Extended Mean Field Study of Complex 4-Theory at Finite ... fileMotivation ’4-TheoryMean Field...

Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook

Extended Mean Field Study of Complex φ4-Theory

at Finite Density and Temperature

Oscar Åkerlund (ETH Zurich)Advisor: Philippe de Forcrand (ETHZ & CERN)Lattice 2013, MainzAugust 2, 2013

Oscar Åkerlund ETH Zurich

EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013

Motivation

� More matter than antimatter→ chemical potential

� Temperature is not zero in the real world

� Sign problem

� Toy models give valuable insight and experience beforeapplying new methods to QCD

Motivation

� Sign problem

Motivation

� Sign problem

Motivation

� Sign problem

In Euclidian space the Lagrangian of complex ϕ4-theoryreads,

L[ϕ(x)] = |∂νϕ(x)|2 + m20|ϕ(x)|2 + λ|ϕ(x)|4.

Invariant under global U(1) transformation of the fields:

ϕ(x)→ ϕ(x)e iθ, ϕ∗(x)→ ϕ∗(x)e−iθ.

Leads to conserved Noether current and charge,

jν = i(ϕ∗(x)∂νϕ(x)− ∂νϕ∗(x)ϕ(x)), Q =

∫d3~x j0(~x)

L[ϕ(x)] = |∂νϕ(x)|2 + m20|ϕ(x)|2 + λ|ϕ(x)|4.

jν = i(ϕ∗(x)∂νϕ(x)− ∂νϕ∗(x)ϕ(x)), Q =

∫d3~x j0(~x)

L[ϕ(x)] = |∂νϕ(x)|2 + m20|ϕ(x)|2 + λ|ϕ(x)|4.

jν = i(ϕ∗(x)∂νϕ(x)− ∂νϕ∗(x)ϕ(x)), Q =

∫d3~x j0(~x)

Chemical potential, µ couples to conserved charge, Q.After discretizing, x = a(nx , ny , nz , nt), we obtain thestandard lattice action:

S =∑x

(η|ϕx |2 + λ|ϕx |4 −

4∑ν=1

[e−µδν,tϕ∗xϕx+ν̂ + eµδν,tϕ∗xϕx−ν̂

with η = m20 + 8.

Sign problem!

Chemical potential, µ couples to conserved charge, Q.After discretizing, x = a(nx , ny , nz , nt), we obtain thestandard lattice action:

S =∑x

(η|ϕx |2 + λ|ϕx |4 −

4∑ν=1

with η = m20 + 8.

Sign problem!

Let us analyze the model using mean field theory.

S =∑x

(η|ϕx |2 + λ|ϕx |4 −

4∑ν=1

� ϕx 6=0 = φ ∈ R� Action is purely real.� Second order phase transition at critical chemical

potential,

3 + coshµc(η, λ) =

2 Exp(−K2)√πErfc(K)

− 2K,

K = η/(

2√λ)

SMF = η|ϕ0|2 + λ|ϕ0|4 −4∑

[e±µδν,tϕ∗0φ + e±µδν,tϕ0φ

]� ϕx 6=0 = φ ∈ R

� Action is purely real.� Second order phase transition at critical chemical

potential,

− 2K,

K = η/(

2√λ)

SMF = η|ϕ0|2 + λ|ϕ0|4 − 4Re[ϕ0]φ(3 + cosh(µ))

� ϕx 6=0 = φ ∈ R� Action is purely real.

� Second order phase transition at critical chemicalpotential,

− 2K,

K = η/(

2√λ)

SMF = η|ϕ0|2 + λ|ϕ0|4 − 4Re[ϕ0]φ(3 + cosh(µ))

� ϕx 6=0 = φ ∈ R� Action is purely real.� Second order phase transition at critical chemical

potential,

− 2K,

K = η/(

2√λ)

Shortcomings of Mean Field Theory

� Only qualitatively correct answer, numerical values arenot accurate.

� Only local variables are accessible, we get no informationon for example the Green’s function.

� No dependence on system size, i.e. we are restricted tozero temperature.

� We can do better!

