Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schrödinger system. (English) Zbl 1185.81071

Authors’ abstract: We study a nonlinear system of partial differential equations in which a complex field (the Higgs field) evolves according to a nonlinear Schrödinger equation, coupled to an electromagnetic field whose time evolution is determined by a Chern-Simons term in the action. In two space dimensions, the Chern-Simons dynamics is a Galileo invariant evolution for \(A\), which is an interesting alternative to the Lorentz invariant Maxwell evolution, and is finding increasing numbers of applications in two dimensional condensed matter field theory. The system we study, introduced by Manton, is a special case (for constant external magnetic field, and a point interaction) of the effective field theory of Zhang, Hansson and Kivelson arising in studies of the fractional quantum Hall effect. From the mathematical perspective the system is a natural gauge invariant generalization of the nonlinear Schrödinger equation, which is also Galileo invariant and admits a self-dual structure with a resulting large space of topological solitons (the moduli space of self-dual Ginzburg-Landau vortices). We prove a theorem describing the adiabatic approximation of this system by a Hamiltonian system on the moduli space. The approximation holds for values of the Higgs self-coupling constant \(\lambda \) close to the self-dual (Bogomolny) value of 1. The viability of the approximation scheme depends upon the fact that self-dual vortices form a symplectic submanifold of the phase space (modulo gauge invariance). The theorem provides a rigorous description of slow vortex dynamics in the near self-dual limit.


81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
81V10 Electromagnetic interaction; quantum electrodynamics
81V22 Unified quantum theories
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[1] Abraham R., Marsden J., Ratiu T.: Manifolds, Tensor Analysis and Applications. Springer-Verlag, New York (1988) · Zbl 0875.58002
[2] Arovas D., Schrieffer R., Wilczek F., Zee A.: Statistical mechanics of anyons. Nucl. Phys. B 251, 117–126 (1985)
[3] Aitchison I.J.R., Ao P., Thouless D., Zhu X.: Phys Rev B. 51, 6531 (1995)
[4] Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Providence, RI: American Mathematical Society, 2000 · Zbl 0946.46002
[5] Berge L., de Bouard A., Saut J.: Blowing up time-dependent solutions of the planar Chern-Simons gauged nonlinear Schrödinger equation. Nonlinearity 8, 235–253 (1995) · Zbl 0822.35125
[6] Bethuel F., Riviere T.: Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12(3), 243–303 (1995)
[7] Bogomolny E.: Stability of Classical Solutions. Sov. J. Nucl. Phys. 24, 861–870 (1976)
[8] Bradlow S.: Vortices in holomorphic line bundles and closed Kaehler manifolds. Commun. Math. Phys. 118, 1–17 (1990) · Zbl 0717.53075
[9] Bradlow S., Daskalopoulos G.: Moduli of stable pairs for holomorphic bundles over Riemann surfaces. Internat. J. Math. 2, 477–513 (1991) · Zbl 0759.32013
[10] Brezis H., Gallouet T.: Nonlinear Schrödinger evolution equation. Nonlin. Anal. T.M.A. 4(4), 677–681 (1980) · Zbl 0451.35023
[11] Deser S., Jackiw R., Templeton S.: Topologically massive gauge theories. Ann. Phys. 140, 372–411 (1982)
[12] Demoulini S., Stuart D.: Gradient flow of the superconducting Ginzburg-Landau functional on the plane. Commun. Anal. Geom. 5(1), 121–198 (1997) · Zbl 0894.35107
[13] Demoulini S.: Global existence for a nonlinear Schrödinger-Chern-Simons system on a surface. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(2), 207–225 (2007) · Zbl 1166.35035
[14] Demoulini S., Stuart D.M.A.: Existence and regularity for generalised harmonic maps associated to a nonlocal polyconvex energy of Skyrme type. Calc. Var. PDE 30(4), 523–546 (2007) · Zbl 1210.35075
[15] Froehlich J., Marchetti P-A.: Commun. Math. Phys. 121, 177–221 (1989) · Zbl 0819.58045
[16] Froehlich J., Studer U.M.: U(1) {\(\times\)} SU(2) - gauge invariance of non-relativistic quantum mechanics and generalized Hall effects. Commun. Math. Phys. 148, 553–600 (1992) · Zbl 0758.53048
[17] Dunne, G.: Aspects of Chern-Simons theory. In: Les Houches Lectures on Topological Aspects of Low Dimensional Systems EDP Sci., Les Ulis, 1998. Available online at http://arxiv.org/abs/hep-th/9902115v1 , 1991
[18] Girvin, S.: The Quantum hall effect: novel excitations and broken symmetries. In: Les Houches Lectures on Topological Aspects of Low Dimensional Systems EDP Sci., Les Ulis, 1998. Available online at http://arxiv.org/abs/cond-mat/9907002v1[cond-mat.mes-hall] , 1999
