Glass transition of quantum hard spheres in high dimensions

We study the equilibrium thermodynamics of quantum hard spheres in the infinite-dimensional limit, determining the boundary between liquid and glass phases in the temperature-density plane by means of the Franz-Parisi potential. We find that as the temperature decreases from high values, the effective radius of the spheres is enhanced by a multiple of the thermal de Broglie wavelength, thus increasing the effective filling fraction and decreasing the critical density for the glass phase. Numerical calculations show that the critical density continues to decrease monotonically as the temperature decreases further, suggesting that the system will form a glass at sufficiently low temperatures for any density. The methods used in this paper can be extended to more general potentials, and also to other transitions such as the Kauzman/Replica Symmetry Breaking (RSB) transition, the Gardner transition, and potentially even jamming.

Figure 1

Phase diagram of quantum hard spheres in high dimensions as a function of inverse temperature $\beta$ and rescaled filling fraction $\widehat{\varphi }$ (see Sec. 1a for the precise definitions of $\widehat{\varphi }$ and $\widehat{M}$). The relevant qualitative physics is sketched above—as $\beta$ increases (left to right), quantum fluctuations increase the effective size of the spheres (see, also, Ref. [31]) and thus decrease the number of spheres per unit volume needed for glassiness.

Figure 2

Equation (34) (left-hand minus right-hand side) as a function of $A$ for different values of $\widehat{\varphi }$, specifically in the infinite-temperature (i.e., classical) limit. At the critical value ${\widehat{\varphi }}_c\approx 4.8067$, the curve acquires a zero near $A\approx 1.15$, indicating that the system is in a glass phase.

Figure 3

Comparison of ${\widehat{\varphi }}_c\left(\beta \right)$ estimated analytically at high temperatures (bottom orange curve—see Sec. 3) to that obtained numerically (top blue curve—see Sec. 4).

Figure 4

An example of a trajectory drawn from a period-1 Brownian bridge (dark green) and its reflected counterpart used to calculate the average of $r_{\text{max}}$ (light orange). At the first time where the green trajectory crosses $r=0.2$ (middle horizontal line), we reflect about $r=0.2$—while the green trajectory ends at $r=0$ (lower line), the orange trajectory ends at $r=0.4$ (upper line).

Figure 5

$G\left(\tau \right)$ for various values of $\widehat{\varphi }$ at $\beta /\widehat{M}=10$.

Figure 6

Distribution of $r_{\text{max}}$ for two choices of $\widehat{\varphi }$, representative of small and large filling fractions, both at $\beta /\widehat{M}=10$. Orange curves are best-fit Gumbel distributions. Left: Small $\widehat{\varphi }$, namely, the noninteracting limit. Right: Large $\widehat{\varphi }$.