The statistics of Curie-Weiss models

LetS _n denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofS _n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S _n-nm)/n^1–1/2k has a limiting distribution with density proportional to exp[-_s ^2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.Research supported in part by National Science Foundation Grants MPS 76-06644 (to RSE) and MPS 74-04870 A01 (to CMN).     

Two-Tape Finite Automata with Quantum and Classical States

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and two-way two-tape deterministic finite automata (2TFA) were introduced by Rabin and Scott. In this paper we study 2TFA and propose a new computing model called two-way two-tape finite automata with quantum and classical states (2TQCFA). First, we give efficient 2TFA algorithms for identifying languages which can be recognized by 2QCFA. Second, we give efficient 2TQCFA algorithms to recognize several languages whose status vis-a-vis 2QCFA have been posed as open questions, such as L_square={aⁿbⁿ² | n ? N}L_{\mathit{square}}=\{a^{n}b^{n^{2}}\mid n\in \mathbf{N}\}. Third, we show that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by (k+1)-tape deterministic finite automata ((k+1)TFA). Finally, we introduce k-tape automata with quantum and classical states (kTQCFA) and prove that {aⁿb^nk | n ? N}\{a^{n}b^{n^{k}}\mid n\in \mathbf{N}\} can be recognized by kTQCFA.     

Temperature dependence of muonium spin exchange with O2 in the range 88 K to 478 K

Our group at TRIUMF reported earlier a study of the spin exchange reactions of Mu with O₂ and NO in the range 295 K to 478 K. We have extended the measurement with O₂ to a low temperature region down to 88 K. From 135 K to 296 K, the spin depolarization rate constantk _d(T) was found to vary according to the relative velocity of the colliding species,T ^1/2, which indicates that the spin exchange cross section of Mu-O₂ is temperature independent in this temperature range. The value ofk _d(T) at 296 K is in good agreement with our earlier study. However, it was found that below 105 K and above 400 K,k _d(T) tends to have stronger temperature dependences (T ⁿ , withn>1/2). This deviation fromT ^1/2-behavior can be attributed to the velocity (energy) dependence of the spin exchange cross section.     

Spin substructure of space-time

Space-time is provided with an underlying SL(2,C) spin space with complex noncommutative spinor coordinatesC ^A which satisfy . It is shown that the orbital angular momentum operator has a realization in this space as a derivative operator which can take on half-integral spin values, and the graded Lie algebra which describes the structure of the spin space is discussed. The spin-space translations mix fermi and bose fields and produce space-time translations which are not nil-potent. A hermitean metric with a line element which is invariant under such localx-dependent translations is introduced.     

Coherence transfer between nuclear spins in paramagnetic systems: effects of nucleus—electron dipole—dipole cross-correlation

In nuclear magnetic resonance of paramagnetic systems, cross-correlations between the fluctuations of a nucleus—nucleus dipole—dipole coupling I^k I^l and a nucleus—electron dipole coupling I^kS induces cross-relaxation and makes it possible to generate bilinear terms in the density matrix of the type 2I^k _xI^l _z from coherence I^k _x that can lead to ‘relaxation-allowed’ coherence transfer between two nuclei I^k and I^l . In this paper these effects are demonstrated in a complex involving a fragment of double-stranded DNA and two chromomycin molecules complexing a paramagnetic cobalt ion. Analytical expressions are given for the cross-correlation rates in particular conditions, while the extension to anisotropic g tensors or zero field splittings are addressed. It is shown that relaxation-allowed coherence transfer leads to characteristic signals in double-quantum filtered correlation spectroscopy (DQF—COSY), but not in total correlation spectroscopy (TOCSY). Analytical expressions are unable to reproduce the observed cross-peak patterns. A careful numerical study reveals that in the high spin Co(II) complex studied here, the cross-correlation dynamic shift contribution is of the same order of magnitude as the cross-correlation rate, a value much larger than what can be computed assuming isotropic Brownian motion and complete separation between the electron spin and the lattice.     

