Quantum Monodromy Research Articles

Monodromy of the quantum 1:1:2 resonant swing spring

We describe the qualitative features of the joint spectrum of the quantum 1:1:2 resonant swing spring. The monodromy of the classical analogue of this problem is studied in Dullin et al. [Physica D 190, 15–37 (2004)]. Using symmetry arguments and numerical calculations we compute its three-dimensional (3D) lattice of quantum states and show that it possesses a codimension 2 defect characterized by a nontrivial 3D-monodromy matrix. The form of the monodromy matrix is obtained from the lattice of quantum states and depends on the choice of an elementary cell of the lattice. We compute the quantum monodromy matrix, that is the inverse transpose of the classical monodromy matrix. Finally we show that the lattice of quantum states for the 1:1:2 quantum swing spring can be obtained—preserving the symmetries—from the regular 3D-cubic lattice by means of three “elementary monodromy cuts.”

Loop Operators and the Kondo Problem

We analyse the renormalisation group flow for D-branes in WZW models from the point of view of the boundary states. To this end we consider loop operators that perturb the boundary states away from their ultraviolet fixed points, and show how to regularise and renormalise them consistently with the global symmetries of the problem. We pay particular attention to the chiral operators that only depend on left-moving currents, and which are attractors of the renormalisation group flow. We check (to lowest non-trivial order in the coupling constant) that at their stable infrared fixed points these operators measure quantum monodromies, in agreement with previous semiclassical studies. Our results help clarify the general relationship between boundary transfer matrices and defect lines, which parallels the relation between (non-commutative) fields on (a stack of) D-branes and their push-forwards to the target-space bulk.

CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy.

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.

Quantum monodromy for diatomic molecules in combined electrostatic and pulsed nonresonant laser fields

We investigate the classical and quantum mechanics of diatomic molecules in combined electrostatic and pulsed laser fields. The integrable case of collinear static and linearly polarized laser fields exhibits both classical and quantum monodromy, and energy–momentum diagrams are presented for several physically relevant field combinations. Although the tilted field problem is nonintegrable, a quantum 〈 H ̂ 〉 : 〈m 2〉 lattice for eigenstates retains much of the structure of the integrable limit and is organized by classical periodic orbits. The regular structure of the quantum lattice in the integrable limit is disrupted in an energy range associated with the onset of chaotic classical motion.

Quantum monodromy in the two-centre problem

Using modern tools from the geometric theory of Hamiltonian systems it is shown that electronic excitations in diatoms which can be modelled by the two-centre problem exhibit a complicated case of classical and quantum monodromy. This means that there is an obstruction to the existence of global quantum numbers in these classically integrable systems. The symmetric case of H+2 and the asymmetric case of H He++ are explicitly worked out. The asymmetric case has a non-local singularity causing monodromy. It coexists with a second singularity which is also present in the symmetric case. An interpretation of monodromy is given in terms of the caustics of invariant tori.

Algebraic Bethe ansatz for a quantum integrable derivative nonlinear Schrödinger model

We find that the quantum monodromy matrix associated with a derivative nonlinear Schrödinger (DNLS) model exhibits U(2) or U(1,1) symmetry depending on the sign of the related coupling constant. By using a variant of quantum inverse scattering method which is directly applicable to field theoretical models, we derive all possible commutation relations among the operator valued elements of such monodromy matrix. Thus, we obtain the commutation relation between creation and annihilation operators of quasi-particles associated with DNLS model and find out the S-matrix for two-body scattering. We also observe that, for some special values of the coupling constant, there exists an upper bound on the number of quasi-particles which can form a soliton state for the quantum DNLS model.

Quantum Monodromy in the Spectrum of Schrödinger Equation with a Decatic Potential

In this study the spectral problem of the two-dimensional Schrodinger equation with the cylindrically symmetrical decatic potential is carried out. The concept of quantum monodromy is introduced to give insight into the energy levels of system with this potential. It is shown that quantum monodromy occurs at ρ = 0 in the distribution of eigenstates around a critical point on the spectrum at E = 0 with zero angular momentum, such that there can be no smoothly valid assignment of quantum number. Cases with the three-well and four-well potentials are presented to give rise to the double degeneracies with respect to energy except for the angular momentum m = 0.

