Spin-liquid and spin-glass behavior in quantum spin models with all-to-all p-spin interactions

Abstract

Spin-liquid and spin-glass states represent two distinct phases of disordered quantum spin systems. These states are, in principle, distinguished by quantum-entangled fluctuations and spin freezing, but identifying each state and characterizing the transition between them remains challenging. Here, we systematically explore the relationship between the spin-liquid and spin-glass states using a model with all-to-all random interactions among p spins, which interpolates between the Ising-like one-component, XY-like two-component, and isotropic three-component cases. By analyzing the system-size N dependence of the Edwards-Anderson order parameter and the density of states, we identify the transition from the spin liquid to the spin glass for various values of p. We show that the phase diagrams for different p can be unified through a scaling with N/p2, revealing that increasing anisotropy in the interactions systematically suppresses the spin-liquid phase and extends the spin-glass regime. Furthermore, we examine the competition between multiple-spin interactions and anisotropy under an external magnetic field in the isotropic case, and find that the spin-liquid phase transitions into the spin-glass phase before entering a quantum paramagnetic phase. Our findings provide insights into quantum disordered phases and the transitions between them.

Figures (12)

• Figure 1: Schematic illustrations of $p$-spin interactions in the model in Eq. \ref{['eq:H_general']} for two cases: (a) $(\varepsilon_x, \varepsilon_y) = (0, 0)$, corresponding to an Ising-like one-component model and (b) $(\varepsilon_x, \varepsilon_y) = (1, 1)$, corresponding to an isotropic three-component model. Blue spheres represent $N$ sites, and colored arrows indicate spins involved in a $p$-spin interaction highlighted by the purple region in each illustration ($p=4$ in this case). Spins are defined on all blue spheres, but for clarity, they are explicitly shown only in the purple-highlighted region.
• Figure 2: Setup of $(\varepsilon_x, \varepsilon_y)$. The cases $(\varepsilon_x, \varepsilon_y) = (1, 1)$, $(1, 0)$, and $(0, 0)$ are discussed in Secs. \ref{['sec:isotropic']}–\ref{['sec:Ising']}, respectively. The connections between these points, represented by the three arrows, are analyzed in Sec. \ref{['sec:diagram']}.
• Figure 3: EA order parameter $q_{\mathrm{EA}}$ for the isotropic $p$-spin model with $(\varepsilon_x, \varepsilon_y) = (1, 1)$ for (a) $p = 3$, (b) $p = 4$, (c) $p = 5$, and (d) $p = 6$. The blue, green, and red dots indicate decreasing trends, minima, and increasing trends with respect to $N$, respectively. For even $p$, these trends are determined separately for even and odd $N$. The data are obtained by averaging over 300 random samples.
• Figure 4: Density of states for the isotropic $p$-spin model with $(\varepsilon_x, \varepsilon_y) = (1, 1)$ for (a) $p = 3$, (b) $p = 4$, (c) $p = 5$, and (d) $p = 6$. The data are obtained by averaging over $2^{24 - N}$ random samples.
• Figure 5: Corresponding plots to Fig. \ref{['fig:qEA_isotropic']} for the $XY$$p$-spin model with $(\varepsilon_x, \varepsilon_y)=(1, 0)$. The data are obtained by averaging over 300 random samples.