Laplace,Bäcklund and gauge transformations in the two-dimensional Toda lattice

The linear Bäcklund transformation (LBT) associated with the two-dimensional Toda lattice is shown to be equivalent to a sequence of Laplace transformations of a hyperbolic linear differential equation. When the Toda lattice is cut at a point, the corresponding Laplace invariant vanishes and the LBT can be integrated.     

The Toda Equations and the Gromov–Witten Theory of the Riemann Sphere

Consequences of the Toda equations arising from the conjectural matrix model for the Riemann sphere are investigated. The Toda equations determine the Gromov–Witten descendent potential (including all genera) of the Riemann sphere from the degree 0 part. Degree 0 series computations via Hodge integrals then lead to higher-degree predictions by the Toda equations. First, closed series forms for all 1-point invariants of all genera and degrees are given. Second, degree 1 invariants are investigated with new applications to Hodge integrals. Third, a differential equation for the generating function of the classical simple Hurwitz numbers (in all genera and degrees) is found – the first such equation. All these results depend upon the conjectural Toda equations. Finally, proofs of the Toda equations in genus 0 and 1 are given.     

Nonlinear normal modes and quanta in the β Fermi-Pasta-Ulam lattice

We enumerate simple periodic orbits for the well-known Fermi-Pasta-Ulam (FPU) problem. Using these solutions as simple modes for the problem, we construct quantum solutions of the corresponding problem. These studies present a natural generalization of the concept of phonon in the nonlinear realm.     

Error analysis and lattice improvement for the C-ADS Injector-Ⅰ

The injector Scheme-(or Injector-) of the C-ADS linac is a 10 mA 10 MeV proton linac working in CW mode. It is mainly comprised of a 3.2 MeV room-temperature 4-vane RFQ and twelve superconducting single-spoke cavities housed in a long cryostat. Error analysis including alignment and field errors, and static and dynamic ones for the injector are presented. Based on detailed numerical simulations, an orbit correction scheme has been designed, which shows that with correction the rms residual orbit errors can be controlled within 0.3 mm and a beam loss rate of 1.7×10-6is obtained. To reduce the beam loss rate further, an improved lattice design for the superconducting spoke cavity section has been studied.     

Error analysis and lattice improvement for the C-ADS Injector-Ⅰ

The injector Scheme-Ⅰ (or Injector-Ⅰ) of the C-ADS linac is a 10 mA 10 MeV proton linac working in CW mode. It is mainly comprised of a 3.2 MeV room-temperature 4-vane RFQ and twelve superconducting single-spoke cavities housed in a long cryostat. Error analysis including alignment and field errors, and static and dynamic ones for the injector are presented. Based on detailed numerical simulations, an orbit correction scheme has been designed, which shows that with correction the rms residual orbit errors can be controlled within 0.3 mm and a beam loss rate of 1.7× 10^-6 is obtained. To reduce the beam loss rate further, an improved lattice design for the superconducting spoke cavity section has been studied.     

Recursion operator and bäcklund transformations for the Ruijsenaars-Toda lattice

We exhibit the recursion operator and the whole class of Bäcklund transformations for a relativistic version of the Toda lattice recently introduced by Ruijsenaars. These results allow us to prove the complete integrability of the system.     

Status of the lattice and τ decay determinations of αs

The two highest precision determinations of α s （M 2 Z ）,that based on the analysis of short-distancesensitive lattice observables,and that based on an analysis of hadronic τ decay data,have,until very recently,given results which are not in good agreement.I review new versions of these analyses which bring the two determinations into excellent agreement,and discuss prospects for additional future improvements.     

Connection formulae for the Kametake and Bäcklund transformation for the toda lattice

It is shown that the Kametake transformation for the Toda lattice is equivalent to the Bäcklund transformation in its standard form.     

Status of the lattice and τ decay determinations of α_s

The two highest precision determinations of α s (M 2 Z ),that based on the analysis of short-distancesensitive lattice observables,and that based on an analysis of hadronic τ decay data,have,until very recently,given results which are not in good agreement.I review new versions of these analyses which bring the two determinations into excellent agreement,and discuss prospects for additional future improvements.     

Bäcklund transformation for the first flows of the relativistic Toda hierarchy and associated properties

In this communication we study a class of one parameter dependent auto-Bäcklund transformations for the first flow of the relativistic Toda lattice and also a variant of the usual Toda lattice equation. It is shown that starting from the Hamiltonian formalism such transformations are canonical in nature with a well defined generating function. The notion of spectrality is also analyzed and the separation variables are explicitly constructed.     

Cure to the Landau–Pomeranchuk and associated long-wavelength Fermi-surface instabilities on the lattice

The cure to the ??=?1 Landau–Pomeranchuk instabilities in translationally invariant fermions is shown to be a state with an anisotropic gap at the Fermi surface. For higher ? and for fermions on a lattice, the general criteria for long-wavelength instabilities and their cure are found in terms of the derivatives of the single particle self-energy with respect to momentum for spin-symmetric instabilities and with respect to magnetic field for spin-antisymmetric instabilities. The results may be relevant to identifying hidden order parameters found in many metals.     

