Critical behavior of the single-crystal helimagnet MnSi

The critical behavior of the single-crystal helimagnet MnSi is investigated by means of bulk dc magnetization at the boundary between the conical state and paramagnetic phase. We obtain the critical exponents $(\beta =0.242\pm 0.006,\gamma =0.915\pm 0.003$, and $\delta =4.734\pm 0.006)$, where the self-consistency and reliability are verified by the Widom scaling law and the scaling equation. The critical exponents of MnSi belong to the universality class of tricritical mean-field theory, which unambiguously indicates a tricritical phenomenon at the boundary between the first-order phase transition and the second-order one induced by the external magnetic field. The tricritical point (TCP) is determined as $H_{\mathrm{TCP}}\approx 3200$ Oe at the critical temperature, consistent with the previous report [A. Bauer, M. Garst, and C. Pfleiderer, Phys. Rev. Lett. 110, 177207 (2013)]. The critical behavior suggests a long-range magnetic coupling with the exchange distance decaying as $J\left(r\right)\approx r^{-4.3}$ in MnSi.

Figure 1

(a) The temperature dependence of magnetization $\mathrm{[}M\left(T\right)\text{]}$ for MnSi (the inset shows the magnification of the transition); (b) the isothermal magnetization $\mathrm{[}M\left(H\right)\text{]}$ at 4 K (the inset gives that in the lower field region).

Figure 2

(a) The initial magnetization around $T_C$ for MnSi; (b) the Arrott plots of $M^2$ vs $H/M$ (the $M\left(H\right)$ curves are measured at interval $\Delta T=1$ K, and $\Delta T=0.5$ K approaching $T_C)$.

Figure 3

The isotherms of $M^{1/\beta }$ vs ${(H/M)}^{1/\gamma }$ with (a) 3D-Heisenberg model; (b) 3D-Ising model; (c) tricritical mean-field model; (d) the normalized slopes $\mathrm{[}\mathrm{NS}=S\left(T\right)/S\left(T_C\right)\text{]}$ as a function of temperature.

Figure 4

(a) The temperature dependence of $M_S$ and ${\chi }_0^{-1}$ for MnSi; (b) the isothermal $M\left(H\right)$ at $T_C$ with the inset plane on the $lg-lg$ scale (the solid curves are fitted).

Figure 5

Effective exponents (a) ${\beta }_{\mathrm{eff}}$ and (b) ${\gamma }_{\mathrm{eff}}$ as a function of the reduced temperature $\varepsilon$ for MnSi (dashed curves are guided on eye).

Figure 6

(a) Scaling plots of renormalized magnetization $m$ vs renormalized field $h$ around the critical temperatures for MnSi (only several typical curves are shown); (b) $m^2$ vs $h/m$ below $T_C$ in lower field region (the inset plot the $H_C$ vs $T$ with the linearly fitting solid line).