Inhomogeneous phasesin the QCD phase diagram

Michael Buballa

TU Darmstadt, Germany

Joint Seminar “Hadron Physics” and “Theory of Hadronic Matter Under Extreme Conditions”

Bogoliubov Laboratory of Theoretical Physics, JINR Dubna, Russia, July 27, 2012

July 27, 2012 | Michael Buballa | 1

Motivation

I QCD phase diagram :

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Motivation

I QCD phase diagram (NICA version):

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Motivation

I QCD phase diagram :

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Motivation

I QCD phase diagram :

T

.µ

.

0

<qq> <qq>

<qq> = 0<qq> = 0

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Motivation

I QCD phase diagram :

T

.µ

.

0

<qq> <qq>

<qq> = 0<qq> = 0

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Motivation

I QCD phase diagram :

T

.µ

.

0

<qq> <qq>

<qq> = 0<qq> = 0

I question marks:I critical pointI color superconductors

I frequent assumption:

〈qq〉, 〈qq〉 spatially constant

I How aboutinhomogeneous phases?

July 27, 2012 | Michael Buballa | 2

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

I blue solid line:

1st-order when restricting tohomogeneous condensates

I orange shaded region:

inhomogeneous phase

I critical point disappeared!

I more details later ...

July 27, 2012 | Michael Buballa | 3

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

I blue solid line:

1st-order when restricting tohomogeneous condensates

I orange shaded region:

inhomogeneous phase

I critical point disappeared!

I more details later ...

July 27, 2012 | Michael Buballa | 3

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

I blue solid line:

1st-order when restricting tohomogeneous condensates

I orange shaded region:

inhomogeneous phase

I critical point disappeared!

I more details later ...

July 27, 2012 | Michael Buballa | 3

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

I blue solid line:

1st-order when restricting tohomogeneous condensates

I orange shaded region:

inhomogeneous phase

I critical point disappeared!

I more details later ...

July 27, 2012 | Michael Buballa | 3

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

I blue solid line:

1st-order when restricting tohomogeneous condensates

I orange shaded region:

inhomogeneous phase

I critical point disappeared!

I more details later ...

July 27, 2012 | Michael Buballa | 3

Inhomogeneous phases:(incomplete) historical overview

I 1960s:I spin-density waves in nuclear matter

(Overhauser)

I crystalline superconductors(Fulde, Ferrell, Larkin, Ovchinnikov)

I 1970s – 1990s:I p-wave pion condensation (Migdal)

I chiral density wave (Dautry, Nyman)

I after 2000:I 1+1 D Gross-Neveu model (Thies et al.)

I crystalline color superconductors(Alford, Bowers, Rajagopal)

I quarkyonic matter(Kojo, McLerran, Pisarski, ... )

July 27, 2012 | Michael Buballa | 4

Inhomogeneous phases:(incomplete) historical overview

I 1960s:I spin-density waves in nuclear matter

(Overhauser)

I crystalline superconductors(Fulde, Ferrell, Larkin, Ovchinnikov)

I 1970s – 1990s:I p-wave pion condensation (Migdal)

I chiral density wave (Dautry, Nyman)

I after 2000:I 1+1 D Gross-Neveu model (Thies et al.)

I crystalline color superconductors(Alford, Bowers, Rajagopal)

I quarkyonic matter(Kojo, McLerran, Pisarski, ... )

Broniowski et al. (1991)

July 27, 2012 | Michael Buballa | 4

Inhomogeneous phases:(incomplete) historical overview

I 1960s:I spin-density waves in nuclear matter

(Overhauser)

I crystalline superconductors(Fulde, Ferrell, Larkin, Ovchinnikov)

I 1970s – 1990s:I p-wave pion condensation (Migdal)

I chiral density wave (Dautry, Nyman)

I after 2000:I 1+1 D Gross-Neveu model (Thies et al.)

I crystalline color superconductors(Alford, Bowers, Rajagopal)

I quarkyonic matter(Kojo, McLerran, Pisarski, ... )

Thies, Urlichs (2003)

July 27, 2012 | Michael Buballa | 4

Inhomogeneous phases:(incomplete) historical overview

I 1960s:I spin-density waves in nuclear matter

(Overhauser)

I crystalline superconductors(Fulde, Ferrell, Larkin, Ovchinnikov)

I 1970s – 1990s:I p-wave pion condensation (Migdal)

I chiral density wave (Dautry, Nyman)

I after 2000:I 1+1 D Gross-Neveu model (Thies et al.)

