of inhomogeneous chiral phases . Tong-Gyu Lee Kochi U Nov. 21 - 24 - - PowerPoint PPT Presentation

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. .

Fluctuation aspects† and Multidimensionality‡

• f 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 iCPs

. . . . . . 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 iCPs

. . . . . . 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 iCPs

. . . . . . 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 iCPs

. . . . . . 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 iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Inhomogeneous chiral condensates (NJL formalism)

▶ NJL-model Lagrangian (chiral limit):

LNJL = ¯ ψiγµ∂µψ + G [( ¯ ψψ )2 + ( ¯ ψiγ5τaψ )2]

▶ Mean-field approximation (space-dependent condensates):

¯ ψ(⃗

x)ψ(⃗ x) → ⟨ ¯

ψψ⟩(⃗ x), ¯ ψ(⃗

x)iγ5τaψ(⃗ x) → ⟨ ¯

ψiγ5τaψ⟩(⃗ x)δa3

▶ General order parameter: M(⃗

x) ≡ −2G[⟨ ¯ ψψ⟩(⃗ x) + i⟨ ¯ ψiγ5τ3ψ⟩(⃗ x)] ▶ Gap equations (minimizing thermodynamic potential w.r.t. order parameter):

∂VMF(T, µ; M(⃗

x))

∂M(⃗

x)

= 0 ▷ need to know the quark energy spectrum: VMF=−T NfNc/V ∑

n ln

( 2 cosh ( En−µ

2T

)) + ...

▷ need to solve the Dirac eq: [i∂

/ +2G(⟨ ¯ ψψ⟩(⃗

x)+iγ5τ3⟨ ¯

ψiγ5τ3ψ⟩(⃗

x))]ψ = 0

(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 VMF w.r.t. variational parameters (∆, q, ν)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Typical examples (1D modulations)

▶ DCDW modulation: M(z) = ∆eiqz 【Nakano-Tatsumi 2005, cf. Dautry-Nyman 1979; Flude-Ferrell 1964】

⟨ψψ⟩(z) = ∆ cos(qz), ⟨ψiγ5τ3ψ⟩(z) = ∆ sin(qz)

▶ RKC modulation: M(z) = ∆(z) 【Nickel 2009, cf. Schnetz et al. 2004; Thies 2006; Larkin-Ovchinnikov 1964】

⟨ψψ⟩(z) = 2∆√ν 1 + √ν sn ( 2∆z 1 + √ν |ν ) , ⟨ψiγ5τ3ψ⟩(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 iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Inhomogeneous chiral condensates (NJL formalism)

▶ NJL-model Lagrangian (chiral limit):

LNJL = ¯ ψiγµ∂µψ + G [( ¯ ψψ )2 + ( ¯ ψiγ5τaψ )2]

▶ General chiral order parameter:

M(⃗ x) = −2G[⟨ ¯ ψψ⟩(⃗ x) + i⟨ ¯ ψiγ5τ3ψ⟩(⃗ x)]

▶ DCDW/RKC condensate: M(⃗

x) = ∆eiqz or 2∆√ν

1+√ν sn( 2∆z 1+√ν |ν)

▶ Gap equations (minimizing thermodynamic potential w.r.t. variational parameters):

∂VMF(T, µ; M(⃗

x))

∂∆ = 0 and ∂VMF(T, µ; M(⃗

x))

∂q(ν) = 0 ▷ need to know the quark energy spectrum: VMF=−T NfNc/V ∑

n ln

( 2 cosh ( En−µ

2T

)) + ...

▷ need to solve the Dirac eq: [i∂

/ +2G(⟨ ¯ ψψ⟩(⃗

x)+iγ5τ3⟨ ¯

ψiγ5τ3ψ⟩(⃗

x))]ψ = 0

(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 VMF w.r.t. variational parameters (∆, q, ν)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (njl mean-field results in the chiral limit)

▶ DCDW: M(⃗

x) = ∆ exp(iqz) ▷ RKC: M(⃗ x) = ∆′√ ν′sn(∆′z|ν′) ▷ 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 iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (njl mean-field results in the chiral limit)

▶ DCDW: M(⃗

x) = ∆ exp(iqz) ▷ RKC: M(⃗ x) = ∆′√ ν′sn(∆′z|ν′)

▶ Dyson-Schwinger study:【M¨

uller et al. 2013】

▷ QM model:【Carignano-Buballa-Schaefer 2014】 ⇒ 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 iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (njl mean-field results in the chiral limit)

▶ DCDW: M(⃗

x) = ∆ exp(iqz) ▷ RKC: M(⃗ x) = ∆′√ ν′sn(∆′z|ν′) ▷ 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 iCPs

. . . . . . 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 (mc ̸= 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 iCPs

. . . . . . 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 (mc ̸= 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 iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (Beyond the mean-field level)

e.g.) DCDW ground state: ϕT

0 = (∆ cos qz,0,0,∆ sin qz)

(SSB: trans.(rot.) and chiral sym.)

