Quantum phase transitions in networks of Lipkin-Meshkov-Glick models

We study the quantum critical behavior of networks consisting of Lipkin-Meshkov-Glick models with an anisotropic ferromagnetic coupling. We focus on the low-energy properties of the system within a mean-field approach and the quantum corrections around the mean-field solution. Our results show that the weak-coupling regime corresponds to the paramagnetic phase when the local field dominates the dynamics, but the local anisotropy leads to the existence of an exponentially degenerate ground state. In the strong-coupling regime, the ground state is twofold degenerate and possesses long-range magnetic ordering. Analytical results for a network with the ring topology are obtained.

Figure 1

Ring network of LMG models. The coupling between neighboring sites is determined by parameter $\kappa$, the strength of interaction within a single site is determined by $\gamma$, and $g$ models an external field.

Figure 2

The phase diagram for the ring network of LMG models. The classical energy surfaces $E_g\left(\alpha \right)$ are shown in each of the characteristic regions.

Figure 3

Numerically calculated ground-state energy along the path ABCDEFGA in the $\gamma \kappa$ space (see contour plots) for the LMG rings with the number of sites $N=1$ (a) and $N=3$ (b); and the star-type network of $N=4$ sites (c). Different single-node total angular momentum values are drawn with thin lines; the expected thermodynamic limits are drawn with thick red lines. Contour plots show the ground-state energy for different $\gamma$ and $\kappa$ values. Orange (light) lines and circles denote the first-order QPT, while brown (dark) ones denote the second-order QPT.

Figure 4

The excitation energy dispersion $\varepsilon \left(k\right)$ across the I-II boundary (a), across the I-III boundary (b), and across the II-III boundary (c) as well as the energy gap $\varepsilon (k=0)$ as a function of $\gamma$ (d). Panels (e) and (f) show the speed of sound ${(d\varepsilon /dk)}_{k=0}$ along the phase boundaries.

Figure 5

Microscopic parts of correlation functions $C_{\xi \xi }\left(r\right)-j{M}_g$ in region I with $\gamma /g=0.5$ and $\kappa /g=0.5$ (a), region II with $\gamma /g=0.5$ and $\kappa /g=1.5$ (b), and region III with $\gamma /g=1.5$ and $\kappa /g=0.5$ (c).