Extended Mean Field Theory (EMFT)

0 1 2 3 4 5

<φ> = 0

<φ> ≠ 0

Monte Carlo

Mean field

Second order perturbation theory

0 0.005 0.01 0.015 0.02 0.025

(κ, λ)-phasediagram ofreal ϕ4-theoryin four dimen-sions1.

1O. Akerlund, P. de Forcrand, A. Georges and P. Werner,hep-lat/1305.7136

EMFT equations

Again focus on a single lattice site (x = 0),

S = η|ϕ0|2 +λ|ϕ0|4−4∑

[e∓µδν,tϕ∗0ϕ±ν̂ + e±µδν,tϕ∗0ϕ∓ν̂

]+Sext

EMFT equations

Again focus on a single lattice site (x = 0),

2~ϕ†0~ϕ0 +

4(~ϕ†0~ϕ0)2 −

4∑ν=1

~ϕ†±ν̂E (±µδν,t)~ϕ0︸︷︷︸−∆S

+ Sext

Introducing,

~ϕ† = (ϕ∗, ϕ), E (x) =

(e−x 0

), G =

(〈ϕ∗ϕ〉c 〈ϕϕ〉c〈ϕ∗ϕ∗〉c 〈ϕϕ∗〉c

Expanding ~ϕ around its (real) mean,

~ϕ† = δ~ϕ† + (φ, φ)

yields (up to a constant),

2~ϕ†0~ϕ0 +

4(~ϕ†0~ϕ0)2 − 2~φ†~ϕ0(3 + cosh(µ))

−∑±ν

δ~ϕ†ν̂E (±µδν,t)~ϕ0︸︷︷︸−δS

+ Sext

Formal integration over the field at all lattice sites exceptthe origin results in replacing −δS by its cumulantexpansion w.r.t. Sext.

The lowest nontrivial contribution is,

⟨∑±ν

δ~ϕ†ν̂E (±µδν,t)δ~ϕ0

∑±ρ

δ~ϕ†ρ̂E (±µδρ,t)δ~ϕ0

⟩Sext

2δ~ϕ†0∆δ~ϕ0,

where ∆ =

(∆11 ∆12

∆12 ∆11

)is a real, symmetric, matrix.

Formal integration over the field at all lattice sites exceptthe origin results in replacing −δS by its cumulantexpansion w.r.t. Sext.The lowest nontrivial contribution is,

⟨∑±ν

δ~ϕ†ν̂E (±µδν,t)δ~ϕ0

∑±ρ

δ~ϕ†ρ̂E (±µδρ,t)δ~ϕ0

⟩Sext

2δ~ϕ†0∆δ~ϕ0,

where ∆ =

(∆11 ∆12

∆12 ∆11

)is a real, symmetric, matrix.

Putting all together we obtain the EMFT action,

SEMFT =1

2~ϕ† (ηI−∆) ~ϕ +

4(~ϕ†0~ϕ0)2

− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).

∆22 = ∆11 , ∆21 = ∆12

φ and ∆ are to be self-consistently determined.

Putting all together we obtain the EMFT action,

SEMFT =1

2~ϕ† (ηI−∆) ~ϕ +

4(~ϕ†0~ϕ0)2

− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).

∆22 = ∆11 , ∆21 = ∆12

φ and ∆ are to be self-consistently determined.

Self-consistency conditions

� Translation-invariant solution implies

φ = 〈ϕ〉SEMFT.

� The self-consistency condition for ∆ is more involved.

� ZEMFT =∫dϕ exp (−SEMFT) is the generator of all local

diagrams (involving ϕ0).

� G (~r = ~0, t = 0) = GEMFT.

φ = 〈ϕ〉SEMFT.

� The self-consistency condition for ∆ is more involved.

� ZEMFT =∫dϕ exp (−SEMFT) is the generator of all local

� G (~r = ~0, t = 0) = GEMFT.

φ = 〈ϕ〉SEMFT.

� The self-consistency condition for ∆ is more involved.� ZEMFT =

∫dϕ exp (−SEMFT) is the generator of all local

� G (~r = ~0, t = 0) = GEMFT.

