\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u516c\u5f0f | \u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3068\u5fdc\u7528<\/title><br \/>\n<\/head><br \/>\n<body><\/p>\n<h2>\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u516c\u5f0f | \u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3068\u5fdc\u7528<\/h2>\n<p>\n\u7269\u7406\u5b66\u3068\u5de5\u5b66\u306e\u5206\u91ce\u3067\u306f\u3001\u30ef\u30cb\u30a8\u95a2\u6570\uff08Wannier Function\uff09\u306f\u56fa\u4f53\u7269\u7406\u5b66\u3067\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u307e\u3059\u3002\u7279\u306b\u3001\u7d50\u6676\u4e2d\u306e\u96fb\u5b50\u72b6\u614b\u3092\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002\u672c\u8a18\u4e8b\u3067\u306f\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u8a08\u7b97\u65b9\u6cd5\u3001\u305d\u3057\u3066\u5b9f\u969b\u306e\u5fdc\u7528\u306b\u3064\u3044\u3066\u308f\u304b\u308a\u3084\u3059\u304f\u8aac\u660e\u3057\u307e\u3059\u3002\n<\/p>\n<h2>\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u30d6\u30ed\u30c3\u30db\u95a2\u6570\uff08Bloch Function\uff09\u3092\u57fa\u306b\u69cb\u7bc9\u3055\u308c\u308b\u5c40\u5728\u5316\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u3067\u3059\u3002\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u306f\u5e73\u9762\u6ce2\u3068\u7d50\u6676\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u306e\u76f8\u4e92\u4f5c\u7528\u3092\u8003\u616e\u306b\u5165\u308c\u305f\u3082\u306e\u3067\u3059\u304c\u3001\u3053\u308c\u3092\u5c40\u5728\u5316\u3055\u305b\u305f\u3082\u306e\u304c\u30ef\u30cb\u30a8\u95a2\u6570\u3067\u3059\u3002\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u6b21\u306e\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\uff1a\n<\/p>\n<p>\n\\[ w_n(\\vec{R}) = \\frac{1}{\\sqrt{N}} \\sum_{\\vec{k}} e^{-i \\vec{k} \\cdot \\vec{R}} \\psi_{\\vec{k}, n} \\]\n<\/p>\n<p>\n\u3053\u3053\u3067\u3001\\( \\psi_{\\vec{k}, n} \\)\u306f\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3001N\u306f\u5358\u4f4d\u30bb\u30eb\u306e\u6570\u3001\\( \\vec{R} \\)\u306f\u683c\u5b50\u306e\u4f4d\u7f6e\u3092\u8868\u3057\u307e\u3059\u3002\n<\/p>\n<h2>\u8a08\u7b97\u65b9\u6cd5<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u307e\u305a\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u8a18\u8ff0\u3055\u308c\u307e\u3059\uff1a\n<\/p>\n<p>\n\\[ \\psi_{\\vec{k}, n} (\\vec{r}) = e^{i \\vec{k} \\cdot \\vec{r}} u_{\\vec{k}, n} (\\vec{r}) \\]\n<\/p>\n<p>\n\u3053\u3053\u3067\u3001\\( u_{\\vec{k}, n} (\\vec{r}) \\)\u306f\u5468\u671f\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u95a2\u3059\u308b\u95a2\u6570\u3067\u3059\u3002\u3053\u306e\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092Fourier\u5909\u63db\u3057\u3001\u5c40\u5728\u5316\u3059\u308b\u3053\u3068\u3067\u30ef\u30cb\u30a8\u95a2\u6570\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\n<\/p>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u6570\u5024\u7684\u306b\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u65b9\u6cd5\u3068\u3057\u3066\u306f\u3001\u4ee5\u4e0b\u306e\u624b\u9806\u304c\u4e00\u822c\u7684\u3067\u3059\uff1a\n<\/p>\n<ol>\n<li>\u307e\u305a\u3001Brillouin\u30be\u30fc\u30f3\u5185\u306e\u5404 \\( k \\)-\u70b9\u3067\u30d6\u30ed\u30c3\u30db\u95a2\u6570 \\( \\psi_{\\vec{k}, n} \\) \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\n<li>\u8a08\u7b97\u3057\u305f\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u4e0a\u8a18\u306e\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9\u5f0f\u306b\u5f93\u3044Fourier\u5909\u63db\u3092\u884c\u3044\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u5f97\u307e\u3059\u3002<\/li>\n<li>\u5f97\u3089\u308c\u305f\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u57fa\u306b\u3001\u7279\u5b9a\u306e\u7269\u7406\u91cf\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\n<\/ol>\n<h2>\u5fdc\u7528<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u7269\u8cea\u306e\u96fb\u5b50\u69cb\u9020\u3084\u7269\u6027\u3092\u8abf\u3079\u308b\u969b\u306b\u975e\u5e38\u306b\u6709\u7528\u3067\u3059\u3002\u4f8b\u3048\u3070\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5fdc\u7528\u304c\u3042\u308a\u307e\u3059\uff1a\n<\/p>\n<ul>\n<li><strong>\u96fb\u5b50\u72b6\u614b\u306e\u89e3\u6790<\/strong>\uff1a\u534a\u5c0e\u4f53\u3084\u91d1\u5c5e\u306e\u30d0\u30f3\u30c9\u69cb\u9020\u89e3\u6790\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n<li><strong>\u8d85\u4f1d\u5c0e\u306e\u7814\u7a76<\/strong>\uff1a\u9ad8\u6e29\u8d85\u4f1d\u5c0e\u4f53\u306e\u96fb\u5b50\u76f8\u95a2\u3092\u8abf\u3079\u308b\u305f\u3081\u306b\u5229\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n<li><strong>\u6750\u6599\u8a2d\u8a08<\/strong>\uff1a\u65b0\u3057\u3044\u6a5f\u80fd\u6750\u6599\u306e\u958b\u767a\u3084\u7279\u6027\u8a55\u4fa1\u306b\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u307e\u3059\u3002<\/li>\n<\/ul>\n<p>\n\u5b9f\u969b\u306e\u7814\u7a76\u3067\u306f\u3001\u30ef\u30cb\u30a8\u5316\uff08Wannierization\uff09\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u3092\u7528\u3044\u3066\u3001\u8a08\u7b97\u6a5f\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3084\u5b9f\u9a13\u30c7\u30fc\u30bf\u3068\u6bd4\u8f03\u3059\u308b\u3053\u3068\u304c\u4e00\u822c\u7684\u3067\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u7269\u8cea\u306e\u6027\u8cea\u3092\u5b9a\u91cf\u7684\u306b\u8a55\u4fa1\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u3068\u306a\u308a\u307e\u3059\u3002\n<\/p>\n<h2>\u307e\u3068\u3081<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u7d50\u6676\u4e2d\u306e\u96fb\u5b50\u72b6\u614b\u3092\u5c40\u5728\u5316\u3057\u3066\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306e\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u3067\u3059\u3002\u305d\u306e\u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3001\u305d\u3057\u3066\u591a\u5c90\u306b\u308f\u305f\u308b\u5fdc\u7528\u306b\u3064\u3044\u3066\u7406\u89e3\u3059\u308b\u3053\u3068\u3067\u3001\u7269\u7406\u3084\u5de5\u5b66\u306e\u5206\u91ce\u3067\u65b0\u305f\u306a\u767a\u898b\u3084\u6280\u8853\u9769\u65b0\u304c\u671f\u5f85\u3055\u308c\u307e\u3059\u3002\u672c\u8a18\u4e8b\u3092\u901a\u3058\u3066\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u57fa\u790e\u7684\u306a\u7406\u89e3\u304c\u6df1\u307e\u3063\u305f\u3053\u3068\u3092\u9858\u3063\u3066\u3044\u307e\u3059\u3002\n<\/p>\n<p><\/body><br 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\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3001\u5fdc\u7528\u306b\u3064\u3044\u3066\u8aac\u660e\u3002\u96fb\u5b50\u72b6\u614b\u306e\u89e3\u6790\u3084\u8d85\u4f1d\u5c0e\u306e\u7814\u7a76\u3001\u6750\u6599\u8a2d\u8a08\u306b\u5f79\u7acb\u3064\u7269\u7406\u306e\u57fa\u790e\u77e5\u8b58\u3092\u308f\u304b\u308a\u3084\u3059\u304f\u89e3\u8aac\u3002<\/p>\n

