Solution of the sign problem in the Potts model at fixed ...Solution of the sign problem in the...

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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∏

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)

withQ±

t = BtP± + P∓ .

Fugacity expansion yields with NmaxQ = 2 ⋅Nc ⋅ L

detMa(µ) =Nmax

∑NQ=−Nmax

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 =∏

det [I ± (2κe+µ)LtPx]2

det [I ± (2κe−µ)LtP†x ]

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

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) ⋅∑

For NQ = 2 quarks:

detMHDNmax

Q−2/Ω ∝ 2∑

TrPx∑y

⎝4∑

TrPx∑y

TrPy − 3∑x

(TrPx)2+ 2 TrP†

Both determinants vanish under global Z(3)-transformations.

Canonical determinant for NQ = 3 quarks:

detMHDNmax

Q−3/Ω = h3 ⋅

⎝4∑

TrP†x∑

TrPy − 3∑x

TrPx TrP†x + 2L3

⎝4∑

TrP†x∑

TrPy + 2∑x

(TrPx)2∑y

+4∑x

TrPx ∑y≠x

TrPy∑z

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

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) ≃∞∑

eµNQZC(NQ) for h = eµ ≪ 1

Canonical partition function

ZC(NQ) =∑n∫ Dz exp(γ ∑

⟨xy⟩

δzx ,zy )∏x

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)∏

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),

∏x∈C

3NC=∏

q∏x∈C

W0(C) =q∏x∈C

qz∑x∈C nx

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

(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 ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

First calculate weight for cluster C containing source z∗y :

q∏w∈C

znw−δw,yw

qz∑w∈C nw−δw,y

1 if ∑w∈C nw = 1 mod 30 else

= δnC ,1

Calculate the subensemble average including zxz∗

∗y ∏

3NC= δCx ,Cy ⋅

+ (1 − δCx ,Cy ) ⋅

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

. . . and the weight for cluster C containing source zx :

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

3NC= δCx ,Cy ⋅

+ (1 − δCx ,Cy ) ⋅

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

3NC= δCx ,Cy ⋅

∗y ∏

+ (1 − δCx ,Cy ) ⋅qzxz

∗y ∏

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

3NC= δCx ,Cy ⋅

∗y ∏

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

3NC= δCx ,Cy ⋅∏

δnC ,0

∗y ∏

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

δnC ,0

∗y ∏

⟨zxz∗y ⟩NQ

≡ ⟪zxz∗y ∏

znww ⟫NQ/⟪∏

w∈Cznww ⟫NQ

q∏w∈C

znw+δw,xw

qz∑w∈C nw+δw,x

= δnC ,2

∗y ∏

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

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

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

ρB×103

log⟨ei

γ: 0.5508

v: 403

v: 503

v: 643

0.00 0.05 0.10 0.15

ρB×103

log⟨ei

γ: 0.5480

γ=0.5480

γ=0.5496

γ=0.5508

0.005 0.010 0.050 0.100 0.500

ρB×103

Canonical simulation results in the deconfined phase:

description in terms of a gas of (free) quarks

Results from below the deconfinement transition:

transition from the confined into the deconfined phase

typical signature of a 1st order phase transition

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

ρ×103

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

ρ×103

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

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

⟨z0z r*⟩

(Anti)Quark-(anti)quark potentials at low temperature:

zz* zz z*z*

0 2 4 6 8 10 12 14

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

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

Solution of the sign problem in the Potts model at fixed ...Solution of the sign problem in the...

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