The properties of fermions in one dimension are markedly different from those in higher dimensions. In two or three dimensions, the many-body behaviour of fermions is described by the Fermi liquid model, whose most prominent result is that, instead of studying the behaviour of individual particles, it is better to describe the emergent phenomena of the whole system, known as quasiparticles. These are collective modes of excitations born out of particle-particle interactions, which resemble the constituent single particles, albeit with renormalised mass. In one dimension, however, the formation of such a correlated state is unfavourable and the quasiparticles are completely absent. Instead, 1D fermions are described by a model known as the Tomonaga-Luttinger liquid. Central to this model is the process of bosonisation, a mathematical technique which maps the excitation of fermions to bosonic modes of spin-density and charge-density waves. As a prerequisite of the bosonisation technique, the fermionic single-particle dispersion is approximated as a linear function. Consequently, the Luttinger liquid model is limited in its scope of applicability. Specifically, it is valid only in the vicinity of the Fermi points, where the excitation energy is low.

Recent advancement in the theoretical field has expanded our understanding of 1D interacting fermions beyond the limit of linearity. The mobile-impurity model, proposed by L. Glazman and his co-authors, is a generalised model of 1D fermions which extends the scope of applicability of the Tomanaga-Luttinger model, accounting for the curvature of single-particle dispersion relations. It extends the region of validity of the tradition TLL theory from the neighbourhood of the Fermi points to a band which follows the single-particle dispersion. This thesis presents the results of an experimental work which provides direct evidence of non-linear behaviour predicted by the new model. It revolves around the measurement of the spectral function of electrons in surface-gate-defined quantum wires on a double-quantum-well, GaAs/AlGaAs device, and the comparison and fitting of the result to the non-linear theory. The spectral function is a theoretical concept frequently used in discussions on many-body problems. Its physical interpretation makes it closely related to experimental observables, such as the tunnelling current between the double quantum wells in this work.

This thesis consists of three parts. Part one concerns the theories of the various many-body models and the experimental technique: Chapter 1 briefly introduces the concept of Fermi liquid, before focusing on a detailed account of the Luttinger-liquid model. The mobile-impurity model will be introduced in the framework of the Luttinger liquid model afterwards, followed by a review of experimental works reported in the literature on the study of TLL physics. In Chapter 2, the principles of the experimental tunnelling spectroscopy technique are described. The second part of the thesis details the structure of the test device used in the experiment and its fabrication. In particular, we will discuss the intricacies of the three-dimensional EBL fabrication process developed specifically for this project. The setup and running of the measurement circuit will also be discussed in this part of the thesis, in the context of a prerequisite sample characterisation measurement. In the final part of the work, data from the main experiment will be presented and analysed, in search for the electron-electron interaction effects which are predicted by the theoretical discussions in the first chapter.