Paired electron pockets in the hole-doped cuprates

Talk online: sachdev.physics.harvard.edu

Paired electron pockets in the hole-doped cuprates

Hole dynamics in an antiferromagnet across a deconfined quantum critical point, R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, Physical Review B 75 , 235122 (2007).Algebraic charge liquids and the underdoped cuprates, R. K. Kaul, Y. B. Kim, S. Sachdev, and T. Senthil, Nature Physics 4, 28 (2008).

Paired electron pockets in the underdoped cuprates, V. Galitski and S. Sachdev, arXiv:0901.0005

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).

Ribhu KaulUCSB

Victor GalitskiMaryland

Cenke XuHarvard

Outline

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

Outline

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

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

dSC

Pseudogap: Temperature-independent energy gap exists T>>Tc

Ch. Renner et al, PRL 80, 149 (1998)Ø. Fischer et al, RMP 79, 353 (2007)

PG

Δ1

+Δ1−Δ1

dSC

Loeser et al, Science 273 325 (1996)Ding et al, Nature 382 51, (1996)Norman et al, Nature 392 , 157 (1998)Shen et al Science 307, 902 (2005)Kanigel et al, Nature Physics 2,447 (2006) Tanaka et al, Science 314, 1912 (2006)

Pseudogap: Temperature-independent energy gap near k~(π,0)

kxky

E

PG

Δ1 Δ1

!

dSC

Loeser et al, Science 273 325 (1996)Ding et al, Nature 382 51, (1996)Norman et al, Nature 392 , 157 (1998)Shen et al Science 307, 902 (2005)Kanigel et al, Nature Physics 2,447 (2006) Tanaka et al, Science 314, 1912 (2006)

Pseudogap: Temperature-dependent energy gap near node

kxky

E

PG

Δ1 Δ1

Δ0

Δ0

!

Development of Fermi arc with underdoping

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

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)

!

Nodal-anti-nodal dichotomy in the underdoped cuprates

S. Hufner, M.A. Hossain, A. Damascelli, and G.A. Sawatzky, Rep. Prog. Phys. 71, 062501 (2008)

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

!

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

Attractive phenomenological model,

but theoretical and microscopic basis is unclear

!

Outline

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

Outline

Increasing SDW order

Spin density wave theory in electron-doped cuprates

Increasing SDW order

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).

!

Increasing SDW order

Spin density wave theory in electron-doped cuprates

Increasing SDW order

!!

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).

Increasing SDW order

Spin density wave theory in electron-doped cuprates

Increasing SDW order

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

to the Fermi liquid.

!!!

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).

N. P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002).

Photoemission in NCCO (electron-doped)

! ! !

Increasing SDW order

Spin density wave theory in electron-doped cuprates

Increasing SDW order

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

to the Fermi liquid.

!!!!

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).

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).

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).

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).

Increasing SDW order

Spin density wave theory in hole-doped cuprates

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

to the Fermi liquid.

!!!!

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).

Spin density wave theory in hole-doped cuprates

N. Harrison, arXiv:0902.2741.

Incommensurate order in YBa2Cu3O6+x

!!

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.

Nature 450, 533 (2007)

Outline

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

1. Nodal-anti-nodal dichotomy in the cupratesSurvey of recent experiments

2. Spin density wave theory of normal metal

From a “large” Fermi surface to electron and hole pockets

3. Algebraic charge liquids Pairing by gauge forces, d-wave superconductivity, and the nodal-anti-nodal dichotomy

Outline

Increasing SDW order

Spin density wave theory in hole-doped cuprates

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

to the Fermi liquid.

!!!!

Increasing SDW order

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

Increasing SDW order

!

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

!

Increasing SDW orderElectronoperator

c1!

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

!

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

c2"c2#

"= Rz

!g+

!g$

"; Rz "

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

".

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

Increasing SDW orderElectronoperator

c2!

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

!

Increasing SDW orderElectronoperator

c2!

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

c2"c2#

"= Rz

!g+

!g$

"; Rz "

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

".

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

!

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

L = Lz + Lg

Lz =1t

!|(!" ! iA" )z!|2 + c2|"! 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"

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 + c2|"! 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"

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.

