Skip to main content

K-theory of C*—Algebras in solid state physics

  • Conference paper
  • First Online:
Statistical Mechanics and Field Theory: Mathematical Aspects

Part of the book series: Lecture Notes in Physics ((LNP,volume 257))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

Monographs

  1. S. AGMON, Lectures on Exponential Decay of Solutions of Second-Order Elliptic Equations. Princeton University Press. (1982). Princeton

    Google Scholar 

  2. T. ANDO, A.B. FOWLER, F STERN. Electronic properties of two-dimensional systems. Rev. Mod. Phys., 54, 437–672. (1982).

    Google Scholar 

  3. M. ATIYAH, K-Theory, Benjamin. New-York Amsterdam. (1967)

    Google Scholar 

  4. H. BOIIR, Almost Periodic functions. Chelsea Publishing Company. New-York, (1947)

    Google Scholar 

  5. I.P. CORNFELD, S.V. FORMIN, Ya.G. SINAI, Ergodic Theory. Grundlerhen Bd245. (1982). Springer Verlag. Berlin. Heidelberg. New York.

    Google Scholar 

  6. J. DIXMIER, Les algèbres d'opérateurs dans l'espace Hilbertien (Algèbres de Von Neumann). Gauthiers-Villars, Paris, (1969).

    Google Scholar 

  7. M. EASTHAM, The Spectral Theory of Periodic Differential Equations. Scottish Academic Press. Edimburgh, (1973).

    Google Scholar 

  8. L. FORD. Automorphic Functions. Chelsea Publishing Company. New-York, (1951).

    Google Scholar 

  9. G. FREUD. Orthogonal Polynomials. Pergamon Press. (1971).

    Google Scholar 

  10. J. GLIMM. A. JAFFE. Quantum Physics, a functional integral point of view, Springer-Verlag, Berlin Heidelberg New-York, (1981).

    Google Scholar 

  11. W. GOTTSCHALK. G.A. HEDLUND. Topological Dynamics. A.M.S. Coll. Publ., vol. 36. Providence. (1955).

    Google Scholar 

  12. F.P. GREENLEAF, Invariant means on topological groups, Van Nostrand, New-York Toronto London Melbourne. (1969).

    Google Scholar 

  13. P.R. HALMOS. Lectures on Ergodic Theory. Chelsea Publishing Company. New-York, (1956).

    Google Scholar 

  14. M.R. HERMAN. Sur la conjugaison différentiable des difféomorphismes du cercle i des rotations. Pub. Math. I.H.E.S., 49. 5–234, (1979).

    Google Scholar 

  15. “Recent developments in gauge theory”. G. 't HOOFT, C. ITZYKSON. A. JAFFE. R. STORA, Eds., Plenum Press. New York. (1980)

    Google Scholar 

  16. R.A. JOHNSON. A Review of Recent Works on Almost Periodic Differential and Difference Operators. Acta Appl. Math. 1 (1983), 161–241.

    Google Scholar 

  17. Operator Algebras and Applications. Proceedings of symposia in Pure Mathematics. D. KADISON Ed.. 38 Part I&It, A.M.S., Providence. Rhode Island. (1982).

    Google Scholar 

  18. T. KATO, Perturbation theory for linear operators, Die Grundlehren der Math. Wiss., vol. 132, Springer-Verlag, New-York, (1965).

    Google Scholar 

  19. G. PEDERSEN, C-algebras and their automorphism groups”, Academic Press, London New-York, (1979).

    Google Scholar 

  20. R.S.PEIERLS, Quantum Theory of Solids, Oxford Clarendon Press, (1955)

    Google Scholar 

  21. H. POINCARË, Les Nouvelles Méthodes de la Mécanique Céleste, vol. 1,11,111, Gauthiers-Villars (1892-94-99), Reprinted by Dover, New-York (1957).

    Google Scholar 

  22. M. REED, B. SIMON, Methods of Modern Mathematical Physics, Vol. IV Academic Press, New-York London, (1972).

    Google Scholar 

  23. J. RENAULT, A Groupoid Approach to C-Algebras, Lecture Notes in Mathematics, 793, (1980), Springer-Verlag, Berlin Heidelberg New-York.

