competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · competing...

Competing orders in the underdoped cuprates

Talk online: sachdev.physics.harvard.edu

http://sachdev.physics.harvard.edu

http://sachdev.physics.harvard.edu

Competing orders in the underdoped cuprates

Paired electron pockets in the underdoped cuprates, V. Galitski and S. Sachdev, Physical Review B 79, 134512 (2009).

Destruction of Neel order in the cuprates by electron doping, R. K. Kaul, M. Metlitksi, S. Sachdev, and C. Xu, Physical Review B 78, 045110 (2008).

Competing orders in the underdoped cuprates, Eun Gook Moon and S. Sachdev, to appear

http://qpt.physics.harvard.edu/p160.pdf








Outline1. Survey of experiments and theory

(a) Quantum oscillations(b) Competing orders(c) Nodal-anti-nodal dichotomy

2. Spin-fluctuation exchange mechanism of d-wave superconductivity

Successful at large doping, but cannot account for competing orders, the nodal-anti-nodal dichotomy (and other phenomena) at low doping

3. Superconductivity of electron and hole pockets in a background of fluctuating antiferromagnetism Pairing by gauge forces the unusual d-wave superconductivity of the underdoped cuprates.

The cuprate superconductors

Ground state has long-range spin density wave (SDW) order

Square lattice antiferromagnet

H =!

!ij"

Jij!Si · !Sj

Order parameter is a single vector field !" = #i!Si

#i = ±1 on two sublattices!!"" #= 0 in SDW state.

The cuprate superconductors

! !

!

The SC energy gap has four nodes.Shen et al PRL 70, 3999 (1993)

Ding et al PRB 54 9678 (1996)Mesot et al PRL 83 840 (1999)

Overdoped SC State: Momentum-dependent Pair Energy Gap

! !

!

N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 447, 565 (2007)

Quantum oscillations andthe Fermi surface in anunderdoped high Tc su-perconductor (ortho-II or-dered YBa2Cu3O6.5). Theperiod corresponds to acarrier density ! 0.076.

(a) Quantum oscillations

Nature 450, 533 (2007)

(a) Quantum oscillations

rc r

H

SC

M

AB

D

C

SC+SDW

SDW

"Normal"

E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Competition between superconductivity (SC) and spin-density wave (SDW) order(b) Phenomenological quantum theory of competing orders

Competing order predictions:

• Upper-critical field, Hc2, decreases as SDW is enhanced with decreas-ing doping (r)

• Onset of SDW order occurs at a larger doping in the normal state(line CM) than in the superconductor (point A).

• For r > rc, there is a field-induced quantum phase transition (lineAM) at H = Hsdw involving onset of SDW order.

rc r

H

SC

M

AB

D

C

SC+SDW

SDW

"Normal"



Hc2





rc r

H

SC

M

AB

D

C

SC+SDW

SDW

"Normal"


(b) Phenomenological quantum theory of competing ordersCompetition between superconductivity (SC) and spin-density wave (SDW) order

rc r

H

SC

M

AB

D

C

SC+SDW

SDW

"Normal"



Hsdw





!

H

dSC (Confinement + VBS order) SDW

+ dSC

MetallicSDW

Exotic metal with “ghost”

Fermi pockets

!cE. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Phase diagramas a function of hole density ! ! t and magnetic field H.

Hc2

Hsdw

!

H


+ dSC

MetallicSDW


Fermi pockets

!c


Neutron scattering on La1.9Sr0.1CuO4

B. Lake et al., Nature 415, 299 (2002)

Hc2

Hsdw

!

H


+ dSC

MetallicSDW


Fermi pockets

!c


Neutron scattering on La1.855Sr0.145CuO4

J. Chang et al., arXiv:0902.1191

Hc2

Hsdw

J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann,

H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess,

M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and

J. Mesot, arXiv:0902.1191

J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow, D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono, M. Oda, M. Ido, and

J. Mesot, Physical Review B 78, 104525 (2008).

!

