Embedded Cluster Expansions (eCE)

• Embedded Cluster Expansions (eCE) are generalized cluster models that incorporate spatial, geometric, and chemical embedding constraints to extend traditional expansion methods.
• They employ rigorous convergence criteria and activity bounds, enabling efficient modeling of complex systems such as multicomponent alloys, quantum spin models, and condensed phases.
• By using dimensionality reduction and hypergraph techniques, eCE streamlines the representation of many-body interactions, facilitating the creation of accurate surrogate Hamiltonians.

Embedded Cluster Expansions (eCE) generalize the formalism of the standard cluster expansion by incorporating spatial, geometric, and chemical embedding constraints into the definition of clusters and their contributions to thermodynamic and statistical quantities. This framework provides both rigorous convergence criteria and efficient parameterization schemes, enabling applications to complex systems such as multicomponent alloys, quantum spin models, Gibbs point processes, and condensed phases. eCE models underpin accurate surrogate Hamiltonians for materials with many degrees of freedom and strong local correlations.

1. Mathematical Foundations and Generalization from Abstract Cluster Expansions

The foundational idea of cluster expansions is the representation of the logarithm of the partition function as a convergent power series over connected clusters (polymers or graphs) equipped with combinatorial weights that encode compatibility and interaction structure (Miracle-Sole, 2012, &&&1&&&). For an abstract polymer system with polymers $P$ and incompatibility relation $\gamma \dashv \gamma'$, the partition function is

$$Z(A) = \sum_{\substack{x \subset A\ x \text{ compatible}}} \prod_{\gamma \in x} \phi(\gamma)$$

with compatibility inherited from the embedding of polymers in space or chemical environments. The expansion

$\ln Z(A) = \sum_{X \in M(A)} \Phi^T(X)$

organizes terms via multi-indices and connected graphs.

In eCE, the abstract set $P$ is replaced or extended to include polymers with explicit geometric, chemical, or site attributes, and the compatibility function $f(\gamma, \gamma')$ depends not just on combinatorial constraints but also on whether clusters overlap in space, violate chemical embedding restrictions, or interact nonlocally due to chemical composition (Miracle-Sole, 2012, Müller et al., 9 Sep 2024). The structure of the coefficients $a^T(\gamma_1, ..., \gamma_n)$ generalizes accordingly, with summation over graphs respecting the additional embedding.

2. Convergence, Bounds, and Embedding-Induced Constraints

Rigorous convergence of both standard and embedded cluster expansions relies on bounding the activities (weights), often via an exponential decay controlled by geometric or chemical embedding (Miracle-Sole, 2012, Jansen, 2018). Theorems establish that, for a positive control function $u$, the sufficient condition

$$|\phi(y_0)| \leq (e^{u(y_0)} - 1) \exp\left( - \sum_{y \sim y_0} u(y) \right)$$

guarantees absolute convergence of the expansion, with suppressions that are amplified by stricter embedding (i.e., fewer allowed overlaps) (Miracle-Sole, 2012). In continuum systems such as Gibbs point processes, the embedding is into measure spaces and the sufficient condition for uniqueness and convergence takes the form (FP$_t$)

$$\sum_{k=1}^\infty \frac{e^{tk}}{k!} \int \prod_j |f(x_0, x_j)| \prod_{i<j} (1 + f(x_i, x_j)) \prod_j z(x_j) e^{a(x_j)} d\lambda(x_j) \leq e^{a(x_0)} - 1$$

(Jansen, 2018), with $f(x,y)$ a Mayer function encoding embedded pair potentials.

Embedding typically reduces the number of allowed clusters, thereby enhancing decay and convergence. In practice, embedding constraints should be incorporated into the construction of the activity bounds and in the verification of the expansion's convergence in the intended regime (e.g., high/low temperature, dilute/dense, or chemically correlated limits).

3. eCE Formalism for Multicomponent and High-Dimensional Systems

In systems with many chemical species (e.g., high-entropy alloys, multi-site quantum spin models), a direct cluster expansion leads to a combinatorial explosion of basis functions and ECI parameters (Müller et al., 9 Sep 2024). The embedded cluster expansion (eCE) formalism addresses this by compressing site basis functions into a lower-dimensional learned chemical space. For instance, with $c$ species, the site function vector $\varphi(\sigma_i)$ can be linearly transformed via a learnable matrix $\mathcal{T}$ to a $k$-dimensional embedded space: $$\tilde{\varphi}(\sigma_i) = \mathcal{T} \varphi(\sigma_i)$$ Cluster functions are then formed as products over embedded site functions, and energies parameterized (often via neural networks) in terms of these symmetrized, embedded cluster descriptors.

This dimensionality reduction enables eCE models to interpolate and extrapolate energetics for alloys not sampled in the training data, leveraging learned chemical similarities (e.g., merging Mo and W) (Müller et al., 9 Sep 2024). Simultaneous training of both embedding ($\mathcal{T}$) and regression weights ($w$) via loss minimization

$$\mathcal{L} = \arg\min_{w,\mathcal{T}} \sum_{\sigma} \left[E^{\text{DFT}}(\sigma) - \sum_i E_i^{\text{eCE}}(\sigma, w, \mathcal{T})\right]^2 + \mathcal{L}_{\text{reg}}$$

leads to models that robustly capture ground-state phase stability and finite-temperature ordering with far fewer parameters than conventional CE.

