Notes on index of quantum integrability

A quantum integrability index was proposed in Komatsu et al(2019 SciPost Phys. 7 065). It systematizes the Goldschmidt and Witten's operator counting argument(Goldschmidt and Witten 1980 Phys. Lett. B 91 392) by using the conformal symmetry. In this work we compute the quantum integrability indexes for the symmetric coset models SU(N)/SO(N) and SO(2N)/SO(N)×SO(N). The indexes of these theories are all non-positive except for the case of SO(4)/SO(2)×SO(2). Moreover we extend the analysis to the theories with fermions and consider a concrete theory: the CP~N model coupled with a massless Dirac fermion. We find that the indexes for this class of models are non-positive as well.     

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.     

Manifestation of destruction of quantum integrability with expectation and uncertainty values of quantum observables

A method to demonstrate destruction of quantum integrability is presented. The method is based on the calculation (for systems with two degrees of freedom) of the expectation and uncertainty values of energy and another quantum observable which is conservative when the system is integrable. Two kinds of avoided crossings can be shown clearly with this method.     

Quantum NOT operation and integrability in two-level systems

We demonstrate the surprising integrability of the classical Hamiltonian associated to a spin 1/2 system under periodic external fields. The one-qubit rotations generated by the dynamical evolution is, on the one hand, close to that of the rotating wave approximation (RWA), on the other hand to two different “average” systems, according to whether a certain parameter is small or large. Of particular independent interest is the fact that both the RWA and the averaging theorem are seen to hold well beyond their expected region of validity. Finally, we determine conditions for the realization of the quantum NOT operation by means of classical stroboscopic maps.     

More on the quantum integrability for the one-dimensional Bariev chain

The transfer matrix, i.e. the generating functional for an infinite number of conserved currents, is written down for the integrable model of arbitrary s interacting XY chains proposed by Bariev. This is achieved by identifying the first non-trivial converved current next to the Hamiltonian. Further, the corresponding quantum R matrix is explicitly constructed in the case of s = 3.     

Effective field theory and integrability in two-dimensional Mott transition

We study the Mott transition in a two-dimensional lattice spinless fermion model with nearest neighbors density–density interactions. By means of a two-dimensional Jordan–Wigner transformation, the model is mapped onto the lattice XXZ spin model, which is shown to possess a quantum group symmetry as a consequence of a recently found solution of the Zamolodchikov tetrahedron equation. A projection (from three to two space–time dimensions) property of the solution is used to identify the symmetry of the model at the Mott critical point as , with deformation parameter q = −1. Based on this result, the low-energy effective field theory for the model is obtained and shown to be a lattice double Chern–Simons theory with coupling constant k = 1 (with the standard normalization). By further employing the effective filed theory methods, we show that the Mott transition that arises is of topological nature, with vortices in an antiferromagnetic array and matter currents characterized by a d-density wave order parameter. We also analyze the behavior of the system upon weak coupling, and conclude that it undergoes a quantum gas–liquid transition which belongs to the Ising universality class.     

Integrability of the S-matrix versus integrability of the Hamiltonian

This report reviews the relations between the integrability properties of the S-matrix and of the Hamiltonian. Particular emphasis is put on the situation where the Hamiltonian has a conserved quantity which is not compatible with the asymptotics and where correspondingly the integrability does not transfer to the S-matrix. As questions of integrability are more readily handled in classical dynamics, all developments are first performed classically. Several examples are discussed to illustrate the main points. The quantum mechanical discussion reveals that the eigenphase statistics of the S-matrix depends principally on the chaoticity of the scattering map while basis dependent quantities such as the distribution of matrix elements tend to have random matrix behaviour only in the presence of topological chaos. The relevance of these considerations to the evaluation of scattering data is discussed.     

Breakdown of integrability in a quasi-1D ultracold bosonic gas

We demonstrate that virtual excitations of higher radial modes in an atomic Bose gas in a tightly confining waveguide result in effective three-body collisions that violate integrability in this quasi-one-dimensional quantum system and give rise to thermalization. The estimated thermalization rates are consistent with recent experimental results in quasi-1D dynamics of ultracold atoms.     

