Dimensional crossover of quantum critical dynamics in many-body phase transitions

The dimensional crossover induces varieties of quantum phenomena. In this paper, we demonstrate the quantum critical dynamics under dimensional crossover involving many-body phase transitions by continuously suppressing correlations and tunnelings along one direction of bulk materials, which provides a smooth connection from higher dimensions to lower ones based on the intrinsic correlations rather than geometry tailoring. By measuring the nonadiabatic excitations, the critical scaling laws in both three and two dimensions are observed and are consistent with predictions. In addition, we find scaling behaviors for intermediate regimes with noninteger dimensions. This study provides insights to extend the descriptions of critical exponents into more general or complex scenarios.

Figure 1

The incoherent fraction for anisotropic optical lattices. (a) The incoherent fraction ${\gamma }_{\text{inc}}$ vs the trap depth $V_{xy}$ and $V_z$. The color bar from dark to light corresponds to the incoherent fraction from large to small. When ${\gamma }_{\text{inc}}$ equals 1, the quasimomentum distribution is exactly a flat plateau. The dashed line is a label for the data shown in panel (b). (b) ${\gamma }_{\text{inc}}$ vs $V_z$ at $V_{xy}=17E_r$ (blue circles with error bars of one standard deviation). The solid line is a fitting guideline. The square and the diamond correspond to the data with $V_z=17E_r$ and $30E_r$, respectively, with their quasimomentum distribution in panels (c) and (e). The inset shows the ramping curves of $V_{xy}$ and $V_z$. (c) The 2D quasimomentum distribution with $V_{xy}=V_z=17E_r$. The color bar corresponds to the relative atomic density obtained from the absorption image along $z$. (d) The quasimomentum distribution along the $y$-direction by integrating the $x$-direction with $V_{xy}=V_z=17E_r$. The plateau labeled by a gray shadow corresponds to the incoherent component, and the peak corresponds to the coherent component. The dashed line labels the first Brillouin zone. (e) The quasimomentum distribution with $V_{xy}=17E_r$ and $V_z=30E_r$.

Figure 2

The excitation vs the ramping. (a) Excitation fraction $n_{\text{ex}}$ vs ramping speed, where $n_{\text{ex}}$ characterizes the topological defects or excitation due to the nonadiabatic ramping. The detailed ramping protocol is shown in panel (b), where the black, blue, and yellow circles correspond to the data of the two-dimensional ramping with different fixed $V_z=17E_r, 19E_r$, and $25E_r$. The red triangles correspond to the data of the three-dimensional ramping. The solid lines are the fit of scaling laws $n_{\text{ex}}\propto k^{\alpha }$. The dashed lines are the extensions of fitting outside of the data range. Based on the fit, $\alpha$ equals 0.94(10), 0.74(4), and 0.63(9) for the 2D ramping with $V_z=17$, 19, and $25E_r$, and $\alpha$ equals 0.93(11) for the 3D ramping. The error bars correspond to one standard deviation, and the shaded area corresponds to the 95% confidence interval of the fitting. (b) The ramping protocol of $V_{xy}$ and $V_z$ for the two-dimensional and three-dimensional schemes, where the horizontal axis is not plotted in scale. In both protocols, two gray lines correspond to the initial value $V_i$ and the final value $V_f$ of ramping. (c) The quasimomentum distribution. Left 1: the adiabatic states at $V_{xy}=5E_r, V_z=19E_r$; Left 2: the adiabatic states at $V_{xy}=17E_r, V_z=19E_r$; Left 3: the dynamical states at $k=1.6E_r$/ms, $V_{xy}=V_f=17E_r, V_z=19E_r$; and Left 4: the dynamical states at $k=3.1E_r$/ms, $V_{xy}=V_f=17E_r, V_z=19E_r$.

Figure 3

The critical exponent $\alpha$ vs the third dimension suppression. (a) The critical exponent $\alpha$ vs the trap depth $V_z$ of the third dimension. The hollow circles correspond to the experimentally obtained $\alpha$ by fitting $n_{\text{ex}}\propto k^{\alpha }$ for different $V_z$. The blue dashed line is a guideline of data. The purple triangles correspond to the critical exponent $\alpha$ under the 2D ramping calculated by the Gutzwiller mean-field theory. The yellow hollow square corresponds to the measured $\alpha$ for the 3D ramping. The theoretically predicted $\alpha$ for the 3D system is 1, while for the 2D systems it is 2/3, both marked by red dashed lines. (b) $n_{\text{ex}}$ vs the ramping speed $k$ for different $V_z$ and 3D ramping procedures. The blue circles correspond to the measured data, and the red squares correspond to the calculation from the Gutzwiller mean-field theory. The solid lines are the fitting results. All the error bars correspond to the same standard deviation. The light-shaded areas correspond to a $95\%$ fitting confidence interval.

