Absence of Luttinger s Theorem

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1 Absence of Luttinger s Theorem Kiaran dave Charlie Kane

2 n = µ N(ω)dω µ IR UV

3 n = µ N(ω)dω µ IR UV can n be deduced entirely from the IR(lowenergy) scale?

4 Luttinger s theorem n =2 k Θ(G(k,ω = 0))

5 Luttinger s theorem n =2 k Θ(G(k,ω = 0))

6 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) E>ε p ε p E E<ε p counting poles (qp)

7 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing E>ε p ε p E ε p E E<ε p counting poles (qp)

8 Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing DetG(ω =0,p)=0 ε p E>ε p E ε p E E<ε p counting poles (qp)

9 singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) poles+zeros (all sign changes)

10 singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) n =2 k Θ(G(k,ω = 0)) poles+zeros (all sign changes)

11 simple problem: n=1 SU(2) U

12 simple problem: n=1 SU(2) µ U

13 simple problem: n=1 SU(2) µ U U 2

14 simple problem: n=1 SU(2) U 2 µ U U 2

15 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2

16 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 =0 if ω =0

17 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 n =2θ(0) = 1 =0 if ω =0

18

19 Is this famous theorem from 1960 correct?

20 SU(N) H = U 2 (n 1 + n N ) 2 2E 25 N =

21 compute Green function exactly (Lehman formula) G αβ (ω) = 1 Z ab exp βk a Q ab αβ Q ab αβ = a c α bb c β a ω K b + K a + a c β bb c α a ω K a + K b do sum explicitly

22 G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N

23 G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N } > 0

24 Luttinger s theorem n = NΘ(2n N)

25 Luttinger s theorem n = NΘ(2n N) { 0, 1, 1/2

26 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2

27 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3

28 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3

29 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3

30 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd

31 Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution

32 does the degeneracy matter? e t t t t t t t t t =0 +

33 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω H c a ρ(0 + )

34 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω H c a ρ(0 + )

35 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + )

36 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) G ab (ω) = ωδ ab Uρ ab ω(ω U)

37 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U)

38 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state

39 G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place

40 Problem

41 Problem G=0

42 Problem G=0 G = 1 E ε p =0??

43 Problem G=0 G = 1 E ε p =0?? G = 1 E ε p Σ

44 Problem G=0 G = 1 E ε p =0?? G = 1 E ε p Σ

45 Problem G=0 G = 1 E ε p =0?? G = 1 E ε p Σ G=0

46 if Σ is infinite

47 if Σ is infinite lifetime of a particle vanishes

48 if Σ is infinite lifetime of a particle Σ < p vanishes

49 if Σ is infinite lifetime of a particle Σ < p vanishes no particle

50 what went wrong?

51 what went wrong? δi[g] = dωσδg

52 what went wrong? δi[g] = dωσδg if Σ

53 what went wrong? δi[g] = dωσδg if Σ integral does not exist

54 what went wrong? δi[g] = dωσδg if Σ integral does not exist No Luttinger theorem!

55 Luttinger s theorem

56

57 are zeros important?

58 Fermi Arcs

59 Fermi Arcs

60 Fermi Arcs Re G Changes Sign across An arc

61 Fermi Arcs Re G Changes Sign across An arc Must cross A zero line (DetG=0)!!! Fermi arcs necessarily imply zeros exist.

62 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS)

63 what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen infinities not seen zeros E F k

64 experimental data (LSCO) `Luttinger count k F 1 x FS zeros do not affect the particle density

65 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density

66 experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density each hole = a single k-state

67 how to count particles?

68 how to count particles? some charged stuff has no particle interpretation

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