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Symmetry-Breaking De Novo Crystal Generation via Markovian Jump Diffusion

Van Khoa Nguyen, Alexandros Kalousis

arXiv:2608.13457v1cs.LG

TL;DR

Existing crystal generators often rely on empirically sampled space groups and do not produce complete crystallographic specifications. SbCD models space-group transitions through symmetry-breaking Markovian jump diffusion, outperforming its symmetry-preserving counterpart on MP20 and MPTS-52.

  • Problem

    Existing methods often rely on empirically sampled space groups, limiting direct generation of complete crystallographic specifications and global symmetry dependencies.

  • Method

    SbCD uses Markovian jump diffusion to model space-group distributions and adaptive symmetry-breaking constraints across continuous and discrete crystal representations.

  • Results

    SbCD outperforms its symmetry-preserving counterparts across most evaluation criteria on MP20 and achieves the best thermodynamic-stability performance on MPTS-52.

  • Takeaways & Limitations

    SbCD provides a proof of concept for generating full crystallographic structure specifications from minimal symmetry assumptions.

Abstract

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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 \emph{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 \emph{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.

1 Introduction

The introduction motivates crystal-generation models by the importance and costly discovery of crystals, identifies limitations in empirical space-group modeling, and presents SbCD’s symmetry-breaking diffusion contributions.

  • Motivation: Crystals support advances in energy storage, semiconductor hardware, and pharmaceuticals, while discovering useful targeted-property crystals has traditionally required years or decades of experimentation.The introduction frames accelerated crystal discovery as a route to superior technologies.
  • Limitations: Many generation methods rely on empirically distributed space groups and Wyckoff positions, limiting their capture of global symmetry and structural dependencies.The introduction notes that models not explicitly learning Wyckoff positions generalize poorly to unseen configurations.
  • Contributions: The paper derives a variational-bound objective, uses Markovian jump diffusion for space-group distributions, and develops symmetry-breaking processes on continuous and discrete state spaces.These processes adaptively enforce space-group constraints and admit analytical forms.

2 Markovian jump diffusion

This section formulates crystal generation as a continuous-time Markov process with abrupt state transitions, then derives a learned reverse process for generation. The reverse dynamics are trained with a variational bound and sampled efficiently using τ-leaping.

  • Forward jump process: The framework models states that persist for random durations before transitioning abruptly, contrasting discrete jump processes with incrementally evolving diffusion processes.The instantaneous rate matrix specifies how quickly the process exits a state and the transition probabilities determine its destination.
  • Reverse process: Reverse transition rates are defined through time reversal of the forward process, but their computation is generally intractable because it requires marginalizing over the initial state.A parametric neural network estimates the required marginal term and the reverse transition rate.
  • Reverse-process learning: The negative log-likelihood admits a variational upper bound for learning the reverse transition-rate matrix.Its terms regularize the total exit rate to prevent excessive jumping and reward the correct reverse transition pair.
  • Sampling: Sampling uses τ-leaping algorithms with the learned reverse rates, including independent Poisson draws for transitions in finite-state chains.For a finite-state chain with K ordinal states, the next state after a leap of size Δt is obtained from Poisson random variables.

3 Space-group-aware crystal generation

This section formalizes space-group-aware crystal representations and constrains diffusion to preserve crystal-family lattice symmetries. It also describes symmetry-preserving site-symmetry diffusion that restricts candidate Wyckoff positions.

  • Crystal symmetry and representation: A space group is a discrete subgroup of E(3) leaving a crystal invariant, with 230 space groups and 32 crystallographic point groups.Point groups are obtained by projecting space-group operations onto their rotational components.
  • Space group constraints: The 230 space groups are classified into 6 crystal families, whose lattice constraints are represented using the polar decomposition L = T exp(U).Here, T is orthogonal and U is symmetric, and the decomposition provides an invariant lattice representation.
  • Space group constraints: A six-dimensional invariant lattice representation k is constrained by crystal family, and diffusion remains valid through space-group-informed masking and bias.The mask–bias pair is induced by the space group and enforces the specified lattice conditions during diffusion.
  • Symmetry preserving: Symmetry-preserving diffusion models site symmetries conditioned on space group G, ensuring generated states remain symmetry-consistent and restricting representative atoms to candidate Wyckoff positions.When multiple candidates share a site-symmetry state, the selected Wyckoff position has symmetry-equivalent sites closest to the generated fractional coordinate.

