Insensitivity of Quantized Hall Conductance to Disorder and Interactions

A two-dimensional quantum Hall system is studied for a wide class of potentials including single-body random potentials and repulsive electron–electron interactions. We assume that there exists a nonzero excitation gap above the ground state(s), and then the conductance is derived from the linear perturbation theory with a sufficiently weak electric field. Under these two assumptions, we prove that the Hall conductance _xy and the diagonal conductance _yy satisfy | _xy+e ² /h|const·L ^–1/2 and | _yy|const·L ^–1/12. Here e ²/h is the universal conductance with the charge –e of the electron and the Planck constant h; is the filling factor of the Landau level; and L is the linear dimension of the system. In the thermodynamic limit, our results show _xy=–e ² /h and _yy=0. The former implies that integral and fractional filling factors with a gap lead to, respectively, integral and fractional quantizations of the Hall conductance.     

Some remarks on variable range hopping—in particular in the quantum Hall regime

An expression for the heat conductivity _xx is derived in the effective medium approximation. Mott type formulas are obtained for _xx and the Peltier coefficient _xx . Using percolation theory in a three-dimensional system the Wiedemann-Franz ratio was found to depend on the temperature like . The Mott type formulas were evaluated in a similar way for a two-dimensional system in the quantum Hall regime within the high-field percolation model. In contrast to previous calculations of the high field hopping conductivity _xx , the results are fully consistent with the experimental data on _xx and the density of states at the Fermi level. Finally, _xx is estimated which together with _xx and _xy =ie ²/h(i=0,1,2,...), determines both thermopower coefficients _xx and _xy .Dedicated to Professor W. Brenig on the occasion of his 60th birthday     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

Interchangeability and bounds on the effective conductivity of the square lattice

The effective conductivity ^* of an infinitely interchangeable two-component random medium is considered. This class of media includes cell materials in the continuum and the bond lattice on ^d , where the cells or bonds are randomly assigned the conductivities ₁ and ₂ ( ₁, ₂ne0) with probabilitiesp ₁ andp ₂=1–p ₁. A rigorous basis for the very old and widely used low volume fraction expansion of ^* is established, by proving that ^* is an analytic function ofp ₂ in a suitable domain containing [0, 1]. In the case of the bond lattice ind=2, rigorous fourth-order upper and lower bounds on ^* valid for allp ₂, ₁, and ₂ are derived. The four perturbation coefficients entering into the bounds are obtained from the first-order volume fraction coefficient using the method of infinite interchangeability.     

Infinite hierarchy of exponents in a two-component random resistor network

We have studied the voltage distribution for a two-component random mixture of conductances _a and _b. A scaling theory is developed for the moments of the distribution, which predicts, for small values ofh=_a/_b, an infinite number of crossover exponents, one for each moment, for Euclidean dimensiond >2, and only one crossover exponent ford=2. Monte Carlo results on the square lattice confirm this prediction.     

Exact expressions for row correlation functions in the isotropicd=2 ising model

We present exact explicit expressions for the row spin-spin correlation functions _{00 n}0 in the isotropicd= 2 Ising model, in terms of elliptic integrals, forn 5. We also give a general structural formula for _{00 n}0.     

Asymptotics of decay of correlations for lattice spin fields at high temperatures. I. The Ising model

We find the asymptotic decrease of correlations < _{A +y} , _B >,yZ ^v +1, |y|, in the Ising model at high temperatures. For the case when monomials _A and _B both are odd, using the saddle-point method, we find the asymptotics of the correlations for any dimension . For even monomials _A , _B we formulate a general hypothesis about the form of the asymptotics and confirm it in two cases: (1) =1 and the vectory has an arbitrary direction, (2)y is directed along a fixed axis and arbitrary . Here we use besides the saddle-point method, some arguments from scattering theory.     

The modular group and super-KMS functionals

Every normal, faithful, self-adjoint functional on a von Neumann algebraA canonically determines a one-parameter-weakly continuous *-automorphism group (the analog of the modular group) and a canonical ₂ grading onA, commuting with . We show that the functional satisfies the weak super-KMS property with respect to and Furthermore, we prove that and are the unique pair of a-weakly continuous one-parameter *-automorphism group and a grading of the algebra, commuting with each other, with respect to which is weakly super-KMS. The above results thus provide a complete extension of the theory of Tomita and Takesaki to the nonpositive case.Supported in part by the National Science Foundation under Grant DMS-8922002.     

