Detecting unparticles and anomalous dimensions in the Strange Metal

Transcription

1 Detecting unparticles and anomalous dimensions in the Strange Metal Thanks to: NSF, EFRC (DOE) Andreas Karch Brandon Langley Garrett Vanacore Kridsangaphong Limtragool Gabriele La Nave

2 Drude metal

3 Drude metal ṗ = e(e + p B m ) p momentum relaxation

4 Drude metal ṗ = e(e + p B m ) p momentum relaxation (!) = 0 1 i!

5 Drude metal ṗ = e(e + p B m ) p momentum relaxation (!) = 0 1 i! lim <!1!0

6 Drude metal ṗ = e(e + p B m ) p momentum relaxation (!) = 0 1 i! lim <!1!0 µ T µi = 1 T ti

7 Drude metal ṗ = e(e + p B m ) p momentum relaxation (!) = 0 1 i! lim <!1!0 µ T µi = 1 T ti massive graviton

8 standard metals resistivity / T 2 Weidemann-Franz law apple xx = 2 T xx 3 optical conductivity < / 1/! 2

9 standard metals resistivity / T 2 Weidemann-Franz law apple xx = 2 T xx 3 optical conductivity < / 1/! 2

10 strange metal: experimental facts T-linear resistivity

11 strange metal: experimental facts T-linear resistivity (!) =C! 2 3 n e 2 m 1 1 i!

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

13 / at

14 is the strange metal important? / at

15

16 Theories of cuprates Theories of strange metal = 1 = en O(0)

17 Theories of cuprates Theories of strange metal = 1 = en O(0)

18 Theories of cuprates Theories of strange metal = 1 = en O(0)

19 Theories of cuprates Theories of strange metal = 1 = en O(0) why is the problem hard?

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

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

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

23 single-parameter scaling / z 1 / T (2 d)/z (!, T) /! (d 2)/z 2/3 C v / T d/z anomalous dimension 2! 2d A

24 single-parameter scaling / z 1 / T (2 d)/z (!, T) /! (d 2)/z 2/3 C v / T d/z anomalous dimension 2! 2d A

25 new length scale?

26 strange metal

27 strange metal multi-scale sector

28 strange metal multi-scale sector [J] =

29 strange metal multi-scale sector [J] = fractional not d-1

30 strange metal multi-scale sector [J] = fractional not d-1 [A µ ]=d A 6=1

31 strange metal multi-scale sector [J] = fractional not d-1 [A µ ]=d A 6=1 probe by fractional Aharonov-Bohm effect

32 optical conductivity

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

34 N e ( ) = 2mV cell e 2 Uchida, et al. Z 0 (!)d! Cooper, et al. x

35 N e ( ) = 2mV cell e 2 Uchida, et al. Z 0 (!)d! Cooper, et al. x low-energy model for N e >x??

36 Drude conductivity n e 2 m 1 1 i! (!) =C! 2 e i (1 /2) =1.35

37 Drude conductivity n e 2 m 1 1 i! (!) =C! 2 e i (1 /2) =1.35

38 Drude conductivity n e 2 m 1 1 i! (!) =C! 2 e i (1 /2) =1.35 tan 2 / 1 = p 3 = 60

39 criticality scale invariance power law correlations

40 criticality scale invariance power law correlations

41 criticality scale invariance power law correlations

42 Anderson: use Luttinger Liquid propagators G R / 1 (! v s k) (!) / 1! Z dx Z dtg e (x, t)g h (x, t)e i!t / (i!) 1+2

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

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

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

46 1997 Anderson

47 1997 Anderson 2012 string theory AdS/CFT

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

49 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

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

51 not so fast!

52 Donos and Gauntlett (Q-lattice) Drude conductivity n e 2 m 1 1 i!

53 Donos and Gauntlett (Q-lattice) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0

54 Donos and Gauntlett (Q-lattice) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0 B! 2/3

55 Donos and Gauntlett (Q-lattice) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0 B! 2/3 no power law!!

56 who is correct?