φ = 〈ϕ〉SEMFT.

� The self-consistency condition for ∆ is more involved.� ZEMFT =

∫dϕ exp (−SEMFT) is the generator of all local

� G (~r = ~0, t = 0) = GEMFT.

Green’s functions

� G (~r = ~0, t = 0) =∫ π−π

d4k(2π)4 G̃ (k)

� G̃−1(k) = G̃−10 (k)+Σ̃(k)︸︷︷︸self-energy due to interaction

= (η−2∑

ν cos(kν−iµδν,t))I+Σ̃(k)

� Likewise, G−1EMFT = G−1

0,EMFT + ΣEMFT = ηI−∆ + ΣEMFT

� Crucial approximation: Neglect k-dependence of Σ̃(k)and set it equal to ΣEMFT

� G̃−1(k) ≈ G−1EMFT + ∆− 2

∑ν cos(kν − iµδν,t)I

Green’s functions

� G (~r = ~0, t = 0) =∫ π−π

d4k(2π)4 G̃ (k)

= (η−2∑

� G̃−1(k) ≈ G−1EMFT + ∆− 2

Green’s functions

� G (~r = ~0, t = 0) =∫ π−π

d4k(2π)4 G̃ (k)

= (η−2∑

� G̃−1(k) ≈ G−1EMFT + ∆− 2

Green’s functions

� G (~r = ~0, t = 0) =∫ π−π

d4k(2π)4 G̃ (k)

= (η−2∑

� G̃−1(k) ≈ G−1EMFT + ∆− 2

Green’s functions

� G (~r = ~0, t = 0) =∫ π−π

d4k(2π)4 G̃ (k)

= (η−2∑

� G̃−1(k) ≈ G−1EMFT + ∆− 2

Self-consistency equations

� φ = 〈ϕ〉SEMFT

� GEMFT =

∫d4k[G−1

EMFT + ∆− 2∑ν

cos(kν − iµδν,t)I]−1

GEMFT =

(〈ϕ∗ϕ〉SEMFT

〈ϕϕ〉SEMFT

〈ϕ∗ϕ∗〉SEMFT〈ϕϕ∗〉SEMFT

)− ~φ~φ†

SEMFT =1

2~ϕ† (ηI−∆) ~ϕ +

4(~ϕ†0~ϕ0)2

− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).

Self-consistency equations

� φ = 〈ϕ〉SEMFT

� GEMFT =

∫d4k[G−1

EMFT + ∆− 2∑ν

cos(kν − iµδν,t)I]−1

GEMFT =

(〈ϕ∗ϕ〉SEMFT

〈ϕϕ〉SEMFT

〈ϕ∗ϕ∗〉SEMFT〈ϕϕ∗〉SEMFT

)− ~φ~φ†

SEMFT =1

2~ϕ† (ηI−∆) ~ϕ +

4(~ϕ†0~ϕ0)2

− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).

Finite temperature

Finite temperature (volume) is conceptually trivial:

kν ∈ [−π, π]→ 2π

Nνnν , nν ∈ {−Nν/2, . . . ,Nν/2− 1}

∫ π

dkν2π→ 1

Nν/2−1∑nν=−Nν/2

Finite volume effects

V (r) ∝ exp(−mr)

0.0001

0 5 10 15 20

µc(∞) L

η = 7.41

fit, rmax = L

fit, rmax = 2L

Critical chemical potential at T = 0 and λ = 1

η 9.00 7.44MF 1.12908 -EMFT 1.14582 0.17202Monte Carlo2 1.146(1) 0.170(1)Complex Langevin3 1.15(?) -

2C. Gattringer and T. Kloiber, Nucl.Phys. B869, 56-73, (2013),hep-lat/1206.2954

3G. Aarts, PoS, LAT2009, hep-lat/0910.3772Oscar Åkerlund ETH Zurich

Density4

n = TV∂Z∂µ

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

µ/µc

EMFT, T/µc = 0

EMFT, T/µc = 0.1745

EMFT, T/µc = 0.4364

MC, T/µc = 0.0087

MC, T/µc = 0.1745

MC, T/µc = 0.4362

η = 94Monte Carlo data from: C. Gattringer and T. Kloiber, Nucl.Phys.

