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Scanning tunneling spectroscopy of a 1DES
Onedimensional electron systems (1DES) are particularly interesting since the electronelectron interaction in a 1DES is pathological. Theory predicts that a singleparticle description of the system breaks down (at least close to the Fermi level) and manyparticle excitations are the only excitations. They are spin and charge density waves with different Fermi velocities. Due to the different velocities, one speaks about spincharge separation and the system is called a Luttinger liquid to distinguish it from Fermi liquids still described within the singleparticle picture.
Fig. 1: from left to right: first: STM image of a step edge; next: same image after subtracting the average step edge showing the disorder as bright and dark spots; next: resulting disorder potential; remaining: LDOS of the 1DES at different energies as indicated. 
We investigated several 1DESs which are confined below charged step edges One step edge is shown in the left column of Fig. 1. This 1DES exhibits twosubbands, but others exhibit only one subband. The determined subband energies in the displayed case correspond to an average kinetic energy of 20 meV and to an average electronelectron interaction strength of 25 meV within the 1DES. The 1DES is moreover subject to disorder, which has been determined and is shown in he third column of Fig. 1. It exhibits fluctuations of a rmsvalue of 10 meV. The other columns of Fig. 1 display the measured LDOS at the step edge at different energies. Alignement of the LDOS with the potential disorder can be observed e.g. at the potential valley marked by an arrow, where a bright spot is found at 40 mV (resonant state).
Fig. 2: left: Fourrier transforms of the measured LDOS of two different 1DES; right: corresponding calculated Fourrier transforms. 
Surprisingly the LDOS could be largely reproduced by a singleparticle calculation. This is demonstrated in Fig. 2, where Fourrier transforms representing the kspace distribution of the states are shown for two different step edgese with one and two subbands, respectively. The yellow lines show the expected singleparticle dispersion of a not disordered system. Obviously the kspace distribution largely follows the undisturbed dispersion with additional intensity within the parabola. More importantly, the measured kspace distrĂbution is very similar to the calculated one. In particular, the kvalue expected for the chargedensity wave within the Luttinger model, also indicated, exhibits no intensity. This intriguing result is currently not completely understood, but might be a good starting point for further investigations of the 1DES case.
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