▶ introduce fluctuations around ϕ0: 【TGL-Nakano-Tsue-Tatsumi-Friman 2015】 ϕ(z) = U(βi)ϕ0(z) =   

∆ cos(qz+β3) cos β2 cos β1 ∆ cos(qz+β3) cos β2 sin β1 ∆ cos(qz+β3) sin β2 ∆ sin(qz+β3)

   (= ϕ0 + δϕ) ▶ dispersion relation for gapless modes: ω2 ∼ ak2

z + b(⃗

k2

⊥)2

⇒ spatially anisotropic (owing to the lack of ⃗

k2

⊥-term)

⇒ symmetric under rotations about x-y (transverse) directions, as in sm-A liquid crystals

▶ impacts of fluctuations: ⟨ϕ(z)⟩ = ⟨U(βi)ϕ0(z)⟩ ≃      ∆ cos(qz)e− ∑

i⟨β2 i ⟩/2

∆ sin(qz)e−⟨β2

3 ⟩/2

    

IR

− − → 0 (destroyed) where Gaussian fluctuations are logarithmically divergent at long-wavelength (IR) limit ⟨β2

i=1,2,3⟩ ≃ 1 2∆2

∫

d3k (2π)3 T ω2 IR

− − → ∞ (log div)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (Beyond the mean-field level)

▷ Landau-Peierls instability

【Landau 1937; Peierls 1934】

▶ The DCDW phase is not expected to exist due to thermal fluctuations

⟨ϕ(z)⟩ = 0 ⇒ but exhibits quasi-long-range order (QLRO)

with algebraically decaying correlation function with the nonzero power depending on T

⟨ϕ(z⃗ ez)·ϕ∗(0)⟩ ∼

1 2 ∆2 cos qz(z/z0)−T/T0 z0 = 2q/Λ2

⟨ϕ(x⊥⃗ e⊥)·ϕ∗(0)⟩ ∼

1 2 ∆2(x⊥/x0)−2T/T0 x0 = 1/Λ, T0 = 32πa6,1∆2q

may be practically realized as a quasi-1D phase as in LCs

【TGL-Nakano-Tsue-Tatsumi-Friman 2015】

▶ The same applies to the RKC phase

【Hidaka-Kamikado-Kanazawa-Noumi 2015】

▶ There is no true LRO for inhomogeneous chiral phases with 1D modulations

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . 1D modulations (Beyond mean-field level)

▷ Possibilities to get rid of LP instability

▶ T = 0 limit ⟨ϕ⟩ = ⟨ϕ0 + δϕ⟩ ̸= 0 (LRO)

⇒ stable against quantum fluctuations

(not diverge: ⟨β2⟩∝

∫ d3kω−1̸=0)

▶ External magnetic fields ω2 ∼ ak2

z + b⃗

k2 + O((⃗ k2)2) for B ̸= 0 (b ∝ B)

(cf. ω2 ∼ ˜ ak2

z + O((⃗

k2)2) for B = 0)

⇒ modified dispersion

(explicit rotational symmetry breaking: k2

t -term)

⇒ could be stabilized

(improved: ⟨β2⟩∝ T (Λuv+O(ΛIR)) ̸= 0) 【Hidaka et al.’15, Brauner-Yamamoto‘16】

▶ Finite-size effects long wave-length fluctuations are cutoff by the system size

(effectively stabilized)

IR cutoff as system size: ΛIR = L−1

(no log div: ⟨β2⟩∝ T ln(O(1/ΛIR))∼T ln(O(L)) ̸= 0)

⇒ QLRO can effectively mimic a true LRO (depending on L or experimental resolutions)

【cf. Als-Nielsen et al. 1980; Baym-Friman-Grinstein 1982; Hidaka-Kamikado-Kanazawa-Noumi 2015】

▶ Two- and three-dimensional modulations

(inferred from Landau-Peierls theorem)

similar suppression of IR div can be expected

⇒ stabilization could occur

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▷ no known analytic solutions for 2+1D or 3+1D systems

(unlike purely 1+1D systems)

▷ assume some ans¨ atze for 2D/3D and compare their free energies with 1D cases ▷ there are a few studies for 2D/3D modulations 【Abuki et al. 2012; Carignano et al. 2012】

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▶ one knows only the area around the LP or at T = 0

⇒ multidimensional modulation may be realized in different areas

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▶ some remarkable results in different contexts

⇒ formation of multidimensional crystalline structures is predicted

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▶ another possible way to find the multidimensional crystalline structures

