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 ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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)
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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)
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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)
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Let us analyze the model using mean field theory.
S =∑x
(η|ϕx |2 + λ|ϕx |4 −
4∑ν=1
[e−µδν,tϕ∗xϕx+ν̂ + eµδν,tϕ∗xϕx−ν̂
])
� ϕx 6=0 = φ ∈ R� Action is purely real.� Second order phase transition at critical chemical
potential,
3 + coshµc(η, λ) =
√λ
2 Exp(−K2)√πErfc(K)
− 2K,
K = η/(
2√λ)
.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Let us analyze the model using mean field theory.
SMF = η|ϕ0|2 + λ|ϕ0|4 −4∑
ν=1
[e±µδν,tϕ∗0φ + e±µδν,tϕ0φ
]� ϕx 6=0 = φ ∈ R
� Action is purely real.� Second order phase transition at critical chemical
potential,
3 + coshµc(η, λ) =
√λ
2 Exp(−K2)√πErfc(K)
− 2K,
K = η/(
2√λ)
.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Let us analyze the model using mean field theory.
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,
3 + coshµc(η, λ) =
√λ
2 Exp(−K2)√πErfc(K)
− 2K,
K = η/(
2√λ)
.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Let us analyze the model using mean field theory.
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,
3 + coshµc(η, λ) =
√λ
2 Exp(−K2)√πErfc(K)
− 2K,
K = η/(
2√λ)
.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Extended Mean Field Theory (EMFT)
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0 1 2 3 4 5
κ
λ
<φ> = 0
<φ> ≠ 0
Monte Carlo
EMFT
Mean field
Second order perturbation theory
0.125
0.126
0.127
0.128
0.129
0.13
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
EMFT equations
Again focus on a single lattice site (x = 0),
S = η|ϕ0|2 +λ|ϕ0|4−4∑
ν=1
[e∓µδν,tϕ∗0ϕ±ν̂ + e±µδν,tϕ∗0ϕ∓ν̂
]+Sext
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
EMFT equations
Again focus on a single lattice site (x = 0),
S =η
2~ϕ†0~ϕ0 +
λ
4(~ϕ†0~ϕ0)2 −
4∑ν=1
~ϕ†±ν̂E (±µδν,t)~ϕ0︸ ︷︷ ︸−∆S
+ Sext
Introducing,
~ϕ† = (ϕ∗, ϕ), E (x) =
(e−x 0
0 ex
), G =
(〈ϕ∗ϕ〉c 〈ϕϕ〉c〈ϕ∗ϕ∗〉c 〈ϕϕ∗〉c
)
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Expanding ~ϕ around its (real) mean,
~ϕ† = δ~ϕ† + (φ, φ)
yields (up to a constant),
S =η
2~ϕ†0~ϕ0 +
λ
4(~ϕ†0~ϕ0)2 − 2~φ†~ϕ0(3 + cosh(µ))
−∑±ν
δ~ϕ†ν̂E (±µδν,t)~ϕ0︸ ︷︷ ︸−δS
+ Sext
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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,
1
2
⟨∑±ν
δ~ϕ†ν̂E (±µδν,t)δ~ϕ0
∑±ρ
δ~ϕ†ρ̂E (±µδρ,t)δ~ϕ0
⟩Sext
=1
2δ~ϕ†0∆δ~ϕ0,
where ∆ =
(∆11 ∆12
∆12 ∆11
)is a real, symmetric, matrix.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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,
1
2
⟨∑±ν
δ~ϕ†ν̂E (±µδν,t)δ~ϕ0
∑±ρ
δ~ϕ†ρ̂E (±µδρ,t)δ~ϕ0
⟩Sext
=1
2δ~ϕ†0∆δ~ϕ0,
where ∆ =
(∆11 ∆12
∆12 ∆11
)is a real, symmetric, matrix.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Putting all together we obtain the EMFT action,
SEMFT =1
2~ϕ† (ηI−∆) ~ϕ +
λ
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Putting all together we obtain the EMFT action,
SEMFT =1
2~ϕ† (ηI−∆) ~ϕ +
λ
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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.
