Diagrammatic Monte Carlo for Strong Coupling LQCD

Wolfgang Unger, ETH Zürich

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

Yukawa Institute, Kyoto
20.02.2012()

DMC for SC-LQCD

Kyoto, 20.02.2012

1 / 27

Motivation for SC-LQCD

1 Motivation for Strong Coupling LQCD in Continuous TimeContinous Time Limit and a/at = f (γ)
Continous Time Partition Function Z (β)

2 Diagrammatic Monte Carlo
General Idea and Motivation
Metropolis-Hastings
Comments on the Updating Rules

3 DiscussionLimitations
Generalizations

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

2 / 27

Motivation for SC-LQCD

The QCD (µ, T ) phase diagram:
T [MeV ]
C
LH
IC,
RH

Cross

〈
  〉≃0
FA
IR

over

CP
&
ent n
em tiofin nsicon Traer
De iralst ordCh〈  〉≠0
Hadronic Matter

VacuumQuark
Gluon
Plasma1

100

Early Universe

200

Nuclear
Matter

Baryonic
Crystal ?

Neutron Stars

0

Color
Superconductor?

1

 B [GeV ]rich phase structure conjectured
chiral and deconfinement transition
QGP at high temperatures
exotic matter at high density

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

3 / 27

Motivation for SC-LQCD

QCD has a severe sign problem for finite
chemical potential µ = 13 µB :fermions anti-commute:

The QCD (µ, T ) phase diagram:
T [MeV ]〈  〉≃0

200

Cross

100

γ5 (i p/ +m+µγ0 )γ5 = (i p/ +m+µγ0 )†

Quark
Gluon
Plasmaover

the fermion determinant det M(µ)
becomes complex!

〈  〉≠0e −Sf = det M(µ) = det M(−µ̄)

Hadronic Matter

Vacuum0

Nuclear
Matter1

 B [GeV ]because of the sign problem: very
little is known
only recently: agreement on
crossover temperature Tc at zero
baryon chemical potential

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

little hope that it can be
circumvented:- Taylor expansion,- imaginary µ with analytic
continuation,- reweighting method
are all limited to small µ/T . 1

Kyoto, 20.02.2012

3 / 27

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 =

P
µδν0
−µδν0
†
1
η
(x
)
e
χ̄(x
)U
(x
)χ(x
+
ν̂)
−
e
χ̄(x
+
ν̂)U
(x
)χ(x
)
+
am
ν
ν
q χ̄χ
ν
ν 2

P
†
β
2

Wolfgang Unger, ETH Zürich

− 2Nc

()

DMC for SC-LQCD

P

trUP + trUP + O(a )Kyoto, 20.02.2012

4 / 27

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 =

P
µδν0
−µδν0
†
1
η
(x
)
e
χ̄(x
)U
(x
)χ(x
+
ν̂)
−
e
χ̄(x
+
ν̂)U
(x
)χ(x
)
+
am
ν
ν
q χ̄χ
ν
ν 2
((
(

P
(
(
†
β
( + O(a2 )
(trU
trU
− 2Nc
(
P +
(
P
P
(
(((
take limit of infinite gauge coupling:cg → ∞, β = 2N→0
g2

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

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

4 / 27

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 =

P
µδν0
−µδν0
†
1
η
(x
)
e
χ̄(x
)U
(x
)χ(x
+
ν̂)
−
e
χ̄(x
+
ν̂)U
(x
)χ(x
)
+
am
ν
ν
q χ̄χ
ν
ν 2
((
(

P
(
(
†
β
( + O(a2 )
(trU
trU
− 2Nc
(
P +
(
P
P
(
(((
take limit of infinite gauge coupling:cg → ∞, β = 2N→0
g2

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, 20.02.2012

4 / 27

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, 20.02.2012

5 / 27

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
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, 20.02.2012

5 / 27

Motivation for SC-LQCD

SC-QCD Partition Function (1)Partition function for SU(Nc ):
Z =

Z Y

d χ̄dχe2amq M(x )

x

Y
µ

N

{

Nc
X
(Nc − kµ (x ))!

Nc !kµ (x )!

