A Quarter Century of High Temperature Superconductors
T. Maurice Rice
ETH Zurich,
Hong
Kong U
., K.

Y. Yang,
F.C
.
Zhang
& W.

Q. Chen
Brookhaven
Natl. Lab
. R.
Konik
, A.
Tsvelik
&
A.James
Introduction to the
Cuprates
and the exotic
Pseudogap
Phase
Lower sights

> Phenomenological YRZ propagator in
Pseudogap
Phase
Comparison to
E
xperiment ARPES ,
Andreev
T
unneling Spectra
etc.
Excitations in
Pseudogap
Phase
–
Spin Response
PSI 16 March 2012
High Temperature Superconductivity
CuO
2
plane
Copper

oxide compounds
1986:
J.G. Bednorz & K.A. Müller
La
2

x
Ba
x
CuO
4
T
c
=35 K
AF
SC
T
x
T
N
T
c
T*
Doped antiferromagnetic
Mott insulator
under optimally over
doped
pseudogap
strange
metal
Tc up to 133K Schilling & Ott ‘93
They are
unconventional d

wave
superconductors
and materials with many anomalous properties
Generic Phase Diagram
Cu
2+
spin S = 1/2
CuO
2
plane electronically relevant
Parent compound: La
2
CuO
4
e
g
t
2g
x
2

y
2
3z
2

r
2
yz
zx
xy
1 hole in 3d

e
g
Cu
2+
3d
9
Single ½

filled Band of Cu

O hybridized States
Strong Onsite Coulomb
interaction
Single Band Hubbard Model on a 2D square lattice
CuO
2

plane
O
Cu
2D square
lattice of Cu

ions
O

octahedra
Mott limit

electrons localized in real space
Example: Lattice of H

Atoms
:
a
B
<< d
Onsite e

e

repulsion:
U
= E(H
+
) + E(H

)
Electrons localized:
Mott Insulator at 1 el/site
Low

energy physics purely due
to electron spins
H
Heisenberg
J
S
i
S
j
i
,
j
antiferromagnetic spin order
generally
at low T
H
+
H

H
2a
B
d

t
S=1/2
•
U
STRONG
Fundamentally different from a band
insulator
No breaking of translational symmetry involved
U > 2zt
Hubbard Model Kin.
Energy
+ U
•
U
MODERATE
U < 2zt
z
:
nn
number
Electrons
itinerant
:
metallic
state
at
1el./
site













General Remarks
Cuprates
described by a ‘ simple’ model
–
Hubbard model
2D
square lattice
Cuprates
are
highly quantum
with only a single relevant orbital e.g. Cu
2+
& Cu
3+

Favors Superconductivity ?
Contrast to

d
1

oxides (Ti2O3, VO2
etc
forming singlet dimer lattices ),

Nickelates
[
Jahn

Teller
polarons
when doped Ni
2+

> Ni
3+
] . .

Also Fe

pnictides
less symmetric [leading to a lower
Tc
? ]
Is this why the
cuprates
are such special superconductors ?
Full
Metal
with
Large Fermi
Surface
Mott Insulator
No
Fermi
Surface
Charge Gap
Pseudogap
Phase : Hall
effect
as
a
doped
Mott
insulator
Translational
s
ymmetry
preserved
in
clean
underdoped
samples
: 124 & YBCO6.5
unlike
Cr

alloys
:
Comm
. AF
Incomm
AF
para.metal
v
F

nodal
–
>
const
.
as
n
h

>
0
unlike
3
He
AF
D

SC
T
X
: doping
T
N
T
c
T*
How
do
cuprates
crossover
from
full
metal
to
Mott
insulator
?
X
0.2
QCP ?
OVERDOPED
UNDERDOPED
Vignolle
et al Nature 2008
G
Breakdown
of
Landau Fermi Liquid in
Overdoped
Cuprates
Recent
Experiments on Single Layer
Overdoped
Tl
2
Ba
2
CuO
6+x
N.Hussey
and
collaborators
Angular
Dependent
Magnetoresistance
(ADMR) in High
Magnetic
Fields
Superconductivity
Suppressed
( B = 45T )

> Normal State
at
low
T
Strongly
Anisotropic
Scattering
rate
around
the
Fermi
surface
Largest
between
antinodal
regions
:
Grows
as
doping
decreases
Agrees
with
results
of
RG
calculations
for
2D Hubbard
model
Honerkamp
et al PRB 2001,
Ossadnik
et al PRL 2009
Single
2D
Band
Full
Metallic Fermi
Surface
Also
seen
in Quantum
Oscillations
Fermi
Surface
agrees
with
LDA
M
antinodal
nodal
G
Full
Metal
with
Large Fermi
Surface
Mott Insulator
No
Fermi
Surface
Charge Gap
Pseudogap
Phase : Hall
effect
as
a
doped
Mott
insulator
•
Translational
s
ymmetry
preserved
in clean
underdoped
samples
: YBa
2
Cu
4
O
8
& YBCO6.5
AF
D

