[Paper Review] Breit-Wigner to Gaussian transition in strength functions

V. K. B. Kota, R. Sahu|ArXiv.org|Jun 30, 2000

Statistical Mechanics and Entropy1 references3 citations

TL;DR

This paper investigates the transition from Breit-Wigner to Gaussian strength functions in interacting fermionic systems using embedded Gaussian orthogonal ensemble (EGOE(1+2)) Hamiltonians. It demonstrates numerically that for six- and seven-fermion systems, the strength function evolves from Breit-Wigner to Gaussian form as the two-body interaction strength λ increases, with the transition onset occurring at λ ≈ 0.2, well beyond the onset of level statistics chaos (λc ≈ 0.06–0.08), confirming a generic feature in chaotic many-body systems including atomic nuclei.

ABSTRACT

Employing hamiltonians defined by two-body embedded Gaussian orthogonal ensemble of random matrices(EGOE(2)) plus a mean-field producing one-body part, strength functions (for states defined by the one-body part) are constructed for various values of the strength of the chaos generating two-body part. Numerical calculations for six and seven fermion systems clearly demonstrate Breit-Wigner to Gaussian transition, in the chaotic domain, in strength functions as found earlier in nuclear shell model and Lipkin-Meshkov-Glick model calculations.

Motivation & Objective

• To establish the generality of the Breit-Wigner to Gaussian transition in strength functions across interacting many-body systems.
• To investigate how the strength function shape evolves with increasing two-body interaction strength λ in fermionic systems.
• To determine the critical interaction strength λF_k at which the transition from Breit-Wigner to Gaussian form occurs, relative to the onset of quantum chaos λc.
• To compare the transition threshold λF_k with the chaos threshold λc derived from level statistics, assessing their relative positions in the chaotic domain.
• To examine the dependence of λF_k on system size (N, m), particularly whether it varies weakly with particle number m compared to λc.

Proposed method

• Employed embedded Gaussian orthogonal ensemble (EGOE(1+2)) Hamiltonians of the form H = h(1) + λV(2), where h(1) is a one-body mean-field and V(2) is a two-body interaction drawn from a GOE with unit variance.
• Constructed strength functions F_k(E) as the spectral density of the overlap |⟨ψ_E|φ_k⟩|², where |φ_k⟩ are mean-field states defined by h(1), and used ensemble averaging over random realizations.
• Used a normalized energy variable Ê = (E − ε)/σ to center and scale the energy axis, ensuring comparability across systems.
• Applied a histogram-based method with adaptive binning width Δ (0.025 for λ ≤ 0.1, 0.1 for λ > 0.1) to compute F_k(Ê) for |φ_k⟩ at Ê_k = 0.
• Defined a quantitative measure R(λ) to track the transition: R = Σ|F_k^(λ) − F_BW|² / Σ|F_BW − F_ED|², where F_ED includes Edgeworth corrections for skewness and kurtosis.
• Used R(λ) = 0.7 as the threshold for the onset of the Gaussian transition, defining λ_F_k accordingly.

Experimental results

Research questions

• RQ1 At what interaction strength λ does the strength function F_k(E) transition from Breit-Wigner to Gaussian form in interacting fermionic systems?
• RQ2 How does the transition threshold λ_F_k compare to the chaos threshold λ_c derived from level spacing statistics?
• RQ3 Is the transition to Gaussian form in F_k(E) a generic feature of chaotic many-body systems, independent of system size?
• RQ4 Does the transition occur in the thermalization regime, and is λ_F_k weakly dependent on the number of particles m?
• RQ5 Can the EG OE(1+2) model reproduce the same transition behavior observed in nuclear shell model and Lipkin-Meshkov-Glick model calculations?

Key findings

• For both six- and seven-fermion systems (N=12 and N=14), the strength function F_k(E) transitions from Breit-Wigner to Gaussian form as λ increases.
• The transition onset is quantified at λ_F_k ≈ 0.2, defined by R(λ_F_k) = 0.7, with R = 0.70 for the 6-fermion system and R = 0.75 for the 7-fermion system.
• The chaos threshold λ_c is significantly lower, at λ_c ≈ 0.08 for six fermions and λ_c ≈ 0.06 for seven fermions, indicating the transition occurs well after the onset of level fluctuations following GOE statistics.
• The value of λ_F_k shows weak dependence on system size (N, m), suggesting it may be nearly independent of m for larger systems.
• The transition occurs in the second layer of complexity, after the onset of chaos, confirming that the Breit-Wigner form persists in the chaotic domain.
• The results confirm the robustness of the Breit-Wigner to Gaussian transition across different models, including nuclear shell model and Lipkin-Meshkov-Glick model, establishing it as a generic feature of interacting many-body systems.