Testing Mottness/unparticles and anomalous dimensions in the Cuprates

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1 Testing Mottness/unparticles and anomalous dimensions in the Cuprates Thanks to: NSF, EFRC (DOE) Brandon Langley Garrett Vanacore Kridsangaphong Limtragool

2 / at

3 what is strange about the strange metal? / at

4 experimental facts T-linear resistivity

5 experimental facts T-linear resistivity (!) =C! 2 3 n e 2 m 1 1 i!

6 experimental facts T-linear resistivity L xy = apple xy /T xy 6=#/ T (!) =C! 2 3 n e 2 m 1 1 i!

7 strange metal

8 strange metal multi-scale sector (unparticles) [J] =

9 strange metal multi-scale sector (unparticles) [J] = fractional not d-1

10 strange metal multi-scale sector (unparticles) [J] = fractional not d-1 probe by Aharonov-Bohm effect on underdoped cuprates

11 optical conductivity

12

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

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

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

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

17 Uchida, et al. Cooper, et al. x

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

19 excess carriers?

20 excess carriers? charge carriers have no particle content?

21 dynamics are not exhausted by counting electrons alone

22 dynamics are not exhausted by counting electrons alone what s the extra stuff?

23 is there anything else?

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

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

26 criticality scale invariance power law correlations

27 criticality scale invariance power law correlations

28 criticality scale invariance power law correlations

29 scale-invariant propagators 1 p 2

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

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

32 problems

33 problems 1.) cuprates are not 1- dimensional

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

35 / 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

36 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

37 1997 Anderson

38 1997 Anderson 2012 string theory AdS/CFT

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

40 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

41 power law? Could string theory be the answer?

42 power law? Could string theory be the answer? momentum relaxation??

43 UV QF T

44 UV IR?? QF T

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

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

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

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

49 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

50 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!

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

52 not so fast!

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

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

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

56 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!!

57 who is correct?

58 who is correct? let s redo the calculation

59 model

60 model action = gravity + EM + lattice

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

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

63 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 ( )]

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

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

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

67 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

68 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

69 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

70 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

71 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

72 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

73 gravitational crystals g µ! g µ + h µ

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

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

76 gravitational crystals g µ! g µ + h µ 1 2 h µ 2m 2 h µ + m 2 / V ( )) graviton has a mass! momentum t T ti 6=0= 1 T ti

77 gravitational crystals g µ! g µ + h µ 1 2 h µ 2m 2 h µ + m 2 / V ( )) graviton has a mass! momentum t T ti 6=0= 1 T ti in Drude formula

78 HST vs. DG Horowitz, Santos, Tong (HST) DG V ( ) = 2 /L 2 V 2 (x) =A 0 cos(kx) inhomogeneous in x (z,x) = (z)e ikx no inhomogeneity in x

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

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

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

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

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

84 charge density

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

86 is there a power law?

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

88 Results

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

90 No

91 Anderson AdS/CFT

92 scale invariance Anderson AdS/CFT

93 single-parameter scaling / z / T (2 d)/z (!, T) /! (d 2)/z C v / T d/z

94 single-parameter scaling / z 1 / T (2 d)/z (!, T) /! (d 2)/z C v / T d/z

95 single-parameter scaling / z 1 / T (2 d)/z (!, T) /! (d 2)/z 2/3 C v / T d/z

96 beyond single-parameter scaling multiple scale sector

97 beyond single-parameter scaling multiple scale sector unparticles no well-defined mass incoherent stuff (all energies)

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

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

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

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

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

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

104 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)

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

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

107 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

108 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!!

109 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

110 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

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

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

113 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

114 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

115 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

116 use unparticle propagators to calculate conductivity

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

118 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

119 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

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

121 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 )

122 what went wrong?

