Solution of the sign problem in the Potts model at fixed ...Solution of the sign problem in the Potts model at xed baryon number Urs Wenger Albert Einstein Center for Fundamental Physics

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


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]


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


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.


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.


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


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 ) ⋅



⟨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=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy

3NC= δCx ,Cy ⋅

+ (1 − δCx ,Cy ) ⋅



⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

w

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ


q∏w∈C

znw+δw,xw

y3=

qz∑w∈C nw+δw,x

y3=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy

3NC= δCx ,Cy ⋅

qzxz

∗y ∏

w

znwwy

3NC

+ (1 − δCx ,Cy ) ⋅qzxz

∗y ∏

w

znwwy

3NC



⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

w

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ


q∏w∈C

znw+δw,xw

y3=

qz∑w∈C nw+δw,x

y3=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy

3NC= δCx ,Cy ⋅

qzxz

∗y ∏

w

znwwy

3NC


∗y ∏

w

znwwy

3NC



⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

w

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ


q∏w∈C

znw+δw,xw

y3=

qz∑w∈C nw+δw,x

y3=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy

3NC= δCx ,Cy ⋅∏

C

δnC ,0


∗y ∏

w

znwwy

3NC



⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

w

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ


q∏w∈C

znw+δw,xw

y3=

qz∑w∈C nw+δw,x

y3=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy


C

δnC ,0


∗y ∏

w

znwwy

3NC



⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

w

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ


q∏w∈C

znw+δw,xw

y3=

qz∑w∈C nw+δw,x

y3=


= δnC ,2


y :

qzxz

∗y ∏

w

znwwy


C

δnC ,0

+ (1 − δCx ,Cy ) ⋅ δnCx ,2 ⋅ δnCy ,1 ∏C≠CxC≠Cy

δnC ,0


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))



line of first order phase transitions from T to E



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


Canonical simulation results in the deconfined phase:

= = =

= = =

-

-

-

-

-

-

ρ×

μ

γ=

description in terms of a gas of (free) quarks


Results from below the deconfinement transition:

= = =

= = =

-

-

-

-

-

-

ρ×

μ

γ=

transition from the confined into the deconfined phase



= = =

= = =

-

-

-

-

-

-

ρ×

μ

γ=

typical signature of a 1st order phase transition



= =

-

-

-

-

-

-

ρ×

μ

γ=

Maxwell construction yields critical µc



-

-

-

-

-

-

ρ×

μ

Maxwell construction yields critical µc


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


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


(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*⟩


(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*⟩


(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


(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

Documents

Solution of the sign problem in the Potts model at fixed ...Solution of the sign problem in the Potts model at xed baryon number Urs Wenger Albert Einstein Center for Fundamental Physics