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Use the sparse-ir-doc notation in the tutorials
Bring the text, formulas and plot labels in line with https://spm-lab.github.io/sparse-ir-doc/src/notation.html: - Matsubara frequencies are written iν with an upright i and defined through the reduced frequency n (ν = nπ/β, odd n for fermions, even n for bosons). The unreduced ω_n = (2n+1)πT definitions are gone, and ω is left for real frequencies. Where both statistics appear (FLEX, TPSC, GW, Eliashberg) they are written iν^F and iν^B, each defined once. - The lowest Matsubara frequency is written by its value (iπT, or iν^B = 0) instead of iω_0/iν_0, since ω_0 is the phonon frequency in the Eliashberg notebook. The DMFT renormalization factors use Σ(3iπT) and Σ(iπT), which is what the code evaluates. - The hat is kept only on the Matsubara transform Û_l(iν). Green's functions, P, W and Σ lose theirs (GW). The SpM and BdG matrices, and the Eliashberg λ matrices, are bold. GW's hat on a τ function is dropped. - IR coefficients are G_l in text and labels (gl stays in code), singular values S_l, the index l (not ℓ), the basis size L (GW used N, BdG N_IR), and the l axis starts at 0 (DLR_jl). The lattice size in second_order_perturbation, which was also called L, is now N_lin. - Endpoints are written 0⁺, 0⁻ and β⁻ (SpM, BdG, GW, migration guide). - The method is called DLR, never SPR (DLR_py). In transformation_py the pole model uses residues a_p, so that c_p are the weights of ρ for both statistics, as in the DLR. The poles are ω̄_p, and the unbalanced parenthesis in ρ_l is gone. - BdG: the normal average ⟨c†c⟩ gets the convergence factor e^{iν0⁺} that a sum with a 1/(iν) tail needs. - Eliashberg: ζ(iν) = iν + μ − Σ is renamed ξ (as `xi_iv` in the code), since ζ denotes the parity of the statistics. DMFT_IPT_py was the only notebook that stored outputs, against the README's "Remove all outputs". Its plot labels change here, so the stale outputs are stripped. Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>
1 parent 9a35886 commit c32b46a

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Lines changed: 259 additions & 445 deletions

‎src/BgG_jl.ipynb‎

Lines changed: 34 additions & 34 deletions
Original file line numberDiff line numberDiff line change
@@ -14,29 +14,29 @@
1414
"We consider a superconductivity in a mean-field level.\n",
1515
"\n",
1616
"## Basic theory of superconductivity in real space\n",
17-
"Let us consider a Hamiltonian for a fermion system given as $H = \\psi^{\\dagger} \\hat{\\cal H} \\psi/2$. \n",
17+
"Let us consider a Hamiltonian for a fermion system given as $H = \\psi^{\\dagger} \\boldsymbol{\\cal H} \\psi/2$. \n",
1818
"The column vector $\\psi$ is composed of $N$ fermionic annihilation $c_i$ and creation operators $c_i^{\\dagger}$ $(i = 1,2,\\cdots,N)$, $\\psi = (\\{ c_i \\}^T, \\{ c_i^{\\dagger} \\}^T)$, where $\\{ c_i \\} = (c_1,c_2,\\cdots,c_N)^T$ and $\\{ c_i^{\\dagger} \\} = (c_1^{\\dagger},c_2^{\\dagger},\\cdots,c_N^{\\dagger})$. The row vector $\\psi^{\\dagger}$ is also defined as $\\psi^{\\dagger} = (\\{c_i^{\\dagger} \\}^T, \\{ c_i \\}^T)$.\n",
19-
"The subscription $i$ in $c_i$ or $c_i^{\\dagger}$ indicates a quantum index depending on spatial site, spin, orbital, etc. The \"Hamiltonian\" matrix $\\hat{\\cal H}$ is a $2N \\times 2N$ Hermitian matrix given as \n",
19+
"The subscription $i$ in $c_i$ or $c_i^{\\dagger}$ indicates a quantum index depending on spatial site, spin, orbital, etc. The \"Hamiltonian\" matrix $\\boldsymbol{\\cal H}$ is a $2N \\times 2N$ Hermitian matrix given as \n",
2020
"\n",
2121
"$$\n",
22-
"\\hat{\\cal H} = \\left( \\begin{matrix} \\hat{H} & \\hat{\\Delta} \\\\\n",
23-
"\\hat{\\Delta}^{\\dagger} & - \\hat{H}^{\\ast}\n",
22+
"\\boldsymbol{\\cal H} = \\left( \\begin{matrix} \\boldsymbol{H} & \\boldsymbol{\\Delta} \\\\\n",
23+
"\\boldsymbol{\\Delta}^{\\dagger} & - \\boldsymbol{H}^{\\ast}\n",
2424
"\\end{matrix}\n",
2525
"\\right),\n",
2626
"$$\n",
2727
"\n",
28-
"where $\\hat{H}$ is a normal state Hamiltonian and $\\hat{\\Delta}$ is a superconducting order parameter."
