superconductivity and mottness: exact results
TRANSCRIPT
Superconductivity and Mottness: Exact Results
Luke YeoEdwin Huang
Cooper instability
px
py
�gb†pbp0
Eb / e1/g
�(g) = g2
FL ! BCS
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
ε
µ
gap with no symmetry breaking!!
charge gap
U � tU
solve the Hubbard Model!!
Cooper instability??
Progress thus far?
DMFT QMC disputes
Sept. 1997 Nov. 1997
� = Mottness
zeros
DetReG(! = 0, p)
6= 0 = 0
counting particles
12
34
5 6
7 8
is there a more efficient way?
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))
are zeros important?
Fermi Arcs
Fermi arcs necessarily imply zeros exist.
Must cross A zero line (DetG=0)!!!
Re G Changes Sign across An arc
+
�
SC in Doped Mott Insulator: Cooper Instability
for a Luttinger (zero surface)??
C. Setty 2019
LS+ Pair fluctuations=
SYK
Is this true in general?
simplest model with
zeros
atomic limit of Hubbard
model
Ht=0 = �µ
X
i�
ni� +X
i
Uni"ni#
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
Can the same physics be
retained in the presence of a
kinetic energy?
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
| G; {�k}i =Y
k2⌦1
c†k�k
Y
k2⌦2
c†k"c†k#|0i,
metal is degenerate: SU(2) invariance
entropy across MIT
LY, PWP, PRD, 99, 094030 (2019)
what does the HK model leave out??
dynamical spectral weight transfer
[Ht, HU ] 6= 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
Eb = �E ⇠ W (1� (U/W )2)e�⇡Wp
1�(U/W )2/g
Cooper Instability
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
Tc = (W � U)4/5U1/5 e�
⇡e�
45
Wg .
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 equation
1 =g
Wsinh�1(
W � U
2�) +
↵g
Wsinh�1(
U
2�)
� ⌧ U,W
� = (W � U)1/2U1/2e�W2g
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!
gap opening: superconducting state
Huangon band
spectral function: superconducting state
Doped MI half-filled metal
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%
condensation energy
UV-IR mixing
Bontemps, et al: FGT sum rule at 2eV
condensation energy:HK model
1.00
>1
�WL / O(2� 3%) as in experiments
why?
H = HHK +Hp
[HHK, Hp] 6= 0
dynamical spectral weight
transfer
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
is this the general
mechanism of the color
change?
IR theory?
Ne↵ = N0 +N1x�
H =
Zddr
†(r)⇥K(�@2)� µ
⇤ (r) +Hother
cp2↵
Caffarelli-Silvestre extension theorem
(2006)
(��)�
Rn+1 g(x, y)
Dirichlet
Rn f(x)
Neumann
connection to holography
HK model
[Ht, HV ] = 0
Mott gap
[HHK, Hp] 6= 0
non-BCS superconductivity
�
Tc/ 1
�lk� /
p2xk⇣
†k� � �zk⇣�k�
�uk� / zk⌘
†k� � �
p2yk⌘�k�
WL(g)
WL(g = 0)> 1