N+ κ ρ(x , y ) c B̄(x )B(x + µ̂) + (−ρ(y , x )) c B̄(x + µ̂)B(x )Wolfgang Unger, ETH Zürich

()k (x )(M(x )M(x + µ̂)) µkµ (x )=0

DMC for SC-LQCD

}Kyoto, 20.02.2012

6 / 27

Motivation for SC-LQCD

SC-QCD Partition Function (1)Partition function for SU(Nc ):
Z =

Z Y

d χ̄dχe2amq M(x )

x

Y
µ

{

Nc
X
(Nc − kµ (x ))!

Nc !kµ (x )!

k (x )(M(x )M(x + µ̂)) µkµ (x )=0NN

+ κ ρ(x , y ) c B̄(x )B(x + µ̂) + (−ρ(y , x )) c B̄(x + µ̂)B(x )}Only confined, colorless degrees of freedom remain after link integration (Nc = 3):
Mesons M(x ) = χ̄(x )χ(x ), represented by monomers
and dimers (non-oriented meson hoppings):

n(x ) ∈ {0, 1, . . . , 3}

Baryons B(x ) = 3!1 i1 ...i3 χi1 (x ) . . . χi3 (x )form self-avoiding oriented loops:kµ (x ) ∈ {0, 1, . . . , 3}

ρ(x , y ) = ηµ̂ (x ) exp(±aτ µ)δµ̂0̂B̄(x )B(y ) ∈ {0, 1}

Note: baryons transform under gauge trafo Ω ∈ U(3) as B(x ) → B(x ) det Ω⇒
not gauge invariant, only SU(3) contains baryons (κ = 1),
U(3) purely mesonic (κ = 0)

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

6 / 27

Motivation for SC-LQCD

SC-QCD Partition Function (2)Exact rewriting of SC-QCD partition function (no approximation!):X

Z(mq , µ, Nτ ) =Y (3 − kb )! Y 3!

{k,n,l} b=(x ,µ̂)

3!kb !

xnx !

(2amq )nx

Yw (`, µ)l

Grassmann 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

1

x ∈`

3!

σ(`)e 3Nτ rl aτ µin the following: restrict to chiral limit,
where monomer density hni = 0
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

7 / 27

Motivation for Continuous Time SC-LQCD

SC-LQCD at finite temperature
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:
LQCD =

P
µ

δ

γ µ0
ην (x )
2e µδµ0 χ̄(x )Uν (x )χ(x + µ̂) − e −µδµ0 χ̄(x + µ̂)Uµ† (x )χ(x )

Should we expect a/aτ = γ, as suggested at weak coupling?
c +1)(Nc +2)
No: meanfield predicts a/aτ = γ 2 , since γc2 = Nτ (d−1)(N
6(Nc +3)⇒

sensible, Nτ -independent definition of the temperature:2

aT ' NγτMoreover: SC-QCD partition function is a function of γ 2 :Z(mq , µ, γ, Nτ ) =PQ{k,n,l}b=(x ,µ)(3−kb )! 2kb δµ0
γ
3!kb !

Q

3!
(2amq )nx
x nx !

Qlw (`, µ)However: precise correspondence between a/aτ and γ 2 not known
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

8 / 27

Motivation for Continuous Time SC-LQCD

SC-LQCD at finite Temperature and Continuous Time:

Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit):Nτ → ∞,

γ → ∞,

γ 2 /Nτ

fixedsame as in analytic studies: aτ = 0, aT = β −1 ∈ R

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

9 / 27

Motivation for Continuous Time SC-LQCD

SC-LQCD at finite Temperature and Continuous Time:

Strategy for unambiguous answer: the continuous Euclidean time limit (CT-limit):Nτ → ∞,

γ → ∞,

γ 2 /Nτ

fixedsame 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 → ∞allows to estimate critical temperatures more precisely, with a faster algorithm(about 10 times faster than Nt = 16 at Tc )baryons become static in the CT-limit, the sign problem is completely absent!

