Semiclassical Trace Formula Research Articles

Symmetry reduction and semiclassical analysis of axially symmetric systems

We derive a semiclassical trace formula for a symmetry-reduced part of the spectrum in axially symmetric systems. The classical orbits that contribute are closed in $(\ensuremath{\rho}{,z,p}_{\ensuremath{\rho}}{,p}_{z})$ and have ${p}_{\ensuremath{\varphi}}=m\ensuremath{\Elzxh}$, where $m$ is the azimuthal quantum number. For $m\ensuremath{\ne}0$, these orbits vary with energy and almost never lie on periodic trajectories in the full phase space in contrast to the case of discrete symmetries. The transition from $m=0$ to $m>0$, however, is not as dramatic as our numerical results indicate, suggesting that contributing orbits occur in topologically equivalent families within which ${p}_{\ensuremath{\varphi}}$ varies smoothly.

Periodic orbits, breaktime and localization

The main aim of this paper is to realize that it is feasible to construct a `periodic orbit theory' of localization by extending the idea of classical action correlations. This possibility had been questioned by many researchers in the field of `quantum chaos'. Starting from the semiclassical trace formula, we formulate a quantal-classical duality relation that connects the spectral properties of the quantal spectrum to the statistical properties of lengths of periodic orbits. By identifying the classical correlation scale it is possible to extend the semiclassical theory of spectral statistics, in case of a complex systems, beyond the limitations that are implied by the diagonal approximation. We discuss the quantal dynamics of a particle in a disordered system. The various regimes are defined in terms of time-disorder `phase diagram'. As expected, the breaktime may be `disorder limited' rather than `volume limited', leading to localization if it is shorter than the ergodic time. Qualitative agreement with scaling theory of localization in one to three dimensions is demonstrated.

Berry’s phase, chaos, and the deformations of Riemann surfaces

Parametrized families of Landau Hamiltonians on compact Riemann surfaces corresponding to classically chaotic families of geodesic motion are investigated. The parameters describe deformations of such surfaces with genus $gg~1$. It is shown that the adiabatic curvature (responsible for Berry's phase) of the lowest Landau level for $gg1$ is the sum of two terms. The first term is proportional to the natural symplectic form on deformation space, and the second is a fluctuating term reflecting the chaos of the geodesic motion for $gg1$. For $g=1$ (integrable motion on the torus) we have no fluctuating term. We propose our result to be interpreted as a curvature analog of the well-known semiclassical trace formulas. Connections with the viscosity properties of quantum Hall fluids on such surfaces are also pointed out. An interesting possibility in this respect is the fractional quantization of certain components of the viscosity tensor of such fluids.

The role of singularities in chaotic spectroscopy

We review the status of the semiclassical trace formula with emphasis on the particular types of singularities that occur in the Gutzwiller-Voros zeta function for bound chaotic systems. To understand the problem better, we extend the discussion to include various classical zeta functions and we contrast properties of Axiom-A scattering systems with those of typical bound systems. Singularities in classical zeta functions contain topological and dynamical information, concerning, for example, anomalous diffusion, phase transitions among generalized Lyapunov exponents, power law decay of correlations. Singularities in semiclassical zeta functions are artifacts and enter because one neglects some quantum effects when deriving them, typically by making saddle point approximation when the saddle points are not separated enough. The discussion is exemplified by the Sinai billiard where intermittent orbits associated with neutral orbits induce a branch point in the zeta functions. This singularity is responsible for a diverging diffusion constant in Lorentz gases with unbounded horizon. In the semiclassical case there is interference between neutral orbits and intermittent orbits. The Gutzwiller-Voros zeta function exhibits a branch point because it does not take this effect into account. Another consequence is that individual states, high up in the spectrum, cannot be resolved by the Berry-Keating technique.

Periodic orbit theory of a circular billiard in homogeneous magnetic fields

We present a semiclassical description of the level density of a two-dimensional circular quantum dot in a homogeneous magnetic field. We model the total potential (including electron-electron interaction) of the dot containing many electrons by a circular billiard, i.e., a hard-wall potential. Using the extended approach of the Gutzwiller theory developed by Creagh and Littlejohn, we derive an analytic semiclassical trace formula. For its numerical evaluation we use a generalization of the common Gaussian smoothing technique. In strong fields orbit bifurcations, boundary effects (grazing orbits) and diffractive effects (creeping orbits) come into play, and the comparison with the exact quantum mechanical result shows major deviations. We show that the dominant corrections stem from grazing orbits, the other effects being much less important. We implement the boundary effects, replacing the Maslov index by a quantum-mechanical reflection phase, and obtain a good agreement between the semiclassical and the quantum result for all field strengths. With this description, we are able to explain the main features of the gross-shell structure in terms of just one or two classical periodic orbits.

