Hofstadter butterflies in a modulated magnetic field - APS Link ...
Hofstadter butterflies in a modulated magnetic field - APS Link ...
Hofstadter butterflies in a modulated magnetic field - APS Link ...
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IYE et al. PHYSICAL REVIEW B 70, 144524 (2004)FIG. 4. Square lattice subjected to a spatially vary<strong>in</strong>g <strong>magnetic</strong><strong>field</strong> <strong>in</strong> a stripe pattern and a uniform <strong>field</strong> . The assignmentof the Peierls phase factor is <strong>in</strong>dicated by the arrows.at =0 for = 1 2, while the splitt<strong>in</strong>g occurs at <strong>in</strong>teger for the checkerboard <strong>field</strong>, as seen earlier. It is evidentthat the spectra for stripe <strong>field</strong>s are considerably “darker”(more bands and fewer gaps) as compared with thosefor checkerboard <strong>field</strong>s. This is <strong>in</strong> l<strong>in</strong>e with a generaltrend that the <strong>Hofstadter</strong> spectrum becomes “darker”as one <strong>in</strong>troduces rectangular anisotropy <strong>in</strong> the squarelattice. 9D. Comparison with a honeycomb latticeAlthough it is a k<strong>in</strong>d of excursion to a side road,it is <strong>in</strong>terest<strong>in</strong>g to compare the spectra for the checkerboardFIG.FIG. 5. <strong>Hofstadter</strong> spectra for different values of , the strengthof the stripe pattern <strong>field</strong>. (a) = 1 8 and (b) = 1 4. The spectra for=0 and = 1 2are identical with Figs. 3(a) and 3(e), respectively.The spectrum for = 3 8 is the same as the one for = 1 8except it isshifted by 1 2along the horizontal axis.6. (a) A square lattice with different transfer <strong>in</strong>tegralst and t. When t=0 [panel (b)], it is topologically equivalentto a honeycomb lattice [panel (c)]. Panel (d) shows unit “brick” ofthis lattice and the one for a square lattice under checkerboard <strong>field</strong>(Fig. 2).<strong>field</strong> shown <strong>in</strong> Fig. 3 with that for a honeycomb latticeunder a uniform <strong>magnetic</strong> <strong>field</strong>. Let us consider a squarelattice shown <strong>in</strong> Fig. 6(a), <strong>in</strong> which the transfer <strong>in</strong>tegralfor every other vertical bond t (dotted l<strong>in</strong>es) can take adifferent value from that for the rest of the bonds t (solidl<strong>in</strong>es).The relevant Schröd<strong>in</strong>ger equation reads n,m = t n−1,m + t n+1,m + te −2<strong>in</strong> n,m−1 + te 2<strong>in</strong> n,m+1 .7Sett<strong>in</strong>g t=t reduces the system to a simple square lattice.On the other hand, sett<strong>in</strong>g t=0 [Fig. 6(b)] makes it topologicallyequivalent to a honeycomb lattice [Fig. 6(c)].Therefore, by chang<strong>in</strong>g the parameter t from t to 0, thesystem evolves cont<strong>in</strong>uously from a square to a honeycomblattice.Figure 7 shows calculated spectra for (a) t=0.5t and (b)t=0 (honeycomb lattice). It is seen that they exhibit a resemblance<strong>in</strong> overall shape with the spectra for = 1 8 and 1 4for the checkerboard <strong>field</strong> [panels (b) and (c) of Fig. 3]. Theresemblance actually runs over the whole range of the twomodels, i.e., between the checkerboard <strong>field</strong> [Eq. (6)] with chang<strong>in</strong>g from 0 to 1 2and the square-to-honeycomb evolution[Eq. (7)] with t chang<strong>in</strong>g from t through 0 to −t. Thespectra at both ends are <strong>in</strong>deed identical: =0⇔t=t and=1/2⇔t=−t.In retrospect, this resemblance is someth<strong>in</strong>g one shouldhave anticipated, because <strong>in</strong> both cases the structural unitis a rectangle consist<strong>in</strong>g of two adjacent unit squares, asshown <strong>in</strong> Fig. 6(d). Both Fig. 2 and Fig. 6(a) are thenconstructed by bricklay<strong>in</strong>g these unit rectangles <strong>in</strong> the144524-4