4 SbCD: Symmetry-breaking crystal diffusion

SbCD unifies continuous- and discrete-state diffusion while explicitly modeling space-group transitions through a Markovian jump-diffusion process. Its symmetry-breaking formulation adapts lattice constraints and site-symmetry priors as the space group evolves.

  • Unified diffusion formulation: SbCD unifies diffusion-based training objectives across continuous and discrete time and state spaces for the asymmetric-unit representation M = (F, A, k, S, G).The representation explicitly includes atom features, lattice parameters, site symmetries, and space group.
  • Unified diffusion formulation: The variational objective decomposes into atom types and fractional coordinates, space group, lattice, and site-symmetry components.Existing frameworks often neglect the space-group term and sample space groups from data during inference, whereas SbCD models G within generation.
  • Space-group jump diffusion: SbCD mirrors spontaneous symmetry breaking with Markovian jump-diffusion, transitioning from a given space group eG to a lower-symmetry group G and reversing these transitions with learned rates.Every jump process terminates at G = P1, the lowest-symmetry triclinic space group containing only the identity operation.
  • Space-group jump diffusion: The symmetry-breaking time window is computed as w = t − τ, where τ is sampled from a truncated-exponential posterior under a constant transition rate.The posterior describes the holding time before the observed transition from eG to G.
  • Lattice diffusion: SbCD introduces masking that preserves lattice constraints before and after symmetry breaking while generalizing the fixed-space-group framework as the ambient group evolves.The prior framework is recovered when no symmetry breaking occurs, eG = G.
  • Site-symmetry diffusion: For site symmetries, SbCD interpolates between the marginal priors of eG and G because space-group transitions change the induced prior over site-symmetry states.The interpolated trajectory lies in the convex hull of the two priors and moves monotonically along their connecting line segment.

5 Experiments

Experiments on MP-20 and the more challenging MPTS-52 show that SbCD improves crystal-generation quality, space-group distribution modeling, and scalability over symmetry-preserving and other baseline methods. Additional analyses examine validity–density trade-offs and robustness to symmetry-breaking transition rates.

  • Task and metrics: SbCD is evaluated for de novo generation on MP-20 and MPTS-52 using validity, coverage, and stability-oriented metrics.MPTS-52 contains up to 52 atoms per primitive unit cell; validity includes minimum pairwise distance and charge neutrality requirements.
  • MP-20 results: SbCD♦ outperforms baselines in thermodynamically stable crystal generation, with higher charge-neutrality and closer unique-element distributions to test materials.SbCD♦ remains competitive with DiffCSP and DiffCSP++ on the S.U.N. metric.
  • Space-group distributions: SbCD variants narrow the space-group-distribution gap with empirical conditioning while outperforming methods without space-group conditioning.The result indicates that SbCD captures a wide range of space-group distributions.
  • MPTS-52 results: On MPTS-52, SbCD scales across model sizes and achieves the best performance on thermodynamic stability metrics.Table 3 reports SbCD variants at S, L, and XL sizes.
  • Symmetry-breaking time window: With w = 1, SbCD variants generate approximately 98% structurally valid crystals, but their mean density ratio relative to training data decreases dramatically.The passage attributes this validity–density trade-off to low-density crystals more readily exhibiting valid structures.
  • Transition rate λ and NFEs: SbCD is more robust than SbCD♯ at high transition rates, supporting the simplified site-symmetry representation’s ability to alleviate the transition-rate challenge.The transition rate controls movement toward the least-constrained space group P1 and must balance denoising in original space groups with convergence to prior distributions.