Quantum Monodromy in Prolate Ellipsoidal Billiards

This is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal boundaries. In the present paper we show that the integrable billiard inside a prolate ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits classical and quantum monodromy. We derive the monodromy matrix from the requirement of smoothness for the action variables for zero angular momentum. The smoothing procedure is illustrated in terms of energy surfaces in action space including the corresponding smooth frequency map. The spectrum of the quantum billiard is computed numerically and the expected change in the basis of the lattice of quantum states is found. The monodromy is already present in the corresponding two-dimensional billiard map. However, the full three degrees of freedom billiard is considered as the system of greater relevance to physics. Therefore, the monodromy is discussed as a truly three-dimensional effect.

Quantum monodromy and semi-classical trace formulæ

We consider an h-pseudodifferential operator whose symbol has a closed Hamiltonian trajectory. There exists a Fourier integral operator which quantizes in a natural way the Poincaré map. With the help of this monodromy operator, we give a trace formula which leads to a new proof of the trace formula of Duistermaat–Guillemin and Gutzwiller.

Monodromy as topological obstruction to global action-angle variables in systems with coupled angular momenta and rearrangement of bands in quantum spectra

We continue to study the relation between the redistribution of levels in quantum energy spectra and monodromy of the corresponding classical system [see Phys. Lett. A 256, 235 (1999)]. A system with two coupled angular momenta and $\mathrm{SO}(2)$ symmetry is compared to a system with three momenta and $\mathrm{SO}(3)$ symmetry.

From the braided to the usual Yang-Baxter relation

Quantum monodromy matrices coming from a theory of two coupled (m)KdV equations are modified in order to satisfy the usual Yang-Baxter relation. As a consequence, a general connection between braided and {\it unbraided} (usual) Yang-Baxter algebras is derived and also analysed.

Quantum Monodromy and Bohr–Sommerfeld Rules

This article is based on a talk given in Dijon for the 2000 summer school ‘Topological and Geometrical Methods: Applications to Dynamical Systems’. The standard definitions of the monodromy invariant for completely integrable classical systems are reviewed, and the link to the quantum monodromy observed in the joint spectrum of commuting operators is explained. The mathematical treatment relies on a modern and attractive version of the Bohr–Sommerfeld quantisation rules.

Spectroscopic signatures of bond-breaking internal rotation. II. Rotation-vibration level structure and quantum monodromy in HCP

The rotation-vibration level structure of ground electronic state HCP is investigated at vibrational energies approaching and exceeding that of the linear CPH saddle point. With respect to energies above the saddle point, we investigate possible spectroscopic manifestations of strong Coriolis interactions between the hindered, bond-breaking internal rotation of the hydrogen about the CP core and the rotation of the molecule in the space-fixed axis system. With respect to energies below the saddle point, we provide new interpretations, from quantum and semiclassical points of view, of previously observed anomalously large B (rotational) and g22 (energy dependence on the vibrational angular momentum) constants for the large-amplitude pure bending states of HCP (referred to elsewhere as “isomerization” or saddle node states). We also predict similar anomalies in other spectroscopic constants, including the “centrifugal distortion” constant D and the “rotational l-resonance” parameter q2. These changes in the effective spectroscopic rotation-vibration constants are shown to be a direct consequence of the spherical pendulum topology of the HCP bend/internal rotor system, which is associated with a phenomenon called quantum monodromy, defined as the absence of a smoothly valid set of quantum numbers for all states. Our semiempirical model for the HCP bend/internal rotor mode is derived using principles of semiclassical inversion and provides new insights into the breakdown in the ability of rovibrational effective Hamiltonians to model highly vibrationally excited states of HCP.

Quantum states of a sextic potential: hidden symmetry and quantum monodromy

A known terminating polynomial ansatz for a finite sub-set of states of the sextic central potential V(a,b,c;r) = ar2 + br4 + cr6, c>0, subject to integer related parameter constraints, is confirmed to apply in two as well as three dimensions. A hidden symmetry relating the eigenstates of V(a,b,c;r) to those of V(a,-b,c;r) is used to establish the boundary between the ansatz sub-set and the infinite set of higher states. Connections between the radial wavefunctions Rn(a,b,c;r) and Rn'(a,-b,c;r) also provide semiclassical insight into the origin of the parameter constraint. Finally the ansatz eigenvalue dispositions are related to the phenomenon of quantum monodromy.

