Direct Mapping of the Finite Temperature Phase Diagram of Strongly Correlated Quantum Models

We propose a general method for obtaining the finite temperature phase diagram of strongly correlated quantum models emulated by optical lattice experiments using only the density profile of atoms in the trap. We illustrate the procedure explicitly for the Bose-Hubbard model by using “exact” quantum Monte Carlo simulations in a trap with up to ${10}^6$ bosons. We show that kinks in the local compressibility, arising from critical fluctuations, demarcate the boundaries between superfluid and normal phases in the trap. The temperature of the bosons in the optical lattice is determined from the density profile at the edge. Our method can be applied to other phase transitions even when reliable numerical results are not available.

Figure 1

Hubbard and Heisenberg-type models form a bridge between strongly correlated materials and optical lattice emulators.

Figure 2

Phase diagram of BHM. (a) $\mu -t/U$ plane at $T=0$ (schematic); (b) $\mu -t/U$ plane at $T\ne 0$ (schematic). The superfluid (S) phase defined by order parameter $⟨b^{†}⟩\ne 0$ and superfluid density ${\rho }_s\ne 0$, both of which vanish in the normal phase. The Mott insulator at $T=0$, characterized by integer filling and a finite gap to excitations ${\Delta }_0$, crosses over to the normal (N) state for $T\approx {\Delta }_0$. At finite $T$, the critical ${\mu }_c(T,t/U)$ demarcates the S-N phase boundary. The local $\mu$ decreases from the center of the trap to the edge (brown vertical line). (c) QMC simulations and finite size scaling to calculate the phase diagram in the $\mu -T$ plane for a fixed $t/U$.

Figure 3

Left panels: Interference pattern after expansion for $N\approx {10}^5$ bosons in a trap. Right panels: Local density profile $\rho (r)$ (red boxes) obtained using QMC data as a function of the radial coordinate in the trap. Note excellent agreement with ${\rho }^h\mathbf{(}\mu (r)\mathbf{)}$ calculated within local density approximation (LDA) (purple diamonds). The superfluid density ${\rho }_s^h\mathbf{(}\mu (r)\mathbf{)}$ is nonzero in a significant portion of the trap only at sufficiently low $T$ in panel (d).

Figure 4

Kinks in the local compressibility: (a) The local density $\rho (r)$ (blue stars) agrees with ${\rho }^h(r)$ (purple diamonds) calculated for a homogeneous system. The compressibility ${\kappa }_{\mathrm{diff}}(r)$ (red boxes) directly obtained from the density profile agrees with ${\kappa }^h\mathbf{(}\mu (r)\mathbf{)}$ (pink triangles) calculated within LDA. The temperature $T/t=0.1$. (b) Determination of $T$ by fitting (lines) density profile at edge of trap to Eq. (6) (symbols are QMC results). The inset shows the fitting to angular integrated densities at the corresponding $T$ with larger signals and better accuracy. $T/t$ are 0.67, 0.57, and 0.44 from top to bottom. In both (a) and (b), $t/U=0.15$, $N={10}^5$, and $\omega =2\pi \times 30\mathrm{Hz}$.

Figure 5

Origin of kinks: The local density ${\rho }^h\mathbf{(}\mu (r)\mathbf{)}$ (purple diamonds), compressibility ${\kappa }^h\mathbf{(}\mu (r)\mathbf{)}$ (red boxes) and superfluid density ${\rho }_s^h\mathbf{(}\mu (r)\mathbf{)}$ (blue triangles) as functions of the radial coordinate in the trap. Panel (a) shows when ${\rho }_s=0$ in the trap, $\kappa (r)$ shows a smooth variation. As $T$ is reduced or $t/U$ is increased in (b),(c),(d) and a finite ${\rho }_s$ develops in some portion of the trap, $\kappa (r)$ shows sharp kinks. The location of these kinks coincides with the S-N boundary in the trap.