Unparticles in High T_c Superconductors

Transcription

1 Unparticles in High T_c Superconductors Thanks to: NSF, EFRC (DOE) Kiaran dave Charlie Kane Brandon Langley J. A. Hutasoit

2 Correlated Electron Matter

3 Correlated Electron Matter What is carrying the current?

4 fractional quantum Hall effect current-carrying excitations e = e q (q odd)

5 fractional quantum Hall effect current-carrying excitations e = e q (q odd)

6 fractional quantum Hall effect current-carrying excitations

7 fractional quantum Hall effect current-carrying excitations e =?

8 How Does Fermi Liquid Theory Breakdown? TRSB / at? LaSrCuO

9 can the charge density be given a particle interpretation?

10 can the charge density be given a particle interpretation? does charge density= total number of electrons?

11 if not?

12 if not? charge stuff= particles + other stuff

13 if not? charge stuff= particles + other stuff unparticles!

14 counting particles

15 counting particles 1

16 counting particles 2 1

17 counting particles 2 3 1

18 counting particles

19 counting particles is there a more efficient way?

20 µ IR 1 UV

21 n = Z µ 1 N(!)d! µ IR 1 UV

22 n = Z µ 1 N(!)d! µ IR 1 UV can n be deduced entirely from the IR(lowenergy) scale?

23 Luttinger s `theorem

24 Each particle has an energy

25 Each particle has an energy particles energy E = " p

26 Each particle has an energy particles energy E = " p G(E) = E 1 " p

27 Each particle has an energy particles energy E = " p G(E) = E 1 " p particle=infinity

28 Each particle has an energy particles energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity

29 Each particle has an energy particles particle count is deducible from Green function energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity

30 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) " p E

31 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E

32 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p

33 Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p particle density= number of sign changes of G

34 Counting sign changes?

35 Counting sign changes? (x)

36 Counting sign changes? (x) = { 0 x<0

37 Counting sign changes? (x) = { 0 x<0 1 x>0

38 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0

39 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes

40 Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 X k (<G(k,! = 0)) Luttinger Theorem for electrons

41 How do functions change sign? E>" p " p E E<" p

42 How do functions change sign? E>" p " p E E<" p divergence

43

44 Is there another way?

45

46 Yes

47 zero-crossing " p E DetG = 0

48 zero-crossing " p E DetG = 0 no divergence is necessary

49 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0))

50 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros

51 closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros how can zeros affect the particle count?

52 what are zeros?

53 what are zeros? Im G=0 µ =0

54 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig

55 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap

56 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0

57 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 0) =0 (single band)

58 what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 0) =0 (single band) strongly correlated gapped systems

59 NiO insulates d 8? Mott mechanism (not Slater) t

60 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t

61 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t

62 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0

63 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone

64 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0

65 NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0 =Mottness

66 U/t = 10 1 interactions dominate: Strong Coupling Physics

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

68 simple problem: n=1 SU(2) µ U

69 simple problem: n=1 SU(2) µ U U 2

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

71 simple problem: n=1 SU(2) U 2 µ U U 2 G = 1! + U/2 + 1! U/2

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

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

74 Fermi Liquids Mott Insulators Luttinger s theorem

75

76 Is this famous theorem from 1960 correct?

77 A model with zeros but Luttinger fails

78 A model with zeros but Luttinger fails e t t t t t t t t

79 A model with zeros but Luttinger fails e t t t t t t t t

80 A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)

81 A model with zeros but Luttinger fails e N flavors of e- t t t t t t t t no hopping=> no propagation (zeros)

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

83 SU(N) for N=3: no particle-hole symmetry

84 SU(N) for N=3: no particle-hole symmetry

85 SU(N) for N=3: no particle-hole symmetry lim µ(t ) T!0

86 G (! = 0) = # 2n N N

87 G (! = 0) = # 2n N N

88 Luttinger s theorem n = N (2n N)

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

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

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

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

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

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

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

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

97 No

98 No as long as SU(N) symmetry is intact ab =Tr c ac b (0 + ) =1/N diag(1, 1, 1,...) mixed state pure state

99 Problem DetG =0 G = 1 E " p =0??

100 Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1

101 Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1 G=0

102 if is infinite

103 if is infinite lifetime of a particle vanishes

104 if is infinite lifetime of a particle = < p vanishes

105 if is infinite lifetime of a particle = < p vanishes no particle

106 what went wrong?

107 what went wrong? I[G] = Z d! G

108 what went wrong? I[G] = Z d! G if!1

109 what went wrong? I[G] = Z d! G if!1 integral does not exist

110 what went wrong? I[G] = Z d! G if!1 integral does not exist No Luttinger theorem!

111 zeros and particles are incompatible

112 Luttinger s theorem

113

114 are zeros important?

