Introduction 1D modulations Beyond 1D Bonus Summary . Fluctuation aspects † and Multidimensionality ‡ of inhomogeneous chiral phases . Tong-Gyu Lee （ Kochi U ） Nov. 21 - 24 , 2016 at Graduate School of Science, Tohoku University, Sendai, Japan ‡ w/ N.Yasutake (CIT), T.Maruyama (JAEA), K.Nishiyama and T.Tatsumi (Kyoto U) † w/ R.Yoshiike and T.Tatsumi (Kyoto U) . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Nonvanishing baryon density . ▶ Dense QCD phase diagram Finite-density regime is still less well understood. BESs at RHIC and upcoming facilities (FAIR, NICA, J-PARC, etc) have attracted attention. One might expect a transition to exotic phases due to high densities. Recent theoretical studies predict inhomogeneous phases. Cold and dense area may be relevant for compact stars. . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Inhomogeneous chiral phase . ▶ Conventional picture (focusing on the chiral symmetry) ⇒ chiral order parameter is constant in space (spatially homogeneously condensed) ⇒ what if one allows for the spatial dependence? 【 Nakano-Tatsumi 2005; Nickel 2009; M¨ uller et al. 2013 】 . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Inhomogeneous chiral phase . ▶ Possible/New picture (focusing on the chiral symmetry) ⇒ chiral-transition region is extended (chiral restoration is delayed) ⇒ chiral-restoration picture may be changed (not a conventional 1st-order) 【 cf. Nakano-Tatsumi 2005; Nickel 2009; M¨ uller et al. 2013, etc. 】 . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Outline . . 1 Introduction 2 1D modulations . . . . 3 Beyond 1D . . 4 Bonus 5 Summary . . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Inhomogeneous chiral condensates (NJL formalism) . ▶ NJL-model Lagrangian (chiral limit) : [( ¯ ( ¯ ) 2 + ) 2 ] L NJL = ¯ ψiγ µ ∂ µ ψ + G ψψ ψiγ 5 τ a ψ ▶ Mean-field approximation (space-dependent condensates) : ¯ x ) → ⟨ ¯ ¯ x ) → ⟨ ¯ ψ ( ⃗ x ) ψ ( ⃗ ψψ ⟩ ( ⃗ x ) , ψ ( ⃗ x ) iγ 5 τ a ψ ( ⃗ ψiγ 5 τ a ψ ⟩ ( ⃗ x ) δ a 3 ▶ General order parameter: M ( ⃗ x ) ≡ − 2 G [ ⟨ ¯ x ) + i ⟨ ¯ ψψ ⟩ ( ⃗ ψiγ 5 τ 3 ψ ⟩ ( ⃗ x )] ▶ Gap equations (minimizing thermodynamic potential w.r.t. order parameter) : ∂ V MF ( T, µ ; M ( ⃗ x ) ) = 0 ∂M ( ⃗ x ) ( E n − µ ( )) ▷ need to know the quark energy spectrum: V MF = − T N f N c /V ∑ n ln 2 cosh + ... 2 T ▷ need to solve the Dirac eq: [ i∂ / +2 G ( ⟨ ¯ x ) + iγ 5 τ 3 ⟨ ¯ x ) )] ψ = 0 ψψ ⟩ ( ⃗ ψiγ 5 τ 3 ψ ⟩ ( ⃗ ( H ( M ) ψ = Eψ ) ▷ assume the condensate shape based on known analytic solutions for 1+1D systems ⇒ use possible ans¨ atze for 1D modulations in 3+1D systems 【 cf. Nickel 2009; Ba¸ sar-Dunne-Thies 2009 】 ▷ obtain gap solutions by minimizing V MF w.r.t. variational parameters ( ∆ , q, ν ) . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Typical examples (1D modulations) . ▶ DCDW modulation: M ( z ) = ∆ e iqz 【 Nakano-Tatsumi 2005, cf. Dautry-Nyman 1979; Flude-Ferrell 1964 】 ⟨ ψψ ⟩ ( z ) = ∆ cos( qz ) , ⟨ ψiγ 5 τ 3 ψ ⟩ ( z ) = ∆ sin( qz ) ▶ RKC modulation: M ( z ) = ∆( z ) 【 Nickel 2009, cf. Schnetz et al. 2004; Thies 2006; Larkin-Ovchinnikov 1964 】 2∆ √ ν ( 2∆ z ) ⟨ ψψ ⟩ ( z ) = 1 + √ ν sn 1 + √ ν | ν , ⟨ ψiγ 5 τ 3 ψ ⟩ ( z ) = 0 ( ∆ : amplitude, q : wavenumber, ν ∈ [0 , 1] : elliptic modulus) ▷ Dual chiral density wave (DCDW) ▷ Real kink crystal (RKC) chiral spirals in 3+1D systems periodic domain walls in 3+1D systems . