Excitons and biexcitons in semiconductor quantum wires

We study the transport of electrons through a long quantum wire connecting two bulk leads. As the electron density in the wire is lowered, the Coulomb interactions lead to short-range crystalline ordering of electrons. In this Wigner crystal state the spins of electrons form an antiferromagnetic Heisenberg spin chain with exponentially small exchange coupling $J$. Inhomogeneity of the electron density due to the coupling of the wire to the leads results in violation of spin-charge separation in the device. As a result the spins affect the conductance of the wire. At zero temperature the low-energy spin excitations propagate freely through the wire, and its conductance remains $2{e}^{2}∕h$. Since the energy of the elementary excitations in the spin chain (spinons) cannot exceed $\ensuremath{\pi}J∕2$, the conductance of the wire acquires an exponentially small negative correction $\ensuremath{\delta}\phantom{\rule{0.2em}{0ex}}G\ensuremath{\propto}\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}J∕2T)$ at low temperatures $T⪡J$. At higher temperatures, $T⪢J$, most of the spin excitations in the leads are reflected by the wire, and the conductance levels off at a new universal value ${e}^{2}∕h$.

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Conductance of a quantum wire at low electron density

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The capabilities of the functional–analytic and of the functional–integral approach for the construction of the Hamiltonian as a self‐adjoint operator on Hilbert space are compared in the context of non‐relativistic quantum mechanics. Differences are worked out by taking the one‐dimensional hydrogen atom as an example, that is, a point mass on the Euclidean line subjected to the inverse–distance potential. This particular choice is made with the intent to clarify a long‐lasting discussion about its spectral properties. In fact, for the four‐parameter family of possible Hamiltonians the corresponding energy‐dependent Green functions are derived in closed form. The multiplicity of Hamiltonians should be kept in mind when modeling certain experimental situations as, for instance, in quantum wires.

The functional–analytic versus the functional–integral approach to quantum Hamiltonians: The one‐dimensional hydrogen atoma)

We review recent advances in the theory of trapped fermions using techniques borrowed from random matrix theory (RMT) and, more generally, from the theory of determinantal point processes. In the presence of a trap, and in the limit of a large number of fermions $N \gg 1$, the spatial density exhibits an edge, beyond which it vanishes. While the spatial correlations far from the edge, i.e. close to the center of the trap, are well described by standard many-body techniques, such as the local density approximation (LDA), these methods fail to describe the fluctuations close to the edge of the Fermi gas, where the density is very small and the fluctuations are thus enhanced. It turns out that RMT and determinantal point processes offer a powerful toolbox to study these edge properties in great detail. Here we discuss the principal edge universality classes, that have been recently identified using these modern tools. In dimension $d=1$ and at zero temperature $T=0$, these universality classes are in one-to-one correspondence with the standard universality classes found in the classical unitary random matrix ensembles: soft edge (described by the "Airy kernel") and hard edge (described by the "Bessel kernel") universality classes. We further discuss extensions of these results to higher dimensions $d\geq 2$ and to finite temperature. Finally, we discuss correlations in the phase space, i.e., in the space of positions and momenta, characterized by the so called Wigner function.

Noninteracting fermions in a trap and random matrix theory

In quantum mechanics of one dimension it is shown for potentials which become infinite at a point but are continuous elsewhere, that the singularity acts as an impenetrable barrier if the potential is not integrable up to the singularity, but if the potential is integrable the behavior is not essentially different from that of a potential which does not become infinite.

M. Andrews

Papers

Singular potentials in one dimension