Ginzburg-Landau approach to inhomogeneous chiral phases of QCD

Transcription

1 Ginzburg-Landau approach to inhomogeneous chiral phases of QCD Hiroaki Abuki (a) thanks to Daisuke Ishibashi (b) and Katsuhiko Suzuki (a) (a) Tokyo University of Science, (b) Water Service of Tokyo City Office Ref: Phys. Rev. D85, 74 (1)

2 Inhomogeneous chiral condensate? T CP µ

3 Inhomogeneous chiral condensate? What?: Halfway state = femtoscale ordered phase separation) T D. Nickel, PRL9, PRD9 S. Carignano, M. Buballa, D.Nickel 1 CP µ

4 Inhomogeneous chiral condensate? What?: Halfway state = femtoscale ordered phase separation) Why?: Compromise of two conflicting effects: net quark density q-qbar pair density q + q qq T D. Nickel, PRL9, PRD9 S. Carignano, M. Buballa, D.Nickel 1 CP µ

5 Inhomogeneous chiral condensate? What?: Halfway state = femtoscale ordered phase separation) Why?: Compromise of two conflicting effects: net quark density q-qbar pair density Hierarchical chiral restoration at T= q + q qq T D. Nickel, PRL9, PRD9 S. Carignano, M. Buballa, D.Nickel 1 CP µ

6 Inhomogeneous chiral condensate? What?: Halfway state = femtoscale ordered phase separation) T D. Nickel, PRL9, PRD9 S. Carignano, M. Buballa, D.Nickel 1 Why?: Compromise of two conflicting effects: net quark density q-qbar pair density Hierarchical chiral restoration at T= Similar example: FFLO state q + q qq CP Pauli paramagnetism vs superconductivity µ

7 Two types of 1D Chiral Crystal Spatial modulation in the real space Larkin-Ovchinnikov (LO) type Spatial modulation (LO) qq Thies 6, Nickel 9 Anisotropy in the momentum space Flude-Ferrel (FF) type q + q c.f. Carignano, Buballa, Nickel 1 z Dual chiral density wave; Nakano, Tatsumi (4) Quarkyonic chiral spiral; Kojo, Hidaka, et al. (1)

8 Aims 1. Multidimensional modulations near CP? Low dimensional modulation in 3D might be unstable at finite T c.f. Landau-Peierls Theorem: Baym, Friman, Grinstein, NPB198 Several works related: Quarkyonic chiral spiral; Kojo et al.(11), NJL model analysis; Carignoano, Buballa(11).. How does the phase structure change going away from the critical point? Rich phase structure even in the chiral limit?

9 Rich phase structure? New state may appear at low T region far from LP Two types of FFLO states & New critical point 8th order terms in GL are needed! Lifshitz Point (LP) Matsuo, Higashitani, Nagato, Nagai, JPSJ 1997

10 Aims 1. Multidimensional modulations near CP?

11 Ginzburg-Landau approach Expansion of Free energy w.r.t. the condensate and its spatial derivative D. Nickel, PRL9 GL = M(x) M(x) M(x) ( M) M ( M) ( M) p i M i j M p j M M M M ij Based on the symmetry & model independent in the vicinity of CP; Also applicable for multidimensional modulations

12 GL phase diagram (1D structure) D. Nickel, PRL9 sgn( 4 ) SB -(8/3) -(36/5) -(16/3) Wigner (symmetric) M(z) [ 6 ]

13 GL phase diagram (1D structure) D. Nickel, PRL9 sgn( 4 ) SB -(8/3) -(36/5) -(16/3) Wigner (symmetric) M(z) [ 6-1 ] M sn (z) = qsn(qz, ) Solitonic chiral condensate Thies (6), Nickel (9)

14 Multidimensional modulations? D square lattice : M D-LO = M ave (sin(qx) + sin(qy)) egg-carton ansatz ; Carignano, Buballa (11) 3D cubic lattice : M 3D-LO = 3 M ave(sin(qx) + sin(qy) + sin(qz)) D square lattice contour plot 3D cubic lattice contour plot

