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TN.html
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TN.html
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<!doctype html>
<html>
<head>
<meta charset='UTF-8'><meta name='viewport' content='width=device-width initial-scale=1'>
<title>TN</title><link href='https://fonts.loli.net/css?family=PT+Serif:400,400italic,700,700italic&subset=latin,cyrillic-ext,cyrillic,latin-ext' rel='stylesheet' type='text/css' /><style type='text/css'>html {overflow-x: initial !important;}:root { --bg-color:#ffffff; --text-color:#333333; --select-text-bg-color:#B5D6FC; --select-text-font-color:auto; --monospace:"Lucida Console",Consolas,"Courier",monospace; }
html { font-size: 14px; background-color: var(--bg-color); color: var(--text-color); font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; -webkit-font-smoothing: antialiased; }
body { margin: 0px; padding: 0px; height: auto; bottom: 0px; top: 0px; left: 0px; right: 0px; font-size: 1rem; line-height: 1.42857; overflow-x: hidden; background: inherit; tab-size: 4; }
iframe { margin: auto; }
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<div id='write' class = 'is-node'><h1><a name='header-n0' class='md-header-anchor '></a>Model</h1><p>Anitferromagnetic Ising model Hamiltonian</p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n4" cid="n4" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-1-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="106.076px" height="46.7535px" viewBox="0 -1007.2 5709.5 2517" role="img" focusable="false" style="vertical-align: -3.507ex; max-width: 100%;"><defs><path stroke-width="0" id="E1-MJMATHI-48" d="M228 637Q194 637 192 641Q191 643 191 649Q191 673 202 682Q204 683 219 683Q260 681 355 681Q389 681 418 681T463 682T483 682Q499 682 499 672Q499 670 497 658Q492 641 487 638H485Q483 638 480 638T473 638T464 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S_{lk}</script></div></div><p><img src="./images/4lattice.png" width="600"></p><h1><a name='header-n19' class='md-header-anchor '></a>变换</h1><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n20" cid="n20" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-8-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="117.639px" height="20.6076px" viewBox="0 -1007.2 6331.6 1109.7" role="img" focusable="false" style="vertical-align: -0.238ex; max-width: 100%;"><defs><path stroke-width="0" id="E8-MJMATHI-53" d="M308 24Q367 24 416 76T466 197Q466 260 414 284Q308 311 278 321T236 341Q176 383 176 462Q176 523 208 573T273 648Q302 673 343 688T407 704H418H425Q521 704 564 640Q565 640 577 653T603 682T623 704Q624 704 627 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x="529" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-62" x="995" y="0"></use></g></g><g transform="translate(7720,0)"><use xlink:href="#E13-MJMATHI-54" x="0" y="0"></use><g transform="translate(584,-155)"><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-62" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-63" x="429" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-66" x="862" y="0"></use></g></g><g transform="translate(9402,0)"><use xlink:href="#E13-MJMATHI-54" x="0" y="0"></use><g transform="translate(584,-155)"><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-64" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-65" x="523" y="0"></use><use transform="scale(0.707)" xlink:href="#E13-MJMATHI-66" x="989" y="0"></use></g></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-13">Z = \sum_{abcdef} T_{adc}T_{aeb} T_{bcf} T_{def}</script></div></div><p>partition functiom</p><div contenteditable="false" spellcheck="false" class="mathjax-block md-end-block md-math-block md-rawblock" id="mathjax-n31" cid="n31" mdtype="math_block"><div class="md-rawblock-container md-math-container" tabindex="-1"><div class="MathJax_SVG_Display" style="text-align: center;"><span class="MathJax_SVG" id="MathJax-Element-14-Frame" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="95px" height="20.6076px" viewBox="0 -806.1 5113.4 1109.7" role="img" focusable="false" style="vertical-align: -0.705ex; max-width: 100%;"><defs><path stroke-width="0" id="E14-MJMATHI-5A" d="M58 8Q58 23 64 35Q64 36 329 334T596 635L586 637Q575 637 512 637H500H476Q442 637 420 635T365 624T311 598T266 548T228 469Q227 466 226 463T224 458T223 453T222 450L221 448Q218 443 202 443Q185 443 182 453L214 561Q228 606 241 651Q249 679 253 681Q256 683 487 683H718Q723 678 723 675Q723 673 717 649Q189 54 188 52L185 49H274Q369 50 377 51Q452 60 500 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62Q367 92 389 158T418 266T424 322Z"></path><path stroke-width="0" id="E14-MJMAIN-5E" d="M112 560L249 694L257 686Q387 562 387 560L361 531Q359 532 303 581L250 627L195 580Q182 569 169 557T148 538L140 532Q138 530 125 546L112 560Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E14-MJMATHI-5A" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-28" x="723" y="0"></use><use xlink:href="#E14-MJMATHI-3B2" x="1112" y="0"></use><use xlink:href="#E14-MJMAIN-29" x="1685" y="0"></use><use xlink:href="#E14-MJMAIN-3D" x="2351" y="0"></use><g transform="translate(3407,0)"><use xlink:href="#E14-MJMAIN-54" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-72" x="722" y="0"></use></g><g transform="translate(4521,0)"><use xlink:href="#E14-MJMATHI-3C1" x="0" y="0"></use><use xlink:href="#E14-MJMAIN-5E" x="91" y="-14"></use></g></g></svg></span></div><script type="math/tex; mode=display" id="MathJax-Element-14">Z(\beta) = \mathrm{Tr}\hat{\rho}</script></div></div><table>