Spin structure of the “Forward” nucleon charge-exchange reaction <Emphasis Type="Italic">n</Emphasis> + <Emphasis Type="Italic">p</Emphasis> → <Emphasis Type="Italic">p</Emphasis> + <Emphasis Type="Italic">n</Emphasis> and the deuteron charge-exchange breakup

The structure of the nucleon charge-exchange process n + p → p + n is investigated basing on the isotopic invariance of the nucleon-nucleon scattering. Using the operator of permutation of the spin projections of the neutron and proton, the connection between the spin matrices, describing the amplitude of the nucleon charge-exchange process at zero angle and the amplitude of the elastic scattering of the neutron on the proton in the “backward” direction, has been considered. Due to the optical theorem, the spin-independent part of the differential cross section of the process n + p → p + n at zero angle for unpolarized particles is expressed through the difference of total cross sections of unpolarized proton-proton and neutron-proton scattering. Meantime, the spin-dependent part of this cross section is proportional to the differential cross section of the deuteron charge-exchange breakup d + p → (pp) + n at zero angle at the deuteron momentum k _d = 2 k _n (k _n is the initial neutron momentum). Analysis shows that, assuming the real part of the spin-independent term of the “forward” amplitude of the process n + p → p + n to be smaller or of the same order as compared with the imaginary part, in the wide range of neutron laboratory momenta k _n > 700 MeV/c the main contribution into the differential cross section of the process n + p → p + n at zero angle is provided namely by the spin-dependent term.     

On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics

The first six coefficients in the expansion of the susceptibility χ, and its inverse, χ ^?1, in ascending powers of the reciprocal temperature, have been determined for the Heisenberg model of a ferromagnetic, for any spin value, S, and any lattice. The first five coefficients appropriate to the magnetic specific heat, C, have also been found. For the body-centred and face-centred cubic lattices, the χ and C coefficients are tabulated for half-integral S from 1/2 to 3.

From these coefficients estimates have been made of the reduced Curie temperatures, θs _c= k T _c/J. It is found that for the simple, body-centred and face-centred cubic lattices the formula reproduces the estimated Curie temperatures fairly accurately. Here X=S(S+1) and z is the lattice coordination-number.

It is found that, suitably scaled, the theoretical curves for inverse susceptibility against temperature above the Curie point are rather insensitive to the spin value and to the precise lattice structure. The ratio of their initial to their final gradients is approximately 0·3. A comparison is made with the experimental values of χ ^?1 for both iron and nickel. If iron is represented by the Heisenberg model with S=1, then the observed Curie temperature corresponds to a J value of 1·19×10^?2 ev.

Brief consideration is given to the use of the tabulated coefficients for antiferromagnetic problems.     

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in ((n/k+logk)n ^2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes (n ^1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in (logn) parallel time, and so is in the complexity class NC.     

Effects of linking chain length and magnetic field on the kinetics of photoprocesses in covalently bonded porphyrin—viologen dyads

The formation and decay kinetics of chain linked triplet radical pairs derived from photo-induced electron transfer reactions in a series of 21 zinc porphyrin-flexible spacer-viologen (ZnP-Sp _n -Vi²⁺) dyads containing 2–138 atoms (n) in the spacer, have been examined by nanosecond laser flash photolysis techniques in an external magnetic field. In non-viscous polar solvents (acetone and CHCl₃ plus CH₃OH = 1:1 v/v), the effect of the spacer length on the rate constant of forward electron transfer can be described by the equation: k _et = k ⁰ _et(n + 6)^?1.5, with k ⁰ _et = 3 × 10¹⁰ s^?1 and 1.2 × 10¹⁰ s^?1 for electron transfer from the singlet and triplet states of ZnP, respectively. In zero magnetic field, the value of the triplet radical pair recombination rate constant, k _r(0) = 0.7 × 10⁶-8 × 10⁶ s^?1, is significantly smaller than k _et. The dependence of k _r(0) on n has an extremum with the maximum near n = 20. In a strong magnetic field (B = 0.21 T), significant retardation of triplet radical pair recombination is observed. In strong magnetic fields the effect of the chain length on triplet radical pair recombination rates is rather small and k _r(B) may vary in the range 0.3 × 10⁶-1 × 10⁷ s^?1. The phenomena observed are discussed in terms of the interplay of molecular and spin dynamics in the limits of slow and fast encounters, taking into account the exchange-interaction.     

