Optical Conductivity in the Cuprates from Holography and Unparticles
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1 Optical Conductivity in the Cuprates from Holography and Unparticles Thanks to: NSF, EFRC (DOE) Brandon Langley Garrett Vanacore Kridsangaphong Limtragool
2 Drude conductivity n e 2 1 m 1 i! (!) =C! 2 3
3 scale-invariant propagators 1 p 2
4 scale-invariant propagators 1 p 2 Anderson: use Luttinger Liquid propagators G R / 1 (! v s k)
5 compute conductivity without vertex corrections (PWA) (!) / 1! Z dx Z dtg e (x, t)g h (x, t)e i!t / (i!) 1+2
6 problems
7 problems 1.) cuprates are not 1- dimensional
8 problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter
9 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
10 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
11 power law?
12 power law? Could string theory be the answer?
13 UV QF T
14 UV IR?? QF T
15 IR?? UV QF T coupling constant g =1/ego
16 IR?? UV QF T coupling constant g =1/ego dg(e) dlne = (g(e)) locality in energy
17 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 R R / p g
18 replace coupled theory with geometry scale-invariant anti de-sitter g( ) 2 AdS 2 R 2 R R / p g AdS 4
19 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
20 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!
21 optical conductivity from a gravitational lattice log-log plots for various parameters for G. Horowitz, Santos, Tong, Journal of High Energy Physics, 2012
22 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, Santos, Tong, Journal of High Energy Physics, 2012
23 new equation! Einstein- Maxwell equations + non-uniform = charge density B! 2/3
24 not so fast!
25 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i!
26 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0
27 Donos and Gauntlett (gravitational crystal) Drude conductivity n e 2 m 1 1 i! 2 3 =1+! 00 0 B! 2/3
28 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!!
29 who is correct?
30 who is correct? let s redo the calculation
31 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q
32 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q A t = µ(1 z)dt =lim z!0 p gf tz
33 conductivity within AdS (g ab,v( ),A t ) (metric, potential, gaugefield) Q A t = µ(1 z)dt =lim z!0 p gf tz
34 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
35 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
36 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
37 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
38 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
39 HST vs. DG
40 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
41 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
42 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
43 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
44 Our Model L =(r 1 ) 2 +(r 2 ) 2 +2V ( 1 )+2V ( 2 )
45 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,.
46 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
47 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
48 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)).
49 Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)). reference metric
50 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.
51 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
52 =0 = 4 = 2
53 =0 = 4 = 2 translational invariance is broken in metric in multiples of 2k
54 charge density
55 conductivity - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form
56 is there a power law?
57 is there a power law? Results 2/3 DG HST
58 Results
59 Results A 1 =0.75,k 1 =2,k 2 =2, =0,µ=1.4, T/µ =0.115
60 No!
61 origin of power law?
62 origin of power law? phenomenology
63 origin of power law? phenomenology no well-defined mass L e = Z 1 0 L(x, m 2 )dm 2
64 origin of power law? phenomenology no well-defined mass L e = Z 1 0 L(x, m 2 )dm 2 incoherent stuff (all energies)
65 L µ (x, m)@ µ (x, m)+m 2 2 (x, m)
66 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 (m 2 )dm 2
67 Z 1 L µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 (m 2 )dm 2 theory with all possible mass!
68 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
69 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!!
70 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
71 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
72 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2
73 propagator Z 1 0 dm 2 m 2 p 2 i m 2 + i 1 / p 2 d U 2
74 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
75 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
76
77 take experiments seriously
78 take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i
79 take experiments seriously i (!) = n ie 2 i i m i 1 1 i! i continuous mass
80 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
81 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
82 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
83 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
84 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
85 a +2b 1 c = 1 3
86 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix
87 a +2b 1 c = 1 3 (!) = 0e M 1! 2 3 Z! 0 0 dx x ix! 0!1
88 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
89 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
90 experiments (!) =C! 2 e i (1 /2) =1.35
91 experiments (!) = 1 3 (p 3+3i) 0e M! 2 3 (!) =C! 2 e i (1 /2) =1.35
92 experiments (!) = 1 3 (p 3+3i) 0e M! 2 3 tan 2 / 1 = p 3 = 60 (!) =C! 2 e i (1 /2) =1.35
93 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
94 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
95 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
96 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
97 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
98 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
99 not necessarily
100 not necessarily but they are a possibility
101 what really is the summation over mass?
102 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
103 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
104 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
105 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
106 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
107 unparticles L e = Z 1 0 L(x, m 2 )dm 2
108 unparticles L e = Z 1 0 L(x, m 2 )dm 2 power-law optical conductivity
109 unparticles (flavors, incoherent stuff) action on AdS
110 emergent gravity Mott problem
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