28+
"where $\\boldsymbol{H}$ is a normal state Hamiltonian and $\\boldsymbol{\\Delta}$ is a superconducting order parameter."
2929
]
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},
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{
3232
"cell_type": "markdown",
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"id": "fd3ba213",
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"metadata": {},
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"source": [
36-
"The BdG equations are regarded as the eigenvalue equation with respect to $\\hat{\\cal H}$ expressed as \n",
36+
"The BdG equations are regarded as the eigenvalue equation with respect to $\\boldsymbol{\\cal H}$ expressed as \n",
3737
"\n",
3838
"$$\n",
39-
"\\hat{\\cal H} \\vec{f}_{\\gamma} = \\epsilon_{\\gamma} \\vec{f}_{\\gamma}\n",
39+
"\\boldsymbol{\\cal H} \\vec{f}_{\\gamma} = \\epsilon_{\\gamma} \\vec{f}_{\\gamma}\n",
4040
"$$\n",
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"\n",
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"$$\n",
@@ -48,10 +48,10 @@
4848
"$$\n",
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"\n",
5050
"The column vectors $\\vec{u}_{\\gamma}$ and $\\vec{v}_{\\gamma}$ are $N$-component complex vetors. \n",
51-
"To solve the BdG equations is equivalent to diagonalization of $\\hat{\\cal H}$ with a unitary matrix $\\hat{U}$, \n",
51+
"To solve the BdG equations is equivalent to diagonalization of $\\boldsymbol{\\cal H}$ with a unitary matrix $\\boldsymbol{U}$, \n",
5252
"\n",
5353
"$$\n",
54-
"\\hat{U}^{\\dagger} \\hat{\\cal H} \\hat{U} = \\hat{D}, \\: \\: \\: \\hat{D} = {\\rm diag} (\\epsilon_1,\\epsilon_2,\\cdots,\\epsilon_{2N})\n",
54+
"\\boldsymbol{U}^{\\dagger} \\boldsymbol{\\cal H} \\boldsymbol{U} = \\boldsymbol{D}, \\: \\: \\: \\boldsymbol{D} = {\\rm diag} (\\epsilon_1,\\epsilon_2,\\cdots,\\epsilon_{2N})\n",
5555
"$$"
5656
]
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},
@@ -64,24 +64,24 @@
6464
"In the mean-field framework, the $2N \\times 2N$ matrix Green's function with a complex frequency is defined as \n",
6565
"\n",
6666
"$$\n",
67-
"\\hat{G}(z) = [z \\hat{I} - \\hat{\\cal H}]^{-1}.\n",
67+
"\\boldsymbol{G}(z) = [z \\boldsymbol{I} - \\boldsymbol{\\cal H}]^{-1}.\n",
6868
"$$\n",
6969
"\n",
70-
"With the unitary matrix $\\hat{U}$, each component of $\\hat{G}(z)$ is epressed as \n",
70+
"With the unitary matrix $\\boldsymbol{U}$, each component of $\\boldsymbol{G}(z)$ is epressed as \n",
7171
"\n",
7272
"$$\n",
7373
"G_{\\alpha \\beta}(z) = \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\frac{1}{z - \\epsilon_{\\gamma}}\n",
7474
"$$\n",
7575
"\n",
76-
"If we set $z = i \\omega_n$ with the Matsubara frequency $\\omega_n = (2n+1)\\pi T$, the above formula corresponds to Matsubara temperature Green's function. \n",
76+
"If we set $z = \\mathrm{i}\\nu$ with the fermionic Matsubara frequency $\\nu = n\\pi T$ (odd $n$), the above formula corresponds to Matsubara temperature Green's function. \n",
