Quantum Biroli-Mézard model: Glass transition and superfluidity in a quantum lattice glass model

We study the quantum version of a lattice model whose classical counterpart captures the physics of structural glasses. We discuss the role of quantum fluctuations in such systems and in particular their interplay with the amorphous order developed in the glass phase. We show that quantum fluctuations might facilitate the formation of the glass at low enough temperature. We also show that the glass transition becomes a first-order transition between a superfluid and an insulating glass at very low temperature, and is therefore accompanied by phase coexistence between superfluid and glassy regions.

Figure 1

Illustration of a cavity iteration for the classical system. The sites $1,\dots ,k$ are described, in the absence of site $0$, by cavity fields ${\psi }_l^{e,u,s}$, $l=1,\dots ,k$. Integrating out these sites leads to the expression of ${\psi }_0^{e,u,s}$ in Eq. (2).

Figure 2

Classical phase diagram in the $(\rho ,T)$ plane for $c=3$ and $\ell =1$. The dynamical ($T_d$) and Kauzmann ($T_K$) temperatures are plotted as a function of the density. See Ref. 47 for a study of a realistic model that displays a similar phase diagram.

Figure 3

Left panel: Average density as a function of the chemical potential, at $\beta =8$ (upper panel) and $\beta =30$ (lower panel). The inset shows a zoom of the $\beta =30$ data around the Kauzmann transition. Right panel: Behavior of the parameter $m^{*}$ as a function of the density at $\beta =8$ and $\beta =30$.

Figure 4

Left panel: Order parameters of the superfluid (${\rho }_c/\rho$, black circles, left scale, RS cavity method) and glass ($q_{\mathrm{EA}}$, red squares, right scale, 1RSB cavity method) phases for $c=3$ and $\ell =1$, for three representative values of the hopping $J$ at the same temperature $\beta V=15$, as functions of the chemical potential $\mu$. The first-order transitions are accompanied by hysteresis; full lines corresponds to following the solution upon increasing $\mu$, dashed lines correspond to decreasing $\mu$. Arrows mark the first-order transition points ${\mu }_c$ as determined by the Maxwell construction. See text for a more detailed description. Right panel: Sketches of the free energy as a function of $\mu$ for the corresponding left panels. From left to right, black lines represent the superfluid, red lines the normal liquid, blue lines the glass; ${\mu }_c$ and ${\mu }_K$ indicate the phase transitions (we do not report data because the jump in the derivative of $f$ at the first-order transition is very small and would be invisible in the figure).

Figure 5

Phase diagram for the quantum model in the $(\mu /V,J/V)$ plane, for $c=3$ and $\ell =1$, for three different temperatures $\beta V=8,15,$ and $30$. The lines divide the phase diagram into three main regions where the system is found in a glass, superfluid, or (normal) liquid phase. Solid lines and circles indicate the second-order superfluid transition. The large dots mark the tricritical point where the superfluid line changes from second to first order. Dash-dotted lines and squares represent the first-order transition between the superfluid phase and the normal (glass or liquid) phase. Dotted lines and triangles indicate the Kauzmann glass transition $J_K(\mu )$. The dot-dot-dashed black line indicates the normal liquid–superfluid transition for hard core bosons at zero temperature. Since below this line $\rho =0$ at $T=0$, the interaction is not relevant around it, so this line is the $T=0$ limit of the normal liquid–superfluid transition also for the model investigated here. The dashed black line serves as a guide to the eye; it has been obtained by interpolating the large $\mu$ superfluid lines at different temperatures (see text for details).

Figure 6

Left panel: Schematic $(J/V,\rho )$ phase diagram of the model. The full line represents the second-order superfluid transition $J_c(\rho )$, separated by a tricritical point (large dot) from a first-order superfluid transition accompanied by phase coexistence (dot-dashed lines). The dotted line represents the Kauzmann transition $J_K(\rho )$. In the limit $T\to 0$, the transition lines have distinct behaviors: The first-order transition line has $J\gg T$, and it has a finite limit for $T\to 0$. On the contrary, the other lines have $J\propto T$ and they shrink to the $J=0$ axis for $T\to 0$. Therefore, at $T=0$ and for $J>0$, the low-$\rho$ part of the phase diagram contains the superfluid phase while the large-$\rho$ part contains the glass. The red lines indicate the behavior of the superfluid-glass transition at $T=0$. Right panel: Data for the $(\beta J,\rho )$ phase diagram, for $c=3$ and $\ell =1$, and for three different temperatures $\beta V=8,15,$ and $30$. The three regions, glass, superfluid, and normal liquid phase, are here reported. The transition lines are plotted using the same styles as in Fig. 5. As a consequence of the first-order phase transition a region of phase coexistence—delimited by dash-dotted lines—is present. The inset shows the reentrant behavior of the glass transition line, at low enough temperature. We plot $\beta J$ on the vertical axis in order to show that the transition lines are proportional to $T$ in the limit $T\to 0$.

Figure 7

A schematic plot of the typical behavior of the complexity as a function of the internal entropy of the states for various densities $\rho$ increasing from top to bottom.

Figure 8

Time-dependent correlation functions obtained with the RS cavity method. Upper panel: Imaginary time advanced Green’s function $G_{>}(\tau )$, for $c=3$ and $\ell =1$, at $\beta V=30$ and $J/V=0.1$, in the superfluid phase at $\mu /V=0.275$ and in the liquid phase just before the Kauzmann transition, at $\mu /V=0.388$. The dotted line indicates the square of the expectation value of the bosonic operator ${\langle \mathrm{â}\rangle }^2$ corresponding to the superfluid phase. Note that the Green’s function $G(\tau )$ is obtained by the periodic image of period $\beta$ of this function for $\tau \in [-\beta /2,\beta /2]$. Bottom panel: Time-dependent density-density correlation for the two same regimes. The dotted line indicates the value of the $q_{\mathrm{EA}}$ for the normal liquid (obtained with the 1RSB cavity method).

Figure 9

Ground state density (a) and $q_{\mathrm{av}}$ (b) as functions of $\mu /V$, with $c=3$ and $\ell =1$, at hopping $J/V=0.3$, obtained by exact diagonalization in the grand-canonical ensemble. The maroon triangles show the result of the cavity method for the same parameters at $\beta V=15$. In correspondence to the jump there is the first-order superfluid-glass phase transition. Upper panels: The black circles, blue squares, and red diamonds represent the ground state density and $q_{\mathrm{av}}$ for some selected realizations of random graphs of size, respectively, $N=20,22,24$. Lower panels: Average of the same quantities over 50 random graphs.

Figure 10

Exact-diagonalization results in the canonical ensemble for $c=3$ and $\ell =1$, for three different sizes $N=20,24,28$, with $N_p=12,14,17$ occupied sites respectively ($\rho ~0.6$), as functions of $J/V$. Results are shown for some representative graphs chosen according to the criterion discussed in the text. (a) The condensate density is seen to decrease with $N$ at small $J$, while it is almost independent of $N$ at large $J$. In the inset we show the average (over 50 accepted graphs) of the largest eigenvalue $n^{\max }$ of the one-body density matrix for two different hopping strengths in the superfluid and glass phase, as a function of $N$. The former increases linearly with $N$; the latter is independent of $N$. (b) Behavior of $q_{\mathrm{av}}$ for the same graphs.