On the degeneracy of <Emphasis Type="Italic">SU</Emphasis>(3)<Subscript><Emphasis Type="Italic">k</Emphasis></Subscript> topological phases

The ground state degeneracy of an SU(N) _k topological phase with n quasiparticle excitations is a relevant quantity for quantum computation, condensed matter physics, and knot theory. It is an open question to find a closed formula for this degeneracy for any N >2. Here we present the problem in an explicit combinatorial way and analyze the case N = 3. While not finding a complete closed-form solution, we obtain generating functions and solve some special cases.     

One-dimensional Ising chain with competing interactions: Exact results and connection with other statistical models

We study the ground state properties of a one-dimensional Ising chain with a nearest-neighbor ferromagnetic interactionJ ₁, and akth neighboranti-ferromagnetic interactionJ _k . WhenJ _k/J₁=–1/k, there exists a highly degenerate ground state with a residual entropy per spin. For the finite chain with free boundary conditions, we calculate the degeneracy of this state exactly, and find that it is proportional to the (N+k–1)th term in a generalized Fibonacci sequence defined by,F _N ^(k) =F _N–1 ^(k) +F _N–k ^(k) . In addition, we show that this one-dimensional model is closely related to the following problems: (a) a fully frustrated two-dimensional Ising system with a periodic arrangement of nearest-neighbor ferro- and antiferromagnetic bonds, (b) close-packing of dimers on a ladder, a 2× strip of the square lattice, and (c) directed self-avoiding walks on finite lattice strips.Work partially supported by grants from AFOSR and ARO.     

Nonminimal Derivative Couplings and Inflation in Generalized Theories of Gravity

We study extended theories of gravity where nonminimal derivative couplings of the form R^klϕ,_kϕ,_l are present in the Lagrangian. We show how and why the other couplings of similar structure may be ruled out and then deduce the field equations and the related cosmological models. Finally, we get inflationary solutions which do follow neither from any effective scalar field potential nor from a cosmological constant introduced “by hand”, and we show the de Sitter space‐time to be an attractor solution.     

Magnetism,exchange and crystal field parameters in the orbitally unquenched Ising antiferromagnet FePS3

FePS₃ is a layered antiferromagnet (T _N=123 K) with a marked Ising anisotropy in magnetic properties. The anisotropy arises from the combined effect of the trigonal distortion from octahedral symmetry and spin-orbit coupling on the orbitally degenerate⁵ T _2g ground state of the Fe²⁺ ion. The anisotropic paramagnetic susceptibilities are interpreted in terms of the zero field Hamiltonian, ℋ=Σ_i [δ(L _iz ² −2)+|λ|L _i .S _i ]−Σ _ij J _ij S _i .S _j . The crystal field trigonal distortion parameter Δ, the spin-orbit coupling λ and the isotropic Heisenberg exchange,J _ij, were evaluated from an analysis of the high temperature paramagnetic susceptibility data using the Correlated Effective Field (CEF) theory for many-body magnetism developed by Lines. Good agreement with experiment were obtained for Δ/k=215.5 K; λ/k=166.5 K;J _nn k=27.7 K; andJ _nnn k=−2.3 K. Using these values of the crystal field and exchange parameters the CEF predicts aT _N=122 K for FePS₃, which is remarkably close to the observed value of theT _N. The accuracy of the CEF approximation was also ascertained by comparing the calculated susceptibilities in the CEF with the experimental susceptibility for the isotropic Heisenberg layered antiferromagnet MnPS₃, for which the high temperature series expansion susceptibility is available.     

The size distribution for theA g RB f−g Model of polymerization

This paper gives the equilibrium distribution of polymer sizes for Flory'sA _g RB _f–g model of polymerization. In this model, the polymers are composed of structural units withg functional groups of the typeA and (f-g) functional groups of the typeB. Reaction is subject to three conditions: (1) Functional groups of the typeA react only with those of typeB, and vice versa. (2) Intramolecular reactions do not occur [and therefore only branched-chain (noncyclic) polymers and formed]. (3) Subject to conditions (1) and (2), all functional groups are equally reactive. The derivation employs Stockmayer's statistical mechanical method (first used on Flory'sRA _f model), coupled with a recursion giving the number of distinct polymers which may be assembled fromk units of theA _g RB _f–g type. We also give distributions for a limiting case of theA _g RB _f–g model, the so-calledA _g RB model. This paper completes the solution of the Smoluchowski coagulation equation (monodisperse case) for the kernelsa _ij =A + B(i +j)+ Cij. The proof will be given in another publication.     

