µ = 0
no change in size of
Brillouin zone
Q q < qc
infall to black hole
q < qcMott
Insulator
use holography: competition between gravity and Coulomb
E > "p
E < "p
G(E) =1
E � "p
"p E
counting poles (qp)
"p E
zero-crossing
Luttinger’s theorem
n = 2X
k
⇥(<G(k,! = 0))
DetG(! = 0, ~p) = 0
strongly correlated gapped systems
what are zeros?
Im G=0µ = 0
= below gap+above gap = 0
DetG(k,! = 0) = 0 (single band)
( ) d!
Z 1
�1!
ReG(0, p) = Kramers -Kronig
poles+zeros (all sign changes)
singularities of lnG
=2i
(2⇡)d+1
Zddp
Z 0
�1d⇠ ln
GR(⇠,p)
G⇤R(⇠,p)
n
n = 2X
k
⇥(<G(k,! = 0))
Fermi Arcs
Fermi arcs necessarily imply zeros exist.
Must cross A zero line (DetG=0)!!!
Re G Changes Sign across An arc
+
�
Exact Green Function
GRi (!) =
1 + x
! � µ+ U2
+1� x
! � µ� U2
,
x = 0
GRi (! = µ = 0) = 0
n = 2⇥
✓�µ
µ2 � U2/4
◆ µ > 0 n = 2
n = 1µ = 0
n = 0µ < 0
LSR violation
Hatsugai-Khomoto Model
HHK = �t
X
hj,li,�
⇣c†j�cl� + h.c.
⌘� µ
X
j�
c†j�cj�
+U
Ld
X
j1..j4
�j1+j3,j2+j4c†j1"cj2"c
†j3#cj4#,
ck� =X
j
eikjcj�
HHK =X
k
Hk =X
k
(⇠k(nk" + nk#) + Unk" nk#) .
⇠k = ✏k � µ
General HK Model
X
k
(⇠k(nk" + nk#) + Unk" nk0#) .
[Ht, HU ] = 0
Solvable Mott transition
Gk�(i!n ! z) =1� hnk�̄iz � ⇠k
+hnk�̄i
z � (⇠k + U)
lower Hubbard band upper Hubbard band
⇣k� = c†k�(1� nk�̄) ⌘k� = c†k�nk�̄
Hubbard band operators
GR� (k,!) =
1
! + i0+ � (⇠k + U/2)� (U/2)2
! + i0+ � (⇠k + U/2)
Fermi liquid QP
{⌃(k,!)
! = ⇠k + U/2
<⌃ = =⌃ = 1
Mott transition: composite excitations
�E = U � 4dt = U �W
insulatorLuttinger Surface
metal
0
.5
1.0
Cooper Instability
H = HHK � gHp
1 = � g
Ld
X
k2⌦0
h1� nk" + n�k#iE � 2⇠k � Uhnk# + n�k"i
| i =X
k2⌦0
↵kb†k|GSi
hnk�i = 0
1 = �g
Z W/2
µd✏
⇢(✏)
E � 2✏+ 2µ,
Eb = hGS|H|GSi � h |H| i 7 0
Pair Susceptibility
�(i⌫n) ⌘1
Ld
Z �
0d⌧ei⌫n⌧ hT�(⌧)�†ig
=�0
1� g�0
�0 = hT�(⌧)�†i0 =X
k,p
hTc�k#(⌧)ck"(⌧)c†p"c
†�p#i0
=X
k
G�k#(⌧)Gk"(⌧),
�0(0) =
Zd! N 0(!)
tanh �!2
2!
LdN 0(!) =X
k2⌦0
�(! � ⇠lk) +X
k2⌦2
�(! � ⇠uk )
Solution for T_c
+1
4
X
k2⌦1
�(! � ⇠lk) + �(! � ⇠uk )
1� g�0(0) = 0
Solve
variational wave function
| BCSi =Y
k
(uk + vkb†k|0i
| BCSi =Y
k>0
(u2k + v2kb
†kb
†�k + ukvk(b
†k + b†�k))|0i
| i =Y
k>0
✓xk + ykb
†kb
†�k +
zkp2(b†k + b†�k)
◆|0i
| i =Y
k>0
✓xk + ykb
†kb
†�k +
zkp2(b†k + b†�k)
◆|0i
HK generalization
|xk|2 + |yk|2 + |zk|2 = 1
three variational parameters
h |H| i =X
k>0
(4⇠k + 2U)|yk|2 + 2⇠k|zk|2
�2gX
k,p>0;k 6=p
(x⇤kxk + y⇤kzk)(x
⇤pzp + z⇤pyp)
simplify with `gap’ equation
1 =g
2
Zd!
N 00(!)p!2 +�2
.
N 00(!) =X
k2⌦0
�(! � ⇠lk) +X
k2⌦2
�(! � ⇠uk )
+X
k2⌦1
�(! � ⇠lk) + �(! � ⇠uk )
⇠lk = ⇠k ⇠uk = ⇠k + U
gap/T_c ratio
� = (W � U)1/2U1/2e�W2g
Tc = (W � U)4/5U1/5 e�
⇡e�
45
Wg .
limg!0
�
Tc! 1
non-BCS superconductivity
Bogoliubov excitations
�k� = ukck� � �vkc†�k�
Huangons excitations
�lk� /
p2xk⇣
†k� � �zk⇣�k�
�uk� / zk⌘
†k� � �
p2yk⌘�k�
�u/lk� | i = 0
Excitation spectrum
h |�u/lk� H�u/lk� )†| i = h |H| i+ E
u/lk
Eu/lk =
q⇠u/lk
2+�2
superconductivity affects both bands!
can we explain the color change?
Al =
Z ⌦
0�(!)d! ⌦/2⇡c = 10000cm�1
Ah =
Z 2⌦
⌦�(!)d! ⌦/2⇡c = 10000cm�1
�Al
Al/ 3%
atomic limit: x holes
2x1-x
density of
states
PESIPES
} }1 + x 1� x
1 + x + �(t/U, x)� =
t
U
�
ij
�c†i�cj�⇥ > 0
1� x� �
density of
states
beyond the atomic limit: any real system
Harris & Lange, 1967 dynamical spectral weight transfer
Caffarelli-Silvestre extension theorem
(2006)
(��)�
Rn+1 g(x, y)
Dirichlet
Rn f(x)
Neumann
connection to holography