Atomic positions and diffusion paths of h and he in the α-Ti lattice

The solution energy of H and He in various interstitial and substitution positions in the hcp lattice of α-Ti has been calculated based on the method of electron density functional. The lowest solution energy of He corresponds to the basal octahedral position and that of H corresponds to the octahedral position (next in energy is the tetrahedral position). The calculated vibration frequencies of H in various positions are used for identification of lines in the vibration spectrum obtained by the method of neutron inelastic scattering. Taking into account these spectra, it can be concluded that hydrogen atoms occupy in the hcp lattice of Ti both the octahedral and tetrahedral positions even at 600 K. The available experimental data do not contradict the conclusion that the octahedral position is more preferable in α-Ti. The energy barriers are estimated for various diffusion paths of H and He.     

Adomian decomposition method and Padé approximants for solving the Blaszak--Marciniak lattice

The Adomian decomposition method （ADM） and Pade approximants are combined to solve the well-known Blaszak-Marciniak lattice, which has rich mathematical structures and many important applications in physics and mathematics. In some cases, the truncated series solution of ADM is adequate only in a small region when the exact solution is not reached. To overcome the drawback, the Pade approximants, which have the advantage in turning the polynomials approximation into a rational function, are applied to the series solution to improve the accuracy and enlarge the convergence domain. By using the ADM-Pade technique, the soliton solutions of the Blaszak-Marciniak lattice are constructed with better accuracy and better convergence than by using the ADM alone. Numerical and figurative illustrations show that it is a promising tool for solving nonlinear problems.     

On Δβ and the search for asymptotic scaling in lattice gauge theory

An ansatz for the β-function of SU(3) lattice gauge theory in four dimensions whose parameters are determined by Monte Carlo data is used both to compare different sets of data for Δβ and to study systematic errors. The data for Δβ obtained from different values of the block-spin renormalization group scaling factor are shown to be compatible within statistical errors. However the data is easily consistent with sizeable deviations (ca. 30% or more) from the two-loop approximation to the renormalization group scaling formula for physical quantities in the region of coupling for which Δβ essentially takes on its asymptotic value.     

Lieb–Robinson Bounds for the Toda Lattice

We establish locality estimates, known as Lieb–Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb–Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems.     

The distributional Borel summability and the large coupling φ4 lattice fields

Following 't Hooft we extend the Borel sum and the Watson-Nevanlinna criterion by allowing distributional transforms. This enables us to prove that the characteristic function of the measure of anyg ^{–2 4} finite lattice field is the sum of a power series expansion obtained by fixing exponentially small terms in the coefficients. The same result is obtained for the trace of the double well semigroup approximated by then ^th order Trotter formula.     

Topological quantum phase transitions and topological flat bands on the kagomé lattice

We investigate the topological properties of the tight-binding electrons on the two-dimensional kagomé lattice with two kinds of short-range hopping integral and two kinds of staggered magnetic flux. Considering the nearest-neighbor hopping (t(1)) with the staggered flux parameter φ(1) and the next nearest-neighbor hopping (t(2)) with the staggered flux parameter φ(2), we demonstrate a series of topological quantum phase transitions and find some topological bands with high Chern numbers, when tuning one parameter (t(2) or φ(2)) while the others are fixed. We have also found that, in some parameter regions, the system exhibits interesting topological flat bands with Chern number C =± 1 and a large gap above them, and the flatness ratio can reach a high value of about 170.     

Dislocation energy and calculation from Peierls stress： a rigorous the lattice theory

In the classical Peierls--Nabarro (P-N) theory of dislocation, there is a long-standing contradiction that the stable configuration of dislocation has maximum energy rather than minimum energy. In this paper, the dislocation energy is calculated rigorously in the context of the full lattice theory. It is found that besides the misfit energy considered in the classical P-N theory, there is an extra elastic strain energy that is also associated with the discreteness of lattice. The contradiction can be automatically removed provided that the elastic strain energy associated with the discreteness is taken into account. This elastic strain energy is very important because its magnitude is larger than the misfit energy, its sign is opposite to the misfit energy. Since the elastic strain energy and misfit energy associated with discreteness cancel each other, and the width of dislocation becomes wide in the lattice theory, the Peierls energy, which measures the height of the effective potential barrier, becomes much smaller than that given in the classical P-N theory. The results calculated here agree with experimental data. Furthermore, based on the results obtained, a useful formula of the Peierls stress is proposed to fully include the discreteness effects.     

A Tropical Analogue of Fay’s Trisecant Identity and the Ultra-Discrete Periodic Toda Lattice

We introduce a tropical analogue of Fay’s trisecant identity for a special family of hyperelliptic tropical curves. We apply it to obtain the general solution of the ultra-discrete Toda lattice with periodic boundary conditions in terms of the tropical Riemann’s theta function.