Majorana Tutorial

Introduction Introductio n to Majorana fermions part I

Fa Wang ( 王垡 ) I!"M # $c%ool of P%&sics, Pe'in( Uni)ersit&

Tutorial for students part I 6/2/2013@PKU

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*eferences ●

Main references  +  -.- Kitae), Phys.Usp. Kitae), Phys.Usp. 44, 131 2001 ari)condmat/0010440 ari)condmat/0010440  +  5-licea, Rep.Prog.Phys. 5-licea, Rep.Prog.Phys. 75, 06701 2012 ari)1202-1283 ari)1202-1283

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T%eoretical references  +  !-9a&a' et al., al., Rev.Mod.Phys. 80, 10:3 200: ari)00-1::8  +  *-M- ;utc%&n et al., al., Phys.Rev.Lett. 105, 0001 2010 +  .-u # !-;-Kane, Phys.Rev.Lett. !-;-Kane,  Phys.Rev.Lett. 102, 216403 2008Tutorial for students part I 6/2/2013@PKU

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*eferences ●

?perimental references  +  A-Mouri' et al., al., Science 336, 1003 2012 +  -Bas et al., al., Nat. Phys. 8, :: 2012 +  ;-P-*o'%inson et al., al., Nat. Phys. 8, 87 2012-

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*eadin( materials  +  >-CilcDe', EMajorana returnsF, Nat.Phys. returnsF,  Nat.Phys. 5, 614 2008-

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Tri)ia aGout Majorana fermion ●

Gout ?ttore Majorana  +  u(- 7, 1806 Gorn in !atania, $icil&  +  Marc% 183: disappeared in Palermo, $icil&

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Gout Majorana fermion  +  *ef- ?- Majorana, Nuovo Cimento 14, 11 183 +  *eal solutions of Birac eHuation, are t%eir o=n antiparticles +  ?lementar& particles 9eutrinos  +  2B condensed matter s&stems J7/2 >"L? state $r2*uermion creation/anni%ilation operators

 +  nticommutation relation

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Befinin( Majorana fermion ●

Majorana fermion operators from one fermion mode

 +  E*eal)aluednessF  +  nticommutation relation

esp-

 +  >ermion numGer  +  >ermion numGer parit&

Tutorial for students part I 6/2/2013@PKU

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Befinin( Majorana fermion ●

T=o fermion modes  +  4d LilGert space tensor product of t=o 2d LilGert space +  Oasis tensor products of sin(le fermion Gasis

 +  nticommutation relation

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Befinin( Majorana fermion ●

T=o fermion modes

contd

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Befinin( Majorana fermion ●

Majorana fermions from t=o fermion modes

 +  E*eal)aluednessF  +  nticommutation relation  +  >ermion numGer

 +  >ermion numGer parit&

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Befinin( Majorana fermion ●

 N  fermion modes  + 

diml LilGert space

 +  Gasis ●

2 N  Majorana fermions c-f- 5ordanCi(ner transformation

 +  >ermions from Majoranas  +  >ermion numGer parit&

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Properties of Majorana fermion ●

E*eal)aluednessF

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nticommutation relation  +  Majorana %as no E)acuumF

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EOasis c%an(eF of Majorana fermions  +  *eal ort%o(onal transformation  +  Ma&/Ma& not c%an(e fermion parit&  +  Includes particle%ole transformation of fermions eample, t=o Majoranas from one fermion

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Properties of Majorana fermion ●

9onlocalit& eample

 + 

affects/depends on man& suGLilGert spaces-

 +  Products of odd Q of Majoranas %a)e similar propert& +  9