density of states approach with ffa for field theories...

Density of states approach with FFA for fieldtheories with a complex action problem

Mario GiulianiChristof Gattringer, Pascal Törek

Karl Franzens Universität Graz

Mario Giuliani (Universität Graz) Graz, 11th October 2015 1 / 37

Table of contents

1 Sign Problem: a brief review

2 Density of States: general introduction

3 Density of States for SU(3)-spin system

4 Density of States for U(1) with a topological term

5 Conclusions


Sign Problem: a brief review

Quantum field theory is based on two quantities:

Z =

∫D[C ]eS[C ] 〈O〉 =

1Z

∫D[C ]O(C )eS[C ]

Probabilistic interpretation: p(C) = eS[C]

* Monte-Carlo: to produce a set of M configurations {ci} with probability p(C)

* 〈O〉 ≈ O = 1M

M∑i=1O(ci )

Sign problem: Chemical potential µ 6= 0 or a topological term:

* S acquires an imaginary part

* Factor eS[C ] cannot be used as a probability weight


Sign Problem: a brief review

Different approaches to solve this problem:

– Reweighting

– Expansion methods

– Stochastic differential equations

– Mapping to dual variables =⇒ Complete Solution

– Et cetera...

Density of states approach:

– Density of states (DoS). An approach developed by K. Langfeld, B. Lucini, A.Rago. LLR Local Linear Relaxation [1–4]

– Different method FFA Functional Fit Approach [5, 6]


References:

Kurt Langfeld et al. “An efficient algorithm for numerical computations of continuous densities ofstates”. In: (2015). arXiv: 1509.08391 [hep-lat].

Kurt Langfeld e Biagio Lucini. “Density of states approach to dense quantum systems”. In: Phys.Rev. D90.9 (2014), p. 094502. DOI: 10.1103/PhysRevD.90.094502. arXiv: 1404.7187 [hep-lat].

Kurt Langfeld e Jan M. Pawlowski. “Two-color QCD with heavy quarks at finite densities”. In:Phys. Rev. D88.7 (2013), p. 071502. DOI: 10.1103/PhysRevD.88.071502. arXiv: 1307.0455[hep-lat].

Kurt Langfeld, Biagio Lucini e Antonio Rago. “The density of states in gauge theories”. In: Phys.Rev. Lett. 109 (2012), p. 111601. DOI: 10.1103/PhysRevLett.109.111601. arXiv: 1204.3243[hep-lat].

Christof Gattringer e Pascal Törek. “Density of states method for the Z3 spin model”. In: Phys.Lett. B747 (2015), pp. 545–550. DOI: 10.1016/j.physletb.2015.06.017. arXiv: 1503.04947[hep-lat].

Ydalia Delgado Mercado, Pascal Törek e Christof Gattringer. “The Z3 model with the density ofstates method”. In: PoS LATTICE2014 (2015), p. 203. arXiv: 1410.1645 [hep-lat].


http://arxiv.org/abs/1509.08391

http://dx.doi.org/10.1103/PhysRevD.90.094502


http://dx.doi.org/10.1103/PhysRevD.88.071502



http://dx.doi.org/10.1103/PhysRevLett.109.111601



http://dx.doi.org/10.1016/j.physletb.2015.06.017




Density of States Method

Density of states

Z =

∫D[ψ]e−S[ψ] 〈O〉 =

1Z

∫D[ψ]O[ψ]e−S[ψ]

In the density of states approach we divide the action into two parts:

S [ψ] = Sρ[ψ] + cX [ψ]

* Sρ[ψ] and X [ψ] are real functionals of the fields ψ

* Sρ[ψ] is the part of the action that we include in the weighted density ρ

* Here c is purely imaginary: c = iξ


DoS: toy models

Examples discussed here:

Effective Polyakov loop model:

Sρ[P] = Re(S [P])

X [P] = Im(S [P])/ξ

ξ = 2k sinh(µ)

2D U(1) lattice gauge theory with a topological term

Sρ[U] = SG [U] (gauge action)

X [U] = Q[U] (discretization of the topological charge)

ξ = θ



The weighted density is defined as:

ρ(x) =

∫D[ψ]e−Sρ[ψ]δ(X [ψ]− x)

Using ρ(x) we can write Z and 〈O〉s:

