Quantum Order-by-Disorder Near Criticality and the Secret of Partial Order in MnSi

The vicinity of quantum phase transitions has proven fertile ground in the search for new quantum phases. We propose a physically motivated and unifying description of phase reconstruction near metallic quantum-critical points using the idea of quantum order by disorder. Certain deformations of the Fermi surface associated with the onset of competing order enhance the phase space available for low-energy, particle-hole fluctuations and self-consistently lower the free energy. Applying the notion of quantum order by disorder to the itinerant helimagnet MnSi, we show that, near the quantum critical point, fluctuations lead to an increase of the spiral ordering wave vector and a reorientation away from the lattice-favored directions. The magnetic ordering pattern in this fluctuation-driven phase is found to be in excellent agreement with the neutron-scattering data in the partially ordered phase of MnSi.

Figure 1

(a) Planar spiral $\mathbf{M}(\mathbf{r})=M[{\mathbf{n}}_x\cos (\mathbf{Q}\mathbf{r})+{\mathbf{n}}_y\sin (\mathbf{Q}\mathbf{r})]$ with propagation vector $\mathbf{Q}=Q{\mathbf{n}}_z$. The magnetic moments of size $M$ are confined to a plane spanned by ${\mathbf{n}}_x,{\mathbf{n}}_x\perp \mathbf{Q}$. (b) Because of higher-order SO terms in the cubic B20 crystal structure of MnSi, the free energy depends on the direction of $\mathbf{Q}=Q(\sin \theta \cos \varphi {\mathbf{n}}_a+\sin \theta \sin \varphi {\mathbf{n}}_b+\cos \theta {\mathbf{n}}_c)$ with respect to the crystal axes ${\mathbf{n}}_a$, ${\mathbf{n}}_b$, and ${\mathbf{n}}_c$.

Figure 2

(a) Magnetic phase diagram for weak SO interaction $\alpha$ as a function of inverse interaction $u^{-1}=\mu /U$ and temperature $t=T/\mu$. For strong interactions, the system develops helimagnetic order of pitch $Q_0=\alpha$. The transition to the paramagnet is continuous. For smaller $u$, quantum fluctuations stabilize a different spiral ground state which emerges from the tricritical point shown in red. The higher-order SO term $\tilde{\alpha }$, which is responsible for the directional dependencies, has a negligible effect upon the location of the phase boundaries and the pitch $Q$ [17]. (b) Magnetization $M$ and relative pitch $\tilde{Q}=Q-Q_0$ as a function of $u^{-1}$ for $t=0.05,0.1,0.15,0.2$. At the transition between the helimagnet and the partially ordered phase, $\tilde{Q}\sim (u^{-1}-u_c^{-1}{)}^{1/2}$ while $M$ is continuous. The transition to the paramagnet is characterized by a first-order jump in $M$.

Figure 3

(a) Boltzmann weight $I(\varphi ,\theta )=I_0\exp [-V_{\mathrm{eff}}\delta F_2/T]=I_0\exp [cg(\varphi ,\theta )]$ of the term $\delta F_2$ which dominates the directional dependence in the partially ordered phase. The overall prefactor $c$ enters as the only free fitting parameter. Fluctuations favor spirals along [110] and equivalent directions, while almost no intensity is found along the [111] directions of the helimagnet. (b) $I(\varphi ,\theta )$ with $c=4$ along a great circle connecting the high-symmetry directions in quantitative comparison with the experimental data [4].