Ising model on self-avoiding walk chains

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T _c) bounds, the average number of nearest-neighbours (Z _eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p _c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively,Z _eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) andp _c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values ofZ _eff givekT _c/J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising modelT _c's andp _c's for the same lattices, we findkT _c/J0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent =1.024±0.007, for the model on SAWs on two dimensional lattices.     

Phase diagram for three-dimensional correlated site-bond percolation

The Coniglio-Stanley-Klein model is a random bond percolation process between the occupied sites of a lattice gas in thermal equilibrium. Our Monte Carlo simulation for 40³ and 60³ simple cubic lattices determines at which bond thresholdp _Bc , as a function of temperatureT and concentrationx of occupied sites, an infinite network of active bonds connects occupied sites. The curvesp _Bc (x, T) depend only slightly onT whereas they cross over if plotted as a function of the field conjugate tox. Except close toT _c we find 1/p _Bc to be approximated well by a linear function ofx, in the whole interval between the thresholdx _c (T) of interacting site percolation atp _Bc =1 and the random bond percolation limitx=1 atp _Bc =0.248±0.001. Thisx _c (T) varied between 0.22 forT=0.96 (coexistence curve) and 0.3117±0.0003 forT= (random site percolation). At the critical point (T=T andx=1/2) we confirmed quite accurately the predictionp _Bc =1-exp(–2J/k _B T _c ) of Coniglio and Klein. As a byproduct we found 0.89±0.01 for the critical exponent of the correlation length in random percolation.     

Diffusion of lattice gases without double occupancy on three-dimensional percolation lattices

We examined the diffusion of lattice gases, where double occupancy of sites is excluded, on three-dimensional percolation lattices at the percolation thresholdp _c . The critical exponent for the root-mean-square displacement was determined to bek=0.183±0.010, which is similiar to the result of Roman for the problem of the ant in the labyrinth. Furthermore, we found a plateau value fork at intermediate times for systems with higher concentrations of lattice gas particles.     

Similarity of Percolation Thresholds on the HCP and FCC Lattices

Extensive Monte Carlo simulations were performed in order to determine the precise values of the critical thresholds for site (p ^hcp _{c, S} =0.199 255 5±0.000 001 0) and bond (p ^hcp _{c, B} =0.120 164 0±0.000 001 0) percolation on the hcp lattice to compare with previous precise measurements on the fcc lattice. Also, exact enumeration of the hcp and fcc lattices was performed and yielded generating functions and series for the zeroth, first, and second moments of both lattices. When these series and the values of p _c are compared to those for the fcc lattice, it is apparent that the site percolation thresholds are different; however, the bond percolation thresholds are equal within error bars, and the series only differ slightly in the higher order terms, suggesting the actual values are very close to each other, if not identical.     

Scaling of particle trajectories on a lattice

The scaling behavior of the closed trajectories of a moving particle generated by randomly placed rotators or mirrors on a square or triangular lattice is studied numerically. On both lattices, for most concentrations of the scatterers the trajectories close exponentially fast. For special critical concentrations infinitely extended trajectories can occur which exhibit a scaling behavior similar to that of the perimeters of percolation clusters.At criticality, in addition to the two critical exponents =15/7 andd _f=7/4 found before, the critical exponent =3/7 appears. This exponent determines structural scaling properties of closed trajectories of finite size when they approach infinity. New scaling behavior was found for the square lattice partially occupied by rotators, indicating a different universality class than that of percolation clusters.Near criticality, in the critical region, two scaling functions were determined numerically:f(x), related to the trajectory length (S) distributionn _s, andh(x), related to the trajectory sizeR _s (gyration radius) distribution, respectively. The scaling functionf(x) is in most cases found to be a symmetric double Gaussian with the same characteristic size exponent =0.433/7 as at criticality, leading to a stretched exponential dependence ofn _S onS, n_Sexp(–S ^6/7). However, for the rotator model on the partially occupied square lattice an alternative scaling function is found, leading to a new exponent =1.6±0.3 and a superexponential dependence ofn _S onS.h(x) is essentially a constant, which depends on the type of lattice and the concentration of the scatterers. The appearance of the same exponent =3/7 at and near a critical point is discussed.     

Corrections to scaling for branched polymers

We extend exact series enumerations for the mean square radius of gyration for lattice animals up to sizeN=19 on the square andN=12 on the simple cubic lattices. We have reanalysized the corrections to scaling exponents using the new data and found =0.5±0.05 and =0.45±0.1 in 2 and 3 dimensions respectively.     