Self-diffusion of V-48 in the FeV σ-phase and in an Fe-47 wt.% V solid solution

Self-diffusion coefficients of vanadium in the FeV -phase and in the corresponding -solid solution (Fe-47 wt.% V) measured in the temperature ranges 1002–1115 °C (-phase) and 1230–1320 °C are reported. The found results differ fundamentally and significantly from the relations in ordered and disordered solid solutions [9]. The diffusivity in -phase at the transition temperature (T _/=1200 °C) is cca 14 times lower than the diffusivity in the b.c.c. solid solution, the chemical composition of which is the same. The lowering is caused by the different values of frequency factors,D _O=0.11 cm²/s andD _O=45 cm²/s. The effect of the corresponding activation enthalpiesH =252 kJ/mole andH =293 kJ/mole is small and quite opposite. The occurence of higher activation enthalpyH in the -solid solution at temperaturesT>T _/ may be attributed to a certain amount of the f.c.c. phase coexisting in the b.c.c. matrix at concentrationsc _v>27 wt.% at sufficiently high temperatures [7]. A comparison of vanadium self-diffusion characteristics measured in the -phase to the extrapolated values obtained on the basis of the previous measurements [1] in the Fe-V primary solid solutions ₁ shows that the diffusivity ratioD ₁/D (1473 K)=33 and that the activation enthalpyH is by about 3% higher than the valuesH ₁ (eq. (5)) measured in the uniphase b.c.c. solid solutions.     

Resistance to dislocation motion in copper-based solid solutions with a statistically disordered atomic distribution

The stress _F due to friction forces in copper-based solid solutions was determined. Under the conditions of the procedure used to measure _F, on the basis of the half-wave hysteresis with polycrystalline samples, the value of d: _F = _F0 + K_Fd^–(1/2) where _F0 is the resistance to dislocation motion in an alloy having an infinite grain size, and K_F is a constant. It is shown that _F0 is governed by the interaction of moving dislocations with impurity atoms in the case of a statistically disordered atomic distribution. A study was made of the effects of various factors on _F and of the nature of the changes in _F0 caused by alloying.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No.3, pp. 18–23, March, 1971.     

Transverse hall-conductivity in the Landau-subbands of a 2D-lattice

Some numerical results concerning the variation of the transverse hall-conductivity _xy atT=0 within the Landau-subbands of a 2D-tight-binding model with a rational magnetic field are presented.Work supported in part by the Sonderforschungsbereich 123 Stochastic Mathematical Models of the Deutsche Forschungsgemeinschaft     

Electrical and thermal properties of chalcogenide glass system Se75Ge25−x Sb x

The dc and thermal conductivities of five different compositions of the chalcogenide glass system Se₇₅Ge_25–x Sb _x have been studied in a temperature range below T _g. The dc conductivity results indicate that each composition has a single activation energy in the considered temperature range. The coefficient of thermal conductivity increased linearly with temperature below T _g for the compositions investigated. The increase of Sb content in the chalcogenide glass system leads to an increased coefficient of electrical conductivity , an increased coefficient of thermal conductivity , and to a decreased activation energy E and pre-exponential factor ₀. The observed compositional dependencies of and E have been correlated with the increase of weak bond density and the decrease of covalent bond density in the structure of the compositions investigated with increasing Sb content at the expense of Ge content. The decrease in ₀ and the increase in has been, respectively, correlated with the decrease in mobility and the increase in phonon velocity.     

Existence of the critical point in φ4 field theory

We consider the ⁴ quantum field theory in two and three spacetime dimensions. In the single phase region the physical mass (inverse correlation length)m() decreases continuously to zero as the bare mass parameter approaches a critical value _c from above. In three dimensions the critical point _c is in the single phase region and the physical mass vanishes there,m( _c )=0.A consequence of our results is that the critical exponentv governing the approach to infinite correlations is bounded below (rigorously) by its classical value, 1/2.Supported in part by the National Science Foundation under Grant MPS74-13252     

The asymmetric contact process on a finite set

The contact process onZ has one phase transition; let _c be the critical value at which the transition occurs. Let _N be the extinction time of the contact process on {0,...,N}. Durrett and Liu (1988), Durrett and Schonmann (1988), and Durrett, Schonmann, and Tanaka (1989) have respectively proved that the subcritical, supercritical, and critical phases can be characterized using a large finite system (instead ofZ) in the following way. There are constants ₁() and ₂() such that if < _c , lim _{N N} /logN = 1/₁(); if > _c , lim _N log _N /N = ₂(); if = _c , lim _{N N} /N= and lim _{N N} /N ⁴=0 in probability. In this paper we consider the asymmetric contact process onZ when it has two distinct critical values _c1< _c2. The arguments of Durrett and Liu and of Durrett and Schonmann hold for < _c1 and > _c2. We show that for [ _c1< _c2), lim _{N N} /N=-1/, (where _i is an edge speed) and for = _c2, lim _N log _N /logN=2 in probability.     