    Google Scholar 

  24. D. RUELLE, Statistical Mechanics. Rigorous Results, Benjamin, Amsterdam, (1969)

    Google Scholar 

  25. S. SAKAI, C“-Algebras and W-Algebras, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Band 60, Springer-Verleg, Berlin Heidelberg New-York, (1971).

    Google Scholar 

  26. C. SIEGEL, J. MOSER. Lectures on Celestial Mechanics, Grundlehren, Band 187, Springer-Verlag, Berlin Heidelberg New-York, (1971).

    Google Scholar 

  27. B. SIMON, Functional integration and quantum physics, Academic Press, New-York, (1979).

    Google Scholar 

  28. J.B. SOKOLOFF, Unusual Band Structure Wave Functions and Electrical Conductance in Crystals with Incommensurate Periodic Potentials, Preprint Northeastern Univ., Boston Ma, (1984)

    Google Scholar 

  29. M. TAKESAKI, Tomita's theory of modular Hilbert algebras and its applications. Lecture Notes In Mathematics, 128. (1970). Springer-Verlag, Berlin Heidelberg New-York.

    Google Scholar 

Articles

  1. S.ALEXANDER, Some Properties of the Spectrum of the Serpinski Gasket in a Magnetic Field, Preprint 1984.

    Google Scholar 

  2. T. ANDO, Y. MATSUMOTO, Y. UEMURA, Theory of Hall effect in a two dimensional electron system, J. Phys. Soc. Japan, 39 (1975). 279

    Google Scholar 

  3. H. AOKI, T. ANDO, Effect of localization on the Hall conductivity In the two dimensional system in a strong magnetic field, Solid State Commun., 38, (1981), 1079–1082.

    Google Scholar 

  4. W. ARVESON, The harmonic analysis of automorphism groups, in [M171]. (1982), 199–269.

    Google Scholar 

  5. S. AUBRY, The new concept of transition by breaking of analyticity in a crystallographic model, Solid State Sci., 8 (1978), 264.

    Google Scholar 

  6. S. AUBRY, G. ANDRE, Analyticity breaking and Anderson localization in incommensurate lattices. Ann. Israel Phys. Soc., 3 (1980), 133.

    Google Scholar 

  7. S. AUBRY P.Y. Le DAERON, The Discrete Frenkel-Kontorova Model and Its Applications, l, Exact Results for the Ground State, Physics 8D, (1983), 381–422.

    Google Scholar 

  8. S.Aubry, P.Y.Le DAERON, G.ANDRÉ, Classical Ground State of a one dimensional Model for Incommensurate structures, Preprint Saclay, France, (1982).

    Google Scholar 

  9. J. AVRON, R. SEILER, B. SIMON, Homotopy and quantization in condensed matter physics, Phys. Rev. Lett., 51, (1983), 51–53.

    Google Scholar 

  10. J. AVRON, R. SEILER. Quantization of the Hall Conductance for General, Multiparticle Schrödinger Hamiltonians, Preprint Berlin 84

    Google Scholar 

  11. J. AVRON, B. SIMON. Transient and recurrent spectrum, J. Func. Anal., 43, (1981) 1–31.

    Google Scholar 

  12. M.F. BARNSLEY, J.S. GERONIMO, A.N. HARRINGTON, Infinite dimensional Jacobi matrices associated with Julia sets, Proc. A.M.S., 88 (1983). 625–630.

    Google Scholar 

  13. M.F. BARNSLEY, J.S. GERONIM0, A.A. HARRINGTON. Almost periodic Jacobi matrices associated with Julia sets for polynomials, Commun. Math. Phys., 92 (1985), 303–317.

    Google Scholar 

  14. J. BELLISSARD, Schrödinger Operators with an Almost Periodic Potential, In “Mathematical Problems in Theoretical Physics”, R. SCHRADER, R. SEILER Eds., Lecture Notes in Physics, 153. (1982), 356–359, Springer-Verlag, Berlin Heidelberg New-York.

    Google Scholar 

  15. J. BELLISSARD, D. BESSIS, P. MOUSSA, Chaotic states of almost periodic Schrödinger operators, Phys. Rev. Lett., 49, (1982),701–704.

    Google Scholar 

  16. J. BELLISSARD, R. LIMA, A. FORMOSO, D. TESTARD, Quasi Periodic Interactions with a Metal Insulator Transition, Phys. Rev., B26. (1982), 3024–3030.

