Fluctuations in dual chiral density wave / T.-G. Lee et al.

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1 Fluctuations in dual chiral density wave Tong-Gyu Lee JAEA/Kochi Univ (YITP, Kyoto Univ) in collaboration with Eiji Nakano and Yasuhiko Tsue (Kochi Univ) Toshitaka Tatsumi (Kyoto Univ) and Bengt Friman (GSI) <in preparation> Quarks and Compact Stars 2014 (Oct 20-22, 2014 KIAA, PKU, Beijing, China) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

QCD phase diagram So far Assumption: constant order-parameters in space uniform chiral condensated phases Chiral phase boundary 1st-order phase transition a critical point Maybe not so simple

2 QCD phase diagram So far Assumption: constant order-parameters in space uniform chiral condensated phases Chiral phase boundary 1st-order phase transition a critical point Maybe not so simple Schematic QCD phase diagram Recently Ansatz: spatially modulated order-parameters nonuniform chiral condensated phases Chiral phase boundary chiral inhomogeneous phases critical point as a Lifshitz point We focus on the spatially modulated chiral condensates near a Lifshitz point QCS2014 (Oct 20-22, PKU, Beijing, China) Recent conjectured phase diagram Fukushima-Hatsuda (2011) Fluctuations in dual chiral density wave / T-G Lee et al

3 Prediction from chiral effective theories Ginzburg-Landau analysis: model-independent approach near critical point (chiral limit) ω(ϕ(x)) = a ϕ(x) 2 + b ϕ(x) 4 + c ϕ(x) 2 + d 2 ϕ(x) 2 + e ϕ(x) 2 ϕ(x) 2 + f ϕ(x) 6 6th order term with d, e, f (> 0): stabilization of potential/system energetically favored if c < 0 Nickel PRL103(2009) Model calculations within mean field approximation: [Chiral phase boundary] 1st-order critical line Inhomogeneous phase Nickel PRD80(2009) Critical point (CP) (a=b=0) Lifshitz point (LP) (a=c=0) Stable and robust wrt vector interactions (unlike the conventional 1st-order picture) Carignano-Nickel-Buballa PRD82(2010); Fukushima PRD86(2012) Survive against finite U l, H, h s, m c, µ I, π-cond, Dirac sea, etc (expand/shrink but not disappear) Nickel (2009); Tatsumi et al (2014); Moreira et al (2014); Abuki (2014); Carignano et al (2014) etc Support from SDE with QCD Lag, as well as Large N c limit Deryagin-Grigoriev-Rubakov(1992) Müller-Buballa-Wambach PLB727(2013) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

4 Prediction from chiral effective theories Ginzburg-Landau analysis: model-independent approach near critical point (chiral limit) ω(ϕ(x)) = a ϕ(x) 2 + b ϕ(x) 4 + c ϕ(x) 2 + d 2 ϕ(x) 2 + e ϕ(x) 2 ϕ(x) 2 + f ϕ(x) 6 6th order term with d, e, f (> 0): stabilization of potential/system energetically favored if c < 0 Nickel PRL103(2009) Model calculations within mean field approximation: [Chiral phase boundary] 1st-order critical line Inhomogeneous phase Nickel PRD80(2009) Critical point (CP) (a=b=0) Lifshitz point (LP) (a=c=0) Stable and robust wrt vector interactions (unlike the conventional 1st-order picture) Carignano-Nickel-Buballa PRD82(2010); Fukushima PRD86(2012) Survive against finite U l, H, h s, m c, µ I, π-cond, Dirac sea, etc (expand/shrink but not disappear) Nickel (2009); Tatsumi et al (2014); Moreira et al (2014); Abuki (2014); Carignano et al (2014) etc Support from SDE with QCD Lag, as well as Large N c limit Deryagin-Grigoriev-Rubakov(1992) Müller-Buballa-Wambach PLB727(2013) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

