Exact solution of the magnetic breakdown problem in quasi-one-dimensional geometry

We present exact solution of the problem of electronic wave functions of quasi one-dimensional band with an inter-band gap at the Fermi surface and in the presence of magnetic field. The details of the analyzed model are appropriate to the situation in the Bechgaard salt (TMTSF) 2 ClO 4 with the dimerizing anion order in the transverse direction. Limiting the effects of dimerization to the standard dimerization gap only, one obtains the electronic spectrum represented through solutions of a generalized Hill system of equations with simply periodic coefficients. The resulting wave-functions are discussed. In particular, we present the solutions for the case when the electrons spend as much time in the junctions as on their quasi-classical orbits. On the other hand, the limit when the tunnelling approach is valid is identified and the results are confronted with the well-known Slutskin-Kadigrobov solution. Furthermore, taking into account also the presumably finite transverse dimerizing displacements of chains, one encounters the qualitatively more complex problem of a system of equations with two-periodic coefficients. Some qualitatively new properties of electronic spectrum and corresponding one-electron physical quantities in this case will be discussed in detail.