Instantons and the spectrum of Bloch electrons in a magnetic field

Denise Freed, Jeffrey A. Harvey
1990 Physical Review B (Condensed Matter)  
A semiclassical path-integral treatment of the spectrum of a charged particle moving in a twodimensional periodic potential in the presence of a transverse magnetic field is presented. Complex instanton solutions to the Euclidean equations of motion are identified, and it is shown that the dilute-instanton-gas sum reproduces features of the spectrum studied by previous authors. Comparison of the path-integral results to a perturbative calculation of the splitting of the lowest Landau level is
more » ... t Landau level is used to relate the self-similar nature of the spectrum to a type of duality in the parameter a, the number of flux quanta per unit cell. As a byproduct of the instanton calculation the generating functional for a sum over lattice paths of given length and algebraic area modulo a fundamental area determined by the magnetic field is evaluated.
doi:10.1103/physrevb.41.11328 pmid:9993553 fatcat:wc7haqvsmremfkv2pbnegv3ju4