Optical Conductivity in the Cuprates: from Mottness to Scale Invariance

Optical Conductivity in the Cuprates: from Mottness to Scale Invariance Thanks to: NSF, EFRC (DOE) Brandon Langley Garrett Vanacore Kridsangaphong Limtragool

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

excess carriers?

does the charge depend on the scale?

classical (atomic) limit

classical (atomic) limit number of empty sites=x

classical (atomic) limit number of empty sites=x N e / x

classical (atomic) limit 2 states number of empty sites=x N e / x

Mott insulator

Mott insulator U =

Mott insulator N U N U = PES IPES

Mott insulator N U N U = PES IPES no hopping:

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

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

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

atomic limit: x holes density of states 1-x 2x 1-x PES IPES

atomic limit: x holes density of states 1-x 2x 1-x PES IPES 1+x

atomic limit: x holes density of states 1-x 2x 1-x PES IPES 1+x 2

U finite U t c i n i µ U c i (1 n i )

U finite U t c i n i µ U c i (1 n i ) double occupancy in ground state!!

U finite U t c i n i µ U c i (1 n i ) double occupancy in ground state!! W PES > 1 + x

U finite U t c i n i µ U c i (1 n i ) N e 6=#x double occupancy in ground state!! W PES > 1 + x

why is this a problem?

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)

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

determination of N e Hubbard model x=0.1 d=1 x=0.2 x=0.3 x=0.4 x=0.5 x=0.6 x=0.7 x=0.8 x=0.9 Phys. Rev. Lett. 105, 106402 (2010) M. Kohno, PRL, 105, Sawatzky 106402

determination of N e chen/batlogg, 1990

DMFT on Hubbard model

DMFT on Hubbard model Millis, 2008 chakraborty & Phillips, 2007

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

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

problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter / G 2 µ µ }[ [G] =L d+1 2d U ]=L 3 d [ µ ]=L 2d U d [ µ ]=L 2d U d+1

problems 1.) cuprates are not 1- dimensional 2.) vertex corrections matter / G 2 µ µ }[ [G] =L d+1 2d U ]=L 3 d [ µ ]=L 2d U d [ µ ]=L 2d U d+1 independent of d U

power law?

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 R RG=GR R / p g

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

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

optical conductivity from a gravitational lattice G. Horowitz et al., Journal of High Energy Physics, 2012

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

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

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! no power law!!

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

who is correct?

who is correct? let s redo the calculation

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

model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2,

model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice

model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, action = gravity + EM + lattice S = 1 16 G N Z d 4 x p g R 2 12 F 2,

model RNAdS ds 2 = L 2 r 2 f r H r dr 2 + r2 L 2 f rh r dt 2 + dx 2 + dy 2, 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 ( )]

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

Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)).

Einstein-De Turck EOM G H ab = G ab r (a b), a = g cd ( a cd(g) a cd(g)). reference metric

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 2 1 2 z3.

Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x)

Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1

Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1 Newton-Raphson on grid

Dirchlet boundary conditions Q tt (0,x)=Q zz (0,x)=Q xx (0,x)=Q yy (0,x)=1 Q zx (0,x)=0 a t (0,x)=µ (0,x)= (1) (x) regularity at z=1 Newton-Raphson on grid z 2 [0, 1],x2 (0, 2 )

=0 = 4 = 2

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

charge density

perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i

perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0,

perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0, solve equations without mistakes!!

perturb with electric field g ab =ḡ ab + h ab A a = Āa + b a i = i + i gauge invariance g ab + L g ab =0, A a + L A a + r a =e i!t µ J x, +L =0, solve equations without mistakes!! Brandon Langley

dashed: RN-AdS solid: lattice - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form

sample conductivity plots dashed: RN-AdS solid: lattice - high-frequency behavior is identical - low-frequency RN has - low-frequency lattice has Drude form

Results

Results no power law...

Results

origin of power law?

origin of power law? phenomenology

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

use Ward-Takahashi identities to simplify vertices

use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 )

use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 ) response function to an electric field

use Ward-Takahashi identities to simplify vertices iq µ µ (p, q) =G 1 (p + q) G 1 (p) q 1µ µ (p, q 1,q 2 )= (p + q 1,q 2 ) (p, q 2 ) response function to an electric field K µ n, n 0 ( q, q 0 )= (2 )2d T 2 Z 1 2 A µ,n (q) A,n 0(q 0 ) A=0 Z[A]

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

last attempt 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

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

but they are possible!

but they are possible! continuous mass

but they are possible! continuous mass charge nonconservation

but they are possible! continuous mass charge nonconservation unparticles

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

action on AdS 5+2 S = 1 2 Z d 4+2 xdz p g @ a @ a + 2 R 2 p g =(R/z) 5+2

action on AdS 5+2 S = 1 2 Z d 4+2 xdz p g @ a @ a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2

action on AdS 5+2 S = 1 2 Z d 4+2 xdz p g @ a @ a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2 unparticle lives in d =4+2 apple 0

action on AdS 5+2 S = 1 2 Z d 4+2 xdz p g @ a @ a + 2 R 2 ds 2 = L2 z 2 µ dx µ dx + dz 2 p g =(R/z) 5+2 unparticle lives in d =4+2 apple 0 generating functional for unparticles

unparticles action on AdS

emergent gravity Mott problem