7777
"The regarded and advanced Green's functions are, respectively, defined as \n",
7878
"\n",
7979
"$$\n",
80-
"\\hat{G}^{\\rm R}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\hat{G}(\\omega + i \\eta)\n",
80+
"\\boldsymbol{G}^{\\rm R}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\boldsymbol{G}(\\omega + \\mathrm{i} \\eta)\n",
8181
"$$\n",
8282
"\n",
8383
"$$\n",
84-
"\\hat{G}^{\\rm A}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\hat{G}(\\omega - i \\eta)\n",
84+
"\\boldsymbol{G}^{\\rm A}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\boldsymbol{G}(\\omega - \\mathrm{i} \\eta)\n",
8585
"$$\n",
8686
"\n",
8787
"In order to obtain physical quantities (e.g., density of states) from Green's functions, we introduce the following useful $2N$-component unit-vectors $\\vec{e}(i)$ and $\\vec{h}(i)$ $(1 \\le i \\le N)$, which are, respectively, defined as\n",
@@ -93,23 +93,23 @@
9393
"For example, the local density of states with respect to the site $i$ is given as \n",
9494
"\n",
9595
"$$\n",
96-
"N(\\omega,i) = -\\frac{1}{2\\pi i} \\vec{e}(i)^T \\hat{d}(\\omega) \\vec{e}(i),\n",
96+
"N(\\omega,i) = -\\frac{1}{2\\pi \\mathrm{i}} \\vec{e}(i)^T \\boldsymbol{d}(\\omega) \\vec{e}(i),\n",
9797
"$$\n",
9898
"\n",
99-
"with the use of the spectral function $\\hat{d}(\\omega)$ defined as \n",
99+
"with the use of the spectral function $\\boldsymbol{d}(\\omega)$ defined as \n",
100100
"\n",
101101
"$$\n",
102-
"[\\hat{d}(\\omega)]_{\\alpha \\beta} \\equiv [\\hat{G}^{\\rm R}(\\omega) - \\hat{G}^{\\rm A}(\\omega)]_{\\alpha \\beta} = - 2\\pi i \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\delta(\\omega - \\epsilon_{\\gamma})\n",
102+
"[\\boldsymbol{d}(\\omega)]_{\\alpha \\beta} \\equiv [\\boldsymbol{G}^{\\rm R}(\\omega) - \\boldsymbol{G}^{\\rm A}(\\omega)]_{\\alpha \\beta} = - 2\\pi \\mathrm{i} \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\delta(\\omega - \\epsilon_{\\gamma})\n",
103103
"$$\n",
104104
"\n",
105105
"Two types of mean-fields $\\langle c_i^{\\dagger} c_j \\rangle$ and $\\langle c_i c_j \\rangle$ can be expressed as \n",
106106
"\n",
107107
"$$\n",
108-
"\\langle c_i^{\\dagger} c_j \\rangle = - \\frac{1}{2 \\pi i} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\hat{d}(\\omega) \\vec{e}(i), \n",
108+
"\\langle c_i^{\\dagger} c_j \\rangle = - \\frac{1}{2 \\pi \\mathrm{i}} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\boldsymbol{d}(\\omega) \\vec{e}(i), \n",
109109
"$$\n",
110110
"\n",
111111
"$$\n",
112-
"\\langle c_i c_j \\rangle = - \\frac{1}{2 \\pi i} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\hat{d}(\\omega) \\vec{h}(i), \n",
112+
"\\langle c_i c_j \\rangle = - \\frac{1}{2 \\pi \\mathrm{i}} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\boldsymbol{d}(\\omega) \\vec{h}(i), \n",
113113
"$$\n",
114114
"\n",
115115
"with $f(x) \\equiv 1/(e^{x/T} + 1)$."