SC
T
X
: doping
T
N
T
c
T*
Pseudogap
Phase
between
full
metal
&
Mott
insulator
?
X
0.2
QCP ?
OVERDOPED
UNDERDOPED
Vignolle
et al Nature 2008
G
^
T
c
Spin Gap
=>Singlet
Pairing of Cu
2+

Spins in the
Pseudogap
Phase
Well Ordered and
Underdoped
•
Continuous
Onset of Spin Pairing in Normal
Phase
–
no kinks
•
Spin
Susceptibility well below AF value at T ~
T
c
indicating
singlet pairing rather than AF fluctuations
YBa
2
Cu
4
O
8
Knight Shift ~
Spin
Susceptibility
–
Not Pauli like at T
C
< T < T*
Bankay,Mali
et al
PRB (‘94)
2D RVB State which is a
superposition of configurations
with Singlet Pairs
can be written as a projected
BCS

State.
singlet
Resonating Valence Bond Theory
Doping allows
singlets
to move as electron
pairs
Elegant idea but difficult to develop a microscopic theory
for this strongly interacting
fermionic
state with
strong short range correlations.
Proposed
by P.W.
Anderson 1987
Singlet energy gain is 3x Classical energy
–

>
S = 0
Fermi surface in the
pseudogap
phase
ARPES shows full Fermi Surface 4 nodal arcs
Norman,
Campuzano
, H. Ding . . ‘98
Ca
2

x
Na
x
CuCl
2
O
2

Shen
et al ‘05
G
ARPES signal within
±
10
meV
of Fermi energy shows no weight at
antinodal
energy gap
Decrease
Doping
thru
‘ QCP
=>
Pseudogap
Phase
with
an
Antinodal
Gap
??
Full
Fermi
Surface
=>
Nodal
Fermi Pockets [
Arcs
]
•
1

Gap Scenario
: Cooper d

wave Pairing Gap opens on full Fermi Surface at T*
Tc
<< T*
:
strong phase fluctuations ( n
s
~ x (hole density) )
=> near
antinodal
sc
>>
Tc
=> Fermi arcs in ARPES at T>
Tc
•
2

Gap Scenario
:
Pseudogap
is an insulating gap opening near
antinodal
k

points
=> RVB charge gap opens on AFBZ
–
a fixed surface in
k

space
due to
ph
&
pp
umklapp
scattering

Precursor to Mott insulator at zero doping
=>
Various Experiments at T <
Tc
support insulating character of
pseudogap
e. g. Andreev & Giaever Tunneling Scales in over

&
underdoping
[
Deutscher
RMP 2005, Yang et al PRL 2010]
Mott Insulating State viewed in k

space
Real space
k

space
Underlying lattice
Umklapp
scattering processes allowed

> Momentum conserved modulo {
G
}
Band filling 1 el./site Surface in
k

space enclosing
an area of ½

Brillouin
zone.
Conclusion ; look at U

surface which is
a) spanned by elastic U

scattering processes
b) encloses an area of ½

Brillouin
zone.

t
2

Leg Hubbard Ladders

> A simple model with a Mott State at weak coupling
½

Filling

> Unique Insulating D

Mott
Groundstate
:
Dagotto
& TMR Science
’
96
Balents
, Fisher et al PRB . . .
Charge & Spin Gap due to
Umklapp
Scattering Processes at
E
f
e.g. K
f1R
& K
f2R

>
K
f1R
& K
f2R
:
momentum change 2π
Translational Symmetry along legs preserved
Short Range AF & d

wave Pairing correlations
Single Particle Propagator takes a BCS form: G(
k
x
,
w
) =
but without an anomalous component: F = 0.
Konik
& Ludwig PRB (2001)
+..

t
YRZ
Ansatz
for Green
`s Fn. in analogy with coupled ladders
K.

Y. Yang, Rice
& F. C.
Zhang PRB ‘06
see
R.Konik
, Rice
& A.
Tsvelik
PRL
`05
•
RVB Gap
R
(
k
)
opens
on p

p
Umklapp
Surface
( = AF
Brillouin
Zone in 2D)
•
F
ixed
line
of
zeros
in G
RVB
(
k
,0)
on
Umklapp
Surface
:
no
change
with
doping
•
U

Surface
encloses
1

el./
site
and
is
spanned
by
U

scattering
processes
Gutzwiller
Renorm.Factors
nn
nnn
nnnn
hopping
t
(x) =
g
t
(x)t
0
+(3/8)
g
s
(x)
J
c
t’(x)=
g
t
(x)t’
0
0
(x)
–
> 0
at
x =x
c
( = 0.2) : RVB Gap
from
Renorm
.
Mean
Field
Theory

F. C. Zhang
et al
`88
Note: t

J model

a
spinon

holon
boundstate
=> el.
G(
k,
w
)
with
YRZ form
P.A.Lee
2011
Pairing Self Energy in a normal Green’s Fn.
but with energy fixed on the U

surface
infinities
zeros
G(
k
,0) > 0 in shaded area bounded by zeros and infinities
Luttinger

Dzyaloshinskii
Sum Rule => Lines of
zeros
:
1
el./ site
Fermi Pockets
evolve
into
Full
Fermi
Suface
as
doping
x
increases
G
Special Form of Nodal Pocket
QP Spectral Weight
very anisotropic
looks like an Arc
Front Large
Back Small
Pocket arises due to back

bending of the
‘
Bogoliubov
‘
Quasiparticle
dispersion
l
eading to particle
–
hole asymmetry in the pocket
N.B. Pocket ends in a Dirac point
Particle

Hole
Asymmetry
in ARPES

BNL Group
–
Peter Johnson
H.B.Yang
et al Nature
`08
I(
w)
= A(
w
)f(
w
)
ARPES with
E
nhanced Resolution
–
H.