6 Conclusions … B.3 Wyckoff position retrieval

SbCD generates full crystallographic specifications from minimal symmetry assumptions through adaptive constraints and symmetry-breaking diffusion. The paper also outlines responsible-use considerations and experimental procedures for symmetry encoding, stability assessment, and Wyckoff-position retrieval.

  • 6 Conclusions: SbCD samples full crystallographic structure specifications from minimal symmetry assumptions using adaptive constraints over continuous and discrete state spaces.The model is inspired by spontaneous symmetry breaking and is presented as a step toward de novo crystal generation.
  • 6 Conclusions: SbCD empirically outperforms its symmetry-preserving counterpart, supporting its potential as a de novo crystal-generation approach.The passage characterizes this result as a promising proof of concept.
  • A Broader impacts: SbCD could accelerate materials discovery by exploring large chemical and crystallographic design spaces across energy storage, catalysis, semiconductors, and medicine.The passage frames these applications as potential advances rather than established outcomes.
  • A Broader impacts: Because SbCD may also enable harmful-material discovery, the paper calls for responsible use, safety screening, and rigorous experimental validation.These safeguards are presented alongside the model’s potential benefits.
  • B.1 Oriented site-symmetry symbols: The implementation provides oriented site-symmetry symbols for one-hot encoding of site symmetries.This list is described as an experimental detail.
  • B.2 Stable, uniqueness, and novelty (S.U.N) metrics: Stability is assessed by estimating crystal energies with pretrained CHGNet and comparing generated structures against a Materials Project convex hull of lowest-energy materials at each composition.The S.U.N. evaluation also computes the number of unique stable structures among stable generated materials.
  • B.3 Wyckoff position retrieval: Although the site-symmetry representation maps directly to a valid point group, oriented site-symmetry symbols alone cannot uniquely determine the appropriate Wyckoff position.Different Wyckoff positions may share the same oriented site-symmetry symbol while representing different symmetry-equivalent-site sets.

B.4 Architecture … B.7 Hardware usage

The paper uses an equivariant graph-neural-network denoiser and symmetry-specific prediction architectures, with selected settings tuned through hyperparameter search. SbCD samples from the minimal-symmetry prior, and all experiments run on a single RTX 3060 system with 20 CPU cores.

  • B.4 Architecture: SbCD adopts the equivariant graph neural network denoiser used in DiffCSP/++, while its symmetry-prediction modules use distinct site-symmetry representations.SbCD♯ predicts 13 symmetry-element probabilities over 15 axes, whereas SbCD and SbCD♦ predict probabilities over 81 oriented site-symmetry symbols.
  • B.4 Architecture: SbCD♯ follows Levy et al.’s site-symmetry prediction approach, predicting probabilities for 13 symmetry elements across 15 possible axes.This shared representation motivates using the same prediction approach in SbCD♯ and Levy et al.’s model.
  • B.4 Architecture: SbCD and SbCD♦ instead predict probabilities over 81 possible oriented site-symmetry symbols.These models modify the existing architecture to use the oriented-symbol representation rather than the 13-element-over-15-axes formulation.
  • B.5 Hyperparameter search: The hyperparameter search varies numLayer ∈{6, 8}, hiddenDim ∈{512, 1024}, embedDim ∈{256, 512}, learnRate ∈{5 × 10−4, 1 × 10−3}, and λ ∈{0.8 × 10−3, 1 × 10−3, 1.5 × 10−3, 2 × 10−3}.The search covers architectural dimensions, learning rate, and transition rate while keeping diffusion schedules similarly configured to Levy et al.’s setup.
  • B.5 Hyperparameter search: The SymmCD‡ baselines were retrained using Levy et al.’s recommended hyperparameters because official model weights were unavailable.This retraining applies to the baselines reported in Tables 2 and 3.
  • B.6 Training and sampling algorithms: SbCD’s sampling procedure starts from the minimal-symmetry prior, the space group P1 with trivial site symmetry 1.The paper provides SbCD’s training and sampling procedures in Algorithms 2 and 1, respectively.
  • B.7 Hardware usage: All experiments can run on a single NVIDIA GeForce RTX 3060 with 12 GB and 20 CPU cores.The stated hardware configuration covers both GPU memory and CPU resources.

