Flash Talk: Diagrammatic Monte Carlo for StrongCoupling LQCD

Wolfgang Unger, ETH Zürich

Wolfgang Unger, ETH Zürich
with Philippe de Forcrand, ETH Zürich/CERN

Yukawa Institute, Kyoto
17.02.2012()

DMC for SC-LQCD

Kyoto, 17.02.2012

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Motivation for SC-LQCD

1 Motivation for Continuous Time Strong Coupling LQCD
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2 Diagrammatic Monte Carlo
3 Outlook
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Wolfgang Unger, ETH Zürich

T [MeV ]Baryonic
Crystal ?

Nuclear
MatterNeutron Stars01()

DMC for SC-LQCD

Color
Superconductor?

 B [GeV ]Kyoto, 17.02.2012

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Motivation for SC-LQCD

What is Strong Coupling Lattice QCD?
Look at QCD in a regime where the sign problem can be made mild:
This is obtained by changing the nature of integration variables:— no sampling of gauge fields {U}!
— no fermion determinant (no HMC)!
Staggered QCD in the strong coupling limit:
start from the “1-flavor” staggered QCD Lagrangian in Euclidean time:
LQCD =Pµδν0−µδν0†1η(x)eχ̄(x)U(x)χ(x+ν̂)−eχ̄(x+ν̂)U(x)χ(x)+amννq χ̄χνν 2P†β2

Wolfgang Unger, ETH Zürich

− 2Nc()DMC for SC-LQCD

P

trUP + trUP + O(a )Kyoto, 17.02.2012

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Motivation for SC-LQCD

What is Strong Coupling Lattice QCD?
Look at QCD in a regime where the sign problem can be made mild:
This is obtained by changing the nature of integration variables:— no sampling of gauge fields {U}!
— no fermion determinant (no HMC)!
Staggered QCD in the strong coupling limit:
start from the “1-flavor” staggered QCD Lagrangian in Euclidean time:
LQCD =Pµδν0−µδν0†1η(x)eχ̄(x)U(x)χ(x+ν̂)−eχ̄(x+ν̂)U(x)χ(x)+amννq χ̄χνν 2(((P((†β( + O(a2 )(trUtrU− 2Nc(P +(PP((((take limit of infinite gauge coupling:cg → ∞, β = 2N→0g2

allows to integrate out the gauge fields completely, as link integration factorizes

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

3 / 13

Motivation for SC-LQCD

What is Strong Coupling Lattice QCD?
Look at QCD in a regime where the sign problem can be made mild:
This is obtained by changing the nature of integration variables:— no sampling of gauge fields {U}!
— no fermion determinant (no HMC)!
Staggered QCD in the strong coupling limit:
start from the “1-flavor” staggered QCD Lagrangian in Euclidean time:
LQCD =Pµδν0−µδν0†1η(x)eχ̄(x)U(x)χ(x+ν̂)−eχ̄(x+ν̂)U(x)χ(x)+amννq χ̄χνν 2(((P((†β( + O(a2 )(trUtrU− 2Nc(P +(PP((((take limit of infinite gauge coupling:cg → ∞, β = 2N
→0g2

allows to integrate out the gauge fields completely, as link integration factorizes
Drawback: strong coupling limit is converse to asymptotic freedom:
degrees of freedom in SC-QCD live on a crystal
fermions have no spin (taste splitting maximal)
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

3 / 13

Motivation for SC-LQCD

Why Strong Coupling Lattice QCD?
1-flavor strong coupling QCD might appear crude, but
exhibits confinement, i.e. only color singlet degrees of freedom survive:
mesons (rep. by monomers and dimers)baryons (rep. by oriented self-avoiding loops)
and chiral symmetry breaking/restoration:
UA (1):

χ(x ) 7→ e iη5 (x )θA χ(x ), χ̄(x ) 7→ χ̄(x )e iη5 (x )θA ,dη5 (x ) = (−1)Σν=0 xν

is spontaneously broken below Tc ,hence its phase diagram might be similar to the QCD phase diagram
SC-LQCD as effective theory for nuclear matter:derive nuclear interactions between hadrons from (Lattice) QCD
transition at high density is the nuclear transition