4. Graphs, Hypergraphs, and Quantum Many-Body eCE

For quantum spin models and systems with multi-site interactions, standard graph-based expansions are insufficient, and eCE requires generalization to hypergraph techniques (Mühlhauser et al., 2022). Here, interactions spanning $k$ sites are represented as hyperedges, and cluster expansion becomes a decomposition over equivalence classes of hypergraphs, defined via isomorphism in a reduced Kőnig representation. The effective Hamiltonian is clustered as

$$H_{\text{eff}} = H_0 + \sum_{\mathfrak{E}} N_{\mathfrak{E}} V_{\mathfrak{E}}$$

with $N_{\mathfrak{E}}$ the embedding factor for equivalence class $\mathfrak{E}$, and $V_{\mathfrak{E}}$ the computed representative contribution.

This classification is essential for high-order series expansions in ground-state energies and excitation gaps, where computational savings accrue by evaluating each topological class once, multiplied by its embedding factor. This approach is essential for eCE in quantum systems with extensive many-site entanglement.

5. eCE in Condensed Matter and Liquid State Theory

For dense and condensed phases, standard cluster expansions are poorly convergent. The crystalline cell approach modifies the reference system to an ideal crystal, with single-particle cell potentials $\varphi_j(q_{n_i})$ optimized to minimize the remainder in the expansion (Bokun et al., 2018). The partition function is expanded as

$$Z_m = Z_m^{(0)} \exp\left\{ \frac{1}{2} \sum_{i,j} \rho_i \rho_j f_{ij} + \frac{1}{6} \cdots \right\}$$

with renormalized Mayer functions measured from the crystalline embedding.

Single-particle potentials are determined self-consistently from the variational condition

$$\frac{\delta \ln Z_m}{\delta \varphi_k(q_{n_m})} = 0$$

ensuring that dominant two-body correlations are absorbed in the reference, and residual many-body corrections converge rapidly. This eCE approach is essential for describing short-range order and local structure in condensed systems, including solid–solid or solid–liquid phase transitions.

6. Implementation Strategies: Basis, Regularization, Subspace Projection

Efficient practical construction of eCE models employs regression of effective cluster interactions (ECI) using advanced feature selection and regularization—LASSO, ARD, Bayesian compressive sensing—as implemented in CLEASE (Chang et al., 2018), icet (Ångqvist et al., 2019), and tutorial frameworks (Ekborg-Tanner et al., 23 May 2024). Key strategies include:

• Generation of cluster sets via symmetry-adapted orbit decomposition and truncation by distance or body-order;
• Training set generation using uncertainty maximization, condition number minimization, or orthogonalization to span the cluster vector space robustly;
• Subspace projection and augmentation with fractional factorial design to resolve confounding of ECIs and reduce bias error (Tan et al., 2012);
• Bayesian regularization or explicit coupling for complex, low-symmetry systems, enforcing physical similarity between clusters with analogous environments (Ekborg-Tanner et al., 23 May 2024).

In linear atomic cluster expansions, rigorous convergence of extracted two-body interactions requires augmentation of diverse datasets with explicit dimer configurations; unconstrained addition of high body-order descriptors can fail to converge the effective two-body curve to theoretical limits (Tan et al., 22 Feb 2025).

7. Diagrammatic Approaches, Stochastic Calculus, and Statistical Models

eCE techniques extend beyond classical Hamiltonians to stochastic integrals, Gibbs point processes, and spatial statistics. Cluster coefficients are organized as sums over connected multigraphs, with cumulants and moments represented diagrammatically (Jansen, 2018). These methods provide bookkeeping required for advanced eCE schemes and relate cluster expansions to generating functions for trees, branching processes, percolation thresholds, and random connection models.

In statistical mechanics, activities, densities, and thermodynamic potentials are encoded using combinatorial series over embedded trees and 2-connected (irreducible) graphs. Inverted density–activity relations and closure approximations for liquids are formulated via eCE to improve the analytic structure of equations of state and to derive accurate correlation and response functions (Tsagkarogiannis, 2023).

8. Applications and Future Directions

Embedded cluster expansions have enabled:

• Accurate modeling of high-entropy alloys and multicomponent systems through chemical embedding and dimensionality reduction (Müller et al., 9 Sep 2024);
• Efficient computation of phase diagrams, order-disorder transitions, and short-range order for alloys and disordered materials (Chang et al., 2018, Ångqvist et al., 2019, Ekborg-Tanner et al., 23 May 2024);
• Explicit convergence and control of expansion errors and bias via subspace augmentation (Tan et al., 2012);
• Robust quantum many-body calculations via hypergraph decompositions (Mühlhauser et al., 2022);
• Rapidly convergent thermodynamics and correlation functions for condensed phases (Bokun et al., 2018);
• Fundamental insight into the structure and limitations of machine-learned interatomic potentials embedding cluster hierarchy (Tan et al., 22 Feb 2025, Ho et al., 3 Jan 2024);

Prospective research involves coupling eCE models to other degrees of freedom (lattice, magnetic, vibrational), generalizing embeddings for off-lattice and nonlocal systems, and careful hybridization of training sets to ensure correct physics in low body-orders. Improved initialization and regularization of embedding matrices, especially informed by chemical trends, have been shown to yield superior extrapolation and predictive performance.

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Embedded cluster expansions continue to enhance both the rigor and breadth of statistical, quantum, and materials modeling, providing a combinatorial and analytic backbone for the extension of cluster techniques into chemically and geometrically complex systems.