Z =

∫ xmax

xmin

dx ρ(x) e−iξx 〈O〉 =1Z

∫ xmax

xmin

dx ρ(x) e−iξxO[x ]

Usually there is a symmetry ψ −→ ψ′ such that:

Sρ[ψ′] = Sρ[ψ], X [ψ′] = −X [ψ],

∫D[ψ′] =

∫D[ψ]

The symmetry implies that Z is real



The symmetry also implies that ρ(x) is an even function ρ(x) = ρ(−x):

ρ(−x) =

∫D[ψ]e−Sρ[ψ]δ(X [ψ] + x) =

∫D[ψ′]e−Sρ[ψ

′]δ(−X [ψ′] + x) = ρ(x)

The partition function simplifies to:

Z = 2∫ xmax

0dx ρ(x) cos(ξx)

... and for the observables O:

〈O〉 = 2∫ xmax

0dx ρ(x) {cos(ξx)Oeven[x ]− i sin(ξx)Oodd [x ]}

Where Oeven = O[x]+O[−x]2 and Oodd = O[x]−O[−x]

2


First example

SU(3) spin system with chemical potential

Natural evolution of the Z3 model: from discrete d.o.f to continuous

See Pascal Törek Master thesis for the discussion of the Z3 model


SU(3)-spin model

SU(3) spin model is a 3D effective theory for heavy dense QCD

Relevant d.o.f. is the Polyakov loop P(n) ∈ SU(3) (static quark source at n)

The model has a real and positive dual representation ⇒ reference data

We have an action:

S [P] = −τ∑n

3∑ν=1

[TrP(n) TrP(n+ν)†+c .c .

]−k

∑n

[eµ TrP(n) +e−µ TrP(n)†]

The action depends only on the trace ⇒ simple parametrization

P(n) =

e iθ1(n) 0 00 e iθ2(n) 00 0 e−i(θ1(n)+θ2(n))


Imaginary part of the action

Term with chemical potential can be decomposed into real and imaginary parts:

k∑n

[eµ TrP(n) + e−µ TrP(n)

]=

2k cosh(µ)∑n

Re[TrP(n)

]+ i2k sinh(µ)

∑n

Im[TrP(n)

]Definition:

X [P] =∑n

Im[TrP(n)

]=∑n

[sin(θ1(n)) + sin(θ2(n))− sin(θ1(n) + θ2(n))

]

X [P] ∈ [− 3√3

2 V ; 3√3

2 V ]≡

[−xmax , xmax ]

θ1

θ2


Reduced Haar Measure

The partition function is:

Z =

∫D[P]e−S[P] , D[P] =

∏n

dP(n)

dP(n) = sin2(θ1(n)−θ2(n)

2

)sin2(θ1(n)+2θ2(n)

2

)sin2(2θ1(n)+θ2(n)

2

)dθ1(n)dθ2(n)

θ1

θ2


Simplification of the partition function

Charge conjugation symmetry:

θi (n)∗−→ −θi (n) Sρ[P]

∗−→ Sρ[P]

dP(n)∗−→ dP(n) X [P]

∗−→ −X [P]

Z =∫D[P]e−Sρ[P]−iξX [P] ∗

=⇒∫D[P]e−Sρ[P]+iξX [P]

=⇒ Z =∫D[P]e−Sρ[P] cos(ξX [P])


Definition of density of states

We define the weighted density of states:

ρ(x) =

∫D[P] e−Sρ[P] δ(x − X [P]) x ∈ [−xmax , xmax ]

Symmetry P(n)→ P(n)∗ implies ρ(−x) = ρ(x)

This simplifies the partition function:

Z =

xmax∫−xmax

dx ρ(x) cos(2k sinh(µ)x) = 2

xmax∫0

dx ρ(x) cos(2k sinh(µ)x)

〈O[X]〉 =

2Z

xmax∫0

dx ρ(x)[OE [x ] cos(2k sinh(µ)x) + i OO [x ] sin(2k sinh(µ)x)

]


Parametrization of the density ρ(x)

Ansatz for the density: ρ(x) = e−L(x), normalization ρ(0) = 1⇒ L(0) = 0

We divide the interval [0, xmax ] into N intervals n = 0, 1, . . . ,N − 1.

