Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley

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1 Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley

2 Correlated Matter

3 Correlated Matter What is carrying the current?

5 current-carrying excitations?

6 current-carrying excitations? e = e q (q odd)

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

8 fractional quantum Hall effect current-carrying excitations?

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

10 Correlated Electron Matter What is carrying the current? UV IR

11 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e IR

12 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e IR

13 Correlated Electron Matter What is carrying the current? e e e UV e e e e e e e e not particles IR

14 can the charge density be given a particle interpretation?

15 if not?

16 if not? charge stuff= particles + other stuff

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

18 counting particles

19 counting particles 1

20 counting particles 2 1

21 counting particles 2 3 1

22 counting particles

23 counting particles is there a more efficient way?

24 µ

25 n = µ N(ω)dω µ

26 n = µ N(ω)dω µ IR UV

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

28 Luttinger s `theorem

29 Each particle has an energy

30 Each particle has an energy particles energy E = ε p

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

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

33 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p ε p E

34 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E

35 Luttinger s Theorem Green function (propagator) G(E) = G(E) 1 E ε p E>ε p ε p E E<ε p

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

37 Counting sign changes?

38 Counting sign changes? Θ(x)

39 Counting sign changes? Θ(x) = { 0 x<0

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

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

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

43 Counting sign changes? Θ(x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 p Θ(G(E =0, p)) Luttinger Theorem for electrons

44 How do functions change sign? E>ε p ε p E E<ε p

45 How do functions change sign? E>ε p ε p E E<ε p divergence

47 Is there another way?

49 Yes

50 zero-crossing ε p E

51 zero-crossing ε p E no divergence is necessary

52 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p))

53 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros

54 closer look at Luttinger s theorem n =2 p Θ(G(E =0, p)) divergences+ zeros how can zeros affect the particle count?

56 Is this famous theorem from 1960 correct?

57 A model with zeros but Luttinger fails

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

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

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

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

62 e spin SU(2)

63 e spin SU(2) generalization N flavors of spin SU(N)

64 e spin SU(2) generalization N =5 N flavors of spin SU(N)

65 e spin SU(2) generalization N =5 N flavors of spin SU(N)

66 e spin SU(2) generalization 2E 25 N = N flavors of spin SU(N)

67 e spin SU(2) generalization 2E N flavors of spin SU(N) 25 N = H = U 2 (n 1 + n N ) 2

68 SU(2) U

69 SU(2) U G = # ω + U/2 + # ω U/2

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

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

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

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

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

75 < 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

76 < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

77 a zero must exist < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

78 Luttinger s theorem n = NΘ(2n N)

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

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

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

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

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

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

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

86 Luttinger s theorem

88 Problem

89 Problem G=0

90 Problem G=0 G = 1 E ε p =0??

91 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N

92 Problem G=0 G = 1 E ε p =0?? G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N must diverge

93 G = 1 E ε p Σ G=0

94 self energy diverges G = 1 E ε p Σ G=0

95 if Σ is infinite

96 if Σ is infinite lifetime of a particle vanishes

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

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

99 zeros and particles are incompatible

100 zeros and particles are incompatible Luttinger theorem is wrong!

101 how to count particles?

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

103

104 what is the extra stuff?

105 Properties all particles share?

106 Properties all particles share? charge fixed mass!

107 Properties all particles share? charge fixed mass! conserved (gauge invariance)

108 Properties all particles share? charge fixed mass! conserved (gauge not conserved invariance)

109 mass sets a scale

110 mass sets a scale mass breaks scale invariance

111 what is scale invariance? invariance on all length scales

112 f(x) =x 2

113 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change

114 f(x) =x 2 f(x/λ) =(x/λ) 2 scale change scale invariance

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

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

117 f(x) =ax 2 + bx 3 x x/λ

118 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ)

119 f(x) =ax 2 + bx 3 x x/λ f(x) Λ 2 (ax 2 + bx/λ) not scale invariant

120 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L

121 free field theory L = 1 2 µ φ µ φ x x/λ φ(x) φ(x) L Λ 2 L scale invariant

122 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2

123 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x)

124 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x) m 2 φ 2

125 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2

126 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2

127 massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2 no scale invariance

128 how should we think about field theory? QFT

129 how should we think about field theory? g 7 g 6 g 5 g 4 g 3 g 2 g 1 QFT

130 how should we think about field theory? g7 g6 g5 g4 g3 g2 g1 QFT

131 g7 g6 g5 g4 g3 g2 g1 QFT

132 what if m depends on the scale? g7 g6 g5 g4 g3 g2 g1 QFT

133 what if m depends on the scale? QFT

134 what if m depends on the scale? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi QFT

135 sum over all mass L = i L(m i )

136 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m)

137 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2

138 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass!

