Optical Conductivity in the Cuprates: from Mottness to Scale Invariance

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1 Optical Conductivity in the Cuprates: from Mottness to Scale Invariance Thanks to: NSF, EFRC (DOE) Brandon Langley Garrett Vanacore Kridsangaphong Limtragool

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5 N e ( ) = 2mV cell e 2 Z 0 (!)d!

6 optical gap N e ( ) = 2mV cell e 2 Z 0 (!)d!

7 optical gap N e ( ) = 2mV cell e 2 Z 0 (!)d! }x

8 optical gap N e ( ) = 2mV cell e 2 Z 0 (!)d! }x N e / x

9 Uchida, et al. Cooper, et al.

10 Uchida, et al. Cooper, et al. x

11 excess carriers?

12 does the charge depend on the scale?

13

14 classical (atomic) limit

15 classical (atomic) limit

16 classical (atomic) limit number of empty sites=x

17 classical (atomic) limit number of empty sites=x N e / x

18 classical (atomic) limit 2 states number of empty sites=x N e / x

19 Mott insulator

20 Mott insulator U =

21 Mott insulator N U N U = PES IPES

22 Mott insulator N U N U = PES IPES no hopping:

23 Mott insulator N U N U = PES IPES no hopping: N-1 PES

24 Mott insulator N U N U = PES IPES no hopping: N-1 N-1 PES IPES

25 Mott insulator N U N U = PES IPES no hopping: N-1 N-1 PES IPES

26 Mott insulator N U N U = PES IPES no hopping: N-1 2 N-1 PES IPES Sawatzky

27 atomic limit: x holes density of states 1-x 2x 1-x PES IPES

28 atomic limit: x holes density of states 1-x 2x 1-x PES IPES 1+x

29 atomic limit: x holes density of states 1-x 2x 1-x PES IPES 1+x 2

30 U finite U t c i n i µ U c i (1 n i )

31 U finite U t c i n i µ U c i (1 n i ) double occupancy in ground state!!

32 U finite U t c i n i µ U c i (1 n i ) double occupancy in ground state!! W PES > 1 + x

33 U finite U t c i n i µ U c i (1 n i ) N e 6=#x double occupancy in ground state!! W PES > 1 + x

34 why is this a problem?

35 counting electron states need to know: N (number of sites)

36 counting electron states x = n h /N need to know: N (number of sites)

37 counting electron states x = n h /N removal states 1 x need to know: N (number of sites)

38 counting electron states x = n h /N removal states 1 x addition states 1+x need to know: N (number of sites)

39 counting electron states 2x x = n h /N removal states 1 x addition states 1+x need to know: N (number of sites)

40 counting electron states 2x x = n h /N removal states 1 x addition states 1+x low-energy electron states need to know: N (number of sites)

41 counting electron states 2x x = n h /N removal states 1 x addition states 1+x low-energy electron states 1 x +2x =1+x need to know: N (number of sites)

42 counting electron states 2x x = n h /N removal states 1 x addition states 1+x low-energy electron states 1 x +2x =1+x high energy 1 x need to know: N (number of sites)

43 spectral function (dynamics)

44 spectral function (dynamics)

45 spectral function (dynamics) U t density of states 1+x + >1+x Harris and Lange (1967)

46 spectral function (dynamics) U t density of states 1+x + >1+x / t/u 1 x Harris and Lange (1967)

47 spectral function (dynamics) U t density of states 1+x + >1+x / t/u 1 x Harris and Lange (1967) not exhausted by counting electrons alone

48 determination of N e Hubbard model x=0.1 d=1 x=0.2 x=0.3 x=0.4 x=0.5 x=0.6 x=0.7 x=0.8 x=0.9 Phys. Rev. Lett. 105, (2010) M. Kohno, PRL, 105, Sawatzky

49 determination of N e chen/batlogg, 1990

50 DMFT on Hubbard model

51 DMFT on Hubbard model Millis, 2008 chakraborty & Phillips, 2007

52 spectral weight transfer Hubbard Model N e > #x optical conductivity (mid-ir)

53 is there anything else?

