Solution of the sign problem in the Potts modelat fixed baryon number
Urs Wenger
Albert Einstein Center for Fundamental PhysicsUniversity of Bern
Phys.Rev. D97 (2018) 114503 [1712.07585] in collaboration with
A. Alexandru, G. Bergner, D. Schaich
10 September 2018, SIGN 2018, Bielefeld, Germany
Motivation for canonical formulation of QCD
Consider the grand-canonical partition function of QCD:
ZQCDGC (µ) = Tr [e−H(µ)/T ] = Tr∏
t
Tt(µ)
The sign problem of QCD is a manifestation of hugecancellations between different states:
all states are present for any µ and T some states need to cancel out at different µ and T
In the canonical formulation:
ZQCDC (NQ) = TrNQ
[e−H(µ)/T ] = Tr∏t
T(NQ)t
dimension of Fock space tremendously reduced less cancellations necessary e.g. ZQCD
C (NQ) = 0 for NQ ≠ 0 mod Nc
Motivation for canonical formulation of QCD
Canonical transfer matrices can be obtained explicitly!
based on the dimensional reduction of the QCD fermiondeterminant [Alexandru, Wenger ’10; Nagata, Nakamura ’10]
identification of transfer matrices [Steinhauer, Wenger ’14]
Motivation QCD in the heavy-dense limit Canonical formulation Absence of the sign problem at strong coupling
Solution in the 3-state Potts model Canonical formulation Bond formulation and cluster algorithm Solution of the sign problem cf. [Alford, Chandrasekharan, Cox, Wiese 2001]
Motivation for canonical formulation of QCD
Canonical transfer matrices can be obtained explicitly!
based on the dimensional reduction of the QCD fermiondeterminant [Alexandru, Wenger ’10; Nagata, Nakamura ’10]
identification of transfer matrices [Steinhauer, Wenger ’14]
Motivation QCD in the heavy-dense limit Canonical formulation Absence of the sign problem at strong coupling
Solution in the 3-state Potts model Canonical formulation Bond formulation and cluster algorithm Solution of the sign problem cf. [Alford, Chandrasekharan, Cox, Wiese 2001]
Dimensional reduction of QCD
Reduced Wilson fermion determinant is given by
detMp,a(µ)∝∏t
detQ+t ⋅ det [e+µLt ⋅ I ± T ]
where T is a product of transfer matrices given by
T =∏t
Q+t ⋅ Ut ⋅ (Q
−t+1)
−1
withQ±
t = BtP± + P∓ .
Fugacity expansion yields with NmaxQ = 2 ⋅Nc ⋅ L
3s
detMa(µ) =Nmax
Q
∑NQ=−Nmax
Q
eµNQ/T ⋅ detMNQ
Heavy-dense limit of grand-canonical QCD
The heavy-dense approximation in general consists of takingthe limit κ ≡ (2m + 8)−1 → 0, µ→∞ while keeping κe+µ fixed.
Better: just drop the spatial hopping terms, but keep forwardand backward hopping in time:
system of static quarks and antiquarks
Reduced Wilson fermion matrix ⇔ effective action in terms ofPolyakov loops P and P† in d = 3:
detMHDp,a =∏
x
det [I ± (2κe+µ)LtPx]2
det [I ± (2κe−µ)LtP†x ]
2
Heavy-dense limit of canonical QCD
The canonical determinants are given by the trace over theminor matrix M,
detMHDNQ
∝ TrMNQ[((2κ)+Lt ⋅ P+P + (2κ)−Lt ⋅ P−P)]
where P denotes the Polyakov loops Px,y = I4×4 ⊗ Px ⋅ δx,y .
Under Z(Nc)-transformation by zk = e2πi ⋅k/Nc ∈ Z(Nc):
detMHDNQ→ detM ′HD
NQ= z
−NQ
k ⋅ detMHDNQ
and summing over zk yields detMNQ= 0 for NQ ≠ 0 modNc
reduces cancellations by factor of Nc
Heavy-dense limit of canonical QCD
Canonical determinant describing no quarks w.r.t. NmaxQ :
detMHDNmax
Q= 1 ⇔ quenched case
Canonical determinant describing a single quark, i.e. NQ = 1:
detMHDNmax
Q−1 = ((2κ)Lt + (2κ)−Lt) ⋅∑
x
TrPx
For NQ = 2 quarks:
detMHDNmax
Q−2/Ω ∝ 2∑
x
TrPx∑y
TrPy
+⎛
⎝4∑
x
TrPx∑y
TrPy − 3∑x
(TrPx)2+ 2 TrP†
x
⎞
⎠
Both determinants vanish under global Z(3)-transformations.
