Gross-Neveu Condensates, Nonlinear Dirac Equations and Minimal Surfaces

Transcription

1 Gross-Neveu Condensates, Nonlinear Dirac Equations and Minimal Surfaces Gerald Dunne University of Connecticut CAQCD 2011, Minnesota, May 2011 Başar, GD, Thies: PRD 2009, Başar, GD: JHEP 2011,

2 motivation: phase diagram of QCD? chiral symmetry confinement/deconfinement Stephanov, 2009

3 Gross-Neveu Models Gross/Neveu, 1974 Nambu/Jona-Lasinio, 1961 GN 2 χgn 2 NJL 2 L GN = ψ i / ψ + g2 ( ) 2 ψψ 2 L NJL = ψ [ i / ψ + g2 ( ) 2 ( ψψ + ψiγ 5 ψ ) ] 2 2 ψ γ 5 ψ ψ e iαγ5 ψ renormalizable dynamical mass generation asymptotically free large Nf limit chiral symmetry breaking (discrete vs. continuous) (T, µ) phase diagram?

4 uniform condensate Wolff, 1985 GN2 σ = ψψ π = ψiγ 5 ψ = σ iπ inhomogeneous condensate Barducci et al, 1995 NJL2 Thies & Urlichs, 2005 Başar, GD, Thies, 2009

5 Inhomogeneous Condensates functional gap equation: σ = ψψ π = ψiγ 5 ψ σ(x) = (x) = δ δσ(x) δ δ (x) ln det (i / σ(x)) ln det = σ iπ ( i / 1 2 (1 γ5 ) (x) 1 ) 2 (1 + γ5 ) (x) inhomogeneous mean-field approximation Hartree-Fock : H = ( ) i x σ(x) σ(x) i x H ψ E = E ψ E or H = ( ) i x (x) (x) i x such that σ(x) = tr ψe E,T,µ (x)ψ E (x) ( (x) = tr ψe E,T,µ (x)ψ E (x) i ψ E (x)iγ 5 ) ψ E (x)

6 zero temperature and density: Dashen-Hasslacher-Neveu (1975): inverse scattering single kink: σ(x) =m tanh(mx) Shei (1976): inverse scattering twisted kink: (x) =m cosh ( m sin( θ 2 ) x i θ 2 cosh ( m sin( θ 2 ) x) ) reflectionless potential reflectionless Dirac operator nonzero temperature and finite density: Thies,Urlichs (2005) kink crystal: σ(x) =m ν sn(mx ν) finite-gap potential Başar, GD (2008) twisted kink crystal: (x) =m σ(mx+ ik i θ 2 ) σ(mx+ ik ) σ(i θ 2 )eiqx finite-gap Dirac operator

7 solving the gap equation σ = ψψ σ(x) = δ δσ(x) ln det (i / σ(x)) π = ψiγ 5 ψ = σ iπ (x) = δ δ (x) ln det ( i / 1 2 (1 γ5 ) (x) 1 ) 2 (1 + γ5 ) (x) ansatz for Gorkov Green s function consistent with gap equation reduces gap equation to the (soluble) nonlinear Schrödinger equation: i (b 2E) 2 (a Eb) = 0 nonlinear Schrödinger equation (NLSE)

8 twisted kink crystal: general solution of NJL2 gap equation real kink crystal spiral crystal

9 phase diagrams gap equation solution has 4 parameters grand potential: Ψ = 1 β de ρ(e) ln (1+e β(e µ)) minimize Ψ w.r.t. parameters, as function of T and µ Thies & Urlichs, 2005 Başar, GD, Thies, 2009

10 physics: Peierls instability and chiral symmetry Peierls: 1d system prefers to open gap at Fermi level GN: discrete chiral symmetry symmetric spectrum NJL: continuous chiral symmetry non-symmetric spectrum

11 all-orders Ginzburg-Landau expansion ln det ln det (i / σ(x)) = n ( i / 1 2 (1 γ5 ) (x) 1 ) 2 (1 + γ5 ) (x) α n (T, µ) a n [σ(x)] = n α n (T, µ) b n [ (x)] Ginzburg-Landau expansion is an expansion in the conserved quantities of the mkdv (GN 2 ) and AKNS (NJL 2 ) integrable hierarchy magic of integrable hierarchies: δa n [σ] δσ(x) = γ n σ(x) δb n [ ] δ (x) = γ n (x)+ β n (x) Başar, GD, Thies, 2009 Correa, GD, Plyushchay, 2009 n

12 extension to higher dimensions? vortex crystal, Skyrme crystal, baryon crystal,...

13 1+1 dim. physics = Peierls instability + chiral symmetry breaking 1+1 dim. gap equation solutions are special: 1. finite gap ( periodic reflectionless ) 2. Ginzburg-Landau = integrable hierarchies extension to higher dimensions? vortex crystal, Skyrme crystal, baryon crystal,...