B.8 Computational analysis … C.2 Statistical significance

SbCD uses memory-efficient representations and achieves consistent performance across independent evaluations, while ablations show that theoretically derived symmetry-breaking schedules outperform fixed windows in property statistics and S.U.N. scores.

  • B.8 Computational analysis: SbCD variants use asymmetric-unit structures and simplified site-symmetry representations to reduce memory overhead and improve training efficiency.The computational comparison uses batch size 256 on an NVIDIA GeForce RTX 3060.
  • B.8 Computational analysis: Table 6 compares SbCD variants by memory usage, training time per epoch, and architecture size.The comparison uses batch size 256, an NVIDIA GeForce RTX 3060, and 20 CPU cores.
  • C.1 Symmetry-breaking time window: Fixed time windows in SbCD variants achieve high structural validity, indicating that they preserve physically plausible atomic configurations.This finding concerns the ablation results reported in Table 7.
  • C.1 Symmetry-breaking time window: Property statistics and S.U.N. scores are significantly worse with fixed time windows than with the theoretically derived posterior symmetry-breaking time window.The comparison is against results in Table 2.
  • C.2 Statistical significance: Because validation is computationally expensive, several baselines report results from a single evaluation over 10000 samples.The baselines named are FlowMM, DiffCSP, DiffCSP++, CDVAE, and SGFM.
  • C.2 Statistical significance: SbCD is evaluated using three independent sampling runs with 10000 samples per run, and its performance remains consistent across these evaluations.The results are reported in Table 8 on the MP-20 dataset.
  • C.2 Statistical significance: The ablation study uses 3,000 generated crystalline materials because structure relaxation and energy-abovehull evaluation are computationally expensive.This clarification concerns Figure 5.

C.3 Numbers of function evaluations (NFEs)

This section compares SbCD and SymmCD across sampling budgets on MP-20, evaluating structural, compositional, and combined validity over 3,000 generated crystals.

  • Comparison setup: SbCD and SymmCD are compared under different numbers of function evaluations during MP-20 sampling.Because pretrained SymmCD weights were unavailable, SymmCD was retrained using Levy et al.’s training and evaluation protocol for fairness.
  • Evaluation metrics: The evaluation measures structural validity, compositional validity, and their product, termed Validity.Each metric is computed over 3,000 generated crystals.
  • Results presentation: Results are summarized in Figure 4 with corresponding numerical tables covering Structure Validity, Composition Validity, and Validity.The reported de novo generation results are also averaged across the 10 most representative space groups on MP-20.

C.4 Analysis of space-group distributions … D.3 Proof of Corollary 4.3

SbCD produces space-group distributions comparable to methods conditioned on training-data space groups while improving structural validity across both common and less frequent groups. The paper also reports competitive thermodynamic stability and derives the model’s reverse-process likelihood, space-group transitions, and holding-time sampling results.

  • C.4 Analysis of space-group distributions: SbCD variants achieve dsg values comparable to DiffCSP++ and SymmCD while outperforming DiffCSP, FlowMM, and CDVAE.DiffCSP++ and SymmCD directly condition on space groups sampled from the training-data distribution; the other baselines do not explicitly model or use crystal symmetries during inference.
  • C.4 Analysis of space-group distributions: SbCD♯ has higher structural validity on the top 10 most common space groups than its average across all space groups, whereas SbCD♦ and SbCD are more valid across less frequent groups.SbCD♯ uses Levy et al.’s site-symmetry representation, while SbCD♦ and SbCD use the proposed representation.
  • C.5 Histogram of the energy above the convex hull Ehull: SbCD generates a significantly higher proportion of structures with Ehull < 0 or Ehull < 0.1 after CHGNET relaxation than competing methods.These thresholds correspond to thermodynamically stable and metastable structures, respectively.
  • D.1 Proof of Proposition 4.1: The negative log-likelihood derivation defines forward, exact reverse, and learned reverse path measures over noisy asymmetric-unit crystal representations.The proof uses a change of measure, bridge decomposition, and a joint probability measure over terminal states and reverse paths.
  • D.1 Proof of Proposition 4.1: Under factorization of the crystal representation and learned reverse measure, the KL chain rule decomposes the objective into componentwise terms whose discretized Radon–Nikodym derivatives recover diffusion-model evidence-bound terms.The remaining KL term is optimized using CTMC Markov jump diffusion.
  • D.2 Proof of Proposition 4.2: The space-group process is a continuous-time Markov process with time-dependent transition rates governed by a generator matrix and jump probabilities.The forward equation is expressed as a matrix ODE and expanded through an iterated-integral Dyson series.
  • D.2 Proof of Proposition 4.2: With commuting instantaneous rate matrices and constant relative transition probabilities, the transition solution has a matrix-exponential form that forces jumps to the lowest-symmetry space group P1.The construction yields analytical space-group transition probabilities.
  • D.3 Proof of Corollary 4.3: For a two-state system observed after transition, Bayes’ theorem combines the transition likelihood, absorption-time prior, and evidence to derive the posterior holding-time density.The derivation explicitly uses the standard survival distribution for the absorption-time prior.