    Google Scholar 

  17. J. BELLISSARD, E. SCOPPOLA, The Density of States of Almost Periodic Hamiltonians and the Frequency Module: a counter example, Commun. Math. Phys., 85, (1982), 301–308.

    Google Scholar 

  18. J. BELLISSARD, B. SIMON, Cantor spectrum for the Almost Mathieu equation, J. Funct. Anal., 48. (1982), 408–419.

    Google Scholar 

  19. J. BELLISSARD, R. LIMA, D. TESTARD, Almost Periodic Schrödinger Operators, in “Mathematics Physics, Lectures on Recent Results”, Vol. 1, L. STREIT ed., World Scientific, Singapore Philadelphia, (1985),1–64.

    Google Scholar 

  20. J. BELLISSARD, D.R. GREMPEL, F. MARTINELLI. E. SCOPPOLA, Localization of electrons with spin—orbit or magnetic interactions in a two dimensional crystal, to appear in Phys. Rev. Rapid Communication B, (1986).

    Google Scholar 

  21. L. BOLTZMANN. a) Wien Ans. 17 (1880). 12. b) Phil. Mag., 9, (1880), 307.

    Google Scholar 

  22. M. Casdagli, Symbolic Dynamics for the Renormalization Map of a Quasiperiodic Schrödinger Equation, Preprint Univ. of Warwick, July 1985, to appear in Commun. Math. Phys.

    Google Scholar 

  23. F.H. CLARO, G.H. WANNIER, Magnetic Subband Structure of Electrons in Hexagonal Lattices, Phys. Rev., B19, (1979). 6068–6074.

    Google Scholar 

  24. L.A. COBURN, R.D. MOYER, I.M. SINGER, C-algebras of almost periodic pseudo-differential operators, Acta Math., 139, (1973), 279–307.

    Google Scholar 

  25. A. CONNES, Une classification des facteurs de type 111, Ann. Sci. Ecole Norm. Sup., 40 série, 6 (1973), 133–252.

    Google Scholar 

  26. A. CONNES, The Von Neumann algebra of a foliation, Lecture Notes in Physics, 80 (1978), 145–151, Springer-Verlag, Berlin Heidelberg New-York.

    Google Scholar 

  27. A. CONNES, Sur la théorie non commutative de l'intégration, in “Algébres d'opérateurs”, P. de la HARPE ed., Lecture Notes in Mathematics, 725, (1980), 19–143, Springer-Verlag, Berlin Heidelberg New-York.

    Google Scholar 

  28. A. Connes, C' algèbres et géométrie différentielle, C.R.A.S., 290, (1980)

    Google Scholar 

  29. A. CONNES, An analog of the Thom homomorphism for crossed products of a C*-algebra by an action of R, Adv. Math., 39, (1981), 31–55.

    Google Scholar 

  30. A. CONNES, A survey of foliations and operator algebras, in [M171, (1982), 521–628.

    Google Scholar 

  31. A. CONNES, G. SKANDALIS, Le théorème de l'index pour les feuilletages, C.R.A.S. I, 292 (1981), 871–876.

    Google Scholar 

  32. A. CONNES. Non Commutative Differential Geometry, Part 1: The Chern Character in K-Homology, Preprint IHES, (1982)

    Google Scholar 

  33. A. CONNES, Non Commutative Differential Geometry, Part Il: De Rham Cohomology and Non Commutative Algebra, Preprint IRES, (1983)

    Google Scholar 

  34. A. CONNES, Cohomologie cyclique et foncteur Extn, C.R.A.S. I, 296 (1983). 953–958.

    Google Scholar 

  35. F. DELYON, B. SOUILLARD, Remark on the continuity of the density of states of ergodic finite difference operators, Commun. Math. Phys., 24, (1984), 289–291.

    Google Scholar 

  36. F. DELYON, B. SOUILLARD, The rotation number for finite difference operators and its properties, Commun. Math. Phys., 94, (1983). 415.

    Google Scholar 

  37. A. DENJOY, Sur les courbes définies parles équations différentielles à la surface du tore, J. de Math. Pures et Appi., (9)11, (1932), 333–375.