15 . sgn( 4 ) SB [ 6-1 ] -(8/3) -(36/5) -(16/3) Wigner M(z) [ 4 3 / 6 ] D D 1D Homogeneous Wigner SN M sn (z) = qsn(qz, ) SN Wigner LO (1D) LO (D) LO (3D) aα [aα 6-1 ] M 1D-LO = M ave sin(qx) M D-LO = M ave (sin(qx) + sin(qy)) M 3D-LO = 3 M ave(sin(qx) + sin(qy) + sin(qz))

16 sgn( 4 ) SB [ 6-1 ] -(8/3) -(36/5) -(16/3) Wigner M(z) Averaged mass as an order parameter [ 4 3 / 6 ] LO (1D) aα [aα 6-1 ] LO (D) M 1D-LO = M ave sin(qx) M ave [ 4 1/ / 6 1/ ] M D-LO.4= M ave (sin(qx) + sin(qy)).3. M 3D-LO =.1 3D D 1D Homogeneous Wigner SN M sn (z) = qsn(qz, ) M ave = M(x) aα [aα 6-1 ].6.5 1D D 3D SN LO (1D) LO (D) LO (3D) SN LO (3D) SN (solitonic wave) Homogeneous state Wigner 3 M ave(sin(qx) + sin(qy) + sin(qz))

17 GL expansion at the critical point Optimizing over wavevector q, the potential can be expanded in powers of M 1D = a 3a 4 16 M ave + 6 a 4 4 M 4 ave + O(M 6 ave) D = a 3a 4 16 M ave + 9 a 4 4 M 4 ave + O(M 6 ave) 3D = a 3a 4 16 M ave + 1 a 4 4 M 4 ave + O(M 6 ave)

18 GL expansion at the critical point Optimizing over wavevector q, the potential can be expanded in powers of M 1D = a 3a 4 16 M ave + 6 a 4 4 M 4 ave + O(M 6 ave) D = a 3a 4 16 M ave + 9 a 4 4 M 4 ave + O(M 6 ave) 3D = a 3a 4 16 M ave + 1 a 4 4 M 4 ave + O(M 6 ave)

19 GL expansion at the critical point Optimizing over wavevector q, the potential can be expanded in powers of M 1D = a 3a 4 16 M ave + 6 a 4 4 M 4 ave + O(M 6 ave) D = a 3a 4 16 M ave + 9 a 4 4 M 4 ave + O(M 6 ave) 3D = a 3a 4 16 M ave + 1 a 4 4 M 4 ave + O(M 6 ave) quadratic coeff. critical point is shared

20 GL expansion at the critical point Optimizing over wavevector q, the potential can be expanded in powers of M 1D = a 3a 4 16 M ave + 6 a 4 4 M 4 ave + O(M 6 ave) D = a 3a 4 16 M ave + 9 a 4 4 M 4 ave + O(M 6 ave) 3D = a 3a 4 16 M ave + 1 a 4 4 M 4 ave + O(M 6 ave) quadratic coeff. critical point is shared

21 GL expansion at the critical point Optimizing over wavevector q, the potential can be expanded in powers of M 1D = a 3a 4 16 M ave + 6 a 4 4 M 4 ave + O(M 6 ave) D = a 3a 4 16 M ave + 9 a 4 4 M 4 ave + O(M 6 ave) 3D = a 3a 4 16 M ave + 1 a 4 4 M 4 ave + O(M 6 ave) quadratic coeff. critical point is shared quartic coeff. WΩ 1D < WΩ D < WΩ 3D

22 Aims. How does the phase structure change going away from the critical point?

23 Need to go beyond 6th order Derivative expansion straightforwardly applied Abuki, Ishibashi, Suzuki (11) GL = M(x) M(x) M(x) M(x) ( M) M ( M) ( M) + 8 M 4 ( M) + 4a ( M) 4 + 4b M M( M) 8 + 4c M ( M) + 6 ( M) Quark loop diagrams yield: = 14, 4a = 1 5, 4b = 18 5, 4c = 14 5, 6 = 1 5 Only one additional parameter aα 8 (>)!