Spatial structure and stability based on random walks

A general expression for a recursion formula which describes a random walk with coupled modes is given. In this system, the random walker is specified by the jumping probabilities P⁺ and P^– which depend on the modes. The transition probability between the modes is expressed by a jumping probabilityR ^(ij) (orr _ij). With the aid of this recursion formula, spatial structures of the steady state of a coupled random walk are studied. By introducing a Liapunov function and entropy, it is shown that the stability condition for the present system can be expressed as the principle of the extremum entropy production.On leave of absence from Tohoku University, Department of Applied Science, Faculty of Engineering, Sendai, 980 Japan.     

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.     

Exact analytical solution for the change of the photon statistics due tok-photon absorption

For unsaturatedk-photon absorption the exact solution of the master equation for the diagonal elements is presented. This solution is found by Laplace-transformation and recursion. It has the form of a double sum over the initial values. Commuting the order of summation we then show, that fork2 the photon distribution does not depend on the initial distribution if the initial mean photon number is sufficiently high and the square of the resulting photon number is small against the initial photon number.     

Integrable Nonlinear Evolution Equations on the Half-Line

 A rigorous methodology for the analysis of initial-boundary value problems on the half-line, is applied to the nonlinear §(NLS), to the sine-Gordon (sG) in laboratory coordinates, and to the Korteweg-deVries (KdV) with dominant surface tension. Decaying initial conditions as well as a smooth subset of the boundary values are given, where n=2 for the NLS and the sG and n=3 for the KdV. For the NLS and the KdV equations, the initial condition q(x,0) = q ₀ (x) as well as one and two boundary conditions are given respectively; for the sG equation the initial conditions q(x,0) = q ₀ (x), q _t (x,0) = q ₁ (x), as well as one boundary condition are given. The construction of the solution q(x,t) of any of these problems involves two separate steps: (a) Given decaying initial conditions define the spectral (scattering) functions {a(k),b(k)}. Associated with the smooth functions , define the spectral functions {A(k),B(k)}. Define the function q(x,t) in terms of the solution of a matrix Riemann-Hilbert problem formulated in the complex k-plane and uniquely defined in terms of the spectral functions {a(k),b(k),A(k),B(k)}. Under the assumption that there exist functions such that the spectral functions satisfy a certain global algebraic relation, prove that the function q(x,t) is defined for all , it satisfies the given nonlinear PDE, and furthermore that . (b) Given a subset of the functions as boundary conditions, prove that the above algebraic relation characterizes the unknown part of this set. In general this involves the solution of a nonlinear Volterra integral equation which is shown to have a global solution. For a particular class of boundary conditions, called linearizable, this nonlinear equation can be bypassed and {A(k),B(k)} can be constructed using only the algebraic manipulation of the global relation. For the NLS, the sG, and the KdV, the following particular linearizable cases are solved: , respectively, where χ is a real constant. Received: 22 October 2001 / Accepted: 22 March 2002 Published online: 22 August 2002     

The isotope dependence of diatomic Dunham coefficients

The isotope dependence of the Dunham vibration-rotation coefficients Y_kl of a diatomic molecule is studied. Rovibronic interactions between different electronic states are taken into account by transformation to an effective vibration-rotation Hamiltonian for each electronic state. This contains modified vibrational and rotational reduced masses as well as the adiabatic correction to the potential energy. The effects of these contributions on the vibration-rotation energies are expressed in terms of two functions and for each atom i. The resultant formula for Y_kl is Y_kl=μ_c^−(k+2l)/2U_kl{1+m_eΔ_kl^a/M_a+m_eΔ_kl^b/M_b+O(m_e²/M_i²)}, where U_kl, Δ_kl^a, and Δ_kl^b are isotopically invariant, M_a and M_b are the atomic masses, and μ_c = M_aM_b/(M_a + M_b − Cm_e) is the atomic reduced mass, modified by the molecular charge number C for charged species. The U_kl with l ≥ 2 can be calculated from those with l = 0 and 1. The corrections U_klΔ_klⁱ are related to the functions and and to the Dunham corrections. Recent data for the CO molecule are discussed, and it is suggested that some large Δ_klⁱ values are associated with accidentally small U_kl values, since the size of U_klΔ_klⁱ is not directly related to that of U_kl.     