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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 +
λ
4(~ϕ†0~ϕ0)2
− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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 +
λ
4(~ϕ†0~ϕ0)2
− ~φ†~ϕ0(2(3 + cosh(µ))−∆11 −∆12).
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Finite temperature
Finite temperature (volume) is conceptually trivial:
kν ∈ [−π, π]→ 2π
Nνnν , nν ∈ {−Nν/2, . . . ,Nν/2− 1}
∫ π
−π
dkν2π→ 1
Nν
Nν/2−1∑nν=−Nν/2
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Finite volume effects
V (r) ∝ exp(−mr)
r3/2
1e-10
1e-08
1e-06
0.0001
0.01
1
0 5 10 15 20
µc(N
s)
- µ
c(∞
)
µc(∞) L
η = 7.41
fit, rmax = L
fit, rmax = 2L
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
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
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density4
n = TV∂Z∂µ
0
0.2
0.4
0.6
0.8
1
1.2
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
n
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
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density4
n = TV∂Z∂µ
0
0.001
0.002
0.003
0.004
0.005
0.006
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
n
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.
B869, 56-73, (2013), hep-lat/1206.2954Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density4
n = TV∂Z∂µ
0
0.001
0.002
0.003
0.004
0.005
0.006
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
n
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.
B869, 56-73, (2013), hep-lat/1206.2954Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density4
n = TV∂Z∂µ
0.0016
0.0018
0.002
0.0022
0.0024
0.0026
0.0028
0.003
1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18
n
µ/µc
η = 7.444Monte Carlo data from: C. Gattringer and T. Kloiber, Nucl.Phys.
B869, 56-73, (2013), hep-lat/1206.2954Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
(µ/T )-Phase diagram5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 1.05 1.1 1.15 1.2
T/µ
c
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
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Outlook
� Multi-component fields, e.g. Gaugeless SU(2) Higgsmodel
� Gauge fields, e.g. Gauged U(1) Higgs model
� QCD?
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Outlook
� Multi-component fields, e.g. Gaugeless SU(2) Higgsmodel
� Gauge fields, e.g. Gauged U(1) Higgs model
� QCD?
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Outlook
� Multi-component fields, e.g. Gaugeless SU(2) Higgsmodel
� Gauge fields, e.g. Gauged U(1) Higgs model
� QCD?
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Thank you for yourattention!
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Efficient k-integration
G (0) =
∫ddk
(2π)dG̃−1(k).
G̃ (k) =
[G−1
EMFT + ∆− 2d∑ν=1
cos (kν − iµδν,t) IN×N
]−1
= [A− ε(k , µ)IN×N ]−1 ,
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
1
a − ε(k , µ)=
∫ ∞0
dτ exp (−τa)d∏ν=1
exp (2τ cos(kν − iµδν,t))
I0(x) =
∫ π
−π
dk
2πexp(x cos(k + z))
∫ddk
(2π)d1
a − ε(k , µ)=
∫ ∞0
dτ exp (−τa) I0(2τ)d .
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
G̃−1(k) =Adj(A− ε(k , µ)I)
det (A− ε(k , µ)I),
Partial fractions decomposition gives,
G̃−1(k)ij =N∑
n=1
B ijn
λn − ε(k , µ),
λ are eigenvalues of A.
G (0)ij =N∑
n=1
B ijnCn, Cn =
∫ ∞0
dτ exp (−τλn) I0(2τ)d
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density
n =∂Z
∂µ
= 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
∑ν
cos(kν − iµδν,t)I
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
Density
n =∂Z
∂µ
= 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
∑ν
cos(kν − iµδν,t)I
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
First order jump
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
1.058 1.06 1.062 1.064 1.066 1.068 1.07
<φ>
µ/µc
EMFT, T/µc = 0.581
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013
Motivation ϕ4-Theory Mean Field Theory Extended Mean Field Theory Results Outlook
First order jump
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 0.5 1 1.5 2 2.5 3 3.5 4
<φ>
J/µ
c(0
)
T/µc(0)
η = 9
η = 7.44
η = 7.41
Oscar Åkerlund ETH Zurich
EMFT Study of Complex ϕ4-Theory Lattice 2013, Mainz, August 2013