<tbody><tr>
<td><img src="./images/4latticeS.png" width="600" onerror="this.style.display = 'none';"></td>
<td><img src="./images/sqrtS.png" width="600" onerror="this.style.display = 'none';"></td>
<td><img src="./images/T.png" width="600" onerror="this.style.display = 'none';"></td>
</tr>
</tbody></table><p> </p><p> </p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">"""</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">This is a triangle lattice of Ising model.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">Using tensor network method to get the partition, ground state energy</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">and degeneracy of ground state energy.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">In addition, also calculate the exact partition function.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">"""</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">torch</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">from</span> <span class="cm-variable">scipy</span> <span class="cm-keyword">import</span> <span class="cm-variable">gradient</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">from</span> <span class="cm-variable">scipy</span>.<span class="cm-property">linalg</span> <span class="cm-keyword">import</span> <span class="cm-variable">sqrtm</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">from</span> <span class="cm-variable">torch</span>.<span class="cm-property">autograd</span> <span class="cm-keyword">import</span> <span class="cm-variable">Variable</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">from</span> <span class="cm-variable">matplotlib</span> <span class="cm-keyword">import</span> <span class="cm-variable">pyplot</span> <span class="cm-keyword">as</span> <span class="cm-variable">plt</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># define a function: compute the value of derivetive of function f(x)</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># at x = a. Number of setps is N, and compute in the region [a-l, a+l]</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">mydiff</span>(<span class="cm-variable">f</span>,<span class="cm-variable">a</span>, <span class="cm-variable">N</span> = <span class="cm-number">1e3</span>, <span class="cm-variable">l</span> = <span class="cm-number">1e-2</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">s</span> = <span class="cm-number">2</span><span class="cm-operator">*</span><span class="cm-variable">l</span><span class="cm-operator">/</span><span class="cm-variable">N</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">x</span> = <span class="cm-variable">np</span>.<span class="cm-property">linspace</span>(<span class="cm-variable">a</span><span class="cm-operator">-</span><span class="cm-variable">l</span>,<span class="cm-variable">a</span><span class="cm-operator">+</span><span class="cm-variable">l</span>,<span class="cm-variable">N</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">fun</span> = <span class="cm-variable">np</span>.<span class="cm-property">linspace</span>(<span class="cm-number">0</span>,<span class="cm-number">0</span>,<span class="cm-variable">N</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-builtin">int</span>(<span class="cm-variable">N</span>)):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">fun</span>[<span class="cm-variable">i</span>] = <span class="cm-variable">f</span>(<span class="cm-variable">x</span>[<span class="cm-variable">i</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">d</span> = <span class="cm-variable">gradient</span>(<span class="cm-variable">fun</span>,<span class="cm-variable">s</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">d</span> = <span class="cm-variable">d</span>[<span class="cm-builtin">int</span>(<span class="cm-variable">N</span>)<span class="cm-operator">//</span><span class="cm-number">2</span>]</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">d</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">beta</span> = <span class="cm-number">100</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># compute the partition function Z</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">getZ</span>(<span class="cm-variable">beta</span>,<span class="cm-variable">mu</span>=<span class="cm-number">0</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">B</span> = <span class="cm-variable">np</span>.<span class="cm-property">array</span>([ [<span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">beta</span>), <span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span>)],</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> [<span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span>), <span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">beta</span>)] ])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">B</span> = <span class="cm-variable">B</span><span class="cm-operator">*</span><span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span><span class="cm-operator">*</span><span class="cm-variable">mu</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">A1</span> = <span class="cm-variable">torch</span>.