Ground state of a spin-phonon system. I. Variational estimates

A study is made of the ground-state energy of a spin-one-half particle in a fieldB and interacting with a phonon bath. The infrared-sensitive case of acoustic phonons with point coupling in three dimensions is characterized by two parameters, a coupling constant andB. Units are used where the high-momentum phonon cutoff is unity. There is a curve (B) separating a symmetry-breaking region with a long-range phonon field from a normal region. Two simple, well-known, approximations are compared. The source theory yields discontinuities in the first derivatives of the energy with respect toB and whenB>e ^–1 and an infinite-order transition whenB<e ^–1, but is trivial in the large- region. The classical theory yields discontinuities in the second derivatives but is trivial in the small- region. An improved variationally fixed ground-state wave function is analyzed. It gives a new (B) curve with an infinite-order transition with continuous energy derivatives whenB<e/(e ²–1/4) and with discontinuous derivatives whenB is larger than this value. It is nontrivial in the entire (B) plane. The crossover to classical behavior occurs near =1/2 forB1. But the wave function does not describe quantum fluctuations in the large- phase. A second way of combining source and classical effects is described. It yields a second-order transition (near =1/2 forB1) everywhere. These theories are special cases of a symmetry-breaking transformation together with a one-mode treatment of quantum fluctuations. The transition is viewed in terms of a single mode with a variable length, coupled dynamically to the spin.     

Ground state of a spin-phonon system. II. Adiabatic limit

The phase transition for a spin in a magnetic fieldB coupled to acoustic phonons by a coupling constant is studied. The caseB1 with an upper cutoff of unity for the phonons is studied systematically by using an adiabatic canonical transformation. In leading order the transition line is at =2/B=1. In the normal phase (<1) the ground-state energy is –B/2 plus a function of that is given explicitly as the solution of a pair theory. In the broken symmetry phase (>1) the energy is the classical energy plus the same function of =1/². It is found that the first derivatives of the energy with respect to and with respect toB have finite jumps across the transition line. Quantum fluctuations in both phases are treated. Higher-order terms are a series of powers of 1/B times functions of . The case of a small transverse fieldB is also studied. The sharp transition disappears and is replaced by rapid variation in a region of order (B₁/B)^2/3 about =1.     

Experimental investigation of the nonlinear properties of the surface helical instability in germanium

The results of an experimental investigation of the parameters of the surface helical instability (SHI) in germanium are presented. The dependence of the amplitude of the current oscillations of the helical wave and frequency on the electric E and magnetic B fields with E and B significantly exceeding the threshold values E_t and B_t are presented for the case of nonequilibrium (for temperatures T < 343°K) and equilibrium (T 343°K) plasma. It is shown that for small supercriticality (_E = (E-E_t)/Et 1, = ( -_t)/_t 1) > the square of the amplitude of the current oscillations and the frequency shift are proportional to the supercriticality, i.e., the regime for excitation of SHI is soft.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 18–21, May, 1981.     

Microscopic theory of gas-surface interaction

A kinetic theory for inelastic scattering, trapping and desorption of gas molecules by surfaces is described. The theory is valid if the time scale _l = 1/r introduced by the relaxation ratesr in the kinetic equations (which is of the order of the life time of vibrational states of adsorbates) is sufficiently large compared to the vibrational period ₀. For sufficiently large activation energies of the adsorbates another time constant _res, the residence time of adsorbed particles, can be determined from the theory. One thus may distinguish four different partly overlapping regimes defined by the time scalest _{I l} , ₀t_II, _l t_III and _rest_IV. Regime I is governed by the Schrödinger equation regime II by the kinetic equations. In the region where both regimes overlap the kinetic coefficients can be expressed in terms of microscopic quantities which have been calculated previously. The relevant quantities in the other regimes are introduced and discussed from a unified point of view thus providing a link between the regimes I and IV which have been treated in detail before.     

Fluctuation of the free energy in the Hopfield model

We prove that in the ergodic region [T>J ²(1 + r)] the deviation of the total free energy of the Hopfield neural network converges in distribution asN to a (shifted) Gaussian variable. Moreover, the free energy per site converges in probability to lim(1/N)ln _N .     

Asymptotics of heavy atoms in high magnetic fields: II. Semiclassical regions

The ground state energy of an atom of nuclear chargeZe in a magnetic fieldB is exactly evaluated to leading order asZ in the following three regions:BZ ^4/3,BZ ^4/3 andZ ^4/3BZ³. In each case this is accomplished by a modified Thomas-Fermi (TF) type theory. We also analyze these TF theories in detail, one of their consequences being the nonintuitive fact that atoms are spherical (to leading order) despite the leading order change in energy due to theB field. This paper complements and completes our earlier analysis [1], which was primarily devoted to the regionsBZ ³ andBZ³ in which a semiclassical TF analysis is numerically and conceptually wrong. There are two main mathematical results in this paper, needed for the proof of the exactitude of the TF theories. One is a generalization of the Lieb-Thirring inequality for sums of eigenvalues to include magnetic fields. The second is a semiclassical asymptotic formula for sums of eigenvalues that isuniform in the fieldB.Work partially supported by U.S. National Science Foundation grant PHY90-19433 A02Work partially supported by U.S. National Science Foundation grant DMS 92-03829Work partially supported by the Heraeus Stiftung and the Research Fund of the University of Iceland.     

