Quantum tunneling with dissipation and the Ising model over ℝ

We consider a quantum particle in a double-well potential, for simplicity in the two-level approximation, coupled to a phonon field. We show that static and dynamical ground state correlations of the particle and of the field are expressible through expectations in an Ising model over (rather than ). Its free measure is a spin flip process with flip rate , the difference in energy between the ground state and the first excited state. The Ising model has a ferromagnetic pair interaction whose form depends on the couplings to the phonon field and on the dispersion relation of the phonon field. In physical applications the interaction is long ranged and decays ast ^–2 for large distances. In this case we prove that for sufficiently strong coupling the particle becomes localized in one of the wells. The effective tunnel rate is zero. The transition to localization is associated with the generation of an infinite number of low momentum phonons. We apply the Ising technology to our problem and discuss the phase diagram in some detail.     

Interface motion in a two-dimensional Ising model with a field

We determine by Monte Carlo simulations the width of an interface between the stable phase and the metastable phase in a two-dimensional Ising model with a magnetic field, in the case of nonconversed order parameter (Glauber dynamics). At zero temperature, the width increases ast with–1/3, as predicted by earlier theories. As temperature increases, the value of the effective exponent that we measure decreases toward the value 1/4, which is the value in the absence of magnetic field.     

Spectral duality for planar billiards

For a bounded open domain with connected complement inR ² and piecewise smooth boundary, we consider the Dirichlet Laplacian- on and the S-matrix on the complement ^c . We show that the on-shell S-matricesS _k have eigenvalues converging to 1 askk ₀ exactly when-- has an eigenvalue at energyk ₀ ² . This includes multiplicities, and proves a weak form of transparency atk=k ₀. We also show that stronger forms of transparency, such asS _{k 0} having an eigenvalue 1 are not expected to hold in general.     

Quadrupole interaction studies of Hg in Zn and lattice location of Hg and Au in Zn

¹⁹⁷Hg,¹⁹⁹Tl and¹⁹⁹Au ions were implanted into Zn single crystals. From time differential perturbed e^–- and - angular correlation experiments with the 5^–/2 states in¹⁹⁷Hg and¹⁹⁹Hg, respectively, the quadrupole interaction for Hg in Zn is reinvestigated. Only the data for the¹⁹⁹Au sample exhibit a single unique quadrupole precession pattern in agreement with the result of the lattice location studies by use of the channeling technique, which show Hg to occupy random lattice sites in Zn while Au is substitutional.From the measured interaction frequency, sign and magnitude of the effective electric field gradient for Hg on the substitutional site in Zn is determined asV _zz ^eff = +15.2(1.8)×10¹⁷V·cm^–2. By comparison of this value to the corresponding lattice field gradient contribution an electronic enhancement factor ofK(HgZn)=2.6 is deduced in disagreement with a previous derivation of this quantity, but which supports the proposal that the electronic field gradient depends linearly upon the valence of the impurity probe atom.     

Theoretical calculations and Raman spectrum of intercalation modes in LixInSe

By using an extended linear-chain model which includes the interlayer forces, we have calculated the new vibrational modes, of Li intercalated InSe. The dispersion curves along thek _z wavevector perpendicular to the layers for the -polytype are determined in the first Brillouin zone. Assuming that the interlayer interaction is not modified upon intercalation and the interaction between lithium atom and adjacent layers in the van der Waals plane has the same value than the interlayer one, the new modes are determined with the force constant given by the rigid layer mode of the, -polytype at 18 cm^–1. This model gives the variation of the acoustic branches and the appearance of two optical intercalation modes at higher frequencies. The Brillouin zone boundary modes of the acoustic branches at 18 and 41 cm^–1 in the pure material are calculated to be 22 and 50 cm^–1 respectively forx=1/2. The dispersion of the new optical branches is flat along thez-direction and frequencies are obtained at 96 cm^–1 for the Li mode perpendicular tok _z and at 218 cm^–1 for the Li mode parallel tok _z. We compare also our results with the Li mode frequencies obtained in a total energy calculation. Raman scattering experiments have been performed in intercalated sample in order to verify the proposed model.     