I crystalline color superconductors(Alford, Bowers, Rajagopal)

I quarkyonic matter(Kojo, McLerran, Pisarski, ... )

CFLliq

QGP

T

µ

crystal?

nuclear

gas

superconducting

= color

compact star

RHIC

Alford (2003)

July 27, 2012 | Michael Buballa | 4

Inhomogeneous phases:(incomplete) historical overview

I 1960s:I spin-density waves in nuclear matter

(Overhauser)

I crystalline superconductors(Fulde, Ferrell, Larkin, Ovchinnikov)

I 1970s – 1990s:I p-wave pion condensation (Migdal)

I chiral density wave (Dautry, Nyman)

I after 2000:I 1+1 D Gross-Neveu model (Thies et al.)

I crystalline color superconductors(Alford, Bowers, Rajagopal)

I quarkyonic matter(Kojo, McLerran, Pisarski, ... )

Kojo et al. (2011)

July 27, 2012 | Michael Buballa | 4

Contents

1. Introduction

2. Inhomogeneous chiral symmetry breaking in the NJL model

3. One-dimensional modulations

4. Two-dimensional modulations

5. Conclusions

July 27, 2012 | Michael Buballa | 5

Model

I NJL model:L = ψ(i∂/−m)ψ + GS

[(ψψ)2 + (ψiγ5~τψ)2]

I bosonize: σ(x) = ψ(x)ψ(x), ~π(x) = ψ(x)iγ5~τψ(x)

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

I mean-field approximation:

σ(x)→ 〈σ(x)〉 ≡ S(~x), πa(x)→ 〈πa(x)〉 ≡ P(~x) δa 3

I S(~x), P(~x) time independent classical fieldsI retain space dependence !

I mean-field thermodynamic potential:

ΩMF (T ,µ) = −TV

ln∫DψDψ exp

(∫x∈[0, 1

T ]×V(LMF + µψγ0ψ)

)

July 27, 2012 | Michael Buballa | 6

Model

I NJL model:L = ψ(i∂/−m)ψ + GS

[(ψψ)2 + (ψiγ5~τψ)2]

I bosonize: σ(x) = ψ(x)ψ(x), ~π(x) = ψ(x)iγ5~τψ(x)

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

I mean-field approximation:

σ(x)→ 〈σ(x)〉 ≡ S(~x), πa(x)→ 〈πa(x)〉 ≡ P(~x) δa 3

I S(~x), P(~x) time independent classical fieldsI retain space dependence !

I mean-field thermodynamic potential:

ΩMF (T ,µ) = −TV

ln∫DψDψ exp

(∫x∈[0, 1

T ]×V(LMF + µψγ0ψ)

)

July 27, 2012 | Michael Buballa | 6

Model

I NJL model:L = ψ(i∂/−m)ψ + GS

[(ψψ)2 + (ψiγ5~τψ)2]

I bosonize: σ(x) = ψ(x)ψ(x), ~π(x) = ψ(x)iγ5~τψ(x)

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

I mean-field approximation:

σ(x)→ 〈σ(x)〉 ≡ S(~x), πa(x)→ 〈πa(x)〉 ≡ P(~x) δa 3

I S(~x), P(~x) time independent classical fieldsI retain space dependence !

I mean-field thermodynamic potential:

ΩMF (T ,µ) = −TV

ln∫DψDψ exp

(∫x∈[0, 1

T ]×V(LMF + µψγ0ψ)

)

July 27, 2012 | Michael Buballa | 6

Model

I NJL model:L = ψ(i∂/−m)ψ + GS

[(ψψ)2 + (ψiγ5~τψ)2]

I bosonize: σ(x) = ψ(x)ψ(x), ~π(x) = ψ(x)iγ5~τψ(x)

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

I mean-field approximation:

σ(x)→ 〈σ(x)〉 ≡ S(~x), πa(x)→ 〈πa(x)〉 ≡ P(~x) δa 3

I S(~x), P(~x) time independent classical fieldsI retain space dependence !