5 Prediction from chiral effective theories Ginzburg-Landau analysis: model-independent approach near critical point (chiral limit) ω(ϕ(x)) = a ϕ(x) 2 + b ϕ(x) 4 + c ϕ(x) 2 + d 2 ϕ(x) 2 + e ϕ(x) 2 ϕ(x) 2 + f ϕ(x) 6 6th order term with d, e, f (> 0): stabilization of potential/system energetically favored if c < 0 Nickel PRL103(2009) Chiral inhomogeneous phases vs [Chiral phase boundary] 1st-order critical line Inhomogeneous phase Nickel PRD80(2009) Critical point (CP) (a=b=0) Lifshitz point (LP) (a=c=0) Stable and robust wrt vector interactions (unlike the conventional 1st-order picture) Carignano-Nickel-Buballa PRD82(2010); Fukushima PRD86(2012) Survive against finite U l, H, h s, m c, µ I, π-cond, Dirac sea, etc (expand/shrink but not disappear) Nickel (2009); Tatsumi et al (2014); Moreira et al (2014); Abuki (2014); Carignano et al (2014) etc Support from SDE with QCD Lag, as well as Large N c limit Deryagin-Grigoriev-Rubakov(1992) Müller-Buballa-Wambach PLB727(2013) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

6 Prediction directly from qcd Ginzburg-Landau analysis: model-independent approach near critical point (chiral limit) ω(ϕ(x)) = a ϕ(x) 2 + b ϕ(x) 4 + c ϕ(x) 2 + d 2 ϕ(x) 2 + e ϕ(x) 2 ϕ(x) 2 + f ϕ(x) 6 6th order term with d, e, f (> 0): stabilization of potential/system energetically favored if c < 0 Nickel PRL103(2009) Possible emergence of inhom phases : [Chiral phase boundary] 1st-order critical line Inhomogeneous phase Nickel PRD80(2009) Critical point (CP) (a=b=0) Lifshitz point (LP) (a=c=0) Stable and robust wrt vector interactions (unlike the conventional 1st-order picture) Carignano-Nickel-Buballa PRD82(2010); Fukushima PRD86(2012) Survive against finite U l, H, h s, m c, µ I, π-cond, Dirac sea, etc (expand/shrink but not disappear) Nickel (2009); Tatsumi et al (2014); Moreira et al (2014); Abuki (2014); Carignano et al (2014) etc Support from SDE with QCD Lag, as well as Large N c limit Deryagin-Grigoriev-Rubakov(1992) Müller-Buballa-Wambach PLB727(2013) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

7 Lifshitz point Schematic phase diagram with LP Lifshitz(1941); Hornreich et al(1975); Diehl(2002) LP : a point with anisotropic scale invariance a disordered phase meet at a uniform ordered phase a modulated ordered phase with q ( LP arises in systems with modulated structures ) g : Nonordering field (intensive variables) (eg, magnetic field, pressure, chemical potential, etc) Candidates for nonuniform chiral condensates in QCD Review: Buballa-Carignano(2014) Chiral Density Waves (CDWs) or Chiral Spirals (CSs) Deryagin-Grigoriev-Rubakov(1992) FF type: Dual Chiral Density Waves (DCDWs) Nakano-Tatumi(2005) LO type: Real Kink Crystals (RKCs) Nickel(2009) Solitonic Chiral Condensates (SCCs) Analytic solutions in 1+1D systems within lower-d models Schön-Thies(2000); Başar-Dunne(2008); Başar-Dunne-Thies(2009) Quarkyonic Chiral Spirals (QyCSs) Kojo-Hidaka-McLerran-Pisarski(2010) Interweaving chiral spirals in 2+1D systems Kojo-Hidaka-Fukushima-McLerran-Pisarski(2012) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

8 Lifshitz point Schematic phase diagram with LP Lifshitz(1941); Hornreich et al(1975); Diehl(2002) LP : a point with anisotropic scale invariance a disordered phase meet at a uniform ordered phase a modulated ordered phase with q ( LP arises in systems with modulated structures ) g : Nonordering field (intensive variables) (eg, magnetic field, pressure, chemical potential, etc) Candidates for nonuniform chiral condensates in QCD Review: Buballa-Carignano(2014) Chiral Density Waves (CDWs) or Chiral Spirals (CSs) Deryagin-Grigoriev-Rubakov(1992) FF type: Dual Chiral Density Waves (DCDWs) Nakano-Tatumi(2005) LO type: Real Kink Crystals (RKCs) Nickel(2009) Solitonic Chiral Condensates (SCCs) Analytic solutions in 1+1D systems within lower-d models Schön-Thies(2000); Başar-Dunne(2008); Başar-Dunne-Thies(2009) Quarkyonic Chiral Spirals (QyCSs) Kojo-Hidaka-McLerran-Pisarski(2010) Interweaving chiral spirals in 2+1D systems Kojo-Hidaka-Fukushima-McLerran-Pisarski(2012) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

