Unparticles in High T_c Superconductors

Unparticles in High T_c Superconductors 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 / 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?

µ IR 1 UV

n = Z µ 1 N(!)d! µ IR 1 UV

n = Z µ 1 N(!)d! µ IR 1 UV can n be deduced entirely from the IR(lowenergy) scale?

Luttinger s `theorem

Each particle has an energy

Each particle has an energy particles energy E = " p

Each particle has an energy particles energy E = " p G(E) = E 1 " p

Each particle has an energy particles energy E = " p G(E) = E 1 " p particle=infinity

Each particle has an energy particles energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity

Each particle has an energy particles particle count is deducible from Green function energy E = " p Green function (propagator) 1 G(E) = E " p particle=infinity

Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) " p E

Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E

Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p

Luttinger s Theorem Green function (propagator) G(E) = E 1 " p G(E) E>" p " p E E<" p particle density= number of sign changes of G

Counting sign changes?

Counting sign changes? (x)

Counting sign changes? (x) = { 0 x<0

Counting sign changes? (x) = { 0 x<0 1 x>0

Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0

Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes

Counting sign changes? (x) = { 0 x<0 1 x>0 1/2 x =0 counts sign changes n =2 X k (<G(k,! = 0)) Luttinger Theorem for electrons

How do functions change sign? E>" p " p E E<" p

How do functions change sign? E>" p " p E E<" p divergence

Is there another way?

Yes

zero-crossing " p E DetG = 0

zero-crossing " p E DetG = 0 no divergence is necessary

closer look at Luttinger s theorem n =2 X k (<G(k,! = 0))

closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros

closer look at Luttinger s theorem n =2 X k (<G(k,! = 0)) divergences+ zeros how can zeros affect the particle count?

what are zeros?

what are zeros? Im G=0 µ =0

what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig

what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap

what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0

what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 0) =0 (single band)

what are zeros? Im G=0 Z 1 µ =0 ReG(0,p)= 1 ( ) d!! Kramers -Kronig = below gap+above gap = 0 DetG(k,! = 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

U/t = 10 1 interactions dominate: Strong Coupling Physics

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! + U/2 + 1! U/2

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

simple problem: n=1 SU(2) U 2 µ U U 2 G = 1! + U/2 + 1! U/2 n =2 (0) = 1 =0 if! =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 e N flavors of e- t t t t t 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

G (! = 0) = # 2n N N

Luttinger s theorem n = N (2n N)

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

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

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

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

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

Luttinger s theorem n = N (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 +

No as long as SU(N) symmetry is intact ab =Tr c ac b (0 + ) =1/N diag(1, 1, 1,...) mixed state pure state

Problem DetG =0 G = 1 E " p =0??

Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1

Problem DetG =0 G = 1 E " p =0?? G = 1 E " p 1 G=0

if is infinite

if is infinite lifetime of a particle vanishes

if is infinite lifetime of a particle = < p vanishes

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

what went wrong?

what went wrong? I[G] = Z d! G

what went wrong? I[G] = Z d! G if!1

what went wrong? I[G] = Z d! G if!1 integral does not exist

what went wrong? I[G] = Z d! G if!1 integral does not exist No Luttinger theorem!

zeros and particles are incompatible

Luttinger s theorem

are zeros important?

photoemission

photoemission particles energy E = " p

photoemission particles energy E = " p p 2

photoemission particles energy E = " p p 2 two crossings: closed surface of excitations

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

Problem poles ReG>0 Re G<0

Problem How to account for the sign change without poles? poles ReG>0 Re G<0

Problem How to account for the sign change without poles? poles ReG>0 Re G<0 Only option: DetG=0! (zeros )

Problem How to account for the sign change without poles? poles 00000 Re G<0 0 0 00 ReG>0 0 0 0 0 0000000 Only option: DetG=0! (zeros )

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

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

f(x) =x 2

f(x) =x 2 f(x/ )=(x/ ) 2 scale change

f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance

f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{

f(x) =x 2 f(x/ )=(x/ ) 2 scale change scale invariance f(x) =x 2 2 g( ) 1{ g( )= 2

what s the underlying theory?

free field theory L = 1 2 @µ @ µ x! x/ (x)! (x) L! 2 L

free field theory L = 1 2 @µ @ µ x! x/ (x)! (x) L! 2 L scale invariant

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

mass sets a scale

unparticles from a massive theory?

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

n massless particles G / (E " p ) n 2

n massless particles G / (E " p ) n 2 G / (E " p ) d U 2

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

n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros

n massless particles G / (E " p ) n 2 G / (E " p ) d U 2 unparticles=fractional number of massless particles d U > 2 zeros no simple sign change

no particle interpretation mass-distribution U = Z B(m 2 ) (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

high energy (UV) QFT low energy (IR) related to sum over mass

high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass

implement E-scaling with an extra dimension high energy (UV) QFT low energy (IR) dg(e) = (g(e)) dlne locality in energy related to sum over mass

implement E-scaling with an extra dimension QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass

gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension UV IR gravity QF T QFT dg(e) dlne = (g(e)) locality in energy related to sum over mass

gauge-gravity duality (Maldacena, 1997) implement E-scaling with an extra dimension IR gravity UV no particles QFT (conserved currents) QF T dg(e) dlne = (g(e)) locality in energy related to sum over mass

what s the geometry?

what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous)

what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ /

what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations

what s the geometry? dg(e) dlne = (g(e)) =0 scale invariance (continuous) E! E x µ! x µ / solve Einstein equations ds 2 = u L 2 µ dx µ dx + L E 2 de 2 anti de-sitter space

mass=extra dimension fixed by metric

can something more exact be done?

can something more exact be done? Z 1 L = @ µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2

can something more exact be done? Z 1 L = @ µ (x, m)@ µ (x, m)+m 2 2 (x, m) 0 m 2 dm 2 m = z 1

can something more exact be done? 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 something more exact be done? 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

generating functional for unparticles action on AdS 5+2 S = 1 2 Z d 4+2 xdz p g @ a @ a + 2 R 2

generating functional for unparticles 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

generating functional for unparticles 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

scaling dimension is fixed m = 1 z

scaling dimension is fixed m = 1 z m 2 AdS = d U (d U d) R 2

scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2

scaling dimension is fixed m = 1 z 1=d m 2 AdS = d U (d U d) U (d U d) R 2 d U = d 2 + p d2 +4 2

G U (p) / p 2(d U d/2) G U (0) = 0

unparticle (AdS) propagator has zeros! G U (p) / p 2(d U 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 i d U 6= 1, 0

interchanging unparticles fractional (d_u) number of massless particles e i d U 6= 1, 0 fractional statistics in d=2+1

d U = d 2 + p d2 +4 2 > d 2 e i d U 6= e i d U time-reversal symmetry breaking from unparticle (zeros=fermi arcs) matter

d ln g d ln = 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