Integrable Nonlinear Evolution Equations on a Finite Interval

Let q(x,t) satisfy an integrable nonlinear evolution PDE on the interval 0<x<L, and let the order of the highest x-derivative be n. For a problem to be at least linearly well-posed one must prescribe N boundary conditions at x=0 and n−N boundary conditions at x=L, where if n is even, N=n/2, and if n is odd, N is either (n−1)/2 or (n+1)/2, depending on the sign of ∂ⁿ_xq. For example, for the sine-Gordon (sG) equation one must prescribe one boundary condition at each end, while for the modified Korteweg-de Vries (mKdV) equations involving q_t+q_xxx and q_t−q_xxx one must prescribe one and two boundary conditions, respectively, at x=0. We will refer to these two mKdV equations as mKdV-I and mKdV-II, respectively. Here we analyze the Dirichlet problem for the sG equation, as well as typical boundary value problems for the mKdV-I and mKdV-II equations. We first show that the unknown boundary values at each end (for example, q_x(0,t) and q_x(L,t) in the case of the Dirichlet problem for the sG equation) can be expressed in terms of the given initial and boundary conditions through a system of four nonlinear ODEs. We then show that q(x,t) can be expressed in terms of the solution of a 2×2 matrix Riemann-Hilbert problem formulated in the complex k-plane. This problem has explicit (x,t) dependence in the form of an exponential; for example, for the case of the sG this exponential is exp {i(k−1/k)x+i(k+1/k)t}. Furthermore, the relevant jump matrices are explicitly given in terms of the spectral functions {a(k),b(k)}, {A(k),B(k)}, and , which in turn are defined in terms of the initial conditions, of the boundary values of q and of its x-derivatives at x=0, and of the boundary values of q and of its x-derivatives at x=L, respectively. This Riemann-Hilbert problem has a global solution.     

Fermion Fields in Theories of Gravitation

After an introduction to the formalism used throughout the paper there follows a concise presentation of the theory of fermion fields in one-tetrad gravitational theories. That presentation gives a hint to the construction of a bi-tetrad theory, the two tetrad fields being denoted by h^A_k and h?^A_k. The tetrad field h^A_k. gives the Riemannian metric g_kl while the tetrad field h?h^A_k is orthonormalized with respect to the flat metric a_kl. Specializing h?^A_k in such a way that they have the form δ^A_k in the preferred coordinates of Minkowski space and using a matter Lagrangian which contains these h?^A_k only by the anholonomic components of the metric Christoffel symbols, we obtain a dynamical energy momentum tensor which is equal to the canonical one. Then we consider the relations of the bi-tetrad theory to other theories which are only covariant with respect to global Lorentz transformations from the beginning. As an example we formulate the main relations of the two-component neutrino theory.     

Variance of the range of a random walk

T _n, the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn, is obtained from its relation to the dual first occupancy probabilityF _ij(x, x), and the latter quantity is obtained from a recursion with the first occupancy probabilityF _k (x). The varianceV _n of the number of distinct sites occupied is calculated directly from T_n; the procedure is illustrated by the calculation ofV _n (4096 /n) and the derivation of asymptotic expansions forV _n for a particular random walk in dimensions 1 through 3.Work completed under the auspices of the United States Department of Energy.     

D-dimensional massless particle with extended gauge invariance

We propose the model ofD-dimensional massless particle whose Lagrangian is given by theN-th extrinsic curvature of world-line. The system hasN+1 gauge degrees of freedom constitutingW-like algebra; the classical trajectories of the model are space-like curves which obey the conditionsk _N+a=k_N−a, k_2N =0,a=1, ...,N−1,N≤[(D−2)/2], while the firstN curvaturesk _i remain arbitrary. We show that the model admits consistent formulation on the anti-DeSitter space. The solutions of the system are the massless irreducible representations of Poincaré group withN nonzero helicities, which are equal to each other. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000.     