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

4 / 13

Motivation for SC-LQCD

Why Strong Coupling Lattice QCD?
1-flavor strong coupling QCD might appear crude, butexhibits confinement, i.e. only color singlet degrees of freedom survive:
mesons (rep. by monomers and dimers)baryons (rep. by oriented self-avoiding loops)
and chiral symmetry breaking/restoration:
UA (1):

χ(x ) 7→ e iη5 (x )θA χ(x ), χ̄(x ) 7→ χ̄(x )e iη5 (x )θA ,dη5 (x ) = (−1)Σν=0 xν

is spontaneously broken below Tc ,hence its phase diagram might be similar to the QCD phase diagram
SC-LQCD as effective theory for nuclear matter:derive nuclear interactions between hadrons from (Lattice) QCD
transition at high density is the nuclear transition
Just another effective model for QCD?
yes and no: think of SC-LQCD rather as a 1-parameter deformation of QCD

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

4 / 13

Motivation for SC-LQCD

SC-QCD Partition Function
Exact rewriting of SC-QCD partition function (no approximation!):XZ(mq , µ, Nτ ) =Y (3 − kb )! Y 3!

{k,n,l} b=(x ,µ̂)

3!kb !

xnx !

(2amq )nxYw (`, µ)lGrassmann constraint:color neutral states at each sitenx +X∀x ∈ Nσ 3 × Nτkµ̂ = 3µ̂=±0̂,...±d̂

weight for baryon loop l (sign σ(`)):
w (`, µ) = Q

Wolfgang Unger, ETH Zürich

1x ∈`3!

σ(`)e 3Nτ rl aτ µ()

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Motivation for Continuous Time SC-LQCD

SC-LQCD at finite Temperature and Continuous Time:How to vary the temperature?
aT = 1/Nτ is discrete with Nτ evenaTc ' 1.5⇒

we cannot address the phase transition!

Solution: introduce an anisotropy γ in the Dirac couplings:
Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit):Nτ → ∞,

γ → ∞,γ 2 /Nτ

fixeda/at = γ 2 inspired by mean field, in contrast to weak coupling a/at = γ
same as in analytic studies: aτ = 0, aT = β −1 ∈ RWolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

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Motivation for Continuous Time SC-LQCD

SC-LQCD at finite Temperature and Continuous Time:How to vary the temperature?
aT = 1/Nτ is discrete with Nτ evenaTc ' 1.5⇒

we cannot address the phase transition!

Solution: introduce an anisotropy γ in the Dirac couplings:
Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit):Nτ → ∞,

γ → ∞,

γ 2 /Nτ

fixeda/at = γ 2 inspired by mean field, in contrast to weak coupling a/at = γ
same as in analytic studies: aτ = 0, aT = β −1 ∈ R
Several advantages of continuous Euclidean time approach:ambiguities arising from the functional dependence of observables on the anisotropy
parameter will be circumvented, only one parameter setting the temperature
no need to perform the continuum extrapolation Nτ → ∞baryons become static in the CT-limit, the sign problem is completely absent!

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

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Continuous Time Partition Function

Continuous Time Partition Function
Final goal: partition function in β = 1/aT :expansion in powers of γ −2 , i. e. in total numberof spatial hoppingsκ = 12Px ∈VM(nL (x ) + nT (x ))sum over all spatial hopping positions ∼ Nτ /2⇒expansion in inverse temperature
β = Nτ /γ 2 :Z(β) =X (β/2)κ Xκ∈2N

κ!

n (C)

vTTC∈ΓκΓκ is the set of equivalence classes ofconfigurations with κ spatial hoppings and C ∈ Γκ
only differ in their topology

successfully used so far: Worm algorithm in continuous time
here: attempt diagrammatic MC based on the diagrammatic expansion in κ
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

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Continuous Time Partition Function