I mean-field thermodynamic potential:

ΩMF (T ,µ) = −TV

ln∫DψDψ exp

(∫x∈[0, 1

T ]×V(LMF + µψγ0ψ)

)July 27, 2012 | Michael Buballa | 6

Mean-field model

I mean-field Lagrangian:

LMF = ψ(x)S−1(x)ψ(x)− GS[S2(~x) + P2(~x)

]I bilinear in ψ and ψ ⇒ quark fields can be integrated out!

I inverse dressed propagator:

S−1(x) = i∂/−m + 2GS(S(~x) + iγ5τ3P(~x)

)

≡ γ0 (i∂0 −HMF )

I effective Hamiltonian (in chiral representation):

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

)I constituent mass functions: M(~x) = m − 2G [S(~x) + iP(~x)]

I HMF hermitean ⇒ can (in principle) be diagonalized ( eigenvalues Eλ)I HMF time-independent ⇒ Matsubara sum as usual

July 27, 2012 | Michael Buballa | 7

Mean-field model

I mean-field Lagrangian:

LMF = ψ(x)S−1(x)ψ(x)− GS[S2(~x) + P2(~x)

]I bilinear in ψ and ψ ⇒ quark fields can be integrated out!

I inverse dressed propagator:

S−1(x) = i∂/−m + 2GS(S(~x) + iγ5τ3P(~x)

)≡ γ0 (i∂0 −HMF )

I effective Hamiltonian (in chiral representation):

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

)I constituent mass functions: M(~x) = m − 2G [S(~x) + iP(~x)]

I HMF hermitean ⇒ can (in principle) be diagonalized ( eigenvalues Eλ)I HMF time-independent ⇒ Matsubara sum as usual

July 27, 2012 | Michael Buballa | 7

Mean-field model

I mean-field Lagrangian:

LMF = ψ(x)S−1(x)ψ(x)− GS[S2(~x) + P2(~x)

]I bilinear in ψ and ψ ⇒ quark fields can be integrated out!

I inverse dressed propagator:

S−1(x) = i∂/−m + 2GS(S(~x) + iγ5τ3P(~x)

)≡ γ0 (i∂0 −HMF )

I effective Hamiltonian (in chiral representation):

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

)I constituent mass functions: M(~x) = m − 2G [S(~x) + iP(~x)]

I HMF hermitean ⇒ can (in principle) be diagonalized ( eigenvalues Eλ)I HMF time-independent ⇒ Matsubara sum as usual

July 27, 2012 | Michael Buballa | 7

Mean-field model

I mean-field Lagrangian:

LMF = ψ(x)S−1(x)ψ(x)− GS[S2(~x) + P2(~x)

]I bilinear in ψ and ψ ⇒ quark fields can be integrated out!

I inverse dressed propagator:

S−1(x) = i∂/−m + 2GS(S(~x) + iγ5τ3P(~x)

)≡ γ0 (i∂0 −HMF )

I effective Hamiltonian (in chiral representation):

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

)I constituent mass functions: M(~x) = m − 2G [S(~x) + iP(~x)]I HMF hermitean ⇒ can (in principle) be diagonalized ( eigenvalues Eλ)I HMF time-independent ⇒ Matsubara sum as usual

July 27, 2012 | Michael Buballa | 7

Mean-field thermodynamic potential

I thermodynamic potential:

ΩMF (T ,µ; S, P) = −TV

Tr ln(

1T

(i∂0 −HMF + µ))

+GS

V

∫V

d3x(

S2(~x) + P2(~x))

= − 1V

∑λ

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

T

)]+

1V

∫V

d3x|M(~x)−m|2

4Gs

I remaining tasks:I Calculate eigenvalue spectrum Eλ[M(~x)] ofHMF for given mass function M(~x).

I Minimize ΩMF w.r.t. M(~x)

I general case: extremely difficult!

July 27, 2012 | Michael Buballa | 8

Mean-field thermodynamic potential

I thermodynamic potential:

ΩMF (T ,µ; S, P) = −TV

Tr ln(

1T

(i∂0 −HMF + µ))

+GS

V

∫V

d3x(

S2(~x) + P2(~x))

= − 1V

∑λ

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

T

)]+

1V

∫V

d3x|M(~x)−m|2

4Gs

I remaining tasks:I Calculate eigenvalue spectrum Eλ[M(~x)] ofHMF for given mass function M(~x).

