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arXiv:1204.0722 · Advanced Studies in Theoretical Physics

Quaternionic vector coherent states for spin-orbit interactions

Isiaka Aremua, Mahouton Norbert Hounkonnou, Ezinvi Baloïtcha

Published Jan 1, 2013• 2 citations
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Abstract

This work addresses the study of two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit (SO) interactions, with effective Zeeman coupling. Exact analytical expressions are found for the eigenvalue problem giving the Landau levels and the associated eigenstates, thanks to an appropriate method of parametrization used to quantize the physical system. Such a system exhibits interesting properties of quaternions for which vector coherent states are built. Similarity with the Weyl-Heisenberg group and some relevant properties of these vectors are discussed.

DOI
10.12988/astp.2013.13024

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•Jan 1, 2013•0 citations
This work addresses the study of two-dimensional electron gas in a perpendicular magnetic field in the presence of both Rashba and Dresselhaus spin-orbit (SO) interactions, with effective Zeeman coupling. Exact analytical expressions are found for the eigenvalue problem giving the Landau levels and the associated eigenstates, thanks to an appropriate method of parametrization used to quantize the physical system. Such a system exhibits interesting properties of quaternions for which vector coherent states are built. Similarity with the Weyl-Heisenberg group and some relevant properties of these vectors are discussed.
Motivated by proposals to employ RKKY-coupled spins as building blocks in a solid-state quantum computer, we analyze how the RKKY interaction in a 2D electron gas is influenced by spin-orbit interactions. Using a two-impurity Kondo model with added Dresselhaus and Rashba spin-orbit interactions we find that spin-rotational invariance of the RKKY interaction — essential for a well-controllable two-qubit gate — is restored when tuning the Rashba coupling to have the same strength as the Dresselhaus coupling. We also discuss the critical properties of the two-impurity Kondo model in the presence of spin-orbit interactions, and extract the leading correction to the block entanglement scaling due to these interactions.
In this Thesis we discussed ultracold Fermi gas with an s-wave interaction and synthetic spin-orbit coupling under a variety of conditions. We considered the system in both three and two spatial dimensions, with equal-Rashba-Dresselhaus type or Rashba-only type of spin-orbit-coupling, and with or without an artificial Zeeman field. We found competing effects on Fermionic superfluidity from spin-orbit coupling and Zeeman fields, and topologically non-trivial states in the presence of both fields. We gave an outlook on the many-body physics in the last.
The z axis magnetization of a two dimensional electron gas with contact repulsive interaction and in presence of a Rashba potential is computed by means of quantum field theory at second order. A striking effect of the pure Rashba interaction is that of hindering spin alignment along the z direction. Evidence of transition at critical repulsive interaction coupling constant is still found. The degree of magnetization, however, shows a clear dependence on the spin-orbit interaction strength. Furthermore, the transition to magnetized state appears to be smoothed by the presence of the Rashba interaction.
We present a theoretical approach to the problem of two-dimensional fermion pairing under spin-orbit (SO) coupling and Zeeman interaction with an external magnetic field. We introduce a generic pairing Hamiltonian operating in the Hilbert space spanning free pairs and bare two-body bound states. The SO coupling of Rashba or Dresselhaus type provides ppp-wave dressing of an sss-wave spin-singlet bound state by continua in the triplet scattering channels. In the strong-coupling (Bose-Einstein condensate) limit of the many-body fermion system at zero temperature our theory maps onto the two-channel Fano-Anderson model of a spinless resonant ppp-wave superfluid.
We studied the influence of spin-orbit interaction (SOI) in interacting low dimensional quantum systems at zero temperature within the framework of the functional renormalization group (fRG). Among the several types of spin-orbit interaction the so-called Rashba spin-orbit interaction is especially intriguing for future spintronic applications as it may be tuned via external electric fields. We investigated its effect on the low energy physics of an interacting quantum wire in an applied Zeeman field which is modeled as a generalization of the extended Hubbard model. To this end we performed a renormalization group study of the two particle interaction, including the SOI and the Zeeman field exactly on the single particle level. Considering the resulting two band model, we formulated the RG equations for the two particle vertex keeping the full band structure as well as the non trivial momentum dependence of the low energy two particle scattering processes. In order to solve these equations numerically we defined criteria that allowed us to classify whether a given set of initial conditions flows towards the strongly coupled regime. We found regions in the models parameter space where a weak coupling method as the fRG is applicable and it is possible to calculate additional quantities of interest. Furthermore we analyzed the effect of the Rashba SOI on the properties of an interacting multi level quantum dot coupled to two semi innite leads. Of special interest was the interplay with a Zeeman field and its orientation with respect to the SOI term. We found a renormalization of the spin-orbit energy which is an experimental quantity used to asses SOI effects in transport measurements, as well as renormalized effective g factors used to describe the Zeeman field dependence. In particular in asymmetrically coupled systems the large parameter space allows for rich physics which we studied by means of the linear conductance obtained via the generalized Landauer-Büttiker formalism from the renormalized self energy. The multi level quantum dot exhibits Kondo physics at finite Zeeman fields which can be suppressed if the field and the SOI term are not aligned in parallel. Due to broken particle hole symmetry we observed a gate voltage dependence for finite field conductance plateaus. The computed dependencies have been observed in experiments and thus highlight the flexibility of the considered model. Finally we computed the linear conductance for a multi level quantum dot model that has been proposed to describe SOI and impurity scattering effects in carbon nanotubes (CNT). The SOI was seen to affect the already complicated level structure of the CNT and thus profoundly changes the linear conductance, especially the location of conductance resonances with respect to an applied magnetic field and gate voltage. Again asymmetries influence the conductance and the effects are explained by means of the effective tunnel couplings to the eigenstates of the CNT. We finished with an improved analysis of a reduced version of this model that includes two particle vertex renormalization. The computed linear conductance is in excellent agreement with experimental data.