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Tom Hodson 2022-11-25 12:03:36 +01:00
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_thesis/2_Background/2.2_HKM_Model.html
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@ -118,7 +118,7 @@ image:
<p>The mapping is defined in terms of four Majoranas per site <span class="math inline">\(b_i^x,\;b_i^y,\;b_i^z,\;c_i\)</span> such that</p>
<p><span id="eq:bg-kh-mapping"><span class="math display">\[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]</span></span></p>
<p>The tildes on the spin operators <span class="math inline">\(\tilde{\sigma_i^\alpha}\)</span> emphasise that they live in this new extended Hilbert space and are only equivalent to the original spin operators after applying a projector <span class="math inline">\(\hat{P}\)</span>. The form of the projection operator can be understood in a few ways. From a group-theoretic perspective, before projection, the operators <span class="math inline">\(\{\tilde{\sigma}^x, \tilde{\sigma}^y, \tilde{\sigma}^z\}\)</span> form a representation of the gamma group <span class="math inline">\(G_{3,0}\)</span>. The gamma groups <span class="math inline">\(G_{p,q}\)</span> have <span class="math inline">\(p\)</span> generators that square to the identity and <span class="math inline">\(q\)</span> that square (roughly) to <span class="math inline">\(-1\)</span>. The generators otherwise obey standard anticommutation relations. The well known gamma matrices <span class="math inline">\(\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}\)</span> represent <span class="math inline">\(G_{1,3}\)</span> the quaternions <span class="math inline">\(G_{0,3}\)</span> and the Pauli matrices <span class="math inline">\(G_{3,0}\)</span>.</p>
<p>The Pauli matrices, however, have the additional property that the <em>chiral element</em> <span class="math inline">\(\sigma^x \sigma^y \sigma^z = \pmi\)</span> is not fully determined by the group properties of <span class="math inline">\(G_{3,0}\)</span>, but it is equal to <span class="math inline">\(i\)</span> in the Pauli matrices. Therefore, to fully reproduce the algebra of the Pauli matrices, we must project into the subspace where <span class="math inline">\(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = +i\)</span>. The chiral element of the gamma matrices for instance <span class="math inline">\(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\)</span> is of central importance in quantum field theory. See <span class="citation" data-cites="petitjeanChiralityDiracSpinors2020"> [<a href="#ref-petitjeanChiralityDiracSpinors2020" role="doc-biblioref">16</a>]</span> for more discussion of this group theoretic view.</p>
<p>The Pauli matrices, however, have the additional property that the <em>chiral element</em> <span class="math inline">\(\sigma^x \sigma^y \sigma^z = \pm i\)</span> is not fully determined by the group properties of <span class="math inline">\(G_{3,0}\)</span>, but it is equal to <span class="math inline">\(i\)</span> in the Pauli matrices. Therefore, to fully reproduce the algebra of the Pauli matrices, we must project into the subspace where <span class="math inline">\(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = +i\)</span>. The chiral element of the gamma matrices for instance <span class="math inline">\(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\)</span> is of central importance in quantum field theory. See <span class="citation" data-cites="petitjeanChiralityDiracSpinors2020"> [<a href="#ref-petitjeanChiralityDiracSpinors2020" role="doc-biblioref">16</a>]</span> for more discussion of this group theoretic view.</p>
<p>So the projector must project onto the subspace where <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde \sigma^z = i\)</span>. If we work this through, we find that in general <span class="math inline">\(\tilde \sigma^x \tilde \sigma^y \tilde\sigma^z = iD\)</span> where <span class="math inline">\(D = b^x b^y b^z c\)</span> must be the identity for every site. In other words, we can only work with <em>physical states</em> <span class="math inline">\(|\phi\rangle\)</span> that satisfy <span class="math inline">\(D_i|\phi\rangle = |\phi\rangle\)</span> for all sites <span class="math inline">\(i\)</span>. From this we construct an on-site projector <span class="math inline">\(P_i = \frac{1 + D_i}{2}\)</span> and the overall projector is simply <span class="math inline">\(P = \prod_i P_i\)</span>.</p>
<p>Another way to see what this is doing physically is to explicitly construct the two intermediate fermionic operators <span class="math inline">\(f\)</span> and <span class="math inline">\(g\)</span> that give rise to these four Majoranas. Denoting a fermion state by <span class="math inline">\(|n_f, n_g\rangle\)</span> the Hilbert space is the set <span class="math inline">\(\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}\)</span>. We can map these to Majoranas with, for example, this definition</p>
<p><span class="math display">\[\begin{aligned}