Few-body approaches to one-dimensional many-body systems
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This thesis presents results regarding one dimensional many-body quantum systems, obtained by considering the few-body physics of their constituent particles, and through use of traditional quantum mechanical techniques such as scattering theory and the variational principle. Choosing a perspective from which the connection between the microscopic behaviour of the systems’ constituents and its macroscopic properties is apparent, we investigate two one-dimensional many-body systems: a ﬂat-banded optical lattice and a fermionic Luttinger liquid. Our choice of approach allows us to give a transparent description of the low-energy physics of both systems. For the former, we ﬁnd that the low-energy eigenstates may be written down directly in terms of position space creation operators, and that they admit a simple and intuitive interpretation in terms of the position space behaviour of the atoms occupying the lattice. For the latter, we employ few-body scattering theory to investigate a long-held but (until now) untested belief about the parameters appearing in Luttinger’s model, a general eﬀective low-energy description of one-dimensional quantum systems. We ﬁnd this interpretation to be untenable, and give arguments as to how the parameters should correctly be regarded.