Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities

The eight-vertex model is equivalent to a solid-on-solid (SOS) model, in which an integer heightl _i is associated with each sitei of the square lattice. The Boltzmann weights of the model are expressed in terms of elliptic functions of period 2K, and involve a variable parameter . Here we begin by showing that the hard hexagon model is a special case of this eight-vertex SOS model, in which =K/5 and the heights are restricted to the range 1l _i4. We remark that the calculation of the sublattice densities of the hard hexagon model involves the Rogers-Ramanujan and related identities. We then go on to consider a more general eight-vertex SOS model, with =K/r (r an integer) and 1l _ir–1. We evaluate the local height probabilities (which are the analogs of the sublattice densities) of this model, and are automatically led to generalizations of the Rogers-Ramanujan and similar identities. The results are put into a form suitable for examining critical behavior, and exponents, , are obtained.Supported by the Guggenheim Foundation and in part by the National Science Foundation, grant MCS 8201733.     

Grassmann-Cartan algebra and Klein-Gordon equations

Given a Riemannian structure (M, g), a hypothesis is investigated that if= _p=0 ⁿ _p (M) is submitted to the differential condition (g++)=0, =mc/—which implies that each component of fulfills the Klein-Gordon equation (- ²) _p =0, ought to be interpreted as a natural complex of the bosonic fields. Then it is found that the complex admits the interpretation in the sense of first quantization with (M) being a convex set of states, with the structure of a Hilbert space over . The definite spin states of bosons are then pure states which are not conserved by the temporal evolution.     

Long-range order in the XXZ model

The existence of long-range order is proved under certain conditions for the antiferromagnetic quantum spin system with anisotropic interactions (XXZ model) on the simple cubic or the square lattice. In three dimensions (the simple cubic lattice), finite long-range order exists at sufficiently low temperatures for any anisotropy(0) ifS1, and for 0<0.29 (XY-like) or>1.19 (Ising-like) ifS=1/2. In two dimensions (the square lattice), ground-state long-range order exists under the following conditions: for any anisotropy (0) ifS3/2; 0<0.032 (XY-like) or 0.67<<1.34 (almost isotropic) or>1.80 (Ising-like) ifS=1;>1.93 (Ising-like) ifS=1/2. We conjecture that the two-dimensional spin-1/2XY model (=0) has finite ground-state long-range order. Numerical evidence supporting this conjecture is given.     

Experimental Test of a Time-Temperature Formulation of the Uncertainty Principle Via Nanoparticle Fluorescence

No Heading The uncertainty in the measured fluorescence decay lifetimes of 30 nm particles of YAG:Cc was used to evaluate the predictions of a novel form of the Heisenberg uncertainty principle suggested by de Sabbata and Sivaram, T t h/k. The worst-case uncertainty in temperature of 4.5 °K (as derived from the relationship between temperature and lifetime) and the measured uncertainty in decay lifetime, 0.45 ns, yielded an internal estimate of T t = 2.0 × 10^–9 °K s, which is 263 times larger than /k = 7.6 × 10^–12 °K s. An external estimate of T t = 4.5 × 10¹¹ °K s (which is = 6 times /k) is derived from the independently measured uncertainty in the temperature of the sample and the experimentally determined uncertainty in lifetime. These results could be low by a factor of 5.6 if signal averaging must be taken into account. If valid, the findings are consistent with the predictions of this version of the uncertainty principle and they imply the existence of a type of thermal quantum limit.     

Stationary states of an electron in the field of a hole in an ionic crystal

An approximate solution is given of the Schrödinger equation for S-states of an electron in the field of a hole, when the potential energy of the electron has the form — e²/r [1+exp (–qr].

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s- , –e²/r [1+exp (–qr].

     

Effect of Nonhomogeneous Broadening on the Free Induction Signal for a Double-Level Spin System

An investigation of the evolution of the free induction signal as a function of /₁ is made, where is the nonhomogeneous width of the magnetic resonance line of a double-level spin system and ₁ = B ₁, where is the gyromagnetic ratio and B ₁ is the amplitude of a radio-frequency field. It is shown that two stages can be distinguished in the evolution of the free induction signal on excitation of the spin system by a large-area pulse (₁ t ₁ >> 1) of length t ₁. In the first stage (t t ₁), reversible relaxation leads to an oscillatory and exponential dip. With a fixed ₁, an increase in practically does not change the frequency of oscillations and the signal amplitude decreases. In the second stage (t > t ₁), there is only an exponential decay of the signal at a rate equal to . It is demonstrated that, with nonresonant excitation, the coherent properties of the free induction signal manifest themselves in the formation of a single-pulse echo. The theoretical results are in good agreement with the experimental data obtained in nuclear magnetic resonance for protons in glycerin. The conclusion is drawn that the theorem of coherent transients in double-level systems with a finite nonhomogeneous width of the line is valid.     

ЭЛЕКТРИЧЕСКАЯ ЭКВИВ АЛЕНТНАЯ СХЕМА ЗАТУХАЮЩИХ КОЛЕБАНИ Й ИЗГИБА У ПБЕЗОЭЛЕКТРИЧЕСКИ Х СТЕРЖНЕЙ

- .     

Small perturbations ofC*-dynamical systems

It is shown that if is the generator of a strongly continuous oneparameter group of *-automorphisms of aC*-algebraA and is an unbounded *-derivation ofA with the same domain as , then + is also a generator for all sufficiently small real numbers .     

A lower bound for the memory capacity in the Potts-Hopfield model

We consider Potts-Hopfield networks of sizeN. We prove the result: _c >0 such that for all 0<< _c we can find, >0 in such a way that, whenN, we can store N patterns, all of them being sorrounded by -energy barriers at distance.     

