Optimized confinement of fermions in two dimensions

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, a recently proposed “off-diagonal confinement”, as well as a trap which produces uniform density in the lattice. At constant entropy and for currently accessible temperatures, we show that (1) diagonal confinement results in the strongest signature of magnetic order, primarily because of its judicious use of entropy sinks at the trap edge and that (2) for d-wave pairing, a trap with uniform density is optimal and can be effectively implemented via off-diagonal confinement. This feature is important to any prospective search for superconductivity in optical lattices.

Figure 1

The density as a function of $U/t$ and $\mu /t$ for the homogeneous fermion Hubbard model. Data were obtained on $8\times 8$ lattices with $T/t=0.5$. The dotted line is a constant density path ($\rho =.80$) used in the trap comparisons. Using the LDA, density profiles of inhomogeneous models can be determined by following an appropriate $(U/t,\mu /t)$ path of local parameters as the lattice position is changed.

Figure 2

(a) Entropy per site, (b) n.n. spin correlation, and (c) n.n.n. $d$-wave pairing correlation shown as functions of $U/t$ and $\mu /t$ for the homogeneous Hubbard Hamiltonian. Data were obtained on $8\times 8$ lattices with $T/t=0.5$. Paths for optimal DC and ODC traps are shown as solid lines, with the constant density trap (CD) as a dashed line. The ridge of prominent $d$-wave pairing (c) occurs at $\rho \sim 0.80$ and is nearly linear, so that an ODC path is close to the constant density one.

Figure 3

(a) n.n. spin correlation, (c) density ($\rho$), and (d) entropy per site ($s$) profiles are shown as a function of the distance ($r$) from the trap center for two different trap types: DC and ODC, using optimal trap parameters. ODC trap with $\mu =0.0$ is also a constant density trap ($\rho =1$). The average n.n. spin correlation is larger for DC trap (0.14) than ODC (0.08) at the same entropy (0.75) and number of fermions (6600). Panel (b) shows average n.n. spin correlation as a function of entropy per fermion ($S$/$N$) for optimal DC and ODC traps.

Figure 4

(a) average n.n.n. $d$-wave pairing correlation, (c) density ($\rho$), and (d) entropy per site ($s$) profiles are shown as a function of the distance ($r$) from the trap center for three different trap types: DC, ODC, and constant density (CD) using optimal trap parameters for each type. The average pairing values for CD and ODC traps are 0.0052 and 0.0051, with the DC trap at 0.0046. Entropy per fermion (0.95) and number of fermions (6600) is the same for each trap. Panel (b) shows average $d$-wave response as a function of entropy per fermion ($S$/$N$) for optimal DC, ODC, and CD traps.