Periodic Toda Lattice Research Articles

Quantum integrability and chaos in a periodic Toda lattice with balanced loss-gain.

We consider an equal-mass quantum Toda lattice with balanced loss-gain for two and three particles. The two-particle Toda lattice is integrable, and two integrals of motion that are in involution have been found. The bound-state energy and the corresponding eigenfunctions have been obtained numerically for a few low-lying states. The three-particle quantum Toda lattice with balanced loss-gain and velocity-mediated coupling admits mixed phases of integrability and chaos depending on the value of the loss-gain parameter. We have obtained analytic expressions for two integrals of motion that are in involution. Although an analytic expression for the third integral has not been found, the numerical investigation suggests integrability below a critical value of the loss-gain strength and chaos above this critical value. The level spacing distribution changes from the Wigner-Dyson to the Poisson distribution as the loss-gain parameter passes through this critical value and approaches zero. An identical behavior is seen in terms of the gap-ratio distribution of the energy levels. The existence of mixed phases of quantum integrability and chaos in the specified ranges of the loss-gain parameter has also been confirmed independently via the study of level repulsion and complexity in higher order excited states.

Balanced loss-gain induced chaos in a periodic Toda lattice

We consider equal-mass periodic Toda oscillators with balanced loss-gain for two and three particles. The two-particle system is integrable. The three-particle equal-mass periodic Toda lattice is considered in presence of balanced loss-gain and velocity mediated coupling. The system is Hamiltonian for specific values of the coupling constants. The generalized momentum is the second integral of motion and study of Poincaré section indicates its integrability in certain region of the parameter-space. The generic model admits both non-chaotic and chaotic solutions. The chaos in the system with vanishing velocity mediated coupling is solely induced due to the presence of balanced loss-gain, since undriven equal-mass Toda lattice is non-chaotic. The chaotic behavior is studied in detail for the generic system.

Chern Numbers Associated with the Periodic Toda Lattice

The periodic Toda lattice is solved by exploiting the spectral properties of the Lax operator, in which the boundary states play an important role. We show that these boundary states have a topological origin similar to that of the edge states in topological insulators, and consequently, that the bulk wave functions of the Lax operator yield nontrivial Chern numbers. This implies that the periodic Toda lattice belongs to the same topological class as the Thouless pump. We demonstrate that the cnoidal wave of the Toda lattice exhibits a Chern number of $-1$ per period.

Topological Properties of the Periodic Toda Lattice: Analogy with the Thouless Pump

We find that the periodic Toda lattice belongs to the same topological class as the Thouless pump.

ON THE INTEGRATION OF A HIRERARCHY FOR THE PEREODIC TODA LATTICE WITH A FREE TERM

In this article, we have explored the Toda lattice hierarchy in the class of periodic functions with a free term. We have given an effective method of constructing of the periodic Toda lattice hierarchy with a free term. We have discussed the complete integrability of the constructed systems that is based on the inverse spectral problem of an associated discrete Hill`s equation with periodic coefficients. In particular, Dubrovin-type equations are derived for the time-evolution of the spectral data corresponding to the solutions of any system in the hierarchy.

Space of isospectral periodic tridiagonal matrices

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized Shr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. We describe the topology of its orbit space and, in particular, show that whenever the isospectral space is a manifold, its orbit space is homeomorphic to $S^4\times T^{n-3}$. There is a classical dynamical system: the flow of the periodic Toda lattice, acting on $X_{n,\lambda}$. Except for the degenerate locus $X_{n,\lambda}^0$, the Toda lattice exhibits Liouville--Arnold behavior, so that the space $X_{n,\lambda}\setminus X_{n,\lambda}^0$ is fibered into tori. The degenerate locus of the Toda system is described in terms of combinatorial geometry: its structure is encoded in the special cell subdivision of a torus, which is obtained from the regular tiling of the euclidean space by permutohedra. We apply methods of commutative algebra and toric topology to describe the cohomology and equivariant cohomology modules of $X_{n,\lambda}$.

Mathcal{N} = 2 supersymmetric extensions of relativistic Toda lattice

mathcal{N} = 2 supersymmetric extensions of both the periodic and non-periodic relativistic Toda lattice are built within the framework of the Hamiltonian formalism. A geodesic description in terms of a non-metric connection is discussed.

An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice

Consider the generalized flag manifold $G/B$ and the corresponding affine flag manifold $\mathcal{Fl}_G$. In this paper we use curve neighborhoods for Schubert varieties in $\mathcal{Fl}_G$ to construct certain affine Gromov-Witten invariants of $\mathcal{Fl}_G$, and to obtain a family of "affine quantum Chevalley" operators $\Lambda_0, \ldots, \Lambda_n$ indexed by the simple roots in the affine root system of $G$. These operators act on the cohomology ring $\mathrm{H}^*(\mathcal{Fl}_G)$ with coefficients in $\mathbb{Z}[q_0, \ldots,q_n]$. By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for $G= \mathrm{SL}_n(\mathbb{C})$. The first quantum ring is a deformation of the subalgebra of $\mathrm{H}^*(\mathcal{Fl}_G)$ generated by divisors. The second ring, denoted $\mathrm{QH}^*_{\mathrm{af}}(G/B)$, deforms the ordinary quantum cohomology ring $\mathrm{QH}^*(G/B)$ by adding an affine quantum parameter $q_0$. We prove that $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of $\mathrm{QH}^*_{\mathrm{af}}(G/B)$ by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of $G$.

On the periodic Toda lattice hierarchy with an integral source

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice hierarchy with an integral type source. The effective method is presented of constructing the periodic Toda lattice hierarchy with an integral source.