Monte Carlo evidence for the deviation from the Alexander-Orbach rule in three-dimensional percolation

Computer simulation is used to study the diffusion at the percolation threshold on large simple cubic lattices. The exponentk for the rms displacementr witht inr t_k is found to be smaller than 0.2, while the Alexander-Orbach 4/3 rule for the spectral dimension predictsk=0.201 ± 0.002.on leave from Minnesota Supercomputer Institute and School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455.     

Diffusion in three-dimensional random systems at their percolation thresholds

Extensive Monte Carlo simulations of theant-in-the-labyrinth problem on randomL* L* L simple cubic lattices are performed, forL up to 960 on a CRAY-YMP supercomputer. The exponentk for the rms displacementr witht inrt ^k is found to bek=0.190±0.003. As a second approach, large percolation clusters with chemical shells up to 300 are generated on a simple cubic lattice at criticality. The diffusion equation is then solved by using the exact enumeration technique. The corresponding critical exponentd ^w is found to be 1/d ^w =0.250±0.003.On leave from I. Institut für Theoretische für Physik, Universität Hamburg, D-2000 Hamburg, Federal Republic of Germany.     

Numerical studies on the anderson localization problem

Our study of Anderson's tight binding model for strongly disordered electronic systems is extended to a numerical treatment of thed c-conductivity atT=0. For 100 × 100 square lattices, 129 × 129 triangular lattices, and for diamond lattices with 27,000 sites, the behaviour of is studied as a function of the Fermi energy and the disorder. The calculations are based on the exact eigenfunction representation of the Kubo formula, which is evaluated by the systematic application of recursion algorithms. Our results are in favour of Mott's original suggestion of a minimum metallic conductivity _min, both in two and three dimensions. In two dimensions we find the universal value of _min=(0.11 ±0.02)e ²/. Based on the thesis of J. Stein, Regensburg 1979     

Random walks on bond percolation clusters: ac hopping conductivity below the threshold

The frequency dependence of the ac hopping conductivity in two and three dimensional lattices with random interruptions is calculated by Monte Carlo simulation of random walks on bond percolation clusters. At low frequencies the real and imaginary parts of the ac conductivity vanish linearly and quadratically with the frequency, respectively. The critical behaviour of the imaginary part of the ac conductivity below the percolation threshold is shown to depend on the long time limit of the mean square displacement of random walksR ² , while the real part depends on the time constant of the system as well.R ² is found to diverge with an exponentu=2- according to the conjecture of Stauffer.     

Hypergeometric series in a series expansion of the directed-bond percolation probability on the square lattice

The asymmetric directed-bond percolation (ADBP) problem with an asymmetry parameterk is introduced and some rigorous results are given concerning a series expansion of the percolation probability on the square lattice. It is shown that the first correction term,d _n,1 (k) is expressed by Gauss' hypergeometric series with a variablek. Since the ADBP includes the ordinary directed bond percolation as a special case withk=1, our results give another proof for the Baxter-Guttmann's conjecture thatd _n,1(1) is given by the Catalan number, which was recently proved by Bousquet-Mélou. Direct calculations on finite lattices are performed and combining them with the present results determines the first 14 terms of the series expansion for percolation probability of the ADBP on the square lattice. The analysis byDlog Padé approximations suggests that the critical value depends onk, while asymmetry does not change the critical exponent of percolation probability.     

Diffusion of selenium and tellurium in silicon

Diffusion of selenium and tellurium in silicon has been investigated in the temperature range 1000°C to 1310°C by sheet conductivity. For SiSeD ₀= 0.3±0.1 cm²/s andh=2.6±0.1 eV, and for SiTeD ₀=0.9±0.3 cm²/s and h=3.3±0.1eV have been obtained. The surface concentrations for both dopants were of the order of 5 × 10¹³ to 6×10¹⁶cm^–3. The Hall coefficient leads to an energy level of 300±15meV for selenium and 200±20meV for tellurium.     

Corrections to cluster-radius scaling for branched polymers and percolation

We report analyses of series enumerations for the mean radius of gyration for isotropic and directed lattice animals and for percolation clusters, in two and three dimensions. We allow for the leading correction to the scaling behaviour and obtain estimates of the leading correction-to-scaling exponent . We find -0.640±0.004 and =0.87±0.07 for isotropic animals in 2d, and =0.64±0.06 in 3d. For directed lattice animals we argue that the leading correction has= or= ; we also estimate =0.82±0.01 and 0.69 ±0.01 ind=2, 3 respectively. For percolation clusters at and abovep _c, we find (p _c) =0.58±0.06 and (p>p _c)=0.84±0.09 in 2d, and (p _c)=0.42±0.11 and (p>p _c)=0.41 ±0.09 in 3d.     