<span class="cm-property">tensor</span>(<span class="cm-variable">B</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">torch</span>.<span class="cm-property">einsum</span>(<span class="cm-string">'ij,jk,ki->'</span>,<span class="cm-variable">A1</span>,<span class="cm-variable">A1</span>,<span class="cm-variable">A1</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">Z</span>.<span class="cm-property">numpy</span>()</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">Z</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">print</span>(<span class="cm-string">'Z='</span>,<span class="cm-variable">getZ</span>(<span class="cm-variable">beta</span>))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># compute lnZ</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">lnZ</span>(<span class="cm-variable">beta</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">getZ</span>(<span class="cm-variable">beta</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-variable">Z</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">Z</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># when 1 is negleticable compare to beta, E is the ground state energy.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">E0</span> = <span class="cm-operator">-</span><span class="cm-variable">mydiff</span>(<span class="cm-variable">lnZ</span>, <span class="cm-variable">beta</span>, <span class="cm-variable">N</span> = <span class="cm-number">1e2</span>, <span class="cm-variable">l</span> = <span class="cm-number">1e-4</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#E0 = np.around(E0)</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">print</span>(<span class="cm-string">'E0 = '</span>,<span class="cm-variable">E0</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment">#print('zzdaf', lnZ(.999))</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">mu</span> = <span class="cm-operator">-</span><span class="cm-number">1</span><span class="cm-operator">/</span><span class="cm-number">3</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># whe mu is equal to groud state energy E0/number of interaction</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">Deg</span>(<span class="cm-variable">beta</span>,<span class="cm-variable">mu</span>=<span class="cm-variable">mu</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Deg</span> = <span class="cm-variable">getZ</span>(<span class="cm-variable">beta</span>,<span class="cm-variable">mu</span>=<span class="cm-variable">mu</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">Deg</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">print</span>(<span class="cm-string">'degeneracy is'</span>,<span class="cm-variable">Deg</span>(<span class="cm-variable">beta</span>))</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">ZZ</span> = (<span class="cm-number">2</span><span class="cm-operator">*</span><span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-number">3</span><span class="cm-operator">*</span><span class="cm-variable">beta</span>) <span class="cm-operator">+</span> <span class="cm-number">6</span><span class="cm-operator">*</span><span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span>) )<span class="cm-comment">#*np.exp(-100)</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">print</span>(<span class="cm-string">'exact partition function is'</span>, <span class="cm-variable">ZZ</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1728px;"></div><div class="CodeMirror-gutters" style="display: none; height: 1728px;"></div></div></div></pre><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded md-focus" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 72px; left: 399.594px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="" style="position: relative;"><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">Z= 1.6128702850896815e+44</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">E0 = -1.0101010126106758</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">degeneracy is 6.000000000000087</span></pre><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">exact partition function is 1.6128702850896815e+44</span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 96px;"></div><div class="CodeMirror-gutters" style="display: none; height: 96px;"></div></div></div></pre><table>
<tbody><tr>
<td><img src="./images/C60.png" width="600" onerror="this.style.display = 'none';"></td>
<td><img src="./images/icosahedron.png" width="600" onerror="this.style.display = 'none';"></td>
</tr>