H


+ dSC

MetallicSDW


Fermi pockets

!c


Neutron scattering on YBa2Cu3O6.45

D. Haug et al., arXiv:0902.3335

Hc2

Hsdw

0.3 0.4 0.5 0.6 0.7

0

500

1000

1500

0.3 0.4 0.5 0.6 0.7

0

500

1000

1500

0.4 0.5 0.6

1500

1600

1700

0.3 0.4 0.5 0.6 0.7

-500

0

500

0.4 0.5 0.6

0.4

0.5

0.6

0.7

QH (r.l.u.)

I(14.9T,2K) ! I(0T,2K)

(d)

(c)

H = 14.9T

(b)

(a)

ba

I (14.9T, 2K) ! I (0T, 2K)

H = 0T

Inte

nsity (

counts

/ ~

10 m

in)

QH (r.l.u.)

H = 14.9T

2K

50K

QH (r.l.u.)

QK (r.l.u.)

Inte

nsity (

counts

/ ~

1 m

in)

Inte

nsity (

counts

/ ~

10 m

in)

Inte

nsity (

counts

/ ~

10 m

in)

QK

QH

D. Haug, V. Hinkov, A. Suchaneck, D. S. Inosov, N. B. Christensen, Ch. Niedermayer, P. Bourges, Y. Sidis, J. T. Park, A. Ivanov, C. T. Lin, J. Mesot, and B. Keimer, arXiv:0902.3335.

!

H


+ dSC

MetallicSDW


Fermi pockets



Hc2

Hsdw

!

H


+ dSC

MetallicSDW


Fermi pockets


Exhibit quantum oscillations without Zeeman splitting


Hc2

Hsdw

Increasing SDW order

Spin density wave theory in hole-doped cuprates

!

S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).



!


!



!!!




!!!


Electron pockets

Hole pockets



!!!


!

Hole pockets



SDW order parameter is a vector, !",whose amplitude vanishes at the transition

to the Fermi liquid.

!!!


!


A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, arXiv:0902.2741.

Incommensurate order in YBa2Cu3O6+x

!!

Competition between the pseudogap and superconductivity in the high-Tc copper oxides

T. Kondo, R. Khasanov, T. Takeuchi, J. Schmalian, A. Kaminski, Nature 457, 296 (2009)

!

(c) Nodal-anti-nodal dichotomy in the underdoped cuprates

Y. Kohsaka et al., Nature 454, 1072, (2008)


V. B. Geshkenbein, L. B. Io!e, and A. I. Larkin, Phys. Rev. B 55, 3173 (1997).

!

-2e bosons at antinodes,+e fermion “arcs” at nodes,and proximity “Josephson”

coupling



Attractive phenomenological model,

but theoretical and microscopic basis is unclear

!






!!!!


Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates

!!!!

David Pines, Douglas Scalapino

Fermions at the large Fermi surface exchangefluctuations of the SDW order parameter !".

!"

! !

Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates

!"

!

Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

Approaching the onset of antiferromagnetism in the spin-fluctuation theory

Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

Approaching the onset of antiferromagnetism in the spin-fluctuation theory

:

• Tc increases upon approaching the SDW transition.SDW and SC orders do not compete, but attract each other.

• No simple mechanism for nodal-anti-nodal dichotomy.





!!!!



Begin with SDW ordered state, and focus onfluctuations in the orientation of !",

by using a unit-length bosonic spinor z!

!" = z!!!#!"z"

Fermi pockets in hole-doped cuprates

!!!!

Charge carriers in the lightly-doped cuprates with Neel order

Electron pockets Hole

pockets


!

For a spacetime dependent SDW order, !" = z!!!#!"z" ,!

c1"c1#

"= Rz

!g+

g$

"; Rz !

!z" "z!#z# z!"

".

So g± are the “up/down” electron operatorsin a rotating reference frame defined by the local SDW order

Increasing SDW orderElectronoperator

c1!