Macroscopic short-time electrodynamics for conductors

In the macroscopic electrodynamics (MED) of good conductors (metals) based on Ohm's law j=E, the momentum relaxation time =m/ne² of the electrons limits the application to electromagnetic (EM) processes with characteristic timest. An interesting physical difficulty occurs in MED since the EM field damping time _R=/ of metals is very small compared with the minimum macroscopic time scale, _R. Consequently, the damping and propagation of EM waves and pulses in good conductors cannot be correctly described within the frame of conventional MED. New hyperbolic EM wave equations with relaxation and memory are proposed, which no longer exhibit the _R deficiency. The latter is caused by Ohm's law, which breaks down for short-time processes, due to neglect of electron inertia. The advantages of the proposed and the disadvantages of the conventional EM wave equations for good conductors are discussed in applications.     

Electrical properties of doped single crystals of Zn4Sb3

Conductivity , thermo-emf , and Hall constant R are presented as functions of temperature for pure Zn₄Sb₃ single crystals and for ones doped with Ag, Au, Ga, In, Sn, Pb, and Te. Here Ag, Au, Sn, and Pb act as acceptors. Increased hole concentration and , together with the reduced , are produced by In, Te, and Ga, on account of formation of solid solutions of InSb, GaSb, and ZnTe in Zn₄Sb₃.     

On X-ray measurements of residual stresses in materials with lattice strain gradient

The non-linear distribution of lattice strain ³¹⁰vs sin² observed on the surface of a polished steel specimen can be interpreted by tri-axial stress state with surface components of stress tensor ₁₁(0)= ₂₂(0) and gradientsg ₁₁= ₁₁/T, g ₃₃= ₃₃/T. The distribution of experimental values is duscussed from the viewpoint of various ways of calculating ₁₁.The authors would like to thank Dr. J. Musil, D. Sc. of the Institute of Physics of Czechoslovak Academy of Sciences for kindly providing the samples which made possible this study.     

An algebraic approach to quantum field theory and commutation relations for energy momentum in the framework of general relativity

We present a consistent set of commutation relations (C.R.) for a quantum system immersed in a classical gravitational field. The gravity field is described by metric tensorg _ik (x) andg ₀₀(x) with coordinate gaugeg _i0=0. The Hamiltonian of the system is found to be a linear function of [–g ₀₀(x)]^1/2. Its properties we define by C.R. avoiding explicit expression in terms of fields, as well as its splitting into free and interaction parts. In this way a consistent set of C.R., which are equally simple for a flat and curvilinear space, can be established. To stress the main idea of our approach, we consider the simple but still nontrivial example of a scalar electrodynamics immersed in a gravity field. The electromagnetic current operator we define by its C.R. and not explicitly. An interesting feature of this approach is that the Poisson equation follows from the consistency of the C.R. The C.R. for the energy and momentum operators of the system in a gravity field are established which generalize the usual Poincare group generators C.R. For example, we find (i/hc ²)[H _(x) ,H _(x) ]=P _– , whereH _(x) is the Hamiltonian of the system, which is a linear functional of (x)[–g ₀₀(x)]^1/2 andP _s(x) represents the momentum-density operator [averaged with the classical functions(x)].     

пРИМЕНЕНИЕ МЕтОДА ОтНОшЕНИь к ОпРЕДЕДЕНИУ сОстАВльУЩЕИ σx НАпРьжЕНИь

In this paper the possibility is analyzed of using the ratio method to determine the biaxial stress component _x in an arbitrary direction from two photographs, one taken with the primary beam incident perpendicularly and rotating film holder and the second photograph with the primary beam incident obliquely and film holder stacionary or oscillating in a small angular range. A relation is derived for the value of _x based on measurements of the diameters of the diffraction circles in the corresponding azimut, and a relation for estimating the magnitude of the percentual error of the value of _x thus determined as compared with the real value of _x .It is shown in the case of aluminium in three typical stress states (circularly symmetrical stress distribution, unaxial state, torsional stress) that the error in determining _x , caused by using the ratio method, is quite unimportant for application in technical practice and is much smaller than when applying the ratio methods to the determination of the sum of the two principal stresses from one photograph taken with the primary beam incident perpendicularly. The application of the ratio method for determining _x gives simultaneously the possibility of using this method for solving the general problem, i.e. of determining the two principal stresses as to their magnitude and direction.     

Pauli-Dirac matrix generators of Clifford Algebras

This article presents a Pauli-Dirac matrix approach to Clifford Algebras. It is shown that the algebra C₂ is generated by two Pauli matrices i₂ and i₃; C₃ is generated by the three Pauli matrices ₁, ₂, ₃; C₄ is generated by four Dirac matrices ₀, ₁, ₂, ₃ and C₅ is generated by five Dirac matrices i₀, i₁, i₂, i₃, i₅. The higher dimensional anticommuting matrices which generate arbitrarily high order Clifford algebras are given in closed form. The results obtained with this Clifford algebra approach are compared with the vector product method which was described in a recent article [Found. Phys. 10, 531–553 (1980) by Poole, Farach and Aharonov] and with the Dirac, Rashevskii and Ramakrishnan methods of matrix generation.Supported by the National Science Foundation under Grant ISP-80-11451.