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arXiv:hep-th/9512164 · Physical Review Letters

Geometric Phase in Quantum Billiards with a Pointlike Scatterer

Taksu Cheon, Takaomi Shigehara

Published Mar 11, 1996• 11 citations
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Abstract

We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.

DOI
10.1103/physrevlett.76.1770

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We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.
We study the low-energy quantum spectra of two-dimensional rectangular billiards with a small but finite-size scatterer inside. We start by examining the spectral properties of billiards with a single pointlike scatterer. The problem is formulated in terms of the self-adjoint extension theory of functional analysis. The condition for the appearance of so-called wave chaos is clarified. We then relate the pointlike scatterer to a finite-size scatterer through an appropriate truncation of the basis. We show that the signature of wave chaos in low-energy states is most prominent when the scatterer is weakly attractive. As an illustration, numerical results of a rectangular billiard with a small rectangular scatterer inside are exhibited. \textcopyright{} 1996 The American Physical Society.
We derive closed analytical expressions for the complex Berry phase of an open quantum system in a state which is a superposition of resonant states and evolves irreversibly due to the spontaneous decay of the metastable states. The codimension of an accidental degeneracy of resonances and the geometry of the energy hypersurfaces close to a crossing of resonances differ significantly from those of bound states. We discuss some of the consequences of these differences for the geometric phase factors. For example, instead of a diabolical point singularity there is a continuous closed line of singularities formally equivalent to a continuous distribution of `magnetic' charge on a diabolical circle, there are different classes of topologically inequivalent non-trivial closed paths in parameter space, the topological invariant associated with the sum of the geometric phases, dilations of the wavefunction due to the imaginary part of the Berry phase and others.
A wave function picks up, in addition to the dynamic phase, the geometric (Berry) phase when traversing adiabatically a closed cycle in parameter space. We develop a general multidimensional theory of the geometric phase for (double) cycles around exceptional degeneracies in non-Hermitian Hamiltonians. We show that the geometric phase is exactly $\ensuremath{\pi}$ for symmetric complex Hamiltonians of arbitrary dimension and for nonsymmetric non-Hermitian Hamiltonians of dimension 2. For nonsymmetric non-Hermitian Hamiltonians of higher dimension, the geometric phase tends to $\ensuremath{\pi}$ for small cycles and changes as the cycle size and shape are varied. We find explicitly the leading asymptotic term of this dependence, and describe it in terms of interaction of different energy levels.
The quantum mechanical Berry phase is a topological contribution to the adiabatic response as system parameters are varied. It is the root of many fascinating topological phenomena in physics. Here, the authors consider spatiotemporal adiabatic response for many-body quantum systems in any dimension, and find analogous topological contributions they dub the ``higher Berry phase''. These contributions constrain the global structure of the phase diagrams of these systems and imply the existence of boundary modes at special parameter values.
Abstract The phase of a quantum state changes rapidly as parameters X = (X1, X2,...) are varied near a degeneracy X*, reflecting the monopole singularity of the underlying phase 2-form V(X) at X*. The singularities may be sources or sinks of V. We study them numerically and display them graphically for two families of hamiltonians whose degeneracy structure is typical. First is a particle moving along a line segment with kinetic energy quartic in the momentum (‘quartic-momentum square well’); the X are incorporated into the boundary conditions. Second is a charged particle moving in a domain D of the plane which is threaded by a magnetic flux line of strength α, with wavefunction vanishing on the boundary ∂D (‘Aharonov-Bohm billiards’); the X are α and parameters specifying ∂D; V is not invariant under gauge transformations of the vector potential generating the flux. For Aharonov-Bohm billiards we study how the spatial patterns of phase of wavefunctions change round circuits near degeneracies; these patterns also have singularities (wavefront dislocations) that appear and disappear by colliding with each other and with ∂D.