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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?
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 2 / 27
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!
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 3 / 27
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]
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 4 / 27
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 )
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 5 / 27
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).
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 6 / 27
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
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.
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 8 / 27
Revisiting Link Integration forSU(N)
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 9 / 27
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
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 10 / 27
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 )
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 11 / 27
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
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 13 / 27
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)
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 14 / 27
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).
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 15 / 27
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.
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 16 / 27
Dualization of pure Yang-Millstheory
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 17 / 27
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
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 19 / 27
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.
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 20 / 27
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.
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 21 / 27
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?
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 22 / 27
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.
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 23 / 27
Backup slides
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 24 / 27
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
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 25 / 27
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
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 26 / 27
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 .
G. Gagliardi, W. Unger Dual representation of LQCD Sign Workshop 2018, Bielefeld 27 / 27