Heavy-dense limit of canonical QCD
Canonical determinant for NQ = 3 quarks:
detMHDNmax
Q−3/Ω = h3 ⋅
⎛
⎝4∑
x
TrP†x∑
y
TrPy − 3∑x
TrPx TrP†x + 2L3
s
⎞
⎠
+ h1
⎛
⎝4∑
x
TrP†x∑
y
TrPy + 2∑x
(TrPx)2∑y
TrPy
+4∑x
TrPx ∑y≠x
TrPy∑z
TrPz
⎞
⎠
describes the propagation of mesons and baryons
Invariant under global Z(3)-transformations
Description in terms of (anti-)quark occupation numbers nx
Suffers from a severe sign problem, unless all Px align ⇐⇒ deconfined phase global Z(3) is promoted to a local one ⇐⇒ strong coupling
The 3-state Potts model in d = 3 dimensions
We use the 3-state Potts model as a proxy for the effectivePolyakov loop action of heavy-dense QCD.
Grand-canonical partition function of the Potts model:
ZGC(h) = ∫ Dz exp(−S[z] + h∑x
zx)
Polyakov loops are represented by the Potts spins zx ∈ Z(3) standard nearest-neighbour interaction
S[z] = −γ ∑⟨xy⟩
δzx ,zy
external ’magnetic’ field h = (2κeµ)β ⇒ breaks Z(3)
There is a sign problem for h ≠ 0, i.e. at finite quark density.
The 3-state Potts model in d = 3 dimensions
Canonical partition function for NQ quarks:
ZC(NQ) = ∑n,∣n∣=NQ
∫ Dz exp(−S[z]) ⋅∏x
f [zx ,nx]
local quark occupation number nx ≤ nmaxx with ∣n∣ = NQ
use the simple local fermionic weights
f [z ,n] = zn
equivalent to the grand-canonical partition function for small hi.e. small density:
ZGC(h) ≃∞∑
NQ=0
eµNQZC(NQ) for h = eµ ≪ 1
The 3-state Potts model in d = 3 dimensions
Canonical partition function
ZC(NQ) =∑n∫ Dz exp(γ ∑
⟨xy⟩
δzx ,zy )∏x
znxx
Action is manifestly complex ⇒ fermion sign problem!
Global Z(3) symmetry ensures ZC(NQ ≠ 0 mod 3) = 0: projection onto integer baryon numbers
In the limit γ → 0, the global Z(3) becomes a local one: projection onto integer baryon numbers on single sites
nx = 0 mod 3 (limitγ → 0)
sign problem is absent
Bond formulation and cluster algorithm
Introduce bonds to express the action as
eγ⋅δzx ,zy =1
∑bxy=0
(δzx ,zy δbxy ,1 (eγ− 1) + δbxy ,0)
The canonical partition function now becomes
ZC(NQ) =∑n∑b∫ Dz∏
⟨xy⟩(δzx ,zy δbxy ,1 (e
γ− 1) + δbxy ,0)∏
x
znxx
sum over all bond configurations b
Define the sum of bond weights over n,b,Dz as ⟪ ⋅⟫NQ:
ZC(NQ) = ⟪∏x
znxx ⟫NQ
Bond formulation and cluster algorithm
NQ = 0 gives the usual Swendsen-Wang cluster construction: bond bxy is occupied with probability p(bxy = 1) = (eγ − 1),
if zx = zy weight of bond configuration is W (b) = (eγ − 1)Nb ,
with Nb = ∑⟨xy⟩ bxy
Summation over Z(3) spins within connected cluster yields
ZC(NQ = 0) = ⟪1⟫NQ=0 =∑b
(eγ − 1)Nb ⋅ 3NC
total number of clusters NC
cluster algorithm requires Euler-tour trees to achieve dynamicconnectivity in O(ln2 V ) instead of O(V lnV ) or O(V 2)
Solution of the sign problem in the canonical formulation
In the canonical formulation the cluster algorithm solves thesign problem:
include the fermionic contribution with an improved estimator similar to idea in the grand canonical formulation
[Alford, Chandrasekharan, Cox, Wiese 2001]
Average ∏x znxx over the subensemble of the 3NC configurations
related by the Z(3) transformations: total weight can be factorized into individual cluster weightsW0(C),
q∏x
znxxy
3NC=
q∏C
∏x∈C
znxxy
3NC=∏
C
q∏x∈C
znxxy
3=∏
C
W0(C)
where
W0(C) =q∏x∈C
znxxy
3=
qz∑x∈C nx
y3=
1 if ∑x∈C nx = 0 mod 3,0 else.