L(x) is continuous and linear on each of the intervals, with a slope kn:

0

1

2

3

4

5

∆0 ∆1∆2 ∆3 ∆4∆5 ∆6 ∆N−2 ∆N−1

k0

k1k2

k3

k4 k5k6

kN−2

kN−1

xmax

L(x)


Determination of the slopes kn

How do we find the slopes kn?Restricted expectation values which depend on a parameter λ ∈ R:

〈〈O〉〉n(λ) =1

Zn(λ)

∫D[P] e−Sρ[P]+λX [P]O

[X [P]

]θn[X [P]

]Zn(λ) =

∫D[P] e−Sρ[P]+λX [P] θn

[X [P]

]θn[x]

=

{1 for x ∈ [xn, xn + 1]

0 otherwise

Update with a restricted Monte Carlo

Vary the parameter λ to fully explore the density


Functional Fit Approach FFA

Expressed in terms of the density:

Zn(λ) =

xmax∫−xmax

dx ρ(x) eλx θn[x]

=

xn+1∫xn

dx ρ(x) eλx = c

xn+1∫xn

dx e(−kn+λ)x

= ce(λx−kn)xn+1 − e(λx−kn)xn

λ− kn

For computing the slopes we use as observable X [P]:

〈〈X [P]〉〉n(λ) =1

Zn(λ)

xn+1∫xn

dx ρ(x) eλx x =∂

∂λln[Zn(λ)

]


Functional Fit Approach FFA

Explicit expression for restricted expectation values:

1∆

[〈〈X [P]〉〉n(λ)− n∆

]− 1

2= h((λ− kn)∆n

)h(r) =

11− e−r

− 1r− 1

2

Strategy to find kn:

1 Evaluate 〈〈X [P]〉〉n(λ) for different values of λ

2 Fit these Monte Carlo data h((λ− kn)∆n)

3 kn are obtained from simple one parameter fits

All Monte Carlo data are used in the process


Properties of the h(r) function

The universal function h(r) behaves like that:

-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

-15 -12 -9 -6 -3 0 3 6 9 12 15

h((λ−

k n)∆

n)

λ

kn = 3

∆n = 1∆n = 2∆n = 4


Overview on the Monte Carlo

λ allows to explore the density:Example: 83, N = 120, n = 70

λ = −1, 30000 steps λ = 10, 30000 steps

X [θ1, θ2]

Haar(θ1, θ2)eX [θ1,θ2]


Fit of slopes ⇒ density ρ(l)

Example: 83, k = 0.005, µ = 0.0

λ4− 2− 0 2 4

21)

n

λ(n

>>

F[P

]<

<∆1

0.5−

0.4−

0.3−

0.2−

0.1−

0

0.1

0.2

0.3

0.4

0.5

0

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

-12000

-10000

-8000

-6000

-4000

-2000

0

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400ln(ρ(x))

x

τ=0.025τ=0.050τ=0.075τ=0.100τ=0.125τ=0.150

kn L(x) ρ(x) = e−L(x)

τ = 0.075


Observables

1 Particle number density n:

n =1V

12k

∂

∂ sinh(µ)lnZ =

1V

2Z

xmax∫0

dx ρ(x) sin(2k sinh(µ)x) x

2 ... and the corresponding susceptibility χn:

χn =12k

∂

∂ sinh(µ)n

=1V

{2Z

xmax∫0

dx ρ(x) cos(2k sinh(µ)x) x2 +( 2Z

xmax∫0

dx ρ(x) sin(2k sinh(µ)x)2}


Comparison with dual approach

Particle number density n

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3 3.5

n

µβ

Lattice:83, τ=0.066, k=0.005Density of StatesDual Formulation

For dual results see Y.D.Mercado, C. Gattringer Nuclear Physics B862 (2012) 737-750We find good agreement for chemical potential up to µβ ≈ 3


Comparison with dual approach

Susceptibility χn

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 3.5

χn

µβ

Lattice:83,τ=0.066,k=0.005

Density of StatesDual formulation

We find a good agreement for chemical potential up to µβ ≈ 2.5

We need to reduce the size of the intervals ∆n to explore larger µβ


Second example

2D U(1) LGT with a topological term

One of the simplest system with a topological term

We have analytical results for this system


U(1) LGT in 2D with a topological term

The partition function is:

Z =

∫D[U]e−SG [U]−iθQ[U]

Where:

