Decomposing Fractional Quantum Hall Wave Functions via Operator Contraction Multiplication

Abstract: We develop a general algebraic scheme to decompose fractional quantum Hall (FQH) wave functions based on the operator contraction multiplication. By introducing fermionic and bosonic operators and establishing three fundamental contraction rules, we achieve an exact decomposition of Laughlin states. This approach naturally extends to multi-component systems by factorizing coupled Jastrow factors via resultants and elementary symmetric polynomials, enabling the first complete decomposition of Halperin states. For Halperin ($2,2,1$) state, we explicitly derive its basic expansion, identify root configurations, and reveal intra- and inter-color squeezing operators, thereby uncovering the underlying generalized Pauli principle. Using this method, we compute orbital entanglement spectra for up to $16$ particles with decomposition dimensions exceeding $10^{11}$, obtaining edge excitation sequences that precisely match chiral Luttinger liquid theory. Our framework breaks through the longstanding limitations of Jack polynomials, provides a unified decomposition for both single- and multi-component FQH states, and opens a new avenue for exploring wave functions for more complex FQH states.

• The paper introduces an operator contraction multiplication framework that decomposes FQH wave functions into Slater or permanent bases while preserving particle statistics.
• It systematically applies contraction operations—fermion-fermion, boson-boson, and fermion-boson—to yield exact expansions for both Laughlin and Halperin states.
• Numerical results validate that the method recovers edge excitation spectra and clarifies topological entanglement, advancing trial wave function construction for FCIs.

Decomposition of FQH Wave Functions via Operator Contraction Multiplication

Algebraic Framework for FQH State Decomposition

The paper introduces a robust algebraic scheme for decomposing fractional quantum Hall (FQH) wave functions grounded in operator contraction multiplication. The formalism articulates decomposition rules for Laughlin states using antisymmetric (fermionic) and symmetric (bosonic) operators, defined respectively as $\mathbf{F}_\lambda$ and $\mathbf{B}_\lambda$. Crucially, three fundamental contraction operations—fermion-fermion, boson-boson, and fermion-boson—are derived, allowing systematic decomposition of FQH states into Slater or permanent bases indexed by angular momentum partitions.

For Laughlin states at filling $\nu=1/m$, the operator formalism directly yields exact expansions by mapping the $m$th power of the Vandermonde determinant to combinations of symmetric and antisymmetric basis states. The operational rules preserve particle statistics and ensure valid occupation multiplicity counting for higher-order bosonic configurations. The scheme unifies the decomposition process for both bosonic and fermionic states, substantially generalizing prior approaches based on Jack polynomials.

Extension to Multi-Component Systems: Halperin State Decomposition

The decomposition framework is naturally extended to multi-component FQH states. The expansion of Halperin states, such as $(m,m,n)$ for two components, is achieved by factorizing coupled Jastrow factors via resultants and elementary symmetric polynomials. The resultant structure, realized through Sylvester matrices and Viete formulas, allows for exact basis expansions of previously analytically intractable wave functions—most notably Halperin states, which resisted Jack polynomial decomposition.

In this formalism, symmetric operators for distinct components (e.g., $\mathbf{U}_\lambda$ and $\mathbf{V}_\lambda$) are combined through direct products, while same-component operators are contracted following the established rules. For the Halperin $(2,2,1)$ case, explicit decomposition outcomes are presented. The expansion reveals root configurations and squeezing operations, delineating intra- and inter-color squeezing while manifesting the generalized Pauli principle (GPP) governing occupation constraints in the multicomponent context.

Numerical Results and Topological Implications

Utilizing this decomposition scheme, orbital entanglement spectra for large systems—up to $16$ particles and decomposition dimensions exceeding $10^{11}$—are computed for both Laughlin and Halperin states. The edge excitation sequences extracted from these spectra reproduce those predicted by chiral Luttinger liquid theory, confirming congruence with topological order as encoded in the entanglement structure.

The decomposition formalism also enables analytic construction of root configurations and hierarchy of basis states via squeezing operations. The occupation language developed provides nuanced insight into color-entangled orbitals, distinguishing inter- from intra-component squeezing, and reinforces GPP as a universal principle across both single- and multi-component FQH states.

Theoretical and Practical Implications

The operator contraction multiplication framework supersedes Jack polynomial-based schemes by facilitating unified decomposition for a broad class of FQH states, including those with complex coupled Jastrow factors and multi-component structure. This advances the algebraic characterization of FQH wave functions, allowing explicit basis construction and entanglement quantification, which are central for probing topological order.

Practically, the methodology enables the systematic construction of trial wave functions for fractional Chern insulators (FCIs), especially color-entangled and multi-component states that were inaccessible through previous techniques. The algebraic expansion and root configuration analysis provide a foundation for modeling intricate FCI phenomena observed in moiré superlattice materials and topological flat band models. The formalism is well-positioned to accommodate novel trial wave function proposals and enforce GPP constraints in emergent multi-component systems.

Future Directions

Anticipated extensions of this work include the application of contraction multiplication to decompositions of more exotic FQH and FCI states with higher-order symmetries, non-Abelian statistics, and more intricate inter-component coupling. Furthermore, this operator-centric approach may facilitate analytical treatments of topological entanglement and edge mode characterization in recently observed multicomponent quantum Hall and Chern systems. The algebraic structure developed offers a pathway toward algorithmically efficient and scalable wave function construction and decomposition for large-particle-number systems.

Conclusion

The operator contraction multiplication framework presents a unified, exact, and scalable method for decomposing FQH wave functions, notably including Laughlin and Halperin states. The approach circumvents the limitations of Jack polynomial expansions, enables analytic access to multi-component states, and elucidates the occupation-based generalized Pauli principle. This decomposition method provides both practical tools for constructing trial wave functions in FCIs and theoretical advances in understanding topological order and entanglement structure in strongly correlated quantum systems ["Decomposing Fractional Quantum Hall Wave Functions via Operator Contraction Multiplication", (2604.21434)].