A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier it has been observed that this kind objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that ordinary difference equation defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive ordinary difference and differential equations, for the roots of the polynomials. Efficiency of the method is approved by some illustrative examples.
Using the orbit method we attempt to reveal geometric and algebraic meaning of separation of variables for the integrable systems on coadjoint orbits in an $\mathfrak{sl}(3)$ loop algebra. We consider two types of generic orbits embedded into a common manifold, endowed with two nonsingular Lie-Poisson brackets. We prove that separation of variables on orbits of both types is realized by the same variables of separation. We also construct the integrable systems on these orbits: a coupled 3-component nonlinear Schrödinger equation and an isotropic SU(3) Landau-Lifshitz equation.
Separation of variables by means of the orbit method is implemented to integrable systems on coadjoint orbits in an $\mathfrak{sl}(4)$ loop algebra. This is a development and a kind of explanation for Sklyanin's procedure of separation of variables. It is shown that points on a spectral curve serve as variables of separation for two integrable systems living on two generic orbits embedded into a common manifold. These orbits are endowed with different nonsingular Lie-Poisson brackets. Explicit expressions for the case of $\mathfrak{sl}(4)$ loop algebra are given.
The partition function of the Symmetric Matrix Ensemble is identified with the $τ-$function of a particular solution of the Pfaff Lattice. We show that in the case of even power interactions, in the thermodynamic limit, the $τ-$function corresponds to the solution of an integrable chain of hydrodynamic type. We prove that the hydrodynamic chain so obtained is diagonalisable and admits hydrodynamic reductions in Riemann invariants in an arbitrary number of components.
New variables of separation for few integrable systems on the two-dimensional sphere with higher order integrals of motion are considered in detail. We explicitly describe canonical transformations of initial physical variables to the variables of separation and vice versa, calculate the corresponding quadratures and discuss some possible integrable deformations of initial systems.
We endow Ruijsenaars' open difference Toda chain with a one-sided boundary interaction of Askey-Wilson type and diagonalize the quantum Hamiltonian by means of deformed hyperoctahedral $q$-Whittaker functions that arise as a $t=0$ degeneration of the Macdonald-Koornwinder multivariate Askey-Wilson polynomials. This immediately entails the quantum integrability, the bispectral dual system, and the $n$-particle scattering operator for the chain in question.
This note is a review of the recently revealed intriguing connection between integrable quantum spin chains and integrable many-body systems of classical mechanics. The essence of this connection lies in the fact that the spectral problem for quantum Hamiltonians of the former models is closely related to a sort of inverse spectral problem for Lax matrices of the latter ones. For simplicity, we focus on the most transparent and familiar case of spin chains on N sites constructed by means of the GL(2)-invariant R-matrix. They are related to the classical Ruijsenaars-Schneider system of N partic…
We present a general method of folding an integrable spin chain, defined on a line, to obtain an integrable open spin chain, defined on a half-line. We illustrate our method through two fundamental models with sl(2) Lie algebra symmetry: the Heisenberg XXX and the Inozemtsev hyperbolic spin chains. We obtain new long-range boundary Hamiltonians and demonstrate that they exhibit Yangian symmetries, thus ensuring integrability of the models we obtain. The method presented provides a "bottom-up" approach for constructing integrable boundaries and can be applied to any spin chain model.
Integrable open elliptic Toda chain with boundaries
Steklov Mathematical Institute of Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University（莫斯科国立大学理论与数学物理研究所）
In this paper we study the spectrum of the spin-1 Temperley-Lieb spin chain with integrable open boundary conditions. We obtain the eigenvalue expressions as well as its associated Bethe ansatz equations by means of the coordinate Bethe ansatz. These equations provide the complete description of the spectrum of the model.
In this paper we consider a differential--difference system which is equivalent to the commutativity condition of two differential--difference operators. We study the rank two algebro--geometric solutions of this system.
We study the correlation functions of the integrable $O(n)$ spin chain in the thermodynamic limit. We addressed the problem of solving functional equations of the quantum Knizhnik Zamolodchikov type for density matrix related to the $O(n)$ spin chain. We give the explicit solution for two-sites density matrix elements for the $O(n)$ which are then evaluated for the $n=3,4,\dots,8$ cases at zero temperature.
We investigate the relation between the local variables of a discrete integrable lattice system and the corresponding separation variables, derived from the associated spectral curve. In particular, we have shown how the inverse transformation from the separation variables to the discrete lattice variables may be factorised as a sequence of canonical transformations, following the procedure outlined by Kuznetsov.
We propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e. certain Stäckel systems with quadratic in momenta integrals of motion.
We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(λ,λ^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schrödinger, and Heisenberg magnetic equations.
We present certain classical continuum long wave-length limits of prototype integrable quantum spin chains, and define the corresponding construction of classical continuum Lax operators. We also provide two specific examples, i.e. the isotropic and anisotropic Heisenberg models.
We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construc…
For integrable inhomogeneous supersymmetric spin chains (generalized graded magnets) constructed employing Y(gl(N|M))-invariant R-matrices in finite-dimensional representations we introduce the master T-operator which is a sort of generating function for the family of commuting quantum transfer matrices. Any eigenvalue of the master T-operator is the tau-function of the classical mKP hierarchy. It is a polynomial in the spectral parameter which is identified with the 0-th time of the hierarchy. This implies a remarkable relation between the quantum supersymmetric spin chains and classical many…
An integrable Heisenberg spin chain with nearest-neighbour couplings, next-nearest-neighbour couplings and Dzyaloshinski-Moriya interacton is constructed. The integrability of the model is proven. Based on the Bethe Ansatz solutions, the ground state and the elementary excitations are studied. It is shown that the spinon excitation spectrum of the present system possesses a novel triple arched structure. The method provided in this paper is general to construct new integrable models with next-nearest-neighbour couplings.
Exact solution of the quantum integrable $D^{(2)}_2$ spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the $D^{(2)}_2$ model are derived via off-diagonal Bethe ansatz.
We show that the equations of motion in one partial integrable case of Goryachev in the rigid body dynamics can be separated by the appropriate change of variables, the new variables x, y being hyperelliptic functions of time. The natural phase variables (components of coordinates and momenta) are expressed via x,y explicitly in elementary algebraic functions.
We examine certain classical continuum long wave-length limits of prototype integrable quantum spin chains. We define the corresponding construction of classical continuum Lax operators. Our discussion starts with the XXX chain, the anisotropic Heisenberg model and their generalizations and extends to the generic isotropic and anisotropic gl_n magnets. Certain classical and quantum integrable models emerging from special "dualities" of quantum spin chains, parametrized by c-number matrices, are also presented.
A new approach for derivation of Benney-like momentum chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed, all their reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.
There are two fundamental problems studied by the theory of hamiltonian integrable systems: integration of equations of motion, and construction of action-angle variables. The third problem, however, should be added to the list: separation of variables. Though much simpler than two others, it has important relations to the quantum integrability. Separation of variables is constructed for the $SL(3)$ magnetic chain --- an example of integrable model associated to a nonhyperelliptic algebraic curve.
The ground state of Shor-Movassagh chain can be analytically described by the Motzkin paths. There is no analytical description of the excited states, the model is not solvable. We prove the integrability of the model without interacting part in this paper [free Shor-Movassagh]. The Lax pair for the free Shor-Movassagh open chain is explicitly constructed. We further obtain the boundary $K$-matrices compatible with the integrability of the model on the open interval. Our construction provides a direct demonstration for the quantum integrability of the model, described by Yang-Baxter algebra. D…
We study integrable open boundary conditions for d(2,1;α)^2 and psu(1,1|2)^2 spin-chains. Magnon excitations of these open spin-chains are mapped to massive excitations of type IIB open superstrings ending on D-branes in the AdS_3 x S^3 x S^3 x S^1 and AdS_3 x S^3 x T^4 supergravity geometries with pure R-R flux. We derive reflection matrix solutions of the boundary Yang-Baxter equation which intertwine representations of a variety of boundary coideal subalgebras of the bulk Hopf superalgebra. Many of these integrable boundaries are matched to D1- and D5-brane maximal giant gravitons.
In this paper we take further steps towards developing the separation of variables program for integrable spin chains with gl(N) symmetry. By finding, for the first time, the matrix elements of the SoV measure explicitly we were able to compute correlation functions and wave function overlaps in a simple determinant form. In particular, we show how an overlap between on-shell and off-shell algebraic Bethe states can be written as a determinant. Another result, particularly useful for AdS/CFT applications, is an overlap between two Bethe states with different twists, which also takes a determin…
In this paper we consider integrable dispersive chains associated with the so called Energy Dependent Schrodinger operator. In a general case multi component reductions of these dispersive chains are new integrable systems, which are characterised by two arbitrary natural numbers. Also we show that integrable three dimensional linearly degenerate quasilinear equations of a second order possess infinitely many differential constraints. Corresponding dispersive reductions are integrable systems associated with the Energy Dependent Schrodinger operator.