139 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2

140 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!!

141 L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

142 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

143 scale-invariant stuff is weird particles energy E = ε p Green function (propagator) 1 G(E) = E ε p particle=infinity

144 scale-invariant stuff is weird

145 scale-invariant stuff is weird unparticles

146 scale-invariant stuff is weird unparticles G (E ε p ) d U 2 d U > 2 unparticles=zero propagator

147 n massless particles G (E ε p ) n 2 n>2 unparticles=fractional number of massless particles

148 n massless particles G (E ε p ) n 2 n>2 n<2 unparticles=fractional number of massless particles unparticles=luttinger liquids

149 n massless particles G (E ε p ) n 2 n>2 n<2 unparticles=fractional number of massless particles unparticles=luttinger liquids common ingredient:scale invariance

150 Where do we expect to find zeros? UV IR

151 Where do we expect to find zeros? e e e UV e e e e e e e e IR

152 Where do we expect to find zeros? e e e UV e e e e e e e e IR

153 Where do we expect to find zeros? e e e UV e e e e e e e e not particles IR

154 G(p, ω) = 0 UV no physical quark or electron states G(p L,ω = 0) = 0 IR G BCS (ω, p) = ω + v( p p F ) ω 2 2 v 2 ( p p F ) 2 G 1+1QCD = ik m 2 g 2 /π +2k + k + g 2 k /πλ i

155 where else do we expect to see zeros?

156 where else do we expect to see zeros? strongly correlated systems

157 where else do we expect to see zeros? strongly correlated systems Mott insulators

158 NiO insulates d 8? What is a Mott Insulator?

159 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL

160 NiO insulates d 8? What is a Mott Insulator? EMPTY STATES= METAL band theory fails!

161 NiO insulates d 8? perhaps this costs energy What is a Mott Insulator? EMPTY STATES= METAL band theory fails!

162 Mott Problem: NiO (Band theory failure) (N rooms N occupants) t U t

163

164

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

166 photoemission

167 photoemission particles energy E = ε p

168 photoemission particles energy E = ε p p 2

169 photoemission particles energy E = ε p p 2

170 photoemission particles energy E = ε p p 2 two crossings: closed surface of excitations

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

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

173 zeros do not affect the particle density

174 zeros do not affect the particle density no Luttinger theorem

175 zeros do not affect the particle density experimental data no Luttinger theorem

176 where do these zeros come from?

177 classical (atomic) limit

178 warm-up no hopping: x holes density of states 1+x 1 x } }

179 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }

180 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x }

181 warm-up no hopping: x holes density of states 1+x 1 x } 1-x 2x } spectral weight: x- dependent G = 1+x ω + µ + U/2 + 1 x ω + µ U/2

182 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ )

183 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!!

184 quantum Mottness: U finite U t c iσ n i σ µ U c iσ (1 n i σ ) double occupancy in ground state!! W PES > 1 + x

185 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α

186 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer

187 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x

188 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states

189 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band

190 beyond the atomic limit: any real system density of states 1 + x + α(t/u, x) α = t U c iσ c jσ > 0 ij 1 x α Harris & Lange, 1967 dynamical spectral weight transfer Intensity>1+x # of charge e states # of electron states in lower band not exhausted by counting electrons alone?

191 Wilsonian analysis density of states d(uhb)

192 Wilsonian analysis density of states 1 + x + α(t/u, x) charge 2e stuff d(uhb)

193 Wilsonian analysis density of states 1 + x + α(t/u, x) charge 2e stuff d(uhb) zeros n< conserved charge

194 do particles +unparticles=infinities+zeros?

195 what really is the summation over mass? Lmi+7 Lmi+6 Lmi+5 Lmi+4 Lmi+3 Lmi+2 Lmi+1 Lmi

196 mass=energy

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

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

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

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

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

202 what s the geometry?

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

204 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous)

205 what s the geometry? dg(e) dlne = β(g(e)) E λe x µ x µ /λ =0 scale invariance (continuous) solve Einstein equations

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

207 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2

208 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime

209 unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 classical gravity in a d+1 curved spacetime gauge theory in AdS 4

210 High T_c unparticles variable mass UPt_3

211 emergent gravity High T_c unparticles variable mass UPt_3

Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley

Is Strongly Correlated Electron Matter Full of Unparticles? Kiaran dave Charlie Kane Brandon Langley 2=3 goal 2 = 3 Properties all particles share? Properties all particles share? charge fixed mass! Properties