I Minimize ΩMF w.r.t. M(~x)

I general case: extremely difficult!

July 27, 2012 | Michael Buballa | 8

Mean-field thermodynamic potential

I thermodynamic potential:

ΩMF (T ,µ; S, P) = −TV

Tr ln(

1T

(i∂0 −HMF + µ))

+GS

V

∫V

d3x(

S2(~x) + P2(~x))

= − 1V

∑λ

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

T

)]+

1V

∫V

d3x|M(~x)−m|2

4Gs

I remaining tasks:I Calculate eigenvalue spectrum Eλ[M(~x)] ofHMF for given mass function M(~x).

I Minimize ΩMF w.r.t. M(~x)

I general case: extremely difficult!

July 27, 2012 | Michael Buballa | 8

Mean-field thermodynamic potential

I thermodynamic potential:

ΩMF (T ,µ; S, P) = −TV

Tr ln(

1T

(i∂0 −HMF + µ))

+GS

V

∫V

d3x(

S2(~x) + P2(~x))

= − 1V

∑λ

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

T

)]+

1V

∫V

d3x|M(~x)−m|2

4Gs

I remaining tasks:I Calculate eigenvalue spectrum Eλ[M(~x)] ofHMF for given mass function M(~x).

I Minimize ΩMF w.r.t. M(~x)

I general case: extremely difficult!

July 27, 2012 | Michael Buballa | 8

Periodic structures

I crystal with a unit cell spanned by vectors ~ai , i = 1, 2, 3

→ periodic mass function: M(~x + ~ai ) = M(~x)

I Fourier decomposition: M(~x) =∑~qk

M~qkei~qk ·~x

I reciprocal lattice: ~qk ·~ai2π ∈ Z

I mean-field Hamiltonian in momentum space:

H~pm ,~pn=

−~σ · ~pm δ~pm ,~pn

∑~qk

M~qkδ~pm ,~pn+~qk∑

~qk

M∗~qkδ~pm ,~pn−~qk

~σ · ~pm δ~pm ,~pn

I different momenta coupled by M~qk

⇒ H is nondiagonal in momentum space!I ~qk discrete ⇒ H is still block diagonal

July 27, 2012 | Michael Buballa | 9

Periodic structures

I crystal with a unit cell spanned by vectors ~ai , i = 1, 2, 3

→ periodic mass function: M(~x + ~ai ) = M(~x)

I Fourier decomposition: M(~x) =∑~qk

M~qkei~qk ·~x

I reciprocal lattice: ~qk ·~ai2π ∈ Z

I mean-field Hamiltonian in momentum space:

H~pm ,~pn=

−~σ · ~pm δ~pm ,~pn

∑~qk

M~qkδ~pm ,~pn+~qk∑

~qk

M∗~qkδ~pm ,~pn−~qk

~σ · ~pm δ~pm ,~pn

I different momenta coupled by M~qk

⇒ H is nondiagonal in momentum space!I ~qk discrete ⇒ H is still block diagonal

July 27, 2012 | Michael Buballa | 9

Periodic structures

I crystal with a unit cell spanned by vectors ~ai , i = 1, 2, 3

→ periodic mass function: M(~x + ~ai ) = M(~x)

I Fourier decomposition: M(~x) =∑~qk

M~qkei~qk ·~x

I reciprocal lattice: ~qk ·~ai2π ∈ Z

I mean-field Hamiltonian in momentum space:

H~pm ,~pn=

−~σ · ~pm δ~pm ,~pn

∑~qk

M~qkδ~pm ,~pn+~qk∑

~qk

M∗~qkδ~pm ,~pn−~qk

~σ · ~pm δ~pm ,~pn

I different momenta coupled by M~qk

⇒ H is nondiagonal in momentum space!I ~qk discrete ⇒ H is still block diagonal

July 27, 2012 | Michael Buballa | 9

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qk

I choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

Periodic structures: minimum free energy

I general procedure:

I choose a unit cell ~ai ⇒ ~qkI choose Fourier components M~qk

I diagonalizeHMF → ΩMF

I minimize ΩMF w.r.t. M~qk

I minimize ΩMF w.r.t. ~ai

→ still very hard!

