Classical and Quantum Liquids Induced by Quantum Fluctuations

Geometrically frustrated interactions may render classical ground states macroscopically degenerate. The connection between classical and quantum liquids and how the degeneracy is affected by quantum fluctuations is, however, not completely understood. We study a simple model of coupled quantum and classical degrees of freedom, the so-called Falicov-Kimball model, on a triangular lattice and away from half-filling. For weak interactions the phase diagram features a charge disordered state down to zero temperature. We provide compelling evidence that this phase is a liquid and show that it is divided by a crossover line that terminates in a quantum critical point. Our results offer a new vantage point to address how quantum liquids can emerge from their classical counterparts.

Figure 1

(a) Phase diagram of the FK model on the triangular lattice in the $T\text{−}U$ plane for $x_f=1/3$ and $x_c=2/3$. The red star indicates the quantum phase transition between the liquid states. The inset depicts a sketch of the phases in the thermodynamic limit. Band structure assuming CDW order for $U=2$ (b) and $U=5$ (c).

Figure 2

Specific heat as a function of $T$ for $U=2$ (a), $U=5$ (b), and $U=7$ (c). Momentum resolved susceptibility of the $f$ charges $\chi (q)$ for $U=5$ (d) and $U=7$ (f). (e) Finite size scaling of $\chi (Q_{\mathrm{Max}})$ with $L$. PCA yielding (g) the 3 most important components for $U=5$ and (h) 2 for $U=7$, plotted for a range of $T$ for $L=20$ and $L=21$, respectively, see Supplemental Material (s4)[25].

Figure 3

$c$-electron properties: DOS for $U=1.6$ (a) and $U=2.7$ (c) for $T=0.015$. Inset of (c) shows $\mathrm{DOS}(\omega =0)$ vs $U$ for different $T$. IPR as a function of energy for different system sizes (b). (d) Scaling of the $\mathrm{IPR}(\omega =0)$ as a function of $U$. Data obtained from simulations with 100 000 Monte Carlo steps with correlation times of at most 20 steps.

Figure 4

Order parameters ${\varphi }_{1/3}$ and ${\varphi }_{1/4}$ for (a) the FK model, (b) the effective classical model truncated to 4th order, (c) an effective variational model, with couplings determined by a linear regression. The inset of (a) and (c) show the scaling of the ${\varphi }_{1/4}$ with system sizes for $U=5$, respectively, for the FK and variational models. The inset in (b) depicts the schematic couplings between sites obtained up to 4th order.