@@ -124,23 +124,23 @@
124124
"With the use of the analytic continuation, the mean-fields are rewritten as \n",
125125
"\n",
126126
"$$\n",
127-
"\\langle c_i^{\\dagger} c_j \\rangle = T \\sum_{n=-\\infty}^{\\infty} \\vec{e}(j)^T \\hat{G}(i \\omega_n) \\vec{e}(i)\n",
127+
"\\langle c_i^{\\dagger} c_j \\rangle = T \\sum_{\\nu} \\vec{e}(j)^T \\boldsymbol{G}(\\mathrm{i}\\nu) \\vec{e}(i)\\, e^{\\mathrm{i}\\nu 0^+}\n",
128128
"$$\n",
129129
"\n",
130130
"$$\n",
131-
"\\langle c_i c_j \\rangle = T \\sum_{n=-\\infty}^{\\infty} \\vec{e}(j)^T \\hat{G}(i \\omega_n) \\vec{h}(i)\n",
131+
"\\langle c_i c_j \\rangle = T \\sum_{\\nu} \\vec{e}(j)^T \\boldsymbol{G}(\\mathrm{i}\\nu) \\vec{h}(i)\n",
132132
"$$\n",
133133
"\n",
134134
"By solving the linear equations defined as \n",
135135
"\n",
136136
"$$\n",
137-
"(i \\omega_n \\hat{I} - \\hat{\\cal H}) \\vec{x}(i,\\omega_n) = \\vec{h}(i), \n",
137+
"(\\mathrm{i}\\nu \\boldsymbol{I} - \\boldsymbol{\\cal H}) \\vec{x}(i,\\nu) = \\vec{h}(i), \n",
138138
"$$\n",
139139
"\n",
140140
"the superconducting mean-field is expressed as \n",
141141
"\n",
142142
"$$\n",
143-
"\\langle c_i c_j \\rangle = T \\sum_{n = - \\infty}^{\\infty} \\hat{e}(j)^T \\vec{x}(i,\\omega_n).\n",
143+
"\\langle c_i c_j \\rangle = T \\sum_{\\nu} \\hat{e}(j)^T \\vec{x}(i,\\nu).\n",
144144
"$$\n",
145145
"\n",
146146
"For example, s-wave superconducting order parameter is defined as \n",
@@ -152,7 +152,7 @@
152152
"The self-consistent cycle has the schematic form \n",
153153
"\n",
154154
"$$ \n",
155-
"\\hat{\\Delta} \\rightarrow \\hat{G}(i \\omega_n) \\rightarrow \\langle c_i c_i \\rangle \\rightarrow \\hat{\\Delta}\n",
155+
"\\boldsymbol{\\Delta} \\rightarrow \\boldsymbol{G}(\\mathrm{i}\\nu) \\rightarrow \\langle c_i c_i \\rangle \\rightarrow \\boldsymbol{\\Delta}\n",
156156
"$$"
157157
]
158158
},
@@ -174,13 +174,13 @@
174174
"The superconducting mean-fielads are expressed as \n",
175175
"\n",
176176
"$$\n",
177-
"\\langle c_i c_j \\rangle = \\vec{e}(j)^T \\hat{G}(\\tau = \\beta) \\vec{h}(i) = \\sum_{l=0}^{N_{\\rm IR}-1} U_l(\\beta) \\vec{e}(j)^T \\hat{G}_l \\vec{h}(i)\n",
177+
"\\langle c_i c_j \\rangle = \\vec{e}(j)^T \\boldsymbol{G}(\\tau = \\beta^-) \\vec{h}(i) = \\sum_{l=0}^{L-1} U_l(\\beta^-) \\vec{e}(j)^T \\boldsymbol{G}_l \\vec{h}(i)\n",
178178
"$$\n",
179179
"\n",
180180
"Since the Matsubara Green's function is expressed as \n",
181181
"\n",
182182
"$$\n",
183-
"G(i \\omega_n) = \\sum_{l=0}^{N_{\\rm IR}-1} G_l U_l(i \\omega_n),\n",
183+
"G(\\mathrm{i}\\nu) = \\sum_{l=0}^{L-1} G_l \\hat{U}_l(\\mathrm{i}\\nu),\n",
184184
"$$\n",
185185
"\n",
186186
"one can obtain $G_l$ by fitting the above equation."
@@ -202,10 +202,10 @@
202202
"The matrix form of the above Hamiltonain is expressed as \n",
203203
"\n",
204204
"$$\n",
205-
"\\hat{H} = -\\mu \\hat{I} -t (\\hat{T}_{+x} + \\hat{T}_{-x} + \\hat{T}_{+y} + \\hat{T}_{-y}),\n",
205+
"\\boldsymbol{H} = -\\mu \\boldsymbol{I} -t (\\boldsymbol{T}_{+x} + \\boldsymbol{T}_{-x} + \\boldsymbol{T}_{+y} + \\boldsymbol{T}_{-y}),\n",
206206
"$$\n",
207207
"\n",
208-
"where $\\hat{T}_{+x}$ is a hopping matrix with respect to $+x$ direction."
