Journal of High Energy Physics

, 2011:127 | Cite as

Gross-Neveu models, nonlinear Dirac equations, surfaces and strings

Article

Abstract

Recent studies of the thermodynamic phase diagrams of the Gross-Neveu model (GN2), and its chiral cousin, the NJL2 model, have shown that there are phases with inhomogeneous crystalline condensates. These (static) condensates can be found analytically because the relevant Hartree-Fock and gap equations can be reduced to the nonlinear Schrödinger equation, whose deformations are governed by the mKdV and AKNS integrable hierarchies, respectively. Recently, Thies et al. have shown that time-dependent Hartree-Fock solutions describing baryon scattering in the massless GN2 model satisfy the Sinh-Gordon equation, and can be mapped directly to classical string solutions in AdS3. Here we propose a geometric perspective for this result, based on the generalized Weierstrass spinor representation for the embedding of 2d surfaces into 3d spaces, which explains why these well-known integrable systems underlie these various Gross-Neveu gap equations, and why there should be a connection to classical string theory solutions. This geometric viewpoint may be useful for higher dimensional models, where the relevant integrable hierarchies include the Davey-Stewartson and Novikov-Veselov systems.

Keywords

Field Theories in Lower Dimensions Integrable Hierarchies Spontaneous Symmetry Breaking 

