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83 lines
5.4 KiB
HTML
---
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title: Particle-Hole Symmetry
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excerpt:
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---
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<title>Particle-Hole Symmetry</title>
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<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<ul>
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<li><a href="#particle-hole-symmetry" id="toc-particle-hole-symmetry">Particle-Hole Symmetry</a></li>
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<li><a href="#bibliography" id="toc-bibliography">Bibliography</a></li>
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<p>Appendices</p>
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<section id="particle-hole-symmetry" class="level1">
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<h1>Particle-Hole Symmetry</h1>
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<p>The Hubbard and FK models on a bipartite lattice have particle-hole (PH) symmetry <span class="math inline">\(\mathcal{P}^\dagger H \mathcal{P} = - H\)</span>, accordingly they have symmetric energy spectra. The associated symmetry operator <span class="math inline">\(\mathcal{P}\)</span> exchanges creation and annihilation operators along with a sign change between the two sublattices. In the language of the Hubbard model of electrons <span class="math inline">\(c_{\alpha,i}\)</span> with spin <span class="math inline">\(\alpha\)</span> at site <span class="math inline">\(i\)</span> the particle hole operator corresponds to the substitution of new fermion operators <span class="math inline">\(d^\dagger_{\alpha,i}\)</span> and number operators <span class="math inline">\(m_{\alpha,i}\)</span> where</p>
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<p><span class="math display">\[d^\dagger_{\alpha,i} = \epsilon_i c_{\alpha,i}\]</span> <span class="math display">\[m_{\alpha,i} = d^\dagger_{\alpha,i}d_{\alpha,i}\]</span></p>
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<p>the lattices must be bipartite because to make this work we set <span class="math inline">\(\epsilon_i = +1\)</span> for the A sublattice and <span class="math inline">\(-1\)</span> for the even sublattice <span class="citation" data-cites="gruberFalicovKimballModel2005"> [<a href="#ref-gruberFalicovKimballModel2005" role="doc-biblioref">1</a>]</span>.</p>
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<p>The entirely filled state <span class="math inline">\(\ket{\Omega} = \sum_{\alpha,i} c^\dagger_{\alpha,i} \ket{0}\)</span> becomes the new vacuum state <span class="math display">\[d_{i\sigma} \ket{\Omega} = (-1)^i c^\dagger_{i\sigma} \sum_{j\rho} c^\dagger_{j\rho} \ket{0} = 0.\]</span></p>
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<p>The number operator <span class="math inline">\(m_{\alpha,i} = 0,1\)</span> counts holes rather than electrons <span class="math display">\[ m_{\alpha,i} = c_{\alpha,i} c^\dagger_{\alpha,i} = 1 - c^\dagger_{\alpha,i} c_{\alpha,i}.\]</span></p>
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<p>With the last equality following from the fermionic commutation relations. In the case of nearest neighbour hopping on a bipartite lattice this transformation also leaves the hopping term unchanged because <span class="math inline">\(\epsilon_i \epsilon_j = -1\)</span> when <span class="math inline">\(i\)</span> and <span class="math inline">\(j\)</span> are on different sublattices: <span class="math display">\[ d^\dagger_{\alpha,i} d_{\alpha,j} = \epsilon_i \epsilon_j c_{\alpha,i} c^\dagger_{\alpha,j} = c^\dagger_{\alpha,i} c_{\alpha,j} \]</span></p>
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<p>Defining the particle density <span class="math inline">\(\rho\)</span> as the number of fermions per site: <span class="math display">\[
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\rho = \frac{1}{N} \sum_i \left( n_{i \uparrow} + n_{i \downarrow} \right)
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\]</span></p>
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<p>The PH symmetry maps the Hamiltonian to itself with the sign of the chemical potential reversed and the density inverted about half filling: <span class="math display">\[ \text{PH} : H(t, U, \mu) \rightarrow H(t, U, -\mu) \]</span> <span class="math display">\[ \rho \rightarrow 2 - \rho \]</span></p>
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<p>The Hamiltonian is symmetric under PH at <span class="math inline">\(\mu = 0\)</span> and so must all the observables, hence half filling <span class="math inline">\(\rho = 1\)</span> occurs here. This symmetry and known observable acts as a useful test for the numerical calculations.</p>
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<p>Next Section: <a href="../6_Appendices/A.2_Markov_Chain_Monte_Carlo.html">Markov Chain Monte Carlo</a></p>
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</section>
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<section id="bibliography" class="level1 unnumbered">
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<h1 class="unnumbered">Bibliography</h1>
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<div id="refs" class="references csl-bib-body" role="doc-bibliography">
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<div id="ref-gruberFalicovKimballModel2005" class="csl-entry" role="doc-biblioentry">
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<div class="csl-left-margin">[1] </div><div class="csl-right-inline">C. Gruber and D. Ueltschi, <em><a href="http://arxiv.org/abs/math-ph/0502041">The Falicov-Kimball Model</a></em>, arXiv:math-Ph/0502041 (2005).</div>
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</div>
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</section>
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