Hyperfine coupling in singlet ground state magnets

TL;DR

This work analyzes how hyperfine coupling to nuclear spins and nuclear quadrupolar splitting modify induced moment order in non-Kramers singlet-ground-state magnets. Using a nuclear-spin extended singlet-singlet model with an Ising-type electronic sector and mean-field plus RPA treatment, the authors derive an implicit equation for the induced ordering temperature $T_m$ and show that finite hyperfine coupling eliminates the quantum critical point at $\xi_e=1$, replacing it with a crossover to nuclear-dominated order. The study reveals a characteristic three-peak structure in the zero-field specific heat, evolves into a two-peak form under strong hyperfine coupling or high fields, and predicts a reentrant magnetic order in the AFM case driven by hyperfine interactions. Incorporating nuclear quadrupole splitting further tunes $T_m$ via an effective temperature-dependent Curie term, with the sign of $ riangle_Q$ determining whether $T_m$ is suppressed or enhanced. Collectively, the results provide a comprehensive framework for understanding thermodynamics and phase behavior in singlet-singlet magnets, with implications for Pr- and Yb-based non-Kramers compounds.

Abstract

The influence of hyperfine coupling to nuclear spins and of their quadrupolar splitting on the induced moment order in singlet ground state magnets is investigated. The latter are found among non-Kramers f electron compounds. Without coupling to the nuclear spins these magnets have a quantum critical point (QCP) separating paramagnetic and induced moment regime. The hyperfine interaction suppresses the QCP and leads to a gradual crossover between induced electronic and nuclear hyperfine coupling dominated magnetic order. It is shown how the critical temperature depends on the electronic and nuclear control parameters including the nuclear spin size and its possible nuclear quadrupole splitting. In particular the dependence of the specific heat on the control parameters and applied field is investigated for ferro- and antiferromagnetic order. It is shown that the three peak structure in the electronic induced moment regime gradually changes to a two-peak structure in the hyperfine coupling dominated nuclear moment order regime or for increasing field strength. Most importantly the possibility of a reentrance behaviour of magnetic order or likewise nonmonotonic critical fields due to hyperfine coupling influence is demonstrated. Finally the systematic evolution of the phase diagram under the influence of nuclear quadrupole coupling is clarified.

Figures (12)

• Figure 1: Critical temperature T$_m$ for induced magnetic order (FM or AFM) as function of electronic control parameter $\xi_e$ for fixed nuclear $\xi_n$ and three different sizes of nuclear spin I. The black line corresponds to vanishing hyperfine coupling when induced order appears only above the QCP at $\xi_e=1$ (Eq. (\ref{['eq:Tm0']})). For finite hyperfine coupling $\xi_n$ also T$_m$ is finite for any $\xi_e$. This leads to a suppression of the QCP which becomes more pronounced with increasing nuclear spin size. The inset is a schematic view of the repulsion and splitting of electronic and nuclear levels (I=5/2) in the ordered regime.
• Figure 2: (a) Dependence of (normalized) electronic and nuclear order parameters on temperature for hyperfine coupling $\xi_n=0.04$. Left panel: For a value of $\xi_e$ above the QCP $\xi_e^c=1$ (without hyperfine coupling), shown for various nuclear spin size. The small differences in $T_m$ for the three cases corresponds to Fig. \ref{['fig:fig1']} in the region above the QCP. Right panel: For a value of $\xi_e$ below the QCP where order is established only due the finite hyperfine coupling control parameter $\xi_n$. Here the electronic and nuclear order parameters exhibit similar T- dependence in contrast to (a) but $T_m$ now depends strongly on the size of the nuclear spin I. (b) Electronic and nuclear saturation moments $(T/\Delta=0.001)$ as function of control parameters. When $T<T_m(\xi_e,\xi_n)$ (Fig. \ref{['fig:fig1']}) $\langle S_x\rangle_0$ is finite and $\langle I_z\rangle\simeq -I$. Otherwise both drop to zero.
• Figure 3: Homogeneous electronic RPA susceptibility $\chi^J$( q=0) (Eqs. (\ref{['eq:susRPA']},\ref{['eq:RPAhomsusz']})) in paramagnetic and AFM ordered regimes for control parameter $\xi_e$ above $(> 1)$ and below $(<1)$ critical value. For the former a clear cusp at the induced magnetic order occurs. For subcritical values the cusp at $T_m$ is much diminished but a clear low temperature depression at $T^*\approx \xi_n\Delta$ corresponding to nuclear splitting energies remains. The upper curve is the reference for $\xi_n=0$ for the paramagnetic CEF vanVleck susceptibility.
• Figure 4: (a) Specific heat of the hyperfine-coupled singlet-singlet CEF and nuclear spin $I=\frac{3}{2}$ for various sub- and above critical electronic control parameters $\xi_e$. The typical three-peak appearance is displayed: i) the broad underlying CEF Schottky peak (dashed line) at $T_{max}$ due to the CEF splitting, ii) for $\xi_e >1$ a superposed induced ordering peak at T$_m$ and iii) a low temperature nuclear specific heat peak with maximum at $T^*\simeq \xi_n\Delta$ due to the nuclear spin splitting caused by the order at $T_m$. (b) For subcritical $\xi_e$ but finite $\xi_n$ the ordering at $T_m$ shows crossover into the region dominated by the hyperfine coupling around $T^*$ (see Fig. \ref{['fig:fig1']}). Inset shows evolution of specific heat jump $\delta C_V(T_m)$ with nonmonotonic change from induced moment regime $\xi_e >1$ (corresponding to (a)) to hyperfine dominated regime $\xi_e <1$ (associated with (b)).
• Figure 5: Specific heat curves for fixed control parameters $\xi_e,\xi_n$ and various nuclear spin size I which strongly influences the peak height due to entropy $\sim\ln(2I+1)$ contained in it. The dashed line corresponds to the CEF specific heat for $\xi_n=0$ (Eq. (\ref{['eq:sCV']})).