</tbody></table><p> </p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang="python" style="break-inside: unset;"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">"""</span></span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">This is a python3.7 code to calculate the ground state energy of a</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">Ising model of C60 using tensor network method</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">copyright: Copyright 2019 by ZQW.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">license: MIT</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string">"""</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">import</span> <span class="cm-variable">numpy</span> <span class="cm-keyword">as</span> <span class="cm-variable">np</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">from</span> <span class="cm-variable">scipy</span>.<span class="cm-property">linalg</span> <span class="cm-keyword">import</span> <span class="cm-variable">sqrtm</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">mydiff</span>(<span class="cm-variable">f</span>,<span class="cm-variable">a</span>,<span class="cm-variable">b</span>,<span class="cm-variable">N</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-string">"""</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> get the derivative of function</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> f: the input function</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> N: calculate the derivatives of function f at N points in [a, b]</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> return: N derivatives of function f</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> """</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">fx</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable">N</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">s</span> = (<span class="cm-variable">b</span><span class="cm-operator">-</span><span class="cm-variable">a</span>)<span class="cm-operator">/</span>(<span class="cm-variable">N</span><span class="cm-operator">-</span><span class="cm-number">1</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">d</span> = <span class="cm-variable">np</span>.<span class="cm-property">zeros</span>(<span class="cm-variable">N</span><span class="cm-operator">-</span><span class="cm-number">1</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">for</span> <span class="cm-variable">i</span> <span class="cm-keyword">in</span> <span class="cm-builtin">range</span>(<span class="cm-variable">N</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">x</span> = <span class="cm-variable">a</span><span class="cm-operator">+</span><span class="cm-variable">i</span><span class="cm-operator">*</span><span class="cm-variable">s</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">fx</span>[<span class="cm-variable">i</span>] = <span class="cm-variable">f</span>(<span class="cm-variable">x</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-builtin">print</span>(<span class="cm-string">'f('</span>,<span class="cm-variable">x</span>,<span class="cm-string">')='</span>,<span class="cm-variable">fx</span>[<span class="cm-variable">i</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">if</span> <span class="cm-variable">i</span><span class="cm-operator">></span><span class="cm-number">0</span>:</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">d</span>[<span class="cm-variable">i</span><span class="cm-operator">-</span><span class="cm-number">1</span>] = ( <span class="cm-variable">fx</span>[<span class="cm-variable">i</span>] <span class="cm-operator">-</span> <span class="cm-variable">fx</span>[<span class="cm-variable">i</span><span class="cm-operator">-</span><span class="cm-number">1</span>] ) <span class="cm-operator">/</span><span class="cm-variable">s</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-builtin">print</span>(<span class="cm-string">'f\' between'</span>,<span class="cm-variable">x</span>,<span class="cm-string">'and'</span>,<span class="cm-variable">x</span><span class="cm-operator">-</span><span class="cm-variable">s</span>,<span class="cm-string">'is'</span>,<span class="cm-variable">d</span>[<span class="cm-variable">i</span><span class="cm-operator">-</span><span class="cm-number">1</span>])</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">d</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-keyword">def</span> <span class="cm-def">getlnZ</span>(<span class="cm-variable">beta</span>):</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">S</span> = <span class="cm-variable">np</span>.<span class="cm-property">array</span>([ [<span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">beta</span>), <span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span>)],<span class="cm-error">\</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> [<span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-variable">beta</span>), <span class="cm-variable">np</span>.<span class="cm-property">exp</span>(<span class="cm-operator">-</span><span class="cm-variable">beta</span>)]]) <span class="cm-comment"># the S matrix</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-comment"># between two lattice</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">S</span> = <span class="cm-variable">sqrtm</span>(<span class="cm-variable">S</span>)</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">T</span> = <span class="cm-variable">np</span>.<span class="cm-property">einsum</span>(<span class="cm-string">'ij,ik,il->jkl'</span>,<span class="cm-variable">S</span>,<span class="cm-variable">S</span>,<span class="cm-variable">S</span>) <span class="cm-comment"># the 3-tensor at each lattice</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">T</span> = <span class="cm-variable">np</span>.<span class="cm-property">einsum</span>(<span class="cm-string">'iae,jab,kbc,lcd,mde->ijklm'</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>) <span class="cm-comment"># the</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-comment"># 5-tensor at each lattice of a icosahedron after contraction of 12 pentagon</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">np</span>.