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) n = te −ik x a n−1 + te ik x a n+1 +2t cosk y a −2n n .3When is a rational number p/q, p and q be<strong>in</strong>gmutually prime, the size of the <strong>magnetic</strong> unit cell becomesqa,a, namely, x+qa,y=x,y and x,y+a=x,y.Figure 1 shows the tight-b<strong>in</strong>d<strong>in</strong>g square lattice withthe choice of gauge appropriate to = p/q. The energyeigenvalues are obta<strong>in</strong>ed by diagonaliz<strong>in</strong>g the follow<strong>in</strong>gmatrix:2t cosk y a −2 te ik x a 0 ¯ 0 e −ik x ate −ik x a 2t cosk y a −4 te ik x a 0 ¯ 00 4 0 00 ¯ 0 te −ik x a 2t cosk y a −2q −1 te ik x ate ik x a 0 ¯ 0 te −ik x a 2t cosk y a −2q.The result is the well-known butterfly spectrum, shown <strong>in</strong>the topmost panel of Fig. 3.B. Checkerboard <strong>magnetic</strong> <strong>field</strong>We consider a spatially vary<strong>in</strong>g <strong>magnetic</strong> <strong>field</strong> whichconsists of a uniform component and a component vary<strong>in</strong>g<strong>in</strong> a checkerboard pattern, as shown <strong>in</strong> Fig. 2. Here, denotes the flux per plaquette of the uniform componentof the <strong>magnetic</strong> <strong>field</strong>, and denotes the flux per plaquettewhich alternates <strong>in</strong> sign <strong>in</strong> the checkerboard pattern. Theassignment of the Peierls phase factor for this flux patternis shown <strong>in</strong> the figure. For = p/q, the system is <strong>in</strong>variantunder translation 2qa,0 or a,a. The relevant Schröd<strong>in</strong>gerequation reads n,m = t n−1,m + t n+1,m + te −2<strong>in</strong>−m+1 n,m−1+ te 2<strong>in</strong>−m e 2i n,m+1 n − m odd, 5 n,m = t n−1,m + t n+1,m + te −2<strong>in</strong>−m+1 e 2i n,m−1+ te 2<strong>in</strong>−m n,m+1 n − m even.Spectra obta<strong>in</strong>ed by diagonalization of the correspond<strong>in</strong>g2q2q matrix are shown <strong>in</strong> Fig. 3. The spectra are symmetricwith respect to transformations, →1±, so that calculationover the range =0– 1 2suffices. Five panels <strong>in</strong> Fig. 3correspond to =0, 1 8 , 1 4 , 3 8 , and 1 2, respectively. The topmostpanel =0 is the orig<strong>in</strong>al <strong>Hofstadter</strong> butterfly spectrum.Introduction of nonzero deforms the spectrum <strong>in</strong> such away that, for <strong>in</strong>stance, the spectral weight at the band center(van Hove s<strong>in</strong>gularity at =0) for =0 is smeared, and aquasigap develops there with <strong>in</strong>creas<strong>in</strong>g . The bottommostpanel = 1 2is identical to the topmost one except that thespectrum is shifted by 1 2along the horizontal axis. Thatthis should be so can be readily understood by recall<strong>in</strong>g thefollow<strong>in</strong>g: At = 1 2 , two adjacent cells enclose + 1 2 and − 1 2flux, respectively. Addition of a uniform flux = 1 2to thesystem changes them to 1 and 0, which is equivalent to the=0,=0 configuration. The same relation (shift by halfperiod)holds between the spectra for = 3 8[panel (d)] andfor = 1 8 [panel (b)].At= 1 4[panel (c)], the periodicity <strong>in</strong> becomes half the orig<strong>in</strong>al one. In other words, the states at=<strong>in</strong>teger and at =half-<strong>in</strong>teger become equivalent for= 1 4. Aga<strong>in</strong>, this can be easily understood by recall<strong>in</strong>g thatthe flux configuration of two adjacent cells is +34 ,+1 4at= 1 4 ,= 1 2, which is equivalent to −14 ,+1 4, and hence to +14 ,−1 4at = 1 4 ,=0 .FIG. 1. Tight-b<strong>in</strong>d<strong>in</strong>g square lattice with assignment of thePeierls phase factor to each bond, for a uniform external <strong>magnetic</strong>flux = p/q.FIG. 2. Square lattice subjected to a spatially vary<strong>in</strong>g <strong>magnetic</strong><strong>field</strong> <strong>in</strong> a checkerboard pattern and a uniform <strong>field</strong> .The assignment of the Peierls phase factor is <strong>in</strong>dicated bythe arrows.144524-2