Figure 4

Correlation functions along the $z$-direction. The main panel shows the Green functions vs distance $z$ at $V_{xy}=15E_r$. The colored dots are numerical results by QMC simulations for different $V_z$ ranging from $13E_r$ to $27E_r$. The solid lines are obtained by fitting each set of dots with $G(z,0)=G_{0}exp(-b\times z^c)$. The black solid line in the inset shows the correlation length $\xi$ vs $V_z$. The correlation length $\xi /a$ decreases smoothly from 2.4(4) to much less than 1. The error bars mainly come from fitting errors. The red dashed line indicates a downtrend of relative tunneling amplitude $t_z/t_{xy}$ as $V_z$ increases.

Figure 5

${\gamma }_{\text{inc}}\left(\text{adiabatic}\right)$ vs $V_z$ at $V_{xy}=17E_r$. The incoherent fraction increases with $V_z$, showing the same trend as experiments.

Figure 6

The defect density $n_{\text{defect}}$ vs $n_{\text{ex}}$. We ramp $V_{xy}$ from $5E_r$ to $17E_r$ with different speed $k$ at $V_z=17E_r, 21E_r$, and $25E_r$. Then we apply linear fitting on three sets of data to characterize the dependence of $n_{\text{ex}}$ on $n_{\text{defect}}$. The hollow circles are GMFT results and the dashed lines are the fitting results. For blue circles, $V_z=17E_r$ and $n_{\text{defect}}=0.32\left(1\right)\times n_{\text{ex}}+0.029\left(4\right)$. For yellow circles, $V_z=21E_r, n_{\text{defect}}=0.30\left(1\right)\times n_{\text{ex}}+0.020\left(6\right)$. For green circles, $V_z=25E_r, n_{\text{defect}}=0.31\left(1\right)\times n_{\text{ex}}+0.016\left(1\right)$. The slopes are consistent with each other.

Figure 7

The $Z$-configuration (upper panel) and the $G$-configuration (bottom panel). The $Z$-configuration contains only closed worldlines, whereas the $G$-configuration allows open worldlines with $Ira$ (${\tau }_I\ne 0$) and $Ira$ (${\tau }_M\ne \beta$) at the beginning and end of the worldlines. The update of the system is accomplished by simple operations on the worm pair of $Masha$ and $Ira$.

Figure 8

The particle distribution at $V_{xy}=15E_r$, and $V_z=15E_r, 21E_r$, and $27E_r$. We apply two numerical methods to simulate the atom number distribution in the $x-y$ plane with $z=0$, and $r$ is defined as $\sqrt{x^2+y^2}$. The smooth lines are the GMFT results, which show consistencies with the zigzag lines by QMC. As $V_z$ gets bigger, the center plateau extends as the incoherent fraction gets larger in the Mott-insulator regime.

Figure 9

The correlation length $\xi$ vs $V_z$ at $V_{xy}=13E_r, 15E_r$, and $17E_r$. The blue and yellow circles are simulated data for $V_{xy}=13E_r, 15E_r$, and $17E_r$, and the corresponding curves are fitted lines. The correlation length $\xi$ decreases as $V_{xy}$ or $V_z$ increases.

Figure 10

The tunneling amplitudes $t_z$ and on-site interaction energy $U$ vs trap depth $V_z$ along the $z$-direction. The blue line with the left axis shows that $t_z$ decays as $V_z$ grows. The red line with the right axis shows the ratio $t_z/U$. The inset shows $U$ vs $V_z$.

Figure 11

Calibration of the ramping adiabaticity. The insets show two time sequences of ramping trap depth. The left one is a simple 80 ms ramping to $V_i=10E_r$ and a 120 ms holding. The right panel is an anisotropic ramping up (80 ms), a holding (20 ms), a ramping down (80 ms), and a holding (20 ms). The main panel shows the measurement results. The red belt shows ${\gamma }_{\text{inc}}$ in the left inset with one standard deviation. The black dots with error bars (one standard deviation) show ${\gamma }_{\text{inc}}$ obtained from the right inset.