Abstract
We explore a new numerical method for studying one-dimensional quantum systems in a trapping potential. We focus on the setup of an impurity in a fermionic background, where a single distinguishable particle interacts through a contact potential with a number of identical fermions. We can accurately describe this system, for various particle numbers, different trapping potentials and arbitrary finite repulsion, by constructing a truncated basis containing states at both zero and infinite repulsion. The results are compared with matrix product states methods and with the analytical result for two particles in a harmonic well.
Highlights
In this paper we explored a new method for studying strongly coupled one-dimensional systems where an impurity interacts with a background of identical fermions, a method that generalizes that of [51]
Our results compare well both with analytical methods for two particles and with numerical methods based on matrix product states
Most numerical methods would perform worse the stronger the interaction strength is, but our method is exact at infinite interaction and works well both for small and strong interactions, with a peak of slower convergence at some intermediate interaction strength
Summary
The investigation of one-dimensional quantum systems of interacting particles has, in the last decades, attracted renewed interest due to striking advances in experiments with cold atoms in optical traps [1]. The degree of control over several experimental parameters, including interactions between the atoms [9,10,11,12] and trapping geometries opens up the possibility of using such experiments as quantum simulators for a multitude of interesting models [13], even beyond usual condensed matter systems [14]. While it may be difficult to reach a regime of strong interactions with usual methods, our approach is exact in the zero and infinite interaction limits. This method is an extension of Ref. We compare our results for spatial densities and momentum distributions to simulations of the continuum obtained with Matrix Product States (MPS) [52, 53], as well as the known analytical results for two atoms in a harmonic trap [54]
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