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Chern numbers, quaternions, and Berry’s phases in Fermi systems. (English) Zbl 0830.57020

A common property of most systems studied in the context of the Berry phase is their oddness under time reversal. The standard example is the quantum Hamiltonian \(H_B= {\mathbf B} \cdot {\mathbf J}\) modelling a particle with \(\text{spin } J\) in an external magnetic field \({\mathbf B}\). If \(\Theta\) denotes the anti-unitary operator effecting the time reversal, then \(\Theta H_B= -H_B \Theta\). In this paper, Hamiltonians of the form \(H_Q= \sum_{i,j} Q_{ij} J_i J_j\) that are even under time reversal are analyzed. Physically, if \(Q\) is a real \(3\times 3\) symmetric matrix with \(\text{trace} (Q) =0\), such Hamiltonians describe systems with \(\text{spin } J\) in a quadrupole electric field \(Q\). Since for systems with integer spin, i.e. bosonic systems, the Berry phase resulting from \(H_Q\) is trivial, only Fermi systems with half-odd-integer spin show interesting topological phenomena.
Removing the point \(Q=0\) of level crossings from the parameter set, the eigenspaces of \(H_Q\) give rise to a canonical bundle structure, and in the adiabatic limit the evolution induced by the (time-dependent) Schrödinger equation leads to an “adiabatic” connection for these bundles. The detailed study of the resulting adiabatic curvature and holonomy, i.e. the generalization of the Berry phase to bundles that are not necessarily line bundles, constitutes the central theme of the paper. In addition, the Chern numbers labelling the spectrum of the operator family \(H_Q\) are computed. Quaternionic vector spaces enter into the analysis via the fact that the time reversal can be formulated as a quaternionic structure map, i.e. an antilinear map \(\Theta\) on a complex vector space satisfying \(\Theta^2 =\pm 1\).
As it turns out, systems with \(\text{spin } J= {3\over 2}\) show a behaviour somewhat different from those with \(J> {3\over 2}\). For \(J= {3\over 2}\), the set of \(H_Q\) is identical to the set of all Hermitian quaternionic matrices with vanishing trace. Furthermore, for unit \(Q\) all \(H_Q\) are unitarily equivalent, whereas for \(J> {3\over 2}\) there are one-dimensional families not unitarily related. Also, For \(J= {3\over 2}\) the adiabatic curvature is shown to be self-dual (with respect to the Hodge duality); on the other hand, the authors prove that for \(J> {3\over 2}\) no bundles with self-dual or anti-self-dual curvatures exist. The second Chern numbers \(C_2\) are calculated as \(C_2= \pm 1\) for \(J= {3\over 2}\), and \(C_2= (J+ {1\over 2}) (2m_T- J- {1\over 2})/2\) for the \(m_T\)th level of \(H_Q\) if \(J> {3\over 2}\) (the latter result was also published separately [cf. the second and third author, Chern numbers for fermionic quadruple systems, J. Phys. A 22, No. 4, L 111–L 115 (1989)]). Finally, explicit expressions for the time evolution operator and spectral projections are derived for \(J= {3\over 2}\) and \(J= {5\over 2}\), respectively, implying expressions for the associated holonomies. The resulting formulas for \(J= {3\over 2}\) are identical to the corresponding ones found previously by the third author [Non-abelian Berry’s phase, accidental degeneracy, and regular momentum, J. Math. Phys. 28, No. 9, 2102-2114 (1987)].

MSC:

