Shinobu Hikami

Spin-Orbit Interaction and Magnetoresistance in the Two Dimensional Random System

Effect of the spin-orbit interaction is studied for the random potential scattering in two dimensions by the renormalization group method. It is shown that the localization behaviors are classified in the three different types depending on the symmetry. The recent observation of the negative magnetoresistance of MOSFET is discussed. In recent experiments on MOSFET by Kawaguchi et al.,u it was found that electrons confined in the MOS inversion layer exhibit the negative magnetoresistxad ance. This effect is closely related to the localization problem in a random potential. In two dimensions, the quantum interxad ference is important and, if the impurity scattering is spin-independent, the conxad ductivity vanishes at zero temperature even when the scattering is very weak. 2>

Anderson Localization in Nonlinear σ Model Representation

It is shown that the Anderson localization problem of electron impurity scattering is described by a special type of the field theoretical model, which is called a nonlinear σ model. The diagrammatic l/EFτ expansion has one-to-one correspondence with small coupling t-expansion in the nonlinear σ model. The renormalization problems in the Anderson localization are solved by the equivalence of this problem to the nonlinear σ model, which is a renormalizable theory in two and three dimensions in the sense of small t and small e=d−2 double expansions. The β-function is calculated up to three-loop order and it shows the asymptotic free behavior in two dimensions when the system has time-reversal and space-inversion invariance. The diffusion constant D is obtained as a function of frequency ω. The results of the unitary and the symplectic case are also presented. The lattice nonlinear σ model in one dimension is considered for the comparison with the β-function of the one dimension electron scattering problem.

Renormalization Group Functions of CPN-1 Non-Linear σ-Model and N-Component Scalar QED Model

The renormalization group functions of CPN-I non-linear O-model are studied in 2+ 365. By the 1/ N-expansion, the relation of CPN- 1 model and scalar QED model will be studied. In the statistical mechanics, the superconductor phase transition or smectic A~nematic phase transition may be formulated by the scalar QED model. This problem is also discussed in this paper.

Non-Linear σ Model of Grassmann Manifold and Non-Abelian Gauge Field with Scalar Coupling

The renormalization group functions of the non-abelian gauge field with scalar coupling are studied near four dimensions. The 1/ N expansion is considered for Z<d<4. It is shown that this model is related to the non-linear !5 model of the real Grassmann manifold O(N)/O(p) xO(N-p) for the large N case. The case of a complex field is also investigated. Recently, the generalized non-linear o models have been investigated. The cpN-! modelv~sl is an example. In a previous paper,<) the renormalization group functions of cpN-! model in 2 + E: dimensions were calculated and the N-comxad ponent scalar QED model was considered as a linearized model of CPN-! model for the large N near four dimensions. In the 1/ N expansion, the consistency of the expression for the critical exponent v has been shown for both models in the large N limit. The generalized non-linear o models of the Grassmann manifold have been calculated for the renormalization group functions near two dimensions.) In cyv-l model, the gauge field is abelian. However, the gauge symmetry becomes nonxad abelian for the non-linear u model of the Grassmann manifold. *1 In this article we will further examine the relation of the non-abelian gauge field with scalar coupling to the Grassmann manifold model by calculating the renormalization group functions. This article is organized as follows: First the non-linear (J model of the real Grassmann manifold is explained. The relevant gauge field model is presented and renormalization group funtions are calculated in § 3. It is shown that there appears an infrared stable fixed point for the large N case. The 1/ N expansion is considered for the fixed value of p and the conxad sistency with the previous result of the non-linear u model is checked in § 4. The parallel analysis for the complex field is given in the Appendix.

