Super-WZNW with reductions to supersymmetric and fermionic integrable models

A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to half integer graded affine Kac–Moody algebras. Explicit examples of the N=1,2 super-sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets sl(p,1)/sl(p)u(1) when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when p=2 leads to the constrained Bukhvostov–Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross–Neveu models.     

Ground state of an arbitrary triangle with a Calogero-Sutherland-Moser potential

We construct the expression for the ground state eigenfunction of the Schrödinger equation for a particle inside an arbitrary planar triangle under the influence of a potential.     

Noncommutative geometry and a class of completely integrable models

We introduce a Hodge operator in a framework of noncommutative geometry. The complete integrability of 2-dimensional classical harmonic maps into groups (-models or principal chiral models) is then extended to a class of noncommutative harmonic maps into matrix algebras.     

Families of commuting transfer matrices and integrable models with disorder

A family of commuting transfer matrices is shown to be associated to each symmetry transformation of a given Yang-Baxter algebra. This applies in lattices models and field theory.The Yang-Baxter algebra remains unchanged when an arbitrary parameter μ_l is associated to each lattice site. We generate in this way integrable one-dimensional hamiltonians with long-range couplings and disorder given by the <{;μ₁<};. These operators are lattice versions of the non-local charges in sigma models. As a simple example we get a Dzialozhinski-Moriya interaction with an arbitrary coupling per site from the six-vertex model. A similar model with a disordered magnetic field follows too. Their exact solution by an algebraic Bethe ansatz is presented. We derive the excitations spectrum in terms of the density of parameters (μ).As another application, the total spin S² is computed for a XXZ Heisenberg chain (μ_l ≡ 0) as a function of the anisotropy Δ (− ∞ < Δ < + ∞).     

On an integrable discretization of integrable systems with a separatrix

An integrable time-discretization of integrable Hamiltonian systems with a separatrix is considered being based on Hirota's bilinear formalism. It is proved that a discrete-time simple pendulum has a complete set of exact solutions, a conserved quantity and a separatrix. The value of the conserved quantity which characterizes the separatrix is remarkably congruent with the value of the continuous-time Hamiltonian. A discrete-time anharmonic oscillator is also shown to have the same property.     

Quantum integrable models of interacting bosons and classical r-matrices with spectral parameters

Using the technique of classical $r$-matrices with spectral parameters we construct a general form of quantum Lax operators of interacting boson systems corresponding to an arbitrary simple (or reductive) Lie algebra. We prove quantum integrability of these models in the physically important case of $\mathfrak{g}=gl\left(n\right)$ and “diagonal” in the root basis classical $r$-matrices. We consider in detail two classes of non-skew-symmetric classical $r$-matrices with spectral parameters and obtain the corresponding quantum Lax operators and quantum integrable many-boson hamiltonians that generalize Bose–Hubbard dimer hamiltonians.     

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.     

Dynamical simulation of integrable and nonintegrable models in the Heisenberg picture

The numerical simulation of quantum many-body dynamics is typically limited by the linear growth of entanglement with time. Recently numerical studies have shown that for 1D Bethe-integrable models the simulation of local operators in the Heisenberg picture can be efficient. Using the spin-1/2 XX chain as generic example of an integrable model that can be mapped to free fermions, we provide a simple explanation for this. We show furthermore that the same reduction of complexity applies to operators that have a high-temperature autocorrelation function which decays slower than exponential, i.e., with a power law. Thus efficient simulability may already be implied by a single conservation law as we will illustrate numerically for the spin-1 XXZ model.     

Constraints on the currents in principal models giving rise to integrable nonlinear systems

Quadratic constraints on the current in the principal SU(2n) model stand at the origin of an extended reduction mechanism. We derive local conservation laws both for the reduced model and for a class of solutions of the principal SU(2n) model.     

Coulomb system equivalent to the energy spectrum of the Calogero-Sutherland-Moser (CSM) model

The purpose of this paper is to prove an equivalence between the energy spectrum of the CSM model and the electrostatic energy of a one-dimensional lattice of quantized point charges interacting via Coulomb potential with Dirichlet boundary conditions. This paper is dedicated to B. Jancovici in honor of his 65th birthday.     

On the geometry of classically integrable two-dimensional non-linear sigma models

A master equation expressing the zero curvature representation of the equations of motion of a two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. Special attention is paid to those representations possessing a spectral parameter. Furthermore, a closer connection between integrability and T-duality transformations is emphasised. Finally, new integrable non-linear sigma models are found and all their corresponding Lax pairs depend on a spectral parameter.     

Threshold singularities in the dynamic response of gapless integrable models

We develop a method of an asymptotically exact treatment of threshold singularities in dynamic response functions of gapless integrable models. The method utilizes the integrability to recast the original problem in terms of the low-energy properties of a certain deformed Hamiltonian. The deformed Hamiltonian is local; hence, it can be analyzed using the conventional field theory methods. We apply the technique to spinless fermions on a lattice with nearest-neighbors repulsion, and evaluate the exponent characterizing the threshold singularity in the dynamic structure factor.     

On baxterized solutions of reflection equation and integrable chain models

Nonpolynomial baxterized solutions of reflection equations associated with affine Hecke and affine Birman–Murakami–Wenzl algebras are found. Relations to integrable spin chain models with nontrivial boundary conditions are discussed.     

Two-way quantum communication: Generalization of secure quantum information exchange to quantum network

The idea of secure quantum information exchange (SQIE) [J. Phys. B: At. Mol. Opt. Phys.44, 115504 (2011)] is introduced for the secure exchange of single qubit information states between two legitimate users, Alice and Bob. In the present paper, we extend this original SQIE protocol by presenting a scheme, which enables the secure exchange of n-single qubit information states among the n nodes of a quantum network, with the aid of a special kind of 4n-qubit entangled state and the classical assistance of an extra participant Charlie. For experimental realization of our extended SQIE protocol, we suggest an efficient scheme for the generation of a special kind of 4n-qubit entangled state using the interaction between highly detuned Λ-type three-level atoms and optical coherent field. Further, by discussing the various experimental parameters, we show that the special kind 4n-qubit entangled state can be generated with the presently available technology.     

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator \widetilde{J} is presented by constructing a subalgebra \widetilde{G } of the loop algebra \widetilde A₂. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.     

Numerical investigation of multidimensional cosmological models based on the Weyl integrable geometry

The dynamics of multidimensional cosmological models based on the Weyl integrable geometry are investigated by means of numerical methods. Models are considered in space and in the presence of matter, the latter modeled by an ideal liquid and a nonminimal scalar field. Sufficient conditions are obtained under which cosmological singularity is absent and the scenario of dynamic dimensional reduction is realized.Scientific Research Center of PV. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 107–113, May, 1995.     

一类S-mKdV方程族及其扩展可积模型

由loop代数的一个子代数出发，构造了一个线性等谱问题，再利用屠格式计算出了一类Liouvelle意义下的可积系统及其双Hamilton结构，作为该可积系统的约化，得到了著名的Schrdinger方程和mKdV方程，因此称该系统为S-mKdV方程族.根据已构造的的子代数，又构造了维数为5的loop代数的一个新的子代数，由此出发设计了一个线性等谱形式，再利用屠格式求得了S-mKdV方程族的一类扩展可积模型.利用这种方法还可以求BPT方程族、TB方程族等谱系的扩展可积模型.因此本方法具有普遍应用价值.最后作为特例，求得了著名的Schrdinger方程和mKdV方程的可积耦合系统.