!

HMetallic

SDWACL

!c

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

!

HMetallic

SDWACL

!c

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

!z!" #= 0

Conventional metalwith SDW order and

electron and hole pockets

!

HMetallic

SDWACL

!c

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

!z!" = 0

A non-Fermi liquidalgebraic chargeliquid (ACL)

with “ghost pockets”of spinless fermions

!

HMetallic

SDWACL

!c

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

“Deconfined” O(4) critical pointwith monopoles in !" suppressed.

!

HMetallic

SDWACL

!c

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

!z!" #= 0

Conventional metalwith SDW order and

electron and hole pockets

!z!" = 0

A non-Fermi liquidalgebraic chargeliquid (ACL)

with “ghost pockets”of spinless fermions

“Deconfined” O(4) critical pointwith monopoles in !" suppressed.

N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys. Rev. Lett. 77, 3009 (1996).I. Ussishkin and A. Stern, Phys. Rev. Lett. 81, 3932 (1998).

Strong pairing of the g± electron pockets

• Problem is similar to double layer quantum Hall systems attotal filling fraction ! = 1. At large layer spacing we have2 composite fermion Fermi surfaces each at filling fraction! = 1/2. At small layer spacing, there is a paired stateformed by attractive interaction mediated by antisymmetricgauge field.

• Gauge forces lead to a s-wave paired state with a Tc of orderthe Fermi energy of the pockets. Inelastic scattering from lowenergy gauge modes lead to very singular g± self energy, butis not pair-breaking.

!g+g!" = !

Strong pairing of the g± electron pockets

• Problem is similar to double layer quantum Hall systems attotal filling fraction ! = 1. At large layer spacing we have2 composite fermion Fermi surfaces each at filling fraction! = 1/2. At small layer spacing, there is a paired stateformed by attractive interaction mediated by antisymmetricgauge field.

• Gauge forces lead to a s-wave paired state with a Tc of orderthe Fermi energy of the pockets. Inelastic scattering from lowenergy gauge modes lead to very singular g± self energy, butis not pair-breaking.

!g+g!" = !

Increasing SDW orderElectronoperator

c2!

Electronoperator

c1!

Strong pairing of the g± electron pockets

!

Increasing SDW orderElectronoperator

c2!

Electronoperator

c1!

Strong pairing of the g± electron pockets

• 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.

!

Increasing SDW orderElectronoperator

c2!

Electronoperator

c1!

Strong pairing of the g± electron pockets

• 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.

!

!

HMetallic

SDWACL

!c

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

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!cInterplay of dSC and SDW order is similar to theory of competing orders

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

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

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!c

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

Neutron scattering on La1.9Sr0.1CuO4

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

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!c

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

Neutron scattering on La1.855Sr0.145CuO4

J. Chang et al., arXiv:0902.1191

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!c

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

Neutron scattering on YBa2Cu3O6.45

D. Haug et al., arXiv:0902.3335

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!cInterplay of dSC and SDW order is similar to theory of competing orders

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

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

Exhibit quantum oscillations without Zeeman splitting

Strong e-pocket pairing removes Fermi surface signatures from H=0 photoemission experiments

Increasing SDW order

g±

!

Increasing SDW order

g±

f±v !

Increasing SDW order

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

%

!

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

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,

Increasing SDW order

f±v ++_

_

!

g±

f±v ++_

_

Increasing SDW order

d-wave pairing of the electrons is associated with

• Strong s-wave pairing of g±

• Weak p-wave pairing of f±v.

!

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!cInterplay of dSC and SDW order is similar to theory of competing orders

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

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

!

H

dSC

Confinement

+VBS order SDW

+dSC

Metallic

SDWACL

!c

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

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

Exhibit quantum oscillations without Zeeman splitting

Neutron scattering on LSCO and YBCO

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

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

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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

★ Non-Landau-Ginzburg theory for loss of spin density wave order in a metal

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

★ New metallic state, the ACL with “ghost” electron and hole pockets, is a useful starting point for building field-doping phase diagram

★ 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