# For which the eigenvalues of t e are complex

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IV.4 The Anderson Tight Binding ModelWhen|tr(T(E))| ≤2 the quantity under the square root sign is negative, and so the eigen-values have a non-zero imaginary part.Let’s use MATLAB/Octave to plot the values of tr(T(E)) as a function ofE. For conve-nience we first define a function that computes the matricesA(z). To to this we type thefollowing lines into a file calledA.min our working directory.function A=A(Z)A=[Z -1; 1 0];endNext we start with a range ofEvalues and define another vectorTthat contains the corre-sponding values of tr(T(E)).N=100;E=linspace(-1,6,N);T=[];for e = ET=[T trace(A(4-e)*A(3-e)*A(2-e)*A(1-e))];endFinally, we plotTagainstE. At the same time, we plot the constant functionsE= 2 andE=-2.plot(E,T)hold onplot(E,2*ones(1,N));plot(E,-2*ones(1,N));axis([-1,6,-10,10])On the resulting picture the energies whereT(E) lies between-2 and 2 have been highlighted.133
IV Eigenvalues and EigenvectorsWe see that there are four conduction bands for this crystal.Summary: Math ConceptsSummary: MATLAB/Octave Concepts134
IV.5 Markov ChainsIV.5 Markov ChainsIV.5.1 Random walkIn the diagram below there are three sites labelled 1, 2 and 3.Think of a walker movingfrom site to site. At each step the walker either stays at the same site, or moves to one ofthe other sites according to a set of fixed probabilities. The probability of moving to theithsite from thejth site is denotedpi,j. These numbers satisfy0pi,j1because they are probabilities (0 means “no chance” and 1 means “for sure”).On thediagram they label the arrows indicating the relevant transitions. Since the walker has togo somewhere at each step the sum of all the probabilities leaving a given site must be one.Thus for everyj,Xipi,j= 1Letxn,ibe the probability that the walker is at siteiafternsteps.We collect theseprobabilities into a sequence of vectors called state vectors. Each state vector contains theprobabilities for thenth step in the walk.