References

  1. [1]
    D.J. Gross and A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974) 3235 [SPIRES].ADSGoogle Scholar
  2. [2]
    R.F. Dashen, B. Hasslacher and A. Neveu, Semiclassical bound states in an asymptotically free theory, Phys. Rev. D 12 (1975) 2443 [SPIRES].MathSciNetADSGoogle Scholar
  3. [3]
    J. Feinberg, All about the static fermion bags in the Gross-Neveu model, Annals Phys. 309 (2004) 166 [hep-th/0305240] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  4. [4]
    M. Thies and K. Urlichs, Revised phase diagram of the Gross-Neveu model, Phys. Rev. D 67 (2003) 125015 [hep-th/0302092] [SPIRES].ADSGoogle Scholar
  5. [5]
    M. Thies, Analytical solution of the Gross-Neveu model at finite density, Phys. Rev. D 69 (2004) 067703 [hep-th/0308164] [SPIRES].MathSciNetADSGoogle Scholar
  6. [6]
    M. Thies, From relativistic quantum fields to condensed matter and back again: Updating the Gross-Neveu phase diagram, J. Phys. A 39 (2006) 12707 [hep-th/0601049] [SPIRES].MathSciNetADSGoogle Scholar
  7. [7]
    V. Schon and M. Thies, Emergence of Skyrme crystal in Gross-Neveu and ’t Hooft models at finite density, Phys. Rev. D 62 (2000) 096002 [hep-th/0003195] [SPIRES].ADSGoogle Scholar
  8. [8]
    V. Schon and M. Thies, 2D model field theories at finite temperature and density, in At the frontier of particle physics, 3 World Scientific, Singapore (2000) 1945 hep-th/0008175 [SPIRES].Google Scholar
  9. [9]
    G. Basar, G.V. Dunne and M. Thies, Inhomogeneous condensates in the thermodynamics of the chiral NJL 2 model, Phys. Rev. D 79 (2009) 105012 [arXiv:0903.1868] [SPIRES].ADSGoogle Scholar
  10. [10]
    P. de Forcrand and U. Wenger, New baryon matter in the lattice Gross-Neveu model, PoS LAT 2006 (2006) 152 [hep-lat/0610117] [SPIRES].Google Scholar
  11. [11]
    D. Nickel and M. Buballa, Solitonic ground states in (color-) superconductivity, Phys. Rev. D 79 (2009) 054009 [arXiv:0811.2400] [SPIRES].ADSGoogle Scholar
  12. [12]
    D. Nickel, How many phases meet at the chiral critical point?, Phys. Rev. Lett. 103 (2009) 072301 [arXiv:0902.1778] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    D. Nickel, Inhomogeneous phases in the Nambu-Jona-Lasino and quark-meson model, Phys. Rev. D 80 (2009) 074025 [arXiv:0906.5295] [SPIRES];MathSciNetADSGoogle Scholar
  14. [14]
    L.Y. Glozman and R.F. Wagenbrunn, Chirally symmetric but confining dense and cold matter, Phys. Rev. D 77 (2008) 054027 [arXiv:0709.3080] [SPIRES].ADSGoogle Scholar
  15. [15]
    L.Y. Glozman and R.F. Wagenbrunn, Second order chiral restoration phase transition at low temperatures in quarkyonic matter, arXiv:0805.4799 [SPIRES].
  16. [16]
    T. Kojo, Y. Hidaka, L. McLerran and R.D. Pisarski, Quarkyonic chiral spirals, Nucl. Phys. A 843 (2010) 37 [AIP Conf. Proc. 1257 (2010) 732] [arXiv:0912.3800] [SPIRES].ADSGoogle Scholar
  17. [17]
    T. Kojo, R.D. Pisarski and A.M. Tsvelik, Covering the Fermi surface with patches of quarkyonic chiral spirals, Phys. Rev. D 82 (2010) 074015 [arXiv:1007.0248] [SPIRES].ADSGoogle Scholar
  18. [18]
    I.E. Frolov, V.C. Zhukovsky and K.G. Klimenko, Chiral density waves in quark matter within the Nambu-Jona-Lasinio model in an external magnetic field, Phys. Rev. D 82 (2010) 076002 [arXiv:1007.2984] [SPIRES].ADSGoogle Scholar
  19. [19]
    K. Pohlmeyer, Integrable Hamiltonian systems and interactions through quadratic constraints, Commun. Math. Phys. 46 (1976) 207 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  20. [20]
    F. Lund and T. Regge, Unified approach to strings and vortices with soliton solutions, Phys. Rev. D 14 (1976) 1524 [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    F. Lund, Note on the geometry of the nonlinear σ-model in two-dimensions, Phys. Rev. D 15 (1977) 1540 [SPIRES].ADSGoogle Scholar
  22. [22]
    F. Lund, Solitons and Geometry, in the proceedings of Nonlinear equations in physics and mathematics, Istanbul (1977), A.O. Barut Ed., D. Reidel, Boston (1978).Google Scholar
  23. [23]