57R99 Differential topology
58J90 Applications of PDEs on manifolds
57R57 Applications of global analysis to structures on manifolds
57R20 Characteristic classes and numbers in differential topology
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[1] Berry, M.V.: Proc. Roy. Soc. London A392, 45 (1984) · Zbl 1113.81306
[2] Simon, B.: Phys. Rev. Lett.51, 2167 (1983)
[3] Wilczek, F., Zee, A.: Phys. Rev. Lett.52, 2111 (1984)
[4] Mead, C. A.: Phys. Rev. Lett.59, 161 (1987)
[5] Avron, J., Sadun, L., Segert, J., Simon, B.: Phys. Rev. Lett.61, 1329 (1988)
[6] Avron, J. E., Seiler, R., Yaffe, L. G.: Commun. Math. Phys.110, 110 (1987) · Zbl 0626.58033
[7] Kato, T.: J. Phys. Soc. J. Jpn.5, 435 (1950)
[8] Choquet-Bruhat, Y., DeWitt-Morette, C., Dillard-Bleick, M.: Analysis, Manifolds and Physics, rev ed. Amsterdam: North Holland 1982 · Zbl 0641.58001
[9] Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I and II. New York: John Wiley 1963, 1969 · Zbl 0119.37502
[10] Atiyah, M.F.: Geometry of Yang-Mills fields, Lezioni Fermiane. Pisa: Accademia Nazionale dei Lincei & Scuola Normale Superiore 1979 · Zbl 0435.58001
[11] Chern, S. S.: Complex manifolds without potential theory, sec. ed. Berlin, Heidelberg, New York: Springer 1979 · Zbl 0444.32004
[12] Zee, A.: Phys. Rev. A38, 1 (1988) · Zbl 1230.81049
[13] Tinkham, M.: Group theory and quantum mechanics. New York: McGraw-Hill 1964 · Zbl 0176.55102
[14] Bjorken, J. D., Drell, S. D.: Relativistic quantum mechanics. New York: McGraw-Hill 1964; · Zbl 0184.54201
[15] Itzykson, C., Zuber, J. B.: Quantum field theory. New York: McGraw-Hill 1980 · Zbl 0453.05035
[16] Wigner, E. P.: Göttinger Nachr.31, 546 (1932); Wigner, E. P.: Group Theory. New York: Academic Press 1959
[17] Frobenius, Schur: Berl. Ber., p. 186 (1906)
[18] Dyson, F.: J. Math. Phys.3, 140 (1964) · Zbl 0105.41604
[19] Mehta, M. L.: Random matrices and the statistical theory of energy levels. New York: Academic Press 1967 · Zbl 0925.60011
[20] Kramers, H.A.: Proc. Acad. Amsterdam33, 959 (1930)
[21] Adams, J.F.: Lectures on Lie Groups. Chicago: University of Chicago Press 1969 · Zbl 0206.31604
[22] Bröcker, T., tom Dieck, T.: Representations of compact Lie groups. Berlin, Heidelberg, New York: Springer 1985 · Zbl 0581.22009
[23] Sadun, L., Segert, J.: J. Phys. A22, L 111 (1989)
[24] von Neumann, J., Wigner, E.P.: Phys. Zeit.30, 467 (1929)
[25] Friedland, S., Robbin, J. W., Sylvester, J. H.: On the crossing rule. Commun. Pure Appl. Math.37, 19 (1984) · Zbl 0523.15011
[26] Mermin, N. D.: Rev. Mod. Phys.51, No. 3 (1979)
[27] Avron, J., Seiler, R., Simon, B.: Phys. Rev. Lett.51, 51 (1983)
[28] Steenrod, N.: The topology of fibre bundles. Princeton: Princeton University Press 1951 · Zbl 0054.07103
[29] Bott, R., Tu, L. W.: Differential forms in algebraic topology. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0496.55001
[30] Herzberg, G., Longuet-Higgins, H. C.: Discuss. Faraday Soc.55, 77 (1963)
[31] Milnor, J. W., Stasheff, J. D.: Characteristic classes. Princeton, NJ: Princeton University Press 1974 · Zbl 0298.57008
[32] Thouless D., et. al.: Phys. Rev. Lett.49, 405 (1982)
[33] Kohmoto, K.: Ann. Phys.160, 343 (1985)
[34] Chang, L. N., Liang, Y.: Mod. Phys. L A3, 1839 (1988)
[35] Atiyah, M. F., Hitchin, N. J., Singer, I. M.: Proc. R. Soc. Lond. A362, 425 (1978) · Zbl 0389.53011
[36] Karoubi, M., Leruste, C.: Algebraic topology via differential geometry. Cambridge: Cambridge University Press 1987 · Zbl 0627.57001
[37] Wu. Y. S., Zee, A.: Phys. Lett. B207, 39 (1988)
[38] Chern, S.S., Simons, J.: Ann. Math.99, 48 (1974) · Zbl 0283.53036
[39] Moody, J., Shapere, A., Wilczek, F.: Phys. Rev. Lett.56, 893 (1986)
[40] Tycko R.: Phys. Rev. Lett.58, 2281 (1987)
[41] Segert, J.: Math. Phys.28, 2102 (1987)
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