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Recently preparing mixtures of Bose and Fermi of
ultracold atomic gases in an optical lattice has become experimentally achievable. Such systems are very promising for studying strongly correlated many-body physics and indeed
several novel phenomena have been theoretically predicted
to appear in different parameter regimes.
We consider a mixture of light fermionic and heavy bosonic atoms confined to a quasi-one-dimensional optical lattice showing that such asystem will exhibit a Peierls instability, which is the instability towards a charge density wave (CDW) with wave number equal to twice the Fermi wave length kF. We start by considering an effective Hubbard model for the boson-fermion mixture in a sufficiently strong optical lattice potential. We examine the case in which the bosonic atoms are much heavier than the fermionic atoms which corresponds to the adiabatic limit and employ the mean field approximation regarding the bosonic field justified for the weak interaction case. We show that the above model maps to the Holstein model which is known in the adiabatic limit to exhibit a Peierls instability. This instability demonstrates that it is favorable for the system to reduce the one dimensional translation symmetry by enlarging the effective unit cell, e.g., for half filling the unit cell doubles opening a gap in the fermionic spectrum at the zone boundary of the folded Brillouin zone. Moreover due to the Peierls instability a bosonic density wave with a wave number of twice the fermionic Fermi wave number will appear in the quasi one-dimensional system. We present numerical results, demonstrating the Peierls instability, in which we display the energy of the fermionic atoms as a function of a given bosonic density modulation showing that the energy minimum appears for modulation of 2kF we present the fermionic spectrum for such a system. Without the trap it is always energetically preferable for the system to exhibit a bosonic density modulation since the energy gain for the fermionic gas is linear in the modulation amplitude whereas the energetic "price" for the bosonic atoms in quadratic in the modulation's amplitude. Considering the trapping potential we demonstrate that there is a threshold for the appearance of the Peierls instability. We find numerically the optimal amplitude of such a modulation by minimizing the total energy of the system. We augment our numerical results by an analytical continuum model in which we explicitly deal with the trap confining potential. |