Kinetic equation in the kinetic region of the dilute and nonuniform electron plasma

Mori's scaling method is used to derive the kinetic equation for a dilute, nonuniform electron plasma in the kinetic region where the space-time cutoff (b, t _c) satisfies _Dbl _f , _D t _{c f} , with _D the Debye length, _D ^–1= _p the plasma frequency, andl _f and _f the mean free path and time, respectively. The kinetic equation takes account of the nonuniformity of the order ofl _f and _D for the single-and the two-particle distribution function, respectively. Thus the Vlasov term associated with the two-particle distribution function is retained. This kinetic equation is deduced from the kinetic equation in the coherent region obtained by Morita, Mori, and Tokuyama, where the space-time cutoff of the coherent region satisfies _Dbr ₀, _Dt _{c 0}, withr ₀ the Landau length and ₀ the corresponding time scale.     

Thermodynamics of the re-entrant regime of type II two-dimensional superconductivity in a strong transverse magnetic field

The thermodynamics (including energy gap and critical temperatureT _c ) of a two-dimensional type II BCS superconductor subjected to a strong transverse magnetic fieldH in the re-entrant regime is studied. There are qualitative differences with the re-entrant 3D case; in particularT _c grows unboundedly likeB/lnB asB (filling factor 1) for fixedN. The thermodynamics atT=0 is completely described by the evaluation of the magnetic Gibbs free energy as a function of andH. A first order magnetic phase transition (failure of H/ B>0 for around each lowlying Landau level) found for the normal phase for unbroadened Landau levels, is washed out in the superconducting phase.     

On the one-dimensional Hubbard model away from half-filling

We study the spectrum of the one-dimensional Hubbard model near the half-filled band, i.e. for electronic densityn=1–, 1, and also in the low-density limit,n1. Ground state and excited states are described, by integral equations which are derived from the Bethe Ansatz. These equations, which can be solved analytically only in the case of half-filling (=0), are investigated systematically in both limits 1,n1. Appropriate expansions of the momenta and energies of the ground state and the excited states as well as correlation exponents are determined.Work performed within the research program of the Sonderforschungsbereich 341, Köln-Aachen-JülichSimilar equations have been derived by Matveenko [3] using a different method     

Optical bistability in a bidirectional ring cavity

We investigate the behaviour of a bistable absorber in a two-directional ring cavity. A hierarchy derived from the Maxwell and Bloch equations is used throughout without any truncation. In the stationary situation we recover the solution found by Bonifacio and Lugiato in the unidirectional ring cavity. Next, we consider the dynamics of the system when it is close to steady state. We find both the self-pulsing instability predicted by the afore-mentioned authors for the field propagating in the same direction of the injected signal, and a new instability originated by the counterpropagating field. Numerical results are given in the mean-field limit L1,T1.     

Note on the Wiedemann-Franz law and the Mott rule in two-dimensional systems in strong magnetic fields

Modivated by a contradiction in the literature, the range of validity of the Wiedemann-Franz law and the Mott rule has been reexamined for a 2d electron-impurity system in a strong magnetic field. The condition for these laws to be fulfilled is shown to be thermal energy level broadening rather than thermal energycyclotron energy.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Effects of the blackbody radiation on the harmonically bound electron-energy level shifts and consistency conditions

The influence of the vacuum radiation field on the harmonically bound electron (frequency ₀) is considered. The electron is minimally coupled to the blackbody radiation field. The dynamics of the system is exactly solvable. The high (k _B T₀) and low (k _B T₀) temperature expansions of the kinetic and potential energy are given. In the high temperature regime theT ²-dependent dynamic Stark shift is found whereas in the low temperature regime there is no temperature dependent shift. The position correlation function of the electron shows in the low temperature regime a unclassical algebraic decay (t ^–4,t/k _B T).     

β-Decay of an unpolarized neutron in an intense electromagnetic field

This study will consider the effect of a circularly polarized electromagnetic field on- decay of a neutron for values of the parameter ea/m 1 and ea/m 1, where e and m are the charge and mass of the electron; anda is the amplitude of the field potential. Within the framework of the V=A variant of the weak interaction theory, in the first-order perturbation theory, it is shown that with increase in field intensity the probability of neutron decay increases. The energy distributions of the electrons generated are calculated for ea/m = 0.5 and ea/m=1, and the results obtained are evaluated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 115–120, April, 1974.The author thanks S. V. Izmailov and R. Yu. Volkovyskii for their evaluation of the study.     