On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results

We consider the model of a 2D surface above a fixed wall and attracted toward it by means of a positive magnetic fieldh in the solid-on-solid (SOS) approximation when the inverse temperature is very large and the external fieldh is exponentially small in . We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular, we show, using the Pirogov-Sinai scheme as given by Zahradník, that there exists a unique critical valueh _k ^* () in the interval (1/4e ^–4k , 4e ^–4k ) such that, for allh(h _k+1 ^* ,h _k ^* ) and large enough, there exists a unique infinite-volume Gibbs state. The typical configurations are small perturbation of the ground state represented by a surface at heightk+1 above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. Whenh=h _k ^* () we prove instead the convergence of the cluster expansion for bothk andk+1 boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external fieldh. Using the results proven in this paper, we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent).     

Chern-Simons-Witten invariants of lens spaces and torus bundles,and the semiclassical approximation

We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS ¹, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), though some are for more general compact groups. We explicitly exhibit agreement of the limiting values of these formulas ask with the semiclassical approximation predicted by the Chern-Simons path integral.Partially supported by an NSF Graduate FellowshipAddress as of September 1, 1991: School of Natural Science, Institute for Advanced Study, Princeton, NJ 08540; USA     

Estimates for Non-Resonant Normal Forms in Hamiltonian Perturbation Theory

We make a remark about an estimate of the rest for the non-resonant action-angle normal forms and exhibit a simple example suggesting the optimality of this estimate when there are no small divisors. Given a polynomial perturbation of degree P and an integer k, calling the size of the small denominators up to order k, we prove that the kth order remainder is bounded by (2/ ₀) ^k+1 with ₀=const ²/(kP ²). Thus, fixing the degree of the perturbation, if is independent of k (i.e., if there are no small divisors), we obtain a rest bounded by (const k) ^k+1. These estimates are also applied to the case in which the small divisors are absent, and they are conjectured to be optimal in this context. To support this idea we present a simplified model problem with no small denominators, formally related to the above calculations, and we show that it indeed has factorial divergence of its Birkhoff series. We also obtain Nekhoroshev's Theorem for harmonic oscillators. We hope that our simple approach makes more accessible to a general audience this important (although quite technical) topic.     

Rayleigh-Taylor instability of a composite medium with variable viscosity in hydromagnetics

The frictional effect of collisions of ionized with neutral atoms on the Rayleigh-Taylor instability of a composite medium with variable viscosity is considered in the presence of a horizontal magnetic field. It is found that the simultaneous presence of viscosity, magnetic field and collisions has a stabilizing effect and completely stabilizes the wave-number bandk _* wherek _*=(k _x ² V²L/g). The collisions have no effect as such on the stratification, i.e., stable configuration remains stable and unstable configuration remains unstable. However the growth rate, under either of conditions (25), decreases with the increase of collisions.     

Inertial range spectrum of a fractal model for turbulence

The present study deals with a model for 3D turbulence which is based on a fractal set ink-space (massM(k)logk, i.e.d _f=0). The energy spectrumE(_k) is calculated by a method similar to the Monte Carlo renormalization technique of Ma [1]. To apply this we decouple the set into next-neighbour pairs of energy shells.E _k is calculated iteratively by simulating the pairs. Their rough scales are forced and their fine scales are damped by an eddy viscosity ^e. We start in the viscous range where ^e is neglected. An intertial range shows up with a spectrumE(k)k^–, =1.68±0.07, and an eddy viscosity ^e(k)k ^-,=1.3±. The possible benefit for the simulation of turbulent flows is pointed out.     

