BCS_figures: A Mathematica script for visualizing BCS superconducting gaps via the Fermion–Boson Duality framework

BCS_figures: A Mathematica script for visualizing BCS superconducting gaps via the Fermion–Boson Duality framework

This repository contains a self-contained Mathematica script that generates two figures arising when the Fermion–Boson Duality (FBD) framework [Maruyama, Frontiers in Physics 13, 1618853 (2025)] is applied to the phenomenology of BCS superconductivity.

Overview

The script visualizes the structure of the superconducting gap near the transition temperature as a difference between B-type (bosonic) and F-type (fermionic) occupation probabilities. It produces the following two figures.

1. BCS_with_Delta.pdf (Figure 1) — Exact decomposition of the Mühlschlegel empirical formula

The Mühlschlegel empirical formula

Δ(T)/Δ0≈tanh⁡ ⁣(1.74Tc/T−1)\Delta(T)/\Delta_0 \approx \tanh\!\left(1.74\sqrt{T_c/T - 1}\right)Δ(T)/Δ0≈tanh(1.74Tc/T−1)

can be rewritten exactly as a difference between the B-type and F-type occupation probabilities of the matter (electronic) sector,

Δ(T)/Δ0=LB(e)(T)−LF(e)(T),\Delta(T)/\Delta_0 = L_B^{(e)}(T) - L_F^{(e)}(T),Δ(T)/Δ0=LB(e)(T)−LF(e)(T),

by means of the tanh identity

tanh⁡(x)=L(−2x)−L(2x),L(y)=1/(1+ey).\tanh(x) = L(-2x) - L(2x), \qquad L(y) = 1/(1+e^y).tanh(x)=L(−2x)−L(2x),L(y)=1/(1+ey).

A numerical verification cell embedded in the script automatically confirms that the two sides agree to the IEEE 754 double-precision machine epsilon 2.22×10−162.22 \times 10^{-16} 2.22×10−16. This is not a "fit" but a mathematical fact that follows from the tanh identity.

2. BCS_ideal_normalized.pdf (Figure 2) — High-temperature superconductor (HTSC) gap via the FBD four-degrees-of-freedom model

The four FBD degrees of freedom {LB(e),LF(e),LB(γ),LF(γ)}\{L_B^{(e)}, L_F^{(e)}, L_B^{(\gamma)}, L_F^{(\gamma)}\} {LB(e),LF(e),LB(γ),LF(γ)} are represented as Hill–Wheeler-type sigmoid functions, parametrized by the sector transition centers μe,μγ\mu_e, \mu_\gamma μe,μγ and the transition sharpness parameters βe,βγ\beta_e, \beta_\gamma βe,βγ. With the illustrative parameter choice (βe,μe)=(2.0,5.0)(\beta_e, \mu_e) = (2.0, 5.0) (βe,μe)=(2.0,5.0) and (βγ,μγ)=(1.5,3.0)(\beta_\gamma, \mu_\gamma) = (1.5, 3.0) (βγ,μγ)=(1.5,3.0), the condition μγ<μe\mu_\gamma < \mu_e μγ<μe yields an observed gap with a staircase structure

$$1\to 1/2\to 0,$$

exhibiting a pseudogap plateau in the intermediate region.

Changes in version 2 

This version extends the original two-figure script into a unified four-figure script that covers not only the BCS regime but also its smooth continuation to the BCS-BEC crossover. The two v1 figures (BCS_with_Delta.pdf and BCS_ideal_normalized.pdf) are retained without modification.

What is new:

1. Unified script. The previous BCS_figures_en.wls and the extension script have been merged into a single self-contained file, BCS_all_figures_en.wls, which generates all four figures in one execution.

2. Figure A — BCS_cooperative_vs_competing.pdf. Side-by-side comparison of two qualitatively distinct two-sector FBD gaps: (a) cooperative sectors (same-sign sigmoids) producing a staircase / pseudogap profile, and (b) competing sectors (opposite-sign sigmoids) producing a V-shaped observable.

3. Figure B — BCS_BEC_crossover_mu_g.pdf. Quantitative application of the competing two-sector mechanism to the unitary Fermi gas: the chemical potential mu(g)/epsilon_F across the BCS-BEC crossover is reproduced with parameters beta_g = 4.0 and g_* = -0.127 calibrated against the Bertsch parameter xi_B = 0.376. The FBD prediction agrees with QMC and experimental data across the entire crossover, including the mu sign-flip at g_0 ≈ 0.45.

Interpretation. The BCS pseudogap and the BCS-BEC crossover are not separate phenomena: they share the same FBD sigmoid building blocks, distinguished only by whether the two sectors act cooperatively or competitively.

Changes in version 3

This version corrects the deep-BEC asymptote of the chemical potential in Figure B (BCS_BEC_crossover_mu_g.pdf). Figures 1, 2 and A are unchanged.

Corrected asymptote. In the deep BEC limit the single-fermion chemical potential approaches half the dimer binding energy, μ → −ε_b/2 with ε_b = ℏ²/(m a²), i.e. μ/ε_F → −1/(k_F a)² = −g². The model term and the plotted asymptote were changed from −g²/2 to −g² accordingly.

Calibration unchanged. The unitarity value μ(0)/ε_F = ξ_B = 0.376 (β_g = 4.0, g_* = −0.127) is unaffected, because the BEC term vanishes at g = 0. The chemical-potential sign-flip shifts from g_0 ≈ 0.45 to g_0 ≈ 0.37.

Reference data. The BEC-side comparison points were updated to literature-consistent values (≈ −g²).