Quaternion Formulation for Jain-Kamilla Projection

• The paper presents a quaternion approach that unifies projection theory and operator algebra for robust LLL projection of composite-fermion states.
• It details an algorithmic correction using polar decomposition to minimally adjust almost commuting projections while preserving time-reversal and Kramers symmetries.
• The method enables significant computational scaling improvements, allowing accurate simulation of large systems and evaluation of static structure factors in quantum Hall systems.

A quaternion formulation for the Jain–Kamilla (JK) projection provides a mathematically robust, symmetry-aware, and computationally efficient approach to projecting composite-fermion (CF) wave functions into the lowest Landau level (LLL) in fractional quantum Hall systems. This technique unites deep results in linear algebra and operator theory—specifically, quaternionic projection theory, principal angle analysis, and C*-algebraic frameworks—with modern large-scale numerical implementation. It is now central to computations that determine subtle long-wavelength properties of CF states, such as the static structure factor, by enabling system sizes that were out of reach with conventional methods.

1. Mathematical Foundations: Quaternionic Projections and Principal Angles

The quaternionic generalization of projection theory, as developed in (Loring, 2013), begins by extending Jordan’s concept of principal angles between pairs of subspaces to quaternionic vector spaces, i.e., spaces where the Hilbert structure incorporates symplectic or antiunitary symmetries (Kramers degeneracy and time-reversal). This is implemented by working in $\mathbb{M}_n(\mathbb{H})$, the space of $n \times n$ matrices over the quaternionic field.

Given two orthogonal projection operators $P$ and $Q$, there exists a unitary transformation that block-diagonalizes them into canonical $2 \times 2$ forms parameterized by principal angles $\theta$ as

$$P_\theta = \begin{pmatrix} 1 & 0 \ 0 & 0 \end{pmatrix}, \quad Q_\theta = \begin{pmatrix} \cos^2\theta & \cos\theta\,\sin\theta \ \cos\theta\,\sin\theta & \sin^2\theta \end{pmatrix}.$$

The principal angle $\theta$ quantifies the geometric “noncommutativity” between the ranges of $P$ and $Q$. In practical scenarios such as numerics for the JK projection, actual projection operators may only approximately commute (i.e., $\|PQ - QP\|$ is small), making these canonical forms and their angles central for quantifying deviations and understanding the effect of projection “errors” or numerical artifacts.

For systems with specific symmetry requirements—e.g., time-reversal invariance enforced via an antiunitary symmetry $\mathcal{T}$ satisfying $\mathcal{T}^2 = -I$ (Kramers structure)—the quaternionic formalism provides a natural embedding: complex Hilbert space representations with $\mathcal{T}$ map into quaternionic modules.

2. Algorithmic Correction: Minimally Displacing Almost Commuting Projections

The technical challenge in both theoretical and practical LLL projection in CF theory is that projection operators constructed numerically or perturbatively often fail to commute exactly. The framework of (Loring, 2013) gives an optimal, algebraically justified correction algorithm to render such projection pairs exactly commuting with minimal displacement:

1. Define $X = QP + (I-Q)(I-P)$.
2. Compute the unitary factor $U$ in the polar decomposition $X = U|X|$. In numerical practice, $U$ can be obtained via iterative methods such as Newton-Raphson schemes.
3. The spectrum and eigenvectors of $PQP$ (the “overlap” of the two projections) are analyzed. Each $2\times2$ block gives

$$PQP = \begin{pmatrix} \cos^2\theta & 0 \ 0 & 0 \end{pmatrix},$$

with $\theta$ the principal angle.

1. For each eigenvector $v$ associated to eigenvalues $>1/2$, the pair $(v, Uv)$ forms a basis for the principal vectors.

The minimal adjustment to achieve commutation is quantified as

$$\|P - P'\| = \|Q - Q'\| = \sin\left(\frac{1}{4} \arcsin (2\delta)\right),$$

where $\delta = \|PQ - QP\|$ is the noncommutativity measure. For $\delta \ll 1$, this correction is small, ensuring the physical content of the state is preserved to high accuracy. In practical quantum Hall numerics, this “clean-up” is applied directly to finite-dimensional projection matrices involved in LLL projection.

3. Universal Real C*-Algebraic Structure

The universal real C*-algebra generated by two projections $p$ and $q$—satisfying $p^2 = p^* = p$, $q^2 = q^* = q$—underpins this mathematical apparatus. This algebra is universal in the sense that every pair of projections in any real, complex, or quaternionic Hilbert space with the same commutation relations admits a representation in this algebra.

This universality offers two main theoretical advantages:

• Field Uniformity: The same mathematical paradigm (including projection correction and principal angle analysis) applies across $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$.
• Continuity and Functoriality: Algorithms developed in this context are independent of basis choices, and thus immune to representation-dependent numerical pathologies or coordinate artifacts.

