Density of states approach for field theories with a ... ·...

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 2 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 3 / 21

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 )

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 4 / 21

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)

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 5 / 21

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)

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 6 / 21

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 )

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 7 / 21

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)

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 8 / 21

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)

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 9 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 10 / 21

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)

]

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 11 / 21

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)

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 12 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 14 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 15 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 16 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 17 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 18 / 21

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

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 19 / 21

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.

Mario Giuliani (Universität Graz) SIGN 2017, 24th March 2017 20 / 21

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.

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