→ further simplifications necessary

July 27, 2012 | Michael Buballa | 10

One dimensional modulations

I consider only one-dimensional modulations: M(~x) = M(z) =∑qk

Mk eikqz

I popular choice: M(z) = M1eiqz (chiral density wave)I ⇔ S(~x) = ∆ cos(qz) , P(~x) = ∆ sin(qz)I HCDW can be diagonalized analytically

I important observation: [D. Nickel, PRD (2009)]

The general problem with 1D modulations in 3+1Dcan be mapped to the 1 + 1 dimensional case

I 1 + 1D solutions known analytically: [M. Thies, J. Phys. A (2006)]

M(z) =√ν∆ sn(∆z|ν) (chiral limit), sn(ξ|ν): Jacobi elliptic functions

I remaining task:I minimize w.r.t. 2 parameters: ∆, νI (almost) as simple as CDW, but more powerfulI m 6= 0: 3 parameters

July 27, 2012 | Michael Buballa | 11

One dimensional modulations

I consider only one-dimensional modulations: M(~x) = M(z) =∑qk

Mk eikqz

I popular choice: M(z) = M1eiqz (chiral density wave)I ⇔ S(~x) = ∆ cos(qz) , P(~x) = ∆ sin(qz)I HCDW can be diagonalized analytically

I important observation: [D. Nickel, PRD (2009)]

The general problem with 1D modulations in 3+1Dcan be mapped to the 1 + 1 dimensional case

I 1 + 1D solutions known analytically: [M. Thies, J. Phys. A (2006)]

M(z) =√ν∆ sn(∆z|ν) (chiral limit), sn(ξ|ν): Jacobi elliptic functions

I remaining task:I minimize w.r.t. 2 parameters: ∆, νI (almost) as simple as CDW, but more powerfulI m 6= 0: 3 parameters

July 27, 2012 | Michael Buballa | 11

One dimensional modulations

I consider only one-dimensional modulations: M(~x) = M(z) =∑qk

Mk eikqz

I popular choice: M(z) = M1eiqz (chiral density wave)I ⇔ S(~x) = ∆ cos(qz) , P(~x) = ∆ sin(qz)I HCDW can be diagonalized analytically

I important observation: [D. Nickel, PRD (2009)]

The general problem with 1D modulations in 3+1Dcan be mapped to the 1 + 1 dimensional case

I 1 + 1D solutions known analytically: [M. Thies, J. Phys. A (2006)]

M(z) =√ν∆ sn(∆z|ν) (chiral limit), sn(ξ|ν): Jacobi elliptic functions

I remaining task:I minimize w.r.t. 2 parameters: ∆, νI (almost) as simple as CDW, but more powerfulI m 6= 0: 3 parameters

July 27, 2012 | Michael Buballa | 11

One dimensional modulations

I consider only one-dimensional modulations: M(~x) = M(z) =∑qk

Mk eikqz

I popular choice: M(z) = M1eiqz (chiral density wave)I ⇔ S(~x) = ∆ cos(qz) , P(~x) = ∆ sin(qz)I HCDW can be diagonalized analytically

I important observation: [D. Nickel, PRD (2009)]

The general problem with 1D modulations in 3+1Dcan be mapped to the 1 + 1 dimensional case

I 1 + 1D solutions known analytically: [M. Thies, J. Phys. A (2006)]

M(z) =√ν∆ sn(∆z|ν) (chiral limit), sn(ξ|ν): Jacobi elliptic functions

I remaining task:I minimize w.r.t. 2 parameters: ∆, νI (almost) as simple as CDW, but more powerfulI m 6= 0: 3 parameters

July 27, 2012 | Michael Buballa | 11

One dimensional modulations

I consider only one-dimensional modulations: M(~x) = M(z) =∑qk

Mk eikqz

I popular choice: M(z) = M1eiqz (chiral density wave)I ⇔ S(~x) = ∆ cos(qz) , P(~x) = ∆ sin(qz)I HCDW can be diagonalized analytically

I important observation: [D. Nickel, PRD (2009)]

The general problem with 1D modulations in 3+1Dcan be mapped to the 1 + 1 dimensional case

I 1 + 1D solutions known analytically: [M. Thies, J. Phys. A (2006)]

