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     

Microwave relaxation and phenomenological damping in thin films

The relation between relaxation timeT, frequency swept resonance linewidth , and phenomenological damping is given by =2/T=(_x+_y), where _x,y = (H ₀+(N _x,y –N _z ) 4M _s ).N _x,y,z are sample demagnetizing factors,H ₀ is the effectivez-directed static field, 4M _s is the saturation induction, and is the gyromagnetic ratio. This fairly simple but general relation shows that the numerical relation between damping and relaxation at a given frequency can be quite different for in-plane and normally magnetized thin films. For thesame loss processes, so thatT andT are equal, is larger than . For permalloy films at 1 GHz, =15 . In addition, the conventional field swept linewidth, H=/, is simply related to only forN _x =N _y . Both and H are geometry dependent and do not provide an intrinsic measure of the relaxation. These results are confirmed by both resonance and transient response experiments. The large values of for large angle switching may also be partially explained by this analysis because the relevant magnetization motion is due to a demagnetizing field normal to the film plane.Visiting scientist on leave fromRaytheon Company, U.S.A. Supported by the Japan Society for the Promotion of Science.     

Quantum supports and modal logic

LetA be a quasi-manual with finite operations. Associate to each E = {e ₁ ,..., e_n} A the set _E of modal formulas: (e ₁ ··· e_n), e_i (e ₁ ··· e_i–1 e_i+1 ··· e_n), i=1,..., n. Set _A = {_E|E A}. We show that supports ofA are in one-to-one correspondence with certain Kripke models of _A where the supports are given by {x |A x is true}.On leave from the Department of Mathematics, Pontificia Universidade Catolica, Rio de Janeiro, Brazil.     

The Schrödinger equation and canonical perturbation theory

LetT ₀(, )+V be the Schrödinger operator corresponding to the classical HamiltonianH ₀()+V, whereH ₀() is thed-dimensional harmonic oscillator with non-resonant frequencies =(₁, ... , _d ) and the potentialV(q ₁, ... ,q _d) is an entire function of order (d+1)^–1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT ₀(, ) admits a convergent expansion in powers of of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues _n (, ) ofT ₀+V such thatn coincides with a KAM torus are given, up to order , by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in .     

ФИЗИЧЕСКИЕ ОСНОВЫ ИЗ МЕРЕНИЯ НАПРАВЛЕНИЯ ВОЗДУШНОГО ПОТОКА МЕ ТОДОМ НАКАЛЕННОЙ ПРО ВОЛОКИ ПРИ ВЫСОКНХ ПОДЗВУКО ВЫХ СКОРОСТЯХ

d - l - t _d R _max - J , ( ) - - , . . - ( ) - - R R _max —R. ,      

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Experimental verification of the theory of the successive production of striations in a glow discharge

The properties of the wave of stratification in the plasma of the positive column in a glow discharge in neon are measured. The measurements are compared with the results of the theory of the successive production of striations and good agreement is found.

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In conclusion the author thanks M. Novák for help in measuring and evaluating the experimental results, and F. Kroupa and V. Krejí for carefully reading this paper and for valuable remarks.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

Energy level crossings in quantum mechanics

From the eigenvalue equationH \ _n () =E _n ()\ _n () withH H ₀ +V one can derive an autonomous system of first order differential equations for the eigenvaluesE _n () and the matrix elementsV _mn () where is the independent variable. To solve the dynamical system we need the initial valuesE _n ( = 0) and \ _n ( = 0). Thus one finds the motion of the energy levelsE _n (). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H ₀ +V ₁ + ² V ₂ and give an application to a supersymmetric Hamiltonian.     

Four-baryon weak interactions with ΔS = 1 and SU (6) symmetry

Relations are found among the amplitudes for four-baryon processes with S = 1 under the assumption that the Lagrangian for s-wave weak interactions has the transformation properties of a component of the SU(6) 35-plet. In particular, it is found that the amplitudes for the¹S₀ ¹S₀ and³S₁ ³S₁ transitions for the p pn reaction are the same, within a sign.Translated from Izvestiya VUZ. Fizika, No. 10, pp. 70–77, October, 1969.     

Invariants of the length spectrum and spectral invariants of planar convex domains

This paper is concerned with a conjecture of Guillemin and Melrose that the length spectrum of a strictly convex bounded domain together with the spectra of the linear Poincaré maps corresponding to the periodic broken geodesics in determine uniquely the billiard ball map up to a symplectic conjugation. We consider continuous deformations of bounded domains _s ,s[0, 1], with smooth boundaries and suppose that ₀ is strictly convex and that the length spectrum does not change along the deformation. We prove that ₀ is strictly convex for anys along the deformation and that for different values of the parameters the corresponding billiard ball maps are symplectically equivalent to each other on the union of the invariant KAM circles. We prove as well that the KAM circles and the restriction of the billiard ball map on them are spectral invariants of the Laplacian with Dirichlet (Neumann) boundary conditions for suitable deformations of strictly convex domains.Supported by Alexander von Humboldt foundation     

A continuum analogue of the lattice gas

We construct a Hilbert space , spanned by vectors , where is a bounded measurable set in ^v (=dimension of space), and interpret as at state where all pointsx are occupied by an incompressible fluid, andx unoccupied. is generated by applying unitary filling operatorsU( ) to a cyclic vector |, the completely unoccupied state. The operatorsU( ) generate a commutativec*-algebra, of which the hermitian elements are interpreted as the observables of the theory.All the -divisible representations of the symmetric group of order 2 are found. We give a generalization to a theory with any number of particle types.     