    A. Neveu and N. Papanicolaou, Integrability of the classical scalar and symmetric scalar-pseudoscalar contact Fermi interactions in two-dimensions, Commun. Math. Phys. 58 (1978) 31 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  24. [24]
    G. Basar and G.V. Dunne, Self-consistent crystalline condensate in chiral Gross-Neveu and Bogoliubov-de Gennes systems, Phys. Rev. Lett. 100 (2008) 200404 [arXiv:0803.1501] [SPIRES].CrossRefADSGoogle Scholar
  25. [25]
    G. Basar and G.V. Dunne, A twisted kink crystal in the chiral Gross-Neveu model, Phys. Rev. D 78 (2008) 065022 [arXiv:0806.2659] [SPIRES].ADSGoogle Scholar
  26. [26]
    F. Correa, G.V. Dunne and M.S. Plyushchay, The Bogoliubov/de Gennes system, the AKNS hierarchy and nonlinear quantum mechanical supersymmetry, Annals Phys. 324 (2009) 2522 [arXiv:0904.2768] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  27. [27]
    F. Gesztesy and H. Holden, Soliton equations and their algebro-geometric solutions, Cambridge University Press (2003).Google Scholar
  28. [28]
    M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, The Inverse scattering transform fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974) 249.MathSciNetGoogle Scholar
  29. [29]
    A. Klotzek and M. Thies, Kink dynamics, sinh-Gordon solitons and strings in AdS 3 from the Gross-Neveu model, J. Phys. A 43 (2010) 375401 [arXiv:1006.0324] [SPIRES].MathSciNetGoogle Scholar
  30. [30]
    C. Fitzner and M. Thies, Exact solution of an N baryon problem in the Gross-Neveu model, arXiv:1010.5322 [SPIRES].
  31. [31]
    A. Jevicki, K. Jin, C. Kalousios and A. Volovich, Generating AdS string solutions, JHEP 03 (2008) 032 [arXiv:0712.1193] [SPIRES].
  32. [32]
    A. Jevicki and K. Jin, Solitons and AdS string solutions, Int. J. Mod. Phys. A 23 (2008) 2289 [arXiv:0804.0412] [SPIRES].
  33. [33]
    E. Antonyan, J.A. Harvey and D. Kutasov, The Gross-Neveu model from string theory, Nucl. Phys. B 776 (2007) 93 [hep-th/0608149] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  34. [34]
    A. Basu and A. Maharana, Generalized Gross-Neveu models and chiral symmetry breaking from string theory, Phys. Rev. D 75 (2007) 065005 [hep-th/0610087] [SPIRES].MathSciNetADSGoogle Scholar
  35. [35]
    J.L. Davis, M. Gutperle, P. Kraus and I. Sachs, Stringy NJLS and Gross-Neveu models at finite density and temperature, JHEP 10 (2007) 049 [arXiv:0708.0589] [SPIRES];CrossRefMathSciNetADSGoogle Scholar
  36. [36]
    A.I. Bobenko, Integrable surfaces, Func. Anal. Appl. 24 (1990) 227.MATHCrossRefMathSciNetGoogle Scholar
  37. [37]
    L. P. Eisenhart, A treatise on the differential geometry of curves and surfaces, Ginn Co., Boston (1909).Google Scholar
  38. [38]
    H. Hopf, Differential geometry in the large, Lect. Notes Math. 1000, Springer, Berlin (1983).MATHGoogle Scholar
  39. [39]
    K. Kenmotsu, Weierstrass formula for surfaces of prescribed mean curvature, Math. Ann. 245 (1979) 89.MATHCrossRefMathSciNetGoogle Scholar
  40. [40]
    A.I. Bobenko, Integrable surfaces, Func. Anal. Appl. 24 (1990) 227.MATHCrossRefMathSciNetGoogle Scholar
  41. [41]
    A.I. Bobenko, Constant mean curvature surfaces and integrable equations, Russ. Math. Surv. 46 (1991) 1.CrossRefMathSciNetGoogle Scholar
  42. [42]
    A.I. Bobenko, All constant mean curvature tori in R3, S3 and H3 in terms of theta-functions, Math. Ann. 290 (1991) 209.MATHCrossRefMathSciNetGoogle Scholar
  43. [43]
    A.I. Bobenko, Exploring Surfaces through Methods from the Theory of Integrable Systems. lectures on the Bonnet problem, in Surveys on geometry and integrable systems, Adv. Stud. Pure Math. 51 (2008) 1 [math/9909003].
  44. [44]
    B.G. Konopelchenko, Induced surfaces and their integrable dynamics, Stud. Appl. Math. 96 (1996) 9 [math/9810138].MATHMathSciNetGoogle Scholar
  45. [45]
    B.G. Konopelchenko and I. A. Taimanov, Constant mean curvature surfaces via integrable dynamical system, J. Phys. A 29 (1996) 1261 [dg-ga/9505006].MathSciNetADSGoogle Scholar
  46. [46]
    A.S. Fokas and I.M. Gel’fand, Surfaces on Lie groups, on Lie algebras, and their integrability, Comm. Math. Phys. 177 (1996) 203.MATHCrossRefMathSciNetADSGoogle Scholar