Isotope Effects by Comparing 1H‐ and 2D‐NMR Spectra of 9,9′‐Bisbicyclo[4.3.0]‐cyclonona‐2,4,7‐triene

Abstract

Inverse secondary kinetic isotope effects are determined for the dimerization of all‐cis‐cyclononatetraenyl radical, 1, to its corresponding dimer, all‐cis‐9,9′‐bicyclonona‐1,3,5,7‐tetraene, 2, (step 1, k _H/k _D=0.5), and cyclization of the latter to 9,9′‐bisbicyclo[4.3.0]cyclonona‐2,4,7‐triene, 3 (step 2, k _H/k _D=0.75). These results are obtained by comparison of ¹H‐ and ²D‐NMR spectra of 3 and employment of a simple statistical method for acquiring kinetic data. This new strategy appears superior to conventional methods in being fast, simple, and less expensive.     

Numerical results for ground states of spin glasses on Bethe lattices

The average ground state energy and entropy for ±J spin glasses on Bethe lattices of connectivities k + 1 = 3..., 26 at T = 0 are approximated numerically. To obtain sufficient accuracy for large system sizes (up to n = 2¹²), the Extremal Optimization heuristic is employed which provides high-quality results not only for the ground state energies per spin e_k+1 but also for their entropies s_k+1. The results indicate sizable differences between lattices of even and odd connectivities. The extrapolated ground state energies compare very well with recent one-step replica symmetry breaking calculations. These energies can be scaled for all even connectivities k + 1 to within a fraction of a percent onto a simple functional form, e _{k + 1} = E _SK - (2E _SK + )/, where E _SK = - 0.7633 is the ground state energy for the broken replica symmetry in the Sherrington-Kirkpatrick model. But this form is in conflict with perturbative calculations at large k + 1, which do not distinguish between even and odd connectivities. We also find non-zero entropies per spin s_k+1 at small connectivities. While s_k+1 seems to vanish asymptotically with 1/(k + 1) for even connectivities, it is numerically indistinguishable from zero already for odd k + 1 ≥ 9. Received 9 August 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: sboettc@emory.edu www.physics.emory.edu/faculty/boettcher     

Muon addition to nitrogen,oxygen and other atoms: Considerations for level crossing resonance studies

The purpose of this note is to examine the conditions under which muon level crossing resonance with quadrupolar nuclei may be used to characterise the elusive diamagnetic fraction which is formed when positive muons are stopped in various media and associated chemically with the host molecules. A potential difficulty is identified for nuclei having integral spin, which may explain why cross polarisation to¹⁴N has not yet been detected. The general suitability of nuclei with half-integral spin (I≥3/2) is illustrated with the case of¹⁷O, and suggestions are made for future studies with other nuclei, including species such as molecular ions and defect complexes (muon-impurity pairs).     

Effect of an electric field on the surface tension of a liquid at low temperatures

An external electric field changes the dispersion law of waves on the surface of a liquid. Besides the usual capillary term (∝k ³, k is the wave number) and gravitational term (∝k), a term quadratic in the wave vector appears in the expression for the square of the frequency in a homogeneous field. These excitations are associated with the variation of the coefficient of surface tension of the liquid at low temperatures. In the case of a large field tangent to the surface, the correction is proportional to T ^8/3, unlike the T ^7/3 correction in the absence of a field. Zh. éksp. Teor. Fiz. 111, 1369–1372 (April 1997)     

量子Generalized Reed-Solomon码

构造出了一族量子纠错码,这族码具有参数［［n,n-2k,k+1］］_q,是q维量子系统上的码,q是任意素数的幂.这族码的最小距离达到了理论上限,因此,以码距来说,它是最优的.证明了当2≤n≤q或者q²-q+2≤n≤q²时,码都是存在的. 关键词： 量子Generalized Reed-Solomon码 量子MDS码 量子纠错码 量子信息     

Gentile statistics and restricted partitions

In a recent paper (Tranet al, Ann. Phys.311, 204 (2004)), some asymptotic number theoretical results on the partitioning of an integer were derived exploiting its connection to the quantum density of states of a many-particle system. We generalise these results to obtain an asymptotic formula for therestricted or coloured partitionsp _k ^s (n), which is the number of partitions of an integern into the summand of sth powers of integers such that each power of a given integer may occur utmostk times. While the method is not rigorous, it reproduces the well-known asymptotic results fors = 1 apart from yielding more general results for arbitrary values ofs.     

