Find in Library

We provide a detailed study of the properties of a few interacting spin <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula> 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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics> </math> </inline-formula> 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.