Optical Conductivity in the Cuprates from holography and unparticles

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

Properties all particles share? charge fixed mass!

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

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

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

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

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

unconventional superconductivity? / at

unconventional superconductivity? unfermi liquid (unmetal) / at

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

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

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

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

Uchida, et al. Cooper, et al.

Uchida, et al. Cooper, et al. x

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

excess carriers?

excess carriers? charge carriers have no particle content?

Mott insulator 1-x

Mott insulator U = 1-x

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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)

why is this a problem?

spectral function (dynamics)

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

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

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

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

is there anything else?

yes Drude conductivity n e 2 1 m 1 i!

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

YBCO thin films

criticality scale invariance power law correlations

scale-invariant propagators 1 p 2

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

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

problems

problems 1.) cuprates are not 1- dimensional

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

/ 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

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

1997 Anderson

1997 Anderson 2012 string theory AdS/CFT

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

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

power law? Could string theory be the answer?

UV QF T

UV IR?? QF T

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

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

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

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

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

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!

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

not so fast!

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

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

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

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!!

who is correct?

who is correct? let s redo the calculation

model

model action = gravity + EM + lattice

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

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

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

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

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

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

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

HST vs. DG

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

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

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

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

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

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,.

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

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

=0 = 4 = 2

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

charge density

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

is there a power law?

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

Results

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

No!

Anderson AdS/CFT

scale invariance Anderson AdS/CFT

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

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

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)

massive free theory mass L = 1 2 @µ @ µ + m 2 2

massive free theory mass L = 1 2 @µ @ µ + m 2 2 x! x/ (x)! (x)

massive free theory mass L = 1 2 @µ @ µ + m 2 2 x! x/ (x)! (x) m 2 2

massive free theory mass L = 1 2 @µ @ µ + m 2 2 2 1 2 @µ @ µ x! x/ (x)! (x) m 2 2

massive free theory mass L = 1 2 @µ @ µ + m 2 2 2 1 2 @µ @ µ x! x/ (x)! (x) m 2 2 no scale invariance

L = @ µ (x, m)@ µ (x, m)+m 2 2 (x, m)

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

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

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

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!!

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

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

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

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

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

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

use unparticle propagators to calculate conductivity

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

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

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

gauge unparticles

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

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

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

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

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 )

what went wrong?

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

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

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

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

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 )

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)

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

last attempt

last attempt take experiments seriously

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

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

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

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

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 2 0 0 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

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 2 0 0 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

a +2b 1 c = 1 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

what really is the summation over mass?

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

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

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

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

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

no gravity?

Cooper, et al.

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

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

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

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

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

what theories have <1?

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

< 0 non-local action with noncanonical kinetic energy

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

emergent `massive gravity Mott problem