Two methods of numerically computing the inverse localization length in one dimension

I will present two independent methods that I have found to compute numerically the inverse localization length (the Lyapunov exponent) of the one-dimensional random Anderson model. They indeed give consistent results for the random-potential model.

I proposed the first method as a by-product of studying the spectrum of the non-Hermitian random Anderson model [1]. We obtain the inverse localization length of the Hermitian model as the edges of the bubble of the complex spectrum of the non-Hermitian model.

The second method is the kernel-polynomial expansion [2] of the inverse localization length. I transform the Chebyshev-polynomial expansion of the density of states [3] to that of the localization length, using the Thouless formula[4]. The expansion produces a smoother dataset than the expansion of the density of states because the expansion coefficients become smaller in high orders. I noticed this method during the collaboration with A. Amir and D.R. Nelson [5].

References: 

[1] N. Hatano and D.R. Nelson, Phys. Rev. Lett. 77, 570 (1996); Phys. Rev. B 56, 8651 (1997).

[2] A. Weiße, G. Wellein, A. Alvermann and H. Fehske, Rev. Mod. Phys. 78, 275 (2006).

[3] R.N. Silver and H. R ̈oder, Int. J. Mod. Phys. C 5, 735 (1994); R.N. Silver, H. Roeder, A.F. Voter and J.D. Kress, J. Comp. Phys. 124, 115 (1996); R.N. Silver and H. R ̈oder, Phys. Rev. E 56, 4822 (1997).

[4] D.J. Thouless, J. Phys. C 5, 77 (1972); B. Derrida, J.L. Jacobsen and R. Zeitak, J. Stat. Phys. 98, 31 (2000)

[5] A. Amir, N. Hatano and D.R. Nelson, in preparation.