Revisiting Jahn--Teller Transitions in Correlated Oxides with Monte Carlo Modeling

Jahn–Teller (JT) distortions are a key driver of physical properties in many correlated oxide materials. Cooperative JT distortions, in which long-range orbital order reduces the symmetry of the average structure macroscopically, are common in JT-distorted materials at low temperatures. This long-range order will often melt on heating, \textit{via} a transition to a high-temperature state without long-range orbital order. The nature of this transition has been observed to vary with different materials depending on crystal structure; in LaMnO$_3$ the transition has generally been interpreted as order-disorder, whereas in layered nickelates $A$NiO$_2$ ($A$=Li,Na) there is a displacive transition. Alternatively, recent theoretical work has suggested that previous attributions of order-disorder may in fact be a consequence of phonon anharmonicity, rather than persistence of JT distortions, which would suggest that the displacive transition may be more common than currently believed. In this work, we run Monte Carlo simulations with a simple Hamiltonian which is modified to include terms dependent on the JT amplitude $ρ$, which is allowed to vary within the simulation \textit{via} the Metropolis algorithm. Our simulations yield distributions of JT amplitudes consistent with displacive rather than order-disorder behaviour for both perovskites and layered nickelates, suggesting that displacive-like JT transitions may be more common than previously assumed in both perovskites and layered nickelates. We also find significant differences between the transition observed for perovskites compared with layered nickelates, which we attribute to differing extensivity of configurational entropy on the two lattices, showing the crucial role of lattice geometry in determining behaviour.

💡 Research Summary

In this paper the authors revisit the long‑standing question of how cooperative Jahn–Teller (JT) distortions disappear upon heating in correlated oxides. Historically, perovskite compounds such as LaMnO₃ have been interpreted as undergoing an order‑disorder transition: the long‑range orbital order melts, but local JT distortions persist, giving rise to a high‑temperature state that still contains finite‑amplitude octahedral elongations. By contrast, layered nickelates (NaNiO₂, LiNiO₂) have been reported to display a displacive transition, where the JT amplitude continuously shrinks to zero and the high‑temperature phase is truly undistorted. Recent theoretical work has suggested that the apparent order‑disorder behavior may be an artifact of phonon anharmonicity rather than genuine persistence of JT distortions, implying that displacive transitions could be more common than previously thought.

To address this, the authors construct a minimalist yet physically motivated Hamiltonian that explicitly includes a variable JT amplitude ρ on each JT‑active octahedron. The total energy is written as H = E_single‑ion + E_geometry. The single‑ion term follows a Landau‑type expansion, E_single‑ion = Σ_i (α ρ_i² + β ρ_i⁴), with α>0 and β<0 chosen (α = 1, β = ‑½) so that a finite ρ lowers the enthalpy but the system is not forced into a fixed amplitude. The geometry term differs for the two families of materials.

For perovskites the geometry term reproduces the anisotropic Potts model of Ahmed and Gehring. Each octahedron carries an orbital state S_i that can point along one of three Cartesian axes. Neighboring sites contribute an interaction J₁ if the orbital direction aligns with the bond vector, J₂ if it is perpendicular, and zero otherwise. A weight factor w(ρ_i,ρ_j)=2ρ_iρ_j/(1+ρ_iρ_j) attenuates the contribution of undistorted sites, preventing an artificial drive toward infinite distortion.

For layered nickelates the geometry term penalizes “oxygen under‑bonding”: if a single oxygen is simultaneously elongated toward more than one Ni, an energy penalty K_oxy max(0, s_O‑1)² is incurred, where s_O counts the number of elongated Ni‑O bonds incident on that oxygen. This term captures the chemical instability of having multiple elongated bonds share the same anion and stabilizes the experimentally observed collinear ordering.

Monte Carlo simulations are performed using the Metropolis algorithm, allowing simultaneous updates of ρ and S_i. The authors employ simulated annealing: starting from a high‑temperature random configuration, the temperature is gradually reduced to T = 0.001 (in reduced units) over 10⁷ MC steps. For each temperature they monitor the average JT amplitude ⟨ρ⟩ and the full distribution P(ρ).

The results are striking. In both perovskites and layered nickelates the low‑temperature state exhibits ⟨ρ⟩≈1 and a sharply peaked distribution at finite ρ, confirming a robust cooperative JT distortion. Upon heating, ⟨ρ⟩ decreases continuously, but the high‑temperature distribution becomes essentially flat: there is no pronounced peak at a non‑zero ρ, nor a delta‑function at ρ = 0. Instead, the system explores the entire (Q₂,Q₃) space, consistent with a dynamic, displacive picture rather than a static order‑disorder mixture of distorted and undistorted octahedra.

A key insight emerges from comparing the two lattice geometries. The three‑dimensional perovskite network allows many more configurational possibilities for the orbital orientations, leading to a larger configurational entropy that grows rapidly with temperature. Consequently, the JT transition occurs at a higher temperature and the reduction of ⟨ρ⟩ is more gradual. In the two‑dimensional triangular lattice of the nickelates, configurational entropy is sub‑extensive; the system cannot gain as much entropy by disordering, so the transition temperature is lower and the collapse of ⟨ρ⟩ is sharper. The authors therefore attribute the differing transition characteristics to the extensivity of configurational entropy dictated by lattice geometry.

The study also revisits earlier Monte Carlo work that built the order‑disorder transition into the model by fixing ρ = 1. By allowing ρ to fluctuate, the authors demonstrate that the previously reported order‑disorder signatures arise naturally from a model that, when constrained, artificially forces a bimodal distribution. Their findings align with recent experimental total‑scattering analyses (e.g., Batnaran et al., 2025) that could reproduce LaMnO₃ data without invoking persistent local JT elongations, supporting the view that many JT transitions are fundamentally displacive.

In conclusion, the paper provides compelling computational evidence that JT transitions in both perovskite and layered nickelate oxides are better described as displacive, with the high‑temperature phase lacking a well‑defined JT amplitude. The lattice dimensionality and associated configurational entropy play a decisive role in shaping the transition temperature and the sharpness of the amplitude collapse. The work showcases how a simple Hamiltonian, augmented with a variable JT amplitude, can capture essential physics that previously required more elaborate DFT‑based Monte Carlo schemes. It opens the door for systematic exploration of JT behavior across a broader class of correlated oxides, and suggests that many historical assignments of order‑disorder transitions merit re‑examination.