9 Lifshitz point Schematic phase diagram with LP Lifshitz(1941); Hornreich et al(1975); Diehl(2002) LP : a point with anisotropic scale invariance a disordered phase meet at a uniform ordered phase a modulated ordered phase with q ( LP arises in systems with modulated structures ) g : Nonordering field (intensive variables) (eg, magnetic field, pressure, chemical potential, etc) Candidates for nonuniform chiral condensates in QCD Review: Buballa-Carignano(2014) Chiral Density Waves (CDWs) or Chiral Spirals (CSs) Deryagin-Grigoriev-Rubakov(1992) FF type: Dual Chiral Density Waves (DCDWs) Nakano-Tatumi(2005) LO type: Real Kink Crystals (RKCs) Nickel(2009) Solitonic Chiral Condensates (SCCs) Analytic solutions in 1+1D systems within lower-d models Schön-Thies(2000); Başar-Dunne(2008); Başar-Dunne-Thies(2009) Quarkyonic Chiral Spirals (QyCSs) Kojo-Hidaka-McLerran-Pisarski(2010) Interweaving chiral spirals in 2+1D systems Kojo-Hidaka-Fukushima-McLerran-Pisarski(2012) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

structures ) g : Nonordering field (intensive variables) (eg, magnetic field, pressure, chemical potential, etc) 2D orders or multi-d

10 Lifshitz point Schematic phase diagram with LP Lifshitz(1941); Hornreich et al(1975); Diehl(2002) LP : a point with anisotropic scale invariance a disordered phase meet at a uniform ordered phase a modulated ordered phase with q ( LP arises in systems with modulated structures ) g : Nonordering field (intensive variables) (eg, magnetic field, pressure, chemical potential, etc) 2D orders or multi-d modulations (not only 1D orders) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

11 Motivation For the inhomogeneous chiral condensates Many studies have been performed so far: possible ground states, most favored shape, fates under various environments There is almost no discussion on fluctuations and stabilities in such phases: stability against quantum and thermal fluctuations are still poorly understood In the context of pion condensation, it has been semi-classically discussed ( 1D structure is unstable due to Landau-Peierls instability at T >0 Baym-Friman-Grinstein(1982) ) similar discussions in liquid crystals, FFLO superconductors/superfluids, etc We check whether or not the chiral inhomogeneous structure is stable to fluctuations QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

Intro Lagrangian SSB and NG modes Low-energy collective excitations Impacts of low energy fluctuations Summary Motivation I Inhomogeneous chiral phases could be realized in QCD or

We investigate the low-energy collective excitations (Nambu-Goldstone modes) in a spatially modulated chiral condensed phase (eg, DCDW phase) First of all, we introduce ﬂuctuations

12 Intro Lagrangian SSB and NG modes Low-energy collective excitations Impacts of low energy fluctuations Summary Motivation I Inhomogeneous chiral phases could be realized in QCD or Compact star core I What is the physical dof emerging from the inhomogeneous chiral phase? We investigate the low-energy collective excitations (Nambu-Goldstone modes) in a spatially modulated chiral condensed phase (eg, DCDW phase) First of all, we introduce ﬂuctuations of order parameter, and discuss the stabilities of 1D-ordered chiral-condensed phase at zero and ﬁnite temperatures, using eﬀective action for ﬂuctuation ﬁelds QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

13 Outline 1 Intro 2 Lagrangian 3 SSB and NG modes 4 Low-energy collective excitations 5 Impacts of low energy fluctuations 6 Summary QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

Intro Lagrangian SSB and NG modes Low-energy collective excitations Impacts of low energy fluctuations Summary How does the order parameter modulate?

14 Intro Lagrangian SSB and NG modes Low-energy collective excitations Impacts of low energy fluctuations Summary How does the order parameter modulate? ( We consider here the one-dimensional modulation ) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

Two types of ground states with 1D modulations Spiral type complex ground state: Flude-Ferrell (1964); Nakano-Tatsumi (2005); Kojo et al (2010) etc ϕ 1 = e iqz Sinusoidal type real ground state:

15 Two types of ground states with 1D modulations Spiral type complex ground state: Flude-Ferrell (1964); Nakano-Tatsumi (2005); Kojo et al (2010) etc ϕ 1 = e iqz Sinusoidal type real ground state: Larkin-Ovchinnikov (1965); Nickel (2009) etc ϕ 2 = sin(qz) ( : constant amplitude, q: wave-vector modulation) FF-type ground state Flude-Ferrell (1964) We consider the DCDW ground state in the flavor-su(2) case: eg, dual chiral density wave (DCDW) or chiral spiral (CS) = ψe iγ 5 τ 3 q r ψ : finite momentum in z-direction QCS2014 (Oct 20-22, PKU, Beijing, China) ψψ = cos(qz), ψiγ 5τ 3ψ = sin(qz) Fluctuations in dual chiral density wave / T-G Lee et al

