By D. J. Thouless (auth.), Professor Yosuke Nagaoka Ph.D., Professor Hidetoshi Fukuyama Ph.D. (eds.)

This quantity comprises the court cases of the Fourth Taniguchi foreign Symposium at the thought of Condensed topic, which used to be held at Senkari Semi nar apartment of Kwansei Gakuin Universi~y in Sanda-shi, Japan, throughout the interval of 3-8 November 1981. the subject of the symposium was once "Anderson rocalization," some of the most primary difficulties in condensed-matter physics. due to the fact Anderson's vintage paper used to be released in 1958, a lot theoretical and experimental attempt has been played to check the matter of electron localization in a random strength. particularly lately, Abrahams, Anderson, Licciardello, and Ramakrishnan proposed a scaling idea of the Anderson lo calization which made it attainable to accomplish microscopic investigations. speedy development has and we're now getting a coherent photograph of the habit of electrons in disordered structures. after we geared up the symposium, we requested Dr. Anderson to take part in it and to offer a evaluate speak on theoretical facets of the matter. although he kindly approved our invitation, he couldn't come because of a unexpected sickness. A evaluate speak was once given via Professor Thouless who kindly approved our request to take where of Dr. Anderson. thankfully, Dr. Anderson has when you consider that re coated from his illness.

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**Extra resources for Anderson Localization: Proceedings of the Fourth Taniguchi International Symposium, Sanda-shi, Japan, November 3–8, 1981**

**Example text**

16) where Do=E F/2mY, and Vpp,(q,W) is the vertex function that is irreducible with respect to the particle-particle channel. 2(b) is used for Vpp ' (q,w). 18) In the present system there is the time reversal symmetry and M1 and M2 are the same. 4. 1 However they are not the same when the symmetry is broken. Conductivity of the system without the time-reversal symmetry Weak magnetic field We use quasiclassical approximation for the magnetic field. We replace the momentum p by p+eA/c. Since the momentum q in ¢ ,(q,w) is a total momentum pp of particle-hole pair which has no net charge, it suffers no change by the magnetic field.

It has been shown that Upp ' obeys the Ward identity [16] (8) P which is a consequence of particle number conservation. Summing (7) on taking the identity (8) into account, one finds the continuity equation, The current relaxation function ¢j (q,w) = 1 (p. q/m) 4Ip(q,w) ( 10) p -+ obeys, for small w,q, the equation i -+ [16] -+ ( 11) [ w + T K (q, w)] ¢ j (q ,w) The dimensionless current relaxation kernel K(q,w) is defined in terms of the irreducible vertex by K(q,w) = 1 + ~ L. q) (12) 29 where 6G-+ P Here n is the particle density and T is the weak coupl ing transport relaxation time, given by T = 2y + d TIN p2 FF ( 13) From (9) and (11) one obtains for small q, ( 14) = w+iD(if,w)q2 where the q,w dependent diffusion coefficient is given by D (q ,w) and Do = (V~/d)T _D_o_ (15) K(q~w) is the bare diffusion constant (v F = PF/m).

This singularity, the so-called Cooper pole or 2kF pole,arises from a subtle quantum interference effect, which only exists if the system is time reversal invariant. It is easily seen that the singularity is exactly the diffusion pole, with a different momentum variable. E). t-q)/2, (p'-p~)/2 ) p+p',w. (21) Equation (21) is demonstrated diagrammatically in Fig. 3. ~ Diagrammatic demonstration of the crossing symmetry for ¢pp,(q,w) in the case of time reversal invariance; see eq. (21) The singularity in the particle-hole propagator at p+p'=O is obtained in lowest order by summing the maximally crossed diagrams [24] as shown in Fig.