Periodic Orbit Sum Research Articles

Many-Body Quantum Chaos: Analytic Connection to Random Matrix Theory

A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor $K(t)$ from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute $K(t)$ explicitly in the leading two orders in $t$ and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.

An exact periodic-orbit formula for the energy levels of the three-pronged star graph

We derive an exact periodic-orbit expansion formula for the energy levels of the three-pronged star graph. In addition, we present a proof of the ‘one-root-per-root-cell’ property and derive exact periodic-orbit sum rules. A spin-off of our periodic-orbit formulae is novel arithmetic series for π2.

The Trace Formula for Quantum Graphs with General Self Adjoint Boundary Conditions

We consider compact metric graphs with an arbitrary self adjoint realisation of the differential Laplacian. After discussing spectral properties of Laplacians, we prove several versions of trace formulae, relating Laplace spectra to sums over periodic orbits on the graph. This includes trace formulae with, respectively, absolutely and conditionally convergent periodic orbit sums; the convergence depending on properties of the test functions used. We also prove a trace formula for the heat kernel and provide small-t asymptotics for the trace of the heat kernel.

Deterministic diffusion in one-dimensional maps—calculation of diffusion constants by harmonic inversion of periodic orbit sums

A method is proposed for the calculation of diffusion constants for one-dimensional maps exhibiting deterministic diffusion. The procedure is based on harmonic inversion and uses a known relation between the diffusion constant and the periodic orbits of a map. The method is tested on an example map for which results calculated by different other techniques are available for comparison.

Numerical study of length spectra and low-lying eigenvaluespectra of compact hyperbolic 3-manifolds

In this paper, we investigatenumerically the length spectra and thelow-lying eigenvalue spectra of the Laplace-Beltrami operator for alarge number of small compact (closed) hyperbolic(CH) 3-manifolds. The first non-zero eigenvalues have beencomputedsuccessfully using the periodic orbit sum method, which are compared withvarious geometric quantities such asthe volume, diameter and length of theshortest periodic geodesic of the manifolds. The deviation of low-lyingeigenvalue spectra of manifolds converging to a cusped hyperbolicmanifold from the asymptotic distribution has been measuredusing the ζ-function andspectral distance.

Periodic Orbit Quantization: How to make Semiclassical Trace Formulae Convergent

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of the periodic orbit signal are the semiclassical eigenvalues, and are determined by either linear predictor, Pade approximant, or signal diagonalization. The second method is based upon the direct application of the Pade approximant to the periodic orbit sum. The Pade approximant allows the resummation of the, typically exponentially, divergent periodic orbit terms. Both techniques do not depend on the existence of a symbolic dynamics, and can be applied to bound as well as to open systems. Numerical results are presented for two different systems with chaotic and regular classical dynamics, viz. the three-disk scattering system and the circle billiard.

Phase space localization of chaotic eigenstates: violating ergodicity.

The correlation between level velocities and eigenfunction intensities provides a new way of exploring phase space localization in quantized nonintegrable systems. It can also serve as a measure of deviations from ergodicity due to quantum effects for typical observables. This paper relies on two well known paradigms of quantum chaos, the bakers map and the standard map, to study correlations in simple, yet chaotic, dynamical systems. The behaviors are dominated by the presence of several classical structures. These primarily include short periodic orbits and their homoclinic excursions. The dependences of the correlations deriving from perturbations allow for eigenfunction features violating ergodicity to be selectively highlighted. A semiclassical theory based on periodic orbit sums leads to certain classical correlations that are superexponentially cut off beyond a logarithmic time scale. The theory is seen to be quite successful in reproducing many of the quantum localization features.

Spectral Statistics of the Bootstrapped Density of States in the d-Dimensional Integrable Billiard System

The bootstrapped density of states introduced by Bogomolny and Keating for classically chaotic systems is applied to integrable systems, and the condition which leads to the universal nature of correlation between levels is investigated. We consider the two-point correlation function of the d-dimensional billiard system. We find the condition between the dimension and the periodic orbit sum in the semiclassical trace formula under which condition the correlation is well described by the Poisson statistics.

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: Firstly, the periodic orbit quantization will be extended to include higher order hbar corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained.

Periodic-orbit theory of anderson localization on graphs

We present the first quantum system where Anderson localization is completely described within periodic-orbit theory. The model is a quantum graph analogous to an aperiodic Kronig-Penney model in one dimension. The exact expression for the probability to return to an initially localized state is computed in terms of classical trajectories. It saturates to a finite value due to localization, while the diagonal approximation decays diffusively. Our theory is based on the identification of families of isometric orbits. The coherent periodic-orbit sums within these families, and the summation over all families, are performed analytically using advanced combinatorial methods.

