Symmetry-Breaking De Novo Crystal Generation via Markovian Jump Diffusion

The paper introduces SbCD (Symmetry-breaking Crystal Diffusion), a novel diffusion-based generative framework for crystals that produces complete crystallographic structure specifications, including space groups and Wyckoff positions, rather than relying on empirical distributions for these elements.

The core motivation is that existing generative models struggle to produce complete crystallographic specifications, limiting their ability to capture global symmetry and structural dependencies. Current state-of-the-art approaches generate crystals only up to site symmetries and rely on sampling space groups from empirical distributions during generation.

Inspired by spontaneous symmetry breaking in physics, where crystals break symmetries under external conditions, SbCD reverses from the lowest-symmetry priors. The method leverages a Markovian jump-diffusion process to model symmetry-breaking dynamics, enabling it to traverse different space groups in a physically motivated manner.

The paper's contributions are:

• (i) Theoretically deriving a variational bound objective that unifies structural dependencies among crystal components

• (ii) Leveraging Markovian jump diffusion to model space-group distributions

• (iii) Deriving novel symmetry-breaking diffusion processes on continuous and discrete state spaces, which adaptively enforce space-group constraints and admit analytical forms

• (iv) Proposing a simplified representation for site symmetries that facilitates capturing site-symmetry distributions and yields more stable generated crystal structures

SbCD represents the first generative framework capable of producing complete crystallographic structure specifications.

The framework models an asymmetric unit as M = (F, A, k, S, G), where F and A are fractional coordinates and atom types, k is the lattice representation, S is site symmetry, and G is the space group. The variational bound decomposes additively into space group, lattice, site symmetry, atom type, and fractional coordinate components.

For space-group modeling, the forward process simulates spontaneous symmetry breaking, where a crystal transitions from a higher-symmetry space group to lower-symmetry ones, with the terminal state fixed to P1, the lowest-symmetry space group. The transition dynamics are governed by a time-dependent rate matrix, and the paper derives analytical forms for the forward transition probabilities.

The paper also introduces the posterior holding time τ and symmetry-breaking time window w, which are sampled via inverse transform sampling from a truncated exponential distribution.

For the continuous-state lattice space, the framework introduces a symmetry-breaking diffusion process that adaptively enforces lattice constraints of the prevailing crystal family whenever the process transitions to a new space group. The forward and reverse sampling kernels are provided analytically.

For the discrete-state site-symmetry space, the paper proposes a novel one-hot vector representation of oriented site-symmetry symbols, identifying 81 distinctive symbols across all 230 space groups. This replaces the previous 15×13 binary matrix encoding, which did not distinguish Wyckoff positions sharing the same oriented site-symmetry symbol.

The paper derives an interpolated prior that smoothly transitions between site-symmetry priors of different space groups, lying in the convex hull of the two priors and moving monotonically along the straight line segment connecting them.

In experiments on MP-20 and MPTS-52 datasets, SbCD outperforms its symmetry-preserving counterpart (SymmCD) by a substantial margin. Key results include:

• SbCD variants demonstrate clear improvements over symmetry-preserving counterparts across most evaluation criteria

• SbCD generates significantly higher proportion of charge-neutral compositions

• SbCD performs competitively on the S.U.N. metric despite not using space-group constraints on fractional coordinates

• SbCD effectively scales to the more challenging MPTS-52 dataset and achieves the best performance on thermodynamic stability metrics

Ablation studies show:

• SbCD demonstrates greater robustness than SbCD even at high transition rates, underscoring the advantage of the simplified site-symmetry representation

• SbCD significantly outperforms its symmetry-preserving counterpart in inference speed, particularly in low-NFE regimes, highlighting its potential for few-step de novo crystal generation

The paper concludes that SbCD marks an important step for de novo crystal generation, offering a promising proof of concept. Future work could extend SbCD to incorporate space-group constrained fractional coordinates on asymmetric units and to more realistic physical transition paths.

Improvements to AI Systems:

1. Unified Generative Framework for Structured Physical Data: Implement SbCD’s variational bound that jointly models discrete (space group, site symmetry, atom types) and continuous (lattice, fractional coordinates) components. This enables AI systems to generate complete, physically valid crystallographic outputs in a single pass, eliminating the need for post-hoc empirical sampling of missing structural constraints.

2. Symmetry-Aware Diffusion with Adaptive Constraints: Adopt the Markovian jump-diffusion process that transitions between symmetry groups (from high to low symmetry) and adaptively enforces lattice constraints per crystal family. This allows AI systems to generate materials with explicit global symmetry, improving structural realism and reducing invalid outputs (e.g., non-physical lattices).

3. Efficient Discrete-State Representation for Site Symmetry: Replace binary matrix encodings with the proposed one-hot vector of 81 oriented site-symmetry symbols. This reduces dimensionality and improves training stability, enabling AI systems to more accurately capture Wyckoff-position distributions and generate more stable crystal structures with fewer artifacts.

4. Analytical Forward/Reverse Kernels for Few-Step Generation: Use the derived analytical diffusion kernels for both continuous and discrete spaces. This allows AI systems to perform fast, few-step generation (low-NFE) without sacrificing quality, making real-time de novo crystal design feasible for high-throughput screening.

5. Interpolated Priors for Cross-Domain Transfer: Leverage the interpolated site-symmetry priors (convex hull between space-group priors) to enable AI systems to smoothly adapt generation across different symmetry regimes. This improves generalization to unseen or mixed-symmetry datasets, such as scaling from MP-20 to MPTS-52.

6. Physically Motivated Time Dynamics: Incorporate the posterior holding time (τ) and symmetry-breaking time window (w) via inverse transform sampling from truncated exponentials. This allows AI systems to model temporal evolution of symmetry breaking, enabling simulation of phase transitions or metastable states, not just static generation.

What the Improved AI System Can Do:

• Generate complete, experimentally plausible crystal structures (including space group, Wyckoff positions, lattice, and atom types) in one step, with no missing crystallographic information.

• Achieve higher charge neutrality and thermodynamic stability in generated materials, as demonstrated by superior performance on MP-20 and MPTS-52 benchmarks.

• Perform rapid, few-step generation (e.g., 5–10 diffusion steps) while maintaining high validity, enabling interactive or high-throughput materials discovery.

• Scale to large, complex datasets (e.g., MPTS-52) and produce diverse, stable structures across multiple crystal families.

• Simulate symmetry-breaking pathways, allowing AI to explore how materials transform under external conditions (e.g., pressure, temperature) and predict intermediate structural states.

Generating crystals has recently attracted significant interest due to their broad applications in materials science. However, existing generative models struggle to produce complete crystallographic specifications, limiting their ability to capture global symmetry and structural dependencies. In particular, current state-of-the-art approaches generate crystals only up to site symmetries and rely on sampling space groups from empirical distributions during generation. Inspired by spontaneous symmetry breaking in physics, where crystals break symmetries under external conditions, we propose a novel diffusion-based framework that generates full structure specifications by reversing from the lowest-symmetry priors. Our method leverages a Markovian jump-diffusion process to model these symmetry-breaking dynamics, enabling it to traverse different space groups in a physically motivated manner. Our model, dubbed Symmetry-breaking Crystal Diffusion (SbCD), introduces a principled approach to explicitly incorporate inter-space-group transitions into the generative process. In de novo generation experiments on MP20 and MPTS-52, SbCD outperforms its symmetry-preserving counterpart by a substantial margin, offering a promising perspective for generative modeling of crystalline materials.