Paramagnetic electron-nuclear spin entanglement in HoCo2Zn20

TL;DR

This work combines macroscopic thermodynamics with a mean-field crystal-field–hyperfine model to resolve the paramagnetic electron–nuclear state in HoCo$_2$Zn$_{20}$. The authors extract the CEF parameters $(W,x)$, the magnetic exchange $J_ ext{ex}$, and the hyperfine constant $A_ ext{HF}$ from magnetization and specific-heat data, revealing a $ ext{Gamma}_5$ CEF ground state whose hyperfine coupling splits the ground manifold into a $F=5/2$ quasi-sextet at low temperature. They show that the hyperfine-induced level scheme depends on the CEF parameter $x$ and that the hyperfine width exceeds $1 ext{ K}$ at zero field, highlighting the relevance of electron–nuclear entanglement for interpreting low-temperature properties and potential magnetic MCK phenomena in Ho-containing rare-earth materials. The findings establish a thermodynamic approach to map hyperfine entangled states and provide a framework for exploring MCK and related quantum phenomena in spin-active rare-earth systems.

Abstract

We investigated electron-nuclear spin entanglement in the paramagnetic ground state of the Ho-based cubic compound HoCo2Zn20. From analyses of magnetization and specific heat data, we determined the cubic crystalline electric field (CEF) parameters, the magnetic exchange constant, and the hyperfine coupling constant between the 4f magnetic moment and the 165Ho nuclear spin. Our results show that the Gamma5 CEF ground state is split by the hyperfine coupling, with an energy width of 1.3 K at 0 T, and that the true paramagnetic ground state is a quasi-sextet arising primarily from entanglement between the f-electron effective spin S = 1 and the 165Ho nuclear spin I = 7/2. We further demonstrate that, depending on the CEF parameters, the paramagnetic ground state can switch to an electron-nuclear coupled dectet. These findings underscore the importance of accurately identifying the electron-nuclear level scheme for understanding the low-temperature properties of rare-earth compounds containing spin-active nuclei.

Figures (11)

• Figure 1: Temperature dependence of the magnetization divided by the applied magnetic field $\mu_0 H = 0.1$ T, denoted as $\chi$ (left axis), and the inverse one, $(\chi - \chi_0)^{-1}$ (right axis), obtained after subtracting a temperature-independent term $\chi_0$, in HoCo$_2$Zn$_{20}$ for $H \parallel [110]$. Data for $H \parallel [100]$ and $[111]$ are presented in Fig. \ref{['fig:chiT_all']}. The blue solid line represents a fit to the modified Curie--Weiss law, and the gray dashed line indicates the expected slope of the inverse susceptibility for free Ho$^{3+}$ ions. The inset shows an enlarged view of $\chi$ and $(\chi - \chi_0)^{-1}$ below 30 K.
• Figure 2: Temperature dependence of the specific heat for HoCo$_2$Zn$_{20}$ at 0, 1, 4, and 9 T applied along the $[110]$ direction, and for LuCo$_2$Zn$_{20}$ at 0 T. The inset shows the electrical resistivity of HoCo$_2$Zn$_{20}$ at 0 T.
• Figure 3: Phase transition in HoCo$_2$Zn$_{20}$ observed in (a) the specific heat contribution from Ho sites, $C_\mathrm{Ho}$, and (b) the magnetization divided by the applied magnetic field, $M/H$. The transition temperatures, indicated by downward-pointing triangles, are defined as the temperatures at which $C_\mathrm{Ho}(T)$ or $M(T)/H$ exhibits a maximum.
• Figure 4: Magnetization curves $M(H)$ of HoCo$_2$Zn$_{20}$ measured at 0.29 K and 2 K for $H \parallel [110]$. The inset shows an enlarged view of $M(H)$ (lines with symbols, left axis) and the differential magnetization $dM/dH$ (lines without symbols, right axis).
• Figure 5: $H$--$T$ phase diagram of HoCo$_2$Zn$_{20}$ for $H \parallel [110]$. AFM and PM denote the antiferromagnetic and paramagnetic states, respectively. Open circles indicate transition temperatures calculated from Eq. \ref{['eq:Hamil_MF_total']} using the refined parameters given in Secs. \ref{['subsec:cef_refinement']} and \ref{['subsec:hyperfine_refinement']}.