54 yes Drude conductivity n e 2 1 m 1 i!

55 yes Drude conductivity n e 2 1 m 1 i! (!) =C! 2 3

56 criticality scale invariance power law correlations

57 scale-invariant propagators 1 p 2

58 scale-invariant propagators 1 p 2 Anderson: use Luttinger Liquid propagators G R / 1 (! v s k)

59 compute conductivity without vertex corrections (PWA) (!) / 1! Z dx Z dtg e (x, t)g h (x, t)e i!t / (i!) 1+2

60 problems

61 problems 1.) cuprates are not 1- dimensional

62 problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter

63 problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter / G 2 µ µ }[ [G] =L d+1 2d U ]=L 3 d [ µ ]=L 2d U d [ µ ]=L 2d U d+1

64 problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter / G 2 µ µ }[ [G] =L d+1 2d U ]=L 3 d [ µ ]=L 2d U d [ µ ]=L 2d U d+1 independent of d U

65 power law?

66 power law? Could string theory be the answer?

67 UV QF T

68 UV IR?? QF T

69 IR?? UV QF T coupling constant g =1/ego

70 IR?? UV QF T coupling constant g =1/ego dg(e) dlne = (g(e)) locality in energy

71 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 R R / p g

72 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 R RG=GR R / p g

73 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 AdS 2 R 2 R AdS 4 RG=GR R / p g

74 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 AdS 2 R 2 R AdS 4 RG=GR R / p g cannot describe systems at g=0!

75 optical conductivity from a gravitational lattice G. Horowitz et al., Journal of High Energy Physics, 2012

76 optical conductivity from a gravitational lattice G. Horowitz et al., Journal of High Energy Physics, 2012

77 optical conductivity from a gravitational lattice log-log plots for various parameters G. Horowitz et al., Journal of High Energy Physics, 2012

78 optical conductivity from a gravitational lattice log-log plots for various parameters for G. Horowitz et al., Journal of High Energy Physics, 2012

79 optical conductivity from a gravitational lattice log-log plots for various parameters for a remarkable claim! replicates features of the strange metal? how? G. Horowitz et al., Journal of High Energy Physics, 2012

80 new equation! Einstein- Maxwell equations + non-uniform = charge density B! 2/3

81 not so fast!

82 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i!

83 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! no power law!!

84 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! no power law!! B! 2/3

85 who is correct?

86 who is correct? let s redo the calculation

87 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q

88 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q A t = µ(1 z)dt =lim z!0 p gf tz

89 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q A t = µ(1 z)dt =lim z!0 p gf tz

90 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) perturb with electric field Q A t = µ(1 z)dt =lim z!0 p gf tz ~ E

91 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) perturb with electric field Q A t = µ(1 z)dt =lim z!0 p gf tz ~ E g ab =ḡ ab + h ab A a = Āa + b a i = i + i

92 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) perturb with electric field Q A t = µ(1 z)dt =lim z!0 p gf tz ~ E g ab =ḡ ab + h ab A a = Āa + b a A x = E i! + J x(x,!)z + O(z 2 ) i = i + i

93 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) perturb with electric field Q A t = µ(1 z)dt =lim z!0 p gf tz ~ E g ab =ḡ ab + h ab A a = Āa + b a A x = E i! + J x(x,!)z + O(z 2 ) i = i + i solve equations of motion with gauge invariance

94 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) perturb with electric field Q A t = µ(1 z)dt =lim z!0 p gf tz ~ E g ab =ḡ ab + h ab A a = Āa + b a A x = E i! + J x(x,!)z + O(z 2 ) i = i + i solve equations of motion with gauge invariance = J x (x,!)/e

95 model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2,

96 model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice

97 model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2,

98 model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2,

99 model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2, L( )= p 2 V ( )]

