Observation of the Berezinskii-Kosterlitz-Thouless Phase Transition in an Ultracold Fermi Gas

We experimentally investigate the first-order correlation function of a trapped Fermi gas in the two-dimensional BEC-BCS crossover. We observe a transition to a low-temperature superfluid phase with algebraically decaying correlations. We show that the spatial coherence of the entire trapped system can be characterized by a single temperature-dependent exponent. We find the exponent at the transition to be constant over a wide range of interaction strengths across the crossover. This suggests that the phase transitions in both the bosonic regime and the strongly interacting crossover regime are of Berezinskii-Kosterlitz-Thouless type and lie within the same universality class. On the bosonic side of the crossover, our data are well described by the quantum Monte Carlo calculations for a Bose gas. In contrast, in the strongly interacting regime, we observe a superfluid phase which is significantly influenced by the fermionic nature of the constituent particles.

Figure 1

First-order correlation function $g_1(r)$ for different temperatures at $\ln (k_F{a}_{2\mathrm{D}})\simeq -0.5$ (upper left panel) and $\ln (k_F{a}_{2\mathrm{D}})\simeq 0.5$ (lower left panel). The temperature scale used here is $t=T/T_{\mathrm{BEC}}^0$. (a) At high temperatures, correlations decay exponentially as expected for a gas in the normal phase. At low temperatures, we observe algebraic correlations [$g_1(r)\propto r^{-\eta (T)}$] with a temperature-dependent scaling exponent $\eta (T)$. (b) This qualitative change of behavior is clearly visible in the ${\chi }^2$ for both exponential and algebraic fits (right panel), where a small value signals a good fit. In particular, this allows for an accurate determination of the transition temperature $T_c$ (vertical dashed lines) [31].

Figure 2

Power-law scaling exponents across the two-dimensional BEC-BCS crossover. The temperature-dependent scaling exponent $\eta (T)$ in (a) the bosonic limit and (b) the crossover regime is shown. The relevant temperature scales in these cases are given by $T_{\mathrm{BEC}}^0$ and $T_F$, respectively. The crossover parameter $\ln (k_F{a}_{2\mathrm{D}})$ is mildly temperature dependent. For reference, we display the value at the critical temperature. For $\tilde{g}=0.60$ [$\ln (k_F{a}_{2\mathrm{D}})\simeq -7.3$], we show the prediction from QMC calculations for a Bose gas (filled red triangles) and an estimate of the effect of the finite imaging resolution present in the measured data (open red triangles) [31]. We find an exponent which increases with temperature in agreement with the BKT theory. The power-law decay eventually ceases at $T_c$, where a maximal exponent ${\eta }_c$ is reached. (c) The value of ${\eta }_c$ is approximately constant for all $\ln (k_F{a}_{2\mathrm{D}})$ where we have previously observed condensation of pairs [24]. This strongly suggests that the associated phase transitions are within one universality class.

Figure 3

Peak phase space density ${\mathcal{D}}_0=n_0{\lambda }_T^2$ obtained from the central density $n_0$. (a) and (b) show experimental and simulated data (for bosons) for the coupling strengths $\tilde{g}=2.76$ and $\tilde{g}=7.75$, respectively. The vertical dashed lines indicate the corresponding critical temperatures obtained from the measured onset of algebraic order. (a) We find excellent agreement between experiment and QMC for $\tilde{g}=2.76$, providing evidence that we realize a strongly interacting 2D Bose gas. We verify the applicability of ${\mathcal{D}}_c=\ln (380/\tilde{g})$ [39] at this interaction strength (horizontal dashed line). (b) For the stronger coupling $\tilde{g}=7.75$, however, we find the bosonic simulations to deviate from the measured results, indicating fermionic superfluidity.