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

9 / 27

Continuous Time Partition Function

Origin of Continuous Time / Diagrammatic Monte Carlo
Continuous time (CT) methods were introduced in Quantum Monte Carlo:
Suzuki-Trotter decomposition H = H1 + H2 allows mapping onto classical spin systemand sampling of diagrammatic expansion in continuous timeFirst proposed for the Heisenberg quantum antiferromagnet (Beard & Wiese, 1996):

1 persue a world-line formulation

(i. e. rewrite the partition function in termsof transition/decay probabilities)

Phys. Rev. Lett. 77 (1996) 5132

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

10 / 27

Continuous Time Partition Function

Origin of Continuous Time / Diagrammatic Monte Carlo
Continuous time (CT) methods were introduced in Quantum Monte Carlo:
Suzuki-Trotter decomposition H = H1 + H2 allows mapping onto classical spin systemand sampling of diagrammatic expansion in continuous timeFirst proposed for the Heisenberg quantum antiferromagnet (Beard & Wiese, 1996):

1 persue a world-line formulation

(i. e. rewrite the partition function in termsof transition/decay probabilities)

2 make time direction continuous, transitions

may occur at any time

Wolfgang Unger, ETH Zürich

Phys. Rev. Lett. 77 (1996) 5132()

DMC for SC-LQCD

Kyoto, 20.02.2012

10 / 27

Continuous Time Partition Function

Origin of Continuous Time / Diagrammatic Monte Carlo
Continuous time (CT) methods were introduced in Quantum Monte Carlo:
Suzuki-Trotter decomposition H = H1 + H2 allows mapping onto classical spin systemand sampling of diagrammatic expansion in continuous timeFirst proposed for the Heisenberg quantum antiferromagnet (Beard & Wiese, 1996):

1 persue a world-line formulation

(i. e. rewrite the partition function in termsof transition/decay probabilities)

2 make time direction continuous, transitions

may occur at any time

Phys. Rev. Lett. 77 (1996) 5132

Situation today:rich literature for quantum impurity systems (see review by Gull et al 2010)
based on diagrammatic Monte Carlo
now: continuous time method applied to gauge theories
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

10 / 27

Continuous Time Partition Function

Continuous Time Partition Function (1)In the following: restrict to chiral limit mq = 0, where monomers are absent.
Key observation: multiple spatial dimers are suppressed with powers of γ −2 :double/triple spatial dimers become resolved
into single dimers as aτ ' a/γ 2 → 0
single spat. dimers survive (constant density)
baryonic spat. hops are suppressed with γ −1
17

−4

−2

−6
Time

discrete timeno spatial 3-dimers
no spatial 2- and 3-dimers
continuous time

χ

16.5

16

15.5

15
0

0.05

Wolfgang Unger, ETH Zürich

0.1

0.15
1 / Nτ
()

0.2

0.25

DMC for SC-LQCD

Kyoto, 20.02.2012

11 / 27

Continuous Time Partition Function

Continuous Time Partition Function (1)The partition function can be decomposed into spatial and temporal parts, i. e.:Tin spatial dimers (meson hoppings) and two
types of temporal intervals in between: dashed(3-0-3-0-. . . ) and solid intervals (2-1-2-1-. . . )the weight of a configuration only depends on
the kind and number of vertices at which
spatial hoppings are attached to solid/dashedlines:“L”-vertices of weight vL = 1, and√“T”-vertices of weight vT = 2/ 3, wherea solid line emits a spatial dimer.
nL (x ) and nT (x ) denote the number ofT-vertices and L-vertices at spatial position x .

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

A

L
B

B
Time

A

B

T

L

B

Kyoto, 20.02.2012

12 / 27

Continuous Time Partition Function

Continuous Time Partition Function (1)Partition function in the limit of large γ 2 , Nτ with T = γ 2 /Nτ :Z(γ, Nτ ) ' γ

3V

X Y

e

3nB (x )µ/T

3{k,nB (x )} x ∈VB

=

X

e

3nB µ/T{k,nB (x )}

Y Y 1

γ

−2kb

vL
γ

vT
γ

(3 − kb0 )!
3!kb0 !

x ∈VM b=(x ,î)Y  nL (x ) Y
b0 =(x ,0̂)nT (x ),nB (x ) ∈ {−1, 0, 1}

x ∈VM

˙ B = Nσ 3 the mesonic/baryonic subvolumes and nB =with VM ∪V

PxnB (x ) the baryon number

typical 2-dimensional
configurations in discreteand continuous time at
the same temperature
no multiple dimers, only
static baryons incontinuous time

