Hearing shapes of few electrons quantum drums: A configuration–interaction study
Introduction
The isospectrality debate “Can one hear the shape of a drum ?”, launched by Kac [1], received a surprisingly simple and late answer by Gordon et al. [2] more than a quarter century later, when they found the first pair of 2D isospectral shapes, called Bilby and Hawk (see Fig. 1), as well as the general conditions for isospectrality. Mathematical and physical aspects of isospectrality have been recently reviewed in [3].
All physical systems obeying the Helmholtz's equation , are candidates for building and testing isospectral shapes, as it was experimentally realized with electromagnetic modes in cavities [4], vibrating smectic films [5] or, more recently, electron modes in nanostructures [6]. Supplementary to testing isospectrality, the later set-up was shown to allow a robust phase extraction (i.e. the position of the nodal lines) of the eigenfunctions [6], [7].
However, two or more electrons interact electrostatically, raising the legitimate question whether the interaction preserves isospectrality, an issue not addressed before to our knowledge. The unique combination of identical single-particle spectra and different wave functions spatial distribution is challenging for many-body physics. Apart from isospectrality study, interaction and localization effects in shapes without spatial symmetries possess an intrinsic interest as they have been rarely addressed in comparison with shapes having spatial symmetries (difficulties arising from lack of analytical knowledge of the single-particle eigenstates).
Having described the motivations of our study, we have two main choices to make: what method to use for the spectral calculations in the presence of interaction and whether to use a continuous or a discrete model for describing the shapes. In what the method is concerned, the best candidate seems to be the configuration–interaction method, which is known to give reliable results of controlled precision. Exact diagonalization is performed on the full set of Slater determinants, separated in subspaces with the same electrons number, no approximations or truncations being done in this part of calculations. The convergence is assured by expending the single particle basis (used for the Slater determinants) till the main results – spectrum and wave functions coefficients – undergo negligible changes. Also, it is important to mention that the method gives faster convergence for the lowest energy states of each symmetry class (i.e. the first singlet and the first triplet), being therefore particularly suited for ground state analysis, which is our main focus. The configuration–interaction approach for quantum dots has been applied intensively for the parabolic confinement case, where the advantage is the analytical knowledge of the single-particle wave functions, as well as some developed techniques for faster integrating the Coulombian elements (see, e.g. [8], [9], [10], [11], [12]), or in cylindrical geometries (see, e.g. [13] and references therein).
In what concerns the model, the discrete (finite differences) model seems the only computationally feasible at the moment, since the eigenstates of the Bilby–Hawk pair are not analytically known, and the continuous approach would have seemingly unaffordable computational demands (as discussed in Section 2). The discrete model is natural in for describing interaction in atomic lattices (see, e.g. [14], [15], [16], [17], [18], [19], [20], [21]), but has also been used – as we shall in this paper – for the discretization of space (in the context of finite-differences method to solve the Schrodinger equation) either for its advantage of easily tailoring any shape [22], [23], [24], or for realistic description of leads-dot couplings in mesoscopic transport [25], to give some examples. A comparison between the discrete and continuous models for the study of isospectrality was addressed in [26], [7], for the single-particle case.
The outline of the paper is as follows: in Section 2 we describe the model used, Section 3 presents the main spectral results, while 4 Wigner localization study, 5 Phase extraction exercise for two electrons address the incipient Winger localization and the phase extraction procedure in the presence of interaction. Section 6 concludes the paper.
Section snippets
Model
The model we use in this paper consists in describing the Bilby and Hawk shapes by discrete sets of points in a rectangular mash, the approach corresponding to the finite-differences method to solve the Schrodinger equation. As mentioned, the method allows for easy tailoring of any shape, the rectangular geometry of the mash being not mandatory (a hexagonal mash for instance can be considered as well), since we are interested in the first few vibrational modes, with physical significance.
As
Two and three interacting electrons on Bilby and Hawk shapes
In this section we address the main question, of whether isospectrality is preserved in the presence of interaction. The first step is to consider two electrons in both Bilby and Hawk shapes, and perform configuration–interaction spectral calculations, following the model described in the previous section. After calculating the elements, the matrix corresponding to Hamiltonian Eq. (1) is built in the basis of all possible two-electrons occupancies of the single-particle states chosen
Wigner localization study
By increasing the role of interaction (as compared to the level spacing), the electrons tend to localize and form Winger molecules, precursors – or the finite size equivalence – of the Winger crystal. The localization of electrons occurs in separated islands, with reduced overlap.
The total charge density distribution gives an account of electrons localization, however the property is more rigourously – and clearly – illustrated by plotting the pair correlation function which give the
Phase extraction exercise for two electrons
The eigenfunctions of isospectral pairs have a special property, called transplantation [2], [3], which means that all eigenmodes of one shapes can be built by algebraically combining parts from the corresponding eigenmode of the other shape. For instance, the wave function in triangle 1 of Hawk (see Fig. 1) can be built by algebraic summing the Bilby functions (corresponding to the same eigenmode) from triangles A, E and F (the function in E will be taken with the sign “−”). The full
Conclusions
We present accurate configuration–interaction calculations of the energy spectrum for two and three interacting electrons confined on the Bilby and Hawk (see Fig. 1) isospectral shapes, with a focus on the ground state properties.
A discrete model is used to describe the shapes, since a continuous approach seems unfeasible at the moment – due to lack of analytical knowledge of single particle eigenfunctions. Advantages and draw-backs of the model are discussed, in the context of few-body
Acknowledgments
We acknowledge support from PN-II, Contract TE 90/05.10.2011 and Core program 45N/2009.
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