For a uniform SDW order with !" ! (0, 0, 1), write

!


c1!


c1"c1#

"= Rz

!g+

g$

"; Rz !

!z" "z!#z# z!"

".

So g± are the “up/down” electron operatorsin a rotating reference frame defined by the local SDW order

!


c2"c2#

"= Rz

!g+

!g$

"; Rz "

!z" !z!#z# z!"

".

Same SU(2) matrix also rotates electrons in second pocket.


c2!

SDW theory also specifies electronsat second pocket for !" ! (0, 0, 1)

!


c2!


c2"c2#

"= Rz

!g+

!g$

"; Rz "

!z" !z!#z# z!"

".

Same SU(2) matrix also rotates electrons in second pocket.

!

Two Fermi surfaces coupled to theemergent U(1) gauge field Aµ with opposite charges

CP1 field theory for z! and an emergent U(1) gauge field Aµ.Coupling t tunes the strength of SDW orientation fluctuations.

Low energy theory for spinless, charge !e fermions g±,and spinful, charge 0 bosons z!:

L = Lz + Lg

Lz =1t

!|(!" ! iA" )z!|2 + v2|"! iA)z!|2

"

+ Berry phases of monopoles in Aµ.

Lg = g†+

#(!" ! iA" )! 1

2m! ("! iA)2 ! µ

$g+

+ g†"

#(!" + iA" )! 1

2m! ("+ iA)2 ! µ

$g"

Strong pairing of the g± electron pockets

• Gauge forces lead to a s-wave paired state with a Tc of order

the Fermi energy of the pockets. Inelastic scattering from low

energy gauge modes lead to very singular g± self energy, but

is not pair-breaking.

!g+g!" = !


c2!

Electronoperator

c1!


!


c2!

Electronoperator

c1!


• Transforming back to the physical fermions:!

c1!c1"

"=

!z! !z#"z" z#!

" !g+

g$

";

!c2!c2"

"=

!z! !z#"z" z#!

" !g+

!g$

",

we find: "c1!c1"# = !"c2!c2"# $#|z!|2 + |z"|2

$"g+g$# ;

i.e. the pairing signature for the electrons is d-wave.

!


g±

!


g±

f±v !


g±

f±v

Low energy theory for spinless, charge +e fermions f±v:

Lf =!

v=1,2

"f†+v

#(!! ! iA! )! 1

2m! ("! iA)2 ! µ

$f+v

+ f†"v

#(!! + iA! )! 1

2m! ("+ iA)2 ! µ

$f"v

%

!


Weak pairing of the f± hole pockets

LJosephson = iJ!g+g!

" !f+1

!

!x f!1 ! f+1

!

!y f!1

+ f+2

!

!x f!2 + f+2

!

!y f!2

"+ H.c.

Proximity Josephson coupling J to g± fermions leads to p-wavepairing of the f±v fermions. The Aµ gauge forces are pair-breaking,and so the pairing is weak.

"f+1(k)f!1(!k)# $ (kx ! ky)J"g+g!#;"f+2(k)f!2(!k)# $ (kx + ky)J"g+g!#;"f+1(k)f!2(!k)# = 0,

Weak pairing of the f± hole pockets

!f+1(k)f!1("k)# $ (kx " ky)J!g+g!#;!f+2(k)f!2("k)# $ (kx + ky)J!g+g!#;!f+1(k)f!2("k)# = 0,


f±v ++_

_

!

g±

f±v ++_

_


d-wave pairing of the electrons is associated with

• Strong s-wave pairing of g±

• Weak p-wave pairing of f±v.

!

Nature 456, 77 (2008).

