Unparticles and Emergent Mottness

Unparticles and Emergent Mottness Thanks to: NSF, EFRC (DOE) Kiaran dave Charlie Kane Brandon Langley J. A. Hutasoit

Correlated Electron Matter

Correlated Electron Matter What is carrying the current?

fractional quantum Hall effect current-carrying excitations e = e q (q odd)

fractional quantum Hall effect current-carrying excitations

fractional quantum Hall effect current-carrying excitations e =?

How Does Fermi Liquid Theory Breakdown? TRSB ρ at? LaSrCuO

can the charge density be given a particle interpretation?

can the charge density be given a particle interpretation? does charge density= total number of electrons?

if not?

if not? charge stuff= particles + other stuff

if not? charge stuff= particles + other stuff unparticles!

counting particles

counting particles 1

counting particles 2 1

counting particles 2 3 1

counting particles 7 8 5 6 3 2 1 4

counting particles 7 8 5 6 3 2 1 4 is there a more efficient way?

n = µ N(ω)dω µ IR UV

n = µ N(ω)dω µ IR UV can n be deduced entirely from the IR(lowenergy) scale?

Luttinger s `theorem

Luttinger s theorem n =2 k Θ(G(k,ω = 0))

Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) E>ε p ε p E E<ε p counting poles (qp)

Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing E>ε p ε p E ε p E E<ε p counting poles (qp)

Luttinger s theorem G(E) = 1 E ε p n =2 k Θ(G(k,ω = 0)) zero-crossing DetG(ω =0,p)=0 ε p E>ε p E ε p E E<ε p counting poles (qp)

singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) poles+zeros (all sign changes)

singularities of ln G n = 2i (2π) d+1 d d p 0 dξ ln GR (ξ,p) G R (ξ,p) n =2 k Θ(G(k,ω = 0)) poles+zeros (all sign changes)

what are zeros?

what are zeros? Im G=0 µ =0

what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig

what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap

what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0

what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0 DetG(k,ω = 0) =0 (single band)

what are zeros? Im G=0 µ =0 ReG(0,p)= ( ) dω ω Kramers -Kronig = below gap+above gap = 0 DetG(k,ω = 0) =0 (single band) strongly correlated gapped systems

NiO insulates d 8? Mott mechanism (not Slater) t

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0

NiO insulates d 8? perhaps this costs energy Mott mechanism (not Slater) t U t µ =0 no change in size of Brillouin zone DetReG(0,p)=0 =Mottness

simple problem: n=1 SU(2) U

simple problem: n=1 SU(2) µ U

simple problem: n=1 SU(2) µ U U 2

simple problem: n=1 SU(2) U 2 µ U U 2

simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2

simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 =0 if ω =0

simple problem: n=1 SU(2) U 2 µ U U 2 G = 1 ω + U/2 + 1 ω U/2 n =2θ(0) = 1 =0 if ω =0

Fermi Liquids Mott Insulators Luttinger s theorem

Is this famous theorem from 1960 correct?

A model with zeros but Luttinger fails

A model with zeros but Luttinger fails e t t t t t t t t

A model with zeros but Luttinger fails e t t t t t t t t no hopping=> no propagation (zeros)

A model with zeros but Luttinger fails N flavors e t t t t t of e- t t t no hopping=> no propagation (zeros)

SU(N) H = U 2 (n 1 + n N ) 2 2E 25 N =5 16 4 9 1

SU(N) for N=3: no particle-hole symmetry

SU(N) for N=3: no particle-hole symmetry lim µ(t ) T 0

compute Green function exactly (Lehman formula) G αβ (ω) = 1 Z ab exp βk a Q ab αβ Q ab αβ = a c α bb c β a ω K b + K a + a c β bb c α a ω K a + K b do sum explicitly

G αβ (ω = 0) = δ αβ K(n + 1) K(n) 2n N N

G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

< 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

< 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

a zero must exist < 0 > 0 G αβ (ω = 0) = δ αβ K(n + }> 1) K(n) 0 2n N N

Luttinger s theorem n = NΘ(2n N)

Luttinger s theorem n = NΘ(2n N) { 0, 1, 1/2

Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2

Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 2=3

Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3

Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd

Luttinger s theorem n = NΘ(2n N) n =2 N =3 { 0, 1, 1/2 even 2=3 odd no solution

does the degeneracy matter? e t t t t t t t t t =0 +

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω H c a ρ(0 + )

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω H c a ρ(0 + )

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + )

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) G ab (ω) = ωδ ab Uρ ab ω(ω U)

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U)

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place

G ab (ω) =Tr c a 1 ω H c 1 b + c b ω ω HU c a ρ(0 + ) ρ ab =Tr c ac b ρ(0 + ) = u 0 c ac b u 0 G ab (ω) = ωδ ab Uρ ab ω(ω U) 1/N diag(1, 1, 1, ) mixed state as long as SU(N) symmetry is intact zeros in the wrong place limiting procedure: lim lim t 0 ω 0

Problem DetG =0 G = 1 E ε p =0??

Problem DetG =0 G = 1 E ε p =0?? G = 1 E ε p Σ

Problem DetG =0 G = 1 E ε p =0?? G = 1 E ε p Σ G=0

if Σ is infinite

if Σ is infinite lifetime of a particle vanishes

if Σ is infinite lifetime of a particle Σ < p vanishes

if Σ is infinite lifetime of a particle Σ < p vanishes no particle

what went wrong?

what went wrong? δi[g] = dωσδg

what went wrong? δi[g] = dωσδg if Σ

what went wrong? δi[g] = dωσδg if Σ integral does not exist

what went wrong? δi[g] = dωσδg if Σ integral does not exist No Luttinger theorem!

zeros and particles are incompatible

Luttinger s theorem

are zeros important?

Fermi Arcs

Fermi Arcs Re G Changes Sign across An arc

Fermi Arcs Re G Changes Sign across An arc Must cross A zero line (DetG=0)!!! Fermi arcs necessarily imply zeros exist.

what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS)

what is seen experimentally? Fermi arcs: no double crossings (PDJ,JCC,ZXS) seen infinities not seen zeros E F k

experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density

experimental data (LSCO) `Luttinger count k F 1 x FS Bi2212 zeros do not affect the particle density each hole = a single k-state

where do these zeros come from?

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 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) density of states 1+x + α>1+x U t Harris and Lange (1967)

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

spectral function (dynamics) density of states 1+x + α>1+x α t/u U t 1 x α Harris and Lange (1967) not exhausted by counting electrons alone

how to count particles? 7 8 5 6 3 2 1 4 some charged stuff has no particle interpretation

what is the extra stuff?

scale invariance

new fixed point (scale invariance)?

new fixed point (scale invariance)

new fixed point (scale invariance) unparticles (IR) (H. Georgi)

what is scale invariance? invariance on all length scales

spectral function from scale invariance A(Λω, Λ α k k)=λ α A A(ω, k)

spectral function from scale invariance A(Λω, Λ α k k)=λ α A A(ω, k) k A(ω, k)=ω α A f ω α k

what s the underlying theory?

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

massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x)

massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 x x/λ φ(x) φ(x) m 2 φ 2

massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ 2 1 2 µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2

massive free theory mass L = 1 2 µ φ µ φ + m 2 φ 2 Λ 2 1 2 µ φ µ φ x x/λ φ(x) φ(x) m 2 φ 2 no scale invariance

mass sets a scale

unparticles from a massive theory?

L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m)

L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2

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

L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2

L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!!

L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

unparticles L = µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 dm 2 theory with all possible mass! φ φ(x, m 2 /Λ 2 ) x x/λ m 2 /Λ 2 m 2 L Λ 4 L scale invariance is restored!! not particles

propagator 0 dm 2 m 2γ i p 2 m 2 + i 1 p 2 γ

propagator 0 dm 2 m 2γ i p 2 m 2 + i 1 p 2 γ d U 2

n massless particles G (E ε p ) n 2

n massless particles G (E ε p ) n 2 G (E ε p ) d U 2

n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles

n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U < 2 almost Luttinger liquid no pole at p_f

n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U > 2 d U < 2 zeros almost Luttinger liquid no pole at p_f

n massless particles G (E ε p ) n 2 G (E ε p ) d U 2 unparticles=fractional number of massless particles d U > 2 d U < 2 zeros no simple sign change almost Luttinger liquid no pole at p_f

no particle interpretation mass-distribution φ U = B(m 2 )φ(m 2 )dm 2 unparticle field (non-canonical) particle field

d U?

what really is the summation over mass?

what really is the summation over mass? mass=energy

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 can be absorbed with

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 can be absorbed with ds 2 = R2 z 2 ηµν dx µ dx ν + dz 2

L = summation over mass? µ φ(x, m) µ φ(x, m)+m 2 φ 2 (x, m) 0 m 2δ dm 2 m = z 1 L = 0 dz 2R2 z 5+2δ 1 2 z 2 R 2 ηµν ( µ φ)( ν φ)+ φ2 2R 2 anti de Sitter can be absorbed with space! ds 2 = R2 z 2 ηµν dx µ dx ν + dz 2

generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2

generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2 ds 2 = L2 z 2 ηµν dx µ dx ν + dz 2 g =(R/z) 5+2δ

generating functional for unparticles action on AdS 5+2δ a Φ a Φ+ Φ2 S = 1 2 d 4+2δ xdz g R 2 ds 2 = L2 z 2 ηµν dx µ dx ν + dz 2 g =(R/z) 5+2δ unparticle lives in d =4+2δ δ 0

gauge-gravity duality (Maldacena, 1997) UV IR gravity QF T

gauge-gravity duality (Maldacena, 1997) UV IR gravity IR?? QF T

gauge-gravity duality (Maldacena, 1997) UV IR gravity IR?? QF T dg(e) dlne = β(g(e)) locality in energy

gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity IR?? QF T dg(e) dlne = β(g(e)) locality in energy

Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO ))

Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z=

Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U

Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U d U = d 2 + d2 +4 2 > d 2

Claim: Z QFT = e Son shell ADS (φ(φ ADS=JO )) S = 1 2 d d xg zz g Φ(z,x) z Φ(z,x) z= Φ U (x)φ U (x ) = 1 x x 2d U d U = d 2 + d2 +4 2 > d 2 scaling dimension is fixed

G U (p) p 2(d U d/2) d U = d 2 + d2 +4 2 > d 2 G U (0) = 0

unparticle (AdS) propagator has zeros! G U (p) p 2(d U d/2) d U = d 2 + d2 +4 2 > d 2 G U (0) = 0

interchanging unparticles fractional (d_u) number of massless particles

interchanging unparticles fractional (d_u) number of massless particles e iπd U = 1, 0

interchanging unparticles fractional (d_u) number of massless particles e iπd U = 1, 0 fractional statistics in d=2+1

d U = d 2 + d2 +4 2 > d 2 e iπd U = e iπd U time-reversal symmetry breaking from unparticle (zeros=fermi arcs) matter

d ln g d ln β = 4d U d>0 fermions T c unparticles with unparticle propagator BCS g tendency towards pairing (any instability which establishes a gap)

TRSB

TRSB unparticles

TRSB unparticles particles

breaking of scale invariance TRSB unparticles particles

emergent gravity High T_c unparticles variable mass

emergent gravity High T_c unparticles variable mass fractional statistics in d=2+1