Chaotic properties of multipoint correlation functions of an Ising model with long-range interactions on the Sierpiński—Gasket lattice

A class of multispin correlation functions of an Ising model with ferromagnetic nearest neighbor interactionsK and constant (distance-independent) long-range interactionsQ ₁=Q,l=1,2,..., on the Sierpiski-gasket lattice is considered. Using an exact method for calculating thermodynamic functions of hierarchically constructed Ising systems, it is shown that, for a set of values ofQ and for almost all values ofK, someM _k-spin correlation functions, whereM _k=3 ^k +3 withk=1,2,...,n andn=1,2,... being the order of lattice construction, change chaotically asn, k, and therebyM _k increase to infinity. Accordingly, in the thermodynamic limit, these correlation functions prove to be nonanalytic for appropriate values ofQ andK. SinceM _k-point correlation functions withk being finite, i.e., correlation functions involving finite numbers of spins, remain analytic asn tends to infinity, there is a smooth crossover between analytic properties of correlation functions of the two types.     

Subcritical asymptotic behavior in the thermodynamic limit of reversible random polymerization processes

We consider a reversible Markov process as a chemical polymerization model and study the asymptotic behavior (in the thermodynamic limit asN+) of a particular probability distribution on the set ofN-dimensional vectors, thekth component of which is the number ofk-mers. The study establishes the existence of three stages (subcritical, near-critical, and supercritical stages) of polymerization, depending on the value of the strength of the fragmentation reaction. The present paper concentrates on the analysis of the subcritical stage. In the subcritical stages we show that the size of the largest length of polymers of sizeN is of the order logN asN+.     

Multiple trapping of random walkers on periodic lattices

Formulas are obtained for the mean absorption time of a set ofk independent random walkers on periodic space lattices containingq traps. We consider both discrete (here we assume simultaneous stepping) and continuous-time random walks, and find that the mean lifetime of the set of walkers can be obtained, via a convolution-type recursion formula, from the generating function for one walker on the perfect lattice. An analytical solution is given for symmetric walks with nearest neighbor transitions onN-site rings containing one trap (orq equally spaced traps), for both discrete and exponential distribution of stepping times. It is shown that, asN , the lifetime of the walkers is of the form Ta_kN², whereT is the average time between steps. Values ofa _k, 2 Sk 6, are provided.     

The second Gelfand-Dickey bracket as a bracket on a Poisson-Lie Grassmannian

We introduce a Poisson structure on a Grassmannian Gr _k (V) on which the Poisson-Lie group GL(V) acts in a Poisson-Lie way. We discuss the analytic complications connected with the infinite-dimensional caseV=C () and show that an open subset of Gr _k (V) with this Poisson structure is isomorphic to the Gelfand-Dickey manifold of differential operators of orderk with the second Gelfand-Dickey bracket. In fact we introduce as a consequence a Poisson-Lie action of an enormous group on the Gelfand-Dickey manifold generalizing (on the semiclassical level) the Sugavara inclusion.Dedicated to Israel M. Gelfand at his80 ^th birthday     

Shape of the clusters of a percolative system in the presence of tunneling bonds and a finite electric field

We consider a model correlated percolative system on a 2D square lattice with a finite electric field applied accross its two opposite sides. We study the shape of the clusters formed with the addition of a new kind of bond (we call them tunneling bonds) which respond only above a finite threshold voltage. As expected, the clusters do have an overall elongated shape in the-direction of the applied field. Intuitively, one expects the elongation (with the aspect ratio >1) to increased indefinitely with the field. But, in a previous study, we found the model to belong to the same universality clas in the limits of a zero and an infinite electric field. We explain this behavior by studying the change in these elongated shapes as a function of the applied voltage in finite size samples and find that actually the amount of elongation takes on a maximum value at a size (L)-dependent finite voltageV _m (L) and that asV thje overall deviation from isotropy in the field direction tends to zero (i.e., aspect ratio1) again.     

