universit¨at bielefeld crc - tr · g. gagliardi, w. unger dual representation of lqcd sign...

CRC - TR

Towards a Dual Representation of Lattice QCD

G. Gagliardi, W. Unger

Universitat Bielefeld

Sign Workshop 2018, Bielefeld

G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 1 / 27

Motivations

Why to use dual representations forfinite density lattice QCD?


QCD at finite density

0 1

100

200

Vacuum

Quark

Gluon

Plasma

NuclearMatter

CP ???

Hadronic Matter

⟨ ⟩≠0

⟨ ψ ψ ⟩=0

Early U

n iverseEarly U

n ivers e Crossover

μ/T=1

Terra Incognita

μB [GeV ]

T [MeV ]

T c

RHIC,LH

C

RHIC,LH

C

0 1

100

200

Vacuum

Quark

Gluon

Plasma

NuclearMatter

CP

μB [GeV ]

T [MeV ]

ColorSuper

conductor?

QuarkyonicMatter ?

Hadronic Matter

⟨ ⟩≠0

⟨ ⟩=0

1 st order

Early U

n iverseEarly U

n ivers e Crossover

FAIRFAIR

Neutron StarsNeutron Stars

Deconfinem

ent &

Chiral Transition

T c

Simulations at finite µB in the standard determinantal approachhindered by the sign problem. A lot of open questions:

Existence/Location of the critical point.Nature of the cold and dense phase (CFL colour superconductor?).Silver blaze.

Sign problem is representation dependent!=⇒ dual representations can tame it!


Overview of the available dual formulations

Pure Yang-Mills theory:Alternative formulation for SU(3) using Hubbard Stratonovichtransformation: [H. Vairinhos & P. De Forcrand ’14]

Plaquette expansion for pure Yang-Mills SU(2) gauge theory: [Leme,Oliveira,Krein ’17]

Dual formulations with matter fields:Nuclear Physics from lattice QCD at strong coupling: [P. De Forcrand &M. Fromm ’09]

Dual lattice simulation of the U(1) gauge-Higgs model at finitedensity: [Mercado, Gattringer, Schmidt ’13]

Dual simulation of the massless lattice Schwinger model withtopological term and non-zero chemical potential: [Goschl ’17]

Abelian color cycles (ACC): [Gattringer & Marchis ’17]

Dual U(N) LGT with staggered fermions: [Borisenko et al ’17]


Strong Coupling expansion: definition and motivationsStrong coupling expansion is a, brute force, Taylor expansion of thegauge action in the lattice gauge coupling β = 2Nc

g2 .

Z =∫

dχx dχx∏`

∫G

dU`e∑

pβ

Nc Re(TrUp) · eTr[U`M†`+U†

`M`

]

=∫

dχx dχx∑{np ,np}

∏`,p

(β/2Nc)np+np

np!np!

∫G

dU`Tr[Up]np Tr[U†p]np eTr[U`M†`+U†

`M`

]

Strong Coupling Limit: β = 2Ncg2 → 0

Gauge fields can be integrated out!Dual representation via color singlets.Sampling using Worm algorithms.

Zsc =∫

[dU]∫

dχχ e−(��Sg +Sf )


The strong coupling partition functionThe 1-link integral can be performed analytically

J0[M,M†] =∫

GdUeTr[UM†+M†U]

After Grassman integration (Nf = 1,staggered fermions):

ZSC (mq, µ) =∑{k,n,`}

∏b=(x ,µ)

(Nc − kb)!Nc !kb!︸︷︷︸

meson hoppings Mx My

∏x

Nc !nx ! (2amq)nx

︸︷︷︸chiral cond. χχ

∏`

w(`, µ)︸︷︷︸baryon hoppings Bx By

kb ∈ {0, . . .Nc}, nx ∈ {0, . . .Nc}, `b ∈ {0,±1}, QCD: Nc = 3 [Wolff & Rossi, 1984]

Grassman constraint:

nx +∑±µ

(kµ(x) + Nc

2 |`µ(x)|)

= Nc

Residual sign problem in w(`, µ).