In the context of the JK projection, this ensures that numerical projection schemes respect underlying physical (e.g., antiunitary) symmetries and that approximation/clean-up steps do not artificially break these invariants.

4. Integration with the Jain–Kamilla Projection of Composite Fermion States

The LLL projection of CF wave functions, whether for electrons or bosons, is indispensable for both analytic and computational understanding. In the standard JK methodology, one projects an unprojected product wave function,

$$\Psi_{\text{unproj}} = \Phi_n(\{z_i\}) \prod_{i<j} (z_i - z_j)^{2p},$$

by promoting antiholomorphic variables to derivatives or, in the spherical geometry, through replacements such as $\bar{u} \to \partial_u$, $\bar{v} \to \partial_v$.

The quaternionic formulation, made explicit in (Anakru et al., 8 Sep 2025), reorganizes the single-particle (spinor) coordinates—conventionally $u = \cos(\theta/2) e^{i\phi/2}$, $v = \sin(\theta/2) e^{-i\phi/2}$—as quaternionic variables. This permits operations such as projection, rotation, and identification of symmetry partners to be handled at the level of quaternion algebra, substantially accelerating calculations and improving stability for large $N$. Spherical-to-planar mappings, necessary for momentum space analysis, are also handled more accurately by working with angular momentum quantum numbers as $q = \sqrt{l(l+1)}/R$.

This approach allows the efficient LLL projection of CF wave functions in spaces with up to $N\gtrsim 900$ particles, a regime previously inaccessible.

5. Practical Impact: Evaluation of the Static Structure Factor $S(q)$

A primary application motivating the quaternion formulation is the calculation of the static structure factor,

$$S(\mathbf{q}) = \frac{1}{N} \langle 0 | \rho^\dagger(\mathbf{q}) \rho(\mathbf{q}) |0\rangle - N\delta_{\mathbf{q},0},$$

for CF metals at filling factor $\nu=1/2$, $1/4$ (electrons) and $\nu=1$, $1/3$ (bosons). With the quaternion method, it is possible to access the regime $q\ell_B \gtrsim 0.1$ and determine the thermodynamic limiting behavior.

A key result emerging from this technique is the observation that $S(q)$ behaves as $q^3$ in the $q\to0$ limit, with no evidence of the $q^3\ln q$ nonanalyticity predicted by field-theoretic treatments (notably the HLR theory). Microscopically, this is captured by the dipole density operator after projection,

$$\bar{\rho}(\mathbf{q}) \approx e^{-q^2/4} \sum_{\mathbf{k}} i\ell_B^2 (\mathbf{q}\times\mathbf{k}) c^\dagger_{\mathbf{k}+\mathbf{q}}c_{\mathbf{k}},$$

leading to

$\bar{S}(q) = \frac{2k_F}{3\pi}q^3 + O(q^4).$

This result supports a physical picture in which the CFs are best regarded as non-interacting dipolar Fermi seas rather than being dominated by singular gauge fluctuations.

6. Numerical and Theoretical Advantages

The quaternion JK projection confers several robust advantages:

• Computational Scaling: Reduces the computational complexity of LLL projection from $O(N^3)$ (in standard approaches) to much better scaling, enabling calculations at unprecedented system sizes (Anakru et al., 8 Sep 2025).
• Symmetry Preservation: Incorporates antiunitary and spinor symmetries directly, which is essential in systems with time-reversal or Kramers degeneracy.
• Stability: Algorithms based in the universal C*-algebra framework are less sensitive to numerical instabilities.
• Generality: The approach unifies projection across different particle statistics (fermions, bosons) and filling factors, and is naturally extensible to generalizations such as hierarchies and multi-component states.

7. Broader Context and Theoretical Implications

Results obtained using the quaternion JK projection challenge several theoretical expectations. Notably, the absence of the predicted nonanalytic $q^3\ln q$ term in $S(q)$ suggests that the long-wavelength density response of the microscopic CF metal is governed by physics beyond the random phase approximation to gauge field fluctuations.

Moreover, the quaternion formulation places the theory of LLL projection and CF construction on a rigorous, unifying algebraic footing, allowing integration of further symmetries (e.g., in multi-flavored or spinful CF generalizations) and facilitating robust numerical algorithms for entanglement spectra and related observables. In special contexts, such as corrections for numerical noncommutativity of projection operators, these techniques ensure physical symmetries and interpretability are retained during simulations or numerical diagonalizations (Loring, 2013, Anand et al., 2022).

In summary, the quaternion formulation for the Jain–Kamilla projection is both a conceptual and computational advance, enabling large-scale, symmetry-respecting simulation and analysis of CF wave functions in the lowest Landau level, leading to insights that reshape the understanding of quantum Hall metallic phases.