Unparticles and Superconductivity

Kiaran dave Charlie Kane Brandon LangleyBrandon Langley

Thanks to: NSF, EFRC (DOE)

J. A. HutasoitFriday, March 14, 14

Properties all particles share?

Friday, March 14, 14

fixed mass!charge

conserved(gauge

invariance)

fixed mass!charge

conserved(gauge

invariance)not conserved

fixed mass!charge

mass sets a scale

particles

E = "penergy

particles

E = "penergy

G(E) =1

E � "p

particles

E = "penergy

G(E) =1

E � "p

particle=infinity

particles

E = "penergy

Green function (propagator)

G(E) =1

E � "p

particle=infinity

particles

E = "penergy

G(E) =1

E � "p

particle=infinity

=poles in Green function

what would you see in a metal?

e�e�

two crossings:closed surface of excitations

expect to see

closed surface

are there any exceptions?

where is this part?

Fermi arcs: (PDJ,JCC,ZXS)

Fermi Arcs

where are arcs seen?

ceramics

normalstate

what’s the explanation?

strange not strange

Two opposing Viewson Fermi arcs

strange not strange

this dispute has an answer!

strange not strange

this dispute has an answer!unparticles

Right Wingers: Fermi Arcs are not Strange

intensity too small to be seen

G(E) =Zp

! � "p

Zp ! 0pole (particle) exists but

Fermi Arcs are Strange

seen not seen

no pole

Re G<0

Problem for left-wingers (me)

Re G<0

How to account for the sign change without poles?

Fermi arcs

E � ✏p �1 = 0

pole no pole

do poles account for all the charged stuff?

if not?

charge stuff= particles + other stuff

if not?

charge stuff= particles + other stuff

unparticles!

counting particles

is there a more efficient way?

Each particle has an energy

particles

E = "penergy

G(E) =1

E � "p

particle=infinity

Each particle has an energy

particles

E = "penergy

G(E) =1

E � "p

particle=infinity

particle count is deduciblefrom Green function

Luttinger’s Theorem

Green function (propagator) G(E) =1

E � "p

E > "p

E � "p

E > "p

E < "p

E � "p

E > "p

E < "p

particle density= number of sign changes of G

Counting sign changes?

⇥(x)

⇥(x) ={ 0 x < 0

⇥(x) ={ 0 x < 01 x > 0

1/2 x = 0

⇥(x) ={ 0 x < 01 x > 0

1/2 x = 0

counts sign changes

⇥(x) ={ 0 x < 01 x > 0

1/2 x = 0

counts sign changes

Luttinger Theorem for electrons

n = 2X

⇥(<G(k,! = 0))

How do functions

change sign?

E > "p

E < "p

How do functions

change sign?

E > "p

E < "pdivergence

How do functions

change sign?

E > "p

E < "pdivergence

Is there another way?

zero-crossing

no divergence is necessary

closer look at Luttinger’s theorem

n = 2X

⇥(<G(k,! = 0))

divergences (poles)+ zeros

n = 2X

⇥(<G(k,! = 0))

divergences (poles)+ zeros

how can zeros affect the particle count?

n = 2X

⇥(<G(k,! = 0))

what are zeros?

are they (like poles) conserved?

what models have zeros?

Mott mechanism

NiO insulates?d8

Sir Neville

Mott mechanism

NiO insulates?d8

perhaps thiscosts energy

Sir Neville

Mott mechanism

U � t

NiO insulates?d8

Sir Neville

Mott mechanism

U � t

NiO insulates?d8

µ = 0

Sir Neville

Mott mechanism

U � t

NiO insulates?d8

µ = 0 no change insize of

Brillouin zoneSir Neville

Mott Problem

Im G=0µ = 0

( ) d!

ReG(0, p) = Kramers-Kronig

Mott Problem

= below gap+above gap

Im G=0µ = 0

( ) d!

Mott Problem

= below gap+above gap = 0

Im G=0µ = 0

( ) d!

Mott Problem

DetG(k,! = 0) = 0 (single band)

Im G=0µ = 0

( ) d!

Mott Problem

DetG(k,! = 0) = 0 (single band)

Im G=0µ = 0

( ) d!

DetReG(0,p)=0 MottnessFriday, March 14, 14

interactions dominate:Strong Coupling Physics

U/t = 10� 1

how do zeros show up?

Re G<0

0000000

0 0 00 0

Re G<0

Only option: DetG=0! (zeros )

0000000

0 0 00 0

Re G<0

Fermi Arcs

zeros + poles

Luttinger, Dzyaloshinskii, Yang, Rice, Zhang,Tsvelik, Anderson (lots of smart people)...

n=zeros + poles

Fermi Liquids Mott Insulators

Is this famous theorem from 1960 correct?

A model with zerosbut Luttinger fails

e�t t t t t t t t

no hopping=> no propagation (zeros)

e�t t t t t t t t

no hopping=> no propagation (zeros)

N flavorsof e-

e� spin

generalization

N flavors of spin

e� spin

generalization

N flavors of spinN = 5

e� spin

generalization

N flavors of spinN = 5

e� spin

generalization

N flavors of spin2E

1491625 N = 5

e� spin

generalization

N flavors of spin

2(n1 + · · ·nN )2

1491625 N = 5

G↵�(! = 0) = #

✓2n�N

G↵�(! = 0) = #

✓2n�N

n = N⇥(2n�N)

Luttinger’s theorem

n = N⇥(2n�N)

{0, 1, 1/2

n = N⇥(2n�N)

{0, 1, 1/2n = 2

n = N⇥(2n�N)

{0, 1, 1/2n = 2

n = N⇥(2n�N)

{0, 1, 1/2n = 2

n = N⇥(2n�N)

{0, 1, 1/2even

n = N⇥(2n�N)

{0, 1, 1/2even odd

n = N⇥(2n�N)

{0, 1, 1/2even odd

no solution

Problem

E � "p � ⌃

Problem

E � "p � ⌃

lifetime of a particle vanishes

Problem

E � "p � ⌃

=⌃ < ✏p

Problem

E � "p � ⌃

=⌃ < ✏p no particle

what went wrong?

�I[G] =

Zd!⌃�G

what went wrong?

�I[G] =

Zd!⌃�G

if ⌃ ! 1

what went wrong?

�I[G] =

Zd!⌃�G

if ⌃ ! 1

integral does not exist

what went wrong?

�I[G] =

Zd!⌃�G

if ⌃ ! 1

integral does not exist

No Luttinger theorem!Friday, March 14, 14

experimental confirmation of violation?

`Luttinger’ countexperimentaldata (LSCO)

1� xFS

Bi2212Zp ! 0

1� xFS

Bi2212Zp ! 0

violation=> zeros are present

1� xFS

each hole a single k-state6=

Bi2212Zp ! 0

violation=> zeros are present

strange

not strange

strange

how to count particles?

some charged stuffhas no particle interpretation

what is the extra stuff?

scale invariance

new fixed point(scale invariance)

?Friday, March 14, 14

unparticles (IR)(H. Georgi)

what is scale invariance?

invariance on all length scales

f(x) = x

f(x/�) = (x/�)2 scale change

f(x) = x

scale invariance

f(x) = x

�2g(�){1

scale invariance

f(x) = x

g(�) = �2

f(x) = x

�2g(�){1

scale invariance

what’s the underlying theory?

2@µ�@µ�

x ! x/⇤

free field theory

�(x) ! �(x)

L ! ⇤2L

2@µ�@µ�

scale invariant

x ! x/⇤

free field theory

�(x) ! �(x)

L ! ⇤2L

2@µ�@µ�+m2�2

massive free theory

2@µ�@µ�+m2�2

massive free theory

x ! x/⇤�(x) ! �(x)

2@µ�@µ�+m2�2

massive free theory

x ! x/⇤�(x) ! �(x)

m2�2

2@µ�@µ�+m2�2

massive free theory

x ! x/⇤�(x) ! �(x)

m2�2⇤2

2@µ�@µ�

2@µ�@µ�+m2�2

massive free theory

x ! x/⇤�(x) ! �(x)

m2�2⇤2

2@µ�@µ�

2@µ�@µ�+m2�2

massive free theory

no scale invariance

x ! x/⇤�(x) ! �(x)

m2�2⇤2

2@µ�@µ�

unparticlesfrom a massive theory?

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

theory with all possible mass!

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

� ! �(x,m2/⇤2)

x ! x/⇤

m2/⇤2 ! m2

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

scale invariance is restored!!

L ! ⇤4L

� ! �(x,m2/⇤2)

x ! x/⇤

m2/⇤2 ! m2

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

not particles

L ! ⇤4L

� ! �(x,m2/⇤2)

x ! x/⇤

m2/⇤2 ! m2

unparticles

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

not particles

L ! ⇤4L

� ! �(x,m2/⇤2)

x ! x/⇤

m2/⇤2 ! m2

✓Z 1

0dm2m2� i

p2 �m2 + i✏

◆�1

/ p2|�|

propagator

✓Z 1

0dm2m2� i

p2 �m2 + i✏

◆�1

/ p2|�|

propagator

dU � 2

G / (E � "p)n�2

n massless particles

G / (E � "p)n�2

G / (E � "p)dU�2

G / (E � "p)n�2

unparticles=fractionalnumber of massless

particles

G / (E � "p)dU�2

G / (E � "p)n�2

particles

G / (E � "p)dU�2

dU > 2

G / (E � "p)n�2

particles

G / (E � "p)dU�2

dU > 2

no simple sign changeFriday, March 14, 14

what really is the summation

over mass?

mass=energy

what really is the summation

over mass?

high energy(UV)

low energy(IR)

related to sum over mass

high energy(UV)

low energy(IR)

dlnE= �(g(E))

locality in energyFriday, March 14, 14

high energy(UV)

low energy(IR)

implement E-scaling with an extra dimension

dlnE= �(g(E))

gauge-gravity duality(Maldacena, 1997)

QFTIR gravityQFT

dlnE= �(g(E))

QFTIR gravityQFT

dlnE= �(g(E))

locality in energy

no particles(conserved currents)

QFTIR gravityQFT

dlnE= �(g(E))

locality in energy

no particles(conserved currents)

unparticles?

mass=extra dimension

fixed by metric

rewriting the action

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

m2�dm2

m = z�1

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

m2�dm2

m = z�1

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

m2�dm2

z5+2�

R2⌘µ⌫(@µ�)(@⌫�) +

m = z�1

L =�@

µ�(x,m)@µ�(x,m) +m

2(x,m)�Z 1

m2�dm2

z5+2�

R2⌘µ⌫(@µ�)(@⌫�) +

can be absorbed with AdS metric

generating functional for unparticles

action on

4+2�x dz

✓@a�@

a�+�2

AdS5+2�

action on

4+2�x dz

✓@a�@

a�+�2

AdS5+2�

�⌘µ⌫dx

⌫ + dz

2� p

�g = (R/z)5+2�

action on

4+2�x dz

✓@a�@

a�+�2

AdS5+2�

unparticle lives in

d = 4 + 2� � 0

�⌘µ⌫dx

⌫ + dz

2� p

�g = (R/z)5+2�

scaling dimension is fixed

m2AdS = dU (dU�d)

R21 = dU (dU � d)

pd2 + 4

m2AdS = dU (dU�d)

R21 = dU (dU � d)

GU (p) / p2(dU�d/2)

GU (0) = 0

GU (p) / p2(dU�d/2)

GU (0) = 0

unparticle (AdS) propagator has zeros!

interchanging unparticles

fractional (d_U) number of massless particles

ei⇡dU 6= �1, 0

fractional statistics in d=2+1

ei⇡dU 6= e�i⇡dU

pd2 + 4

time-reversal symmetry breakingfrom unparticle (zeros=Fermi arcs) matter

unparticles BCS

d ln g

d ln�= 4dU � d > 0

tendency towards pairing (any instabilitywhich establishes a gap)

fermionswith unparticle propagator

High T_cunparticles

variable massUPt_3

emergentgravity

High T_cunparticles

variable massUPt_3

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