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).

Changes in version 4

This version expands the four-figure script into a unified seven-figure script and renumbers all figures into a single, logically ordered sequence spanning ordinary BCS, the HTSC pseudogap, and the BCS-BEC crossover. The four version-3 figures are retained; in particular the crossover calibration (beta_g = 4.0, g_* = -0.127, xi_B = 0.376) and the corrected deep-BEC asymptote (mu/epsilon_F -> -g^2) are unchanged.

What is new:

1. BCS_synchronized.pdf. The synchronized-sector limit of the four-degrees-of-freedom model: with identical transition centers (mu_e = mu_gamma) and identical sharpness (beta_e = beta_gamma) the two sectors coincide and the observed gap reduces exactly to the ordinary single-step weak-coupling BCS curve, with no pseudogap plateau. This identifies ordinary BCS as the degenerate limit of the multi-sector picture.

2. competing_BCS_BEC_g.pdf. The competing two-sector structure as a function of the coupling g = 1/(k_F a). The complementary B- and F-type occupations L_B, L_F (L_B + L_F = 1, crossing at g = 0) are shown together with the activities delta_BCS = L_F - L_B, delta_BEC = L_B - L_F and their average delta_obs. The vertical axis represents occupation/activity, not gap magnitude, so the minimum of delta_obs does not correspond to a vanishing pairing gap; g < 0 is BCS-like, g = 0 unitary, g > 0 BEC-like.

3. BCS_gap_vs_squared.pdf. In the weak-coupling synchronized limit, the normalized gap Delta/Delta_0 = L_B^(e) - L_F^(e) and its square are plotted against temperature. Near T_c the gap itself drops steeply while its square vanishes linearly, ~ (1 - T/T_c); the squared curve is the quantity followed by the inverse-square penetration depth lambda_0^2/lambda^2 and the spectral weight W.

Unified figure sequence (generated in one execution):

1. BCS_with_Delta.pdf - exact tanh decomposition of the Muhlschlegel gap
2. BCS_synchronized.pdf - synchronized sectors -> ordinary BCS
3. BCS_ideal_normalized.pdf - threshold splitting -> 1 -> 1/2 -> 0 pseudogap staircase
4. BCS_cooperative_vs_competing.pdf - cooperative vs competing sectors
5. competing_BCS_BEC_g.pdf - competing-sector occupation/activity decomposition vs g
6. BCS_BEC_crossover_mu_g.pdf - mu(g)/epsilon_F across the BCS-BEC crossover
7. BCS_gap_vs_squared.pdf - gap and its square vs temperature (penetration depth / spectral weight)

Interpretation. Ordinary BCS, the HTSC pseudogap, and the BCS-BEC crossover form a single continuous family built from the same FBD sigmoid building blocks, distinguished only by the relative position (synchronized vs split) and the relative sign (cooperative vs competing) of the two sectors.