16 3+1D chiral symmetric effective Lagrangian we construct effective Lagrangian using GL expansion of ϕ and QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

17 Effective Lagrangian in terms of ϕ and Chiral symmetric effective Lagrangian density: L = c 2 t ϕ t ϕ V Potential term: V = a 2 (ϕ ϕ) + a 4,1 (ϕ ϕ) 2 + a 4,2 ( ϕ ϕ) + a 6,1 ( 2 ϕ 2 ϕ) + a 6,2 ( ϕ ϕ)(ϕ ϕ) + a 6,3 (ϕ ϕ) 3 + a 6,4 (ϕ ϕ) 2 hϕ ( up to 6th order in space-time derivatives ( t, ) and chiral order-parameter field ϕ T = (σ, π) ) background medium without Lorenz invariance 1st time-derivative term (real field ϕ) ic 1 ϕ t ϕ = 0 6th-order terms with a 6,i > 0 a stable inhomogeneous phase chiral order-parameter field ϕ ( ) T σ ψψ, π ψiγ 5 τψ QCS2014 (Oct 20-22, PKU, Beijing, China) for simplicity c 2 = 1 explicit breaking term h = 0 (chiral limit) isospin symmetric systems 1 2 ϵ ijklµ ij Q kl = 0 isospin density Q = i(ϕ i t ϕ j ϕ j t ϕ i ) Fluctuations in dual chiral density wave / T-G Lee et al

18 Chiral inhomogeneous phase (q 0 and 0) Stationary condition for q: V(ϕ 0 ) q = 0 q 2 ( a 4,2 + 2a 6,1 q 2 + a 6,2 2) = 0 Stationary condition for : V(ϕ 0 ) = 0 [ a 2 + a 4,2 q 2 + a 6,1 q 4 + 2(a 4,1 + a 6,2 q 2 ) 2 + 3a 6,3 4] = 0 (a specific coefficient set depending on thermodynamic environment singles out the energetically favored phase) Normal phase: q = = 0 Chiral condensed phase (q = 0): q = 0, 0 Chiral inhomogeneous phase (q 2 = (a 4,2 + a 6,2 2 )/2a 6,1 ): q 0, 0 QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

19 Symmetry breaking and Nambu-Goldstone modes in the DCDW phase ϕ 0 case QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

20 Symmetry breaking and NG modes Lagrangian density: L = c 2 t ϕ t ϕ a 2 (ϕ ϕ) a 4,1 (ϕ ϕ) 2 a 4,2 ( ϕ ϕ) +a 6,1 2 ϕ 2 ϕ + a 6,2 ( ϕ ϕ)(ϕ ϕ) + a 6,3 (ϕ ϕ) 3 + a 6,4 (ϕ ϕ) 2 Ground state: ϕ 0 = (cos qz, 0, 0, sin qz) T Symmetry breaking in DCDW phase with ϕ 0 finite q and SU(2) SU(2) symmetry in part Translation in z direction Infinitesimal translations: (σ σ β π, π π α π + βσ, s s + z) cos(qz) [qs(t, x) + β 3 (t, x)] sin qz ϕ 0 = 0 0 ϕ 0 + β 1 (t, x) cos qz α 2 (t, x) sin qz β 2 (t, x) cos qz + α 1 (t, x) sin qz sin(qz) [qs(t, x) + β 3 (t, x)] cos qz ( α(t, x), β(t, x): su(2) su(2) rotation-parameters, s(t, x): z-direction displacement-parameter) ϕ 0 is invariant if qs(t, x) + β 3 (t, x) = 0 ( ie, ϕ 0 + δϕ ϕ 0 ) isospin-translation locking symmetry QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

21 Symmetry breaking and NG modes SU(2) SU(2) rotation and translation symmetries Two linear combinations: qs(t, x) + β 3 (t, x) = 0 H (Unbroken symmetry) Isospin-Translation locking symmetry qs(t, x) + β 3 (t, x) 0 G/H (Broken symmetry) NG mode β 3 (:= qs(t, x) + β 3 (t, x)) which is described out-of-phase operation: qs(t, x) + β 3 (t, x) 0 QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

