Signature of Quantum Criticality in the Density Profiles of Cold Atom Systems

In recent years, there has been considerable experimental effort using cold atoms to study strongly correlated many-body systems. One class of phenomena of particular interest is quantum critical (QC) phenomena. While prevalent in many materials, these phenomena pose notoriously difficult theoretical problems due to the vanishing of energy scales in the QC region. So far, there are no systematic ways to deduce the QC behavior of bulk systems from the data of trapped atomic gases. Here, we present a simple algorithm to use the experimental density profile to determine the $T=0$ phase boundary of bulk systems, as well as the scaling functions in the QC regime. We also present another scheme for removing finite-size effects of the trap. We demonstrate the validity of our schemes using exactly soluble models.

Figure 1

Phase diagram of 1D hard-core Bose gas. I and II are the quantum critical points of vacuum (V) to TLL and Mott (M) to TLL transitions. The quantum critical regions associated with quantum critical points I and II are denoted as QCI and QCII, and are given by $|\mu +2J|<T$ and $|\mu -2J|<T$, respectively. The shaded region in the vacuum and the Mott phase represent the low fugacity regime of particles and holes.

Figure 2

Scaled density (top panels) and scaled compressibility (bottom panels) vs $\mu (x)=\mu -\frac{1}{2}M{\omega }^2{x}^2$ at different temperatures $T/J=0.1$, 0.15, 0.2, and 0.25 (red dots, blue boxes, purple diamonds, and brown triangles), for the 1D hard-core Bose gas discussed in the text. The left and right panels are results for transitions I and II, respectively. When $z$ is chosen to be 2 and $\nu =1/2$, all scaled densities (compressibilities) intersect at ${\mu }_{\mathrm{I}}^{*}/J=-1.98(-1.94)$ and ${\mu }_{\mathrm{II}}^{*}/J=1.998(1.991)$. The exact values ${\mu }_c$ of these critical points are $-1$ and 1. For both critical points I and II, we plot the range of chemical potential over the same interval of lattice sites around the location of the transition.

Figure 3

Scaled density (top panels) and scaled compressibility (bottom panels) vs $\mathbf{(}\mu (x)-{\mu }_c\mathbf{)}/T^{z\nu }$. All the data collapse onto a single curve. The parameters are the same as in Fig. 2. The left (right) figures correspond to the critical point I (II). The curves are in the same color and shape scheme as in Fig. 2.

Figure 4

Determining ${\mu }_c$ of bulk systems using a finite-size scaling scheme: the scaled density and scaled compressibility are plotted against $\mu$ for $T/J=0.02$, 0.03, 0.05, and 0.07 (red dots, blue boxes, purple diamonds, brown triangles, and pink inverted triangles) for 1D hard-core bosons in harmonic traps. The value $\mathcal{D}$ defined in the text is set equal to 0.25.

Figure 5

Scaled density (top panels) and scaled compressibility (bottom panels) vs $(\mu -{\mu }_c)/T^{1/z\nu }$ with $\nu z=1$. All curves in Fig. 4 collapse onto a single curve. The parameters are the same as in Fig. 4. The curves are in the same color and shape scheme as in Fig. 4.