SC-QCD Phase Diagram
Studied via CT-Worm algorithm (arXiv:1111.1434 [hep-lat])Comparison of phase diagram with Nτ = 4 data (M. Fromm, 2010):
CT-data compared to Nτ = 4 data for identificationaµ = γ 2 aτ µbehavior at low µ agrees well, location of TCP agrees within errors
no re-entrance is seen at small temperatures

Wolfgang Unger, ETH Zürich

1.4
1.2aT1
Continuous Time:
2nd order
tricritical point
1st order

0.8
0.6

Nτ=4:
2nd order
tricritical point
1st order

0.4
0.2

Nτ=2
1st order

000.1
()

0.2

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0.4

aµ
DMC for SC-LQCD

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Diagrammatic Monte Carlo

Motivation: Generalization of SC-QCD to 2 chiral flavors

At present, no 2-flavor formulation for SC-QCD suitable for the Worm algorithm
present:
already the mesonic sector has a severe (unphysical) sign problem in dimer
representation
new type of 2-flavor dimer give negative sign in mesonic loops already for U(2).
Aim: obtain phase diagram for 2-flavor SC-QCD, where pion exchange may play a
crucial role for nuclear transition
observation: box representation for 2 different flavors can be compose such thatcancellations appear
first step: Monte Carlo for insertion/removal of boxes rather than Wormalgorithm

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCDKyoto, 17.02.2012

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Diagrammatic Monte Carlo

Diagrammatic Monte Carlo (1)In principle, diagrammatic expansion amounts to hopping parameter expansion in κ on Ns3 with timeordering of L- and T-vertices:
label spatial dimers with timeindex i = 1, . . . κΓκ : enumeration of all valid
configurations consistent withlocation not feasible
diagrammatic MC algorithm
in progress

Wolfgang Unger, ETH Zürich

A'A'

A'B'

= A'B'B'B'= B'B'

B'B'

ACA

ADA

AEA

AEA

AFA= AFA

BCB

BDB

BEB

BEB

BFB

BFB

BCB

BDBBEBBEB

BFB

BFB

and more Graphs, generated by (A+B+B)(C+D+E+E+F+F+G+G)(A+B+B)

A2+A2-A2+A2-

B2+B2-B2+B2-

B2+B2-B2+B2-

and more Graphs, generated by(A2++B2++B2+)(A2-+B2-+B2-)(A2++B2++B2+)(A2-+B2-+B2-)()

DMC for SC-LQCD

Kyoto, 17.02.2012

10 / 13Diagrammatic Monte Carlo

Diagrammatic Monte Carlo (2)Combinatorics of diagrams
goverened by assignment ofinΓκemission/absorption events tovertices, or
quivalently: even/odd lengths oftime intervals
this is remnant of Grassmann
constraint
Observation: emission-absorption ordering induces orientation on boxes:
spatial dimers have orientationfrom emission to absorption site
solid lines can be consistentlyoriented (colors for illustration)
SC-QCD partition function can beconceived as composed of staticlines and oriented rectangles
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

11 / 13Diagrammatic Monte Carlo

Diagrammatic Monte Carlo (3)DMC flow chart:
insertion and removal updates for pairsof oppositely oriented spatial dimers

Wolfgang Unger, ETH ZürichBesides insertion/removal update:additional shift and static line
updates improve on ergodicity
for d > 1: swap update neededFirst measurement:
specific heat (can be obtainedfrom spatial dimers) for U(1) on
1+1-dimensional lattice:
0.16
CT-Worm
DMC
0.14
0.12

cV

0.1
0.08
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0.020()DMC for SC-LQCD

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1aT

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212 / 13Diagrammatic Monte CarloConclusions

Prospects:
CT partition function: new formulation for analytic treatment
hope: extend formulation for two flavors (incorporates pion exchange)
extension to SU(3) with finite baryon chem. potential straight forward
Drawbacks
generalization to higher dimensions turns out to be very difficult
not yet possible to study periodic boundary conditions
in contrast to worm: no 2-point function for free

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 17.02.2012

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