Strong coupling QCD

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

aT

aµ

a4 ∆f

2nd

order

1st

ordertricritical

3×10-6

5×10-6

1×10-5

2×10-5

3×10-5

5×10-5

1×10-4

Advantages:Average complex phase 〈eiφ〉pq = e−V

T (fpq−f ) close to one.=⇒ sign problem is mild and phase diagram can be mapped out.

Simulations are fast and no supercomputers are required.Chiral limit even faster than the massive case.

Con: Lattice is maximally coarse.G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 7 / 27

Going beyond strong coupling QCDAt β = 0 just color singlets [mesons & baryons]=⇒MDP formulation.β corrections to strong coupling needed to make the lattice finer.Plaquette excitations produce world sheets bounded by quarkfluxes:

O (β)

Gauge Flux

Baryonic Quark Flux

Mesonic Quark Flux

B B

(MM )3

qq

q

g

g(MM )2

qgqg

Gauge integrals become strongly coupled at β > 0.

Caveat: Dual representation does not solve,per se, the sign problem.Plaquette contributions could reintroduce it.


Revisiting Link Integration forSU(N)


One-Link Integrals

Link Integrals no longer factorizes in presence of plaquettes.The 1-link integrals must be now computed explicitly with opencolor indices. Example at O(β):∏ ∫

dU`eβ

2Nc Tr[Up+U†p ] · eTr[U`M†`+U†

`M`

]≈

∏ ∫dU`(1 + β

2NcTr[Up + U†p]︸︷︷︸

O(β)correction

) · eTr[U`M†`+U†

`M`

]

After the hopping expansion of the fermionic action:∏`∈p

∫dU`

β

2NcTr[Up]eTr

[U`M†`+U†

`M`

]= Tr[

∏`∈p

J`]

(J`[M,M†])nm =

∑κ`,κ`

1κ`!κ`!

κ∏α=1

(M`)iαjα

κ∏β=1

(M†`)kβ`βIk`+1,k`

m+i n+j ,k `︸︷︷︸Gauge integral to compute


One-Link IntegralsThe prototype of Gauge Integral to compute is:

Ia,bi j ,k ` =

∫SU(Nc )

dUa∏

α=1U jα

iα

b∏β=1

(U†)`βkβ | a − b |= q · Nc

Cases a = 0 and q = 0, 1 addressed: [Creutz 1978, Collins ’03,’06, Zuber ’17].We extended their results by computing the generating functional:

Z a,b[K , J ] =∫

SU(N)dU[Tr(UK )

]a [Tr(U†J)]b

n=min{a,b}= (qNc + n)!Nc−1∏i=0

i!(i + q)!(det K )q

︸︷︷︸Baryonic contrib.

∑ρ`n

W n,qg (ρ,Nc)tρ(JK )

︸︷︷︸Mesonic contrib.

W n,qg (ρ,Nc) =

∑λ`n

`(λ)≤Nc

1(n!)2

f 2λ χ

λ(ρ)Dλ,Nc +q

, tρ(A) =∏ρi

Tr(Aρi )


One-Link Integrals: The Weingarten functionsZ a,b[K , J ] expressed as a sum over integer partitions weighted by themodified Weingarten functions Wg .

Group theoretical factors enter theexpression for W n,q

g : fλ,Dλ,Ndimensions of irrep λ of Sn andSU(Nc). χλ(ρ) are the irreduciblecharacters of the symmetric group.

W n,qg take as an argument both Nc

and ρ. Conjugacy classes ofpermutations can also berepresented as integer partitions.

W n,qg (ρ,Nc) =

∑λ`n

`(λ)≤Nc

1(n!)2

f 2λχ

λ(ρ)Dλ,Nc +q

(1 2 32 1 3

)∼= (12)(3) ∼=

Ia,b obtained from Z a,b[K , J ] by differentiating in J ,K .G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 12 / 27

Application to Strong CouplingQCD with O(β) correction


O(β) Corrections: DerivationRelevant gauge integrals at O(β):In,n

i j ,k ` =∑

σ,τ∈Sn

δ`σi δjkτ W n,0

g (∣∣σ ◦ τ−1∣∣,Nc)

INc +1,1i j ,k ` = 1

(Nc !)2

Nc +1∑α=1

∏r=i ,j

εr1,..,rα−1,rα+1,..,rNc +1

δ`1iαδ

jαk1

W 1,1g (id,Nc)

Simplification: depending on the number of fermion hoppings, restrictWg to the irreps consistent with the fermionic content:

W n,qg (ρ,Nc)→ W n,q

g ,Λ (ρ,Nc) =λ∈Λ∑λ`n

`(λ)≤Nc

1(n!)2

f 2λχ

λ(ρ)Dλ,Nc +q

, O(β) : Λ ={

[1n], [21n−1]}

This gives immediately:

(J`[M,M†])ba =

Nc∑k=1

(Nc − k)!Nc !(k − 1)! (Mx Mx+µ)k−1 χx ,aχ

bx+µ + ...

