Semiclassical bifurcations and topological phase transitions in a one-dimensional lattice of coupled Lipkin-Meshkov-Glick models

In this work we study a one-dimensional lattice of Lipkin-Meshkov-Glick models with alternating couplings between nearest-neighbors sites, which resembles the Su-Schrieffer-Heeger model. Typical properties of the underlying models are present in our semiclassical-topological hybrid system, allowing us to investigate an interplay between semiclassical bifurcations at mean-field level and topological phases. Our results show that bifurcations of the energy landscape lead to diverse ordered quantum phases. Furthermore, the study of the quantum fluctuations around the mean-field solution reveals the existence of nontrivial topological phases. These are characterized by the emergence of localized states at the edges of a chain with free open-boundary conditions.

Figure 1

Chain of LMG models with alternating couplings ${\kappa }_1$ and ${\kappa }_2$. Gray-shaded elementary unit cell contains two nodes ($A$ and $B$). Each node is characterized by its pseudospin ${\mathit{J}}_{Pl}$ and parameters ${\omega }_0$ and $\gamma$.

Figure 2

(a) Bifurcation diagram for the system. Disordered phase I and four symmetry-broken phases II–V. The bifurcation from one to two degenerate ground states occurs at the phase boundaries of I. The gray box is for the future reference. (b) Dependence of the phase boundary on the value of $\gamma$ (labels of the contours). For $\gamma ⩾1$ only symmetry-broken phases are present. (c) Bulk band structure for particular points of the phase diagram depicted in panel a as calculated from Eqs. (20) and (21). Parameter values used in panel (c) are: $\gamma =0.5$, ${\kappa }_1=0.15$ and (from top to bottom) ${\kappa }_2=\pm 0.5$, $\pm 0.35$, $\pm 0.1$. The left and right columns of panel (c) correspond to the negative and positive values of the intercell coupling ${\kappa }_2$, respectively.

Figure 3

(a) Topological and quantum phases of the system [cf. gray box in Fig. 2]. Couplings ${\kappa }_{1,2}$ are assumed positive. Primed phases are topologically trivial while doubly primed phases (green-shaded) are nontrivial. (b) Macroscopic magnetization depending on position in the chain for $\gamma =0.5$, ${\kappa }_1=0.35$, ${\kappa }_2=0.25$ (empty squares), and ${\kappa }_2=0.5$ (filled squares). (c, d) Excitation energies for the open chain with $N=20$, $\gamma =0.5$, and $0⩽{\kappa }_2⩽1$ along the lines ${\kappa }_1=0.35$ and ${\kappa }_1=0.15$ respectively. (e–j) Microscopic fluctuations of the magnetization depending on the position in the chain and the number of the band. $\gamma =0.5$, ${\kappa }_2=0.5$. For panels (e–g), ${\kappa }_1=0.35$; for panels (h–j), ${\kappa }_1=0.15$. Each of the plots corresponds to respectively color-coded points in panels (c) and (d).