to be a constant for simplic ity This is the Kondo problem The nonvanishing

To be a constant for simplic ity this is the kondo

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to be a constant for simplic- ity. This is the Kondo problem. The nonvanishing beta-function means that the coupling grows with decreasing energy (for J positive). This is vividly seen in measurements of resistivity as a function of temperature, which in- creases as T decreases rather than showing the simple constant behavior of potential scattering. When the coupling gets strong, a number of behaviors are possible, depending on the value of s , sign of J , and various general- izations. In particular, in some cases one finds fixed points with critical behavior given by rather nontrivial conformal field theories: more examples of the interesting things that can happen when a marginal coupling gets strong! Exercise: Show that if the Fermi surface is right at a van Hove singular- ity, then under scaling of the energy to zero and of the momenta toward the singular point, the four-Fermi interaction is marginal in two space dimensions. In other words, if all electron momenta in a graph lie near the singularity, the graph is marginal: one does not have the usual simplifications of Landau theory. Acknowledgements I would like to thank I. Affleck, B. Blok, N. Bulut, F. de Wette, V. Kaplunov- sky, A. Ludwig, M. Marder, J. Markert, D. Minic, M. Natsuume, D. Scalapino, E. Smith, and L. Susskind for helpful remarks and conversations. I would also like to thank R. Shankar for discussions of his work and S. Weinberg for comments on the manuscript. This work was supported in part by National 31
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Science Foundation grants PHY89-04035 and PHY90-09850, by the Robert A. Welch Foundation, and by the Texas Advanced Research Foundation. Bibliography Lecture 1 Wilson’s approach to the effective action is developed in K. G. Wilson, Phys. Rev. B4 (1971) 3174, 3184. For further developments see K. G. Wilson and J. G. Kogut, Phys. Rep. 12 (1974) 75; F. J. Wegner, in Phase Transitions and Critical Phenomena, Vol. 6, ed. C. Domb and M. S. Green, Academic Press, London, 1976; L. P. Kadanoff, Rev. Mod. Phys. 49 (1977) 267; K. G. Wilson, Rev. Mod. Phys. 55 (1983) 583. For the treatment of perturbative renormalization from Wilson’s point of view see J. Polchinski, Nucl. Phys. B231 (1984) 269; G. Gallavotti, Rev. Mod. Phys. 57 (1985) 471. As I hope is clear from the discussion, these ideas do not depend on pertur- bation theory, and have been used to prove the existence of the continuum limit nonpertubatively in asymptotically free theories. This is done for the Gross-Neveu model in K. Gawedzki and A. Kupiainen, Comm. Math. Phys. 102 (1985) 1, and for D = 4 non-Abelian gauge theories in a series of papers culminating in T. Balaban, Comm. Math. Phys. 122 (1989) 355. The idea that pion physics can be encoded in a Lagrangian appeared in 32
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S. Weinberg, Phys. Rev. Lett. 18 (1967) 188; Phys. Rev. 166 (1968) 1568, and was developed further in S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177 (1969) 2239; C. G. Callan, S. Coleman, J. Wess, and B. Zumino, Phys. Rev. 177 (1969) 2247; S. Weinberg, Physica 96A (1979) 327; J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158 (1984) 142; H. Georgi, Weak Interactions and Modern Particle Theory, Ben- jamin/Cummings, Menlo Park, 1984.
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