Strongly interacting one-dimensional fermions form an effective spin chain inthe absence of an external lattice potential. We show that the exchangecoefficients of such a chain may be locally tuned by properly tailoring thetransversal confinement. In particular, in the vicinity of aconfinement-induced resonance (CIR) the exchange coefficients may havesimultaneously opposite ferromagnetic and antiferromagnetic characters atdifferent locations along the trap axis. Moreover, the local exchanges may beengineered to induce avoided crossings between spin states at the CIR, andhence a ramp across the resonance may be employed to create different spinstates and to induce spin dynamics in the chain. We show that such unusual spinchains have already been realized in the experiment of Murmann et al. [Phys.Rev. Lett. 115, 215301 (2015)].
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