Logical independence in quantum logic

The projection latticesP(₁),P(₂) of two von Neumann subalgebras ₁, ₂ of the von Neumann algebra are defined to be logically independent if A B0 for any 0AP(₁), 0BP(₂). After motivating this notion in independence, it is shown thatP(₁),P(₂) are logically independent if ₁ is a subfactor in a finite factor andP(₁),P(₂ commute. Also, logical independence is related to the statistical independence conditions called C^*-independence W^*-independence, and strict locality. Logical independence ofP(₁,P(₂ turns out to be equivalent to the C^*-independence of (₁,₂) for mutually commuting ₁,₂ and it is shown that if (₁,₂) is a pair of (not necessarily commuting) von Neumann subalgebras, thenP(₁,P(₂ are logically independent in the following cases: is a finite-dimensional full-matrix algebra and ₁,₂ are C^*-independent; (₁,₂) is a W^*-independent pair; ₁,₂ have the property of strict locality.     

Dispersion laws of hot holes in silicon

All possible exact and main approximations for decoupling the Kane equation into three linear or into a linear and a quadratic equation have been studied in the energy region E Eg. As a result of the study, a convenient analytical solution to this equation has been obtained in the plane (110), as well as approximate analytical solutions in the vicinity of <100> and <111> axes. Results obtained are also valid for other crystals with the diamond and sphalerite structure, which have the region E Eg.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 74–78, April, 1986.     

Maximum uniform deformation in tension

An equation is derived for the hyperbola which touches the true stress curve S =f(), where is the contraction of the specimen at the point _p (uniform contraction), S_B (true ultimate strength). With a flat maximum of the tensile force, this hyperbola coincides with the true stress curve at a part corresponding to extension by the maximum force. The use of the tangent hyperbola for determining _p and S_B is demonstrated.It is found that for those metals and alloys which are at present known to have a convex true stress curve in the uniform plasticity range, the uniform contraction _p cannot exceed 0.5, corresponding to a uniform elongation _p 1, while the true (logarithmic) uniform elongation _p 0.693. The limiting values of the hardening modulus and of the ratio S_B/_B are also found.     

The Hamiltonian of the guiding centre in the electromagnetic field

The Hamiltonian of the guiding centre in an adiabatically variable electromagnetic field (for the relative variations of the field 1/H. dH/dt _c , 1/E. dE/dt _c ) is derived in the present paper.The solution of the corresponding canonical equations is given for an axially symmetric stationary electromagnetic field.     

On the direct observation of the gas-dynamics of laser-pulse sputtering of polymers

We bring together a wide range of ideas relating to the gas-dynamic effects that are now recognized to play a leading role in laser-pulse sputtering. The ideas are grouped according to three basic models. (a) In the effusion model one deals with particles which are released from a target surface, form a Knudsen layer (KL), and then enter an unsteady adiabatic expansion (UAE). When the release terminates at time t=_r there is an abrupt change at the surface from positive to zero flow velocity, which means that particles moving towards the surface are reflected. The flow breaks up into three regions and analytical solutions exist for all aspects of this flow. (b) In the recondensation model the comportment of the target is initially like that of the effusion model but when the release terminates at t=_r the change at the surface is from positive to negative flow velocity, meaning that particles which move towards the surface recondense. Only numerical solutions presently exist (due to Sibold and Urbassek) but they are sufficient to show that the flow breaks up into two (not three) regions. (c) The outflow model could be described as the escape of gas from a finite reservoir, a well-known problem since it describes some aspects of guns. In its application to laser sputtering it is assumed that bond-breakage occurs rapidly over a characteristic depth and the resulting gas-like particles then flow out in a UAE; there is no formal KL.For part II, in which explicit photographs of laser-sputtered particles are analyzed, see [1]     

On the Lowest Energy Nucleation Path in a Supersaturated Lattice Gas

A lattice gas with non-conserved spin flip dynamics (of both non-Glauber and Glauber types) is considered at TT _c , the critical temperature. For arbitrary supersaturation, S, a general expression for the inverse of the nucleation rate along the lowest energy path is derived. The exponential part is identical to the one by Neves and Schonmann [Commun. Math. Phys. 137:20 (1991)]. The preexponential can be expressed in terms of elliptic theta-functions for small S, and in the limits, respectively, of ST/ or ST/ (– being the nearest-neighbor interaction energy), elementary versions of the general expression are further obtained. The preexponential has a smooth component, as well as small-scale modulations which are approximately periodic in the inverse supersaturation. For ST/, the smooth part is proportional to , in contrast to the zero-T limit where it is linear in S. The latter limit becomes apparent only at extremely low temperatures which are cubic in S.     

Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques

Under some hypotheses of analyticity and integrability we show the existence and uniqueness of a strong regular solution of the Schrödinger equation using a natural generalisation to the complex case of the Feynman-Kac formula. This explicit representation allows us to study in certain cases the asymptotic behavior of the solution when the Planck constanth tends to zero. The same method can be used for the solution of more general Schrödinger equations.

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Membre du Laboratoire Associé au C.N.R.S., n° 224 Processus Stochastiques et Applications