Classical and quantum integrability for a class of potentials in two dimensions

A method for the construction of the second constant of motion in fourth order is carried out. Correspondingly the fourth order potential equation is obtained whose solutions directly provide the classical integrable systems. Second constant of motion is obtained for a large class of potentials. Quantum invariants are also obtained with second order quantum corrections of the order O(?²) to the corresponding classical invariants. The phase space diagrams for these cases are drawn using a mathematical computer software package in two dimensions.     

Statistical properties of inter-series mixing in helium: from integrability to chaos

The photoionization spectrum of helium shows considerable complexity close to the double-ionization threshold. By analyzing the results from both our recent experiments and ab initio three- and one-dimensional calculations, we show that the statistical properties of the spacings between neighboring energy levels clearly display a transition towards quantum chaos.     

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.     

Hints on integrability in the Wilsonian/holographic renormalization group

The Polchinski equations for the Wilsonian renormalization group in the D-dimensional matrix scalar field theory can be written at large N in a Hamiltonian form. The Hamiltonian defines evolution along one extra holographic dimension (energy scale) and can be found exactly for the subsector of Trϕ ⁿ (for all n) operators. We show that at low energies independently of the dimensionality D the Hamiltonian system in question reduces to the integrable effective theory. The obtained Hamiltonian system describes large wavelength KdV type (Burger-Hopf) equation with an external potential and is related to the effective theory obtained by Das and Jevicki for the matrix quantum mechanics.     

Solution of then-channel Kondo problem (scaling and integrability)

The exact solution of the Kondo model forn-flavours of electrons with the spin 1/2 scattered by theS-spin impurity is presented. Forn=2S=5 the model describes manganese impurities dissolved in a metal. It is shown that atn>2S the effective exchange coupling approaches a finite fixed point as the energy scale decreases. It means that atn>2S the Gell-Mann-Low function turns to zero in this point and the scaling behaviour of physical quantities is observed. The scaling behaviour, first obtained in the 1D quantum field theory, can be analyzed on the basis of the exact solution. In the casen≦2S the effective coupling becomes infinitely strong at low energies.     

Edge states tunneling in the fractional quantum Hall effect: integrability and transport

This is a short review of nonperturbative techniques that have been used in the past 5 years to study transport out of equilibrium in low dimensional, strongly interacting systems of condensed matter physics. These techniques include massless factorized scattering, the generalization of the Landauer Büttiker approach to integrable quaisparticles, and duality. The case of tunneling between edges in the fractional quantum Hall effect is discussed in details. To cite this article: H. Saleur, C. R. Physique 3 (2002) 685–695.     

Exact solution of the multichannel Kondo problem,scaling, and integrability

In this work we give the exact solution of the model describing the scattering of conduction electrons by an impurity in the orbital singlet state (so-calledn- channel Kondo problem). Depending on the relation between the impurity spinS and the number of electron scattering channelsn, the model behaves differently at low energies. At 2S the effective charge increases to infinity at low energies, whereas atn > 2S it tends to a finite fixed point. The model under study is the first example of the one-dimensional quantum field theory exhibiting scaling behavior.     

A new perspective on the integrability of Inozemtsev’s elliptic spin chain

The aim of this paper is studying from an alternative point of view the integrability of the spin chain with long-range elliptic interactions introduced by Inozemtsev. Our analysis relies on some well-established conjectures characterizing the chaotic vs. integrable behavior of a quantum system, formulated in terms of statistical properties of its spectrum. More precisely, we study the distribution of consecutive levels of the (unfolded) spectrum, the power spectrum of the spectral fluctuations, the average degeneracy, and the equivalence to a classical vertex model. Our results are consistent with the general consensus that this model is integrable, and that it is closer in this respect to the Heisenberg chain than to its trigonometric limit (the Haldane–Shastry chain). On the other hand, we present some numerical and analytical evidence showing that the level density of Inozemtsev’s chain is asymptotically Gaussian as the number of spins tends to infinity, as is the case with the Haldane–Shastry chain. We are also able to compute analytically the mean and the standard deviation of the spectrum, showing that their asymptotic behavior coincides with that of the Haldane–Shastry chain.