|
14 | 14 | "We consider a superconductivity in a mean-field level.\n", |
15 | 15 | "\n", |
16 | 16 | "## 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", |
18 | 18 | "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", |
20 | 20 | "\n", |
21 | 21 | "$$\n", |
22 | | - "\\hat{\\cal H} = \\left( \\begin{matrix} \\hat{H} & \\hat{\\Delta} \\\\\n", |
23 | | - "\\hat{\\Delta}^{\\dagger} & - \\hat{H}^{\\ast}\n", |
| 22 | + "\\boldsymbol{\\cal H} = \\left( \\begin{matrix} \\boldsymbol{H} & \\boldsymbol{\\Delta} \\\\\n", |
| 23 | + "\\boldsymbol{\\Delta}^{\\dagger} & - \\boldsymbol{H}^{\\ast}\n", |
24 | 24 | "\\end{matrix}\n", |
25 | 25 | "\\right),\n", |
26 | 26 | "$$\n", |
27 | 27 | "\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." |
29 | 29 | ] |
30 | 30 | }, |
31 | 31 | { |
32 | 32 | "cell_type": "markdown", |
33 | 33 | "id": "fd3ba213", |
34 | 34 | "metadata": {}, |
35 | 35 | "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", |
37 | 37 | "\n", |
38 | 38 | "$$\n", |
39 | | - "\\hat{\\cal H} \\vec{f}_{\\gamma} = \\epsilon_{\\gamma} \\vec{f}_{\\gamma}\n", |
| 39 | + "\\boldsymbol{\\cal H} \\vec{f}_{\\gamma} = \\epsilon_{\\gamma} \\vec{f}_{\\gamma}\n", |
40 | 40 | "$$\n", |
41 | 41 | "\n", |
42 | 42 | "$$\n", |
|
48 | 48 | "$$\n", |
49 | 49 | "\n", |
50 | 50 | "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", |
52 | 52 | "\n", |
53 | 53 | "$$\n", |
54 | | - "\\hat{U}^{\\dagger} \\hat{\\cal H} \\hat{U} = \\hat{D}, \\: \\: \\: \\hat{D} = {\\rm diag} (\\epsilon_1,\\epsilon_2,\\cdots,\\epsilon_{2N})\n", |
| 54 | + "\\boldsymbol{U}^{\\dagger} \\boldsymbol{\\cal H} \\boldsymbol{U} = \\boldsymbol{D}, \\: \\: \\: \\boldsymbol{D} = {\\rm diag} (\\epsilon_1,\\epsilon_2,\\cdots,\\epsilon_{2N})\n", |
55 | 55 | "$$" |
56 | 56 | ] |
57 | 57 | }, |
|
64 | 64 | "In the mean-field framework, the $2N \\times 2N$ matrix Green's function with a complex frequency is defined as \n", |
65 | 65 | "\n", |
66 | 66 | "$$\n", |
67 | | - "\\hat{G}(z) = [z \\hat{I} - \\hat{\\cal H}]^{-1}.\n", |
| 67 | + "\\boldsymbol{G}(z) = [z \\boldsymbol{I} - \\boldsymbol{\\cal H}]^{-1}.\n", |
68 | 68 | "$$\n", |
69 | 69 | "\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", |
71 | 71 | "\n", |
72 | 72 | "$$\n", |
73 | 73 | "G_{\\alpha \\beta}(z) = \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\frac{1}{z - \\epsilon_{\\gamma}}\n", |
74 | 74 | "$$\n", |
75 | 75 | "\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", |
77 | 77 | "The regarded and advanced Green's functions are, respectively, defined as \n", |
78 | 78 | "\n", |
79 | 79 | "$$\n", |
80 | | - "\\hat{G}^{\\rm R}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\hat{G}(\\omega + i \\eta)\n", |
| 80 | + "\\boldsymbol{G}^{\\rm R}(\\omega) = \\lim_{\\eta \\rightarrow 0^+} \\boldsymbol{G}(\\omega + \\mathrm{i} \\eta)\n", |
81 | 81 | "$$\n", |
82 | 82 | "\n", |
83 | 83 | "$$\n", |
84 | | - "\\hat{G}^{\\rm A}(\\omega) = \\lim_{\\tau \\rightarrow 0+} \\hat{G}(\\omega - i \\eta)\n", |
| 84 | + "\\boldsymbol{G}^{\\rm A}(\\omega) = \\lim_{\\eta \\rightarrow 0^+} \\boldsymbol{G}(\\omega - \\mathrm{i} \\eta)\n", |
85 | 85 | "$$\n", |
86 | 86 | "\n", |
87 | 87 | "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 | 93 | "For example, the local density of states with respect to the site $i$ is given as \n", |
