One of the challenging features of studying model Hamiltonians with cold atoms in optical lattices is the presence of spatial inhomogeneities induced by the confining potential, which results in the coexistence of different phases. This paper presents Quantum Monte Carlo results comparing methods for confining fermions in two dimensions, including conventional diagonal confinement (DC), a recently proposed 'off-diagonal confinement' (ODC), as well as a trap which produces uniform density in the lattice. At constant entropy and for currently accessible temperatures, we show that the current DC method results in the strongest magnetic signature, primarily because of its judicious use of entropy sinks at the lattice edge. For d-wave pairing, we show that a constant density trap has the more robust signal and that ODC can implement a constant density profile. This feature is important to any prospective search for superconductivity in optical lattices. PACS numbers: 71.10Fd, 37.10.Jk, 71.27.+a Introduction: Optical lattice emulators (OLE) control ultra-cold atomic gases with lasers and magnetic fields to create experimental realizations of quantum lattice models of bosonic or fermionic particles. For bosons, classic signatures of low temperature correlated states-superfluidity and the Mott transition-have been explored now for a decade 1 . For fermions, quantum degeneracy has been established through the observation of a Fermi surface 2 , as has the Mott transition 3,4 . The observation of magnetic order is the next immediate experimental objective 5-8 . One ultimate goal is resolving the longstanding question of whether the doped 2D fermion Hubbard Hamiltonian has long-range d-wave superconducting order 9 .Optical lattice experiments face at least two major obstacles in simulating the fermion Hubbard model. The first is achieving low enough temperature to pass through phase transitions and into reduced entropy ordered phases. Present limits in experiments are to temperatures T ∼ t (the near-neighbor hopping energy), and to local entropies per atom ∼ 0.77k B10 , values which are at the border for observing short-range magnetic order.The other obstacle, which we will be addressing in this paper, is inhomogeneity arising from the confining potential 11 . The external field conventionally used to trap cold atoms in the lattice, a spatially dependent chemical potential which we refer to as 'diagonal confinement' (DC), causes variations in the density per site ρ i , with more atoms, on average, in the center of the lattice and fewer at the edges. Density plays a key role in determining which correlations are dominant in interacting quantum systems, but this is especially true of the fermion Hubbard Hamiltonian in two dimensions where the magnetic response is very sharply peaked 12 near halffilling (ρ = 1). Various analytic and numerical calculations suggest that pairing order also has a fairly sharp optimal filling, ρ ≈ 0.80 − 0.85.

Experiments on cold atom systems in which a lattice potential is ramped up on a confined cloud have raised intriguing questions about how the temperature varies along isentropic curves, and how these curves intersect features in the phase diagram. In this paper, we study the isentropic curves of two models of magnetic phase transitions-the classical Blume-Capel Model (BCM) and the Fermi Hubbard Model (FHM). Both Mean Field Theory (MFT) and Monte Carlo (MC) methods are used. The isentropic curves of the BCM generally run parallel to the phase boundary in the Ising regime of low vacancy density, but intersect the phase boundary when the magnetic transition is mainly driven by a proliferation of vacancies. Adiabatic heating occurs in moving away from the phase boundary. The isentropes of the half-filled FHM have a relatively simple structure, running parallel to the temperature axis in the paramagnetic phase, and then curving upwards as the antiferromagnetic transition occurs. However, in the doped case, where two magnetic phase boundaries are crossed, the isentrope topology is considerably more complex.