Deformations of quantum hyperplanes

We consider the quantum hyperplanex ⁱ x ^j =q _ij x ^j x ⁱ (i,j = 1..n) and define and consider deformations of the formx ⁱ x ^j =q _ij x ^j x ⁱ + _{k k} ^ij x ^k + ^ij , where _k ^ij and ^ij are complex numbers. We prove that for genericq _ij no nontrivial deformations exist forn 3.     

The Influence of Supplementary Conditions for Papapetrou Equations on the Lyapunov Stability of Motion

The Lyapunov stability of circular motions of spinning test bodies in the Schwarzschild field has been considered. For Papapetrou equations the generalized supplementary conditions S⁰ⁱu₀ = kS^iju_j = k const, i, j = 1, 2, 3, have been chosen. Four of the first integrals of motion have been indicated. With the aid of Lyapunov functions the domains of stable motions have been found.     

Diagonal and Off-Diagonal Recursion Relation for the General Potential V(r) = Arp

A recursion relation is derived for the potential V(r) = Ar ^p. Generally, this connects off-diagonal matrix elements of r ^k–2, r ^k+p, r ^k, and r ^k+2. The diagonal case is obtained by setting m = n in this relation. The relation is derived by elementary methods and without recourse to specific properties of the eigenstates. Finally, this relation is studied for the familiar potentials p = –1, 1, 2.     

Heterodyne measurements of 12C18O, 13C16O,and 13C18 O laser frequencies; mass dependence of Dunham coefficients

Difference frequencies between rare isotope CO lasers and a ¹²C¹⁶O laser have been measured by optical heterodyne techniques. These data for ¹²C¹⁸O, ¹³C¹⁶O, and ¹³C¹⁸O have been used together with the 15 previously reported Dunham coefficients Y_kl for ¹²C¹⁶O to determine a set of mass independent parameters Δ_kl and U_kl defined by Y_kl = μ^-(k/2+l)[1+m_e(Δ^C_kl/M_C + Δ^O_kl/M_O)] U_kl. The 01, 1 0, and 20 correction terms were found to be statistically significant. Line frequencies calculated from the resulting 15 Dunham coefficients for the rare isotopes are accurate to a few MHz in the measured laser bands.     

How does the s(E + N) equation work? Comparisons with a modified Swain–Scott equation (E + sN) and revision of the N1 scale of solvent nucleophilicity

Modifications of the Swain–Scott equation (log k/k₀) = sn) give an equation log k₁ = (E + sN₁′); k₁ is the rate constant, E is an electrophilicity parameter, N₁′ is a solvent nucleophilicity parameter and s is an electrophile‐specific sensitivity parameter. The equation is tested using over 300 published first‐order rate constants (k₁) for decay of a range of benzhydrylium cations in various solvents, on which the published N₁ scale of solvent nucleophilicity is based (S. Minegishi, S. Kobayashi and H. Mayr, J. Am. Chem. Soc. 2004, 126, 5174–5181) using the alternative equation log k = s(E + N₁), in which s is a nucleophile‐specific parameter. The modified (E + sN₁′) equation provides a revised N₁′ scale of solvent nucleophilicity, and a more precise fit, with less than half the number of adjustable parameters. It is found that the sensitivities of the benzhydrylium cations to changes in solvent nucleophilicity decrease slightly as reactivity increases, in contrast to s(E + N) equations, which show no trends in s values. It is proposed that more reliable N scales can be defined using (E + sN), because N is determined directly from definitions, and residual errors (e.g. experimental or due to solvation effects) can be incorporated into the slope and intercept. The complex reasons for the success of equations of the type log k = s(E + N) are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

The Riemann Surface of the Chiral Potts Model Free Energy Function

In a recent paper we derived the free energy or partition function of the N-state chiral Potts model by using the infinite lattice inversion relation method, together with a non-obvious extra symmetry. This gave us three recursion relations for the partition function per site T _pq of the infinite lattice. Here we use these recursion relations to obtain the full Riemann surface of T _pq . In terms of the t _p ,t _q variables, it consists of an infinite number of Riemann sheets, each sheet corresponding to a point on a (2N–1)-dimensional lattice (for N>2). The function T _pq is meromorphic on this surface: we obtain the orders of all the zeros and poles. For N odd, we show that these orders are determined by the usual inversion and rotation relations (without the extra symmetry), together with a simple linearity ansatz. For N even, this method does not give the orders uniquely, but leaves only [(N+4)/4] parameters to be determined.