M(z) =√ν∆ sn(∆z|ν) (chiral limit), sn(ξ|ν): Jacobi elliptic functions

I remaining task:I minimize w.r.t. 2 parameters: ∆, νI (almost) as simple as CDW, but more powerfulI m 6= 0: 3 parameters

July 27, 2012 | Michael Buballa | 11

Phase diagram (chiral limit)[D. Nickel, PRD (2009)]

homogeneous phases only

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I 1st-order line completely covered by the inhomogeneous phase!I all phase boundaries 2nd orderI critical point coincides with Lifshitz point (NJL specific)

July 27, 2012 | Michael Buballa | 12

Phase diagram (chiral limit)[D. Nickel, PRD (2009)]

including inhomogeneous phase

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I 1st-order line completely covered by the inhomogeneous phase!I all phase boundaries 2nd orderI critical point coincides with Lifshitz point (NJL specific)

July 27, 2012 | Michael Buballa | 12

Phase diagram (chiral limit)[D. Nickel, PRD (2009)]

including inhomogeneous phase

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I 1st-order line completely covered by the inhomogeneous phase!I all phase boundaries 2nd orderI critical point coincides with Lifshitz point (NJL specific)

July 27, 2012 | Michael Buballa | 12

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 307.5 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 307.5 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 308 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 309 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 310 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 320 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 330 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 340 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 345 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

0

20

40

60

80

100

240 260 280 300 320 340 360

T (

MeV

)

µ (MeV)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 307.5 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 307.5 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 307.5 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 307.5 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 308 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 308 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 309 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 309 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 310 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 310 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .

July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 320 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 320 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 330 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 330 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 340 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 340 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .July 27, 2012 | Michael Buballa | 13

Mass functions and density profiles (T = 0)[S. Carignano, D. Nickel, M.B., PRD (2010)]

I M(z) =√ν∆ sn(∆z|ν) →

∆ tanh(∆z) for ν → 1√ν∆ sin(∆z) for ν → 0

M(z) (µ = 345 MeV)

-300

-200

-100

0

100

200

300

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 345 MeV)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30 35 40

z (fm)

I Quarks reside in the chirally restored regions, cf. bag model!

I Density gets smoothened with increasing µ and T .July 27, 2012 | Michael Buballa | 13

Free energy difference[D. Nickel, PRD (2009)]

290 300 310 320 330 340 350

-0.15-0.10-0.05

0.000.050.10

Μq @MeVD

HWph

ase-W@0DLB

vac

I homogeneous chirally broken

I solitons

I chiral density wave:

MCDW (z) = ∆ eiqz

I soliton phase favored, when it exists

I δΩsoliton ≈ 2δΩCDW ⇒ CDW never favored

July 27, 2012 | Michael Buballa | 14

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Including vector interactions[S. Carignano, D. Nickel, M.B., PRD (2010)]

I additional vector term: LV = −GV (ψγµψ)2

I additional mean field:I ψγµψ → 〈ψγµψ〉 ≡ n(~x) δµ0 (density !)I 〈ψγ3ψ〉 possible for inhomogeneous phases, but neglected

I mean-field Hamiltonian: HMF − µ = HMF |GV =0 − µ(~x)I µ(~x) = µ− 2GV n(~x) “shifted chemical potential”

I further approximation: n(~x)→ 〈n〉 = const . ⇒ µ = const .

I questionable in the inhomogeneous phase at low µ and T

I ok near the restored phase (including the Lifshitz point)

I advantage: known analytic solutions can still be used

I additional parameter: µ, fixed by constraint ∂ΩMF∂µ

= 0

July 27, 2012 | Michael Buballa | 15

Phase diagram

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

Gv=0

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

20

40

60

80

100

µ (MeV)

Gv=Gs

T -〈n〉 phase diagram:

0 20 40 60 80

100 120 140 160 180

0 0.2 0.4 0.6 0.8 1 1.2

T (

MeV

)

Density (fm-3)

Homogeneous, broken

Inhomogeneous

Restored

I independent of GV !

