TACE: A unified Irreducible Cartesian Tensor Framework for Atomistic Machine Learning

Here, we introduce the Tensor Atomic Cluster Expansion (TACE), a unified framework formulated entirely in Cartesian space, enabling systematic and consistent prediction of arbitrary structure-dependent tensorial properties. TACE achieves this by decomposing atomic environments into a complete hierarchy of irreducible Cartesian tensors, ensuring symmetry-consistent representations that naturally encode invariance and equivariance constraints. Beyond geometry, TACE incorporates universal embeddings that flexibly integrate diverse attributes including computational levels, charges, magnetic moments and field perturbations. This allows explicit control over external invariants and equivariants in the prediction process. Long-range interactions are also accurately described through the Latent Ewald Summation module within the short-range approximation, providing a rigorous yet computationally efficient treatment of electrostatic and dispersion effects. We demonstrate that TACE attains accuracy, stability, and efficiency on par with or surpassing leading equivariant frameworks across finite molecules and extended materials. This includes in-domain and out-of-domain benchmarks, spectra, Hessian, external-field responses, charged and magnetic systems, multi-fidelity training, heterogeneous catalysis, and even superior performance within the uMLIP benchmark. Crucially, TACE bridges scalar and tensorial modeling and establishes a Cartesian-space paradigm that unifies and extends beyond the design space of spherical-tensor-based methods. This work lays the foundation for a new generation of universal atomistic machine learning models capable of systematically capturing the rich interplay of geometry, fields and material properties within a single coherent framework.