Optical properties of individual manganese-doped quantum dots

The magnetic state of a single magnetic ion (Mn²⁺) embedded in an individual quantum dot is optically probed using micro-spectroscopy. The fine structure of a confined exciton in the exchange field of a single Mn²⁺ ion (S=) is analyzed in detail. The exciton–Mn²⁺ exchange interaction shifts the energy of the exciton depending on the Mn²⁺ spin component and six emission lines are observed at zero magnetic field. The emission spectra of individual quantum dots containing a single magnetic Mn atom differ strongly from dot to dot. The differences are explained by the influence of the system geometry, specifically the in-plane asymmetry of the quantum dot and the position of the Mn atom. Depending on both these parameters, one has different characteristic emission features which either reveal or hide the spin state of the magnetic atom. The observed behavior in both zero field and under magnetic field can be explained quantitatively by the interplay between the exciton–Mn²⁺ exchange interaction (dependent on the Mn position) and the anisotropic part of the electron–hole exchange interaction (related to the asymmetry of the quantum dot).     

Discontinuous behavior of effective transport coefficients in quasiperiodic media

We investigate the effective conductivity ^* of a quasiperiodic medium in ^d and the discontinuous dependence, found in ref. 1, of ^* on the wavelengths of the system. It was shown there, for example, that the effective conductivity ^*(k) for a layered medium with a one-dimensional local conductivity _k (x)=A+cosx+coskx, A>2, is discontinuous ink. An explicit class of higherdimensional examples which exhibit the discontinuity is constructed here. The conductivity ^*(k, L) of a sample of lengthL in one dimension asL is also analyzed and shown to have a plateau structure for any irrationalk well approximated by rationals.     

Clustering in a Self-Gravitating One-Dimensional Gas at Zero Temperature

We study a system of gravitationally interacting sticky particles. At the initial time, we have n particles, each with mass 1/n and momentum 0, independently spread on [0, 1] according to the uniform law. Due to the confining of the system, all particles merge into a single cluster after a finite time. We give the asymptotic laws of the time of the last collision and of the time of the kth collision, when n. We prove also that clusters of size k appear at time n ^{–1/2(k–1)}. We then investigate the system at a fixed time t<1. We show that the biggest cluster has size of order logn, whereas a typical cluster is of finite size.     

Monte Carlo simulation of phase separation and clustering in the ABV model

As a model for a binary alloy undergoing an unmixing phase transition, we consider a square lattice where each site can be either taken by an A atom, a B atom, or a vacancy (V), and there exists a repulsive interaction between AB nearest neighbor pairs. Starting from a random initial configuration, unmixing proceeds via random jumps of A atoms or B atoms to nearest neighbor vacant sites. In the absence of any interaction, these jumps occur at jump rates _A and _B, respectively. For a small concentration of vacancies (c _v=0.04) the dynamics of the structure factorS(k,t) and its first two momentsk ₁(t),k ₂ ² (t) is studied during the early stages of phase separation, for several choices of concentrationc _B of B atoms. Forc _B=0.18 also the time evolution of the cluster size distribution is studied. Apart from very early times, the mean cluster sizel(t) as well as the moments of the structure function depend on timet and the ratio of the jump rates (= _B/ _A) only via a scaled timet/(). Qualitatively, the behavior is very similar to the direct exchange model containing no vacancies. Consequences for phase separation of real alloys are briefly discussed.     

High-frequency sum rules for classical one-component plasma in a magnetic field

A high-frequency sum-rule expansion is derived for all elements of a classical plasma dielectric tensor in the presence of an external magnetic field, ₄ ¹³ is found to be the only coefficient of ^–4 that has no correlational and finite-radiation-temperature contributions. The finite-radiation-temperature effect results in an upward renormalization of the frequencies of the modes; it also leads to either reduction of the negative correlational effect on the positive thermal dispersion or, together with correlation, enhancement of the positive thermal dispersion for finitek, depending on the direction of propagation. Further, for the extraordinary mode, the finite-radiation-temperature effect increases the positive refractive dispersion for finitek.