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One-dimensional multi-component Fermi or Bose systems with strong zero-rangeinteractions can be described in terms of local exchange coefficients andmapping the problem into a spin model is thus possible. For arbitrary externalconfining potentials the local exchanges are given by highly non-trivialgeometric factors that depend solely on the geometry of the confinement throughthe single-particle eigenstates of the external potential. To obtain accurateeffective Hamiltonians to describe such systems one needs to be able to computethese geometric factors with high precision which is difficult due to thecomputational complexity of the high-dimensional integrals involved. Anapproach using the local density approximation would therefore be a mostwelcome approximation due to its simplicity. Here we assess the accuracy of thelocal density approximation by going beyond the simple harmonic oscillator thathas been the focus of previous studies and consider some double-wells ofcurrent experimental interest. We find that the local density approximationworks quite well as long as the potentials resemble harmonic wells but breakdown for larger barriers. In order to explore the consequences of applying thelocal density approximation in a concrete setup we consider quantum statetransfer in the effective spin models that one obtains. Here we find that evenminute deviations in the local exchange coefficients between the exact and thelocal density approximation can induce large deviations in the fidelity ofstate transfer for four, five, and six particles.

the european physical journal. d, atomic, molecular and optical physics/the european physical journal. d, atomic, molecular, optical and plasma physics

Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results