Symmetry Analysis of the Multidimensional Polywave Equation

Abstract

We present symmetry classification of the polywave equation □ ^l u = F (u). It is established that the equation in question is invariant under the conformal group C(1, n) iff F (u) = λe ^u, n + 1 = 2l or F (u) = λu ^{(n+1+2l)/(n+1?2l)}, n + 1 6= 2l. Symmetry reduction for the biwave equation □ ² u = λu ^k is carried out. Some exact solutions are obtained.     

Dependence of the critical temperature of high-temperature cuprate superconductors on hoppings and spin correlations between CuO<Subscript>2</Subscript> planes

The influence of interlayer hoppings on the superconducting transition temperature (T _c) in bilayer cuprates has been studied. The parameter of hopping between layers is expressed as t _⊥(k) = t _⊥(cos(k _x ) − cos(k _y ))² and treated as a small perturbation for the states of two CuO₂ planes described by the t-t′-t″-J* model. In the generalized mean field approximation for d_{x² - y²}{d_{{x^2} - {y^2}}} symmetry of the superconducting gap, neither the interlayer hopping or exchange interaction, nor the pair hopping between CuO₂ layers provides an additional mechanism of Cooper pair formation or an increase in T _c. In the concentration dependence of T _c, the bilayer splitting of the upper Hubbard band of quasi-holes is manifested as two peaks with temperatures slightly lower than the maximum T _c for a single-layer cuprate. Interlayer antiferromagnetic spin correlations suppress bilayer splitting.     

Ionic,electronic and ion-diffusion controlled relaxation processes in CaF2, BaF2 and LiBaF3 crystals

Abstract

The ionic, electronic and anion-diffusion controlled thermally stimulated relaxation (TSR) processes at 80—700 K in CaF₂ BaF₂ and LiBaF₃ crystals (X-ray irradiated or non-irradiated) have been investigated by means of ionic conductivity, ionic thermally stimulated (TS) depolarization current (TSDC); as well as current (TSC), luminescence (TSL) and bleaching (TSB) techniques. Above 250—290 K broad and overlapping anion TSDC peaks and correlated TSB stages are detected. The TSB kinetics is initiated and controlled by anion detrapping and interaction with the localized charges, i.e., the anion-diffusion controlled TSR processes take place in fluorides. The TSL and TSC data for LiBaF₃ indicate that the lifetime and drift of electrons at 80—250 K is very small because of deep retrapping. The main TSL peaks at 132K, 170K and 220 K are caused by V_k center detrapping and hole-diffusion controlled tunnel recombination within pairs like {D_n e^?…V_k }.     

Multichannel Kondo problem and some applications

We review analytical and numerical results derived from the Bethe ansatz solution of the n-channel Kondo model of arbitrary spin S as a function of temperature, external field, impurity spin S and the number of channels n. Three situations have to be distinguished: (i) If n = 2S the conduction electrons exactly compensate the impurity spin into a singlet at low temperatures, (ii) if n < 2S the impurity spin is only partially compensated (undercompensated), and (iii) if n > 2S the impurity spin is said to be overcompensated giving rise to critical behaviour. The results are discussed in the context of magnetic impurities, e.g. Fe, Cr and Tm in simple metals, the quadrupolar Kondo effect, an impurity spin embedded in the Takhtajan-Babujian Heisenberg model and electron assisted-tunnelling of an atom in a double-well potential.     

Axial and diagonal anisotropy crossover exponents for cubic systems

Cubic n-component spin systems exhibit two distinct quadratic spin anisotropy crossover exponents, appropriate for axial and diagonal anisotropies. Results for n→∞ and to order ?², and for nm-component systems are given, and relevance to multi-critical points in various systems is discussed.