208+
"where $\\boldsymbol{T}_{+x}$ is a hopping matrix with respect to $+x$ direction."
209209
]
210210
},
211211
{
@@ -445,11 +445,11 @@
445445
"If one wants to use the Matsubara frequencies, one has to introduce the cutoff of the frequencies. Then the mean-fields are expressed as \n",
446446
"\n",
447447
"$$\n",
448-
"\\langle c_i c_j \\rangle = T \\sum_{n = - n_c}^{n_c} \\hat{e}(j)^T \\vec{x}(i,\\omega_n).\n",
448+
"\\langle c_i c_j \\rangle = T \\sum_{|\\nu| < \\nu_c} \\hat{e}(j)^T \\vec{x}(i,\\nu).\n",
449449
"$$\n",
450450
"\n",
451-
"Here, $n_c$ is a cutoff value. \n",
452-
"With the use of the maximum frequency $\\omega_c$, we calculate the set of Matsubara frequencies defined as"
451+
"Here, $\\nu_c$ is a cutoff value (`ωc` in the code). \n",
452+
"We calculate the set of Matsubara frequencies below the cutoff as"
453453
]
454454
},
455455
{
@@ -769,11 +769,11 @@
769769
"The Hamiltonian in momentum space is given as \n",
770770
"\n",
771771
"$$\n",
772-
"\\hat{\\cal H}(\\vec{k}) = \n",
772+
"\\boldsymbol{\\cal H}(\\vec{k}) = \n",
773773
"\\left(\n",
774774
" \\begin{matrix}\n",
775-
" h_0(\\vec{k}) & i \\Delta \\sigma_y \\\\\n",
776-
" - i \\Delta^{\\dagger} \\sigma_y & - h_0^{\\ast}(\\vec{k})\n",
775+
" h_0(\\vec{k}) & \\mathrm{i} \\Delta \\sigma_y \\\\\n",
776+
" - \\mathrm{i} \\Delta^{\\dagger} \\sigma_y & - h_0^{\\ast}(\\vec{k})\n",
777777
"\\end{matrix}\n",
778778
"\\right), \n",
779779
"$$\n",

‎src/DLR_jl.ipynb‎

Lines changed: 7 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -25,12 +25,14 @@
2525
"$$\n",
2626
"\n",
2727
"where the matrix $\\boldsymbol{V}_{lp}~[\\equiv V_l(\\bar{\\omega}_p)]$ is well-conditioned.\n",
28-
"As a result, in DLR, the Green's function is represented as\n",
28+
"As a result, in DLR, the Green's function of fermions is represented as\n",
2929
"\n",
3030
"$$\n",
31-
"G(\\mathrm{i}\\omega_n) = \\sum_{p=1}^L \\frac{c_p}{\\mathrm{i}\\omega_n - \\bar{\\omega}_p}.\n",
31+
"G(\\mathrm{i}\\nu) = \\sum_{p=1}^L \\frac{c_p}{\\mathrm{i}\\nu - \\bar{\\omega}_p},\n",
3232
"$$\n",
3333
"\n",
34+
"where $\\nu = n\\pi/\\beta$ with odd $n$ is a fermionic Matsubara frequency.\n",
35+
"\n",
3436
"[1] J. Kaye, K. Chen, O. Parcollet, Phys. Rev. B 105, 235115 (2022).<br>\n",
3537
"[2] H. Shinaoka et al., arXiv:2106.12685v2.<br>"
3638
]
@@ -49,7 +51,7 @@
4951
"The corresponding Green's function is given by\n",
5052
"\n",
5153
"$$\n",
52-
"G(\\mathrm{i}\\omega_n) = \\int_{-1}^1 \\mathrm{d}\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\omega_n - \\omega}.\n",
54+
"G(\\mathrm{i}\\nu) = \\int_{-1}^1 \\mathrm{d}\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\nu - \\omega}.\n",
5355
"$$\n",
5456
"\n",
5557
"The Green's function is expanded in IR as\n",
@@ -128,7 +130,7 @@
128130
"outputs": [],
129131
"source": [
130132
"rhol = overlap(basis.v, rho)\n",
131-
"ls = collect(1:length(basis))\n",
133+
"ls = collect(0:length(basis)-1)\n",
132134
"plot(ls[1:2:end], abs.(rhol)[1:2:end], marker=:cross, yaxis=:log, ylims=(1e-5,1))"
133135
]
134136
},
@@ -213,7 +215,7 @@
213215
"plot(\n",
214216
" imag.(iv), [imag.(giv_exact), imag.(giv)], marker=[:cross :x], line=(nothing,nothing),\n",
215217
" xlabel=latexstring(\"\\\\nu\"),\n",
216-
" ylabel=latexstring(\"G(\\\\mathrm{i}\\\\omega_n)\")\n",
218+
" ylabel=latexstring(\"\\\\mathrm{Im}\\\\, G(\\\\mathrm{i}\\\\nu)\")\n",