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

13 / 27

Continuous Time Partition Function

Continuous Time Partition Function (3)Final step: partition function in β = 1/aT :expansion in powers of γ −2 ,P
i. e. in total number of spatial hoppings: κ = 12 x ∈V (nL (x ) + nT (x ))Msum over all spatial hopping positions ∼ Nτ /2⇒ expansion in inverse temperature β ' Nτ /γ 2 :P (β/2)κ P

Z(β) =

κ!

n (C) β3µB(C)

vTT

e

,

nT =

P
x

C∈Γκ

κ∈2N

nT (x )

Γκ is the set of equivalence classes of configurations with κ spatial hoppingsC ∈ Γκ is a representative of all configurations obtained from shifts of spatial
hoppings which leave their time ordering unchanged
prefactor is weight for equivalence class of time ordered graph C ∈ Γκ
(weight does not depend on specific time coordinates):

Wolfgang Unger, ETH Zürich

1
2κ

Z β

Z β
dk1

Z β

0

k1

()

n (C)

dkκ vTT

dk2 . . .
kκ−1

DMC for SC-LQCD

=

1
κ!

 κ
β
2

n (C)

vTT

Kyoto, 20.02.2012

14 / 27

Continuous Time Partition Function

Continuous Time Partition Function (3)Final step: partition function in β = 1/aT :expansion in powers of γ −2 ,P
i. e. in total number of spatial hoppings: κ = 12 x ∈V (nL (x ) + nT (x ))Msum over all spatial hopping positions ∼ Nτ /2⇒ expansion in inverse temperature β ' Nτ /γ 2 :Z(β) =P (β/2)κ P

n (C) β3µB(C)

vTT

κ!

e

,

nT =

C∈Γκ

κ∈2N

P
x

nT (x )

Combinatorics governed by constraints concerning assignments of emission/absorption
event to vertices:
state vector χ characterizing time slice, χ(t) = (χ1 , . . . χV ), χx ∈ {|0i, . . . |Nc i}
π(x ) π(y )Jt) with (for Nc = 3)

spatialdimers acts at timeti on χ(t): D<x ,y > = (Jx
J+ =

0
vL

0
vT

0
vL

, J − = (J + )T and π(x ) = ±1 the parity of site x
0

peridicity implies: Mχ(0) = χ(β) ≡ χ(0),and the “Grassmann constraint” is:Wolfgang Unger, ETH Zürich

()i. e. M =

Qκ

D

t ≡1

i=1 <x ,y > i+−
J |Nc i = 0, J |0i = 0(forbidden)

DMC for SC-LQCD

Kyoto, 20.02.2012

14 / 27

Continuous Time Partition Function

Continuous Time Partition Function (3)Final step: partition function in β = 1/aT :expansion in powers of γ −2 ,P
i. e. in total number of spatial hoppings: κ = 12 x ∈V (nL (x ) + nT (x ))Msum over all spatial hopping positions ∼ Nτ /2⇒ expansion in inverse temperature β ' Nτ /γ 2 :Z(β) =P (β/2)κ P
κ!

n (C) β3µB(C)

vTT

e

C∈Γκ

κ∈2N

,

nT =

P
x

nT (x )

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, 20.02.2012

14 / 27

Continuous Time Partition Function

Short Comment on CT-Worm Algorithm
CT Worm algorithm is derived from directed
path Worm algorithm (Adams & Chandrasekharan, 2003):
key step: hopping times are uniformly
distributed and according to thestatistics of a Poisson process:the dashed/solid interval lengths are
exponentially distributed:

absorption events
ps =

1
1kt
pt =

k t
1k tp−t =k −tk −t k t

pt =

k t
k −t k taaremove spatial dimer

choose temporal direction

emission events
p−t =0.5pt =e−2d / 4pt =0.5ee
ps =1−e−2d  / 4P(∆β) = e −λ∆β ,∆β ∈ [0, β = 1/aT ]choose temporal direction

emit spatial dimer

incoming direction
outgoing direction
λ is the “decay constant” for spatial
dimer emissions:P
λ = dM (x )/4,dM (x ) = 2d − µ̂=±i |nB (x + µ̂)|solid/dashed lines

with dM (x ) the number of mesonic neighbors at x
idea of continuous time worm in SC-QCD: use probability distributionp(∆β) for spatial dimer emission after time interval ∆β
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

