatrix
vector
Stencil Operator
wn =
4∑
µ=0
[(
Un,µvn+µ − U†n−µ,µvn−µ
)
+(
Nn,µvn+3µ − N†n−3µ,µvn−3µ
)]
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 9
complex 3-dim vector
complex 3×3 matrix U(3) matrix→֒ reconstruct from 14 floats
ν
µmatrix
vector
w = standard
Stencil Operator
wn =
4∑
µ=0
[(
Un,µvn+µ − U†n−µ,µvn−µ
)
+(
Nn,µvn+3µ − N†n−3µ,µvn−3µ
)]
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 9
complex 3-dim vector
complex 3×3 matrix U(3) matrix→֒ reconstruct from 14 floats
ν
µmatrix
vector
w = standard
+ naik
� precalculated
� three-link term
Stencil Operator
wn =
4∑
µ=0
[(
Un,µvn+µ − U†n−µ,µvn−µ
)
+(
Nn,µvn+3µ − N†n−3µ,µvn−3µ
)]
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 9
complex 3-dim vector
complex 3×3 matrix U(3) matrix→֒ reconstruct from 14 floats
ν
µmatrix
vector
w = standard
+ naik
� precalculated
� three-link term
1146 Flop/site
0.8 Flop/byte→֒ single-precision
Multiple right-hand sides
Memory
Memory
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
Multiple right-hand sides
Memory
Memory
constant matrices
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( ) SO multi3( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
·, , ,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( ) SO multi3( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
·, , ,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( ) SO multi3( )
Multiple right-hand sides
Memory
Memory
constant matrices η0 η1 η2 η3 η4 η5 η6 · · ·
random vectors
,
·, , ,
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 10
SO( ) SO multi3( )
pro: much better arithmetic intensity
con: higher register pressure
Multiple right-hand sides
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
1 2 3 4 5 6 7 8
Flop/byte
#right-hand sides
Stencil Operator
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 11
Single-node Performance
0
500
1000
1500
2000
0 5 10 15 20
Conjugate Gradient
GFlop/s
#right-hand sides
fp32, single node
Volta V100
Knights Landing
Skylake
Haswell
K40
K20X
Ivy Bridge
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 12
Multi-node Performance
short setup phase assigns local problems to each GPU
e.g. inversions of different matrices
performance scales linearly with number of nodes
maximize number of nodes to reduce time to solution
largest Summit job used 2k nodes
achieved 23 PFlop/s
largest Titan job used 14k nodes
achieved 5 PFlop/susing both GPU and CPU
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 13
The QCD phase diagram
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 14
Critical point from Taylor expansions
expansion of QCD pressure
P
T 4=
∑
n
1
n!χB
n µ̂nB , χB
n =1
VT 3
∂n lnZ
∂µ̂nB
∣
∣
∣
∣
µB=0
analysis of convergence radius can determine bound on the
location of a critical point:
rP2n =
∣
∣
∣
∣
∣
(2n + 2)(2n + 1)χB2n
χB2n+2
∣
∣
∣
∣
∣
1/2
only if coefficients are positive for all n ≥ n0
if not → no critical point on real axis
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 15
Radius of convergence
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 16
complex function
f (z) = 11+z2
series expansion
f (z) =∑∞
n (−1)nz2n
convergence determined
by nearest singularity
from the origin
Comparison to experiment
Moment Symbol Experiment Lattice
mean MX 〈NX 〉 VT3χ
X1
variance σ2X
⟨
(δNX )2⟩
VT3χ
X2
skewness SX
⟨
(δNX )3⟩
σ3X
VT 3χX3
(
VT 3χX2
)3/2
kurtosis kX
⟨
(δNX )4⟩
σ4X
− 3VT 3χX
4(
VT 3χX2
)2
volume independent ratios
σ2X
MX=
χX2
χX1
, SXσX =χX
3
χX2
, kXσ2X =
χX4
χX2
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 17
only at freeze-out (µf , Tf )
First results from Summit
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 18
The QCD crossover line
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 19
135
140
145
150
155
160
165
170
175
0 50 100 150 200 250 300 350 400
Tc [MeV]
µB [MeV]
nS = 0, nQ
nB= 0.4
crossover line: O(µ4B)
constant: ǫs
freeze-out: STARALICE
No signs for a QCD critical point along Tc(µB)
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 20
0.10
0.12
0.14
0.16
0.18
0.20
0 50 100 150 200 250
c2s(Tc(µB), µB)
µB [MeV]
µQ = µS = 0
O(µ4B)
O(µ2B)
HRG0.000
0.002
0.004
0.006
0.008
0.010
0.012
0 50 100 150 200 250
Tc(µB)3/cP (Tc(µB), µB)
µB [MeV]
µQ = µS = 0
O(µ6B)
O(µ4B)
HRG
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0 50 100 150 200 250 300
χdisc(Tc(µB), µB)/χdisc(T0, 0)− 1
µB [MeV]
HotQCD preliminary
nS = 0, nQ
nB= 0.4O(µ4
B)
O(µ2B)
15.0
20.0
25.0
30.0
35.0
40.0
45.0
0 50 100 150 200 250
cV (Tc(µB), µB)/Tc(µB)3
µB [MeV]
µQ = µS = 0
O(µ4B)
O(µ2B)
HRG
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0 50 100 150 200 250
1/(Tc(µB)4κT (Tc(µB), µB))
µB [MeV]
µQ = µS = 0
O(µ6B)
O(µ4B)
HRG
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 50 100 150 200 250 300
σ2B(Tc(µB), µB)/σ
2B(T0, 0)− 1
µB [MeV]
HotQCD preliminary
nS = 0, nQ
nB= 0.4O(µ4
B)
O(µ2B)
HRG
0
1
2
3
4
5
6
7
8
9
135 140 145 150 155
r nχ -
- e
stim
ato
r fo
r µ
Bcrit /T
T [MeV]
2017: lower bound for r4χ
estimator r2χ
D’Elia et al., 2016, r4χ
Datta et al., 2016
Fodor, Katz, 2004
0
1
2
3
4
5
6
7
8
9
135 140 145 150 155
r2χ,HRG
disfavored region for thelocation of a critical point
r4χ,HRG
r6χ,HRG
0
20
40
60
80
100
135 145 155 165 175 185 195
nS=0, nQ/nB=0.4
χdisc/fk4
T [MeV]
Nτ=8, O(µB
6)
µB = 0.0 MeV125.0 MeV200.0 MeVno deviations from HRG
no increased fluctuations
no narrowing crossover
Summary
constrains on location of QCD critical point
upper bound from crossover line and radius of convergence
negative 6th and 8th order Taylor coefficients of P
no increased fluctuations along crossover
possible existing critical point may be found only for
µB > 400 MeV and T < (130 − 140) MeV
provides important input for Beam Energy Scan II currently
performed at the Relativistic Heavy Ion Collider
CG sustained 23 PFlop/s on Summit using 2k nodes
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 21
Thank you for your attention!
April 26, 2019 Patrick Steinbrecher, HotQCD Slide 22
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