22 Symmetry breaking and NG modes Lagrangian density: L = c 2 tϕ tϕ a 2 (ϕ ϕ) a 4,1 (ϕ ϕ) 2 a 4,2 ( ϕ ϕ) +a 6,1 2 ϕ 2 ϕ + a 6,2 ( ϕ ϕ)(ϕ ϕ) + a 6,3 (ϕ ϕ) 3 + a 6,4 (ϕ ϕ) 2 Ground state: ϕ 0 = (cos qz, 0, 0, sin qz) T Symmetry breaking in DCDW phase with ϕ 0 finite q and SU(2) SU(2) rotation and spatial translation symmetries Infinitesimal translations: (σ σ β π, π π α π + βσ, s s + z) cos(qz) [qs(t, x) + β 3 (t, x)] sin qz ϕ 0 = 0 0 ϕ 0 + β 1 (t, x) cos qz α 2 (t, x) sin qz β 2 (t, x) cos qz + α 1 (t, x) sin qz sin(qz) [qs(t, x) + β 3 (t, x)] cos qz ( α(t, x), β(t, x): su(2) su(2) rotation-parameters, s(t, x): z-direction displacement-parameter) Rotations with β 1 and α 2 (or β 2 and α 1 ) [ϕ 0 ϕ 0 + δϕ] One of which will be redundant in promoting to local fields Corresponding NG modes: β 1 = β 1 (t, x) and β 2 = β 2 (t, x) (Only β 1 (t, x) or β 2 (t, x) becomes relevant if q = 0) QCS2014 (Oct 20-22, PKU, Beijing, China) Low-Manohar(2002) Fluctuations in dual chiral density wave / T-G Lee et al

23 Symmetry breaking and NG modes Lagrangian density: L = c 2 t ϕ t ϕ a 2 (ϕ ϕ) a 4,1 (ϕ ϕ) 2 a 4,2 ( ϕ ϕ) +a 6,1 2 ϕ 2 ϕ + a 6,2 ( ϕ ϕ)(ϕ ϕ) + a 6,3 (ϕ ϕ) 3 + a 6,4 (ϕ ϕ) 2 Ground state: ϕ 0 = (cos qz, 0, 0, sin qz) T Symmetry breaking in DCDW phase with ϕ 0 finite q and SU(2) SU(2) rotation and spatial translation symmetries Spatial rotation (x- and y-axes rotations) eg) x-axis rotation: e iqz e iq(z cos θ(t, x)+y sin θ(t, x)) (θ(t, x): angle parameter) However NG mode β 3 (t, x) can completely describe the local modes θ(t, x) θ(t, x): Redundant Low-Manohar(2002); Watanabe-Murayama(2013); Hayata-Hidaka( ) NG modes in DCDW with ϕ 0 dim(g/h) = 3 ( 9 3 due to linear combination of generators and redundancy of NG modes ) β = β(t, x) (NG modes can be identified by axial isospin rotations) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

24 Low-energy collective excitations we introduce fluctuation fields in the ϕ 0 phase QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

25 Low-energy collective excitations Fluctuations on the DCDW phase with ϕ 0 = (cos qz, 0, 0, sin qz) T ϕ = ( + δ) ( cos(qz + β3 ) cos β 2 cos β 1 cos(qz + β 3 ) cos β 2 sin β 1 cos(qz + β 3 ) sin β 2 sin(qz + β 3 ) ) = ( + δ)u( β) ( cos(qz) 0 0 sin(qz) ) where δ: amplitude fluctuation, β = {β1, β 2, β 3 }: 4D-sphere of chiral circle (S 3 ), and U( β) := e β 1 L 1 e β 2 L 2 e β 3 L3 with axial isospin generators L 1,2,3 displacement of z-direction = rotation by β 3 under above parametrization ϕ = cos qz 0 0 sin qz + β 3 sin qz β 1 cos qz β 2 cos qz β 3 cos qz + O(β 2 ) we get expected fluctuations of 1st order we can obtain a low-energy effective theory for the fluctuation fields δ and β (NG modes) QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

