Theory of the superglass phase

A superglass is a phase of matter which is characterized at the same time by superfluidity and a frozen amorphous structure. We introduce a model of interacting bosons in three dimensions that displays this phase unambiguously and that can be analyzed exactly or using controlled approximations. Employing a mapping between quantum Hamiltonians and classical Fokker-Planck operators, we show that the ground-state wave function of the quantum model is proportional to the Boltzmann measure of classical hard spheres. This connection allows us to obtain quantitative results on static and dynamic quantum correlation functions. In particular, by translating known results on the glassy dynamics of Brownian hard spheres we work out the properties of the superglass phase and of the quantum phase transition between the superfluid and the superglass phase.

Figure 1

(Color online) Form of the pair potential $v^{\text{pair}}\left(r\right)$ in the mapped quantum problem that derives from the classical potential $V\left(r\right)=V_0\text{exp}\left(-\lambda \left[{\left(r/\sigma \right)}^2-1\right]\right)$. (Notice that we have set $V_0=1$ and $\sigma =1$ in the text.)

Figure 2

(Color online) Classical phase diagram for hard sphere systems (top) and its quantum counterpart (bottom).

Figure 3

(Color online) Condensate fraction as a function of the packing fraction for the hard sphere Jastrow wave function (Ref. 43). Full (black) curve: superfluid phase [Eq. (20)]. Dotted-dashed (blue/dark gray) curve: supercrystal phase [Eq. (24)]. Dashed (red/gray) curve: superglass phase [Eq. (39)]; note that the theory slightly overestimates the random close packing density with respect to the commonly accepted value of $\varphi =0.64$. The blue/dark gray dot marks the location of the melting transition, where liquid-crystal phase coexistence begins, while the red/gray dot marks the glass transition density. Inset: the enhancement factor $n_0/f\left(\rho \right)$ due to the spatial inhomogeneity of the crystal and glass phases. Note that this factor is quite large, of the order of ${10}^2–{10}^3$ at the melting and glass transition densities. Still, the condensate fractions we find are extremely small, probably due to the classical-like nature of solids with the Jastrow hard sphere wave function (Ref. 43).

Figure 4

(Color online) Correlation function of the density (left) and of the condensate wave function (right) for the quantum system corresponding to classical hard spheres at different densities in the glass phase. Details of the computations are in Ref. 33. Note that these quantities can be measured in quantum Monte Carlo numerical simulations (see, e.g., Ref. 20).

Figure 5

(Color online) (Left) Long-time shape of the correlation function [Eq. (49)] $F_{\text{cl}}\left(q,t\right)\sim \text{exp}\left(-\sqrt{t/{\tau }_{\alpha }}\right)$ and $F_Q\left(q,t\right)\sim \text{exp}\left(-\sqrt{t/2{\tau }_{\alpha }}\right)\text{cos}\left(\sqrt{t/2{\tau }_{\alpha }}\right)$ as a function of $t/{\tau }_{\alpha }$ in a linear-logarithmic scale. (Right) Imaginary part of the response function of the quantum problem [Eqs. (48, 49)] as a function of frequency for different values of ${\tau }_{\alpha }$ (in arbitrary units).

Figure 6

Schematic behavior of $e\left(\rho \right)$, $P\left(\mu \right)$, $\mu \left(\rho \right)$, and $P\left(\rho \right)$ across the glass transition. Note that there are values of $P$ and $\mu$ that correspond to the same density ${\rho }_K$.

Figure 7

(Color online) Form of the pair potential $v^{\text{pair}}\left(r\right)$ in the mapped quantum problem that derives from the classical $\alpha =6$ and $\beta =12$ Lennard-Jones potential. ($\epsilon =1$ is set for simplicity.) The potential is repulsive at large and small distances, but to see the repulsion for large $r$ one needs to zoom closer, as shown in the inset.