On Integrability of the Kontsevich Non-Abelian ODE System

We consider systems of ODEs with the right-hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich (private communication). We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries.     

Information recovery from black holes

We argue that if black hole entropy arises from a finite number of underlying quantum states, then any particular such state can be identified from infinity. The finite density of states implies a discrete energy spectrum, and, in general, such spectra are non-degenerate except as determined by symmetries. Therefore, knowledge of the precise energy, and of other commuting conserved charges, determines the quantum state. In a gravitating theory, all conserved charges including the energy are given by boundary terms that can be measured at infinity. Thus, within any theory of quantum gravity, no information can be lost in black holes with a finite number of states. However, identifying the state of a black hole from infinity requires measurements with Planck scale precision. Hence observers with insufficient resolution will experience information loss. First Award in the 2006 Essay Competition of the Gravity Research Foundation.     

Different types of integrability and their relation to decoherence in central spin models

We investigate the relation between integrability and decoherence in central spin models with more than one central spin. We show that there is a transition between integrability ensured by the Bethe ansatz and integrability ensured by complete sets of commuting operators. This has a significant impact on the decoherence properties of the system, suggesting that it is not necessarily integrability or nonintegrability which is related to decoherence, but rather its type or a change from integrability to nonintegrability.     

A Suggestion for an Integrability Notion for Two-Dimensional Spin Systems

We suggest that trialgebraic symmetries might be a sensible starting point for a notion of integrability for two dimensional spin systems. For a simple trialgebraic symmetry we give an explicit condition in terms of matrices which a Hamiltonian realizing such a symmetry has to satisfy and give an example of such a Hamiltonian which realizes a trialgebra recently given by the authors in another paper. Besides this, we also show that the same trialgebra can be realized on a kind of Fock space of q-oscillators, i.e. the suggested integrability concept gets via this symmetry a close connection to a kind of noncommutative quantum field theory, paralleling the relation between the integrability of spin chains and two dimensional conformal field theory.     

Quantum and classical spin clusters: disappearance of quantum numbers and Hamiltonian chaos

We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.     

Integrability and exact solution of correlated hopping multi-chain electron systems

Exact quantum integrability is established for a class of multi-chain electron models with correlated hopping and spin models with interchain interactions, by constructing the related Lax operators and R-matrices through twisting and gauge transformations. Exact solution of the eigenvalue problem for commuting conserved quantities of such systems is achieved through algebraic Bethe ansatz, on the examples of Hubbard and t–J models with correlated hopping. Our systematic construction identifies the integrable subclass of such known solvable models and also generates new systems including the generalized t–J models. At the same time it makes proper correction to a well known model and resolves recent controversies regarding the equivalence and solvability of some known models.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

Double Bracket Equations and Geodesic Flows on Symmetric Spaces

In this paper we consider the geometry of Hamiltonian flows on the cotangent bundle of coadjoint orbits of compact Lie groups and on symmetric spaces. A key idea here is the use of the normal metric to define the kinetic energy. This leads to Hamiltonian flows of the double bracket type. We analyze the integrability of geodesic flows according to the method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity. We demonstrate for example integrability of the geodesic flow on the real and complex Grassmannians. We also consider right invariant systems and the generalized rigid body equations in this setting. Received:23 July 1996 / Accepted: 16 December 1996     

Chaos-assisted tunneling and 1/falpha spectral fluctuations in the order-chaos transition

It has been shown that the spectral fluctuations of different quantum systems are characterized by 1/falpha noise, with 1< or =alpha< or =2, in the transition from integrability to chaos. This result is not well understood. We show that chaos-assisted tunneling gives rise to this power-law behavior. We develop a random matrix model for intermediate quantum systems, based on chaos-assisted tunneling, and we discuss under which conditions it displays 1/falpha noise in the transition from integrability to chaos. We conclude that the variance of the elements that connect regular with chaotic states must decay with the difference of energy between them. We compare the characteristics of the transition modeled in this way with what is obtained for the Robnik billiard.     

Quantum phase transitions in the interacting boson model: integrability, level repulsion, and level crossing

We study the quantum phase transition mechanisms that arise in the interacting boson model. We show that the second-order nature of the phase transition from U(5) to O(6) may be attributed to quantum integrability, whereas all the first-order phase transitions of the model are due to level repulsion with one singular point of level crossing. We propose a model Hamiltonian with a true first-order phase transition for finite systems due to level crossings.     

Topological insulators and C-algebras: Theory and numerical practice

We apply ideas from C^∗-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems.We use this approach to calculate the index for time-reversal invariant systems with spin–orbit scattering in three dimensions, on sizes up to 12³, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an “order parameter” for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C^∗-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.     

Thermodynamic Limit for a Spin Lattice

An integrable spin lattice is a higher dimensional generalization of integrable spin chains. In this paper we consider a special spin lattice related to quantum mechanical interpretation of the three-dimensional lattice model in statistical mechanics (Zamolodchikov and Baxter). The integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables. The significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters. There is a specific bilinear functional equation for the eigenvalues of this transfer matrix.The spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity. The limiting form of bilinear equation is derived. It allows to analyze the distributions of eigenvalues of the whole commuting set. The ground state distribution is obtained explicitly. A structure of excited states is discussed.     