Solution of the sign problem in the canonical formulation
Hence, the canonical partition function becomes
Sign free canonical partition function
ZC(NQ) = ⟪∏x
znxx ⟫NQ=∑n
∑b
(eγ − 1)Nb ⋅ 3NC ⋅∏C
δnC ,0
nC = ∑x∈C nx mod 3 denotes the triality of the cluster C ∏C δnC ,0 projects on sector of configurations with triality-0
clusters only
An intuitive, physical picture emerges: only clusters with integer baryon number contribute
⇒ confinement quarks can move freely within the cluster
⇒ deconfinement within cluster
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
First calculate weight for cluster C containing source z∗y :
q∏w∈C
znw−δw,yw
y3=
qz∑w∈C nw−δw,y
y3=
1 if ∑w∈C nw = 1 mod 30 else
= δnC ,1
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅
+ (1 − δCx ,Cy ) ⋅
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅
+ (1 − δCx ,Cy ) ⋅
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅
qzxz
∗y ∏
w
znwwy
3NC
+ (1 − δCx ,Cy ) ⋅qzxz
∗y ∏
w
znwwy
3NC
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅
qzxz
∗y ∏
w
znwwy
3NC
+ (1 − δCx ,Cy ) ⋅qzxz
∗y ∏
w
znwwy
3NC
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅∏
C
δnC ,0
+ (1 − δCx ,Cy ) ⋅qzxz
∗y ∏
w
znwwy
3NC
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅∏
C
δnC ,0
+ (1 − δCx ,Cy ) ⋅qzxz
∗y ∏
w
znwwy
3NC
Improved estimators for physical quantities
We can define an improved estimator for the quark-antiquarkcorrelator:
⟨zxz∗y ⟩NQ
≡ ⟪zxz∗y ∏
w
znww ⟫NQ/⟪∏
w∈Cznww ⟫NQ
. . . and the weight for cluster C containing source zx :
q∏w∈C
znw+δw,xw
y3=
qz∑w∈C nw+δw,x
y3=
1 if ∑w∈C nw = 2 mod 30 else
= δnC ,2
Calculate the subensemble average including zxz∗
y :
qzxz
∗y ∏
w
znwwy
3NC= δCx ,Cy ⋅∏
C
δnC ,0
+ (1 − δCx ,Cy ) ⋅ δnCx ,2 ⋅ δnCy ,1 ∏C≠CxC≠Cy
δnC ,0
Improved estimators for physical quantities
Similar expressions for zxzy , z∗x z∗
y , zx , z∗x = z2x , . . .
