Michael Buballa TU Darmstadt, Germany

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?

Motivation

I QCD phase diagram (NICA version):

Motivation

<qq> = 0<qq> = 0

Motivation

<qq> = 0<qq> = 0

Motivation

<qq> = 0<qq> = 0

Are inhomogeneities important?

I one highlight example:

Phase diagram in the NJL-model

300 350 400 450 500 0

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 ...

300 350 400 450 500 0

Gv=Gs/5

I blue solid line:

inhomogeneous phase

300 350 400 450 500 0

Gv=Gs/5

I blue solid line:

inhomogeneous phase

300 350 400 450 500 0

Gv=Gs/5

I blue solid line:

inhomogeneous phase

300 350 400 450 500 0

Gv=Gs/5

I blue solid line:

inhomogeneous phase

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, ... )

(Overhauser)

Broniowski et al. (1991)

(Overhauser)

Thies, Urlichs (2003)

(Overhauser)

CFLliq

crystal?

nuclear

superconducting

= color

compact star

Alford (2003)

(Overhauser)

Kojo et al. (2011)

Contents

1. Introduction

2. Inhomogeneous chiral symmetry breaking in the NJL model

3. One-dimensional modulations

4. Two-dimensional modulations

5. Conclusions

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)

[(ψψ)2 + (ψiγ5~τψ)2]

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

ln∫DψDψ exp

(∫x∈[0, 1

[(ψψ)2 + (ψiγ5~τψ)2]

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

ln∫DψDψ exp

(∫x∈[0, 1

[(ψψ)2 + (ψiγ5~τψ)2]

⇒ L = ψ(i∂/−m + 2GS(σ + iγ5~τ · ~π)

)ψ − GS

(σ2 + ~π 2)

ln∫DψDψ exp

(∫x∈[0, 1

)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

Mean-field model

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

)≡ γ0 (i∂0 −HMF )

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

Mean-field model

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

)≡ γ0 (i∂0 −HMF )

HMF = HMF [S, P] =

(−i~σ · ~∂ M(~x)M∗(~x) i~σ · ~∂

Mean-field model

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

)≡ γ0 (i∂0 −HMF )

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

Mean-field thermodynamic potential

I thermodynamic potential:

ΩMF (T ,µ; S, P) = −TV

Tr ln(

(i∂0 −HMF + µ))

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

= − 1V

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

d3x|M(~x)−m|2

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!

Tr ln(

(i∂0 −HMF + µ))

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

= − 1V

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

d3x|M(~x)−m|2

Tr ln(

(i∂0 −HMF + µ))

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

= − 1V

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

d3x|M(~x)−m|2

Tr ln(

(i∂0 −HMF + µ))

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

= − 1V

[Eλ − µ

2+ T ln

(1 + e

Eλ−µ

d3x|M(~x)−m|2

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∑

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

Periodic structures

M~qkei~qk ·~x

H~pm ,~pn=

∑~qk

Periodic structures

M~qkei~qk ·~x

H~pm ,~pn=

∑~qk

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

I choose a unit cell ~ai ⇒ ~qk

I choose Fourier components M~qk

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

Mk eikqz

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

homogeneous phases only

240 260 280 300 320 340 360

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)

including inhomogeneous phase

240 260 280 300 320 340 360

µ (MeV)

including inhomogeneous phase

240 260 280 300 320 340 360

µ (MeV)

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)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

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

I Density gets smoothened with increasing µ and T .

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 307.5 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 308 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 309 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 310 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 320 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 330 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 340 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 345 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

240 260 280 300 320 340 360

µ (MeV)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 307.5 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 307.5 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 307.5 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 307.5 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 308 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

normalized density (µ = 308 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 309 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 310 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 320 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

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

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 330 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 340 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

I M(z) =√ν∆ sn(∆z|ν) →

M(z) (µ = 345 MeV)

0 5 10 15 20 25 30 35 40

z (fm)

0 5 10 15 20 25 30 35 40

z (fm)

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

ase-W@0DLB

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

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

Phase diagram

300 350 400 450 500

300 350 400 450 500 0

Gv=Gs/5

300 350 400 450 500

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

µ (MeV)

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

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

Phase diagram

300 350 400 450 500

300 350 400 450 500 0

Gv=Gs/5

300 350 400 450 500

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

µ (MeV)

0 20 40 60 80

100 120 140 160 180

0 0.2 0.4 0.6 0.8 1 1.2

Density (fm-3)

Homogeneous, broken

Inhomogeneous

Restored

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

Phase diagram

300 350 400 450 500

300 350 400 450 500 0

Gv=Gs/5

300 350 400 450 500

µ (MeV)

Gv=Gs/2

300 350 400 450 500 0

µ (MeV)

0 20 40 60 80

100 120 140 160 180

0 0.2 0.4 0.6 0.8 1 1.2

Density (fm-3)

Homogeneous, broken

Inhomogeneous

Restored

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

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

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

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

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

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

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

I amplitudes and wave numbers:I egg carton:

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

free-energy gain at T = 0:

310 320 330 340

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!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

restored

M(x , y ) =3∑

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

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

restoredhomogen. broken

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1d

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1djacobi 1d

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1djacobi 1d

square 2d

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1djacobi 1d

square 2dhexagon 2d

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1djacobi 1d

square 2dhexagon 2d

I 2D not favored in this regime

I more Fourier components:

M(x , y ) =3∑

no effect!

300 310 320 330 340 350

µ (MeV)

I hexagon:

300 310 320 330 340 350

µ (MeV)

310 320 330 340

µ (MeV)

cos 1djacobi 1d

square 2dhexagon 2d

M(x , y ) =3∑

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)

⇒ “egg carton” local minimum

I higher chemical potentials

400 500 600 700 800 900 1000

µ (MeV)

square-hexjacobi-hex

jacobi-square

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

I µ > 900 MeV: hexagon favored

I free energy:

100 200 300 400

Qx (MeV)

400 500 600 700 800 900 1000

µ (MeV)

jacobi-square

I free energy:

100 200 300 400

Qx (MeV)

400 500 600 700 800 900 1000

µ (MeV)

jacobi-square

I free energy:

100 200 300 400

Qx (MeV)

400 500 600 700 800 900 1000∆

µ (MeV)

jacobi-square

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 ...

Conclusions

Collaborators

Dominik Nickel(INT Seattle→ Siemens)

Stefano Carignano(TU Darmstadt)

Daniel Nowakowski(TU Darmstadt)

Michael Buballa TU Darmstadt, Germany

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