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

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

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

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

127 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

128 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 )

129 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)

130 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

131 last attempt

132 last attempt take experiments seriously

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

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

135 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

136 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

137 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

138 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

139 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

140 a +2b 1 c = 1 3

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

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

143 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

144 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

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

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

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

148 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

149 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

150 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

151 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

152 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 momentum loss

153 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 momentum loss c =1 a +2b =2/3

154 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 momentum loss b =0 a =2/3

155 No

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

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

158 Cooper, et al.

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

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

161 what if? K.E. / (@ 2 µ) f-sum rule W (n, T ) ce 2 = An 1+ 2( 1) d +

162 what if? K.E. / (@ 2 µ) f-sum rule W (n, T ) ce 2 = An 1+ 2( 1) d + W>n if <1

163

164 what theories have <1?

165 < 0 non-local action with noncanonical kinetic energy

166 vector potential has an anomalous dimension

167 Detecting Anomalous dimension of the Current

168 Detecting Anomalous dimension of the Current flux through ring = e ~ I ~A d` = eb r2 ~

169 Detecting Anomalous dimension of the Current Aharonov-Bohm Effect flux through ring = e ~ I ~A d` = eb r2 ~

170 Detecting Anomalous dimension of the Current Aharonov-Bohm Effect flux through ring = e ~ I ~A d` = eb r2 ~ I Stokes theorem ~A d` = Z S B d ~ S

171 what if B has an anomalous dimension? [B] =L 2 6= 1 F (L, )?

172 what if B has an anomalous dimension? [B] =L 2 6= 1 = eb r2 ~ is no longer dimensionless F (L, )?

173 what if B has an anomalous dimension? [B] =L 2 6= 1 = eb r2 ~ is no longer dimensionless = eb r2 ~ F (L, ) non-locality of the current F (L, )?

174 gauge invariance A µ!a µ µ G field strength F µ =2@ [µ A ],

175 gauge invariance A µ!a µ µ G A µ!a µ µ G field strength F µ =2@ [µ A ], F µ µ µ A µ

176 gauge invariance A µ!a µ µ G A µ!a µ µ G field strength F µ =2@ [µ A ], F µ µ µ A µ ~r ~J t =0 continuity eq.

177 gauge invariance A µ!a µ µ G A µ!a µ µ G field strength F µ =2@ [µ A ], F µ µ µ A µ ~r ~J t =0 continuity eq. ~r ~ A = ~ B

178 gauge invariance A µ!a µ µ G A µ!a µ µ G field strength F µ =2@ [µ A ], F µ µ µ A µ ~r ~J t =0 continuity eq. ~r ~ A = ~ B no Stokes theorem I ~A d` 6= Z S B d ~ S

179 physical gauge connection a i [@ i,i i A i ]=@ i I i A i compute AB phase

180 physical gauge connection a i [@ i,i i A i ]=@ i I i A i A µ! A µ µ compute AB phase

181 physical gauge connection a i [@ i,i i A i ]=@ i I i A i A µ! A µ µ a µ! a µ µ compute AB phase

182 physical gauge connection a i [@ i,i i A i ]=@ i I i A i A µ! A µ µ ~ 2 a µ! a µ µ 2m (@ i i e ~ a i) 2 = i~@ t. compute AB phase

183 compute AB phase I = e ~a (r 0 ) ~d`0 ~

184 compute AB phase I = e ~a (r 0 ) ~d`0 ~

185 compute AB phase I = e ~a (r 0 ) ~d`0 ~ use fractional calculus R = eb`2 ~ b 1 d 1 2 ( ) c l, d l

186 compute AB phase I = e ~a (r 0 ) ~d`0 ~ use fractional calculus R = eb`2 ~ b 1 d 1 2 ( ) c l, d l

187 D = e p 2 ~ r2 BR (2 ) (1 2 )! ( ) ( ) sin F 1 (1, 2 ; 2; r2 R 2 )

188 D = e p 2 ~ r2 BR (2 ) (1 2 )! ( ) ( ) sin F 1 (1, 2 ; 2; r2 R 2 )

189 is the correction large? =1+2/3 =5/3

190 is the correction large? =1+2/3 =5/3 R = eb`2 ~ L 5/3 /(0.43) 2 yes!

191 combine AC +DC transport fixes all exponents a,b,c [J] =d U

192 combine AC +DC transport fixes all exponents a,b,c probe with Aharonov-Bohm effect [J] =d U