100 HST vs. DG

101 HST vs. DG Horowitz, Santos, Tong (HST) V ( ) = 2 /L 2 =z (1) + z 2 (2) +, (1) (x) =A 0 cos(kx) inhomogeneous in x m 2 = 2/L 2

102 HST vs. DG Horowitz, Santos, Tong (HST) V ( ) = 2 /L 2 =z (1) + z 2 (2) +, (1) (x) =A 0 cos(kx) inhomogeneous in x m 2 = 2/L 2 de Donder gauge

103 HST vs. DG Horowitz, Santos, Tong (HST) DG V ( ) = 2 /L 2 V 2 =z (1) + z 2 (2) +, (1) (x) =A 0 cos(kx) inhomogeneous in x m 2 = 2/L 2 (z,x) = (z)e ikx no inhomogeneity in x m 2 = 3/(2L 2 ) de Donder gauge

104 HST vs. DG Horowitz, Santos, Tong (HST) DG V ( ) = 2 /L 2 V 2 =z (1) + z 2 (2) +, (1) (x) =A 0 cos(kx) inhomogeneous in x m 2 = 2/L 2 de Donder gauge (z,x) = (z)e ikx no inhomogeneity in x m 2 = 3/(2L 2 ) radial gauge

105 Our Model L =(r 1 ) 2 +(r 2 ) 2 +2V ( 1 )+2V ( 2 )

106 Our Model L =(r 1 ) 2 +(r 2 ) 2 +2V ( 1 )+2V ( 2 ) 1 = z (1) 1 + z 2 (2) 1 +, 2 = z (1) 2 + z 2 (2) 2 +, (1) 1 (x) =A 0 cos (1) 2 (x) =A 0 cos kx kx + 2 2,.

107 Our Model L =(r 1 ) 2 +(r 2 ) 2 +2V ( 1 )+2V ( 2 ) 1 = z (1) 1 + z 2 (2) 1 +, 2 = z (1) 2 + z 2 (2) 2 +, (1) 1 (x) =A 0 cos (1) 2 (x) =A 0 cos kx kx + 2 2,. =0 HST

108 Our Model L =(r 1 ) 2 +(r 2 ) 2 +2V ( 1 )+2V ( 2 ) 1 = z (1) 1 + z 2 (2) 1 +, 2 = z (1) 2 + z 2 (2) 2 +, (1) 1 (x) =A 0 cos (1) 2 (x) =A 0 cos kx kx + 2 2,. =0 = 2 HST DG

109 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)).

110 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)). reference metric

111 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)). metric ansatz reference metric ds 2 = L2 z 2 apple (1 z)p (z)q tt dt 2 + Q zzdz 2 (1 z)p (z) + Q xx(dx + z 2 Q zx dz) 2 + Q yy dy 2, P (z) =1+z + z 2 µ z3.

112 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)). metric ansatz reference metric ds 2 = L2 z 2 apple (1 z)p (z)q tt dt 2 + Q zzdz 2 (1 z)p (z) + Q xx(dx + z 2 Q zx dz) 2 + Q yy dy 2, P (z) =1+z + z 2 µ z3. RN-AdS when Q tt = Q zz = Q yy =1 =0 a t = µ 1 = µ

113 Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x)

114 Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1

115 Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1 Newton-Raphson on grid

116 Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1 Newton-Raphson on grid z 2 [0, 1],x2 (0, 2 )

117 =0 = 4 = 2

118 =0 = 4 = 2 translational invariance is broken in metric in multiples of 2k

119 charge density

120 perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i

121 perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0,

122 perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0, solve equations without mistakes!!

123 perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0, solve equations without mistakes!! Brandon Langley

124 dashed: RN-AdS solid: lattice - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form

125 sample conductivity plots dashed: RN-AdS solid: lattice - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form

126

127 Results

128 Results

129 Results

130 Results no power law...

131

132 Results

133 origin of power law?