I homogeneous phases: strong GV -dependence of the critical point

I inhomogeneous regime: stretched in µ direction, Lifshitz point at constant T

July 27, 2012 | Michael Buballa | 16

Phase diagram

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

Gv=0

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

20

40

60

80

100

µ (MeV)

Gv=Gs

T -〈n〉 phase diagram:

0 20 40 60 80

100 120 140 160 180

0 0.2 0.4 0.6 0.8 1 1.2

T (

MeV

)

Density (fm-3)

Homogeneous, broken

Inhomogeneous

Restored

I independent of GV !

I homogeneous phases: strong GV -dependence of the critical point

I inhomogeneous regime: stretched in µ direction, Lifshitz point at constant TJuly 27, 2012 | Michael Buballa | 16

Phase diagram

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

Gv=0

300 350 400 450 500 0

20

40

60

80

100

Gv=Gs/5

0

20

40

60

80

100

300 350 400 450 500

T (

MeV

)

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

20

40

60

80

100

µ (MeV)

Gv=Gs

T -〈n〉 phase diagram:

0 20 40 60 80

100 120 140 160 180

0 0.2 0.4 0.6 0.8 1 1.2

T (

MeV

)

Density (fm-3)

Homogeneous, broken

Inhomogeneous

Restored

I independent of GV !

I homogeneous phases: strong GV -dependence of the critical point

I inhomogeneous regime: stretched in µ direction, Lifshitz point at constant TJuly 27, 2012 | Michael Buballa | 16

Two-dimensional modulations

I consider two shapes:

I square lattice (“egg carton”)

M(x , y ) = M cos(Qx) cos(Qy ) -8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

I hexagonal lattice

M(x , y ) = M3

[2 cos (Qx) cos

(1√3Qy)

+ cos( 2√3Qy )]

-8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

I minimize both cases numerically w.r.t. M and Q

July 27, 2012 | Michael Buballa | 17

Two-dimensional modulations

I consider two shapes:

I square lattice (“egg carton”)

M(x , y ) = M cos(Qx) cos(Qy ) -8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

I hexagonal lattice

M(x , y ) = M3

[2 cos (Qx) cos

(1√3Qy)

+ cos( 2√3Qy )]

-8 -6 -4 -2 0 2 4 6 8-8

-6-4

-2 0

2 4

6 8

-1

0

1

M(x

,y)

/ M

Q x

Q y

M(x

,y)

/ M

I minimize both cases numerically w.r.t. M and Q

July 27, 2012 | Michael Buballa | 17

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restored

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy)

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restored

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1d

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1djacobi 1d

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1djacobi 1d

square 2d

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1djacobi 1d

square 2dhexagon 2d

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1djacobi 1d

square 2dhexagon 2d

I 2D not favored in this regime

I more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!

July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: results[S. Carignano, M.B., arXiv:1203.5343]

I amplitudes and wave numbers:I egg carton:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

I hexagon:

0

100

200

300

400

500

300 310 320 330 340 350

M,Q

(M

eV

)

µ (MeV)

MQ

free-energy gain at T = 0:

-4

-3

-2

-1

0

310 320 330 340

Ω -

Ωre

st (M

eV

/fm

3)

µ (MeV)

restoredhomogen. broken

cos 1djacobi 1d

square 2dhexagon 2d

I 2D not favored in this regimeI more Fourier components:

M(x , y ) =3∑

n=1Mn cos(nQx) cos(nQy )

no effect!July 27, 2012 | Michael Buballa | 18

Two-dimensional modulations: further results[S. Carignano, M.B., arXiv:1203.5343]

I rectangular lattice:

M(x , y ) = M cos(Qx x) cos(Qy y )

I free energy:

100 200 300 400

Qx (MeV)

100

200

300

400

Qy (

MeV

)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Ωm

in (

MeV

/fm

3)

⇒ “egg carton” local minimum

I higher chemical potentials

-10

-5

0

5

10

400 500 600 700 800 900 1000

∆Ω

(M

eV

/fm

3)

µ (MeV)

square-hexjacobi-hex

jacobi-square

I 450 MeV < µ < 900 MeV:egg carton favored

I µ > 900 MeV: hexagon favored

July 27, 2012 | Michael Buballa | 19

Two-dimensional modulations: further results[S. Carignano, M.B., arXiv:1203.5343]

I rectangular lattice:

M(x , y ) = M cos(Qx x) cos(Qy y )

I free energy:

100 200 300 400

Qx (MeV)

