Optical Conductivity in the Cuprates from holography and unparticles

Transcription

1 Optical Conductivity in the Cuprates from holography and unparticles Thanks to: NSF, EFRC (DOE) Brandon Langley Garrett Vanacore Kridsangaphong Limtragool

2 Properties all particles share? charge fixed mass!

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

4 Properties all particles share? charge fixed mass! mass sets a scale conserved (gauge invariance)

5 Properties all particles share? charge fixed mass! mass sets a scale conserved (gauge invariance) not conserved

6 Properties all particles share? charge fixed mass! mass sets a scale conserved (gauge invariance) not conserved

7 optical conductivity in the cuprates scale invariant sector `unparticles (no definite mass)

8 optical conductivity in the cuprates incoherent metal scale invariant sector `unparticles (no definite mass)

9 unconventional superconductivity? / at

10 unconventional superconductivity? unfermi liquid (unmetal) / at

11

12

13

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

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

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

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

18 Uchida, et al. Cooper, et al.

19 Uchida, et al. Cooper, et al. x

20 Uchida, et al. Cooper, et al. x low-energy model for N e >x??

21 excess carriers?

22 excess carriers? charge carriers have no particle content?

23 Mott insulator 1-x

24 Mott insulator U = 1-x

25 Mott insulator N U N U = PES 1-x IPES

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

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

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

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

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

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

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

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

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

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

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

37 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)

38 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)

39 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)

40 why is this a problem?

41 spectral function (dynamics)

42 spectral function (dynamics)

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

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

45 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

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

47 is there anything else?

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

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

50 YBCO thin films

51 criticality scale invariance power law correlations

52 criticality scale invariance power law correlations

53 criticality scale invariance power law correlations

54 scale-invariant propagators 1 p 2

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

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

57 problems

58 problems 1.) cuprates are not 1- dimensional

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

60 / Z [ problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter d d pg G( µ ) 2, [G] =L d+1 [ µ ]=L 2d U d 2d U µ ]=L 2d U d+1 µ

61 problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter / Z d d pg G( µ ) 2, [G] =L d+1 [ µ ]=L 2d U d [ 2d U µ ]=L 2d U d+1 µ } independent [ ]=L 2 d of d U

62 1997 Anderson

63 1997 Anderson 2012 string theory AdS/CFT

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

65 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

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 AdS 2 R 2 R R / p g AdS 4

73 replace coupled theory with geometry scale-invariant anti de-sitter z =1 z =0 g( ) 2 R AdS AdS 2 R 2 RG=GR 4 R / p g

74 replace coupled theory with geometry scale-invariant anti de-sitter z =1 z =0 g( ) 2 R AdS AdS 2 R 2 RG=GR 4 R / p g cannot describe systems at g=0!

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

76 not so fast!

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

78 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0

79 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0 B! 2/3

80 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0 B! 2/3 no power law!!

81 who is correct?

82 who is correct? let s redo the calculation

83 model

84 model action = gravity + EM + lattice

85 model action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2,

86 model action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2,

87 model action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2, L( )= p 2 V ( )]

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

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

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

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

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 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 = J x (x,!)/e

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 ) Brandon Langley i = i + i solve equations of motion with gauge invariance (without mistakes) = J x (x,!)/e

95 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 ) Brandon Langley i = i + i solve equations of motion with gauge invariance (without mistakes) = J x (x,!)/e

96 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 ) Brandon Langley i = i + i solve equations of motion with gauge invariance (without mistakes) = J x (x,!)/e

97 HST vs. DG

98 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

99 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

100 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

101 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

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

103 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,.

104 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

105 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

106 =0 = 4 = 2

107 =0 = 4 = 2 translational invariance is broken in metric in multiples of 2k

108 charge density

109 conductivity - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form

110 is there a power law?

111 is there a power law? Results 2/3 DG HST

112 Results

113 Results A 1 =0.75,k 1 =2,k 2 =2, =0,µ=1.4, T/µ =0.115

114 No!

115 Anderson AdS/CFT

116 scale invariance Anderson AdS/CFT

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

118 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

119 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)

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

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

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

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

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

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

126 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)

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

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

129 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

130 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!!

131 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

132 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

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

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

135 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

136 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

137 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

138 use unparticle propagators to calculate conductivity

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

140 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

141 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

142 gauge unparticles

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

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

145 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

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

147 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 )

148 what went wrong?

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

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

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

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

153 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

154 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 )

155 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)

156 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

157 last attempt

158 last attempt take experiments seriously

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

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

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

162 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

163 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

164 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

165 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

166 a +2b 1 c = 1 3

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

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

169 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

170 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

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

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

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

174 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

175 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

176 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

177 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

178 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

179 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

180 No

181 No but the Lorenz ratio is not a constant L H = apple xy T xy T T 2 /z

182 No but the Lorenz ratio is not a constant L H = apple xy T xy T T 2 /z Hartnoll/Karch =bz = 2/3

183 combine AC +DC transport fixes all exponents a,b,c

184 combine AC +DC transport fixes all exponents a,b,c current has anomalous dimension (probe with noise measurements)

185 what really is the summation over mass?

186 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

187 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

188 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

189 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

190 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

191 no gravity?

192 Cooper, et al.

193 Cooper, et al. low-energy model for N e >x??

194 Cooper, et al. low-energy model for N e >x?? does such a theory include gravity?

195 f-sum rule K.E. = p 2 /2m N e = x

196 what if? K.E. / (@ 2 µ) f-sum rule W (n, T ) e 2 = Acn 1+ 2( 1) d + B ( 1)(d + 2( 1))T 2 c n 1 2( +1) d

197 what if? K.E. / (@ 2 µ) f-sum rule W (n, T ) e 2 = Acn 1+ 2( 1) d + B ( 1)(d + 2( 1))T 2 c n 1 2( +1) d W>n if <1

198

199 what theories have <1?

200 gravitational crystals g µ! g µ + h µ 1 2 h µ 2m 2 h µ + m 2 / V ( )) graviton has a mass!

201 < 0 non-local action with noncanonical kinetic energy

202 incoherent metals: Mottness unparticles f-sum rule violation `massive gravity

203 emergent `massive gravity Mott problem