Quantum Monte Carlo (QMC) techniques are used to provide an approximation-free investigation of the phases of the one-dimensional attractive Hubbard Hamiltonian in the presence of population imbalance. The temperature at which the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase is destroyed by thermal fluctuations is determined as a function of the polarization. It is shown that the presence of a confining potential does not dramatically alter the FFLO regime, and that recent experiments on trapped atomic gases likely lie just within the stable temperature range.
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