  47. [47]
    A.S. Fokas, I.M. Gel’fand, F. Finkel and Q.M. Liu, A formula for constructing infinitely many surfaces on Lie algebras and integrable equations, Sel. Math., New ser. 6 (2000) 347.MATHCrossRefMathSciNetGoogle Scholar
  48. [48]
    I.A. Taimanov, Two-dimensional Dirac operator and the theory of surfaces, Russ. Math. Surv. 61 (2006) 79.MATHCrossRefMathSciNetGoogle Scholar
  49. [49]
    S.P. Novikov and I.A. Taimanov, Modern geometric structures and fields, AMS, Providence (2006) [SPIRES].MATHGoogle Scholar
  50. [50]
    D.M. Hofman and J.M. Maldacena, Giant magnons, J. Phys. A 39 (2006) 13095 [hep-th/0604135] [SPIRES].MathSciNetGoogle Scholar
  51. [51]
    L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, JHEP 11 (2009) 082 [arXiv:0904.0663] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  52. [52]
    N. Dorey, Notes on integrability in gauge theory and string theory, J. Phys. A 42 (2009) 254001 [SPIRES].MathSciNetADSGoogle Scholar
  53. [53]
    N. Beisert et al., Review of AdS/CFT integrability: an overview, arXiv:1012.3982 [SPIRES].
  54. [54]
    L.S. Da Rios, Moto d’un liquido indefinito con un filetto vorticoso di forma qualunque (On the motion of an unbounded liquid with a vortex filament of any shape), Rend. Circ. Mat. Palermo 22 (1906) 117.MATHCrossRefGoogle Scholar
  55. [55]
    R.L. Ricca, Rediscovery of the Da Rios equations’, Nature 352 (1991) 561.CrossRefADSGoogle Scholar
  56. [56]
    G.L. Lamb, Analytical descriptions of ultrashort optical pulse propagation in a resonant medium, Rev. Mod. Phys. 43 (1971) 99 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    G.L. Lamb, Solitons on moving space curves, J. Math. Phys. 18 (1977) 1654 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  58. [58]
    G.L. Lamb, Solitons and the motion of helical curves, Phys. Rev. Lett. 37 (1976) 235 [SPIRES];CrossRefMathSciNetADSGoogle Scholar
  59. [59]
    R. Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972) 477.MATHCrossRefADSGoogle Scholar
  60. [60]
    A. Sym and J. Corones, Lie group explanation of geometric interpretations of solitons, Phys. Rev. Lett. 42 (1979) 1099 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  61. [61]
    A. Sym, Soliton surfaces, Lett. Nuovo Cim. 332 (1982) 394.CrossRefMathSciNetGoogle Scholar
  62. [62]
    R.K. Dodd, Soliton immersions, Comm. Math. Phys. 197 (1998) 641.MATHCrossRefMathSciNetADSGoogle Scholar
  63. [63]
    A. Calini and T. Ivey, Connecting geometry, topology and spectra for finite-gap NLS potentials, Physica D 152 (2001) 9.MathSciNetADSGoogle Scholar
  64. [64]
    P.G. Grinevich and M.U. Schmidt, Closed curves in R3: a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the filament equation, dg-ga/9703020.
  65. [65]
    U. Wolff, The phase diagram of the infinite-N Gross-Neveu model at finite temperature and chemical potential, Phys. Lett. B 157 (1985) 303 [SPIRES].ADSGoogle Scholar
  66. [66]
    T.F. Treml, Dynamical mass generation in the Gross-Neveu model at finite temperature and density, Phys. Rev. D 39 (1989) 679 [SPIRES];ADSGoogle Scholar
  67. [67]
    A. Barducci, R. Casalbuoni, M. Modugno, G. Pettini and R. Gatto, Thermodynamics of the massive Gross-Neveu model, Phys. Rev. D 51 (1995) 3042 [hep-th/9406117] [SPIRES].ADSGoogle Scholar
  68. [68]
    B.G. Konopelchenko and G. Landolfi, Generalized Weierstrass representation for surfaces in multidimensional Riemann spaces, Journ. Geom. Phys. 29 (1999) 319 [math/9804144].MATHCrossRefMathSciNetADSGoogle Scholar
  69. [69]
    B.G. Konopelchenko, Weierstrass representations for surfaces in 4D spaces and their integrable deformations via DS hierarchy, Ann. Global Anal. Geom. 16 (2000) 61. [math/9807129].CrossRefMathSciNetGoogle Scholar
  70. [70]
    I.A. Taimanov, Surfaces in the four-space and the Davey-Stewartson equations, J. Geom. Phys. 56 (2006) 1235 [math/0401412].MATHCrossRefMathSciNetADSGoogle Scholar
  71. [71]
    B. Rosenstein, B. Warr and S.H. Park, Dynamical symmetry breaking in four Fermi interaction models, Phys. Rept. 205 (1991) 59 [SPIRES].CrossRefADSGoogle Scholar