A Semi-classical Trace Formula for Schrödinger Operators in the Case of a Critical Energy Level

LetĤ=−(ℏ2/2)Δ+V(x) be a Schrödinger operator on Rn, with smooth potentialV(x)→+∞ as |x|→+∞. The spectrum ofĤis discrete, and one can study the asymptotic of the smoothed spectral densityϒ(E,ℏ)=∑kϕEk(ℏ)−Eℏ,as ℏ→0. Here, {Ek(ℏ)}k∈Nis the spectrum ofĤand ϕ∈C∞0(R). We shall investigate the case whereEis a critical value of the symbolHofĤand, extending the work of Brummelhuis, Paul and Uribe in [3], we will prove the existence of a full asymptotic expansion forϒin terms ofℏand lnℏ and compute the leading coefficient. We will consider new Weyl-type estimates for the counting function:NEc,ρ(ℏ)=#{k∈N/|Ek(ℏ)−E|⩽ρℏ}.

Accuracy of trace formulas

Using quantum maps, we study the accuracy of semiclassical trace formulas. The role of chaos in improving the semiclassical accuracy in some systems is demonstrated quantitatively. However, our study of the standard map cautions that this may not be most general. While studying a sawtooth map we demonstrate the rather remarkable fact that at the level of the time one trace even in the presence of fixed points on singularities the trace formula may be exact, and in any case has no logarithmic divergences observed for the quantum bakers map. As a byproduct we introduce fantastic periodic curves akin to curlicues.

Quantum corrections to the semiclassical level density of the circular disk in homogeneous magnetic fields

In the semiclassical trace formula for the level density of a circular disk in homogeneous magnetic fields, quantum corrections to the Maslov phase have been shown to be important in strong fields. In this article further quantum corrections are considered, namely, grazing corrections which are relevant for whispering-gallery orbits, and a uniform approximation to the bifurcation points in strong fields is applied. Both corrections are shown to have a surprisingly small effect on the semiclassical level density. Implementing those corrections requires a technique different from the common Gaussian smoothing in the numerical evaluation of the trace formula. The appropriate generalization is presented. © 1997 Elsevier Science B.V. All rights reserved.

Near-integrable systems: Resonances and semiclassical trace formulas.

Trace formulas relate the quantum density of states to the properties of the periodic orbits of the underlying classical system. The resulting expressions depend critically on the nature of the dynamics and whether the orbits are stable or unstable. Several open questions exist for the class of classical systems that are near integrability. The most important consequence of a generic perturbation to an integrable system is the creation of resonances. We derive generalized expressions appropriate for resonances and apply them to a system that can be taken as a paradigm for the transition from regular to chaotic dynamics. \textcopyright{} 1996 The American Physical Society.

Simple metal clusters in magnetic fields.

The electronic supershell structure of simple metal clusters in a uniform magnetic field is studied with a spherical infinite-well potential as a simple yet realistic model for confining the valence electrons. It is found that there is little perceivable change in the supershells with experimentally achievable field strengths. If yet stronger fields are applied, the supershell structure gets destroyed and then new beat structures come about as the field strength increases further. These phenomena are explained by a recently derived semiclassical trace formula for broken symmetry that has been applied to the present system. \textcopyright{} 1996 The American Physical Society.

Periodic orbits, bifurcation diagrams and the spectroscopy of C2H2 system

The principal families of periodic orbits that emerge from the stationary points of the six-dimensional potential energy surface of the C2H2 molecular system, as well as periodic orbits from saddle-node bifurcations, have been located and propagated for an energy range up to 36 500 cm−1 above the absolute minimum of the potential. The bifurcation diagrams of these periodic orbits reveal the regions of phase space where the dynamics are regular or chaotic (with soft or hard chaos) for acetylene, vinylidene, and the region over these two isomers. An association of the structure of phase space with spectroscopic findings is made by calculating Gutzwiller’s semiclassical trace formula and classical survival probability functions.

The Semi-Classical Trace Formula and Propagation of Wave Packets

We study spectral and propagation properties of operators of the form Sh = ∑Nj=0hjPj where ∀ j P j is a differential operator of order j on a manifold M, asymptotically as h → 0. The estimates are in terms of the flow {φt} of the classical Hamiltonian H(x, p) = ∑Nj=0 σPj(x, p) on T*M, where σPj, is the principal symbol of P j. We present two sets of results. (I) The "semiclassical trace formula", on the asymptotic behavior of eigenvalues and eigenfunctions of Sh in terms of periodic trajectors of H. (II) Associated to certain isotropic submanifolds Λ ⊂ T*M we define families of functions {ψh} and prove that ∀t {exp(− ithSħ)(ψħ)} is a family of the same kind associated to φt(Λ).