Fixed Points and Anomalous Dimensions in a Thirring-Type Model in 2+ε Dimensions

Ve deal with the U(N) symmetric Thirring-type Lagrangian of .N-component fennions in ~+E dimensions. Applying the 1/N expansion technique we show the existence of a nontrivial ultraviolet fixed point in the Callan-Symanzik function ;3(g) between ~ and 4 space-time dimensions. In 2 dimensions 19 (g) vanishes identically as is characteristic of the Thirring model. vVe calculate an anomalous dimension of the fermion field in 1/ N order and find that it is discontinuous in 2 dimensions, i.e., it is 1/4N if a limit to the~ dimensions is taken from above, while it is a coupling dependent parameter as in the Thirring morld if it is evaluated in ~ dimensions. § l. Introduction and summary It is well-known that the renormalizability of the theory go1·erned by a given Lagrangian is closely related to the dimensionality of a coupling constant O. The dimensionality o is a decreasing function of the space-time dimension D and hence, for a gi1·en interaction, the renormalizability requirement o>O sets an upper bound on D which ,,-e call a critical dimension of the interaction, D,. Characteristic features of the theory may be described by the Callan-Symanzik function /3(g). At the critical dimension D the behaYior of 13 (g) for small g may :fall into either of the two types sho,vn in Figs. 1 (a) and (b). The latter case is of the so-called asymptotic freedom. For D--;L D the cot1pling constant has a dimension and we must replace g by gpcdn-n,J to make it dimensionless ·where a>O is a numerical constant and ,1 a

Nonlinear σ Models on Symmetric Spaces and Large N Limit

shown that there exists a third order phase transition in the large N limit of two dimensional lattice U(N) gauge theory which becomes equivalent to one dimensional chiral nonlinear (J model of N = =. In this paper, we investigate various lattice nonlinear (J models defined on symmetric spaces, which are coset spaces C/H where C is the Lie group and H is its maximum compact subgroup. We consider explicitly relevant invariant measure of our symmetric spaces. We consider the system in zero and one space dimension and calculate the energy and correlation function. It will be shown that the large N limit of these models has third order phase transition for the compact case. It will be also shown that for anisotropic Grassmannian model like Cp N 1 model, there exists a phase transition which has a discontinuity of specific heat in the large N limit. These zero and one dimensional studies may give basis of further investigations in higher dimensions. This article will be divided as follows: In § 2, we express the action by the angle variables and determine the Haar measure for various symmetric spaces. In § 3, we calculate energy of the one link and we discuss the large N behavior. In § 4, we consider CPN-l and RpN-l model as anisotropic large N cases. In § 5, two point correlation function in one dimension is considered. In § 6, S-funcxad tion is derived. Section 7 is devoted to discussion. In the Appendix, we present the large N calculation for d-dimensional lattice RpN-l model.

Equation of State in 1/n Expansion n-Vector Model in the Presence of Magnetic Field

region (s~1,n)>1). Concerning critical phenomena, a universal property is known that symmetry and space dimension determine the critical behavior. The n-vector model, which has been introduced by Stanley,2) is a model of spin system with n components. It is a generalized version of Ising (n=1), XY (n=2) and Heisenberg (n=3) models. This n-vector model has been investigated in 1/n expansion.3l~n Indeed these treatments lead to the same results as obtained by the ¢• theory.8l~m However, the n-vector model has a distinct property of the constraint for the spin field 6; the norm of (J is fixed to a certain value. Among field theories, nonlinear 6-modeP3) is known to have the same property. The linear 6-model,14l which has no constraint, corresponds to the ¢ 4 theory. Quite recently, the non linear 6-model has been investigated by several authors15)~17l for d = 2 + s dimension. In this paper, we present a refined treatment of n-vector model by clarifying the renormalization procedure and give the expression for equation of state up to order 1/n. Recently, universal ratio of critical amplitudes has been discussed by s expansion. 18) We also consider the same problem in 1/n expansion.

Study of Specific Heat below Tc in 1/n Expansion

The specific heat below Tc is studied in 1/n expansion, on the basis of the Ginzburgxad Landa,u-Wilson (GLW) Hamiltonian in the pr.esence of 11 constant magnetic field. It is shown that a is equal to aup to 0(1/n). The critical amplitude- ratio A+ I A_ is calculated for 3<d<4 (d: spatial dimension) and is proved to be consistent with the result obtained by E expansion. A divergence of A_ in the lowest order is pointed out at particular dimensions d,.=2+2/m (m=2,3;4,···). The A+/A- at d=3 is shown to approachaconstant as 1/n-70, whereas A+/A- for 3<d<4 is proportional to n in the same limit.

Fixed Points of a Coupled Order System in 1/n Expansion

A critical behavior of a coupled order system in ¢-theory is investigated. Two fields ¢, and ¢, have components, and their numbers are given by n and m, respectively. The interxad action of two fields is represented as g¢,¢, 2• The fixed points and the critical exponent 1J are studied in the large n and large m limit. The relation of a present model to an anisoxad tropic n-vector model is also discussed.