15 / 27

Continuous Time Partition Function

Generalization to arbitrary SU(Ncc)
Continuous time methods can be applied to any gauge group SU(Nc ):baryons only become static forSU(Nc ) with Nc ≥ 3
Nc = 2: diquark loops can havespatial hoppings, no T-vertices
present
large Nc feasible

U(1)
U(2)

U(3)

U(4)

aTcMF=d (Nc+1)(Nc+2) / (6 (Nc+3)).
Nc
1
2
3
4
1-(x)**41

Delta epsilon 4/(3Nc)

0.8

0.6

list of line types:
dashed, single, double, triple. . .

0.4

0.2

0
0

0.5

Wolfgang Unger, ETH Zürich

1

1.5
aT / aTcMF

2

2.5

()

3

for Nc > 3: generalization ofT-vertices change line-type by one
unit or its parity
DMC for SC-LQCD

Kyoto, 20.02.2012

16 / 27

Continuous Time Partition Function

Crosscheck with analytic results for U(1)Analytic solution of the U(1) system on 2x Nτ latties:
1
1
Nτ
tanh
acsch(γ 2 )(1 + γ −4 )− 2 + tanh221
χ(T ) = tanh (1/2T ) (1 + tanh (1/2T ))2χ(Nτ , γ 2 ) =



1.2

Nτ
acsch(γ 2 )
2



1.1

γ
0.25
0.5
1.0
2.0
4.0
8.0
16.0

1



Nτ
4
8
16
32
64
128
cont.
time

1
0.9
0.8

0.8

0.7
χ

0.6
0.6

0.5
0.4

0.4

0.3
0.2
0.2

0.1
0
0.001

0
1

10

100
Nτ

0.01

0.1
aT

1

→ Monte Carlo results on chiral susceptibility agrees well with analytic results

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

17 / 27

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 µ

behavior 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
1.2

aT

1
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

0
0

0.1
()

0.2

0.3

0.4

DMC for SC-LQCD
aµ

0.5

0.6

0.7

0.8
Kyoto, 20.02.2012

18 / 27

Diagrammatic Monte Carlo

What is Diagrammatic Monte Carlo?
Importance sampling of diagrams described in terms of a perturbative series:
each term in the diagrammatic expansion of the partition function is represented
by a world line configuration
perturbative series may not converge, but at any finite volume and temperature,
only a finite number of orders contribute
CT-Worm is one of the DMC techniqes, others are the loop cluster algorithm(Evertz et al.), stochastic series expansion (Sandvik)

Here: make use of insertions/shifts of world line decorations:
Problem: vanishing acceptance rate:p = (tδτ )2 → 0 ⇒ pins = min(1, p) → 0
Solution:over all possible insertions within interval Λ:R Λ integrateRΛ
2 2
p = 0 dτ1 τ dτ2 t 2 = Λ 2t 6= 0 ⇒ pins = min(1, p) 6= 0

Wolfgang Unger, ETH Zürich1()

DMC for SC-LQCD

t1

t2

Kyoto, 20.02.2012

19 / 27

Diagrammatic Monte Carlo

Motivation: Generalization of SC-QCD to 2 chiral flavors!
Aim: obtain phase diagram for 2-flavor SC-QCD,
where pion exchange may play a crucial role for nuclear transition
but: at present, no 2-flavor formulation for
staggered SC-QCD suitable for MC present
already the mesonic sector has a severe(unphysical) sign problem in dimer
representationa = ūux ūuyb = d̄dx d̄dyc = ūdx d̄uy
d = d̄ux ūdy
α = ūdx d̄ux ūuy d̄dy
β = ūux d̄dx ūdy d̄uy