26 Low-energy collective excitations Lagrangian L = d 4 xl, up to 2nd-order of δ and β(x): L = ( 0 δ) ( 0 βu ) ( 0 β 3 ) 2 ( V δ + V δβ + V β ), Euclidean action in Fourier space: dk T n dk 3 /(2π) 3, k = (2πnT ( iω), k) S E = ( δ dk ) T ( (k) S 1 β3 (k) δ0 (k) g(k) g(k) S 1 0 (k) ( β dk ) T ( T (k) S 1 β T 0 (k) G(k) (k + 2qẑ) G(k) S 1 0 (k + 2qẑ) ) ( δ(k) β 3 (k) ) ) ( β T (k) β T (k + 2qẑ) ) Amplitude δ and β 3-fluctuation fields mix for nonzero wavenumber q Fluctuation fields β T (= β 1,2 ) mix among themselves with different momentum k and k + 2qẑ QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

27 Low-energy collective excitations Dispersion relations for mixing δ and β 3 ω+ 2 M 2 + a 6,1 [4q 2 kz 2 + ( k 2 )2 ] + a 6,4 2 k 2 + A k 2 k2 z + Bkz 4 ω 2 a 6,1 [4q 2 kz 2 + ( k 2 )2 ] A k 2 k2 z Bkz 4 (massless mode ω is still soft in x-y directions) Physically, absence of k 2 leads to redundancy of twist modes in liquid crystal nature This absence comes from rotational symmetry of the system QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

28 Low-energy collective excitations Dispersion relations for β T (= β 1,2 ) ω 2 k = a 6,1[4q 2 k 2 z + ( k 2 )2 ] a 6,1 2k 2 z( k 2 )2 4q 2 + 6qk z + 2k 2 z + k 2 2nd term is higher-order correction of O(k 6 ) from the interactions with background modulation QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

29 Impacts of low-energy fluctuations stabilities to the low-energy fluctuations of order parameter, evaluating long-range correlations QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

30 Impacts of low-energy fluctuations Low-energy fluctuations ( + δ)u(β i )ϕ 0 = U(β i )ϕ 0 + δu(β i )ϕ 0 where U(β i )ϕ 0 = δu(β i )ϕ 0 = cos(qz) exp( i β2 i /2) 0 0 sin(qz) exp( β 2 3 /2) sin(qz) δβ 3 exp( i β2 i /2) 0 0 cos(qz) δβ 3 exp( β 2 3 /2) Here, 2nd-order fluctuations δβ 3 = 0, β1, (under harmonic approximation; eg, cos(qz + β 3 ) = cos(qz) exp ( β 2 3 /2)) d 3 k (2π) 3 T ω k 2, β d 3 k (2π) 3 T ω 2 which of all, β 2 1,2,3 T >0, logarithmically diverge due to soft modes in x-y direction QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

31 Impacts of low-energy fluctuations Off-diagonal long range order ( + δ)u(β i )ϕ 0 = 0 low-energy fluctuations wash out the order parameter at T > 0 which, however, does not immediately lead to no existence of DCDW phase which possibly exists as a quasi-1d phase Landau-Lifshitz(1969); Baym-Friman-Grinstein (1982) (QLRO through Kosterlitz-Thouless transition Kosterlitz-Thouless(1973) ) Long-range scalar correlations of the order parameter in the z direction ϕ(zẑ) ϕ (0) cos qz(z/z 0 ) T /T 0 which decays with some power of spatial distance correlation length is effectively finite Baym-Friman-Grinstein (1982) Quantum fluctuations δβ 3 = 0, β1, d 3 k (2π) 3 T, β ω 2 k d 3 k (2π) 3 which of all, β 2 1,2,3 T =0, do not diverge; ( +δ)u(β i )ϕ 0 =0 low-energy fluctuations are not so strong to wash out the order parameter at T = 0 T ω QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

32 Summary QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

33 Summary Remarks Collective excitations in DCDW phase Flavor-translation locking symmetry Would-be NG modes mix with amplitude fluctuations Dispersion relations is spatially anisotropic (x-y is soft) Low-energy fluctuations tend to wash out the order-parameter Long-range correlations algebraically decay DCDW phase can be only realized as a quasi-1d ordered phase Prospects 2D-ordered phase Finite isospin density Under magnetic fields LO-type case Possible phenomenological implications: thermodynamic/transport properties of in-medium QCD or Compact-star core QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

34 Thank you for your attention and patience! QCS2014 (Oct 20-22, PKU, Beijing, China) Fluctuations in dual chiral density wave / T-G Lee et al

Ginzburg-Landau approach to inhomogeneous chiral condensates

Ginzburg-Landau approach to inhomogeneous chiral condensates Hiroaki Abuki (Science University of Tokyo) in collaboration with Daisuke Ishibashi, Katsuhiko Suzuki Ref: Abuki, Ishibashi, Suzuki, arxiv: