Different critical behaviors in perovskites with a structural phase transition from cubic-to-trigonal and cubic-to-tetragonal symmetry

Perovskites like ${\mathrm{LaAlO}}_3$ (or ${\mathrm{SrTiO}}_3$) undergo displacive structural phase transitions from a cubic crystal to a trigonal (or tetragonal) structure. For many years, the critical exponents in both these types of transitions have been fitted to those of the isotropic three-components Heisenberg model. However, field theoretical calculations showed that the isotropic fixed point (FP) of the renormalization group is unstable, and renormalization group iterations flow either to a cubic FP or to a fluctuation-driven first-order transition. Here we show that these two scenarios correspond to the cubic-to-trigonal and to the cubic-to-tetragonal transitions, respectively. In both cases, the critical behavior is described by slowly varying effective critical exponents, which exhibit universal features. For the trigonal case, we predict a crossover of the effective exponents from their Ising values to their cubic values (which are close to the isotropic ones). For the tetragonal case, the effective exponents can have the isotropic values over a wide temperature range before exhibiting large changes en route to the first-order transition. New renormalization group calculations near the isotropic FP in three dimensions are presented and used to estimate the effective exponents, and dedicated experiments to test these predictions are proposed. Similar predictions apply to cubic magnetic and ferroelectric systems.

Figure 1

The cubic (left) and tetragonal (right) unit cells in ${\mathrm{SrTiO}}_3$ (the latter shows only half the cell; neighboring cells rotate in opposite directions). Large, intermediate, and small spheres correspond to Sr, O, and Ti ions, respectively. The dashed lines represent the octahedra, which rotate around the vector $\mathbf{Q}$, lying along the vertical axis (the O ions in the central horizontal plane move as indicated by the arrow). When $\mathbf{Q}$ is along a diagonal of the cube, the octahedra rotate around that diagonal, and the unit cell is stretched along $\mathbf{Q}$, causing a cubic-to-trigonal transition (as in ${\mathrm{LaAlO}}_3$).

Figure 2

Schematic flow diagram and FPs for the cubic model, Eq. (2), adapted from Ref. [24]. $G=\mathrm{Gaussian}, I=\mathrm{isotropic}, D=\mathrm{Decoupled}$ (Ising), and $C=\mathrm{Cubic}$ FPs. S = initial point for ${\mathrm{SrTiO}}_3$. L = initial point for ${\mathrm{LaAlO}}_3$. The dashed lines represent the stability edges, $u+v=0$ (for $v<0$) and $u+v/n=0$ (for $v>0$), below which the free energy in Eq. (2) is stabilized by the terms of order ${\left|\mathbf{Q}\right|}^6$, and the transitions are first order. The shaded areas are the regions of attraction of the stable FPs ($I$ on left and $C$ on right).

Figure 3

Flow trajectories in the $u-v$ plane (blue) for several initial points. The dots indicate integer values of $\ell$. The red line is the universal asymptotic line, Eq. (12). The green line is the asymptotic line in the linear approximation, $v=-\delta u/z_{01}^{}$. The small circles denote the isotropic ($v_I^{*}=0$) and cubic FPs.

Figure 4

Effective exponents $\beta$ and $\gamma$ for the trajectories shown in Fig. 3 as functions of $\ell$. The horizontal axes are at the asymptotic values of the isotropic FP (which are also very close to those at the cubic FP). The exponents corresponding to trajectories with $v\left(0\right)>0$ (III and IV, dashed lines) approach this asymptotic value. In contrast, those with $v\left(0\right)<0$ (I and II, full lines) initially come close to these values but then turn downward to smaller values.