  72. [72]
    G. Gat, A. Kovner and B. Rosenstein, Chiral phase transitions in D = 3 and renormalizability of four Fermi interactions, Nucl. Phys. B 385 (1992) 76 [SPIRES].CrossRefADSGoogle Scholar
  73. [73]
    K.G. Klimenko, Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field, Z. Phys. C 54 (1992) 323 [SPIRES];MathSciNetADSGoogle Scholar
  74. [74]
    K. Urlichs, Baryons and baryonic matter in four-fermon interaction models, Ph.D. Thesis, University of Erlangen, Erlangen, Germany (2007), unpublished (2007).Google Scholar
  75. [75]
    S. Hands, S. Kim and J.B. Kogut, The U(1) Gross-Neveu model at nonzero chemical potential, Nucl. Phys. B 442 (1995) 364 [hep-lat/9501037] [SPIRES].ADSGoogle Scholar
  76. [76]
    S. Hands, Four fermion models at non-zero density, Nucl. Phys. A 642 (1998) 228 [hep-lat/9806022] [SPIRES];ADSGoogle Scholar
  77. [77]
    S.P. Novikov and A.P. Veselov, Finite-zone two-dimensional periodic Shroedinger operators: potential operators, Dokl. Akad. Nauk SSSR 279 (1984) 784.MathSciNetGoogle Scholar
  78. [78]
    S.P. Novikov and A.P. Veselov, Two-dimensional Schroedinger operator: inverse scattering and evolutional equations, Physica D 18 (1986) 267.MathSciNetADSGoogle Scholar
  79. [79]
    S.P. Novikov and A.P. Veselov, Exactly solvable two-dimensional Schrodinger operators and Laplace transformations, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, 179 (1997) 109 [math-ph/0003008v1].MathSciNetGoogle Scholar
  80. [80]
    L.V. Bogdanov, Veselov-Novikov equation as a natural two-dimensional generalization of the Korteweg-De Vries equation, Theor. Math. Phys. 70 (1987) 219.MATHCrossRefMathSciNetGoogle Scholar
  81. [81]
    L.V. Bogdanov, On the two-dimensional Zakharov-Shabat problem, Theor. Math. Phys. 72 (1987) 790 [SPIRES].MATHCrossRefMathSciNetGoogle Scholar
  82. [82]
    I.A. Taimanov, Modified Novikov-Veselov equation and differential geometry of surfaces, Amer. Math. Soc. Transl. Ser. 2, 179 (1997) 133Google Scholar
  83. [83]
    I.A. Taimanov, Modified Novikov-Veselov equation and differential geometry of surfaces, Amer. Math. Soc. Transl. Ser. 2, 179 (1997) 133 [dg-ga/9511005].ADSGoogle Scholar
  84. [84]
    H.J. De Vega and N.G. Sanchez, Exact integrability of strings in D-Dimensional de Sitter space-time, Phys. Rev. D 47 (1993) 3394 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  85. [85]
    W. Bietenholz, A. Gfeller and U.J. Wiese, Dimensional reduction of fermions in brane worlds of the Gross-Neveu model, JHEP 10 (2003) 018 [hep-th/0309162] [SPIRES];CrossRefMathSciNetADSGoogle Scholar
  86. [86]
    L.H. Haddad and L.D. Carr, Relativistic linear stability equations for the nonlinear Dirac equation in Bose-Einstein condensates, arXiv:1006.3893 [SPIRES].
  87. [87]
    L.H. Haddad and L.D. Carr, The Nonlinear Dirac Equation in Bose-Einstein Condensates: Foundation and Symmetries, Physica D 238 (2009) 1413 [arXiv:0803.3039].MathSciNetADSGoogle Scholar
  88. [88]
    R.S. Ward, Integrable and solvable systems, and relations among them, Phil. Trans. Roy. Soc. Lond. A 315 (1985) 451.ADSGoogle Scholar
  89. [89]
    N.J. Hitchin, The self-duality equations On A Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59.MATHCrossRefMathSciNetGoogle Scholar
  90. [90]
    B. Grossman, Hierarchy of soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett. 65 (1990) 3230 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  91. [91]
    G.V. Dunne, R. Jackiw, S.-Y. Pi and C.A. Trugenberger, Selfdual Chern-Simons solitons and two-dimensional nonlinear equations, Phys. Rev. D 43 (1991) 1332 [Erratum-ibid. D 45 (1992) 3012] [SPIRES].MathSciNetADSGoogle Scholar
  92. [92]
    G.V. Dunne, Selfdual Chern-Simons theories, Lect. Notes Phys. M 36, Springer, Heidelberg (1995).CrossRefGoogle Scholar
  93. [93]
    L. Martina, Kur. Myrzakul, R. Myrzakulov and G. Soliani, Deformation of surfaces, integrable systems, and Chern-Simons theory, Journ. Math. Phys. 42 (2001) 1397.MATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of ConnecticutStorrsU.S.A.

Personalised recommendations