SG = −β2

∑n

[Up(n) + U∗p (n)], Q[U] =1i4π

∑n

[Up(n)− U∗p (n)]

The DoS is defined as:

ρ(q) =

∫D[U]e−SG ρ(q − Q[U])

The topological charge is bounded:Q[U] = 1

i4π

∑n[Up(n)− U∗p (n)] = 1

i4π

∑n 2 i ImUp(n) = 1

2π

∑n ImUp(n)

Q[U] ∈ [−Qmax ,Qmax ], Qmax = L1L22π


U(1): Partition function and observables

Symmetry: charge conjugation Uµ(n) −→ U ′µ(n) = U∗µ(n)

ρ(−q) = ρ(q)

Z = 2∫ Qmax

0dr ρ(q) cos(θq)

〈O〉 =2Z

Qmax∫0

dq ρ(q)[Oeven[q] cos(θq) + i Oodd[q] sin(θq)

]

Simpler problem:

* small range of vacuum angle θ ∈ [0; 2π]

* oscillation frequency linear in θ (no exponential growth)...

* ...but more challenging structure of the density ρ


Parametrization of the density ρ(q)

Ansatz for the density: ρ(q) = e−L(q), normalization ρ(0) = 1⇒ L(0) = 0

We divide the interval [0,Qmax ] into N intervals n = 0, 1, . . . ,N − 1

L(q) is continuous and linear on each of the intervals, with a slope kn:

0

1

2

3

4

5

∆0 ∆1∆2 ∆3 ∆4∆5 ∆6 ∆N−2 ∆N−1

k0

k1k2

k3

k4 k5k6

kN−2

kN−1

Qmax

L(q)


Semi-analytical result

We have a theoretical result for the partition function:

Z (θ) =

q=∞∑q=−∞

I|q|

2

√(β

2

)2

−(θ

4π

)2

(β2 −

θ4π

)2(β2

)2−(θ4π

)2

q2NsNt

We can obtain the density ρ(q) from the partition function:

ρ(q) = 2∫ ∞0

Z (θ) cos(θq)

We can study the continuum limit sending β →∞ and L1 = L2 →∞ with βL1L2

constant


Theoretical DoS

The DoS that we want to find looks like:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

ρ(q

)

q

20x20, β = 4.032x32, β = 10.24

The peaks can have a big non linear component, so it can be difficult to obtainthe right shapeMario Giuliani (Universität Graz) Graz, 11th October 2015 31 / 37

A different parametrization for ρ(q)

We tried different parametrizations for this system, for example:

We divide the interval [0,Qmax ] into N intervals n = 0, 1, . . . ,N − 1

ρ(q) = e−L(q), L(q) is constant on each of the intervals, with a value hn

The simulation is done on two neighbouring intervals

0

1

2

3

4

5

∆0 ∆1 ∆2 ∆3 ∆N−2 ∆N−1

h1 h2

Qmax

h3 h4 hN−2 hN−1

h2

h3h4

hN−1

L(q)

We don’t find better results using it


DoS from our simulation

We obtain a reasonable approximation for 202:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

ρ(q

)

q

Lattice:202 β : 4.0 βNsNt

= 0.01

Analytical resultDensity of States



It’s not so simple to get the same for 322:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

ρ(q

)

q


= 0.01




We can try to improve the statistic, for example of a factor 3:

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4

ρ(q

)

q


= 0.01


We need to use considerable smaller intervalsMario Giuliani (Universität Graz) Graz, 11th October 2015 35 / 37

Observables

Observable: expectation value of the topological charge:

〈Q〉 = − 1L1L2

∂

∂θlnZ =

1L1L2

2Z

Qmax∫0

dq ρ(q) q sin(θq)

Analytical result for the topological charge:

-0.02

-0.01

0

0.01

0.02

0 2 4 6 8 10 12

〈Q〉

θ


= 0.01

Analytical result

It’s really difficult to get this sawtooth shapeMario Giuliani (Universität Graz) Graz, 11th October 2015 36 / 37

Conclusions

DoS is a general approach

Crucial: accuracy of ρ

FFA uses restricted Monte Carlo and probes the density with an additionalBoltzmann weight

Tested in SU(3) Spin model: with a modest simulation agreement until µ ≈ 2.5

Interesting applications: Theories with a topological term (Work in progress....)


density of states approach with ffa for field theories...

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