134 origin of power law? phenomenology

135 origin of power law? phenomenology scale-invariant propagators (p 2 ) d U d/2

136 origin of power law? phenomenology scale-invariant propagators (p 2 ) d U d/2 no well-defined mass L e = Z 1 0 L(x, m 2 )dm 2

137 origin of power law? phenomenology scale-invariant propagators (p 2 ) d U d/2 no well-defined mass L e = Z 1 0 L(x, m 2 )dm 2 incoherent stuff (all energies)

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

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

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

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

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

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

144 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)

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

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

147 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

148 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!!

149 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

150 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

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

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

153 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2 continuous mass (x, m 2 ) flavors

154 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2 continuous mass (x, m 2 ) flavors e 2 (m) Karch, 2005 multi-bands

155 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2 continuous mass (x, m 2 ) flavors e 2 (m) Karch, 2005 multi-bands

156 use unparticle propagators to calculate conductivity

157 use unparticle propagators to calculate conductivity assume Gaussian action S = Z d d+1 p U (p)ig 1 (p) U (p)

158 use unparticle propagators to calculate conductivity assume Gaussian action S = Z d d+1 p U (p)ig 1 (p) U (p) U (x) = Z 1 dm 2 f(m 2 ) (x, m 2 ) 0

159 use unparticle propagators to calculate conductivity assume Gaussian action S = Z d d+1 p U (p)ig 1 (p) U (p) U (x) = Z 1 dm 2 f(m 2 ) (x, m 2 ) 0 G(p) i ( p 2 + i ) d+1 2 d U

160 gauge unparticles

161 gauge unparticles S = Z Wilson line d d+1 xd d+1 y U (x)f (x y)w (x, y) U (y),

162 gauge unparticles S = Z Wilson line d d+1 xd d+1 y U (x)f (x y)w (x, y) U (y), vertices

163 gauge unparticles S = Z Wilson line d d+1 xd d+1 y U (x)f (x y)w (x, y) U (y), vertices g µ (p, q) = 3 S A µ (q) (p + q) (p) 1-gauge g 2 µ (p, q 1,q 2 )= 4 S A µ (q 1 ) A (q 2 ) (p + q 1 + q 2 ) (p) 2-gauge

164 use Ward-Takahashi identities to simplify vertices

165 use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 )

166 use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 ) response function to an electric field

167 use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 ) response function to an electric field K µ n, n 0 ( q, q 0 )= (2 )2d T 2 Z 1 2 A µ,n (q) A,n 0(q 0 ) A=0 Z[A]

168 compute conductivity µ 1 (i! n )=lim K q!0! n µ (q)! n

169 compute conductivity µ 1 (i! n )=lim K q!0! n µ (q)! n no power law (i! n )=( d +1 2 d U ) 0 (i! n )

170 what went wrong?

171 what went wrong? free field U (x) = Z 1 dm 2 f(m 2 ) (x, m 2 ) 0

172 what went wrong? free field U (x) = Z 1 dm 2 f(m 2 ) (x, m 2 ) 0 unparticle field has particle content

173 what went wrong? free field U (x) = Z 1 dm 2 f(m 2 ) (x, m 2 ) 0 unparticle field has particle content

174 continuous mass taken seriously S = NX i=1 Z d Z d d x( D µ 2 i + m 2 i i 2 )

175 continuous mass taken seriously NX Z Z S = d d d x( D µ 2 i + m 2 i i 2 ) i=1 X! Z (m)dm i

176 continuous mass taken seriously NX Z Z S = d d d x( D µ 2 i + m 2 i i 2 ) i=1 X! Z (m)dm i (!) = Z M 0 dm (m)e 2 (m)f(!, m, T )

177 continuous mass taken seriously NX Z Z S = d d d x( D µ 2 i + m 2 i i 2 ) i=1 X! Z (m)dm i (!) = Z M 0 dm (m)e 2 (m)f(!, m, T ) /! >0(! <2M)