217219
" )"
218220
]
219221
}

‎src/DLR_py.ipynb‎

Lines changed: 10 additions & 8 deletions
Original file line numberDiff line numberDiff line change
@@ -25,12 +25,14 @@
2525
"$$\n",
2626
"\n",
2727
"where the matrix $\\boldsymbol{V}_{lp}~[\\equiv V_l(\\bar{\\omega}_p)]$ is well-conditioned.\n",
28-
"As a result, in SPR, the Green's function is represented as\n",
28+
"As a result, in DLR, the Green's function of fermions is represented as\n",
2929
"\n",
3030
"$$\n",
31-
"G(\\mathrm{i}\\omega_n) = \\sum_{p=1}^L \\frac{c_p}{\\mathrm{i}\\omega_n - \\bar{\\omega}_p}.\n",
31+
"G(\\mathrm{i}\\nu) = \\sum_{p=1}^L \\frac{c_p}{\\mathrm{i}\\nu - \\bar{\\omega}_p},\n",
3232
"$$\n",
3333
"\n",
34+
"where $\\nu = n\\pi/\\beta$ with odd $n$ is a fermionic Matsubara frequency.\n",
35+
"\n",
3436
"[1] J. Kaye, K. Chen, O. Parcollet, Phys. Rev. B 105, 235115 (2022).<br>\n",
3537
"[2] H. Shinaoka et al., arXiv:2106.12685v2.<br>"
3638
]
@@ -40,7 +42,7 @@
4042
"id": "0248f127",
4143
"metadata": {},
4244
"source": [
43-
"We consider the semi circular DOS\n",
45+
"We consider fermions and the semi circular DOS\n",
4446
"\n",
4547
"$$\n",
4648
"\\rho(\\omega) = \\sqrt{\\frac{2}{\\pi}} \\sqrt{1-\\omega^2}.\n",
@@ -49,7 +51,7 @@
4951
"The corresponding Green's function is given by\n",
5052
"\n",
5153
"$$\n",
52-
"G(\\mathrm{i}\\omega_n) = \\int_{-1}^1 \\mathrm{d}\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\omega_n - \\omega}.\n",
54+
"G(\\mathrm{i}\\nu) = \\int_{-1}^1 \\mathrm{d}\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\nu - \\omega}.\n",
5355
"$$\n",
5456
"\n",
5557
"The Green's function is expanded in IR as\n",
@@ -184,14 +186,14 @@
184186
"metadata": {},
185187
"outputs": [],
186188
"source": [
187-
"# Transform back to IR from SPR\n",
189+
"# Transform back to IR from DLR\n",
188190
"gl_reconst = dlr.to_IR(g_dlr)\n",
189191
"\n",
190192
"plt.semilogy(np.abs(gl), label=\"Exact\", ls=\"\", marker=\"+\")\n",
191193
"plt.semilogy(np.abs(gl_reconst), label=\"Reconstructed from DLR\", ls=\"\", marker=\"x\")\n",
192194
"plt.semilogy(np.abs(gl-gl_reconst), label=\"error\")\n",
193195
"plt.ylim([1e-18,None])\n",
194-
"plt.ylabel(\"$|g_l|$\")\n",
196+
"plt.ylabel(\"$|G_l|$\")\n",
195197
"plt.legend(loc=\"best\", frameon=False)\n",
196198
"plt.show()"
197199
]
@@ -220,9 +222,9 @@
220222
"giv_exact = MatsubaraSampling(basis, v).evaluate(gl)\n",
221223
"\n",
222224
"plt.plot(iv.imag, giv_exact.imag, ls=\"\", marker=\"x\", label=\"Exact\")\n",
223-
"plt.plot(iv.imag, giv.imag, ls=\"\", marker=\"x\", label=\"Reconstructed from SPR\")\n",
225+
"plt.plot(iv.imag, giv.imag, ls=\"\", marker=\"x\", label=\"Reconstructed from DLR\")\n",
224226
"plt.xlabel(r\"$\\nu$\")\n",
225-
"plt.ylabel(r\"Im $G(\\mathrm{i}\\omega_n)$\")\n",
227+
"plt.ylabel(r\"Im $G(\\mathrm{i}\\nu)$\")\n",
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"\n",
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"plt.show()"
228230
]

‎src/DMFT_IPT_jl.ipynb‎

Lines changed: 5 additions & 5 deletions
Original file line numberDiff line numberDiff line change
@@ -22,11 +22,11 @@
2222
"The self-consistent equations to be solved are\n",
2323
"\n",
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"$$\n",
25-
