Workshop on methods of theoretical physics Homework 1 Anderson localization Jan Major 17.04.2014 Localization The Anderson localization (aka. strong localization) is a property of disordered quantum systems. In a periodic potential (for example V pxq “ sin x) a classical particle will either travel forward changing only its velocity (if its energy is higher than potential) or will be bound. On the other hand a quantum particle can tunnel to adjacent sites even if its energy is lower than height of barrier also it can be reflected even if it have high energy. Non-zero tunneling for all finite lattice heights means that after some time wave-function will spread across entire system and become totally delocalized (in other words we can say that the eigenfunctions for a periodic potential is Bloch waves for which the proper quantum number is momentum k, not position). Knowing this, we can expect that if we add some chaos to our system (i.e. change randomly heights of our periodic barriers or depths of wells) particle behavior won’t change. But it will! In a presence of (sufficiently strong) disorder the particle will localize, and probability density ´ |x´x0 | will reach profile of Anderson state e ξloc , where ξloc is the localization length. Better theoretical descriptions can be found in good compendium [4], 7th chapter of [3] or source articles: [1, 2]. -10 -5 0 5 10 Figure 1: Exponential profile of localized state. 1 1.0 0.8 0.6 0.4 0.2 0 50 100 150 200 Figure 2: The wave-function of a particle in the lattice after long evolution in a regular (red) and a disordered (green) potential. Exercises Ex. 1. The simplest modeling A simple system, in which we can observe localization, is the one-dimensional lattice model. The discretized Hamiltonian of such a system has form: ÿ pn ψn ´ Jn ψn`1 ´ Jn ψn´1 q , (1) H“ n where ψn denotes probability that a particle will be found on site n (the whole wave-function is a vector Ψ “ rψ1 . . .s), Jn is tunneling parameter, and n onsite energy. Evolution is governed by the Schroedinger equation (in matrix form): iBt Ψ “ HΨ. (2) Implement the model to calculate the evolution (use whatever you want: Mathematica, C++, fortran ...). By taking a simple initial state (for example ψ “ t0 . . . 0, 1, 0 . . . 0u), you can see that for regular lattice (J and independent on n) the wave-function will spread over all sites. Check, how results will change if you add some randomness to n or Jn . Tips: • As disorder is disordered in some realizations you can get totally different results. The best idea is to do a lot of simulations for different realizations of chaos and then take a mean of them. • As localized function is exponential it’s easiest to see it on a log-plot. Ex. 2. Matrix diagonalization approach From now we will restrict to a diagonal disorder, i.e. only n are randomized. We can attack problem from a different site. The Hamiltonian is time independent so we can search for eigenfunctions diagonalizing it. We have eigenproblem: HΨ “ EΨ. (3) This way we can find eigenenergies (E) and eigenstates. Check that all eigenstates are localized. 2 10- 27 10-57 10-87 10-117 10-147 0 50 100 150 200 Figure 3: The wave-function of a particle in the lattice after long evolution in a regular (red) and a disordered (green) potential presented in logscale – we can clearly see that the localized function have exponential profile. Ex. 3. Transfer matrix (hard) A more rigorous approach to the localization problem is based on transfer matrices. We can write part of (3) for any chosen n: n ψn ´ Jψn`1 ´ Jψn´1 “ Eψn . Using only the one equation (for given n) we can find the transfer matrix: „  „  ψn`1 ψn “ Mn . ψn ψn´1 (4) (5) It is a formula for transport of particles between neighboring sites (both tunneling and reflections). Find the explicit form of Mn . Having the recurrence relation (5) if we choose occupation of two sites (say rψ1 , ψ0 s) we can get values for every pair rψn`1 , ψn s. Check that for most of chosen initial values and energies E you will get exponential growth. You can easily check that determinants for all n are: |Mn | “ 1. Using Furstenberg’s theorem we can get that product of series of such matrices Mm . . . M2 M1 for a large m will have eigenvalues expp˘mγq (γ is Lapunov exponent, ´ γ1 “ ξloc ). For any possible initial condition it will explode at least in one direction. Only for a discrete set of eigenenergies we can have γ ă 0 and localization. Try to find localization length using this method. Is it possible to prove that energy spectrum for disordered system is discrete? References [1] P. W. Anderson. Absence of diffusion in certain random lattices. Phys. Rev., 109:1492–1505, Mar 1958. [2] T. Pendry J.B. Barnes, C. Wei-Chao. The localization length and density of states of 1d disordered systems. J. Phys.: Condens. Matter, 3:5297–5305, Mar 1991. [3] Fritz Haake. Quantum Signatures of Chaos. Springer Series in Synergetics. Springer-Verlag, Berlin, Germany, 2nd edition, 2001. [4] D. Muller, C.A. Delande. Disorder and interference: localization phenomena. arXiv preprint arXiv:1005.0915v2, February 2012. 3