On the quantum inverse scattering problem

A general method for solving the so-called quantum inverse scattering problem (namely the reconstruction of local quantum (field) operators in term of the quantum monodromy matrix satisfying a Yang–Baxter quadratic algebra governed by an R-matrix) for a large class of lattice quantum integrable models is given. The principal requirement being the initial condition ( R(0)= P, the permutation operator) for the quantum R-matrix solving the Yang–Baxter equation, it applies not only to most known integrable fundamental lattice models (such as Heisenberg spin chains) but also to lattice models with arbitrary number of impurities and to the so-called fused lattice models (including integrable higher spin generalizations of Heisenberg chains). Our method is then applied to several important examples like the sl n XXZ model, the XYZ spin- 1 2 chain and also to the spin- s Heisenberg chains.

The reconstruction of local quantum operators for the boundaryXXZspin-½ Heisenberg chain

The method of the reconstruction of local quantum operators in terms of the elements of the quantum monodromy matrix is applied to the XXZ spin-½ Heisenberg chain with integrable boundary conditions.

Form factors of the XXZ Heisenberg [formula omitted] finite chain

Form factors for local spin operators of the XXZ Heisenberg spin-z finite chain are computed. Representation theory of Drinfel'd twists in terms of F-matrices for the quantum affine algebra U q( s ̂ l 2) in finite-dimensional modules is used to calculate scalar products of two states involving one Bethe state (leading to Gaudin formula) and to solve the quantum inverse problem for local spin operators in the finite chain. Hence, we obtain the representation of the n-spin correlation functions in terms of expectation values (in ferromagnetic reference state) of the operator entries of the quantum monodromy matrix satisfying Yang-Baxter algebra. This leads to the direct calculation of the form factors of the XXZ Heisenberg spin-! finite chain as determinants of usual functions of the parameters of the model. A two-point correlation function for adjacent sites is also derived using similar techniques.

Quantum Monodromy in Integrable Systems

Let P1(h),...,Pn(h) be a set of commuting self-adjoint h-pseudo-differential operators on an n-dimensional manifold. If the joint principal symbol p is proper, it is known from the work of Colin de Verdiere [6] and Charbonnel [3] that in a neighbourhood of any regular value of p, the joint spectrum locally has the structure of an affine integral lattice. This leads to the construction of a natural invariant of the spectrum, called the quantum monodromy. We present this construction here, and show that this invariant is given by the classical monodromy of the underlying Liouville integrable system, as introduced by Duistermaat [9]. The most striking application of this result is that all two degree of freedom quantum integrable systems with a focus-focus singularity have the same non-trivial quantum monodromy. For instance, this proves a conjecture of Cushman and Duistermaat [7] concerning the quantum spherical pendulum.

Spectroscopy from first principles: a breakthrough in water line assignments

Variational calculations of the spectrum of both hot and room temperature water vapor have led to a major increase in the number and scope of observed vibration–rotation transitions that have full quantum number assignments. The calculations are performed using high accuracy ab initio potential energy surfaces. As demonstrated, it is also necessary to consider corrections due to adiabatic effects, non-adiabatic effects and the relativistic motions of the electrons. The result of this work has been to double the number and energy range of the measured energy levels for water. New energy levels for the (010), (100), (020), (001), (110), (030), (011), (040), (050) and (060) states are presented here. Analysis of these levels and the associated transitions shows unanticipated features, including rotational difference bands, and an absence of clustering. The prospects of observing quantum monodromy in the spectrum of water are discussed.

Quantum monodromy in the spectrum of H2O and other systems: new insight into the level structure of quasi-linear molecules

The concept of quantum monodromy is introduced to give insight into the energy levels of systems with cylindrically symmetrical potential energy barriers. The K structure of bending progressions of bent molecules and the pendular states of dipolar molecules in strong electric fields are taken as molecular examples. Results are given for a two-dimensional champagne bottle model, for six computed bending progressions for water taken from the compilation of Partridge and Schwenke and for the quantal spherical pendulum. Sharp changes in the energy level patterns around the barrier energy are related to changes in the forms of the relevant classical trajectories. Analytical insight into the mathematical origin of the monodromy are also given.