94 | 94 | "\n", |
95 | 95 | "$$\n", |
96 | | - "N(\\omega,i) = -\\frac{1}{2\\pi i} \\vec{e}(i)^T \\hat{d}(\\omega) \\vec{e}(i),\n", |
| 96 | + "N(\\omega,i) = -\\frac{1}{2\\pi \\mathrm{i}} \\vec{e}(i)^T \\boldsymbol{d}(\\omega) \\vec{e}(i),\n", |
97 | 97 | "$$\n", |
98 | 98 | "\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", |
100 | 100 | "\n", |
101 | 101 | "$$\n", |
102 | | - "[\\hat{d}(\\omega)]_{\\alpha \\beta} \\equiv [\\hat{G}^{\\rm R}(\\omega) - \\hat{G}^{\\rm A}(\\omega)]_{\\alpha \\beta} = - 2\\pi i \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\delta(\\omega - \\epsilon_{\\gamma})\n", |
| 102 | + "[\\boldsymbol{d}(\\omega)]_{\\alpha \\beta} \\equiv [\\boldsymbol{G}^{\\rm R}(\\omega) - \\boldsymbol{G}^{\\rm A}(\\omega)]_{\\alpha \\beta} = - 2\\pi \\mathrm{i} \\sum_{\\gamma=1}^{2N} U_{\\alpha \\gamma} U_{\\beta \\gamma}^{\\ast} \\delta(\\omega - \\epsilon_{\\gamma})\n", |
103 | 103 | "$$\n", |
104 | 104 | "\n", |
105 | 105 | "Two types of mean-fields $\\langle c_i^{\\dagger} c_j \\rangle$ and $\\langle c_i c_j \\rangle$ can be expressed as \n", |
106 | 106 | "\n", |
107 | 107 | "$$\n", |
108 | | - "\\langle c_i^{\\dagger} c_j \\rangle = - \\frac{1}{2 \\pi i} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\hat{d}(\\omega) \\vec{e}(i), \n", |
| 108 | + "\\langle c_i^{\\dagger} c_j \\rangle = - \\frac{1}{2 \\pi \\mathrm{i}} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\boldsymbol{d}(\\omega) \\vec{e}(i), \n", |
109 | 109 | "$$\n", |
110 | 110 | "\n", |
111 | 111 | "$$\n", |
112 | | - "\\langle c_i c_j \\rangle = - \\frac{1}{2 \\pi i} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\hat{d}(\\omega) \\vec{h}(i), \n", |
| 112 | + "\\langle c_i c_j \\rangle = - \\frac{1}{2 \\pi \\mathrm{i}} \\int_{-\\infty}^{\\infty} d\\omega f(\\omega) \\vec{e}(j)^T \\boldsymbol{d}(\\omega) \\vec{h}(i), \n", |
113 | 113 | "$$\n", |
114 | 114 | "\n", |
115 | 115 | "with $f(x) \\equiv 1/(e^{x/T} + 1)$." |
|
124 | 124 | "With the use of the analytic continuation, the mean-fields are rewritten as \n", |
125 | 125 | "\n", |
126 | 126 | "$$\n", |
127 | | - "\\langle c_i^{\\dagger} c_j \\rangle = T \\sum_{n=-\\infty}^{\\infty} \\vec{e}(j)^T \\hat{G}(i \\omega_n) \\vec{e}(i)\n", |
| 127 | + "\\langle c_i^{\\dagger} c_j \\rangle = T \\sum_{\\nu} \\vec{e}(j)^T \\boldsymbol{G}(\\mathrm{i}\\nu) \\vec{e}(i)\\, e^{\\mathrm{i}\\nu 0^+}\n", |
128 | 128 | "$$\n", |
129 | 129 | "\n", |
130 | 130 | "$$\n", |
131 | | - "\\langle c_i c_j \\rangle = T \\sum_{n=-\\infty}^{\\infty} \\vec{e}(j)^T \\hat{G}(i \\omega_n) \\vec{h}(i)\n", |
| 131 | + "\\langle c_i c_j \\rangle = T \\sum_{\\nu} \\vec{e}(j)^T \\boldsymbol{G}(\\mathrm{i}\\nu) \\vec{h}(i)\n", |
132 | 132 | "$$\n", |
133 | 133 | "\n", |
134 | 134 | "By solving the linear equations defined as \n", |
135 | 135 | "\n", |
136 | 136 | "$$\n", |
137 | | - "(i \\omega_n \\hat{I} - \\hat{\\cal H}) \\vec{x}(i,\\omega_n) = \\vec{h}(i), \n", |
| 137 | + "(\\mathrm{i}\\nu \\boldsymbol{I} - \\boldsymbol{\\cal H}) \\vec{x}(i,\\nu) = \\vec{h}(i), \n", |
138 | 138 | "$$\n", |
139 | 139 | "\n", |
140 | 140 | "the superconducting mean-field is expressed as \n", |
141 | 141 | "\n", |
142 | 142 | "$$\n", |
143 | | - "\\langle c_i c_j \\rangle = T \\sum_{n = - \\infty}^{\\infty} \\hat{e}(j)^T \\vec{x}(i,\\omega_n).\n", |