⇒ Thomas-Fermi approximation ⇒ using the expression for Eq obtained from the TFA ⇒ explore the lowest free energy for given 1D, 2D, 3D ans¨ atze ⇒ need to effectively extract derivative terms of the condensate in the model ⇒ but no self-consistency here

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▶ self-consistent way to explore multidimensional structures w/o any ans¨ atz

▷ NJL model:

【cf. Nickel 2009】

LMF = ¯ ψγ0(i∂0 − HD)ψ − M2(r)/4G

(M=−2G(⟨ ¯ ψψ⟩+i⟨ ¯ ψiγ5τ3ψ⟩))

HDψi(r) = Eiψi(r), HD = −iγ0γ · ∇ + 2G ( ⟨ ¯ ψψ⟩ + iγ5τ3⟨ ¯ ψiγ5τ3ψ⟩ )

(isospectral: HD ≡ H+)

= H+ ⊗ H−

(H+(−)=−iγ0γ·∇+γ0M(∗))

Ω = −NfNcTV −1 ∑

Ei ln(2 cosh( Ei−µ 2T

)) +

1 4GV

∫ d3rM2(r) δΩ/δM∗

(r) = 0 ⇒ M (r) = GNfNcV −1 ∑ Eitanh( Ei−µ 2T

)) ¯ ψi(r)(1 − γ5)ψi(r) ▷ finite-difference method:

discretized {Ei} and the corresponding {ψi} can be simultaneously numerically solved

• cf. similar self-consistent way — finite-mode approach

【Wagner 2007; Heinz et al. 2016】

(e.g., ˆ σ(x) = ∑N

i=1⟨ ¯

ψiψi⟩ = ∑10

−10 cneinx)

▷ before investigating arbitrary M(r)

need to correctly reproduce the known result in 1+1D systems 【Ba¸

sar-Dunne-Thies 2009】

(M (r) → M (z) = ∆eiqz or ∆√νsn(∆z|ν): known analytic solutions)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

▶ self-consistent way to explore multidimensional structures w/o any ans¨ atz

▷ 1+1 dimensional model

(GN or NJL2 models) 【cf. Ba¸

sar-Dunne-Thies 2009】

LMF = ¯ ψγ0(i∂0 − HBdG)ψ − M2(x)/4G

(M=−2G(⟨ ¯ ψψ⟩−i⟨ ¯ ψiγ5ψ⟩)≡∆(x)eiqx)

HBdGψi(x) = Eiψi(x) HBdG = −iγ5∂x + γ0 ( 1

2 (1 − γ5)M (x) − 1 2(1 + γ5)M (x)∗)

=

  −i∂x M(x) M(x)∗ i∂x   NLSE

− − − − → M (x) = ∆e2iqz

• r ∆( 2√ν

1+√ν )sn( 2∆x 1+√ν |ν) [ analytic solutions for 1D ]

▷ finite-difference method:

f′

i = 1 2△x (fi+1−fi−1),

f′′

i = 1 △x2 (fi+1+fi−1−2fi)

HBdGψi(x) = Eiψi(x), ψi = {fi(x), gi(x)} ⇔ {−if′

i + Mgi = Eifi

M∗fi + ig′

i = Eigi

− → f′′

i + (Ei − |Mi|2)fi + i M′

i

Mi (Eifi + if′ i ) = 0

both the discretized {Ej} and the corresponding {fj, gj} can be numerically obtained

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Beyond 1D modulations (2D/3D modulations)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

Bonus

Another effect of fluctuations

(if time permits...)

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . brazovskii-dyugaev effect

▷ Boundary features of inhomogeneous chiral phases

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . brazovskii-dyugaev effect

▷ Brazovskii-Dyugaev effect 【Brazovskii ‘75; Dyugaev ‘75; Karasawa-TGL-Tatsumi ‘16; Yoshiike-TGL-Tatsumi in prep.】

21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs

. . . . . . Introduction 1D modulations Beyond 1D Bonus Summary

. . Summary

▶ Inhomogeneous chiral phases should be considered:

▶ intermediate state between HM and QM/CSC

▶ Fluctuation effects are studied: (for both inside and outside the phase)

▶ Landau-Peierls instability ▶ for 1D: no true LRO, while QLRO ▶ stable at T = 0, magnetic fields, 2D/3D, etc. ▶ multidimensional structure of chiral condensates ▶ self-consistent way in progress ▶ Brazovskii-Dyugaev effect ▶ fluctuation-induced 1st-order phase transitions ▶ may lead to a multidimensional structure associated w/ 1st-order tr.?

▶ Implications of compact-star phenomena:

▶ need to consider more:

magnetic fields, β-equilibrium, charge neutrality, strangeness, Coulomb int., etc.

▶ inhomogeneous phases may reside inside CSs 21/11/2016 — NSMAT2016, Sendai Japan — T.-G. Lee (Kochi U) Fluctuation aspects & Multidimensionality of iCPs