57 who is correct? let s redo the calculation

58 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 ( )]

59 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 ( )]

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

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

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

63 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

64 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

65 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

66 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

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 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

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 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

69 HST vs. DG

70 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

71 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

72 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

73 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

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

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

76 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

77 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

78 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

79 charge density

80 is there a power law?

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

82 is there a power law? Results k=2 2/3 k=1 DG HST

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

84 similar results: Rangamani, Rozali, Smyth arxiv:

85 No

86 Anderson AdS/CFT

87 scale invariance Anderson AdS/CFT

88 beyond single-parameter scaling multiple scale sector incoherent stuff (all energies)

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

90 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)

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

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

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

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

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

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

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

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

99 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

100 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, 2015 multi-bands

101

102 take experiments seriously

103 take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i

104 take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i continuous mass

105 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

106 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

107 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 (!) = 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

108 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 (!) = 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

109 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 (!) = 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

110 a +2b 1 c = 1 3

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

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

113 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

114 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

115 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

116 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

117 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

118 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

119 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

120 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

121 No

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

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

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

125 gauge invariance A µ!a µ µ G has no units [A µ ]= 1 L =1 if [A µ ] 6= 1

126 gauge invariance A µ!a µ µ G has no units [A µ ]= 1 L =1 if [A µ ] 6= 1

127 How can A µ have an anomalous dimension?

128 How can A µ have an anomalous dimension? solution A µ!a µ µ G

129 How can A µ have an anomalous dimension? solution A µ!a µ µ G F µ µ µ µ A µ

130 How can A µ have an anomalous dimension? solution A µ!a µ µ G F µ µ µ µ A µ ~r ~ A = ~ B

131 How can A µ have an anomalous dimension? solution A µ!a µ µ G F µ µ µ µ A µ ~r ~ A = ~ B no Stokes theorem I ~A d` 6= Z S B d ~ S

132 Aharonov-Bohm Effect must change flux through ring = e ~ I ~A d` = eb r2 ~ I Stokes theorem ~A d` = Z S B d ~ S

133 Aharonov-Bohm Effect must change flux through ring = e ~ I ~A d` = eb r2 ~ I Stokes theorem ~A d` = Z S B d ~ S

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

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

136 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

137 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

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

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

140 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

141 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

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

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

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

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

146 what is the mechanism for the anomalous dimension?

147 what is the mechanism for the anomalous dimension? is the nonlocality real?

148 bulk-boundary operator correspondence operators O (@ µ ) 2 + m 2 2 IR UV QF T e R d d x O O

149 bulk-boundary operator correspondence operators O (@ µ ) 2 + m 2 2 IR UV QF T e R d d x O O

150 bulk-boundary operator correspondence operators O (@ µ ) 2 + m 2 2 IR UV QF T e R d d x O O smearing function

151 bulk-boundary operator correspondence operators O (@ µ ) 2 + m 2 2 IR UV QF T e R d d x O O see also K. Rehren (2000) smearing function

152 O = C O lim z!0 z (x, z)

153 O = C O lim z!0 z (x, lim za z!0 = C d, ( r) f = 1 a 2 use Caffarelli/ Silvestre

154 O = C O lim z!0 z (x, lim za z!0 = C d, ( r) f = 1 a 2 use Caffarelli/ Silvestre O =( ) 0 the O for massive scalar field = p 4m 2 + d 2 /2

155 mechanism: fractional gauge fields

156 mechanism: fractional gauge fields F µ F µ + m 2 A 2 y dynamical `Higgs mode

157 mechanism: fractional gauge fields F µ F µ + m 2 A 2 y A? µ A?µ dynamical `Higgs mode

158 mechanism: fractional gauge fields F µ F µ + m 2 A 2 y A? µ A?µ dynamical `Higgs mode new length scale

159 N e ( ) = 2mV cell e 2 Uchida, et al. Z 0 (!)d! Cooper, et al. x low-energy model for N e >x??

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

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

162

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

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

165 combine AC +DC transport fixes all exponents a,b,c boundary non-local action probe with fractional Aharonov-Bohm effect [J] =d U