On the complete integrability of the periodic quantum Toda lattice

Abstract We prove that the periodic quantum Toda lattice corresponding to any extended Dynkin diagram is completely integrable. This has been conjectured and proved in all classical cases and E 6 {E_{6}} by Goodman and Wallach at the beginning of the 1980s. As a direct application, in the context of quantum cohomology of affine flag manifolds, results that were known to hold only for some particular Lie types can now be extended to all types.

Mutations of the cluster algebra of type and the periodic discrete Toda lattice

A direct connection between two sequences of points, one of which is generated by seed mutations of the cluster algebra of type and the other by time evolutions of the periodic discrete Toda lattice, is explicitly given. In this construction, each of them is realized as an orbit of a QRT map, and specialization of the parameters in the maps and appropriate choices of the initial points relate them. The connection with the periodic discrete Toda lattice enables us a geometric interpretation of the seed mutations of the cluster algebra of type as an addition of points on an elliptic curve.

Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions

Symmetries of the periodic Toda lattice are expresssed in action-angle coordinates and characterized in terms of the periodic and Dirichlet spectrum of the associated Jacobi matrices. Using these symmetries, the phase space of the lattice with Dirichlet boundary conditions is embedded into the phase space of a higher-dimensional periodic lattice. As an application, we obtain a Birkhoff normal form and a KAM theorem for the lattice with Dirichlet boundary conditions.

On the periodic Toda lattice with a self-consistent source

This work is devoted to the application of inverse spectral problem for integration of the periodic Toda lattice with self-consistent source. The effective method of solution of the inverse spectral problem for the discrete Hill’s equation is presented.

Combinatorial aspects of the conserved quantities of the tropical periodic Toda lattice

The tropical periodic Toda lattice (trop p-Toda) is a dynamical system attracting attention in the area of the interplay of integrable systems and tropical geometry. We show that the Young diagrams associated with trop p-Toda given by two very different definitions are identical. The first definition is given via a Lax representation of the discrete periodic Toda lattice, and the second one is associated with a generalization of the Kerov–Kirillov–Reshetikhin bijection in the combinatorics of the Bethe ansatz. By means of this identification, it is shown for the first time that the Young diagrams given by the latter definition are preserved under time evolution. This result is regarded as an important first step in clarifying the iso-level set structure of this dynamical system in general cases, i.e. not restricted to generic cases.

A geometric realization of the periodic discrete Toda lattice and its tropicalization

An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice (pdTL) is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the pdTL, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the pdTL. Finally, the formulation is lifted to the framework of tropical geometry and a tropical geometric realization of the periodic box–ball system is constructed via tropical curve intersections.

Bäcklund transformations and Hamiltonian flows

In this work we show that, under certain conditions, parametric Bäcklund transformations for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous Hamiltonian. The two systems share the same constants of motion. This observation leads to the identification of the Hamiltonian interpolating the iteration of the discrete map defined by the transformations, which indeed in numerical applications can be considered a linear combination of the integrals appearing in the spectral curve of the Lax matrix. An example with the periodic Toda lattice is given.

Double Bruhat Cells in Kac–Moody Groups and Integrable Systems

We construct a family of integrable Hamiltonian systems generalizing the relativistic periodic Toda lattice, which is recovered as a special case. The phase spaces of these systems are double Bruhat cells corresponding to pairs of Coxeter elements in the affine Weyl group. In the process we extend various results on double Bruhat cells in simple algebraic groups to the setting of Kac-Moody groups. We also generalize some fundamental results in Poisson-Lie theory to the setting of ind-algebraic groups, which is of interest beyond our immediate applications to integrable systems.

Cyclic calorons

The Nahm data of periodic instantons, often called calorons, with spatial CN-symmetries are considered, by applying Sutcliffeʼs ansatz for the monopoles with CN-symmetries. The bulk data of calorons are shown to enjoy the periodic Toda lattice, and the solutions are given in terms of elliptic theta functions. The case of N=3 calorons are investigated in detail. It is found that the “scale parameters” of these calorons have upper bounds in their values, so that they do not have the large scale, or monopole, limits. The instanton limit of the C3-symmetric caloron is obtained.

Commuting Time Evolutions in the Tropical Periodic Toda Lattice

The tropical (ultradiscrete) periodic Toda lattice is a dynamical system derived from a time-discretized version of the periodic Toda lattice through a limiting procedure called tropicalization. We propose a new formulation for this dynamical system with a representation by two-colored strips. Based on this formulation, a family of its commuting time evolutions is constructed.

Long-time asymptotics of the periodic Toda lattice under short-range perturbations

We compute the long-time asymptotics of periodic (and slightly more generally of algebro-geometric finite-gap) solutions of the doubly infinite Toda lattice under a short-range perturbation. In particular, we prove that the perturbed lattice asymptotically approaches a modulated lattice. More precisely, let g be the genus of the hyperelliptic curve associated with the unperturbed solution. We show that, apart from the phenomenon of solitons travelling in a quasi-periodic background, the n/t-pane contains g + 2 areas where the perturbed solution is close to a finite-gap solution on the same isospectral torus. In between there are g + 1 regions where the perturbed solution is asymptotically close to a modulated lattice which undergoes a continuous phase transition (in the Jacobian variety) and which interpolates between these isospectral solutions. In the special case of the free lattice (g = 0), the isospectral torus consists of just one point and we recover the known result. Both the solutions in the isospectral torus and the phase transition are explicitly characterized in terms of Abelian integrals on the underlying hyperelliptic curve. Our method relies on the equivalence of the inverse spectral problem to a vector Riemann–Hilbert problem defined on the hyperelliptic curve and generalizes the so-called nonlinear stationary phase/steepest descent method for Riemann–Hilbert problem deformations to Riemann surfaces.