<span class="cm-property">einsum</span>(<span class="cm-string">'abcde,kfejt,jdirs,ciqph,hongb,mgafl,\</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-string"> ABCDz,Ayrqx,Bxpow,Cwnmv,Dvlku,zutsy->'</span><span class="cm-error">\</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> ,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>,<span class="cm-variable">T</span>) <span class="cm-comment"># contract the 12</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-comment"># lattices of the icosahedron to get the partition function of C60</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">Z</span> = <span class="cm-variable">Z</span>.<span class="cm-property">real</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-variable">lnZ</span> = <span class="cm-variable">np</span>.<span class="cm-property">log</span>(<span class="cm-variable">Z</span>) <span class="cm-comment"># the log of partition of C60</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"> <span class="cm-keyword">return</span> <span class="cm-variable">lnZ</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-variable">E0</span> = <span class="cm-operator">-</span> <span class="cm-variable">mydiff</span>( <span class="cm-variable">getlnZ</span>, <span class="cm-number">1</span>,<span class="cm-number">12</span>,<span class="cm-number">12</span>) <span class="cm-comment"># according to statistical mechanics</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-comment"># to get the energy. when temperature is 0, it is the ground state energy.</span></span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span class="cm-builtin">print</span>(<span class="cm-variable">E0</span>)</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 1296px;"></div><div class="CodeMirror-gutters" style="display: none; height: 1296px;"></div></div></div></pre><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm modeLoaded" lang=""><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang=""><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"><pre><span>xxxxxxxxxx</span></pre></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation" style=""><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 1.0 )= 78.4421074656477</span></pre></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 2.0 )= 142.01437800946442</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 2.0 and 1.0 is 63.57227054381673</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 3.0 )= 207.72503688198492</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 3.0 and 2.0 is 65.7106588725205</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 4.0 )= 273.6863836975408</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 4.0 and 3.0 is 65.96134681555588</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 5.0 )= 339.6811612250026</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 5.0 and 4.0 is 65.99477752746179</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 6.0 )= 405.6804545976738</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 6.0 and 5.0 is 65.99929337267122</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f( 7.0 )= 471.680358971803</span></pre><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;">f' between 7.0 and 6.0 is 65.99990437412919</span></pre></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 312px;"></div><div class="CodeMirror-gutters" style="display: none; height: 312px;"></div></div></div></pre><p> </p><pre spellcheck="false" class="md-fences md-end-block ty-contain-cm" lang="python"><div class="CodeMirror cm-s-inner CodeMirror-wrap" lang="python"><div style="overflow: hidden; position: relative; width: 3px; height: 0px; top: 0px; left: 4px;"><textarea autocorrect="off" autocapitalize="off" spellcheck="false" tabindex="0" style="position: absolute; bottom: -1em; padding: 0px; width: 1000px; height: 1em; outline: none;"></textarea></div><div class="CodeMirror-scrollbar-filler" cm-not-content="true"></div><div class="CodeMirror-gutter-filler" cm-not-content="true"></div><div class="CodeMirror-scroll" tabindex="-1"><div class="CodeMirror-sizer" style="margin-left: 0px; margin-bottom: 0px; border-right-width: 0px; padding-right: 0px; padding-bottom: 0px;"><div style="position: relative; top: 0px;"><div class="CodeMirror-lines" role="presentation"><div role="presentation" style="position: relative; outline: none;"><div class="CodeMirror-measure"></div><div class="CodeMirror-measure"></div><div style="position: relative; z-index: 1;"></div><div class="CodeMirror-code" role="presentation"><div class="CodeMirror-activeline" style="position: relative;"><div class="CodeMirror-activeline-background CodeMirror-linebackground"></div><div class="CodeMirror-gutter-background CodeMirror-activeline-gutter" style="left: 0px; width: 0px;"></div><pre class=" CodeMirror-line " role="presentation"><span role="presentation" style="padding-right: 0.1px;"><span cm-text=""></span></span></pre></div></div></div></div></div></div><div style="position: absolute; height: 0px; width: 1px; border-bottom: 0px solid transparent; top: 24px;"></div><div class="CodeMirror-gutters" style="display: none; height: 24px;"></div></div></div></pre></div>
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