    Google Scholar 

  38. B.A. DOBRUVIN, V.B. MATVEEV, S.P. NOVIKOV, Non linear equations of Korteweg-de Vries type, finite zone linear operators and abellan varieties, Russ. Math. Surveys, 31, (1976), 59–146.

    Google Scholar 

  39. M. DUNEAU, A. KATZ, Quasiperiodic Patterns, Phys. Rev. Letters, 54, (1985). 2688.

    Google Scholar 

  40. M. DUNEAU, A. KATZ, Quasiperiodic Patterns and Icosahedral Symmetry, Ecole Polytechnique Preprint, Palaiseau,(France)(1985).

    Google Scholar 

  41. G. ELLIOTT, Gaps in the spectrum of an almost periodic Schrödinger operator, C.R. Math. Ref. Acad. Sci. Canada, 4., (1982). 255.

    Google Scholar 

  42. G. ELLIOTT, on the K-Theory of the C'-algebra generated by a projective representation of a torsion-tree discrete abelian group, in Proc. of the Coal. on "Operator Algebras and Group Representations”, G. ARSENE Ed., Pitman, London, (1983)

    Google Scholar 

  43. T. FACK, G. SKANDALIS, Structure des idéaux de la C-algèbre associée à un feuilletage, Preprint Univ. Pierre et Marie Curie, Paris (1980).

    Google Scholar 

  44. J. Fröhlich, T. SPENCER, Absence of diffusion for the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys., 35, (1983). 203–245.

    Google Scholar 

  45. J. GLIMM, A. JAFFE, Quantum field theory models, in “Statistical Mechanics and Quantum Field Theory”, C. de WITT & R. STORA, Eds., Les Houches summer school 1970, Gordon and Breach Science Pub., New York-London-Paris, (1971).

    Google Scholar 

  46. A. GROSSMAN, Momentum-Like Constants of Motion, 1971 Europhysics Conference on Statistical Mechanics and Field Theory, Haifa, R.N. SEN & C. WEIL Eds. (1972)

    Google Scholar 

  47. R. HAAG, H. HUGENHOLTZ, M. WINNINK, On the equilibrium states in quantum statistical mechanics, Commun. Math. Phys., 5, (1967), 215.

    Google Scholar 

  48. R. HAAG, D. KASTLER, An algebraic approach to quantum field theory, J. Math. Phys., 5, (1964). 848–851.

    Google Scholar 

  49. E.H. HALL, On a New Action of the Magnet on Electric Currents, Amer. J. of Math., 2 (1879), 287, and Phil. Mag., 9, (1880), 225

    Google Scholar 

  50. B.I. HALPERIN, Quantized Hall conductance, current-carrying edge states, and the existence of extended states In a two dimensional disordered potential, Phys. Rev. B25, (1982). 2185

    Google Scholar 

  51. P.G. HARPER, Single Band Motion of Conduction Electrons in a Uniform Magnetic Field, Proc. Phys. Soc. London, A68, (1955). 874

    Google Scholar 

  52. M.R. HERMAN, Une méthode pour minorer les exposants de Lyapounov, et quelques exemples montrant le caractère local d'un théorème d'Arnold et Moser sur un tore de dimension 2, Commun. Math. Helv., 58, (1983), 453–502.

    Google Scholar 

  53. D.R. HOFSTADTER, Energy levels and wave functions of Bloch electrons in a rational or irrational magnetic field, Phys. Rev., B14 (1976), 2239.

    Google Scholar 

  54. R. JOHNSON, The Recurrent Hill's Equation, J. Diff. Eqns, 46 (1982), 165–194

    Google Scholar 

  55. R. JOHNSON, J. MOSER, The rotation number for almost periodic potentials, Commun. Math. Phys., 84, (1982), 403.

    Google Scholar 

  56. R.A.JOHNSON, Exponential dichotomy, rotation number, and linear differential equations, to appear in J. Diff. Eqns., (1985).

    Google Scholar 

  57. A. Katok, Some Remarks on Birkhoff and Mather Twist Map Theorem, Erg. Theo. and Dyn. Syst., 2 (1982), 185–194.

    Google Scholar 

  58. A.KATOK, More about Birkhoff Periodic Orbits and Masher Sets for Twist Maps, Preprint I.Ihes. (1982), Preprint 11, Univ. Maryland, (1982).

