Induced fermion number in the O(3) nonlinear <span class="aps-inline-formula"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math></span> model

M. Carena; S. Chaudhuri; C. E. M. Wagner

doi:10.1103/PhysRevD.42.2120

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Induced fermion number in the O(3) nonlinear $σ$ model

M. Carena^*

S. Chaudhuri^†

C. E. M. Wagner^‡

Department of Physics, Purdue University, West Lafayette, Indiana 47907

Theoretical Physics Group, Fermi National Accelerator Laboratory, Batavia, Illinois 60510

Department of Physics, Purdue University, West Lafayette, Indiana 47907

^*Electronic address: carena%purdue.hepnet@lbl.bitnet.
^†Electronic address: shyam@fnal.bitnet.
^‡Electronic address: wagner%purdue.hepnet@lbl.bitnet.

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Phys. Rev. D 42, 2120 – Published 15 September, 1990

DOI: https://doi.org/10.1103/PhysRevD.42.2120

Abstract

We study the fermion number induced by nontrivial topological configurations in the O(3) nonlinear $σ$ model in 2+1 dimensions. We consider a scalar background configuration that adiabatically evolves from the normal vacuum to a soliton of winding number unity. The appearance of zero-energy modes is analyzed as a function of the relative magnitudes of the fermion mass scale $m_{f}$ and the inverse of the soliton width, $\frac{1}{ρ_{s}}$ . We find a single energy-level crossing whenever $\frac{m_{f} > 1}{ρ_{s}}$ , implying that, as in the case of the O(4) $σ$ model in four dimensions, the soliton carries the fermion number of any sufficiently heavy fermion. We also show the existence of states with fractional fermion number, $Q_{GS} = \pm \frac{1}{2}$ .

References (31)

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[c1] Y. -H. Chen, F. Halperin, F. Wilczek, and E. Witten, Int. J. Mod. Phys. B 3, 1001 (1989) J. Lykken, J. Sonnenschein, and N. Weiss, Report No. Fermilab-Pub-89/231-T, 1989 (unpublished)

[c2] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983)

[c3] A. J. Legett, Rev. Mod. Phys. 47, 331 (1975)

[c4] G. E. Volovik and V. M. Yakovenko, J. Phys. Condensed Matter 1, 5263 (1989)

[c5] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. Lett. 60, 1057 (1988)

[c6] P. Voruganti and S. Doniach, Phys. Rev. B 41, 9358 (1990)

[c7] A. A. Belavin and A. M. Polyakov, Pis'ma Zh. Eksp. Teor. 22, 503 (1975) [JETP Lett. 22, 245 (1975)]

[c8] Y. -H. Chen and F. Wilczek, Int. J. Mod. Phys. B 3, 117 (1989)

[c9] F. Wilczek and A. Zee, Phys. Rev. Lett. 51, 2250 (1983)

[c10] I. Dzyaloshinski, A. Polyakov, and P. B. Wiegmann, Phys. Lett. A 127, 112 (1988) P. B. Wiegmann, Phys. Rev. Lett. 60, 821 (1988)

[c11] E. Fradkin and M. Stone, Phys. Rev. B 38, 7215 (1988)

[c12] Z. Hlousek, D. Senechal, and S. -H. H. Tye, Phys. Rev. D 41, 3773 (1990)

[c13] R. Jackiw and C. Rebbi, Phys. Rev. D 13, 3398 (1976)

[c14] R. Jackiw, Rev. Mod. Phys. 49, 681 (1977)

[c15] R. Jackiw and J. R. Schrieffer, Nucl. Phys. B190, 253 (1980)

[c16] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979)

[c17] J. Goldstone and F. Wilczek, Phys. Rev. Lett. 47, 986 (1981)

[c18] R. MacKenzie and F. Wilczek, Phys. Rev. D 30, 2260 (1984) ibid.30, 2194 (1984)

[c19] T. Jaroszewicz, Phys. Lett. 146B, 337 (1984)

[c20] A. J. Niemi and G. W. Semenoff, Nucl. Phys. B269, 131 (1986)

[c21] A. J. Niemi and G. W. Semenoff, Phys. Rep. 135, 99 (1986)

[c22] A. J. Niemi, Nucl. Phys. B235, 14 (1985)

[c23] S. Forte, Nucl. Phys. B301, 69 (1988)

[c24] M. Carena, Ph.D. thesis, Report No. DESY 89-084, 1989

[c25] S. Kahana and G. Ripka, Nucl. Phys. A429, 462 (1984)

[c26] E. D'Hoker and E. Farhi, Nucl. Phys. B241, 109 (1984)

[c27] M. Carena, Phys. Lett. B 217, 135 (1989)

[c28] M. Carena, Phys. Lett. B 211, 117 (1988)

[c29] Handbook of Mathmetical Functions, edited by M. Abramowtiz and I. Stegun (Dover, New York, 1965)

[c30] G. Hall and J. M. Watt, Modern Numerical Methods for Ordinary Differential Equations (Clarendon, Oxford, 1976)

[c31] R. Goldflam and L. Wilets, Phys. Rev. D 25, 1951 (1982)

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Induced fermion number in the O(3) nonlinear $σ$ model

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Induced fermion number in the O(3) nonlinear σ model

Abstract

References (31)

Induced fermion number in the O(3) nonlinear $σ$ model