100

200

300

400

Qy (

MeV

)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Ωm

in (

MeV

/fm

3)

⇒ “egg carton” local minimum

I higher chemical potentials

-10

-5

0

5

10

400 500 600 700 800 900 1000

∆Ω

(M

eV

/fm

3)

µ (MeV)

square-hexjacobi-hex

jacobi-square

I 450 MeV < µ < 900 MeV:egg carton favored

I µ > 900 MeV: hexagon favored

July 27, 2012 | Michael Buballa | 19

Two-dimensional modulations: further results[S. Carignano, M.B., arXiv:1203.5343]

I rectangular lattice:

M(x , y ) = M cos(Qx x) cos(Qy y )

I free energy:

100 200 300 400

Qx (MeV)

100

200

300

400

Qy (

MeV

)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Ωm

in (

MeV

/fm

3)

⇒ “egg carton” local minimum

I higher chemical potentials

-10

-5

0

5

10

400 500 600 700 800 900 1000

∆Ω

(M

eV

/fm

3)

µ (MeV)

square-hexjacobi-hex

jacobi-square

I 450 MeV < µ < 900 MeV:egg carton favored

I µ > 900 MeV: hexagon favored

July 27, 2012 | Michael Buballa | 19

Two-dimensional modulations: further results[S. Carignano, M.B., arXiv:1203.5343]

I rectangular lattice:

M(x , y ) = M cos(Qx x) cos(Qy y )

I free energy:

100 200 300 400

Qx (MeV)

100

200

300

400

Qy (

MeV

)

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Ωm

in (

MeV

/fm

3)

⇒ “egg carton” local minimum

I higher chemical potentials

-10

-5

0

5

10

400 500 600 700 800 900 1000∆

Ω (

Me

V/f

m3)

µ (MeV)

square-hexjacobi-hex

jacobi-square

I 450 MeV < µ < 900 MeV:egg carton favored

I µ > 900 MeV: hexagon favored

July 27, 2012 | Michael Buballa | 19

Conclusions

I Inhomogeneous phases must be considered!

I NJL model with one- and two-dimensional modulations of 〈qq〉:I 1st-order line and critical point covered by an inhomogeneous regionI inhomogeneous phase rather stable w.r.t. vector interactionsI number susceptibility always finite (for GV > 0)I 1d modulations favored at “moderate” µI 2d modulations might be favored at higher µI competition with color superconductivity must be taken into account

I experimental signatures?

I to be worked out ...

July 27, 2012 | Michael Buballa | 20

Conclusions

I Inhomogeneous phases must be considered!

I NJL model with one- and two-dimensional modulations of 〈qq〉:I 1st-order line and critical point covered by an inhomogeneous regionI inhomogeneous phase rather stable w.r.t. vector interactionsI number susceptibility always finite (for GV > 0)I 1d modulations favored at “moderate” µI 2d modulations might be favored at higher µI competition with color superconductivity must be taken into account

I experimental signatures?

I to be worked out ...

July 27, 2012 | Michael Buballa | 20

Conclusions

I Inhomogeneous phases must be considered!

I NJL model with one- and two-dimensional modulations of 〈qq〉:I 1st-order line and critical point covered by an inhomogeneous regionI inhomogeneous phase rather stable w.r.t. vector interactionsI number susceptibility always finite (for GV > 0)I 1d modulations favored at “moderate” µI 2d modulations might be favored at higher µI competition with color superconductivity must be taken into account

I experimental signatures?

I to be worked out ...

July 27, 2012 | Michael Buballa | 20

Conclusions

I Inhomogeneous phases must be considered!

I NJL model with one- and two-dimensional modulations of 〈qq〉:I 1st-order line and critical point covered by an inhomogeneous regionI inhomogeneous phase rather stable w.r.t. vector interactionsI number susceptibility always finite (for GV > 0)I 1d modulations favored at “moderate” µI 2d modulations might be favored at higher µI competition with color superconductivity must be taken into account

I experimental signatures?

I to be worked out ...

July 27, 2012 | Michael Buballa | 20

Collaborators

Dominik Nickel(INT Seattle→ Siemens)

Stefano Carignano(TU Darmstadt)

Daniel Nowakowski(TU Darmstadt)

July 27, 2012 | Michael Buballa | 21