Decimation and harmonic inversion of periodic orbit signals

We present and compare three generically applicable signal processing methods for periodic orbit quantization via harmonic inversion of semiclassical recurrence functions. In a first step of each method, a band-limited decimated periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. In a second step, the frequencies and amplitudes of the decimated signal are determined by either Decimated Linear Predictor, Decimated Pade Approximant, or Decimated Signal Diagonalization. These techniques, which would have been numerically unstable without the windowing, provide numerically more accurate semiclassical spectra than does the filter-diagonalization method.

The construction of a quantum Markov partition

We present a method for constructing a quantum Markov partition. Its elements are obtained by quantizing the characteristic function of the classical rectangles. The result is a set of quantum operators which behave asymptotically as projectors over the classical rectangles apart from edge and corner effects. We investigate their spectral properties and different methods of construction. The quantum partition is shown to induce a symbolic decomposition of the quantum evolution operator. In particular, an exact expression for the traces of the propagator is obtained having the same structure as the Gutzwiller periodic orbit sum.

Periodic orbit sum rules for billiards: accelerating cycle expansions

We show that the periodic orbit sums for two-dimensional billiards satisfy an infinity of exact sum rules. We demonstrate their utility by using the flow conservation sum rule to accelerate the convergence of cycle expansions for the overlapping three-disc billiard. The effectiveness of the approach is studied by applying the method on averages, known explicitly by other sum rules. The method is then applied to the Lyapunov exponent.

Semiclassical non-trace-type formulas for matrix-element fluctuations and weighted densities of states.

Densities of states weighted with the diagonal matrix elements of two operators A and B, i.e., rho(A,B)(E)= summation operator(n)<nAn><nBn>delta(E-E(n)), cannot, in general, be written as a trace formula, and therefore no simple extension of semiclassical trace formulas is known for this case. However, from the high resolution analysis of quantum spectra in the semiclassical regime we find strong evidence that weighting the delta functions in the quantum mechanical density of states with the product of diagonal matrix elements, <nAn><nBn>, is equivalent to weighting the periodic orbit contributions in the semiclassical periodic orbit sum with the product of the periodic orbit means, <A>(p)<B>(p), of the classical observables A and B. Results are presented for the hydrogen atom in a magnetic field for both the chaotic and near-integrable regime, and for the circle billiard.

Semiclassical spectra and diagonal matrix elements by harmonic inversion of cross-correlated periodic orbit sums.

Semiclassical spectra weighted with products of diagonal matrix elements of operators A(alpha), i.e., g(alphaalpha')(E)= summation operator(n)<nA(alpha)n><nA(alpha')n>/(E-E(n)), are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.

Periodic orbit action correlations in the Baker map

Periodic orbit action correlations are studied for the piecewise linear, area-preserving Baker map. Semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum Baker map suggest the existence of universal periodic orbit correlations. The calculation of periodic orbit sums for the Baker map can be performed with the help of a Perron-Frobenius-type operator. This makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map. Periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error. Exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation.

Bifurcations in the periodic orbit sum: multiple windings

Periodic orbits that participate in a bifurcation contribute collectively to the periodic orbit sum for the quantum density of states. The contributions of multiple windings of isolated orbits are easily obtained from powers of the stability matrix, but it is generally hard to compose the actions that determine the contributions of higher windings of a bifurcation. Here we derive an approximate relation between the amplitude of the contributions of different windings for the saddle-centre bifurcation and the period-doubling bifurcation.

Bifurcations of periodic orbits and uniform approximations

We derive uniform approximations for contributions to Gutzwiller's periodic-orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller's type would diverge at the bifurcation. New results for the tangent, the period doubling and the period tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results.

A particle-number expansion beyond self-consistent field theory

Semi-classical periodic-orbit theory is applied to classically chaotic systems of interacting identical particles yielding a particle-number expansion of the periodic-orbit sum. Truncating the expansion we obtain effective one- and two-body problems. The corresponding few-body orbits are relevant as some have short periods and small Lyapunov exponents.

On the Quantum Baker′s Map and Its Unusual Traces

The quantum baker′s map is the quantization of a simple classically chaotic system and has many generic features that have been studied over the last few years. While there exists a semiclassical theory of this map, a more rigorous study of the same revealed some unexpected features which indicated that correction terms of the order of log( ħ) had to be included in the periodic orbit sum. Such singular semiclassical behaviour was also found in the simplest traces of the quantum map. In this note we study the quantum mechanics of a baker′s map which is obtained by reflecting the classical map about its edges, in an effort to understand and circumvent these anomalies. This leads to a real quantum map with traces that follow the usual Gutzwiller-Tabor like semiclassical formulae. We develop the relevant semiclassical periodic orbit sum for this map which is closely related to that of the usual baker′s map, with the important difference that the propagators leading to this sum have no anomalous traces.