Periodic orbit asymptotics for intermittent Hamiltonian systems

We address the problems in applying cycle expansions to bound chaotic systems, caused by e.g. intermittency and incompleteness of the symbolic dynamics. We discuss zeta functions associated with weighted evolution operators and in particular a one-parameter family of weights relevant for the calculation of classical resonance spectra, semiclassical spectra and topological entropy. For bound intermittent system we discuss an approximation of the zeta function in terms of probabilities rather than cycle instabilities. This approximation provides a generalization of the fundamental part of a cycle expansion for a finite subshift symbolic dynamics. This approach is particularly suitable for determining asymptotic properties of periodic orbits which are essential for scrutinizing the semiclassical limit of Gutzwiller's semiclassical trace formula. The Sinai billiard is used as model system. In particular we develope a crude approximation of the semiclassical zeta function which turns out to possess non analytical features. We also discuss the contribution to the semiclassical level density from the neutral orbits. Finally we discuss implications of our findings for the spectral form factor and compute the asymptotic behaviour of the spectral rigidity. The result is found to be consistent with exact quantum mechanical calculations.

Chaotic scattering on C4v four-disk billiards: Semiclassical and exact quantum theories.

We present the semiclassical and exact quantum theories of a point particle bouncing in ${\mathit{C}}_{4\mathit{v}}$ four-disk billiards, which exhibit chaotic scattering in their classical dynamics. The ${\mathit{A}}_{1}$ sector of the \ensuremath{\zeta} function in the semiclassical trace formula is calculated by using cycle expansions with periodic orbits up to period three in the number of bounces. Its complex poles provide resonances that are in excellent agreement with the exact quantum resonances in a wide range of wave numbers k. Assemblies of resonances lying parallel to the real k axis indicate that the distribution of the imaginary parts of the resonances presents a threshold, and consists of a sequence of small bands below the threshold. The consequences of these resonances on the dynamical behavior of semiconductor mesoscopic devices are briefly discussed.

Correlations in the actions of periodic orbits derived from quantum chaos.

We discuss two-point correlations of the actions of classical periodic orbits in chaotic systems. For systems where the semiclassical trace formula is exact and the spectral statistics follow random matrix theory, there exist nontrivial correlations between actions, which we express in a universal form. We illustrate this result with the analogous problem of the pair correlations between prime numbers. We also report on numerical studies of three chaotic systems where the semiclassical trace formula is only approximate, but nevertheless these unexpected action correlations are observed.

Shell structure in faceted metal clusters.

We study the quantized electronic energy levels in a three-dimensional icosahedral billiard modeling a faceted metal cluster. The first 2000 levels are determined numerically. The magic numbers are compared with experimental data and with the results for a spherical model. We discuss the supershell structure and propose its study as a test of cluster sphericity. We compare our results with the predictions of the semiclassical trace formula and point out the relevance of diffractive orbits.

Non-generic spectral statistics in the quantized stadium billiard

We consider the effect of a continuous family of neutral (bouncing ball) orbits on the energy spectrum of the quantized stadium billiard. Using a semiclassical approximation we derive analytic expressions for standard two-point spectral measures. The corrections due to the bouncing ball orbits account for some of the non-generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured. Once the bouncing ball contributions are subtracted, the spectrum is shown to be well reproduced by the semi-classical trace formula based on unstable periodic orbits. We also study special patterns in the spectrum which are due to other non-generic features such as edge effects and 'whispering gallery' trajectories.

Influence of intermittency on quantum spectra

Bound chaotic systems typically exhibit intermittency. Some semiclassical properties of intermittency are studied. The hyperbola billiard and the x2y2 model are used as strongly intermittent model systems. The almost integrable motion in the arms of the potential may be treated in the adiabatic approximation. The corresponding adiabatic Hamiltonian may be semiclassically quantized, yielding surprisingly good agreement for the hyperbola billiard. It is then demonstrated, by means of the semiclassical trace formula, how this result may be related to families of periodic orbit exclusively exploring the potential arms. It is discussed how this result implies an integrable component in the spectrum. Possible implications for the resummation problem of the trace formula, as well as for the statistical properties of energy levels, are discussed.

Periodic orbits and semiclassical quantization of dispersing billiards

Periodic orbits in a dispersing billiard system consisting of three circular arcs are studied numerically by using a partial coding rule together with an efficient method for enumerating periodic orbits on the real billiard plane. By examining several statistical measures, it is shown that the length spectrum and the stability exponents are highly uncorrelated. The validity of the semiclassical trace formula is also tested, and a remarkable agreement of the semiclassical and quantum density of states is obtained at least for about the lower 15 levels.

Semiclassical principal symbols and Gutzwiller's trace formula

Gutzwiller's semiclassical trace formula gives an asymptotic expansion of the spectral density for Schrödinger operators as an infinite sum over all periodic orbits of the corresponding classical system. We present a rigorous version of this result, based on concepts from the theory of Fourier We present a rigorous version of this result, based on concepts from the theory of Fourier integral operators.