B. Yang et al PRL 2011
QP dispersion extrapolated form maximum
Evolution of Nodal Pockets with doping
QP dispersion &
spectral weight in YRZ
QP dispersion
spectral
weight
( 0,0)
(π,π)
Angle Integrated PES

Hashimoto et al PRB (2009)
Experiment

0.2

0.1 E
F
+0.1(
eV
)
Theory
K.

Y. Yang et al EPL (2009)
Hole Pocket ends in a Dirac Point
on the nodal line.
DOS ~ E [ ≠ const. ]

0.2

0.1 E
F
+0.1
YRZ
Ansatz 2

Gap
Phenomenology
•
Antinodal
energy
gap
different
to
dSC
gap
{
mainly
on
nodal
pockets
}
YRZ
Ansatz
for
dSC
used
to
model
T & x
dependence
of
:
•
Raman
Scattering
Valenzuela

Bascones
PRB ‘07
;
PRB ‘08
Carbotte
, Nicol &
collaborators
PRB ‘10
•
Specific
Heat
Carbotte
, Nicol &
collaborators
PRB ‘09
•
Infrared
Conductivity
Carbotte
, Nicol &
collaborators
PRB ‘09
•
Penetration
Depth
Carbotte
, Nicol &
collaborators
PRB ‘10
•
Andreev Tunneling
K

Y. Yang et al
PRL
’
10
Review
:

TMR, K.

Y. Yang & F. C. Zhang
Rep.
Prog
. Phys.
75
,
01650 (2012)
Generalized Drude form for optical conductivty
Infrared
optical
conductivity
Illes,Carbotte
& Nicol PRB
`09
London
penetration
length
Carbotte
et al PRB
`10
Tunneling
Experiments
:
Transparent
Barrier
=
> Andreev
Reflection
High
Barrier
=>
Giaever [Single
Electron
]
Tunneling
BCS
S
uperconductors
Voltage
Scales
:
Andreev [ V <
]
& Giaever [V>
]
tunneling
are
equal
Overdoped
Cuprates

>
Equal
Voltages
Scales
are
observed
Pseudogap
Phase

> Different
Voltage
Scales
are
observed
Calculate
tunneling
conductance
using
YRZ
generalized
to
d

wave
sc
in
the
Keldysh
formulation

K

Y.
Yang,
K.Huang,W

Q Chen, TMR &
F.C.Zhang
PRL 2010
Giaever
Andreev
See Deutscher RMP (2005)
Overdoped Cuprates: SC Gap over Full Fermi Surface
Equal
Energy
Scales
Unequal
Voltage
Scales
: Andreev
conductance
s
A
s
; Giaever
pseudogap
peaks
in
s
n
s
s
Tunneling
into
Underdoped
Cuprate
,
[x
=
0.1]
: Andreev vs. Giaever
YRZ
:
SC
pairing
amplitude
and
gap
antinodal
Pseudogap
mainly
on
the
arcs
(
pockets
) in normal [
n
]
and
sc
[s]
states
Spin Response in the YRZ Model
: Andrew James, Robert
Konik
& TMR
Brookhaven Natl. Lab.
arXiv
1112.2676
Spin Response in RPA using YRZ propagators to calculate bare response :
Spin
Excitations
in
Pseudogap
Phase
—
>
Hourglass
Spectrum
—
> Triplet
magnon
in
the
pseudogap
&
qp
transitions
at
low
energies
[
suppressed
below
Tc
]
Hourglass Spectrum rescaled
expts
.
LSCO, LBCO & YBCO

underdoped
Gapped Region
Hole Pockets
Stock et al PRB 2004
Conclusions
: main points of YRZ phenomenological theory
•
Interpolates between Fermi pockets at nodal & insulating gap at
antinodal
without translational symmetry breaking
also found in DMFT calculations
–
Georges,
Kotliar
, Millis, Tremblay + . . . .
•
Superconductivity in
Pseudogap
Phase supported through coupling of
4 near

nodal Fermi pockets and
Cooperon
mode at
antinodal
.
[
Konik,TMR
&
Tsvelik
PRB (2010) ]
•
Magnetic Response has Hourglass Form in
Pseudogap
Phase.
AF order at small x thru’ softening of the Triplet
Magnon
mode
•
Pseudogap
Phase is an Unstable Fixed Point
[a phase with only short range
correlations
analagous
to
the Landau
Fermi
liquid in a metal]
,

unstable against
dSC
etc.
at low T [ Anderson
Physica
B ’02]
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