arXiv subjects

Niall D. Whelan

Publications and source records attributed to Niall D. Whelan.

11 recordsLinked to original sources

arXiv2002

Semiclassical Trace Formulas for Noninteracting Identical Particles

We extend the Gutzwiller trace formula to systems of noninteracting identical particles. The standard relation for isolated orbits does not apply since the energy of each particle is separately conserved causing the periodic orbits to occur in continuous families. The identical nature of the particles also introduces discrete permutational symmetries. We exploit the formalism of Creagh and Littlejohn [Phys. Rev. A 44, 836 (1991)], who have studied semiclassical dynamics in the presence of continuous symmetries, to derive many-body trace formulas for the full and symmetry-reduced densities of states. Numerical studies of the three-particle cardioid billiard are used to explicitly illustrate and test the results of the theory.

nlin.CD↗

arXiv2001

Scarring and the statistics of tunnelling

We show that the statistics of tunnelling can be dramatically affected by scarring and derive distributions quantifying this effect. Strong deviations from the prediction of random matrix theory can be explained quantitatively by modifying the Gaussian distribution which describes wavefunction statistics. The modified distribution depends on classical parameters which are determined completely by linearised dynamics around a periodic orbit. This distribution generalises the scarring theory of Kaplan [Phys. Rev. Lett. {\bf 80}, 2582 (1998)] to describe the statistics of the components of the wavefunction in a complete basis, rather than overlaps with single Gaussian wavepackets. In particular it is shown that correlations in the components of the wavefunction are present, which can strongly influence tunnelling-rate statistics. The resulting distribution for tunnelling rates is tested successfully on a two-dimensional double-well potential.

nlin.CD↗

arXiv2000

Combined Paramagnetic and Diamagnetic Response of YBCO

It has been predicted that the zero frequency density of states of YBCO in the superconducting phase can display interesting anisotropy effects when a magnetic field is applied parallel to the copper-oxide planes, due to the diamagnetic response of the quasi-particles. In this paper we incorporate paramagnetism into the theory and show that it lessens the anisotropy and can even eliminate it altogether. At the same time paramagnetism also changes the scaling with the square root of the magnetic field first deduced by Volovik leading to an experimentally testable prediction. We also map out the analytic structure of the zero frequency density of states as a function of the diamagnetic and paramagnetic energies. At certain critical magnetic field values we predict kinks as we vary the magnetic field. However these probably lie beyond currently accessible field strengths.

cond-mat.supr-con↗

arXiv2000

Semiclassical Trace Formulas for Two Identical Particles

Semiclassical periodic orbit theory is used in many branches of physics. However, most applications of the theory have been to systems which involve only single particle dynamics. In this work, we develop a semiclassical formalism to describe the density of states for two noninteracting particles. This includes accounting properly for the particle exchange symmetry. As concrete examples, we study two identical particles in a disk and in a cardioid. In each case, we demonstrate that the semiclassical formalism correctly reproduces the quantum densities.

nlin.CD↗

arXiv1999

Regular Tunnelling Sequences in Mixed Systems

We show that the pattern of tunnelling rates can display a vivid and regular pattern when the classical dynamics is of mixed chaotic/regular type. We consider the situation in which the dominant tunnelling route connects to a stable periodic orbit and this orbit is surrounded by a regular island which supports a number of quantum states. We derive an explicit semiclassical expression for the positions and tunnelling rates of these states by use of a complexified trace formula.

chao-dyn↗

arXiv1999

The Statistics of Chaotic Tunnelling

We discuss the statistics of tunnelling rates in the presence of chaotic classical dynamics. This applies to resonance widths in chaotic metastable wells and to tunnelling splittings in chaotic symmetric double wells. The theory is based on using the properties of a semiclassical tunnelling operator together with random matrix theory arguments about wave function overlaps. The resulting distribution depends on the stability of a specific tunnelling orbit and is therefore not universal. However it does reduce to the universal Porter-Thomas form as the orbit becomes very unstable. For some choices of system parameters there are systematic deviations which we explain in terms of scarring of certain real periodic orbits. The theory is tested in a model symmetric double well problem and possible experimental realisations are discussed.

chao-dyn↗

arXiv1999

Homoclinic Structure Controls Chaotic Tunnelling

Tunnelling from a chaotic potential well is explained in terms of a set of complex periodic orbits which contain information about the real dynamics inside the well as well as the complex dynamics under the confining barrier. These orbits are associated with trajectories which are homoclinic to a real trajectory emerging from the optimal tunnelling path. The theory is verified by considering a model double-well problem.

chao-dyn↗

arXiv1998

A Matrix Element for Chaotic Tunnelling Rates and Scarring Intensities

It is shown that tunnelling splittings in ergodic double wells and resonant widths in ergodic metastable wells can be approximated as easily-calculated matrix elements involving the wavefunction in the neighbourhood of a certain real orbit. This orbit is a continuation of the complex orbit which crosses the barrier with minimum imaginary action. The matrix element is computed by integrating across the orbit in a surface of section representation, and uses only the wavefunction in the allowed region and the stability properties of the orbit. When the real orbit is periodic, the matrix element is a natural measure of the degree of scarring of the wavefunction. This scarring measure is canonically invariant and independent of the choice of surface of section, within semiclassical error. The result can alternatively be interpretated as the autocorrelation function of the state with respect to a transfer operator which quantises a certain complex surface of section mapping. The formula provides an efficient numerical method to compute tunnelling rates while avoiding the need for the exceedingly precise diagonalisation endemic to numerical tunnelling calculations.

chao-dyn↗

arXiv1996

Complex Periodic Orbits and Tunnelling in Chaotic Potentials

We derive a trace formula for the splitting-weighted density of states suitable for chaotic potentials with isolated symmetric wells. This formula is based on complex orbits which tunnel through classically forbidden barriers. The theory is applicable whenever the tunnelling is dominated by isolated orbits, a situation which applies to chaotic systems but also to certain near-integrable ones. It is used to analyse a specific two-dimensional potential with chaotic dynamics. Mean behaviour of the splittings is predicted by an orbit with imaginary action. Oscillations around this mean are obtained from a collection of related orbits whose actions have nonzero real part.

chao-dyn↗

arXiv1996

Edge Diffraction, Trace Formulae and the Cardioid Billiard

We study the effect of edge diffraction on the semiclassical analysis of two dimensional quantum systems by deriving a trace formula which incorporates paths hitting any number of vertices embedded in an arbitrary potential. This formula is used to study the cardioid billiard, which has a single vertex. The formula works well for most of the short orbits we analyzed but fails for a few diffractive orbits due to a breakdown in the formalism for certain geometries. We extend the symbolic dynamics to account for diffractive orbits and use it to show that in the presence of parity symmetry the trace formula decomposes in an elegant manner such that for the cardioid billiard the diffractive orbits have no effect on the odd spectrum. Including diffractive orbits helps resolve peaks in the density of even states but does not appear to affect their positions. An analysis of the level statistics shows no significant difference between spectra with and without diffraction.

chao-dyn↗

arXiv1995

Semiclassical Quantisation Using Diffractive Orbits

Diffraction, in the context of semiclassical mechanics, describes the manner in which quantum mechanics smooths over discontinuities in the classical mechanics. An important example is a billiard with sharp corners; its semiclassical quantisation requires the inclusion of diffractive periodic orbits in addition to classical periodic orbits. In this paper we construct the corresponding zeta function and apply it to a scattering problem which has only diffractive periodic orbits. We find that the resonances are accurately given by the zeros of the diffractive zeta function.

chao-dyn↗