Hofstadter butterflies in a modulated magnetic field - APS Link ...

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 beingmutually prime, the size of the magnetic unit cell becomesqa,a, namely, x+qa,y=x,y and x,y+a=x,y.Figure 1 shows the tight-binding square lattice withthe choice of gauge appropriate to = p/q. The energyeigenvalues are obtained by diagonalizing the followingmatrix: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 inthe topmost panel of Fig. 3.B. Checkerboard magnetic fieldWe consider a spatially varying magnetic field whichconsists of a uniform component and a component varyingin a checkerboard pattern, as shown in Fig. 2. Here, denotes the flux per plaquette of the uniform componentof the magnetic field, and denotes the flux per plaquettewhich alternates in sign in the checkerboard pattern. Theassignment of the Peierls phase factor for this flux patternis shown in the figure. For = p/q, the system is invariantunder translation 2qa,0 or a,a. The relevant Schrödingerequation reads n,m = t n−1,m + t n+1,m + te −2in−m+1 n,m−1+ te 2in−m e 2i n,m+1 n − m odd, 5 n,m = t n−1,m + t n+1,m + te −2in−m+1 e 2i n,m−1+ te 2in−m n,m+1 n − m even.Spectra obtained by diagonalization of the corresponding2q2q matrix are shown in Fig. 3. The spectra are symmetricwith respect to transformations, →1±, so that calculationover the range =0– 1 2suffices. Five panels in Fig. 3correspond to =0, 1 8 , 1 4 , 3 8 , and 1 2, respectively. The topmostpanel =0 is the original Hofstadter butterfly spectrum.Introduction of nonzero deforms the spectrum in such away that, for instance, the spectral weight at the band center(van Hove singularity at =0) for =0 is smeared, and aquasigap develops there with increasing . 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 recalling thefollowing: 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 in becomes half the original one. In other words, the states at=integer and at =half-integer become equivalent for= 1 4. Again, this can be easily understood by recalling 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-binding square lattice with assignment of thePeierls phase factor to each bond, for a uniform external magneticflux = p/q.FIG. 2. Square lattice subjected to a spatially varying magneticfield in a checkerboard pattern and a uniform field .The assignment of the Peierls phase factor is indicated bythe arrows.144524-2