\n
\n
\n
\n \u30ef\u30cb\u30a8\u95a2\u6570\u306e\u516c\u5f0f | \u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3068\u5fdc\u7528<\/title><br \/>\n<\/head><br \/>\n<body><\/p>\n<h2>\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u516c\u5f0f | \u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3068\u5fdc\u7528<\/h2>\n<p>\n\u7269\u7406\u5b66\u3068\u5de5\u5b66\u306e\u5206\u91ce\u3067\u306f\u3001\u30ef\u30cb\u30a8\u95a2\u6570\uff08Wannier Function\uff09\u306f\u56fa\u4f53\u7269\u7406\u5b66\u3067\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u307e\u3059\u3002\u7279\u306b\u3001\u7d50\u6676\u4e2d\u306e\u96fb\u5b50\u72b6\u614b\u3092\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002\u672c\u8a18\u4e8b\u3067\u306f\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9\u3068\u305d\u306e\u8a08\u7b97\u65b9\u6cd5\u3001\u305d\u3057\u3066\u5b9f\u969b\u306e\u5fdc\u7528\u306b\u3064\u3044\u3066\u308f\u304b\u308a\u3084\u3059\u304f\u8aac\u660e\u3057\u307e\u3059\u3002\n<\/p>\n<h2>\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u30d6\u30ed\u30c3\u30db\u95a2\u6570\uff08Bloch Function\uff09\u3092\u57fa\u306b\u69cb\u7bc9\u3055\u308c\u308b\u5c40\u5728\u5316\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u3067\u3059\u3002\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u306f\u5e73\u9762\u6ce2\u3068\u7d50\u6676\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u306e\u76f8\u4e92\u4f5c\u7528\u3092\u8003\u616e\u306b\u5165\u308c\u305f\u3082\u306e\u3067\u3059\u304c\u3001\u3053\u308c\u3092\u5c40\u5728\u5316\u3055\u305b\u305f\u3082\u306e\u304c\u30ef\u30cb\u30a8\u95a2\u6570\u3067\u3059\u3002\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u6b21\u306e\u5f0f\u3067\u8868\u3055\u308c\u307e\u3059\uff1a\n<\/p>\n<p>\n\\[ w_n(\\vec{R}) = \\frac{1}{\\sqrt{N}} \\sum_{\\vec{k}} e^{-i \\vec{k} \\cdot \\vec{R}} \\psi_{\\vec{k}, n} \\]\n<\/p>\n<p>\n\u3053\u3053\u3067\u3001\\( \\psi_{\\vec{k}, n} \\)\u306f\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3001N\u306f\u5358\u4f4d\u30bb\u30eb\u306e\u6570\u3001\\( \\vec{R} \\)\u306f\u683c\u5b50\u306e\u4f4d\u7f6e\u3092\u8868\u3057\u307e\u3059\u3002\n<\/p>\n<h2>\u8a08\u7b97\u65b9\u6cd5<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u306f\u3001\u307e\u305a\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092\u6c42\u3081\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306b\u8a18\u8ff0\u3055\u308c\u307e\u3059\uff1a\n<\/p>\n<p>\n\\[ \\psi_{\\vec{k}, n} (\\vec{r}) = e^{i \\vec{k} \\cdot \\vec{r}} u_{\\vec{k}, n} (\\vec{r}) \\]\n<\/p>\n<p>\n\u3053\u3053\u3067\u3001\\( u_{\\vec{k}, n} (\\vec{r}) \\)\u306f\u5468\u671f\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u95a2\u3059\u308b\u95a2\u6570\u3067\u3059\u3002\u3053\u306e\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092Fourier\u5909\u63db\u3057\u3001\u5c40\u5728\u5316\u3059\u308b\u3053\u3068\u3067\u30ef\u30cb\u30a8\u95a2\u6570\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\n<\/p>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u6570\u5024\u7684\u306b\u8a08\u7b97\u3059\u308b\u305f\u3081\u306e\u65b9\u6cd5\u3068\u3057\u3066\u306f\u3001\u4ee5\u4e0b\u306e\u624b\u9806\u304c\u4e00\u822c\u7684\u3067\u3059\uff1a\n<\/p>\n<ol>\n<li>\u307e\u305a\u3001Brillouin\u30be\u30fc\u30f3\u5185\u306e\u5404 \\( k \\)-\u70b9\u3067\u30d6\u30ed\u30c3\u30db\u95a2\u6570 \\( \\psi_{\\vec{k}, n} \\) \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\n<li>\u8a08\u7b97\u3057\u305f\u30d6\u30ed\u30c3\u30db\u95a2\u6570\u3092\u7528\u3044\u3066\u3001\u4e0a\u8a18\u306e\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u5b9a\u7fa9\u5f0f\u306b\u5f93\u3044Fourier\u5909\u63db\u3092\u884c\u3044\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u5f97\u307e\u3059\u3002<\/li>\n<li>\u5f97\u3089\u308c\u305f\u30ef\u30cb\u30a8\u95a2\u6570\u3092\u57fa\u306b\u3001\u7279\u5b9a\u306e\u7269\u7406\u91cf\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/li>\n<\/ol>\n<h2>\u5fdc\u7528<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u7269\u8cea\u306e\u96fb\u5b50\u69cb\u9020\u3084\u7269\u6027\u3092\u8abf\u3079\u308b\u969b\u306b\u975e\u5e38\u306b\u6709\u7528\u3067\u3059\u3002\u4f8b\u3048\u3070\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5fdc\u7528\u304c\u3042\u308a\u307e\u3059\uff1a\n<\/p>\n<ul>\n<li><strong>\u96fb\u5b50\u72b6\u614b\u306e\u89e3\u6790<\/strong>\uff1a\u534a\u5c0e\u4f53\u3084\u91d1\u5c5e\u306e\u30d0\u30f3\u30c9\u69cb\u9020\u89e3\u6790\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n<li><strong>\u8d85\u4f1d\u5c0e\u306e\u7814\u7a76<\/strong>\uff1a\u9ad8\u6e29\u8d85\u4f1d\u5c0e\u4f53\u306e\u96fb\u5b50\u76f8\u95a2\u3092\u8abf\u3079\u308b\u305f\u3081\u306b\u5229\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n<li><strong>\u6750\u6599\u8a2d\u8a08<\/strong>\uff1a\u65b0\u3057\u3044\u6a5f\u80fd\u6750\u6599\u306e\u958b\u767a\u3084\u7279\u6027\u8a55\u4fa1\u306b\u91cd\u8981\u306a\u5f79\u5272\u3092\u679c\u305f\u3057\u307e\u3059\u3002<\/li>\n<\/ul>\n<p>\n\u5b9f\u969b\u306e\u7814\u7a76\u3067\u306f\u3001\u30ef\u30cb\u30a8\u5316\uff08Wannierization\uff09\u3068\u547c\u3070\u308c\u308b\u624b\u6cd5\u3092\u7528\u3044\u3066\u3001\u8a08\u7b97\u6a5f\u30b7\u30df\u30e5\u30ec\u30fc\u30b7\u30e7\u30f3\u3084\u5b9f\u9a13\u30c7\u30fc\u30bf\u3068\u6bd4\u8f03\u3059\u308b\u3053\u3068\u304c\u4e00\u822c\u7684\u3067\u3059\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u7269\u8cea\u306e\u6027\u8cea\u3092\u5b9a\u91cf\u7684\u306b\u8a55\u4fa1\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u3068\u306a\u308a\u307e\u3059\u3002\n<\/p>\n<h2>\u307e\u3068\u3081<\/h2>\n<p>\n\u30ef\u30cb\u30a8\u95a2\u6570\u306f\u3001\u7d50\u6676\u4e2d\u306e\u96fb\u5b50\u72b6\u614b\u3092\u5c40\u5728\u5316\u3057\u3066\u8a18\u8ff0\u3059\u308b\u305f\u3081\u306e\u5f37\u529b\u306a\u30c4\u30fc\u30eb\u3067\u3059\u3002\u305d\u306e\u5b9a\u7fa9\u3001\u8a08\u7b97\u65b9\u6cd5\u3001\u305d\u3057\u3066\u591a\u5c90\u306b\u308f\u305f\u308b\u5fdc\u7528\u306b\u3064\u3044\u3066\u7406\u89e3\u3059\u308b\u3053\u3068\u3067\u3001\u7269\u7406\u3084\u5de5\u5b66\u306e\u5206\u91ce\u3067\u65b0\u305f\u306a\u767a\u898b\u3084\u6280\u8853\u9769\u65b0\u304c\u671f\u5f85\u3055\u308c\u307e\u3059\u3002\u672c\u8a18\u4e8b\u3092\u901a\u3058\u3066\u3001\u30ef\u30cb\u30a8\u95a2\u6570\u306e\u57fa\u790e\u7684\u306a\u7406\u89e3\u304c\u6df1\u307e\u3063\u305f\u3053\u3068\u3092\u9858\u3063\u3066\u3044\u307e\u3059\u3002\n<\/p>\n<p><\/body><br 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