Six-vertex model with rotated boundary conditions

We investigate the six-vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes, a model recently introduced by Litvin and Priezzhev. Auxiliary vertices are used to define an inhomogeneous system which leads to a one-parameter family of commuting transfer matrices. A product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system. Using an equation for commuting transfer matrices, we calculate their eigenvalues. Finite-size properties of the model are discussed from the viewpoint of the conformal field theory.     

Integrable spin–boson models descending from rational six-vertex models

We construct commuting transfer matrices for models describing the interaction between a single quantum spin and a single bosonic mode using the quantum inverse scattering framework. The transfer matrices are obtained from certain inhomogeneous rational vertex models combining bosonic and spin representations of SU(2)$\mathit{SU}(2)$, subject to non-diagonal toroidal and open boundary conditions. Only open boundary conditions are found to lead to integrable Hamiltonians combining both rotating and counter-rotating terms in the interaction. If the boundary matrices can be brought to triangular form simultaneously, the spectrum of the model can be obtained by means of the algebraic Bethe ansatz after a suitable gauge transformation; the corresponding Hamiltonians are found to be non-Hermitian. Alternatively, a certain quasi-classical limit of the transfer matrix is considered where Hermitian Hamiltonians are obtained as members of a family of commuting operators; their diagonalization, however, remains an unsolved problem.     

Black holes as generalised Toda molecules

In this note we compare the geodesic formalism for spherically symmetric black hole solutions with the black hole effective potential approach. The geodesic formalism is beneficial for symmetric supergravity theories since the symmetries of the larger target space lead to a complete set of commuting constants of motion that establish the integrability of the geodesic equations of motion, as shown in arXiv:1007.3209. We point out that the integrability lifts straightforwardly to the integrability of the equations of motion with a black hole potential. This construction turns out to be a generalisation of the connection between Toda molecule equations and geodesic motion on symmetric spaces known in the mathematics literature. We describe in some detail how this generalisation of the Toda molecule equations arises.     

Kostka matrices at the level of bases and the complete set of commuting Jucys-Murphy operators

A method for evaluation of Kostka matrices at the level of bases, and determination of related irreducible basis of the Weyl duality is proposed. The method bases on Jucys-Murphy operators which constitute a complete set of commuting Hermitian operators along the general Dirac formalism of quantum mechanics, applied to the algebra of a symmetric group. The way of construction of appropriate projection operators is pointed out, and the combinatorial meaning of the path on the Young graph, corresponding to a standard Young tableau, is made transparent.     

A Remark on the Notion of Independence of Quantum Integrals of Motion in the Thermodynamic Limit

Studies of integrable quantum many-body systems have a long history with an impressive record of success. However, surprisingly enough, an unambiguous definition of quantum integrability remains a matter of an ongoing debate. We contribute to this debate by dwelling upon an important aspect of quantum integrability—the notion of independence of quantum integrals of motion (QIMs). We point out that a widely accepted definition of functional independence of QIMs is flawed, and suggest a new definition. Our study is motivated by the PXP model—a model of N spins 1/2 possessing an extensive number of binary QIMs. The number of QIMs which are independent according to the common definition turns out to be equal to the number of spins, N. A common wisdom would then suggest that the system is completely integrable, which is not the case. We discuss the origin of this conundrum and demonstrate how it is resolved when a new definition of independence of QIMs is employed.

     

Quasi-linearization and stability analysis of some self-dual,dark equations and a new dynamical system†

We describe a class of self-dual dark nonlinear dynamical systems a priori allowing their quasi-linearization, whose integrability can be effectively studied by means of a geometrically based gradient-holonomic approach. A special case of the self-dual dynamical system, parametrically dependent on a functional variable is considered, and the related integrability condition is formulated. Using this integrability scheme, we study a new self-dual, dark nonlinear dynamical system on a smooth functional manifold, which models the interaction of atmospheric magneto-sonic Alfvén plasma waves. We prove that this dynamical system possesses a Lax representation that allows its full direct linearization and compatible Poisson structures. Moreover, for this self-dual nonlinear dynamical system we construct an infinite hierarchy of mutually commuting conservation laws and prove its complete integrability.     

Elliptic Quantum Groups and Ruijsenaars Models

We construct symmetric and exterior powers of the vector representation of the elliptic quantum groupsE _Τ,η(slN). The corresponding transfer matrices give rise to various integrable difference equations which could be solved in principle by the nested Bethe ansatz method. In special cases we recover the Ruijsenaars systems of commuting difference operators.     

Growth and integrability in the dynamics of mappings

The growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed. In the general case such characteristics as complexity by Arnold or the number of the different images for the multiple-valued mappings are growing exponentially. It is shown that the integrability is closely related with thepolynomial growth. The analogies with quantum integrable systems are discussed.