| 143 | + "\\langle c_i c_j \\rangle = T \\sum_{\\nu} \\vec{e}(j)^T \\vec{x}(i,\\nu).\n", |
144 | 144 | "$$\n", |
145 | 145 | "\n", |
146 | 146 | "For example, s-wave superconducting order parameter is defined as \n", |
|
152 | 152 | "The self-consistent cycle has the schematic form \n", |
153 | 153 | "\n", |
154 | 154 | "$$ \n", |
155 | | - "\\hat{\\Delta} \\rightarrow \\hat{G}(i \\omega_n) \\rightarrow \\langle c_i c_i \\rangle \\rightarrow \\hat{\\Delta}\n", |
| 155 | + "\\boldsymbol{\\Delta} \\rightarrow \\boldsymbol{G}(\\mathrm{i}\\nu) \\rightarrow \\langle c_i c_i \\rangle \\rightarrow \\boldsymbol{\\Delta}\n", |
156 | 156 | "$$" |
157 | 157 | ] |
158 | 158 | }, |
|
174 | 174 | "The superconducting mean-fielads are expressed as \n", |
175 | 175 | "\n", |
176 | 176 | "$$\n", |
177 | | - "\\langle c_i c_j \\rangle = \\vec{e}(j)^T \\hat{G}(\\tau = \\beta) \\vec{h}(i) = \\sum_{l=0}^{N_{\\rm IR}-1} U_l(\\beta) \\vec{e}(j)^T \\hat{G}_l \\vec{h}(i)\n", |
| 177 | + "\\langle c_i c_j \\rangle = \\vec{e}(j)^T \\boldsymbol{G}(\\tau = \\beta^-) \\vec{h}(i) = \\sum_{l=0}^{L-1} U_l(\\beta^-) \\vec{e}(j)^T \\boldsymbol{G}_l \\vec{h}(i)\n", |
178 | 178 | "$$\n", |
179 | 179 | "\n", |
180 | 180 | "Since the Matsubara Green's function is expressed as \n", |
181 | 181 | "\n", |
182 | 182 | "$$\n", |
183 | | - "G(i \\omega_n) = \\sum_{l=0}^{N_{\\rm IR}-1} G_l U_l(i \\omega_n),\n", |
| 183 | + "G(\\mathrm{i}\\nu) = \\sum_{l=0}^{L-1} G_l \\hat{U}_l(\\mathrm{i}\\nu),\n", |
184 | 184 | "$$\n", |
185 | 185 | "\n", |
186 | 186 | "one can obtain $G_l$ by fitting the above equation." |
|
202 | 202 | "The matrix form of the above Hamiltonain is expressed as \n", |
203 | 203 | "\n", |
204 | 204 | "$$\n", |
205 | | - "\\hat{H} = -\\mu \\hat{I} -t (\\hat{T}_{+x} + \\hat{T}_{-x} + \\hat{T}_{+y} + \\hat{T}_{-y}),\n", |
| 205 | + "\\boldsymbol{H} = -\\mu \\boldsymbol{I} -t (\\boldsymbol{T}_{+x} + \\boldsymbol{T}_{-x} + \\boldsymbol{T}_{+y} + \\boldsymbol{T}_{-y}),\n", |
206 | 206 | "$$\n", |
207 | 207 | "\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." |
209 | 209 | ] |
210 | 210 | }, |
211 | 211 | { |
|
445 | 445 | "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", |
446 | 446 | "\n", |
447 | 447 | "$$\n", |
448 | | - "\\langle c_i c_j \\rangle = T \\sum_{n = - n_c}^{n_c} \\hat{e}(j)^T \\vec{x}(i,\\omega_n).\n", |
| 448 | + "\\langle c_i c_j \\rangle = T \\sum_{|\\nu| < \\nu_c} \\vec{e}(j)^T \\vec{x}(i,\\nu).\n", |
449 | 449 | "$$\n", |
450 | 450 | "\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" |
453 | 453 | ] |
454 | 454 | }, |
455 | 455 | { |
|
769 | 769 | "The Hamiltonian in momentum space is given as \n", |
770 | 770 | "\n", |
771 | 771 | "$$\n", |
772 | | - "\\hat{\\cal H}(\\vec{k}) = \n", |
| 772 | + "\\boldsymbol{\\cal H}(\\vec{k}) = \n", |
773 | 773 | "\\left(\n", |
774 | 774 | " \\begin{matrix}\n", |
775 | | - " h_0(\\vec{k}) & i \\Delta \\sigma_y \\\\\n", |
776 | | - " - i \\Delta^{\\dagger} \\sigma_y & - h_0^{\\ast}(\\vec{k})\n", |
| 775 | + " h_0(\\vec{k}) & \\mathrm{i} \\Delta \\sigma_y \\\\\n", |
| 776 | + " - \\mathrm{i} \\Delta^{\\dagger} \\sigma_y & - h_0^{\\ast}(\\vec{k})\n", |
777 | 777 | "\\end{matrix}\n", |
778 | 778 | "\\right), \n", |
779 | 779 | "$$\n", |
|
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