24 Departing the critical point (aα 6>) [ 6 4 / 8 ] sgn( 4 ) SB -(8/3) -(36/5) Wigner M(z) [ 6 3 / 8 ]

25 Departing the critical point (aα 6>) -(8/3) -(36/5) [ 6 4 / 8 ] - Wigner sgn( 4 ) = SB 84 + M(z) [ 6 3 / 8 ]

26 Departing the critical point (aα 6>) [ 6 4 / 8 ] - -(8/3) -(36/5) = Wigner sgn( 4 ) = SB M(z) [ 6 3 / 8 ]

27 Going to new regime: aα 6<.4. cχ SB CP T L Wigner aα 6 < CP: (-1/7, 1/3) L: (, 5/18) T: (-.16,.7) 4 [ 6 / 8 ] SB P M(z) P: (.14,.16) [ 6 3 / 8 ]

28 Characterizing the triple point Three different forms of sym.-broken phase compete with each other and coexist at the triple point (T)! M M CP SB Wigner M(z) M.38.. q M(z) T L q L SB P M(z) Model independent universal ratios associated with T

29 Summary

30 Summary Multidimensional crystals, not favored!

31 Summary Multidimensional crystals, not favored! Extended the previous GL analysis to higher order and explored the phase structure away from the Lifshitz point.

32 Summary Multidimensional crystals, not favored! Extended the previous GL analysis to higher order and explored the phase structure away from the Lifshitz point. if aα 6>, non-linear effects; but qualitative phase structure unchanged.

33 Summary Multidimensional crystals, not favored! Extended the previous GL analysis to higher order and explored the phase structure away from the Lifshitz point. if aα 6>, non-linear effects; but qualitative phase structure unchanged. If aα 6<, phase structure becomes rather rich; Existence of Triple Point (T)!

34 Further questions Do we really have no suitable multidimensional chiral crystal? If no, how could one dimensional structure be stabilized against thermal fluctuations? c.f. Landau-Peierls Theorem Triple point in model? How Polyakov loop and/or vector interaction change aα 6 at TCP? How does isospin asymmetry affect the TCP and phase structure of its neighborhood? [see poster contribution by Yuhei Iwata]

35 Backups

36 How does GL map onto NJL phase diagram (µ, T,,G) = 1 G +4N aα 8< d 3 p 1.4 cn f aα 6< T ( ) 3 8 = [(i n n + µ) p ] d 3 p 1 4(µ, T, ) =4N c N f T ( ) 3 aα 8<.3 [(i n + µ) p ] T/.5 4 = 6(8)(µ, T ) =4N c N f T..1 aα 8> aα 6< n n d 3 p 1 ( ) 3 [(i n + µ) p ] 3(4) 6(µ, T )=f(t/µ)/µ 8(µ, T )=g(t/µ)/µ 4 =: g=. g=.5 g= µ/ 6 = aα 6> D. Nickel, PRL9

37 How does GL map onto NJL phase diagram Τ CP χsb L Wigner T E χsb M(z) µ

38 Dimensional analysis and Scaling Introducing dimensionless variables: = [ 4 / 6 ] M = m [ 4 / 6 ] = [ 4 3 / 6] x = x [ 6/ 4 ] Relevant parameters are reduced: = m + sgn( 4) (m 4 +( m) ) m 6 +5m ( m) + 1 ( m) Inhomogeneous phase appears when aα 4 becomes negative

39 One dimensional modulations Chiral Condensate: M(x) = G ( qq + i qi 5 3 q ) Fulde-Ferrell (1964) Chiral spiral Nakano-Tatsumi (5) M FF (z) = e iqz Complex Larkin-Ovchinnikov (1964) CDW; Nickel (9) M LO (z) = sin(qz) Real Solitonic chiral condensate Buzdin, Kachkachi (1997) Thies (6), Nickel (9) M sn (z) = qsn(qz, ) Real Higher harmonics truncated at N=5 M hh (z) = N N ne inqz Complex

40 Ejiri et al. 3 de Forcrand, Philipsen, 3 LR4; Fodor-Katz; µe/tτ E».7 LTE3; Ejiri et al., µe/tτ E».8 LTE4; Gavai, Guputa µe/tτ E».35

41 How does GL map onto (µ, TΤ )-phase diagram (µ, T,,G) = 1.4 G +4N cn f T 4(µ, T, ) =4N c N f T T/.5.3 6(8)(µ, T ) =4N c N f T..1 4 = LR4; F-K µe/tτ E».7 n n n d 3 p 1 ( ) 3 8 = [(i n + µ) p ] d 3 p 1 ( ) 3 [(i n + µ) p ] d 3 p 1 ( ) 3 [(i n + µ) p ] 3(4) 6(µ, T )=f(t/µ)/µ 8(µ, T )=g(t/µ)/µ 4 =: g=. g=.5 g= µ/ 6 = D. Nickel, PRL9