178 continuous mass taken seriously NX Z Z S = d d d x( D µ 2 i + m 2 i i 2 ) i=1 X! Z (m)dm i (!) = Z M 0 dm (m)e 2 (m)f(!, m, T ) /! >0(! <2M) >0 convergence of integral

179 last attempt

180 last attempt take experiments seriously

181 last attempt take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i

182 last attempt take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i continuous mass

183 last attempt take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i continuous mass (!) = Z M 0 (m)e 2 (m) (m) m 1 1 i! (m) dm

184 variable masses for everything (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c Karch, 2015

185 variable masses for everything (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c Karch, 2015 (!) = 0e M a+2b+c Z M 0 dm ma+2b+c 2 1 i! 0 m c M c = 0e2 0 cm 1!(! 0 ) a+2b 1 c Z! 0 0 dx x a+2b 1 c 1 ix

186 variable masses for everything (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c Karch, 2015 (!) = 0e M a+2b+c Z M 0 dm ma+2b+c 2 1 i! 0 m c M c = 0e2 0 cm 1!(! 0 ) a+2b 1 c Z! 0 0 dx x a+2b 1 c 1 ix

187 variable masses for everything (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c Karch, 2015 (!) = 0e M a+2b+c Z M 0 dm ma+2b+c 2 1 i! 0 m c M c = 0e2 0 cm 1!(! 0 ) a+2b 1 c Z! 0 0 dx x a+2b 1 c 1 ix perform integral

188 a +2b 1 c = 1 3

189 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix

190 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix! 0!1

191 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix! 0!1 (!) = 1 3 (p 3+3i) 0e M! 2 3

192 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix! 0!1 (!) = 1 3 (p 3+3i) 0e M! 2 3 tan = p 3 60

193 experiments (!) =C! 2 e i (1 /2) =1.35

194 experiments (!) = 1 3 (p 3+3i) 0e M! 2 3 (!) =C! 2 e i (1 /2) =1.35

195 experiments (!) = 1 3 (p 3+3i) 0e M! 2 3 tan 2 / 1 = p 3 = 60 (!) =C! 2 e i (1 /2) =1.35

196 experiments (!) = 1 3 (p 3+3i) 0e M! 2 3 victory!! tan 2 / 1 = p 3 = 60 (!) =C! 2 e i (1 /2) =1.35

197 are anomalous dimensions necessary a +2b 1 c = 1 3 (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c

198 are anomalous dimensions necessary a +2b 1 c = 1 3 (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c hyperscaling violation

199 are anomalous dimensions necessary a +2b 1 c = 1 3 (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c hyperscaling violation anomalous dimension

200 are anomalous dimensions necessary a +2b 1 c = 1 3 (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 M c hyperscaling violation anomalous dimension c =1 a +2b =2/3

201 are anomalous dimensions necessary a +2b 1 c (m) = 0 m a 1 M a m b e(m) =e 0 M b m c (m) = 0 c =1 = 1 3 M c a +2b =2/3 hyperscaling violation anomalous dimension b =0 a =2/3

202 No

203 but they are possible!

204 but they are possible! continuous mass

205 but they are possible! continuous mass charge nonconservation

206 but they are possible! continuous mass charge nonconservation

207 but they are possible! continuous mass charge nonconservation unparticles

208 what really is the summation over mass?

209 what really is the summation over mass? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2

210 what really is the summation over mass? Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1

211 what really is the summation over mass? 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

212 what really is the summation over mass? 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 ds 2 = L2 z 2 µ dx µ dx + dz 2 anti de Sitter metric

213 what really is the summation over mass? 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 ds 2 = L2 z 2 µ dx µ dx + dz 2 anti de Sitter metric

214 action on AdS 5+2 S = 1 2 Z d 4+2 xdz p a + 2 R 2 p g =(R/z) 5+2

215 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

216 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

217 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 generating functional for unparticles

218 unparticles action on AdS

219 emergent gravity Mott problem