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Kitaev Chain  |  |
The Kitaev chain is a prototype example of the topological superconductor. Here, we demonstrate how to examine its properties using . See Kitaev (2001), Kitaev's original, for technical details of the model.
Number operator | |
Constructs the hopping Hamiltonian terms | |
Constructs the pairing Hamiltonian terms | |
Converts Dirac fermions to Majorana fermions. | |
Converts Majorana fermions to Dirac fermions. | |
Visualizes a non-interacting Hamiltonian or corresponding matrix in a graph. | |
Visualizes the off-diagonal block of a matrix with chiral symmetry. |
Some useful functions to study the Kitaev chain.
Load Q3.
In[1]:=
Needs["QuantumMob`Q3`"]
Consider some local fermion modes.
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Let[Fermion,c]
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$L=5;
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cc=c[Range[1,$L]]cccc=Join[cc,Dagger@cc]
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{,,,,}
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,,,,,,,,,
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Let t, μ, and Δ denote the hopping amplitude, chemical potential, and pairing potential, respectively.
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Let[Real,t,μ,Δ]
Construct the model Hamiltonian.
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H0=-μ*Q[cc]Hhop=-t*PlusDagger@FockHopping[cc]Hpair=Δ*PlusDagger@FockPairing[cc]HH=H0+Hhop+Hpair
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-μ++++
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-t+++++++
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Δ--------
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Take a look at the connectivity of the model. Here, the black thin lines indicate single-particle tunneling and the red thick lines represent the p-wave pairing.
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GraphForm[HH]
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Spectral Properties
To investigate the quasi-particle spectrum by means of the Bogoliubov-de Gennes (BdG) equation, obtain the single-particle BdG Hamiltonian.
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matK=CoefficientTensor[HH,Dagger@cccc,cccc];ArrayShort[matK,"Size"8]
Out[13]//MatrixForm=
- μ 2 | - t 2 | 0 | 0 | 0 | 0 | - Δ 2 | 0 | … |
- t 2 | - μ 2 | - t 2 | 0 | 0 | Δ 2 | 0 | - Δ 2 | … |
0 | - t 2 | - μ 2 | - t 2 | 0 | 0 | Δ 2 | 0 | … |
0 | 0 | - t 2 | - μ 2 | - t 2 | 0 | 0 | Δ 2 | … |
0 | 0 | 0 | - t 2 | - μ 2 | 0 | 0 | 0 | … |
0 | Δ 2 | 0 | 0 | 0 | μ 2 | t 2 | 0 | … |
- Δ 2 | 0 | Δ 2 | 0 | 0 | t 2 | μ 2 | t 2 | … |
0 | - Δ 2 | 0 | Δ 2 | 0 | 0 | t 2 | μ 2 | … |
… | … | … | … | … | … | … | … | … |
Now, study the quasi-particle energy spectrum of the model. Be aware of the finite size effects.
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KK[a_,b_,c_]:=Block[{μ=a,t=b,Δ=c},SparseArray@ArrayRules@matK]
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Manipulate[Module[{eval=Table[Sort@Eigenvalues[KK[μ,1.,del]],{μ,0,4,0.02}]},ListLinePlot[Transpose@eval,AxesNone,FrameTrue,FrameLabel{"μ / t","Eenergy / t"},DataRange{0,4}]],{{del,1,"Δ"},0.5,2,0.1},SaveDefinitionsTrue]
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Majorana Representation
In this section, the Kitaev chain is expressed in terms of Majorana modes. The Majorana representation clearly reveals the chiral symmetry in the Kitaev chain.
Consider Majorana modes twice as many as the Dirac fermion modes in the above.
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Let[Majorana,a]
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aa=a@Range[2*$L]
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{,,,,,,,,,}
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Rewrite the Hamiltonian in terms of the Majorana fermions.
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MM=ToMajorana[HH,ccaa]
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--μ-(t-Δ)+(t+Δ)-μ-(t-Δ)+(t+Δ)-μ-(t-Δ)+(t+Δ)-μ-(t-Δ)+(t+Δ)-μ
5μ
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Examine the connectivity through the single-particle Hamiltonian.
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matM=CoefficientTensor[MM,aa,aa];ArrayShort[matM,"Size"8]
Out[82]//MatrixForm=
0 | - μ 4 | 0 | - t 4 Δ 4 | 0 | 0 | 0 | 0 | … |
μ 4 | 0 | t 4 Δ 4 | 0 | 0 | 0 | 0 | 0 | … |
0 | - t 4 Δ 4 | 0 | - μ 4 | 0 | - t 4 Δ 4 | 0 | 0 | … |
t 4 Δ 4 | 0 | μ 4 | 0 | t 4 Δ 4 | 0 | 0 | 0 | … |
0 | 0 | 0 | - t 4 Δ 4 | 0 | - μ 4 | 0 | - t 4 Δ 4 | … |
0 | 0 | t 4 Δ 4 | 0 | μ 4 | 0 | t 4 Δ 4 | 0 | … |
0 | 0 | 0 | 0 | 0 | - t 4 Δ 4 | 0 | - μ 4 | … |
0 | 0 | 0 | 0 | t 4 Δ 4 | 0 | μ 4 | 0 | … |
… | … | … | … | … | … | … | … | … |
Examine the connectivity.
The above graph clearly shows the chiral (i.e. sublattice) symmetry. To make it clearer, rearrange the Majorana modes.
The block-off diagonal structure of the above matrix demonstrates the chiral symmetry.
Back to General Case
Back to General Case
Motivated by the above special case, we now examine again the general case with the rearranged Majorana modes. The chiral symmetry is clear now.
Connection to the SSH Chain
Consider two sets of Majorana modes.
Construct a two copies of the Kitaev chain.
Now, fuse pairs of Majorana fermions across the two copies of the Kitaev chain to form Dirac fermions.
Fuse the pairs of Majorana fermions across the two Kitaev chains. That is, rewrite the Hamiltonian for the double Kitaev chain in terms of Dirac fermions.