BgG_jl: tighten the RSCG tolerance for the eta = 0.01 local DOS

The two local-DOS figures of the topological superconductor (c55, c57)
now call greensfunctions with eps = 1e-60. RSCG stops when
|r|^2 max_j |rho_j| < eps, so the default eps = 1e-12 stopped long before
convergence at eta = 0.01. Compared with exact diagonalization of the same
1024 x 1024 BdG Hamiltonian (peak height about 0.9):

| eps    | center   | edge     |
|--------|----------|----------|
| 1e-12  | 6.0e-2   | 5.6e-2   |
| 1e-20  | 2.5e-3   | 1.1e-3   |
| 1e-40  | 6.5e-7   | 3.1e-7   |
| 1e-60  | 1.8e-12  | 7.9e-12  |

A relative change of 1e-8 in Delta now changes the RSCG local DOS by at
most 7e-8 (it was 7e-2), the size of the change of the exact local DOS.
The cells take about 1 s each. The self-consistent loops and the s-wave
local DOS (eta = 0.05) are not changed.

Refs #10

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

Author: shinaoka

src/BgG_jl.ipynb

@@ -1094,18 +1094,22 @@
     "end\n",
     "\n",
     "\n",
+    "# RSCG stops when |r|^2 max_j|rho_j| < eps. The default eps = 1e-12 is too loose at eta = 0.01:\n",
+    "# the local DOS then deviates from exact diagonalization by up to 0.06. With eps = 1e-60 it agrees to 1e-11.\n",
+    "eps_rscg = 1e-60\n",
+    "\n",
     "ix = Nx ÷ 2\n",
     "iy = Ny ÷ 2\n",
     "ispin  =1\n",
     "\n",
     "i = (( iy-1)*Nx + ix-1)*2 + ispin\n",
     "j = i\n",
-    "Gij1 = greensfunctions(i,j,σ,Htsc) \n",
+    "Gij1 = greensfunctions(i,j,σ,Htsc; eps=eps_rscg)\n",
     "\n",
     "ispin  =2\n",
     "i = (( iy-1)*Nx + ix-1)*2 + ispin\n",
     "j = i\n",
-    "Gij2 = greensfunctions(i,j,σ,Htsc) \n",
+    "Gij2 = greensfunctions(i,j,σ,Htsc; eps=eps_rscg)\n",
     "plot(real.(σ),(-1/π)*imag.(Gij1 .+ Gij2),\n",
     "    xlabel = \"Energy [t]\",\n",
     "    ylabel = \"Local DOS\",)\n"
@@ -1132,12 +1136,12 @@
     "\n",
     "i = (( iy-1)*Nx + ix-1)*2 + ispin\n",
     "j = i\n",
-    "Gij1 = greensfunctions(i,j,σ,Htsc) \n",
+    "Gij1 = greensfunctions(i,j,σ,Htsc; eps=eps_rscg)\n",
     "\n",
     "ispin  =2\n",
     "i = (( iy-1)*Nx + ix-1)*2 + ispin\n",
     "j = i\n",
-    "Gij2 = greensfunctions(i,j,σ,Htsc) \n",
+    "Gij2 = greensfunctions(i,j,σ,Htsc; eps=eps_rscg)\n",
     "plot(real.(σ),(-1/π)*imag.(Gij1 .+ Gij2),\n",
     "    xlabel = \"Energy [t]\",\n",
     "    ylabel = \"Local DOS\",)"