14 a new perspective

15 1+1 dim. Gross-Neveu models and string theory Kink dynamics, sinh-gordon solitons and strings in AdS 3 from the Gross-Neveu model Andreas Klotzek and Michael Thies Hartree-Fock: (i / σ(x)) ψ k =0 σ(x) = k ψ k (x)ψ k (x) amazingly, mode-by-mode : ψk (x)ψ k (x) =f(k) σ(x) k nonlinear Dirac equation : ( i / l(k) ψk (x)ψ k (x) ) ψ k =0

16 Neveu-Papanicolaou 1977, Lund 1977 nonlinear Dirac equation : ( i / l ψ(x)ψ(x) ) ψ =0 bilinear : σ(x) = ψ(x)ψ(x) σσ (σ ) 2 σ 4 = 1 σ = ψψ = e θ/2 1 dim. Sinh-Gordon θ + 4 sinh θ =0 σ 2σ 3 + σ =0 NLSE

17 exact solutions to time-dependent Hartree-Fock Hartree-Fock: (i / σ(x, t)) ψ k =0 amazingly, mode-by-mode : σ(x, t) = k ψ k (x, t)ψ k (x, t) ψ k (x, t)ψ k (x, t) =f(k) σ(x, t) k nonlinear Dirac equation : ( i / l(k) ψk ) (x, t)ψ k (x, t) ψ k (x, t) =0 σ = ψψ e θ/2 2 dim. Sinh-Gordon µ µ θ + 4 sinh θ =0.

18 ( x vt ) boosted solution: σ(x) σ 1 v 2 breather solution: not real scattering solution : ( ) ( ) σ(x, t) = v cosh(2x/ 1 v 2 ) cosh(2vt/ 1 v 2 ) v cosh(2x/ 1 v 2 ) + cosh(2vt/ 1 v 2 ) Klotzek, Thies, baryon-antibaryon baryon-baryon

19 this means we have a solution to the gap equation: σ(x, t) = δ δσ(x, t) ln det (i / σ(x, t)) suggests we can find a solution to : σ(x, y) = δ δσ(x, y) ln det (i / σ(x, y)) could represent a static 2d inhomogeneous condensate of the 2+1 dimensional Gross-Neveu model

20 a geometric perspective... Başar, GD: JHEP 2011, o we now consider an embedding of the form X(x, t) :Σ R 1,1 R 1,2, with arameterization ds 2 = f 2 ( dt 2 + dx 2 ). The first fundamental form in light-cone immersion of a surface in 3 dimensions N X +. X = f 2 2 X y X(x, y), X+. X + =0, X. X = 0 (38) ds 2 = f 2 (x +,x ) dx + dx e Gauss-Codazzi equations, we define the Minkowski analogues of the Hopf differ- X x ean curvature: X ++. N = iq (+), X.N = iq ( ), X+.N = i 2 Hf2 (39) Gauss-Codazzi equations azzi equations can easily be written by repeating the steps of the Euclidean connkowsi space: H: mean curvature ff + f + f 1 4 H2 f 4 = Q (+) Q ( ) (40) Q (+) = 1 2 f 2 H + (41) Q ( ) + = 1 2 f 2 H (42)

21 N X y spinor representation of surfaces (Weierstrass, Enneper, Eisenhart, Hopf, Neveu-Papanicolaou, Lund, Bobenko, Gel fand, Konopelchenko,...) X x SO(1, 2) SU(1, 1) ( ) X3 X X =(X 1,X 2,X 3 ) X = i 1 ix 2 X 1 + ix 2 X 3 X +, X, N SU(1, 1) spinors ψ

22 Gauss-Codazzi equations spinor representation: Dirac equation: induced metric factor: mean curvature: Hopf differentials: ff + f + f 1 4 H2 f 4 (i/ S)ψ =0 = Q (+) Q ( ) Q (+) = 1 2 f 2 H + Q ( ) + = 1 2 f 2 H f = ψψ S = H ψψ Q (+) = i(ψ1 ψ 1,+ ψ1,+ ψ 1) Q ( ) = i(ψ2 ψ 2, ψ2, ψ 2) constant mean curvature: nonlinear Dirac equation ( i / l ψ(x)ψ(x) ) ψ =0 f 2 = e θ Sinh-Gordon µ µ θ + 4 sinh θ =0.

23 constant mean curvature in flat R 2,1 space zero mean curvature in AdS3 space explicit map between time-dependent solutions to Gross-Neveu gap equation & classical string solutions in AdS3 Pohlmeyer, Lund/Regge, Barbashov et al,... Klotzek, Thies, 2010 Başar, GD, 2010 suggests new geometrical approach to search for inhomogeneous solutions to Gross-Neveu gap equations

24 geometric meaning of static inhomogeneous condensates in 1+1 dimensions for GN 2 immersion of curves into 3 dimensional space Da Rios (1906), student of Levi-Civita vortex filament equations

25 Frenet-Serret equations for moving frame of curve immersion can be written as a Dirac equation d ds t n = b 0 κ 0 κ 0 τ 0 τ 0 t n b potential satisfies NLSE (equiv Sinh-Gordon in 1+1) (spectral) deformations of these curves : mkdv hierarchy mkdv governs thermodynamics of 1+1 GN model

26 proposal/conjecture : Başar, GD, 2010 Gauss-Codazzi equations for surface embedding can be written as a Dirac equation solutions satisfy Sinh-Gordon (spectral) deformations of these surfaces : (m) Novikov-Veselov hierarchy question: does mnv govern the thermodynamics of 2+1 dimensional GN model?

27 lowest nontrivial equation of mkdv : 2 2 = ν lowest nontrivial equation of mnv : [( + ) 2 2 ] = ν TWO-DIMENSIONAL SCHRÖDINGER OPERATOR: INVERSE SCATTERING TRANSFORM AND EVOLUTIONAL EQUATIONS 1986 S. P. NOVIKOV AND A. P. VESELOV localized 1987 dromion solutions

28 Conclusions gap equation for chiral GN/NJL2 full, exact, thermodynamics & phase diagrams physics = Peierls instability + chiral symmetry breaking Ginzburg-Landau expansion = mkdv or AKNS hierarchy geometric picture = curve and surface embedding higher dimensional models : Novikov-Veselov hierarchy? Peierls instability? surface emeddings?

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