An interesting quantity is the the quark chemical potential:
µ(ρ) ≡ −1
3log
ZC(NQ + 3)
ZC(NQ)
quark density ρ ≡ (NQ + 3/2)/V
The expectation value of the phase of Z(NQ):
ln⟨exp(iφ)⟩∣ ⋅ ∣,NQ=
⟪∏x znxx ⟫NQ
⟪1⟫NQ
= −3NQ/3−1
∑k=0
µ(3
2+ 3k) − lnP(NQ ,V )
Physics of the 3-state Potts model
Phase diagram in the (eµ, γ) ≡ (h, κ)-plane:[Alford, Chandrasekharan, Cox and Wiese 2001]
deconfinement phase transition at T = (0,0.550565(10))
Physics of the 3-state Potts model
Phase diagram in the (eµ, γ) ≡ (h, κ)-plane:[Alford, Chandrasekharan, Cox and Wiese 2001]
line of first order phase transitions from T to E
Physics of the 3-state Potts model
Phase diagram in the (eµ, γ) ≡ (h, κ)-plane:[Alford, Chandrasekharan, Cox and Wiese 2001]
critical endpoint E = (0.000470(2),0.549463(13))
Severity of the sign problem
Deconfined phase: Confined phase:
v: 403
v: 503
v: 643
0.00 0.05 0.10 0.15-50
-40
-30
-20
-10
0
ρB×103
log⟨ei
ϕ⟩
γ: 0.5508
v: 403
v: 503
v: 643
0.00 0.05 0.10 0.15
-150
-100
-50
0
ρB×103
log⟨ei
ϕ⟩
γ: 0.5480
γ=0.5480
γ=0.5496
γ=0.5508
0.005 0.010 0.050 0.100 0.500
10
20
50
ρB×103
L 0
Physics of the 3-state Potts model
Canonical simulation results in the deconfined phase:
= = =
= = =
-
-
-
-
-
-
ρ×
μ
γ=
description in terms of a gas of (free) quarks
Physics of the 3-state Potts model
Results from below the deconfinement transition:
= = =
= = =
-
-
-
-
-
-
ρ×
μ
γ=
transition from the confined into the deconfined phase
Physics of the 3-state Potts model
Results from below the deconfinement transition:
= = =
= = =
-
-
-
-
-
-
ρ×
μ
γ=
typical signature of a 1st order phase transition
Physics of the 3-state Potts model
Results from below the deconfinement transition:
= =
-
-
-
-
-
-
ρ×
μ
γ=
Maxwell construction yields critical µc
Physics of the 3-state Potts model
Results from below the deconfinement transition:
-
-
-
-
-
-
ρ×
μ
Maxwell construction yields critical µc
Physics of the 3-state Potts model
Results from below the critical endpoint:
V=203 V=253 V=323
V=403 V=503 V=643
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-8.0
-7.5
-7.0
-6.5
-6.0
-5.5
ρ×103
μ
γ=0.5480
crossover from the confined into the deconfined phase
Physics of the 3-state Potts model
Results from below the critical endpoint:
V=403 V=503 V=643
0.00 0.02 0.04 0.06 0.08-8.0
-7.8
-7.6
-7.4
-7.2
-7.0
-6.8
-6.6
ρ×103
μ
γ=0.5480
crossover from the confined into the deconfined phase
Physics of the 3-state Potts model
(Anti)Quark-(anti)quark potentials at zero density:
γ: 0.3
γ: 0.4
γ: 0.5
γ: 0.6
0 1 2 3 4 5 6 7
-15
-10
-5
0
r
log⟨z0z r*⟩
Physics of the 3-state Potts model
(Anti)Quark-(anti)quark potentials at zero density:
γ: 0.3 γ: 0.4 γ: 0.5 γ: 0.6
0 1 2 3 4 5 6 70.0
0.2
0.4
0.6
0.8
1.0
1.2
r
⟨z0z r*⟩
Physics of the 3-state Potts model
(Anti)Quark-(anti)quark potentials at low temperature:
zz* zz z*z*
0 2 4 6 8 10 12 14
-10
-5
0
5
r
log⟨z0z r*⟩,⟨z0z r⟩,⟨z0*z r*⟩
⟨z⟩⟨z*⟩
⟨z⟩⟨z⟩
⟨z*⟩⟨z*⟩
⟨z⟩'
⟨z*⟩'
confined phase: γ = 0.3 for NQ = 24, V = 163, i.e. ρ = 5.9 ⋅ 10−3
Physics of the 3-state Potts model
(Anti)Quark-(anti)quark potentials at low temperature:
zz* zz z*z*
0 2 4 6 8 10 12 14
-10
-5
0
5
r
log⟨z0z r*⟩,⟨z0z r⟩,⟨z0*z r*⟩
⟨z⟩⟨z*⟩
⟨z⟩⟨z⟩
⟨z*⟩⟨z*⟩
⟨z⟩'
⟨z*⟩'
values at r = 0 and r →∞ match ⟨z⟩, ⟨z∗⟩, ⟨z∗⟩⟨z∗⟩, . . .
Conclusions
We solved the fermion sign problem for the Z(3) Potts model isolate coherent dynamics of the Z(3) spins in clusters cluster subaverages project on nonzero, positive contributions
⇒ increase of statistics exponential in NC
The solution provides an appealing physical picture: quarks confined in clusters, but move freely within at γ → 0 clusters are confined to single sites only deconfinement corresponds to appearance of a percolating
cluster
Good algorithms reflect true physics insight!
Extension to Polyakov loop models could be possible: mechanism at work at β = 0 extend it to β > 0 ⇒ Z(Nc) clusters for gauge fields