New approach to classification of integrable hydrodynamic chains is established. Generating functions of conservation laws are classified by the method of hydrodynamic reductions. N parametric family of explicit hydrodynamic reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new hydrodynamic chains are found.
In this paper we construct multi-phase solutions for integrable dispersive chains associated with the three-dimensional linearly degenerate Mikhalev system of first order. These solutions are parameterized by infinitely many arbitrary parameters. As byproduct we describe multi-phase solutions for finite component dispersive reductions of these integrable dispersive chains.
Using the method of hydrodynamic reductions, we find all integrable infinite (1+1)-dimensional hydrodynamic-type chains of shift one. A class of integrable infinite (2+1)-dimensional hydrodynamic-type chains is constructed.
Chains of interacting non-Abelian anyons with local interactions invariant under the action of the Drinfeld double of the dihedral group $D_3$ are constructed. Formulated as a spin chain the Hamiltonians are generated from commuting transfer matrices of an integrable vertex model for periodic and braided as well as open boundaries. A different anyonic model with the same local Hamiltonian is obtained within the fusion path formulation. This model is shown to be related to an integrable fusion interaction round the face model. Bulk and surface properties of the anyon chain are computed from the…
We study the correlation functions of su(2) invariant spin-s chains in the thermodynamic limit. We derive non-linear integral equations for an auxiliary correlation function $ω$ for any spin s and finite temperature T. For the spin-3/2 chain for arbitrary temperature and zero magnetic field we obtain algebraic expressions for the reduced density matrix of two-sites. In the zero temperature limit, the density matrix elements are evaluated analytically and appear to be given in terms of Riemann's zeta function values of even and odd arguments.
Hydrodynamic reductions of the hydrodynamic chain associated with dispersionless limit of 2+1 Harry Dym equation are found by the Miura type and reciprocal transformations applied to the Benney hydrodynamic chain.
This paper is devoted to a description of integrable Hamiltonian hydrodynamic chains associated with Dorfman Poisson brackets. Three main classes of these hydrodynamic chains are selected. Generating functions of conservation laws and commuting flows are found. Hierarchies of these Hamiltonian hydrodynamic chains are extended on negative moments and negative time variables. Corresponding three dimensional quasilinear equations of the second order are presented.
There being no precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)] the canonical coordinates and the conjugated operators are constructed which satisfy the ``quantum characteristic equation'' (quantum counterpart of the spectral algebraic curve for the L operator). The coordinates constructed provide a local separation of variables. The conditions are enlisted which are necessary for the global separation of variables to take place.
The thermodynamic Bethe ansatz is applied to a quantum integrable spin chain associated with the Lie superalgebra osp(1|2). Using the string hypothesis, we derive a set of infinite number of non-linear integral equations (thermodynamic Bethe ansatz equation), which characterize the free energy. The low temperature limit of the free energy is also discussed.
We classify all fundamental integrable spin chains with two-dimensional local Hilbert space which have regular R-matrices of difference form. This means that the R-matrix underlying the integrable structures is of the form R(u,v)=R(u-v) and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, 8 of which correspond to well-known eight or lower vertex models. The remaining 6 models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R-matrices, we find a bijection betwe…
We derive exactly scalar products and form factors for integrable higher-spin XXZ chains through the algebraic Bethe-ansatz method. Here spin values are arbitrary and different spins can be mixed. We show the affine quantum-group symmetry, $U_q(\hat{sl_2})$, for the monodromy matrix of the XXZ spin chain, and then obtain the exact expressions. Furthermore, through the quantum-group symmetry we explicitly derive the diagonalized forms of the $B$ and $C$ operators in the $F$-basis for the spin-1/2 XXZ spin chain, which was conjectured in the algebraic Bethe-ansatz calculation of the XXZ correlat…
We consider two-site XXX Heisenberg magnets with different boundary conditions, which are integrable systems on so(4) possessing additional cubic and quartic integrals of motion. The separated variables for these models are constructed using the Sklyanin method.
Following the procedure, described in the paper nlin.SI/0003002, for the integrable DST chain we construct Baxter Q-operators as the traces of monodromy of some M-operators, that act in quantum and auxiliary spaces. Within this procedure we obtain two basic M-operators and derive some functional relations between them such as intertwining relations and wronskian-type relations between two basic Q-operators.
A large (infinitely-dimensional) class of completely integrable (possibly non-autonomous) spin chains is discovered associated to an infinite-dimensional Lie Algebra of infinite rank. The complete set of integrals of motion is constructed explicitly, as well as their eigenstates and spectra. As an example we outline kicked Ising model: Ising chain periodically kicked with transversal magnetic field.