115 photoemission

116 photoemission particles energy E = " p

117 photoemission particles energy E = " p p 2

118 photoemission particles energy E = " p p 2

119 photoemission particles energy E = " p p 2 two crossings: closed surface of excitations

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

121 Problem poles ReG>0 Re G<0

122 Problem poles ReG>0 Re G<0

123 Problem How to account for the sign change without poles? poles ReG>0 Re G<0

124 Problem How to account for the sign change without poles? poles ReG>0 Re G<0 Only option: DetG=0! (zeros )

125 Problem How to account for the sign change without poles? poles Re G< ReG> Only option: DetG=0! (zeros )

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

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

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

129

130 what is the extra stuff?

131

132 scale invariance

133 new fixed point (scale invariance)?

134 new fixed point (scale invariance)

135 new fixed point (scale invariance) unparticles (IR) (H. Georgi)

136 what is scale invariance? invariance on all length scales

137 f(x) =x 2

138 f(x) =x 2 f(x/ )=(x/ ) 2 scale change

139 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance

140 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{

141 f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{ g( )= 2

142 what s the underlying theory?

143 free field theory L = µ x! x/ (x)! (x) L! 2 L

144 free field theory L = µ x! x/ (x)! (x) L! 2 L scale invariant

145 massive free theory mass L = µ + m 2 2

146 massive free theory mass L = µ + m 2 2 x! x/ (x)! (x)

147 massive free theory mass L = µ + m 2 2 x! x/ (x)! (x) m 2 2

148 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2

149 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2

150 massive free theory mass L = µ + m µ x! x/ (x)! (x) m 2 2 no scale invariance

151 mass sets a scale

152

153 unparticles from a massive theory?

154 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)

155 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2

156 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!

157 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2

158 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!!

159 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!! not particles

160 unparticles Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 dm 2 theory with all possible mass!! (x, m 2 / 2 ) x! x/ m 2 / 2! m 2 L! 4 L scale invariance is restored!! not particles

161 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2

162 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2

163 n massless particles G / (E " p ) n 2

164 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2

165 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles

166 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros

167 n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros no simple sign change

168 no particle interpretation mass-distribution U = Z B(m 2 ) (m 2 )dm 2 unparticle field (non-canonical) particle field

169 d U?

170 what really is the summation over mass?

171 what really is the summation over mass? mass=energy

172 high energy (UV) QFT low energy (IR) related to sum over mass

173 high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass

174 implement E-scaling with an extra dimension high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass

175 implement E-scaling with an extra dimension QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass

176 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity QF T QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass

177 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension IR gravity UV no particles QFT (conserved currents) QF T dg(e) dlne = (g(e)) locality in energy related to sum over mass

178 gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension IR gravity UV no particles QFT (conserved currents) QF T dg(e) dlne = (g(e)) locality in energy related to sum over mass unparticles?

179 what s the geometry?

180 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous)

181 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ /

182 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations

183 what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations ds 2 = u L 2 µ dx µ dx + L E 2 de 2 anti de-sitter space

184 mass=extra dimension fixed by metric

185 can something more exact be done?

186 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2

187 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1

188 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1 L = Z 1 0 dz 2R2 z 5+2 apple 1 2 z 2 R 2 µ (@ µ )(@ )+ 2 2R 2

189 can something more exact be done? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1 L = Z 1 0 dz 2R2 z 5+2 apple 1 2 z 2 R 2 µ (@ µ )(@ )+ 2 2R 2 can be absorbed with AdS metric

190 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2

191 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2

192 generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2 unparticle lives in d =4+2 apple 0

193 scaling dimension is fixed m = 1 z

194 scaling dimension is fixed m = 1 z m 2 AdS = d U (d U d) R 2

195 scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2

196 scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2 d U = d 2 + p d2 +4 2

197 G U (p) / p 2(d U d/2) G U (0) = 0

198 unparticle (AdS) propagator has zeros! G U (p) / p 2(d U d/2) G U (0) = 0

199 interchanging unparticles fractional (d_u) number of massless particles

200 interchanging unparticles fractional (d_u) number of massless particles

201 interchanging unparticles fractional (d_u) number of massless particles

202 interchanging unparticles fractional (d_u) number of massless particles

203 interchanging unparticles fractional (d_u) number of massless particles e i d U 6= 1, 0

204 interchanging unparticles fractional (d_u) number of massless particles e i d U 6= 1, 0 fractional statistics in d=2+1

205 d U = d 2 + p d > d 2 e i d U 6= e i d U time-reversal symmetry breaking from unparticle (zeros=fermi arcs) matter

206 d ln g d ln = 4d U d>0 fermions T c unparticles with unparticle propagator BCS g tendency towards pairing (any instability which establishes a gap)

207 TRSB

208 TRSB

209 TRSB unparticles

210 TRSB unparticles particles

211 breaking of scale invariance TRSB unparticles particles

212 emergent gravity High T_c unparticles variable mass

213 emergent gravity High T_c unparticles variable mass fractional statistics in d=2+1