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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <blockquote> <p>This page is for developing preliminary notes or making typographical experiments, etc. It may be edited by anybody, anytime. But you don’t necessarily need to delete other people’s ongoing notes here in order to add your own. In any case, overwritten edits may always be recovered from the <a href="https://ncatlab.org/nlab/history/Sandbox">page history</a>.</p> </blockquote> <p><br /></p> <hr /><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mi>h</mi><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mover><mi>ρ</mi><mo>^</mo></mover><mo>′</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{l} \big(\widehat{\rho}'(g^{-1})\widehat{\rho}(g)\big) \rho(h) \big(\widehat{\rho}'(g^{-1})\widehat{\rho}(g)\big)^{-1} \\ \;=\; \widehat{\rho}'(g^{-1})\widehat{\rho}(g) \rho(h) \widehat{\rho}(g^{-1})\widehat{\rho}'(g) \\ \;=\; \widehat{\rho}'(g^{-1}) \rho(g h g^{-1}) \widehat{\rho}'(g) \\ \;=\; \end{array} </annotation></semantics></math></div> <p>Particle-hole symmetry.</p> <p>Original</p> <ul> <li>S. M. Girvin: <em>Particle-hole symmetry in the anomalous quantum Hall effect</em>, Phys. Rev. B <strong>29</strong> (1984) 6012(R) [<a href="https://doi.org/10.1103/PhysRevB.29.6012">doi:10.1103/PhysRevB.29.6012</a>]</li> </ul> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≠</mo><mo>></mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\neq \gt 1</annotation></semantics></math> may be understood as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\nu = 1</annotation></semantics></math> for majority spin polarization with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\nu - 1</annotation></semantics></math> for the minority polarization (Haldane 1983 <a href="https://sci-hub.se/10.1103/PhysRevLett.51.605">doi:10.1103/PhysRevLett.51.605</a>)</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="402.894pt" height="360.551pt" viewBox="0 0 402.894 360.551" version="1.2"> <defs> <g> <symbol overflow="visible" id="3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph0-0"> <path style="stroke:none;" d=""></path> </symbol> <symbol overflow="visible" id="3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph0-1"> <path style="stroke:none;" d="M 2.328125 -5.28125 C 2.328125 -5.296875 2.359375 -5.40625 2.359375 -5.40625 C 2.359375 -5.453125 2.328125 -5.515625 2.234375 -5.515625 C 2.1875 -5.515625 1.953125 -5.5 1.765625 -5.484375 L 1.3125 -5.453125 C 1.140625 -5.4375 1.0625 -5.421875 1.0625 -5.28125 C 1.0625 -5.171875 1.171875 -5.171875 1.265625 -5.171875 C 1.65625 -5.171875 1.65625 -5.125 1.65625 -5.046875 C 1.65625 -5.03125 1.65625 -5.015625 1.609375 -4.875 L 0.484375 -0.34375 C 0.453125 -0.21875 0.453125 -0.171875 0.453125 -0.171875 C 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transform="matrix(0.998756,0,0,-0.998756,291.268144,334.914925)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-1" x="291.268144" y="342.018079"></use> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-2" x="295.497392" y="342.018079"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph1-2" x="319.460036" y="337.900207"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-1" x="345.165026" y="333.199062"></use> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-3" x="349.394273" y="333.199062"></use> </g> <path style="fill:none;stroke-width:0.478;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.000964314 0.000863594 L 8.466599 0.000863594 " transform="matrix(0.998756,0,0,-0.998756,345.165026,334.914925)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-1" x="345.165026" y="342.018079"></use> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-2" x="349.394273" y="342.018079"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-6" x="374.227833" y="333.199062"></use> </g> <path style="fill:none;stroke-width:0.478;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -0.00127231 0.000863594 L 4.234465 0.000863594 " transform="matrix(0.998756,0,0,-0.998756,374.227833,334.914925)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#3RdhebZn4NguOi8wU4a8A_DPDGw=-glyph3-7" x="374.227833" y="342.018079"></use> </g> <g clip-path="url(#3RdhebZn4NguOi8wU4a8A_DPDGw=-clip1)" clip-rule="nonzero"> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -215.142121 158.396225 L 190.956721 158.396225 " transform="matrix(0.998756,0,0,-0.998756,201.394066,180.05078)"></path> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -176.942266 199.22435 L -176.942266 -171.326373 " transform="matrix(0.998756,0,0,-0.998756,201.394066,180.05078)"></path> </g> </svg> <p>Here</p> <ul> <li> <p>“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math>” means that the corresponding combination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,q)</annotation></semantics></math> is not admissible, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">k q</annotation></semantics></math> is odd or the <a class="existingWikiWord" href="/nlab/show/Gauss+sum">Gauss sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mi>e</mi> <mrow><mstyle displaystyle="false"><mfrac><mrow><mi>π</mi><mi mathvariant="normal">i</mi></mrow><mi>k</mi></mfrac></mstyle><mi>q</mi><msup><mi>n</mi> <mn>2</mn></msup></mrow></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{n=0}^{k-1} e^{ \tfrac{\pi \mathrm{i}}{k} q n^2 } = 0</annotation></semantics></math>.</p> </li> <li> <p>“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math>” means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mi>q</mi><mi>k</mi></mfrac></mstyle><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\tfrac{q}{k} \gt 2</annotation></semantics></math></p> </li> </ul> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="85.836pt" height="0.797pt" viewBox="0 0 85.836 0.797" version="1.2"> <defs> <clipPath id="1efyKOgPaINXdY20BXcAZXIATrA=-clip1"> <path d="M 8.648438 0 L 77.058594 0 L 77.058594 0.636719 L 8.648438 0.636719 Z M 8.648438 0 "></path> </clipPath> </defs> <g id="1efyKOgPaINXdY20BXcAZXIATrA=-surface1"> <g clip-path="url(#1efyKOgPaINXdY20BXcAZXIATrA=-clip1)" clip-rule="nonzero"> <path style="fill:none;stroke-width:0.79701;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.000800188 0.00100345 L 85.041382 0.00100345 " transform="matrix(0.797,0,0,-0.797,8.964206,0.317206)"></path> </g> </g> </svg> <p>So:</p> <ol> <li> <p>in the column of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q = 1</annotation></semantics></math> all odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> are excluded and all even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> are admissible,</p> </li> <li> <p>in the column of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">q = 2</annotation></semantics></math> all odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> are excluded and exactly every second even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is admissible,</p> </li> <li> <p>in rows of odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> the odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> are excluded, and for even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>2</mn><mi>r</mi></mrow><annotation encoding="application/x-tex">q = 2 r</annotation></semantics></math> the result is admissible iff the Jacobi symbol <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(r \vert k) \neq 0</annotation></semantics></math>.</p> <p>Since the Jacobi symbol <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(r \vert k)</annotation></semantics></math> vanishes iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>gcd</mi><mo stretchy="false">(</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">gcd(r,k) \neq 1</annotation></semantics></math>, this means that, at least for odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, exactly only the reduced fractions appear.</p> </li> </ol> </body></html> </div> <div class="revisedby"> <p> Last revised on April 1, 2025 at 14:12:19. 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