Flux Quantization in a One-Dimensional Model

Michael Schick

doi:10.1103/PhysRev.166.404

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Flux Quantization in a One-Dimensional Model

Michael Schick^*

Department of Physics, Stanford University, Stanford, California

^*Present address: Department of Physics, Western Reserve University, Cleveland, Ohio.

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Phys. Rev. 166, 404 – Published 10 February, 1968

DOI: https://doi.org/10.1103/PhysRev.166.404

Abstract

A one-dimensional model of interacting electrons is studied to determine whether such a system, in thermal equilibrium, can exhibit flux quantization. The free energy and current of the system are calculated and shown to be periodic functions of the flux enclosed in a ring-shaped sample with period $\frac{hc}{e}$ . The Maxwell equations provide a second relation between the current and flux. It is found that at finite temperatures, the equations for the current $I$ and the flux $Φ_{B}$ have only the trivial solution $I = Φ_{B} = 0$ in the limit of macroscopic systems. Therefore, there is no flux quantization. The free energy is calculated by a generalization of the method of Tomonaga. This method describes the Fermi system in terms of an equivalent set of bosons which represent the collective modes of the Fermi gas. The major results of the generalization are the appearance of trilinear terms in the equivalent boson Hamiltonian and effects of a vector potential.

References (20)

W. A. Little, Phys. Rev. 134, A1416 (1964)
P. C. Hohenberg, Phys. Rev. 158, 383 (1967)
C. N. Yang, Rev. Mod. Phys. 34, 694 (1962)
W. A. Little, Phys. Rev. 156, 396 (1967)
J. S. Langer and V. Ambegaokar, Phys. Rev. (to be published)
[4,5]
F. Bloch, Helv. Phys. Acta 7, 385 (1934)
S. Tomonaga, Progr. Theoret. Phys. (Kyoto) 5, 544 (1950)
R. A. Ferrell, Phys. Rev. Letters 13, 330 (1964)
H. Gutfreund and M. Schick (to be published)
G. Wentzel, Phys. Rev. 83, 168 (1951)
M. Schick, Ph.D. thesis, Stanford University (unpublished)
S. Engelsberg and B. B. Varga, Phys. Rev. 136, A1582 (1964)
H. Fröhlich, Proc. Roy. Soc. (London) A215, 291 (1952)
N. Byers and C. N. Yang, Phys. Rev. Letters 7, 46 (1961)
[13]
J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 108, 1175 (1957)
H. T. Davis, The Summation of Series (Principia Press, Inc., San Antonio, Tex., 1962), p. 234
F. Bloch, Phys. Rev. 137, A787 (1965)
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1963), 4th ed., pp. 464, 470, and 472

[c1] W. A. Little, Phys. Rev. 134, A1416 (1964)

[c2] P. C. Hohenberg, Phys. Rev. 158, 383 (1967)

[c3] C. N. Yang, Rev. Mod. Phys. 34, 694 (1962)

[c4] W. A. Little, Phys. Rev. 156, 396 (1967)

[c5] J. S. Langer and V. Ambegaokar, Phys. Rev. (to be published)

[c6] [4,5]

[c7] F. Bloch, Helv. Phys. Acta 7, 385 (1934)

[c8] S. Tomonaga, Progr. Theoret. Phys. (Kyoto) 5, 544 (1950)

[c9] R. A. Ferrell, Phys. Rev. Letters 13, 330 (1964)

[c10] H. Gutfreund and M. Schick (to be published)

[c11] G. Wentzel, Phys. Rev. 83, 168 (1951)

[c12] M. Schick, Ph.D. thesis, Stanford University (unpublished)

[c13] S. Engelsberg and B. B. Varga, Phys. Rev. 136, A1582 (1964)

[c14] H. Fröhlich, Proc. Roy. Soc. (London) A215, 291 (1952)

[c15] N. Byers and C. N. Yang, Phys. Rev. Letters 7, 46 (1961)

[c16] [13]

[c17] J. Bardeen, L. Cooper, and J. Schrieffer, Phys. Rev. 108, 1175 (1957)

[c18] H. T. Davis, The Summation of Series (Principia Press, Inc., San Antonio, Tex., 1962), p. 234

[c19] F. Bloch, Phys. Rev. 137, A787 (1965)

[c20] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, England, 1963), 4th ed., pp. 464, 470, and 472

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