Universal theory of superconductivity

• Complex, ‘relativistic’bosons z!, with |z!|2 = 1

• Fermions g±

• Gauge field (A" ,A)

L =1t

!|(!! ! iA! )z"|2 + v2|"! iA)z"|2

"

+ g†+

#(!! ! iA! )! 1

2m! ("! iA)2 ! µ

$g+

+ g†"

#(!! + iA! )! 1

2m! ("+ iA)2 ! µ

$g"

• Theory fully characterized by two dimensionless parameters:

– (1/tc ! 1/t)/m! measures distance from SDW orderingquantum transition

– kF /(m!v)

• Characteristic length scale: 1/kF .

• Characteristic energy scale: m!v2.



• Fermions g±


L =1t

!|(!! ! iA! )z"|2 + v2|"! iA)z"|2

"

+ g†+

#(!! ! iA! )! 1

2m! ("! iA)2 ! µ

$g+

+ g†"

#(!! + iA! )! 1

2m! ("+ iA)2 ! µ

$g"



• Fermions g±


L =1t

!|(!! ! iA! )z"|2 + v2|"! iA)z"|2

"

+ g†+

#(!! ! iA! )! 1

2m! ("! iA)2 ! µ

$g+

+ g†"

#(!! + iA! )! 1

2m! ("+ iA)2 ! µ

$g"

• Theory fully characterized by two dimensionless parameters:

– (1/tc ! 1/t)/m! measures distance from SDW orderingquantum transition

– kF /(m!v)

• Characteristic length scale: 1/kF .

• Characteristic energy scale: m!v2.


0 0.005 0.01

0.02

0.04

0.06

0.08

0.1

-0.005-0.01

SDW

Exotic metal with “ghost” Fermi pockets


• Fermions g±


L =1t

!|(!! ! iA! )z"|2 + v2|"! iA)z"|2

"

+ g†+

#(!! ! iA! )! 1

2m! ("! iA)2 ! µ

$g+

+ g†"

#(!! + iA! )! 1

2m! ("+ iA)2 ! µ

$g"

dSC


0 0.005 0.01

0.02

0.04

0.06

0.08

0.1

-0.005-0.01

SDW

Exotic metal with “ghost” Fermi pockets

dSCdSC+SDW

Ez

SDW order is suppressed in the superconductor,i.e. Ez > 0, by enhancement of gauge field fluctuations,which are screened only in the metallic phases.

!

H


+ dSC

MetallicSDW


Fermi pockets



Hc2

Hsdw

!

H


+ dSC

MetallicSDW


Fermi pockets



Exhibit quantum oscillations without Zeeman splitting

Hc2

Hsdw

★ Gauge theory for pairing in the underdoped cuprates, describing “angular” fluctuations of spin-density-wave order

★ Natural route to d-wave pairing with strong pairing at the antinodes and weak pairing at the nodes

★ Explains characteristic “competing order” features of field-doping phase diagram: SDW order is more stable in the metal than in the superconductor.

★ New metallic state, an algebraic charge liquid, with “ghost” electron and hole pockets, could describe the finite temperature “pseudo-gap” regime.

★ Paired electron pockets are expected to lead to valence-bond-solid modulations at low temperature

★ Needed: theory for transition to “large” Fermi surface at higher doping

Conclusions

!

H


+ dSC

MetallicSDW


Fermi pockets



Hc2

Hsdw







Conclusions

12 nmTunneling Asymmetry (TA)-map at E=150meV

Ca1.90Na0.10CuO2Cl2

Bi2.2Sr1.8Ca0.8Dy0.2Cu2Oy

Y. Kohsaka et al. Science 315, 1380 (2007)

Indistinguishable bond-centered TA contrast

with disperse 4a0-wide nanodomains

Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)

4a0

12 nm

TA Contrast is at oxygen site (Cu-O-Cu bond-centered)

Y. Kohsaka et al. Science 315, 1380 (2007)

Ca1.88Na0.12CuO2Cl2, 4 KR map (150 mV)

4a0

12 nm

TA Contrast is at oxygen site (Cu-O-Cu bond-centered)

S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999).

Evidence for a predicted valence bond supersolid







Conclusions

competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · competing...

Documents