    Google Scholar 

  59. S. KAWAJI, J. WAKABAYASHI, Hall conductivity in n-type silicon inversion layers under strong magnetic fields, Surf. Science, 98, (1980), 299–307.

    Google Scholar 

  60. S. KAWAJI, J. WAKABAYASHI, Hall current measurement under a strong magnetic field for silicon MOS inversion layers. J. Phys. Soc. Japan, 48, (1980), 333–334.

    Google Scholar 

  61. M. KLEMAN, DONNADIEU, Use of tessellations in the study of properties of covalent glasses, to appear in Phil. Mag. (1985).

    Google Scholar 

  62. K. von KLITZING, G. DORDA, M. PEPPER, Realization of a resistance standard based on fundamental constants, Phys. Rev. Letters, 45, (1980), 494–497.

    Google Scholar 

  63. M. KOHMOTO, L. KADANOFF, C. TANG, Localization problem in one dimension: mapping and escape, Phys. Rev. Lett., 50 (1983), 1870–1872.

    Google Scholar 

  64. P. KRAMER, R. NERI, Acta Crystallogr., 40 (1984), 580–587.

    Google Scholar 

  65. P. KRAMER, On the theory of a non periodic quasilattice associated with the icosahedral group, Preprint Tübingen, May 1985.

    Google Scholar 

  66. H. KUNZ, B. SOUILLARD, Sur le spectre des opérateurs aux différences finies aléatoires, Commun. Math. Phys., 78 (1980), 201–246.

    Google Scholar 

  67. H. KUNZ, B. SOUILLARD, The localization transition on the Bethe lattice, J. Physique Lett.(Paris), 44 (1983). L411.

    Google Scholar 

  68. L. LANDAU, Z. für Phys., 64 (1930), 629

    Google Scholar 

  69. R.B. LAUGHLIN, Quantized Hall conductivity in two dimensions, Phys. Rev., B23, (1981), 5652–565

    Google Scholar 

  70. D. LEVINE, P. STEINHARDT, Quasicrystals: A New Class of Ordered Structures, Phys. Rev. Letters, 53, (1984). 2477.

    Google Scholar 

  71. G. MACKEY. Ergodic theory, group theory and differential geometry, Proc. Nat. Acad. Sci. U.S.A., 5D (1963). 1184–1191.

    Google Scholar 

  72. G. MACKEY. Ergodic Theory and Virtual Subgroups, Math. Ann., 166, (1966). 187–207, see especially section 4, p. 190 and section 6, p.195.

    Google Scholar 

  73. J.N. MATHER, Existence of Quasi Periodic Orbits for Twist Homeomorphisms of the Annulus. Topology, 21. (1982). 457–467.

    Google Scholar 

  74. J. MATHER. Amount of Rotation About a Point and the Morne Index, Commun. Math. Phys., 94 (1984), 141–153.

    Google Scholar 

  75. J. MOSER. An example of a Schrödinger equation with an almost periodic potential and a nowhere dense spectrum, Commun. Math. Helv., 56, (1981), 198.

    Google Scholar 

  76. R. MOSSERI, J.F. SADOC, in “Structure of non crystalline materials 1982”, Taylor & Francis Ed. (1983), London.

    Google Scholar 

  77. P.MOUSSA, Un Opérateur de Schrödinger Presque Périodique Spectre Singulier Associé aux Itérations d'un Polynome, Proc. RCP 25, CNRS n034. (1984), 43, Strasbourg.

    Google Scholar 

  78. D. NELSON, S. SACHDEV, Statistical Mechanics of pentagonal and icosahedral order in dense liquids, Phys. Rev., B32 (1985). 1480–1502.

    Google Scholar 

  79. S. OSTLUND, R. PANDIT, D. RAND, H. SCHNELLHUBER, E. SIGGIA, One dimensional Schrödinger equation with in almost periodic potential, Phys. Rev. Lett. 50, (1983). 1873–1875.

    Google Scholar 

  80. S.OSTLUNDT, RYANDIT, Renormalization Group Analysis of a Discrete Quasi Periodic Schrödinger Equation, Preprint (1984).

    Google Scholar 

  81. M. PIMSNER, D. YOICULESCU, Exact Sequence for K groups and Ext groups of certain cross-product C*-algebras, J. Operator Theory, 4, (1980). 93–118.

