Supersolid Phase Accompanied by a Quantum Critical Point in the Intermediate Coupling Regime of the Holstein Model

We reveal that electron-phonon systems described by the Holstein model on a bipartite lattice exhibit, away from half filling, a supersolid (SS) phase characterized by coexisting charge order (CO) and superconductivity (SC), and an accompanying quantum critical point (QCP). The SS phase, demonstrated by the dynamical mean-field theory with a quantum Monte Carlo impurity solver, emerges in the intermediate-coupling regime, where the peak of the $T_c$ dome is located and the metal-insulator crossover occurs. On the other hand, in the weak- and strong-coupling regimes the CO-SC boundary is of first order with no intervening SS phases. The QCP is associated with the continuous transition from SS to SC and characterized by a reentrant behavior of the SS around it. We further show that the SS-SC transition is hallmarked by diverging charge fluctuations and a kink (peak) in the superfluid density.

Figure 1

Phase diagram of the Holstein model plotted against (a) the chemical potential $\mu$ and (b) the electron band filling $⟨n⟩$ for ${\omega }_0=4$, $\lambda =3$. Blue areas indicate the supersolid (SS) region, while red diamonds at $T=0$ denote the quantum critical point (QCP). In the normal state, the dc conductivity is displayed with color coding.

Figure 2

SC and CO order parameters against $\mu$ (a) and against $⟨n⟩$ (b). Blue areas indicate the SS region. (c) Evolution of the filling as a function of $\mu$. (d) Inverse of the static charge susceptibility ${\chi }_{\mathbf{Q}}$ at $\mathbf{Q}=(\pi ,\pi )$, and the inverse of the squared London penetration depth ${\lambda }_L$, against the electron band filling. The parameters are ${\omega }_0=4$, $\lambda =3$, $\beta =35$.

Figure 3

(a) $\lambda \text{−}\mu$ phase diagram ($\mu$ is the chemical potential, $\lambda$ the phonon-mediated attraction) for the SS region for ${\omega }_0=4$, $\beta =35$. (b) Transition temperatures for CO and SC against $\lambda$ at half filling for ${\omega }_0=4$. CO is suppressed to obtain the $T_c$ for SC. In the normal state, the dc conductivity (difference from its value at $T=0.25$ for each $\lambda$) is shown with color coding. (c) Transition temperatures for CO and SC as a function of filling for ${\omega }_0=4$ without any restriction on the type of order.

Figure 4

(a) Phase diagram of the mean-field solution for the leading-order effective spin model [Eq. (2)] at ${\omega }_0=4$, $\lambda =4.5$. (b) $⟨n⟩-1$ as a function of $\mu$ at ${\omega }_0=4$, $\lambda =4.5$, $\beta =80$. The green area indicates a hysteretic region for the CO and SC solutions. (c) The phase diagram at $T=0$ of the fourth-order effective spin model. The blue region represents the supersolid state, and the dotted line the SC-CO boundary in the leading-order effective spin model. The inset shows the dependence of the next-nearest neighbor exchange interaction on $\lambda$. (d) Schematic picture of the motion of bipolarons (circles) on the $A$ sublattice (red squares) with those on the $B$ sublattice (blue) forming a CO pattern. Dotted arrows represent next-nearest neighbor hoppings arising from the higher-order terms in the effective spin model.