Mx = χxχx , ` = (x , µ)


O(β) Corrections: Phase Diagram

0 0.2

0.4 0.6

0.8 1

1.2 1.4

1.6

0 0.5

1 1.5

2 2.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

T [lat. units]

2nd order

tricriticalpoint

1st order

nuclear CEP

β

µB [lat. units]

T [lat. units]

[P. De Forcrand & al., ’14]

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

chiral 1st

and 2nd

order, direct sampling

aTc(aµ), 083x4

aT

aµ

1st

order β=0β=0

β=0.3β=0.6β=0.9

1

st order extrap.

w. plaquette surfaces

Tc(µ = 0) moves to lower values at β > 0.Tricritical point slightly changes with β at leading order.Chiral and nuclear transition do not split at O(β).


O(β) Corrections: Sign Problem

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

aµ

average sign on 44x4

β0.00.10.51.02.05.0

0

0.005

0.01

0.015

0.02

0.025

0.03

0 0.2 0.4 0.6 0.8 1

aµ

a4∆f

β0.00.10.51.02.05.0

Complex phase goes quickly to zero for β > 1. Sign problem getsworse at µ = 0 as well.Diagrammatic origin: Frustrated monomers, Nc -fluxes and dimersperpendicular on a plaquette surface.

As a plaquette-induced sign problem is present at µ = 0, try to finda positive representation for pure Yang-Mills theory.


Dualization of pure Yang-Millstheory


Strong Coupling Expansion for Y-M theory

ZYM =∫

SU(N)[dU]e

β2Nc

∑p

[TrUp+TrU†p

]Taylor expand the action in terms of plaquette(anti-plaquette)occupation numbers {np, np}:

ZYM =∑{np ,np}

(β/2Nc)∑

p np+np∏p np!np!

∏`

∏p

∫SU(N)

dU` (TrUp)np(TrU†p

)np

︸︷︷︸W({np,np})

Plaquette constraint: For each link ` = (x , µ) :

∑ν>µ

δnx ,µ,ν − δnx−ν,µ,ν︸︷︷︸δnp=np−np

={

0 U(N)0 mod N SU(N)

d`=(x ,µ)︸︷︷︸dimer number

:= min{ ∑

ν>µ nx ,µ,ν + nx−ν,µ,ν∑ν>µ nx ,µ,ν + nx−ν,µ,ν

}G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 18 / 27

Dual form of the partition function: U(N) case

In,ni j ,k ` =

∑σ,τ∈Sn

W ng (∣∣σ ◦ τ−1∣∣,Nc)δ`σi δ

jkτ

For U(N) only the Ia,b with a = b are non-zero. W ({np, np}) can beevaluated as follows:

Associate a pair of permutation (σ`, τ`) ∈ Sd` to each bond.The delta functions δσ and δτ contract on each vertex.An additional permutation πx sitting on each vertex tells us how tore-orient the color flux:

i.e. how to contract the indices between δ’s associated to links thatjoin the same vertex.

W ({np, np}) =∑

{σ`,τ`∈Sd`}

∏`

W d`g (∣∣σ` ◦ τ−1

`

∣∣,Nc)︸︷︷︸≷≷≷0

∏x

N`(σ◦πx )c︸︷︷︸

from delta contraction


Dual form of the partition function: U(N) case

Problems: Analytic resummation too expensive. Half of the Wg ’sare negative. No possibility of reweighting. E.g. :

W 2,0g (2,Nc) = −1

Nc(N2c − 1)

Idea: Express the Gauge Integrals in a different basis by introductinga new class of operators Pa,b

λ :

In,ni j k` =

∑λ`n

`(λ)≤Nc

1Dλ,Nc

·(

Pa,bλ

)`i

(Pa,bλ

)j

kPa,bλ = 1

fλ∑π∈Sn

Ma,bλ (π)δπ

Ma,bλ (π) are the matrix elements of the irrep. λ of Sn in the

Young-Yamanouchi basis.Mλ(π) is an orthogonal matrix for all π ∈ Sn.Matrix elements computed using computer algebra.