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . Inhomogeneous chiral condensates (NJL formalism) . ▶ NJL-model Lagrangian (chiral limit) : [( ¯ ( ¯ ) 2 + ) 2 ] L NJL = ¯ ψiγ µ ∂ µ ψ + G ψψ ψiγ 5 τ a ψ ▶ General chiral order parameter: x ) = − 2 G [ ⟨ ¯ x ) + i ⟨ ¯ M ( ⃗ ψψ ⟩ ( ⃗ ψiγ 5 τ 3 ψ ⟩ ( ⃗ x )] x ) = ∆ e iqz or 2∆ √ ν ▶ DCDW/RKC condensate: M ( ⃗ 2∆z 1+ √ ν sn( 1+ √ ν | ν ) ▶ Gap equations (minimizing thermodynamic potential w.r.t. variational parameters) : ∂ V MF ( T, µ ; M ( ⃗ x ) ) ∂ V MF ( T, µ ; M ( ⃗ x ) ) = 0 and = 0 ∂ ∆ ∂q ( ν ) ( E n − µ ( )) ▷ need to know the quark energy spectrum: V MF = − T N f N c /V ∑ n ln 2 cosh + ... 2 T ▷ need to solve the Dirac eq: [ i∂ / +2 G ( ⟨ ¯ x ) + iγ 5 τ 3 ⟨ ¯ x ) )] ψ = 0 ψψ ⟩ ( ⃗ ψiγ 5 τ 3 ψ ⟩ ( ⃗ ( H ( M ) ψ = Eψ ) ▷ assume the condensate shape based on known analytic solutions for 1+1D systems ⇒ use possible ans¨ atze for 1D modulations in 3+1D systems 【 cf. Nickel 2009; Ba¸ sar-Dunne-Thies 2009 】 ▷ obtain gap solutions by minimizing V MF w.r.t. variational parameters ( ∆ , q, ν ) . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . 1D modulations (njl mean-field results in the chiral limit) . x ) = ∆ ′ √ ▶ DCDW: M ( ⃗ ν ′ sn(∆ ′ z | ν ′ ) x ) = ∆ exp( iqz ) ▷ RKC: M ( ⃗ ▷ amplitudes for DCDW and RKC condensates ( T = 0 ) ▷ between homo. and inhomo. ⇒ DCDW: discontinuous (1st) ⇒ RKC: continuous (2nd) ▷ between inhomo. and restored ⇒ Both: smoothly to zero (2nd) . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . 1D modulations (njl mean-field results in the chiral limit) . x ) = ∆ ′ √ ▶ DCDW: M ( ⃗ ν ′ sn(∆ ′ z | ν ′ ) x ) = ∆ exp( iqz ) ▷ RKC: M ( ⃗ ▶ Dyson-Schwinger study: 【 M¨ ▷ QM model: 【 Carignano-Buballa-Schaefer 2014 】 uller et al. 2013 】 ⇒ similar features are also obtained from different studies 【 cf. Buballa-Carignano PPNP2015 】 . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . 1D modulations (njl mean-field results in the chiral limit) . x ) = ∆ ′ √ ▶ DCDW: M ( ⃗ ν ′ sn(∆ ′ z | ν ′ ) x ) = ∆ exp( iqz ) ▷ RKC: M ( ⃗ ▷ free energies for DCDW and RKC condensates ( T = 0 ) ⇒ RKC is more favored than DCDW within MFA in the chiral limit 【 Nickel 2009 】 √ ⇒ but the situation is reversed at eB ̸ = 0 【 Frolov et al. 2010; Tatsumi et al. 2014 】 see also Nishiyama et al. 2015, Cao et al. 2016, Abuki 2016. ⇒ what if fluctuation effects are included? . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . 1D modulations (njl mean-field results in the chiral limit) . ▶ massive DCDW: 【 Karasawa-Tatsumi 2015 】 ▷ massive RKC: 【 Nickel 2009 】 ▷ free energies for DCDW and RKC condensates ( T = 0 ) ⇒ RKC is more favored than DCDW within MFA in the chiral limit 【 Nickel 2009 】 ⇒ still qualitatively similar (shrink but survive) in a massive case ( m c ̸ = 0) 【 cf. Nickel 2019; Karasawa et al. 2015 】 ⇒ what if fluctuation effects are included? . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
Introduction 1D modulations Beyond 1D Bonus Summary . 1D modulations (njl mean-field results in the chiral limit) . ▶ DCDW with fluctuations: ▷ RKC with fluctuations: ▷ free energies for DCDW and RKC condensates ( T = 0 ) ⇒ RKC is more favored than DCDW within MFA in the chiral limit 【 Nickel 2009 】 ⇒ still qualitatively similar (shrink but survive) in a massive case ( m c ̸ = 0) 【 cf. Nickel 2019; Karasawa et al. 2015 】 ⇒ what if fluctuation effects are included? . . . . . . 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of i CPs
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