Deployed 5b5dccd with MkDocs version: 1.6.0

index.html

@@ -1294,8 +1294,8 @@ <h3 id="main-problem-and-dependencies">Main Problem and Dependencies</h3>
 </code></pre></div></p>
 <h3 id="subproblems">Subproblems</h3>
 <p><strong>1.1 Write a Haldane model Hamiltonian on a hexagonal lattice, given the following parameters: wavevector components <span class="arithmatex">\(k_x\)</span> and <span class="arithmatex">\(k_y\)</span> (momentum) in the x and y directions, lattice spacing <span class="arithmatex">\(a\)</span>, nearest-neighbor coupling constant <span class="arithmatex">\(t_1\)</span>, next-nearest-neighbor coupling constant <span class="arithmatex">\(t_2\)</span>, phase <span class="arithmatex">\(\phi\)</span> for the next-nearest-neighbor hopping, and the on-site energy <span class="arithmatex">\(m\)</span>.</strong></p>
-<p><strong><em>Scientists Annotated Background:</em></strong>
-Source: Haldane, F. D. M. (1988). Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the" parity anomaly". Physical review letters, 61(18).</p>
+<p><strong><em>Scientists Annotated Background:</em></strong></p>
+<p>Source: Haldane, F. D. M. (1988). Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the" parity anomaly". Physical review letters, 61(18).</p>
 <p>We denote <span class="arithmatex">\(\{\mathbf{a}_i\}\)</span> are the vectors from a B site to its three nearest-neighbor A sites, and <span class="arithmatex">\(\{\mathbf{b}_i\}\)</span> are next-nearest-neighbor distance vectors, then we have</p>
 <div class="arithmatex">\[
 {\mathbf{a}_1} = (0,a),
@@ -1387,8 +1387,8 @@ <h3 id="subproblems">Subproblems</h3>
 <span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">calc_hamiltonian</span><span class="p">(</span><span class="n">kx</span><span class="p">,</span> <span class="n">ky</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">t1</span><span class="p">,</span> <span class="n">t2</span><span class="p">,</span> <span class="n">phi</span><span class="p">,</span> <span class="n">m</span><span class="p">),</span> <span class="n">target</span><span class="p">)</span>
 </code></pre></div>
 <strong>1.2 Calculate the Chern number using the Haldane Hamiltonian, given the grid size <span class="arithmatex">\(\delta\)</span> for discretizing the Brillouin zone in the <span class="arithmatex">\(k_x\)</span> and <span class="arithmatex">\(k_y\)</span> directions (assuming the grid sizes are the same in both directions), the lattice spacing <span class="arithmatex">\(a\)</span>, the nearest-neighbor coupling constant <span class="arithmatex">\(t_1\)</span>, the next-nearest-neighbor coupling constant <span class="arithmatex">\(t_2\)</span>, the phase <span class="arithmatex">\(\phi\)</span> for the next-nearest-neighbor hopping, and the on-site energy <span class="arithmatex">\(m\)</span>.</strong></p>
-<p><strong><em>Scientists Annotated Background:</em></strong>
-Source: Fukui, Takahiro, Yasuhiro Hatsugai, and Hiroshi Suzuki. "Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances." Journal of the Physical Society of Japan 74.6 (2005): 1674-1677.</p>
+<p><strong><em>Scientists Annotated Background:</em></strong></p>
+<p>Source: Fukui, Takahiro, Yasuhiro Hatsugai, and Hiroshi Suzuki. "Chern numbers in discretized Brillouin zone: efficient method of computing (spin) Hall conductances." Journal of the Physical Society of Japan 74.6 (2005): 1674-1677.</p>
 <p>Here we can discretize the two-dimensional Brillouin zone into grids with step <span class="arithmatex">\(\delta {k_x} = \delta {k_y} = \delta\)</span>. If we define the U(1) gauge field on the links of the lattice as <span class="arithmatex">\(U_\mu (\mathbf{k}_l) := \frac{\left\langle n(\mathbf{k}_l)\middle|n(\mathbf{k}_l + \hat{\mu})\right\rangle}{\left|\left\langle n(\mathbf{k}_l)\middle|n(\mathbf{k}_l + \hat{\mu})\right\rangle\right|}\)</span>, where <span class="arithmatex">\(\left|n(\mathbf{k}_l)\right\rangle\)</span> is the eigenvector of Hamiltonian at <span class="arithmatex">\(\mathbf{k}_l\)</span>, <span class="arithmatex">\(\hat{\mu}\)</span> is a small displacement vector in the direction <span class="arithmatex">\(\mu\)</span> with magnitude <span class="arithmatex">\(\delta\)</span>, and <span class="arithmatex">\(\mathbf{k}_l\)</span> is one of the momentum space lattice points <span class="arithmatex">\(l\)</span>. The corresponding curvature (flux) becomes</p>
 <div class="arithmatex">\[
 F_{xy}(\mathbf{k}_l) := \ln \left[U_x(\mathbf{k}_l)U_y(\mathbf{k}_l+\hat{x})U_x^{-1}(\mathbf{k}_l+\hat{y})U_y^{-1}(\mathbf{k}_l)\right]