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Progress of Theoretical Physics

Reduction Relations for the Third and the Fourth Moments of Energy Spectra in a Many-Particle Configuration

Masao Nomura

Published Jul 1, 1972• 15 citations
Publisher

Abstract

The third and fourth moments of energy spectra in a many-particle configuration are completely related to two-particle matrix elements, so that neither c.f.p. nor explicit diagonalization of any matrix is needed to evaluate these moments. Recah algebra and reduction relations for c.f.p. are well used. We confine ourselves, for simplicity, to jn and j1n1j2n2 configurations of identical particles.

DOI
10.1143/ptp.48.110

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The third and fourth moments of energy spectra in a many-particle configuration are completely related to two-particle matrix elements, so that neither c.f.p. nor explicit diagonalization of any matrix is needed to evaluate these moments. Recah algebra and reduction relations for c.f.p. are well used. We confine ourselves, for simplicity, to jn and j1n1j2n2 configurations of identical particles.
A formalism recently developed to obtain a matrix representation of the second-order spin-adapted reduced Hamiltonian [J. Karwowski, W. Duch, and C. Valdemoro, Phys. Rev. A 33, 2254 (1986)] and derived from the symmetric-group approach to configuration-interaction methods has been generalized to the case of the third-order spin-adapted reduced Hamiltonian. As in the second-order case, the matrix elements are linear combinations of generalized two-electron integrals with coefficients explicitly related to traces of products of the orbital occupation-number operators. The third-order spin-adapted reduced Hamiltonian matrix has been block diagonalized according to the irreducible representations of the symmetric group S3{S}_{3}S3​.
A new approach to the one-particle Green's functions G\mathit{G}G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M\mathit{M}M related to G\mathit{G}G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (nthn\mathrm{th}nth-order ADC schemes) that represent infinite partial summations for M\mathit{M}M and (via the Dyson equation) for G\mathit{G}G being complete through nthn\mathrm{th}nth order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.
This research establishes an exact, matrix-free framework for spectral analysis of Fibonacci-constrained quantum systems, demonstrating how exponentially expanding Hilbert spaces can admit computational descriptions whose complexity grows only linearly with chain length at any fixed spectral moment order. The investigation focuses on the open PXP quantum chain, whose admissible configurations obey a nearest-neighbor exclusion constraint and form a Hilbert space of dimension dim(Hₙ) = Fₙ₊₂. By combining Fibonacci combinatorics, Zeckendorf indexing, temporal-mask transfer operators, and exact integer arithmetic, the study develops a unified mathematical approach to spectral moments, constrained quantum dynamics, and gauge-sensitive observables without explicitly constructing the exponentially large Hamiltonian matrix. The central result is a universal rational hierarchy of even spectral moments, expressed through the generating function Σₙ₌₀∞ Tr(Aₙ²ᵐ) zⁿ = Pₘ(z) / (1 − z − z²)ᵐ⁺¹, with deg Pₘ ≤ 3m + 2. This theorem establishes that every fixed-order even spectral moment possesses a rational generating function governed by a power of the Fibonacci characteristic polynomial. Consequently, infinitely many spectral moments across increasing chain lengths can be reconstructed from a finite collection of exact initial values. Explicit rational expressions are derived for the fourth, sixth, and eighth spectral moments, accompanied by integer recurrence relations and independently reproducible computational certificates. The underlying temporal-mask transfer identity reduces fixed-order moment evaluation to O(n) arithmetic operations, even with spatially inhomogeneous hopping amplitudes and local diagonal fields. Its computational constants depend exponentially on moment order, but not on the exponentially growing Hilbert-space dimension. A complementary result introduces an exact gauge-sensitive fourth-moment certificate for quantum chains carrying occupation-dependent hopping phases. The spectral response obeys the identity Tr(Hₙ,φ⁴) − Tr(Hₙ,₀⁴) = 8(cos φ − 1) Sₙ, where Sₙ is obtained through a Fibonacci-based weighted recurrence requiring only O(n) arithmetic operations. Remarkably, this identity remains valid under arbitrary static real diagonal disorder, provided the diagonal energies and hopping magnitudes remain unchanged between phase settings. It therefore isolates a precisely defined gauge-dependent contribution to the fourth spectral moment without requiring diagonalization or knowledge of the complete spectrum. The certificate reconstructs a dimensionless flux response rather than an absolute quantum-action scale. The broader significance lies in demonstrating that exponential state-space growth does not necessarily imply exponential computational complexity for every exact spectral observable. The results connect constrained quantum many-body physics, Fibonacci graph theory, rational generating functions, and finite-state transfer methods within a common algebraic framework. All reported coefficients are supported by exact-integer verification rather than numerical fitting, and the established complexity bounds concern arithmetic operations rather than total bit complexity. The work provides a reproducible mathematical foundation for further investigations of spectral invariants, gauge responses, quantum constraints, and scalable symbolic computation, while restricting its formal claims to the specified finite-dimensional Hamiltonian models. Keywords: Fibonacci-constrained quantum chains · PXP Hamiltonian · Rational spectral hierarchy · Matrix-free computation · Temporal-mask transfer · Zeckendorf indexing · Gauge invariants · Exact spectral moments · Quantum many-body systems · Integer verification