D.4 Proof of Proposition 4.4 … D.6 Proof of Corollary 4.6

The appendices prove that symmetry-constrained diffusion can be represented through lattice masking and that symmetry-breaking priors interpolate validly between space-group distributions. They also derive the reverse sampling formulation, which moves toward higher crystal symmetry while preserving a valid posterior parameterization.

  • D.4 Proof of Proposition 4.4: The lattice-parameter diffusion process is recast as a masking problem across six crystal families ordered by increasing symmetry.This establishes the masking framework used to analyze transitions between space groups.
  • D.4 Proof of Proposition 4.4: For G ≺ eG, the masking axioms specify commutation, annihilation, and scalar-invariance relations between masks and biases.These identities support the inductive proof across symmetry-changing diffusion steps.
  • D.4 Proof of Proposition 4.4: The induction shows that after a jump from eG to G, Gaussian noise compositions retain the required variances at subsequent diffusion times.The proof repeatedly uses m⊙em = em, m⊙mb = 0, and Gaussian variance-addition identities.
  • D.4 Proof of Proposition 4.4: DDIM sampling reverses the process toward increasing symmetry levels, proceeding from G to eG.The biases remain constant across crystal families because Cmb = mb for any scalar C.
  • D.5 Proof of Proposition 4.5: The non-stationary prior mixture changes its target distribution at the symmetry-breaking event τ while assuming G ≺ eG.The symmetry-breaking prior smoothly interpolates between the marginal site-symmetry priors of eG and G.
  • Boundary consistency: π_t = ν_t e g + (1 − ν_t)g with 0 ≤ ν_t ≤ 1, so convexity keeps the forward kernel row-stochastic without renormalization.For t ≤ τ, π_t = e g; as t → T, ν_t → 0 and π_t approaches g continuously.
  • Monotone geometric path: The mixture path moves monotonically toward g, and the reverse posterior formulation handles jumps from G_t = G to G_{t−1} = eG.The model posterior remains valid for any non-negative unnormalized prior and can be parameterized by predicting S_0.

E Future work

Future work targets space-group-constrained coordinates without fixed Wyckoff positions and more realistic, gradual symmetry-breaking transition paths toward P1. SbCD currently uses direct transitions from high-order space groups to P1.

  • Space-group constrained fractional coordinates on asymmetric units: Existing constrained-coordinate approaches typically require fixed Wyckoff positions during training and inference.DiffCSP++ and SGFM use empirical Wyckoff positions and project noisy fractional coordinates onto corresponding Wyckoff subspaces.
  • More realistic physical transition paths: SbCD currently models direct transitions from high-order space groups to the lowest-symmetry space group P1.This is identified as a limitation of its current transition design.
  • More realistic physical transition paths: Future work will redesign the instantaneous rate matrix Λ_t and add adaptive lattice and site-symmetry constraints for gradual multi-space-group transitions toward P1.The goal is to more faithfully reflect symmetry-breaking dynamics.
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