42 GL in the vicinity of critical point M sn (z) = q sn(qz, ) D. Nickel, PRL9 M =1 =.1 z

43 Soliton formation sgn( 4 ) SB -(8/3) -(36/5) -(16/3) Wigner M(z) [ 6 ]

44 Soliton formation sgn( 4 ) SB -(8/3) -(36/5) -(16/3) Wigner M(z) [ 6 ]

45 Soliton formation M [ 4 1/ / 6 1/ ] M(z) z [ 6 1/ 1/ / 4 ] sgn( 4 ) SB -(8/3) -(36/5) -(16/3) Wigner M(z) [ 6 ]

46 Soliton formation a=-.5 M [ 4 1/ / 6 1/ ] [ 4 3 / 6 ] M(z) z [ 6 1/ 1/ / 4 ] sgn( 4 ) SB -.4 -(8/3) -.6 -(36/5) -(16/3) Wigner M(z) z [ 6 1/ / 4 1/ ] [ 6 ]

47 Soliton formation a=-.5 M [ 4 1/ / 6 1/ ] [ 4 3 / 6 ] M(z) z [ 6 1/ 1/ / 4 ] sgn( 4 ) SB -.4 -(8/3) -.6 -(36/5) -(16/3) Wigner M(z) z [ 6 1/ / 4 1/ ] a= [ 6 ]

48 Soliton formation a=-.5 M [ 4 1/ / 6 1/ ] [ 4 3 / 6 ] M(z) z [ 6 1/ 1/ / 4 ] sgn( 4 ) SB -.4 -(8/3) -.6 -(36/5) -(16/3) Wigner M(z) z [ 6 1/ / 4 1/ ] a= at onset [ 6 ]

49 Energy density profile in detail (z) = M(z) M(z) M(z)6 + 4 M 4 + 5M ( M) 6 + ( M) 1 1: no derivative terms : Derivative term at 4 th order 3: Derivative terms at 6 th order.4. [ 4 3 / 6 ] Total z [ 6 1/ 1/ / 4 ]

50 Going to new regime: aα 6<.4 E SB Wigner 4 [ 6 / 8 ].3. T L SB P M(z) [ 6 3 / 8 ]

51 Going to new regime: aα 6<.6.4 [ 6 1/ / 8 1/ ] <M > 1/ cχ SB cχ SB M(z) q/ 4 [ 6 / 8 ].3. E SB T L Wigner [ 6 3 / 8 ] Wigner SB P M(z) [ 6 3 / 8 ]

52 More on the dimensionality d-dimensional extension of 3D-cubic LO M dd-lo = d M ave(sin(qx 1 ) + sin(qx )+ + sin(qx d )) Effective potential dd = a 3a 4 16 M ave d a 4 M 4 ave + O(M 6 ave) Taking the minimum dd = 1 a 4 a 3a /d + lim d dd 1D =.5

53 Dimensional & Scaling analysis again Introducing dimensionless variables: = 6 3 / 8 4 = 4 6 / 8 M = m 6 / 8 x = x 8/ 6 = 4 6 / 3 8 Relevant parameters are reduced: = m (m4 +( m) ) + sgn( 6) 6 m 6 +5m ( m) + 1 ( m) m m 4 ( m) 1 5 ( m) m m( m) m ( m) ( m)

54 GL in the vicinity of critical point D. Nickel, PRL9. Nothing interesting -. sgn( 4 ) 4 happens even if -.4 we go far away from Lifshitz point = SB = (16/3) -(8/3) -(36/5) -(16/3) Wigner M(z) [ 6 ]

55 GL in the vicinity of critical point D. Nickel, PRL9. Nothing interesting -. sgn( 4 ) 4 happens even if -.4 we go far away from Lifshitz point = SB = (16/3) -(8/3) -(36/5) -(16/3) Wigner M(z) Need to go beyond the minimal (6-th order) GL description -1 [ 6 ]