2 new types of flavored dimers give negative
sign in mesonic loops already for U(2).
Observation in continuous time formulation:
flux representation for 2 different flavors canbe composed such that cancellations appear
first step in this direction:
Monte Carlo for insertion/removal of rectangles
rather than worm algorithm
Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD(Michael Fromm,
PhD Thesis 2010)

Kyoto, 20.02.2012

20 / 27

Diagrammatic Monte Carlo

B'B'

B'B'

HH

HH

B'B'

The Hopping parameter Expansion of CT-SC-QCD
t1 t2 t3 t4Diagrammatic expansion for SC-QCD amounts
to hopping parameter expansion in β/2 on
Ns3 with time ordering of L- and T-vertices:HH

t1 t2 t3 t4

ACA

ADA

t1 t2 t3 t4
AFA

AEA

AEA

AGA

BCB

BDB

AGA

label spatial dimers with time index
i = 1, . . . κ and attach at its ends
vertices with weight vL , vT

BEB

BEB

BFB

BCB

BDB

BFB

BEB

BEB

BFB

enumeration of Γκ , i. e. all valid
configurations consistent with parity(e. g. even/odd interval lengths, insketch: even intervals between L- andT-vertices highlighted) not feasible

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

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

t1 t2 t3 t4

t1 t2 t3 t4

t1 t2 t3 t4

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-)

Kyoto, 20.02.2012

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Diagrammatic Monte Carlo

The Hopping parameter Expansion of CT-SC-QCD
Diagrammatic expansion for SC-QCD amounts
to hopping parameter expansion in β/2 on
Ns3 with time ordering of L- and T-vertices:

Some time-ordered diagrams of orderκ = 4:

label spatial dimers with time index
i = 1, . . . κ and attach at its ends
vertices with weight vL , vT
enumeration of Γκ , i. e. all valid
configurations consistent with parity(e. g. even/odd interval lengths, insketch: even intervals between L- andT-vertices highlighted) not feasible
Data structure used in DMC for sampling Γκ :state vector χ(x , t = 0) ∈ {0, . . . Nc }set of time ordered spatial dimers{hx1V1 , y1W1 i, hx2V2 , y2W2 i, . . . hxκVκ , yκWκ i}with Vi , Wi ∈ {L, T }Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

t1 t2 t3 t4

t1 t2 t3 t4

A'A'

A'B'

A'B'

B'B'

B'B'

B'B'

HH

HH

t1 t2 t3 t4HH

t1 t2 t3 t4

ACA

ADA

t1 t2 t3 t4

t1 t2 t3 t4
AFA

AEA

AEA

AGA

BCB

BDB

AGA

BEB

BEB

BFB

BCB

BDB

BFB

BEB

BEB

BFB

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

t1 t2 t3 t4

t1 t2 t3 t4

t1 t2 t3 t4

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-)

Kyoto, 20.02.2012

21 / 27

Diagrammatic Monte Carlo

Flux Representation
Combinatorics of diagrams
goverened by assignment ofinΓκ

emission/absorption events tovertices, or
equivalently: even/odd lengths oftime intervals
Observation: emission-absorption ordering induces orientation on rectangles:
spatial dimers have orientationfrom emission to absorption site
solid lines can be consistently
oriented (colors for illustration)
2-dim. SC-QCD partition function
can be conceived as composed ofstatic lines and oriented
rectangles
Wolfgang Unger, ETH Zürich

()re-expressing a 2-dim. configuration in termsof oriented rectangles and oriented static lines

DMC for SC-LQCD

Kyoto, 20.02.2012

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Diagrammatic Monte Carlo

Algorithmic Details (1)Apply Metropolis-Hastings Alogrithm to SC-QED:
1
configuration with κ spatial dimers has weight w (C (κ) ) = κ!
eveninsertion: pins
removal: prem



(κ)
(κ+2)
C1 → C2





(κ)
(κ−2)
C1 → C2= min 1,= min 1,

(κ+2)

w(C2

(κ)

(κ−2)

w(C2

(κ)

(κ+2)

) Arem (C2

w(C1 )w(C1 )β2κ, κ always

(κ)

→C1 )



(κ)

(κ+2)

(κ−2)

→C1 )

(κ)

(κ−2)

Ains (C1 →C2) Ains (C2)(κ)

