Abstract
We provide a detailed study of the properties of a few interacting spin 1 / 2 fermions trapped in a one-dimensional harmonic oscillator potential. The interaction is assumed to be well represented by a contact delta potential. Numerical results obtained by means of direct diagonalization techniques are combined with analytical expressions for both the non-interacting and strongly interacting regime. The N = 2 case is used to benchmark our numerical techniques with the known exact solution of the problem. After a detailed description of the numerical methods, in a tutorial-like manner, we present the static properties of the system for N = 2 , 3 , 4 and 5 particles, e.g., low-energy spectrum, one-body density matrix, ground-state densities. Then, we consider dynamical properties of the system exploring first the excitation of the breathing mode, using the dynamical structure function and corresponding sum-rules, and then a sudden quench of the interaction strength.
Highlights
The theoretical study of one-dimensional systems has always attracted a lot of attention
They are looking at those systems as a very versatile laboratory where they apply ab-initio techniques, without the complexity of the inter-particle interactions that appear in other fields, as for instance in nuclear physics [18,19]
In this chapter we describe the method named direct diagonalization to find an approximate solution of the many-body Schrödinger equation
Summary
The theoretical study of one-dimensional systems has always attracted a lot of attention. The groundbreaking experiments of Jochim’s group in Heildelberg, have opened new theoretical challenges to study one-dimensional fermionic systems They have been able to precisely control the number of atoms and the strength of the interactions [11]. Mathematics 2020, 8, 1196 phenomena in few-fermion systems [15], the behaviour of two fermions in a double-well potential [16], the realization of an antiferromagnetic spin chain of few cold atoms [17], all belong to the long list of theoretical challenges that the experimental advances are offering to theoreticians They are looking at those systems as a very versatile laboratory where they apply ab-initio techniques, without the complexity of the inter-particle interactions that appear in other fields, as for instance in nuclear physics [18,19].
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