[19] Gustafson S., Sigal I.M.: The stability of magnetic vortices. Commun. Math. Phys. 212, 257–275 (2000) · Zbl 0955.58015
[20] Haskins M., Speight J.M.: The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps. J. Math. Phys. 44, 3470–3494 (2003) · Zbl 1062.58021
[21] Hassaïne M., Horvathy P.: Non-relativistic Maxwell-Chern-Simons vortices. Ann. Phys. 263(2), 276–294 (1998) · Zbl 0920.58075
[22] Horvathy, P., Zhang, P.: Vortices in abelian Chern-Simons gauge theory. http://arxiv.org/abs/0811.2094v3[hep-th] , 2009
[23] Jackiw, R., Pi, So-Young: Self-dual Chern-Simons solitons. In: Low-Dimensional Field Theories and Condensed Matter Physics (Kyoto,1991), Progr. Theoret. Phys. Suppl. 107, 1–40 (1992)
[24] Jost J.: Riemannian Geometry and Geometric Analysis. Springer-Verlag, Berlin-Heidlberg-NewYork (1988) · Zbl 0631.53005
[25] Kato T.: Perturbation Theory for Linear Operators. Springer-Verlag, Berlin-Heidlberg-NewYork (1980) · Zbl 0435.47001
[26] Kato, T.: Quasi-linear Equations of Evolution with Applications to Partial Differential Equations. Springer Lecture Notes in Mathematics 448 Berlin-Heidlberg-NewYork: Springer-Verlag, 1975, pp. 27–50 · Zbl 0315.35077
[27] Krusch S., Sutcliffe P.: Schrödinger-Chern-Simons vortex dynamics. Nonlinearity 19, 1515–1534 (2006) · Zbl 1118.82052
[28] Jaffe A., Taubes C.: Vortices and Monopoles. Birkhauser, Boston (1982)
[29] Majda A., Bertozzi A.: Vorticity and Incompressible Fluid Flow. Cambridge University Press, Cambridge (2001) · Zbl 0983.76001
[30] Manton N.: First order vortex dynamics. Ann. Phys. 256, 114–131 (1997) · Zbl 0932.58014
[31] Manton N., Sutcliffe P.: Topological Solitons. Cambridge University Press, Cambridge (2004) · Zbl 1100.37044
[32] Nagosa N.: Quantum Field Theory in Condensed Matter Physics. Springer, Berlin (1999)
[33] Palais R.: Foundations of Global Nonlinear Analysis. Mathematics lecture note series. NewYork, W.A. Benjamin (1968) · Zbl 0164.11102
[34] Prange R., Girvin S.: The Quantum Hall Effect 2nd edition. Springer-Verlag, New York (1990)
[35] Reed M., Simon B.: Functional Analysis. Academic Press, San Diego CA (1980) · Zbl 0459.46001
[36] Rodnianski, I., Sterbenz, J.: On the formation of singularities in the critical O(3) {\(\sigma\)}-model. http://arxiv.org/abs/math/0605023v3[math.AP] , 2008
[37] Romao N.: Quantum Chern-Simons vortices on a sphere. J. Math. Phys. 42, 3445–3469 (2001) · Zbl 1036.81022
[38] Romao N., Speight J.M.: Slow Schrödinger dynamics of gauged vortices. Nonlinearity 17(4), 1337–1355 (2004) · Zbl 1074.82033
[39] Rubin H., Ungar P.: Motion under a strong constraining force. Commun. Pure Appl. Math. 10, 65–87 (1957) · Zbl 0077.17401
[40] Sandier, E., Serfaty, S.: Vortices in the Magnetic Ginzburg-Landau Model. Progress in Nonlinear Differential Equations and their Applications 70 Basel-Boston: Birkhauser, 2007 · Zbl 1112.35002
[41] Sondhi S.L., Karlhede A., Kivelson S.A., Rezayi E.H.: Skyrmions and the crossover from the integer to fractional quantum Hall effect at small Zeeman energies. Phys. Rev. B 47, 16419 (1993)
[42] Stone, M.: Superfluid dynamics of the fractional quantum Hall state. Phys. Rev. B 42, 1, 212 (1990)
[43] Stone M.: Int. J. Mod. Phys. B 9, 1359 (1995)
[44] Stuart D.: Dynamics of Abelian Higgs vortices in the near Bogomolny regime. Commun. Math. Phys. 159, 51–91 (1994) · Zbl 0807.35141
[45] Stuart D.: The geodesic approximation for the Yang-Mills-Higgs equations. Commun. Math. Phys. 166, 149–190 (1994) · Zbl 0814.53052
[46] Stuart D.: Periodic solutions of the Abelian Higgs model and rigid rotation of vortices. Geom. Funct. Anal. 9, 1–28 (1999) · Zbl 0998.35053
[47] Stuart D.: Analysis of the adiabatic limit for solitons in classical field theory. Proc R Soc A 463, 2753–2781 (2007) · Zbl 1130.70013
[48] Taylor M.: Partial Differential Equations. Applied Mathematical Sciences, Vol 117. Springer-Verlag, Berlin-Heidelberg-NewYork (1996)
[49] Tsvelik A.M.: Quantum Field Theory in Condensed Matter Physics. Cambridge University Press, Cambridge (2003) · Zbl 1059.81005
[50] Wilczek, F.(1982) Quantum mechanics of fractional spin particles. Phys. Rev. Lett. 49, 1, 957
[51] Zhang S.C.: The Chern-Simons-Landau-Ginzburg theory of the fractional quantum Hall effect. Int. J. Mod. Phys. B 6(1), 43–77 (1992)
[52] Zhang, S.C., Hansson, T.H., Kivelson, S.: Effective field theory model for the fractional quantum Hall effect. Phys. Rev. Lett. 62, 82 (1989), Erratum: Phys. Rev. Lett. 62, 980 (1989)
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