Distribution analysis of electronic wavefunction participation ratios in Fibonacci quasicrystals
Author
Jessica Alfonsi
Title
Distribution analysis of electronic wavefunction participation ratios in Fibonacci quasicrystals
Description
Supplementary materials for reproducing paper "Searching for metal-insulator transition in Fibonacci quasicrystals by distribution analysis of electronic wavefunction participation ratios" available at https://doi.org/10.1088/2633-1357/abaa35
Category
Academic Articles & Supplements
Keywords
quasicrystal Fibonacci tight-binding eigenvalues participation ratios distribution fitting
URL
http://www.notebookarchive.org/2020-11-97z1csz/
DOI
https://notebookarchive.org/2020-11-97z1csz
Date Added
2020-11-20
Date Last Modified
2020-11-20
File Size
166.69 kilobytes
Supplements
Rights
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This file contains supplementary code and data accompanying the paper “Searching for metal-insulator transition in Fibonacci quasicrystals by distribution analysis of electronic wavefunction participation ratios” by Jessica Alfonsi, available at
https://iopscience.iop.org/article/10.1088/2633-1357/abaa35
https://iopscience.iop.org/article/10.1088/2633-1357/abaa35
This snippet is just a simple example for coding quickly the construction of the tridiagonal tight-binding Hamiltonian matrix . It is by no means meant to be the most efficient way for treating larger systems (i.e. with matrix size above x ) . However, it may be very helpful at the beginning to check results obtained from codes developed in other languages such as Python or C++.
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In[]:=
(*two-letterFibonaccisubstitutionrule*)Fibonaccirules={L{L,S},S{L}};
In[]:=
(*inflationrule*)wordlist1D[rule_,depth_]:=Flatten[Nest[(#/.rule)&,S,depth]];
In[]:=
(*choiceofrule,nestingdepth*)rule:=Fibonaccirules
In[]:=
nestingdepth:=8
In[]:=
(*Fibonacciwordgeneration*)fw=wordlist1D[rule,nestingdepth]
In[]:=
Length[fw]
(*1Dquasiperiodictight-bindingHamiltonian,tridiagonalmatrixrepresentation*)
In[]:=
(*1DHamiltonian;inverseratio(0<L/S<1):L1,St,τ^-1=2/(Sqrt[5]+1)=0.618...*)ham1Dinv[t_]:=SparseArray[{{i_,j_}/;And[(i-j)1,wordlist1D[rule,nestingdepth][[j]]L]1,{i_,j_}/;And[(i-j)-1,wordlist1D[rule,nestingdepth][[i]]L]1,{i_,j_}/;And[(i-j)1,wordlist1D[rule,nestingdepth][[j]]S]t,{i_,j_}/;And[(i-j)-1,wordlist1D[rule,nestingdepth][[i]]S]t},{Length[fw]+1,Length[fw]+1}];
In[]:=
ham1Dinv[t]//MatrixForm
In[]:=
(*1DHamiltonian;standardratio(1<L/S<∞):Lt,S1,τ=(Sqrt[5]+1)/2=1.618*)ham1Dstd[t_]:=SparseArray[{{i_,j_}/;And[(i-j)1,wordlist1D[rule,nestingdepth][[j]]L]t,{i_,j_}/;And[(i-j)-1,wordlist1D[rule,nestingdepth][[i]]L]t,{i_,j_}/;And[(i-j)1,wordlist1D[rule,nestingdepth][[j]]S]1,{i_,j_}/;And[(i-j)-1,wordlist1D[rule,nestingdepth][[i]]S]1},{Length[fw]+1,Length[fw]+1}];
In[]:=
ham1Dstd[t]//MatrixForm
(*ComputationofalleigenvaluesandeigenvectorsforagivenL/Sratio*)
In[]:=
{vals1,vecs1}=Eigensystem[ham1Dinv[N[2/(1+Sqrt[5])]]];
In[]:=
{vals2,vecs2}=Eigensystem[ham1Dstd[N[(1+Sqrt[5])/2]]];
(*Computationofparticipationratioforalleigenvectors*)
In[]:=
ppratios1=Table[Total[vecs1[[i]]^2]^2/(Length[vecs1[[i]]]*Total[vecs1[[i]]^4]),{i,1,Length[vecs1]}];
In[]:=
ppratios2=Table[Total[vecs2[[i]]^2]^2/(Length[vecs2[[i]]]*Total[vecs2[[i]]^4]),{i,1,Length[vecs2]}];
(*symmetrycheckingatτandτ^-1:samehistogramdistributions*)
In[]:=
HistogramList[ppratios1,{0.1}]
In[]:=
HistogramList[ppratios2,{0.1}]
Fig. 4: data for dynamical histograms
Fig. 4: data for dynamical histograms
Fig. 5: example of data fitting for participation ratios
Fig. 5: example of data fitting for participation ratios
Fig. 6: data of fit parameters plotted against distortion ratio L/S
Fig. 6: data of fit parameters plotted against distortion ratio L/S
Fig. 7: Fibonacci Polya-Aeppli fit results at τ vs n (QC length)
Fig. 7: Fibonacci Polya-Aeppli fit results at τ vs n (QC length)
Related supplementary notebooks at Wolfram Demonstration Project
Related supplementary notebooks at Wolfram Demonstration Project
Cite this as: Jessica Alfonsi, "Distribution analysis of electronic wavefunction participation ratios in Fibonacci quasicrystals" from the Notebook Archive (2020), https://notebookarchive.org/2020-11-97z1csz
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