Fluctuation-induced quadrupole order in magneto-electric materials

Abstract

Phases that go beyond dipolar ordering and into multipolar ordering have recently been observed in magneto-electric materials. The resulting phase diagram is commonly explained using the concept of competing orders and exact microscopic interactions. In contrast, we propose an approach based on composite order emerging from a parent phase to explain quadrupoling above magnetic or electric dipolar orders. We include thermal fluctuations and symmetry and show their influence on the emergence of quadrupolar order. We find an analytical expression for the quadrupolar transition temperature, the critical anisotropy and explain the coupling of the quadrupolar order to mechanical strain, in agreement with experiments. The shift in perspective on quadrupolar ordering from competing to composite order is universal and can be extended to other types of multipolar ordering. This offers the possibility of understanding tunability and material-specific predictions of the related phase transitions without explicit knowledge of the microscopic mechanisms.

Figures (5)

• Figure 1: Free energy density minima for linear and quadratic order parameters. The quadratic order parameter, representing a composite order (a), exhibits a minimum similar to the linear order parameter (b) at low anisotropy. At higher anisotropy, the linear order parameter retains a single minimum (b), while the quadratic order parameter develops two distinct minima (d).
• Figure 2: Gap equation for quadrupole transition. The function $g(T)$ denotes the right-hand side of the gap equation Eq. \ref{['eq:gap-equation']}. Depending on the anisotropy, the gap equation has no solutions ($\beta_- < \beta_-^*$), one solution at $T_q = 2T_c$ ($\beta_- = \beta_-^*$), or two solutions with $T_c < T_q < 2T_c$ ($\beta_- > \beta_-^*$). For a tiny anisotropy, $T_q \sim T_c$.
• Figure 3: Minima of the quadrupole free energy density. The three distinct minima are rotationally invariant. The reduced temperature is $t=-5$, with parameters $a_0=1$, $\gamma_q = 0.4$, and $u_q = 0.6$.
• Figure 4: First-order quadrupole phase transition. The minimum of the free energy density in Eq. \ref{['eq:free-energy-phi']} continuously approaches zero with decreasing relative temperature. At the finite value of $t^*=0.2$, the quadrupole phase is entered when the minimum crosses $f(\Phi_{E_{g;1}}, \Phi_{E_{g;2}})=0$ and becomes negative. Here, $\Phi_{E_{g;1}} = 0$.
• Figure 5: Tetragonal distortion of the cubic lattice symmetry of Ba2ReMgO6 induced by strain during the quadrupole phase transition. The analytical strain curve is calculated using Eq. \ref{['eq:distortion-phi2']}, and the experimental X-ray diffraction data are from Ref. hirai2020detection. The reduced temperature $t_q = t + t^*$ is derived from the experimental data.