Integrable Hamiltonians for higher spin periodic XXZ chains are constructed in terms of the spin generators; explicit examples for spins up to 3/2 are given. Relations between Hamiltonians for some U_q(sl_2)-symmetric and U(1)-symmetric universal r-matrices are studied; their properties are investigated. A certain modification of the higher spin periodic chain Hamiltonian is shown to be an integrable U_q(sl_2)-symmetric Hamiltonian for an open chain.
Bi-presymplectic chains of one-forms of co-rank one are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of dual Poisson-presymplectic pair is used and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of related flow leads directly to the construction of separation coordinates in…
Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations
Center for Joint Quantum Studies and Department of Physics, School of Science, Tianjin University（天津大学理学院物理系联合量子研究中心）；Department of Physics, Guangxi Normal University（广西师范大学物理学院）；Peng Huanwu Center for Fundamental Theory（彭桓武理论物理中心）
A general way to construct chain models with certain Lie algebraic or quantum Lie algebraic symmetries is presented. These symmetric models give rise to series of integrable systems. As an example the chain models with $A_n$ symmetry and the related Temperley-Lieb algebraic structures and representations are discussed. It is shown that corresponding to these $A_n$ symmetric integrable chain models there are exactly solvable stationary discrete-time (resp. continuous-time) Markov chains whose spectra of the transition matrices (resp. intensity matrices) are the same as the ones of the correspon…
We develop a new method to compute the exact overlaps between integrable boundary states and on-shell Bethe states for integrable spin chains. Our method is based on the coordinate Bethe Ansatz and does not rely on the "rotation trick" of the corresponding lattice model. It leads to a rigorous proof of the factorized overlap formulas in a number of cases, some of which were inaccessible to earlier methods. As concrete examples, we consider the compact XXX and XXZ Heisenberg spin chains, and the non-compact $SL(2,\mathbb{R})$ spin chain.
It is shown that a generalization of Bethe Ansatz based on an utilization of {\it degenerative discrete-diffractive} wave functions solves the three-magnon problem for the $S=1$ isotropic ferromagnetic infinite chain. The four-magnon problem is briefly discussed.
We find membrane configurations in AdS_4 x S^7, which correspond to the continuous limit of the SU(2) integrable spin chain, considered as a limit of the SU(3) spin chain, arising in N=4 SYM in four dimensions, dual to strings in AdS_5 x S^5. We also discuss the relationship with the Neumann-Rosochatius integrable system at the level of Lagrangians, comparing the string and membrane cases.
Generic Hamiltonian systems have a mixed phase space where regions of regular and chaotic motion coexist. We present a method for constructing an integrable approximation to such regular phase-space regions including a nonlinear resonance chain. This approach generalizes the recently introduced iterative canonical transformation method. In the first step of the method a normal-form Hamiltonian with a resonance chain is adapted such that actions and frequencies match with those of the non-integrable system. In the second step a sequence of canonical transformations is applied to the integrable …
The discrete symmetry's dressing chains of the nonlinear Schrodinger equation (NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly reversible Backlund auto-transformations for the mNLS and mDS equations are constructed. We demonstrate discrete symmetry's conjugate chains of the KP and DS models. The two-dimensional generalization of the P4 equation are obtained.
We discuss two polynomial bi-Hamiltonian structures for the generalized integrable Chaplygin system on the sphere S^2 with an additional integral of fourth order in momenta. An explicit procedure to find the variables of separation, the separation relations and the transformation of the corresponding algebraic curves of genus two is considered in detail.
Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our resu…
We investigate the exact overlaps between eigenstates of integrable spin chains and a special class of states called "integrable initial/final states". These states satisfy a special integrability constraint, and they are closely related to integrable boundary conditions. We derive new algebraic relations for the integrable states, which lead to a set of recursion relations for the exact overlaps. We solve these recursion relations and thus we derive new overlap formulas, valid in the XXX Heisenberg chain and its integrable higher spin generalizations. Afterwards we generalize the integrabilit…
We present new integrable models of interacting spin-1/2 chains, which can be interpreted as hard rod deformations of the XXZ Heisenberg chains. The models support multiple particle types: dynamical hard rods of length $\ell$ and particles with lengths $\ell'<\ell$ that are immobile except for the interaction with the hard rods. We encounter a remarkable phenomenon in these interacting models: exact spectral degeneracies across different deformations and volumes. The algebraic integrability of these systems is also treated using a recently developed formalism for medium range integrable spin c…
