Uebungsblatt 3

lmaxVal=1;
hmm[k_,V0_]:=Table[((k-2*Pi*l)^2+V0/2)*KroneckerDelta[l,m]+V0/4*(KroneckerDelta[l,m+1]+KroneckerDelta[l+1,m]),{l,-lmaxVal,lmaxVal},{m,-lmaxVal,lmaxVal}];
hmm[k,V0]//MatrixForm

NmaxVal=6;
V0Val=10;
spectrum=Table[Eigenvalues[N[hmm[-Pi+idx*2*Pi/NmaxVal,V0Val]]],{idx,0,NmaxVal}]
spectrum=Table[{-Pi+(idx-1)*2*Pi/NmaxVal,spectrum[[idx,jdx]]},{jdx,1,2*lmaxVal+1},{idx,1,NmaxVal+1}]
ListPlot[spectrum,PlotRange->All,Frame->True]

gapPi=Table[{V0,Sort[Eigenvalues[N[hmm[Pi,V0]]]][[2]] - Sort[Eigenvalues[N[hmm[Pi,V0]]]][[1]]},{V0,0,10,0.5}];
ListPlot[gapPi,Frame->True,FrameLabel->{{"",""},{"","Gap between first and second band at k = π"}}]
gapZero=Table[{V0,Sort[Eigenvalues[N[hmm[0,V0]]]][[3]] - Sort[Eigenvalues[N[hmm[0,V0]]]][[2]]},{V0,0,10,0.5}];
ListPlot[gapZero,Frame->True,FrameLabel->{{"",""},{"","Gap between second and third band at k = 0"}}]

gapPi=Table[{V0,Sort[Eigenvalues[N[hmm[Pi,V0]]]][[2]] - Sort[Eigenvalues[N[hmm[Pi,V0]]]][[1]]},{V0,{1,3,6,10}}]
ListPlot[gapPi,Frame->True,FrameLabel->{{"",""},{"","Gap between first and second band at k = π"}}]
gapZero=Table[{V0,Sort[Eigenvalues[N[hmm[0,V0]]]][[3]] - Sort[Eigenvalues[N[hmm[0,V0]]]][[2]]},{V0,{1,3,6,10}}]
ListPlot[gapZero,Frame->True,FrameLabel->{{"",""},{"","Gap between second and third band at k = 0"}}]

tVal=1;
tpVal=0.05;
spectrumSq[kx_,ky_]:=-2*tVal*(Cos[kx]+Cos[ky])-2*tpVal*(Cos[kx+ky]+Cos[kx-ky])
Plot3D[spectrumSq[kx,ky],{kx,-Pi,Pi},{ky,-Pi,Pi}]
Table[ContourPlot[spectrumSq[kx,ky]==EF,{kx,-Pi,Pi},{ky,-Pi,Pi}],{EF,{-3,0,3}}]

FindMinimum[spectrumSq[kx,ky],{{kx,Pi/2},{ky,0}}]
FindMaximum[spectrumSq[kx,ky],{{kx,Pi},{ky,Pi/2}}]

mmPoly[k_,t_,Δ_,u_]={{u*Δ^2,-t*(1+Δ)*Exp[I*k/2]-t*(1-Δ)*Exp[-I*k/2]},{-t*(1+Δ)*Exp[-I*k/2]-t*(1-Δ)*Exp[I*k/2],u*Δ^2}};
mmPoly[k,t,Δ,u]//MatrixForm
spectrumPoly[k_,t_,Δ_,u_]=FullSimplify[Eigenvalues[mmPoly[k,t,Δ,u]],Assumptions->{k∈Reals&&k>0&&t>0&&Δ>0}]

temp=Integrate[Sqrt[1-(1-Δsq)*Sin[k/2]^2],{k,-Pi,Pi},Assumptions->Δsq<1&&Δsq^2>0]

Series[temp/4,{Δsq,0,1}]//Normal//N

Table[Plot[spectrumPoly[k,1,Δ,0],{k,-Pi,Pi}],{Δ,{0,0.25,0.75,1}}]