B869, 56-73, (2013), hep-lat/1206.2954Oscar Åkerlund ETH Zurich

Density4

n = TV∂Z∂µ

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

µ/µc

EMFT, T/µc = 0

EMFT, T/µc = 0.581

EMFT, T/µc = 0.969

MC, T/µc = 0.059

MC, T/µc = 0.588

MC, T/µc = 0.980

η = 7.444Monte Carlo data from: C. Gattringer and T. Kloiber, Nucl.Phys.

Density4

n = TV∂Z∂µ

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

µ/µc

EMFT, T/µc = 0

EMFT, T/µc = 0.581

EMFT, T/µc = 0.969

MC, T/µc = 0.059

MC, T/µc = 0.588

MC, T/µc = 0.980

Density4

n = TV∂Z∂µ

0.0016

0.0018

0.0022

0.0024

0.0026

0.0028

1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

µ/µc

(µ/T )-Phase diagram5

1 1.05 1.1 1.15 1.2

µ/µc

EMFT, η = 9

EMFT, η = 7.44

Monte Carlo, η = 9

Monte Carlo, η = 7.44

5Monte Carlo data from: C. Gattringer and T. Kloiber, Nucl.Phys.B869, 56-73, (2013), hep-lat/1206.2954

Outlook

� Multi-component fields, e.g. Gaugeless SU(2) Higgsmodel

� Gauge fields, e.g. Gauged U(1) Higgs model

� QCD?

Outlook

� QCD?

Outlook

� QCD?

Thank you for yourattention!

Efficient k-integration

G (0) =

∫ddk

(2π)dG̃−1(k).

G̃ (k) =

[G−1

EMFT + ∆− 2d∑ν=1

cos (kν − iµδν,t) IN×N

= [A− ε(k , µ)IN×N ]−1 ,

a − ε(k , µ)=

∫ ∞0

dτ exp (−τa)d∏ν=1

exp (2τ cos(kν − iµδν,t))

I0(x) =

∫ π

2πexp(x cos(k + z))

∫ddk

(2π)d1

a − ε(k , µ)=

∫ ∞0

dτ exp (−τa) I0(2τ)d .

G̃−1(k) =Adj(A− ε(k , µ)I)

det (A− ε(k , µ)I),

Partial fractions decomposition gives,

G̃−1(k)ij =N∑

λn − ε(k , µ),

λ are eigenvalues of A.

G (0)ij =N∑

B ijnCn, Cn =

∫ ∞0

dτ exp (−τλn) I0(2τ)d

Density

n =∂Z

= 2 sinhµ 〈ϕ〉2 + 2

(sinhµ

∫d4k

(2π)4Re[ 〈ϕ∗(k)ϕ(k)〉c ] cos(k4)

+ coshµ

∫d4k

(2π)4Im[ 〈ϕ∗(k)ϕ(k)〉c ] sin(k4)

〈ϕ∗(k)ϕ(k)〉c = G−1EMFT + ∆− 2

cos(kν − iµδν,t)I

Density

n =∂Z

= 2 sinhµ 〈ϕ〉2 + 2

(sinhµ

∫d4k

(2π)4Re[ 〈ϕ∗(k)ϕ(k)〉c ] cos(k4)

+ coshµ

∫d4k

(2π)4Im[ 〈ϕ∗(k)ϕ(k)〉c ] sin(k4)

〈ϕ∗(k)ϕ(k)〉c = G−1EMFT + ∆− 2

cos(kν − iµδν,t)I

First order jump

1.058 1.06 1.062 1.064 1.066 1.068 1.07

µ/µc

EMFT, T/µc = 0.581

First order jump

0 0.5 1 1.5 2 2.5 3 3.5 4

T/µc(0)

η = 9

η = 7.44

η = 7.41

Extended Mean Field Study of Complex 4-Theory at Finite ... fileMotivation ’4-TheoryMean Field...

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