Dual form of the partition function: U(N) caseProperties of the operators Pa,b

λ

Define hermitian product between operators in (CNc )⊗n:〈A,B〉 := Tr(A†B) =⇒ 〈δπ, δσ〉 = N `(σ◦π−1)

c

Pa,bλ is a complete, orthogonal set, with respect to 〈·〉. We obtain:

W ({np, np}) =∑

{λ``d`}`(λ`)≤Nc

W ({np ,np},{λ`})≥0︷︸︸︷∑a`,b`

∏`

1Dλ`,Nc

∏x

w(x)

w(x) = 〈

⊗`∈nb(x) Pa`,b`

λ`, δπx 〉

Advantages: Quantity in brackets is positive and much faster to compute=⇒ allows for importance sampling. Orthogonality helps us under-standing which configurations have non zero weight. Extension to SU(N)easier in this orthogonal basis.


Numerical costA bottleneck:

Complexity to evaluate Monte Carlo weights still prohibitive.

How to deal with it?Tabularize the weights (β-dependence factorizes).Tensor-network based methods?


Summary

Results:We successfully computed all the Gauge Integrals needed to handle theplaquette contributions at β > 0.We obtained a fully dualized partition function for Yang-Mills theory.We checked the correctness of our approach comparing with knownresults.By resumming a subset of weights we obtained only positiveconfigurations which are labelled by integer partitions.

Outlook:Speedup the computation of the Monte Carlo weights.Implement a Markov Chain Monte Carlo simulation for the dualizedpartition function.Extend our approach to matter fields: scalar QCD, staggered fermions.


Backup slides


Average plaquette for pure Yang-Mills in the dual representation and com-parison with standard heat bath algorithm for various dimensions:

Weights computed using invariants(valid for np − np mod Nc = 0

)Cube updates still missing.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

d=2

U(1) PLAQU(2) PLAQU(3) PLAQ

SU(2) PLAQSU(3) PLAQ

U(1) M/HBU(2) M/HBU(3) M/HB

SU(2) M/HBSU(3) M/HB

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

d=3, PLAQ only valid in strong coupling regime

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8

d=4, PLAQ only valid in strong coupling regime


Observables in the dual representation

Mean plaquette: ⟨Re Tr Uµ,ν(x)Nc

⟩dual=

⟨2nx ,µ,νβ

⟩Scalar glueball JPC = 0++: Extracted from temporal correlator ofspatial plaquettes:

C(t) = 〈ψ(t)ψ(0)〉 − 〈ψ(t)〉〈ψ(0)〉

ψ(t) = 1Nc

∑~x

µ 6=0∑µ<ν

Re Tr Uµ,ν(~x , t) dual=∑~x

µ 6=0∑µ<ν

n(~x ,t),µ,ν + n(~x ,t),µ,νβ


Prospects for MC simulationOur new degrees of freedom are {np, np} and integer partitions λ` ` d`.

Different types of updates to ensure ergodicity without violating theplaquette constraint:

1) Select a plaquette p′ and propose (np′ , np′ )→ (np′ ± 1, np′ ± 1). Randomlychoose new partitions λ′

`′on links `′ ∈ p′ . Accept new configuration with

probability:

Pacc = min{

1, (β/2Nc )±2[(np′±1)·(np′±1)

]±1 ·W({np±δp,p′ ,np±δp,p′ },{λ

′`})

W ({np ,np},{λ`})

}2) Select a plaquette p and propose np → np ± Nc or np → np ± Nc .

3) At fixed {np, np} select a random link `′ andaccept λ`′ → λ

′

`′with probability:

Pacc = min{

1,W({np ,np},{λ

′`})

W ({np ,np},{λ`})

}4) Propose cube updates to change np − np mod Nc .


universit¨at bielefeld crc - tr · g. gagliardi, w. unger dual representation of lqcd sign...

Documents