Gravitation and universal Fermi coupling in general relativity

The generally covariant Lagrangian densityG = + 2K _matter of the Hamiltonian principle in general relativity, formulated by Einstein and Hilbert, can be interpreted as a functional of the potentialsg _ikand of the gravitational and matter fields. In this general relativistic interpretation, the Riemann-Christoffel form _kl ⁱ = _kl ⁱ for the coefficients _kl ⁱ of the affine connections is postulated a priori. Alternatively, we can interpret the LagrangianG as a functional of , g_ik, and the coefficients _kl ⁱ . Then the _kl ⁱ are determined by the Palatini equations. From these equations and from the symmetry _kl ⁱ = _lk ⁱ for all matter fields with /=0 the Christoffel symbols again result. However, for Dirac's bispinor fields, / becomes dependent on the Dirac current, essentially with a coupling factor Khc. In this case, the Palatini equations define a new transport rule for the spinor fields, according to which a second universal interaction results for the Dirac spinors, besides Einstein's gravitation. The generally covariant Dirac wave equations become the general relativistic nonlinear Heisenberg wave equations, and the second universal interaction is given by a Fermi-like interaction term of the V-A type. The geometrically induced Fermi constant is, however, very small and of the order 10^–81erg cm³     

Ultrasound in spin glasses

The change of the sound velocity v(,T) and the damping of sound waves (,T) in spin glasses are calculated in the frame-work of an Ising model with a random distribution of exchange interactions. The calculation is based on linearized equations of motion for the spins and on an improved mean field approximation which includes the Onsager reaction field. Near to the freezing temperatureT _f and at high temperatures v(,T) and (,T) turn out to be approximately proportional to the real and the imaginary parts of the dynamical susceptibility. For the special case of infinite range interactions atT=T _f one has v(, T_f) ( )^1/2 and (, T_f) (/)^1/2 where is the relaxation time of independent spins. However, already slightly aboveT _f the frequency dependence of both quantities becomes rather small for RKKY spin glasses. At high temperatures both, v(,T) and (,T) vary asT ^–1.SFB 125 Aachen-Jülich-Köln     

Low-temperature and long-time anomalies of a damped quantum particle

The time evolution of a damped quantum particle is discussed. Dissipation is modeled by the bilinear coupling to a set of harmonic oscillators. Using a functional integral technique that accounts for initial correlations between the particle and the reservoir, one can express the dynamics of the damped particle entirely in terms of equilibrium correlation functions. The long-time behavior of these correlations is determined for memory damping arising from the coupling to a reservoir with spectral densityI() at low frequencies, where > 0. The time evolution of nonequilibrium initial states of the damped particle is discussed. At finite temperatures an initially localized state is found to spread subdiffusively or superdiffusively, depending on . For > 2 the damping becomes ineffective for long times, and the width of a state grows kinematically. At zero temperature and for < 1, an initially localized state remains localized for all times. For 1 the state spreads, but with a slower rate than at finite temperatures. Study of arbitrary initial states indicates that the process is ergodic at finite temperatures only for 2 and at zero temperature for 1 2.     

Bell's inequality forC *-algebras

Bell's inequality dealing with local hidden variables is given two formulations in terms ofC ^*-algebras. In particular, Bell's inequality holds for all states onAB wheneverA andB are unitalC ^*-algebras at least one of which is Abelian, i.e., at least one corresponds to a classical physical system.     

Variance calculations and the Bessel kernel

In the Laguerre ensembleof n xN Hermitian matrices, it is of interest both theoretically and for applications to quantum transport problems to compute the variance of a linear statistic, denoted var_N f, asN . Furthermore, this statistic often contains an additional parameter a for which the limit is most interesting and most difficult to compute numerically. We derive exact expressions for both lim_N var_N f and lim , lim_N var_N f.     

On the structure of Σ-hypernuclei

We investigate the-nucleus potential in the framework of a relativistic mean field approximation. As input for the elementaryN interaction we supplement the-repulsion by the exchange of an effective, complex-meson to account for theN N conversion. As the main result we find that the strength of ths absorptive central potential is smaller than the corresponding real part, whereas in the spin-orbit interaction, which is expected to be much weaker than for a nucleon, the imaginary component dominates.This work was partially supported by CNPq, FINEP (Brazil) and by the Bilateral Cooperation W. Germany — Brazil (KFA — CNPq).     

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.     

Upper bounds onT c for one-dimensional Ising systems

We present upper bounds on the critical temperature of one-dimensional Ising models with long-range,l/n interactions, where 1<2. In particular for the often studied case of =2 we have an upper bound onT _c which is less than theT _c found by a number of approximation techniques. Also for the case where is small, such as =1.1, we obtain rigorous bounds which are extremely close, within 1.0%, to those found by approximation methods.     

Anderson localization for the almost Mathieu equation: A nonperturbative proof

We prove that for any diophantine rotation angle and a.e. phase the almost Mathieu operator (H()) _n = _n–1 + _n+1 +cos(2(+n)) _n has pure point spectrum with exponentially decaying eigenfunctions for 15. We also prove the existence of some pure point spectrum for any 5.4.     

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.     

The van der Waals limit for classical systems

For a -dimensional system of particles with the two-body potentialq(r)+ ^v K(r) and density , it is proved under fairly weak conditions onq andK that the canonical pressure (, ) and chemical potential (, ) tend to definite limits when 0. The limiting functions are absolutely continuous and are given in terms of the derivative of the limiting free energy density which was found in Part I.