" G_{\\mathrm{loc}}(\\mathrm{i}\\omega_n) = \\int d\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\omega_n-\\omega+\\mu-\\Sigma(\\mathrm{i}\\omega_n)}, \n",
25+
" G_{\\mathrm{loc}}(\\mathrm{i}\\nu) = \\int d\\omega \\frac{\\rho(\\omega)}{\\mathrm{i}\\nu-\\omega+\\mu-\\Sigma(\\mathrm{i}\\nu)}, \n",
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"$$\n",
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"\n",
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"$$\n",
29-
" \\mathcal{G}^{-1}(\\mathrm{i}\\omega_n) = G_{\\mathrm{loc}}^{-1}(\\mathrm{i}\\omega_n) + \\Sigma(\\mathrm{i}\\omega_n),\n",
29+
" \\mathcal{G}^{-1}(\\mathrm{i}\\nu) = G_{\\mathrm{loc}}^{-1}(\\mathrm{i}\\nu) + \\Sigma(\\mathrm{i}\\nu),\n",
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"$$\n",
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"\n",
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"$$\n",
@@ -35,14 +35,14 @@
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"\n",
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"where \n",
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"$G_{\\mathrm{loc}}$ is the local Green function, \n",
38-
"$\\omega_n$ is a fermionic Matsubara frequency, \n",
38+
"$\\nu = n\\pi/\\beta$ with odd $n$ is a fermionic Matsubara frequency, \n",
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"$\\mu$ is the chemical potential ($\\mu=U/2$ at half filling),\n",
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"$U$ is the onsite-Coulomb interaction,\n",
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"$\\Sigma$ is the self-energy,\n",
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"$\\mathcal{G}$ is the cavity Green function defined as\n",
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"\n",
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"$$\n",
45-
"\\mathcal{G}(\\mathrm{i}\\omega_n) = \\frac{1}{\\mathrm{i}\\omega_n - \\mu - \\Delta(\\mathrm{i}\\omega_n)},\n",
45+
"\\mathcal{G}(\\mathrm{i}\\nu) = \\frac{1}{\\mathrm{i}\\nu - \\mu - \\Delta(\\mathrm{i}\\nu)},\n",
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"$$\n",
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"\n",
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"$\\Delta$ is the hybrization function,\n",
@@ -54,7 +54,7 @@
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"The Mott transition can be detected by monitoring the renormalization factor\n",
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"\n",
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"$$\n",
57-
"z = \\frac{1}{1-\\frac{\\partial \\Sigma(\\omega+\\mathrm{i}0^+)}{\\partial \\omega}}\\Big|_{\\omega=0} \\approx \\frac{1}{1 - \\frac{\\Sigma(\\mathrm{i}\\omega_1)-\\Sigma(\\mathrm{i}\\omega_0)}{\\mathrm{i}\\omega_1 - \\mathrm{i}\\omega_0 }}.\n",
57+
"z = \\frac{1}{1-\\frac{\\partial \\Sigma(\\omega+\\mathrm{i}0^+)}{\\partial \\omega}}\\Big|_{\\omega=0} \\approx \\frac{1}{1 - \\frac{\\Sigma(3\\mathrm{i}\\pi/\\beta)-\\Sigma(\\mathrm{i}\\pi/\\beta)}{2\\mathrm{i}\\pi/\\beta}}.\n",
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"$$\n",
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"\n",
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"In the non-interacting limit $U\\rightarrow 0$, we have $z\\rightarrow 1$. Correlation effects reduce $z$ as $z < 1$.\n",

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