    Google Scholar 

  82. M. PIMSNER, D. VOICULESCU, Imbedding the irrational rotation C*-algebra into an AF algebra, J. Operator Theory, 4 (1980), 211–218.

    Google Scholar 

  83. R. PRANGE, Quantized Hall resistance and the measurement of the fine structure constant, Phys. Rev., B23. (1981), 4802–4805.

    Google Scholar 

  84. I. PUTNAM, K. SCHMIDT. C. SKAU, C*-algebras associated with Denjoy homeomorphisms of the circle, Preprint Trondheim. (Norway), (1985).

    Google Scholar 

  85. R. RAMMAL, Spectrum of Harmonic Excitations on Fractals, J. Physique, 45, (1984), 191–206.

    Google Scholar 

  86. M.A. RIEFFEL, Irrational rotation C*-algebras, Short communication at the International Congress of Mathematicians, (1978).

    Google Scholar 

  87. M.A. RIEFFEL,a) Morita equivalence for operator algebras, in [M 17], p.285–298; b) Applications of strong-Morita equivalence to transformation group C'-algebras, in [M171], p. 299–310 see also reference therein.

    Google Scholar 

  88. D. RUELLE, D. Sullivan, Cycles and the dynamical study of foliated manifolds, Invent. Math., 36 (1976), 225–255.

    Google Scholar 

  89. R. SACKEL, G. SELL, Existence of dichotomies and invariant splittings for differential systems I) J. Diff. Eqns, 15, (1974), 429–458. II) J. Diff. Eqns, 22 (1976), 478–496,III) J. Diff. Eqns, 22 (1976), 497–522, IV) J. Diff. Eqns, 27, (1974), 106–137.

    Google Scholar 

  90. R. SACKEL, G. SELL, A spectral theory for linear differential systems, J. Diff. Eqns, 27 (1978), 320–338.

    Google Scholar 

  91. R. SEILER, Private communication

    Google Scholar 

  92. M.A. SHUBIN, The spectral theory and the index of elliptic operators with almost periodic coefficients, Russ. Math. surveys, 34, (1979). 109–157.

    Google Scholar 

  93. B. SIMON, Almost Periodic Schrödinger Operators, Adv. Appl. Math., 3, (1982), 463–490.

    Google Scholar 

  94. M.M. SKRIGANOV, Proof of the Bethe-Sommerfeld conjecture In dimension two, Sov. Math. Dokl., 20, (1979), 956–959.

    Google Scholar 

  95. S.SMALE, Differential Dynamical Systems, Bull. A.M.S., 73, (1967), 747–897, see especially the appendix 11, p. 797.

    Google Scholar 

  96. B. SOUILLARD, G. TOULOUSE, M. VOOS, L'effet Hall en quatre actes, Preprint Palaiseau (France), (1984).

    Google Scholar 

  97. D. THOULESS, Localization and the two-dimensional Hall effect, J. of Phys., C14, (1982), 3475–3480.

    Google Scholar 

  98. D. THOULESS, M. KOHMOTO, M. NIGHTINGALE, M. den NIJS, Quantized Hall conductance in two dimensional periodic potential, Phys. Rev. Lett., 49, 405. (1982).

    Google Scholar 

  99. C.H. WILCOX, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146–167.

    Google Scholar 

  100. This method is well known from the experts. I thank Michael Barnsley for mentioning the Schur complement, and Pierre Duclos for the reference to Feschback. Some informations can be found in: P. LOWDIN, “.The calculation of upper and lower bounds of energy eigenvalues in perturbation theory by mean of partitioning technics” In Int. Summer School on “ Quantum theory of polyatomic molecules”, Menton. (1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

T. C. Dorlas N. M. Hugenholtz M. Winnink

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Bellissard, J. (1986). K-theory of C*—Algebras in solid state physics. In: Dorlas, T.C., Hugenholtz, N.M., Winnink, M. (eds) Statistical Mechanics and Field Theory: Mathematical Aspects. Lecture Notes in Physics, vol 257. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16777-3_74

Download citation

  • DOI: https://doi.org/10.1007/3-540-16777-3_74

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16777-8

  • Online ISBN: 978-3-540-39858-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics