Unparticles and Superconductivity
Transcript of Unparticles and Superconductivity
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Unparticles and Superconductivity
Kiaran dave Charlie Kane Brandon LangleyBrandon Langley
Thanks to: NSF, EFRC (DOE)
J. A. HutasoitFriday, March 14, 14
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Properties all particles share?
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Properties all particles share?
fixed mass!charge
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Properties all particles share?
conserved(gauge
invariance)
fixed mass!charge
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Properties all particles share?
conserved(gauge
invariance)not conserved
fixed mass!charge
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mass sets a scale
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particles
E = "penergy
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particles
E = "penergy
G(E) =1
E � "p
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particles
E = "penergy
G(E) =1
E � "p
particle=infinity
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particles
E = "penergy
Green function (propagator)
G(E) =1
E � "p
particle=infinity
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particles
E = "penergy
Green function (propagator)
G(E) =1
E � "p
particle=infinity
=poles in Green function
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what would you see in a metal?
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what would you see in a metal?
e�
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what would you see in a metal?
e�
e�e�
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e�
e�e�
e�
e�
e�
e�
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what would you see in a metal?
e�
e�e�
e�
e�
e�e�
e�
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two crossings:closed surface of excitations
p
"p
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expect to see
px
py
closed surface
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are there any exceptions?
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where is this part?
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where is this part?
Fermi arcs: (PDJ,JCC,ZXS)
Fermi Arcs
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where are arcs seen?
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ceramics
normalstate
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what’s the explanation?
strange not strange
Two opposing Viewson Fermi arcs
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what’s the explanation?
strange not strange
Two opposing Viewson Fermi arcs
this dispute has an answer!
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what’s the explanation?
strange not strange
Two opposing Viewson Fermi arcs
this dispute has an answer!unparticles
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Right Wingers: Fermi Arcs are not Strange
intensity too small to be seen
G(E) =Zp
! � "p
Zp ! 0pole (particle) exists but
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Fermi Arcs are Strange
EF
seen not seen
k
no pole
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poles
Re G<0
ReG>0
Problem for left-wingers (me)
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poles
Re G<0
ReG>0
Problem for left-wingers (me)
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poles
Re G<0
ReG>0
How to account for the sign change without poles?
Problem for left-wingers (me)
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Fermi arcs
0
##
E � ✏p �1 = 0
pole no pole
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do poles account for all the charged stuff?
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if not?
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if not?
charge stuff= particles + other stuff
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if not?
charge stuff= particles + other stuff
unparticles!
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counting particles
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counting particles
1
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counting particles
12
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counting particles
12
3
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counting particles
12
34
5 6
7 8
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counting particles
12
34
5 6
7 8
is there a more efficient way?
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Each particle has an energy
particles
E = "penergy
Green function (propagator)
G(E) =1
E � "p
particle=infinity
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Each particle has an energy
particles
E = "penergy
Green function (propagator)
G(E) =1
E � "p
particle=infinity
particle count is deduciblefrom Green function
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Luttinger’s Theorem
Green function (propagator) G(E) =1
E � "p
G(E)
"p E
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Luttinger’s Theorem
Green function (propagator) G(E) =1
E � "p
G(E)
"p E
E > "p
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Luttinger’s Theorem
Green function (propagator) G(E) =1
E � "p
G(E)
"p E
E > "p
E < "p
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Luttinger’s Theorem
Green function (propagator) G(E) =1
E � "p
G(E)
"p E
E > "p
E < "p
particle density= number of sign changes of G
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Counting sign changes?
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Counting sign changes?
⇥(x)
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Counting sign changes?
⇥(x) ={ 0 x < 0
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Counting sign changes?
⇥(x) ={ 0 x < 01 x > 0
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Counting sign changes?
⇥(x) ={ 0 x < 01 x > 0
1/2 x = 0
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Counting sign changes?
⇥(x) ={ 0 x < 01 x > 0
1/2 x = 0
counts sign changes
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Counting sign changes?
⇥(x) ={ 0 x < 01 x > 0
1/2 x = 0
counts sign changes
Luttinger Theorem for electrons
n = 2X
k
⇥(<G(k,! = 0))
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How do functions
change sign?
"p E
E > "p
E < "p
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How do functions
change sign?
"p E
E > "p
E < "pdivergence
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How do functions
change sign?
"p E
E > "p
E < "pdivergence
pole
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Is there another way?
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Yes
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"p E
zero-crossing
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"p E
zero-crossing
no divergence is necessary
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closer look at Luttinger’s theorem
n = 2X
k
⇥(<G(k,! = 0))
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closer look at Luttinger’s theorem
divergences (poles)+ zeros
n = 2X
k
⇥(<G(k,! = 0))
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closer look at Luttinger’s theorem
divergences (poles)+ zeros
how can zeros affect the particle count?
n = 2X
k
⇥(<G(k,! = 0))
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what are zeros?
are they (like poles) conserved?
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what models have zeros?
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Mott mechanism
t
NiO insulates?d8
Sir Neville
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Mott mechanism
t
NiO insulates?d8
perhaps thiscosts energy
Sir Neville
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Mott mechanism
t
U � t
NiO insulates?d8
perhaps thiscosts energy
Sir Neville
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Mott mechanism
t
U � t
NiO insulates?d8
perhaps thiscosts energy
µ = 0
Sir Neville
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Mott mechanism
t
U � t
NiO insulates?d8
perhaps thiscosts energy
µ = 0 no change insize of
Brillouin zoneSir Neville
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Mott Problem
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Mott Problem
Im G=0µ = 0
( ) d!
Z 1
�1!
ReG(0, p) = Kramers-Kronig
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Mott Problem
= below gap+above gap
Im G=0µ = 0
( ) d!
Z 1
�1!
ReG(0, p) = Kramers-Kronig
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Mott Problem
= below gap+above gap = 0
Im G=0µ = 0
( ) d!
Z 1
�1!
ReG(0, p) = Kramers-Kronig
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Mott Problem
= below gap+above gap = 0
DetG(k,! = 0) = 0 (single band)
Im G=0µ = 0
( ) d!
Z 1
�1!
ReG(0, p) = Kramers-Kronig
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Mott Problem
= below gap+above gap = 0
DetG(k,! = 0) = 0 (single band)
Im G=0µ = 0
( ) d!
Z 1
�1!
ReG(0, p) = Kramers-Kronig
DetReG(0,p)=0 MottnessFriday, March 14, 14
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interactions dominate:Strong Coupling Physics
U/t = 10� 1
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how do zeros show up?
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poles
Re G<0
ReG>0
How to account for the sign change without poles?
Problem for left-wingers (me)
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poles
0000000
00000
0 0 00 0
0
00
Re G<0
ReG>0
How to account for the sign change without poles?
Problem for left-wingers (me)
Friday, March 14, 14
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Only option: DetG=0! (zeros )
poles
0000000
00000
0 0 00 0
0
00
Re G<0
ReG>0
How to account for the sign change without poles?
Problem for left-wingers (me)
Friday, March 14, 14
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Fermi Arcs
zeros + poles
Luttinger, Dzyaloshinskii, Yang, Rice, Zhang,Tsvelik, Anderson (lots of smart people)...
n=zeros + poles
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Fermi Liquids Mott Insulators
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Is this famous theorem from 1960 correct?
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A model with zerosbut Luttinger fails
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e�t t t t t t t t
A model with zerosbut Luttinger fails
Friday, March 14, 14
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e�t t t t t t t t
A model with zerosbut Luttinger fails
Friday, March 14, 14
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e�t t t t t t t t
A model with zerosbut Luttinger fails
no hopping=> no propagation (zeros)
Friday, March 14, 14
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e�t t t t t t t t
A model with zerosbut Luttinger fails
no hopping=> no propagation (zeros)
N flavorsof e-
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e� spin
Friday, March 14, 14
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e� spin
generalization
N flavors of spin
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e� spin
generalization
N flavors of spinN = 5
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e� spin
generalization
N flavors of spinN = 5
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e� spin
generalization
N flavors of spin2E
1491625 N = 5
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e� spin
generalization
N flavors of spin
H =U
2(n1 + · · ·nN )2
2E
1491625 N = 5
Friday, March 14, 14
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G↵�(! = 0) = #
✓2n�N
N
◆
Friday, March 14, 14
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G↵�(! = 0) = #
✓2n�N
N
◆
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2n = 2
N = 3
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2n = 2
N = 3
2 = 3
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2n = 2
N = 3
2 = 3
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2even
n = 2
N = 3
2 = 3
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2even odd
n = 2
N = 3
2 = 3
Friday, March 14, 14
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n = N⇥(2n�N)
Luttinger’s theorem
{0, 1, 1/2even odd
no solution
n = 2
N = 3
2 = 3
Friday, March 14, 14
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Problem
G=0
Friday, March 14, 14
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Problem
G=0
G =1
E � "p � ⌃
1
Friday, March 14, 14
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Problem
G=0
G =1
E � "p � ⌃
1
lifetime of a particle vanishes
Friday, March 14, 14
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Problem
G=0
G =1
E � "p � ⌃
1
lifetime of a particle vanishes
=⌃ < ✏p
Friday, March 14, 14
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Problem
G=0
G =1
E � "p � ⌃
1
lifetime of a particle vanishes
=⌃ < ✏p no particle
Friday, March 14, 14
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what went wrong?
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what went wrong?
�I[G] =
Zd!⌃�G
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what went wrong?
�I[G] =
Zd!⌃�G
if ⌃ ! 1
Friday, March 14, 14
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what went wrong?
�I[G] =
Zd!⌃�G
if ⌃ ! 1
integral does not exist
Friday, March 14, 14
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what went wrong?
�I[G] =
Zd!⌃�G
if ⌃ ! 1
integral does not exist
No Luttinger theorem!Friday, March 14, 14
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Luttinger’s theorem
Friday, March 14, 14
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Friday, March 14, 14
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experimental confirmation of violation?
Friday, March 14, 14
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Friday, March 14, 14
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`Luttinger’ countexperimentaldata (LSCO)
kF
1� xFS
Bi2212Zp ! 0
Friday, March 14, 14
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`Luttinger’ countexperimentaldata (LSCO)
kF
1� xFS
Bi2212Zp ! 0
violation=> zeros are present
Friday, March 14, 14
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`Luttinger’ countexperimentaldata (LSCO)
kF
1� xFS
each hole a single k-state6=
Bi2212Zp ! 0
violation=> zeros are present
Friday, March 14, 14
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strange
Two opposing Viewson Fermi arcs
this dispute has an answer!
not strange
Friday, March 14, 14
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strange
Two opposing Viewson Fermi arcs
this dispute has an answer!
Friday, March 14, 14
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how to count particles?
12
34
5 6
7 8
Friday, March 14, 14
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how to count particles?
12
34
5 6
7 8
some charged stuffhas no particle interpretation
Friday, March 14, 14
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what is the extra stuff?
Friday, March 14, 14
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Friday, March 14, 14
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scale invariance
Friday, March 14, 14
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new fixed point(scale invariance)
?Friday, March 14, 14
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new fixed point(scale invariance)
Friday, March 14, 14
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new fixed point(scale invariance)
unparticles (IR)(H. Georgi)
Friday, March 14, 14
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what is scale invariance?
invariance on all length scales
Friday, March 14, 14
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f(x) = x
2
Friday, March 14, 14
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f(x) = x
2
f(x/�) = (x/�)2 scale change
Friday, March 14, 14
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f(x) = x
2
f(x/�) = (x/�)2 scale change
scale invariance
Friday, March 14, 14
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f(x) = x
2
f(x/�) = (x/�)2 scale change
f(x) = x
2�
�2g(�){1
scale invariance
Friday, March 14, 14
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f(x) = x
2
f(x/�) = (x/�)2 scale change
g(�) = �2
f(x) = x
2�
�2g(�){1
scale invariance
Friday, March 14, 14
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what’s the underlying theory?
Friday, March 14, 14
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L =1
2@µ�@µ�
x ! x/⇤
free field theory
�(x) ! �(x)
L ! ⇤2L
Friday, March 14, 14
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L =1
2@µ�@µ�
scale invariant
x ! x/⇤
free field theory
�(x) ! �(x)
L ! ⇤2L
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
x ! x/⇤�(x) ! �(x)
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
x ! x/⇤�(x) ! �(x)
m2�2
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
x ! x/⇤�(x) ! �(x)
m2�2⇤2
✓1
2@µ�@µ�
◆
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
x ! x/⇤�(x) ! �(x)
m2�2⇤2
✓1
2@µ�@µ�
◆
Friday, March 14, 14
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L =1
2@µ�@µ�+m2�2
massive free theory
mass
no scale invariance
x ! x/⇤�(x) ! �(x)
m2�2⇤2
✓1
2@µ�@µ�
◆
Friday, March 14, 14
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Friday, March 14, 14
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unparticlesfrom a massive theory?
Friday, March 14, 14
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L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�
Friday, March 14, 14
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L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
Friday, March 14, 14
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theory with all possible mass!
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
Friday, March 14, 14
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theory with all possible mass!
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
� ! �(x,m2/⇤2)
x ! x/⇤
m2/⇤2 ! m2
Friday, March 14, 14
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theory with all possible mass!
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
scale invariance is restored!!
L ! ⇤4L
� ! �(x,m2/⇤2)
x ! x/⇤
m2/⇤2 ! m2
Friday, March 14, 14
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theory with all possible mass!
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
not particles
scale invariance is restored!!
L ! ⇤4L
� ! �(x,m2/⇤2)
x ! x/⇤
m2/⇤2 ! m2
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unparticles
theory with all possible mass!
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0dm2
not particles
scale invariance is restored!!
L ! ⇤4L
� ! �(x,m2/⇤2)
x ! x/⇤
m2/⇤2 ! m2
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✓Z 1
0dm2m2� i
p2 �m2 + i✏
◆�1
/ p2|�|
propagator
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✓Z 1
0dm2m2� i
p2 �m2 + i✏
◆�1
/ p2|�|
propagator
dU � 2
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G / (E � "p)n�2
n massless particles
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G / (E � "p)n�2
n massless particles
G / (E � "p)dU�2
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G / (E � "p)n�2
n massless particles
unparticles=fractionalnumber of massless
particles
G / (E � "p)dU�2
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G / (E � "p)n�2
n massless particles
unparticles=fractionalnumber of massless
particles
G / (E � "p)dU�2
zeros
dU > 2
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G / (E � "p)n�2
n massless particles
unparticles=fractionalnumber of massless
particles
G / (E � "p)dU�2
zeros
dU > 2
no simple sign changeFriday, March 14, 14
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dU?
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what really is the summation
over mass?
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mass=energy
what really is the summation
over mass?
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high energy(UV)
low energy(IR)
related to sum over mass
QFT
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high energy(UV)
low energy(IR)
related to sum over mass
QFT
dg(E)
dlnE= �(g(E))
locality in energyFriday, March 14, 14
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high energy(UV)
low energy(IR)
related to sum over mass
implement E-scaling with an extra dimension
QFT
dg(E)
dlnE= �(g(E))
locality in energyFriday, March 14, 14
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related to sum over mass
implement E-scaling with an extra dimension
QFT
dg(E)
dlnE= �(g(E))
locality in energyFriday, March 14, 14
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related to sum over mass
implement E-scaling with an extra dimension
gauge-gravity duality(Maldacena, 1997)
UV
QFTIR gravityQFT
dg(E)
dlnE= �(g(E))
locality in energyFriday, March 14, 14
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related to sum over mass
implement E-scaling with an extra dimension
gauge-gravity duality(Maldacena, 1997)
UV
QFTIR gravityQFT
dg(E)
dlnE= �(g(E))
locality in energy
no particles(conserved currents)
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related to sum over mass
implement E-scaling with an extra dimension
gauge-gravity duality(Maldacena, 1997)
UV
QFTIR gravityQFT
dg(E)
dlnE= �(g(E))
locality in energy
no particles(conserved currents)
unparticles?
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mass=extra dimension
fixed by metric
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rewriting the action
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rewriting the action
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0
m2�dm2
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rewriting the action
m = z�1
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0
m2�dm2
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rewriting the action
m = z�1
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0
m2�dm2
L =
Z 1
0dz
2R2
z5+2�
1
2
z2
R2⌘µ⌫(@µ�)(@⌫�) +
�2
2R2
�
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rewriting the action
m = z�1
L =�@
µ�(x,m)@µ�(x,m) +m
2�
2(x,m)�Z 1
0
m2�dm2
L =
Z 1
0dz
2R2
z5+2�
1
2
z2
R2⌘µ⌫(@µ�)(@⌫�) +
�2
2R2
�
can be absorbed with AdS metric
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generating functional for unparticles
action on
S =1
2
Zd
4+2�x dz
p�g
✓@a�@
a�+�2
R
2
◆
AdS5+2�
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generating functional for unparticles
action on
S =1
2
Zd
4+2�x dz
p�g
✓@a�@
a�+�2
R
2
◆
AdS5+2�
ds
2 =L
2
z
2
�⌘µ⌫dx
µdx
⌫ + dz
2� p
�g = (R/z)5+2�
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generating functional for unparticles
action on
S =1
2
Zd
4+2�x dz
p�g
✓@a�@
a�+�2
R
2
◆
AdS5+2�
unparticle lives in
d = 4 + 2� � 0
ds
2 =L
2
z
2
�⌘µ⌫dx
µdx
⌫ + dz
2� p
�g = (R/z)5+2�
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scaling dimension is fixed
m =1
z
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scaling dimension is fixed
m =1
z
m2AdS = dU (dU�d)
R2
Friday, March 14, 14
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scaling dimension is fixed
m =1
z
m2AdS = dU (dU�d)
R21 = dU (dU � d)
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scaling dimension is fixed
m =1
z
dU =d
2+
pd2 + 4
2
m2AdS = dU (dU�d)
R21 = dU (dU � d)
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GU (p) / p2(dU�d/2)
GU (0) = 0
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GU (p) / p2(dU�d/2)
GU (0) = 0
unparticle (AdS) propagator has zeros!
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
ei⇡dU 6= �1, 0
Friday, March 14, 14
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interchanging unparticles
fractional (d_U) number of massless particles
ei⇡dU 6= �1, 0
fractional statistics in d=2+1
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ei⇡dU 6= e�i⇡dU
dU =d
2+
pd2 + 4
2>
d
2
time-reversal symmetry breakingfrom unparticle (zeros=Fermi arcs) matter
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g
unparticles BCS
Tc
d ln g
d ln�= 4dU � d > 0
tendency towards pairing (any instabilitywhich establishes a gap)
fermionswith unparticle propagator
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High T_cunparticles
variable massUPt_3
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emergentgravity
High T_cunparticles
variable massUPt_3
Friday, March 14, 14