Percolation and cluster distribution. II. layers,variable-range interactions,and exciton cluster model

Monte Carlo simulations for the site percolation problem are presented for lattices up to 64 x 10⁶ sites. We investigate for the square lattice the variable-range percolation problem, where distinct trends with bond-length are found for the critical concentrations and for the critical exponents and. We also investigate the layer problem for stacks of square lattices added to approach a simple cubic lattice, yielding critical concentrations as a functional of layer number as well as the correlation length exponent. We also show that the exciton migration probability for a common type of ternary lattice system can be described by a cluster model and actually provides a cluster generating function.Supported by NSF Grant DMR76-07832.     

Collapse transition of a two-dimensional lattice animal

The phase diagram and the tricritical point of a collapsing lattice animal are studied through an extended series expansion of the isothermal compressibilityK _T on a square lattice. As a function of the variables of fugacity and Boltzmann weight,K _T is investigated using partial differential approximant techniques. The characteristic flow pattern of partial differential approximant trajectories is determined for a stable fixed point. We obtain satisfactory estimates for the tricritical fugacityx _t =0.024±0.005 and temperatureT _t =0.54±0.04. Taking into account only linear scaling fields, we are also able to get the scaling exponent =1.4±0.2 and the crossover =0.66±0.08. Our results are in reasonable agreement with previous estimates in the literature.     

On the ground-state threshold in random two-dimensional Ising ±J models

For square, triangular, and for hexagonal lattices there is numerical and theoretical support that the ground-state thresholdp _c between ferro- and paramagnetism in random 2D Ising ±J models, withp as the concentration of antiferromagnetic bonds, is identical top ^*which is characterized by minimal matching properties of frustrated plaquettes. From square lattices of size 100×100 we have got p_c,sq<0.117 by simulations which produced average groundstate magnetizations per spin by means of exact minimal matchings. Moreover, from the squareL×L-lattices treated (L = 10, 20, 50, 100) we obtained the estimatep _c,sq 0.1 which is in agreement with the Grinstein estimatep _c,sq 0.099 andp _c,sq 0.105 by Freund and Grassberger.     

Study of the centrally producer ππ andK \bar K systems at 85 and 300 GeV/c

We have studied the andK systems centrally produced in proton proton collisions at 300 GeV/c and ⁺/p proton collisions at 85 GeV/c using the CERN spectrometer. Clear evidence forS ^*/f _o(975) production is observed. An analysis performed on the ^{+ –} mass spectrum in the 1.0 GeV region, using a coupled channel formalism, shows that it is possible to describe theS ^*/f _o(975) effect with one single resonance once interference of theS ^*/f _o(975) with theS-wave background is introduced. The resultingS ^*/f _o(975) parameters arem _o =979±4MeV,g =0.28±0.04,g _K =0.56±0.18 corresponding to a pole position on sheet II at (1001±2)–i(36±4) MeV. Evidence is also found for a structure having a mass of 1472±12 MeV and a width of 195±33 MeV.     

The semileptonic decay fraction ofB-mesons in the light of interfering amplitudes

Consequences of the interference between spectator amplitudes for the lifetimes and semileptonic decay fractions ofB ⁰ andB ⁺ mesons are discussed. Extracting these amplitudes from a fit to 11 exclusive hadronicB decay fractions we finda ₁=1.05±0.03±0.10,a ₂=+0.227±0.012±0.022, an inclusive semileptonic decay fraction of (11.2±0.5±1.7)%, and a lifetime ratio (B ⁺)/(B ⁰)=0.83±0.01±0.01.Supported under DOE grant number DE-FG02-91-ER40690     

Evolution of the homogeneous IR spectrum of a highly vibrationally excited molecule as a result of a change in its vibrational energy

Using a photodissociation technique, we have measured the IR spectrum of thev ₂₁ mode of the (CF₃)₃ CI molecule with a vibrational energy ofE ₂=42500±3500 cm^–1 which is more than two times the dissociation energy. The experimental spectrum of a Lorenzian shape with a halfwidth of ₂=10.8±1.5 cm^-1 has been analyzed simultaneously with the results of the preceding work (₁=8.6±0.6 cm^-1) that were obtained at a lower vibrational energy (E ₁=36500±2500 cm^–1).     

Percolation and scaling on a quasilattice

We report results of a study of percolation on a two-dimensional Penrose quasilattice. After an extensive numerical analysis, we find that two-dimensional universality is obeyed. The scaling exponents and have the values expected,=2.04 and=0.39, consistent with the universality class for percolation on a 2D periodic lattice. But the percolation thresholdp _c=0.483, differs from other 2D lattices with the same average coordination number ¯z=4.We dedicate this paper to the memeory of Marc Kac. One of us (J.L.B.) had the good fortune to known him on many levels: as friend, fellow scientist, and co-worker for human rights. His insight, wisdom, and compassion will remain with us as a lasting legacy of his life.