Density of states approach for field theories with a ... · Densityofstatesapproachforﬁeldtheorieswith acomplexactionproblem Mario Giuliani Christof Gattringer Karl Franzens Universität

Density of states approach for field theories witha complex action problem

Mario GiulianiChristof Gattringer

Karl Franzens Universität Graz

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 1 / 21

Table of contents

1 Density of States: general introduction

2 Example system: SU(3) gauge theory with static color sources

3 Two strategies to compute the density: LLR and FFA

4 Results and conclusions


Some recent DoS papers

Examples of systems treated with modern DoS methods:

LLR method:

U(1) LGT

SU(2) with heavy quarks at finite densities

Relativistic Bose Gas

Heavy Dense QCD

Phys.Rev.Lett. 109 (2012)

Phys.Rev. D88 (2013)

PoS LATTICE2015 (2016)

Eur.Phys.J. C76 (2016)

FFA method:

Z3 spin system at finite µ

SU(3) spin system at finite µ

2D U(1) LGT with θ term

SU(3) LGT with color sources

Phys.Lett. B747 (2015)

Nucl.Phys. B913 (2016)

POS LATTICE2015 (2016)

arXiv:1703.03614


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 real and imaginaryparts:

S [ψ] = Sρ[ψ]− iξX[ψ]

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

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

* ξ is a real valued control parameter, e.g., ξ ∝ sinh(µNT )


Density of States Method

The weighted density is defined as:

ρ(x) =

∫D[ψ]e−Sρ[ψ]δ(X[ψ]− x)

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]

Z is real and ρ(x) is an even function

Key challenge: high precision for ρ(x)


SU(3) LGT with static color sources

SU(3) spin model is a 4D effective theory for heavy dense QCD

The static color sources are represented by Polyakov loops

We have the following action:

S [U] = −SWilson[U]− η[eµNT

∑~n

P(~n) + e−µNT

∑~n

P(~n)∗]

Where the Polyakov loops are:

P(~n) =13Tr

NT−1∏n4=0

U4(~n, n4)


SU(3) static color sources

Decompose the action in real and imaginary parts:

S [U] = Sρ[U]− i2η sinh(µNT )X[U] = Sρ − iξX

where:

Sρ[U] = SWilson[U]− 2η cosh(µNT )∑~n

Re[P(~n)]

X[U] =∑~n

Im[P(~n)]

ξ = 2η sinh(µNT )


Definition of the Density of States

We define the weighted DoS

ρ(x) =

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

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

This simplifies the partition function:

Z =

xmax∫−xmax

dx ρ(x)ei2η sinh(µNT )x = 2

xmax∫0

dx ρ(x) cos(2η sinh(µNT )x)


Parametrization of the density ρ(x)

Ansatz for the density: ρ(x) = e−L(x)

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

L(x) is continuous and linear on each In, 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:

〈〈X〉〉n(λ) =1

Zn(λ)

∫D[U] e−Sρ[U]+λX[U] X[U] θn

[X[U]

]Zn(λ) =

∫D[U] e−Sρ[U]+λX[U] θn

[X[U]

]

θn[x]

=

{1 for x ∈ In

0 otherwise

Update with a restricted conventional Monte Carlo

Vary the parameter λ to fully explore the density


Functional Fit Approach FFA

Closed expression for Zn(λ) 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(λ−kn)xn+1 − e(λ−kn)xn

λ− kn

So for the observable X[U]:

〈〈X〉〉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:

Yn(λ) ≡ 1∆n

[〈〈X〉〉n(λ)− xn

]− 1

2= h((λ− kn)∆n

)h(r) =

11− e−r

− 1r− 1

2Strategy to find kn:

1 Evaluate 〈〈X〉〉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

Alternative approach: directly find the zero of this function using an iterationalgorithm (LLR algorithm)


Properties of the h((λ− kn)∆n) function

-0.40

-0.20

0.00

0.20

0.40

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

kn=3

h((λ-kn)Δn)

λ

Δ=0.25Δ=1.00Δ=4.00

FFA: fitting Monte Carlo data with h((λ− kn)∆n)

LLR: finding the zero of h((λ− kn)∆n) with an iteration algorithmMario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 13 / 21

Fit of slopes ⇒ density ρ(x)

Example: 83 × 4, η = 0.04, µ = 0.150

λ15− 10− 5− 0 5 10 15

0.5−

0.4−

0.3−

0.2−

0.1−

0

0.1

0.2

0.3

0.4

0.5

)λ(nY

0

10

20

30

40

50

60

70

80

90

100

110

120

130

-8000

-6000

-4000

-2000

0

0 50 100 150 200 250 300 350 400 450

ln(ρ(x))

x

β=5.40β=5.50β=5.60β=5.70

kn L(x) ρ(x) = e−L(x)

β = 5.40


Observables

1 Imaginary part of the Polyakov loop 〈Im[P]〉:

〈Im[P]〉 =1V

12η

∂

∂ sinh(µNT )lnZ

2 ... and the corresponding susceptibility χIm[P]:

χIm[P] =12η

∂

∂ sinh(µNT )Im[P]

3 Related to particle number and its susceptibility


A simple check with the susceptibility at µ = 0

For µ = 0 : 〈Im(P)〉 = 0

While χIm(P) 6= 0

Conventional simulation to check consistency our DoS results for χIm(P)

0.02

0.04

0.06

0.08

0.10

0.12

0.14

1.0 2.0 3.0 4.0 5.0 6.0 7.0

χIm[P]

β

Density of States

conventional simulation

Excellent agreement


Results for 〈Im[P]〉 at µ 6= 0

Lattice 83 × 4, and η = 0.04:

0.000

0.005

0.010

0.015

0.020

5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80

⟨Im[P]⟩

β

µ=0.000

µ=0.075

µ=0.150

µ=0.250

µ=0.350


Results for χIm[P] at µ 6= 0

Lattice 83 × 4, and η = 0.04:

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

5.40 5.45 5.50 5.55 5.60 5.65 5.70 5.75 5.80

χIm[P]

β

µ=0.000

µ=0.075

µ=0.150

µ=0.250

µ=0.350


Phase transition affects shape of the density

-30

-25

-20

-15

-10

-5

0

0.00 10.00 20.00 30.00 40.00 50.00

ln(ρ(x))

x

β=5.400β=5.500β=5.600β=5.625β=5.650β=5.700β=5.800

Across the phase transition there is a strong change of the shape of the density


Position of the critical temperature

At µ = 0 there is a transition with some critical TC (0)

The µ− dependence of the pseudo-critical temperature can be parameterized as:

TC (3µ)

TC (µ = 0)= 1− κ

(3µ

TC (3µ)

)2

We can fit the position of the peaks of the cubic fits of the susceptibility:

0.84

0.88

0.92

0.96

1.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

T/TC(0)

3µ/TC(µ)

κ = 0.012(3)

κ = 0.0149(21) WB collaborationκ = 0.020(4) Cea et all.


Conclusions

DoS is a general approach

Crucial: accuracy of ρ to integrate over the rapid oscillating functions

Density ρ is parameterized by the slopes of its exponent

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

LLR: iteratively find the zero of the restricted MC dataFFA: fit all data points produced at different λ

Tested for a theory more similar to QCD: SU(3) LGT with static color sources.Encouraging results.


Documents

Density of states approach for field theories with a ... · Densityofstatesapproachforﬁeldtheorieswith acomplexactionproblem Mario Giuliani Christof Gattringer Karl Franzens Universität