Mixing properties,differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature

For the Ising model with nearest neighbour interaction it is shown that the spin correlations _{A B} - _{A B} decrease exponentially asd(A, B) in a pure phase when the temperature is well belowT _c. This is used to prove that the free energyF(,h) is infinitely differentiable in and has one sided derivatives inh of all orders forh=0. The bounds are also used to prove that the central limit theorem holds for several variables such as e.g. the total energy and the total magnetization of the system, the limit distribution being gaussian with variances determined by the second derivatives ofF(,h).     

Linear and non-linear ac response of the stochastic classical model for sliding charge density waves

A comprehensive study of the ac response properties of the classical stochastic model for sliding charge density waves (CDW) in quasi one-dimensional metals is made by numerically solving the associated Fokker-Planck equation. Above the conductivity threshold a noise mechanism is indispensable to give finite line widths to the resonances of the applied ac signal of frequency with the narrow band noise frequency _OSC inherent in the model. In the present investigation a current noise of strengthT _N proportional to the CDW current is used in the Fokker-Planck equation in order to model the broad band current noise frequently observed above threshold. The present model thus incorporates three characteristic frequency scales: _OSC,T _N ,and a crossover frequency _OSC. Results are evaluated for the ac conductivity (;E ₀,E ) as function of frequency , dc bias electric fieldE ₀ and ac signal field strengthE . ForE 0 the linear ac response is obtained by a separate treatment of the Fokker-Planck equation. The resonances near =_OSC are studied in detail. Strong ac signals reduce the response at the fundamental resonance and lead to a harmonic interference structure nearn=_OSC. The overall agreement of the present results with recent measurements of the linear ac response is not good. In reality our results seem to be superimposed on a background not reproduced by the classical model with one cross over frequency. However, the peak in Im (;E ₀,E =0) vs.E ₀, when the narrow band noise frequency is near , is well reproduced. The spectral width of this peak which corresponds to the inductive dip in the susceptibility is studied as function of current noise strengthT _N .The results stress the need for a complete Fokker-Planck treatment sinceT _N is not simply related to the line width.     

Rational Conformal Field Theories and Complex Multiplication

We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFTs corresponding to the T² target and identify the Cardy branes with geometric branes. The T²s leading to RCFTs admit complex multiplication which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFTs on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.     

Anomalies and curvature ofW manifolds

We study the holomorphic structure of certain complex manifolds associated withW algebras, namely, the flag manifoldsW /T andW ₁₊/T ₁₊, and the spacesW /SL(),R) andW ₁₊/GL(,R), whereT andT ₁₊ are the maximal tori inW andW ₁₊. We compute their Ricci curvature and show how the results are related to the anomaly-freedom conditions forW andW ₁₊. We discuss the relation of these manifolds with extensions of universal Teichmüller space.Supported in part by the U.S. Department of Energy, under grant DE-AS05-81ER40039Supported in part by the U.S. Department of Energy, under grant DE-FG03-84ER40168     

Depth distribution of x-rays in electron microprobe analysis

The dependence of the relative intensityk on the thin film thickness has been measured. This dependence has enabled us to verify the Monte Carlo simulation. From the experimentally determined dependence, distributions of X-rays in the specimen ((z) functions) have been derived. It has been found that the intensities calculated by the present Monte Carlo model are always higher than those measured.The authors wish to thank Prof. R. Kuel of the Faculty of Mathematics and Physics, Charles University, Prague, for the measurement of the sputtered thin films thickness.     

Exact distribution of cluster size and perimeter for two-dimensional percolation

Exact series expansion data of Sykes et al. are used to calculate the average numberc _n and perimeters _n of clusters of sizen20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp _n we find a sharply peaked distribution of perimeterss _n with mean s _n =((1–p _n )/p _c )n+O(n ) and width s _n ² –S _n ²n ^1.6 where1/=0.39. This perimeter s _n should not be interpreted as a cluster surface in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc _n withn is consistent with the postulated asymmetry aboutp _c : logc _n –n forn with1 forp<p _c and1/2 forp>p _c .     

Geometric expansion of the boundary free energy of a dilute gas

We consider a dilute classical gas in a volume ^–1 which tends to ^d by dilation as 0. We prove that the pressurep(^–1) isC ^q in at =0 (thermodynamic limit), for anyq, provided the boundary isC ^q and provided the Ursell functionsu _n (x ₁, ...,x _n) admit moments of degreeq and have nice derivatives.     

Fock representations of the affine Lie algebraA 1 (1)

The aim of this note is to show that the affine Lie algebraA ₁ ⁽¹⁾ has a natural family _, ,v of Fock representations on the spaceC[x _i,y _j;i andj ], parametrized by (,v) C ². By corresponding the highest weight _, of _, to each (,), the parameter spaceC ₂ forms a double cover of the weight spaceC₀C –₁ with singularities at linear forms of level –2; this number is (–1)-times the dual Coxeter number. Our results contain explicit realizations of irreducible non-integrable highest wieghtA ₁ ⁽¹⁾ -modules for generic (,v).