Arem (C1 →C2)

proposal probabilities Ains (C1 → C2 ), Arem (C2 → C1 ) not indepenent of C1 ,C2 , need to be calculated for each update in an efficient way
proposal probabilities depend on the way how insertion, removal is proposedseveral options how to implement rectangle updates, here:
split into two parts, insertion/removal of pair (easy part, local), moving themapart (shift update difficult part, non-local)

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

23 / 27

Diagrammatic Monte Carlo

Algorithmic Details (2)Apply Metropolis-Hastings Alogrithm to
SC-QED:
insertion update: κ 7→ κ + 2:

DMC flow chart:
insertion and removal updates for pairsof oppositely oriented spatial dimers- choose a specific insertion for C1- determine number npins of possible
insertions (cheap: at each bond and ti insertion ispossible if parity agrees)- determine number of distinct insertions
ncins which would gives the same new
config. C2⇒ Ains (C1 → C2 ) = ncins /npins- determine number nprem of possible
removals in C2 (cheap: computed along with possible
insertions)- determine number of distinct removals
ncrem which would gives the same old
config. C1⇒

Arem (C2 → C1 ) = ncrem /nprem

removal update: κ 7→ κ − 2:
similar to the above

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

24 / 27

Diagrammatic Monte Carlo

Algorithmic Details (2)Apply Metropolis-Hastings Alogrithm to
SC-QED:
static line update: change
k0 (x ) 7→ k0 (x ) ± 1, if consitent
with vertices on site x

DMC flow chart:
insertion and removal updates for pairsof oppositely oriented spatial dimers

shift update: needed for ergodicity,and in simplified insertion/removal(only pairs): essential!
- Ashift trivial, but shifts may involve large
number of flow inversions- need to construct clusters which spatial
dimers are affected- demanding in eihter memory or CPU time

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

24 / 27

Diagrammatic Monte Carlo

Preliminary Results
So far: only spatial dimer measurements in 1 spatial dimension possible
First measurement: comparing CT-Worm with DMC
specific heat (can be obtained from spatial dimers) for U(1) (left) and U(3) (right)
on 1+1-dimensional lattice:
0.16

0.2
CT-Worm
DMC

CT-Worm
DMC

0.18

0.14

0.16

0.12

0.14
cV

cV

0.1
0.12

0.08
0.1
0.06

0.08

0.04

0.06

0.02

0.04
0

0.5

1
aT

1.5

2

0

0.5

1
aT

1.5

2

observation: DMC performs better at small volumes, but not so well on large
volumes (improvements on shift update?)also DMC becomes demanding in memory for large volumes (for shift updates)

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

25 / 27

Diagrammatic Monte Carlo

Generalization to higher dimensions:so far: every graph composed of only planar rectangles
in spatial dimensions d > 1, algorithm is not ergodic

x2

Flip update:does not change order in β
but might add/remove
T-vertices → another
Metropolis update . . .

x4

x2
t1

t2

x3

t1 t 2
t3

t4

x1

t3
t4

x4

t2
x2

sufficient to produce any
higher-dimensional closed
loops occuring in hopping
parameter expansion

Wolfgang Unger, ETH Zürich

x1

x3

()

x1

t2

x3
x4

t1

DMC for SC-LQCD

x2

x3

t3

t1

t3

t4

x1

x4

t4

Kyoto, 20.02.2012

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Diagrammatic Monte Carlo

Conclusions

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
(but: cluster-size as a measure for susceptibility?)might turn out that worm algorithm is more efficient at large volumes

Wolfgang Unger, ETH Zürich

()DMC for SC-LQCD

Kyoto, 20.02.2012

27 / 27

Diagrammatic Monte Carlo

Backup Slides

Why Study Strong Coupling QCD on the Lattice?
Two possible scenarios for the relation between
SC-LQCD (back) and the (L)QCD phase diagram for four flavors (front):

T /m BT /m B
=1, N  =2

=1, N  =2
=1, N =4

=1, N  =4

2nd

1st order

1st order

/m B

/m B





Wolfgang Unger, ETH Zürich

2nd
order

order

()

DMC for SC-LQCD

Kyoto, 20.02.2012

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