Theory of the competition between spin density waves and d...

Theory of the competition between spin density waves

and d-wave superconductivity in the underdoped cuprates

HARVARD

Talk online: sachdev.physics.harvard.edu

Where is the quantum critical point in the

cuprate superconductors ?

HARVARD

arXiv:0907.0008

Talk online: sachdev.physics.harvard.edu

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 Cenke Xu, Physical Review B 78, 045110 (2008).

Competition between spin density wave order and superconductivity in the underdoped cuprates, Eun Gook Moon and S. Sachdev, Physical Review B 80, 035117 (2009).

Fluctuating spin density waves in metals S. Sachdev, M. Metlitski, Yang Qi, and Cenke Xu arXiv:0907.3732 HARVARD

Hole dynamics in an antiferromagnet across a deconfined quantum critical point,R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, Phys. Rev. B 75, 235122 (2007).

Algebraic charge liquidsR. K. Kaul, Yong-Baek Kim, S. Sachdev, and T. Senthil, Nature Physics 4, 28 (2008).

The cuprate superconductors

N. E. Hussey, J. Phys: Condens. Matter 20, 123201 (2008)

Crossovers in transport properties of hole-doped cuprates

0 0.05 0.1 0.15 0.2 0.25 0.3

(K)

Hole doping

2

+ 2

or

FL?

coh?

( )S-shaped

*

-wave SC

(1 < < 2)AFM

upturnsin ( )

Classicalspin

waves

Dilutetriplon

gas

Quantumcritical

Neel order Pressure in TlCuCl3Christian Ruegg et al. , Phys. Rev. Lett. 100, 205701 (2008)

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

Canonical quantum critical phase diagram of coupled-dimer antiferromagnet

0 0.05 0.1 0.15 0.2 0.25 0.3

T(K)

Hole doping x

d-wave SC

AFM

Strange metal

xm

Crossovers in transport properties of hole-doped cuprates

Strange metal: quantum criticality ofoptimal doping critical point at x = xm ?

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).

0 0.05 0.1 0.15 0.2 0.25 0.3

T(K)

Hole doping x

d-wave SC

AFM

xs

Strange metal

Only candidate quantum critical point observed at low T

Spin density wave order presentbelow a quantum critical point at x = xs

with xs ! 0.12 in the La series of cuprates

! !

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Theory of quantum criticality in the cuprates

Underlying SDW ordering quantum critical pointin metal at x = xm

R. Daou et al., Nature Physics 5, 31 - 34 (2009)

!

Fermi surfaces in electron- and hole-doped cupratesHole states

occupied

Electron states

occupied

!E!ective Hamiltonian for quasiparticles:

H0 = !!

i<j

tijc†i!ci! "

!

k

!kc†k!ck!

with tij non-zero for first, second and third neighbor, leads to satisfactory agree-ment with experiments. The area of the occupied electron states, Ae, fromLuttinger’s theory is

Ae ="

2"2(1! p) for hole-doping p2"2(1 + x) for electron-doping x

The area of the occupied hole states, Ah, which form a closed Fermi surface andso appear in quantum oscillation experiments is Ah = 4"2 !Ae.

Spin density wave theory

A spin density wave (SDW) is the spontaneous appearanceof an oscillatory spin polarization. The electron spin polar-ization is written as

!S(r, ") = !#(r, ")eiK·r

where !# is the SDW order parameter, and K is a fixed or-dering wavevector. For simplicity we will consider the caseof K = ($, $), but our treatment applies to general K.

Spin density wave theory

In the presence of spin density wave order, !" at wavevector K =(#, #), we have an additional term which mixes electron states withmomentum separated by K

Hsdw = !" ·!

k,!,"

c†k,!!$!"ck+K,"

where !$ are the Pauli matrices. The electron dispersions obtainedby diagonalizing H0 + Hsdw for !" ! (0, 0, 1) are

Ek± =%k + %k+K

2±

"#%k " %k+K

2

$+ "2

This leads to the Fermi surfaces shown in the following slides forelectron and hole doping.

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).

Electron pockets

Hole pockets

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).

!

Hole pockets

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

to the Fermi liquid.

Spin density wave theory in hole-doped cuprates

A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, Physical Review Letters 102, 206405 (2009).

Incommensurate order in YBa2Cu3O6+x

!!

Evidence for connection between linear resistivity andstripe-ordering in a cuprate with a low Tc

Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductorR. Daou, Nicolas Doiron-Leyraud, David LeBoeuf, S. Y. Li, Francis Laliberté, Olivier Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough & Louis Taillefer, Nature Physics 5, 31 - 34 (2009)

Magnetic field of upto 35 T

used to suppress superconductivity

FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Theory of quantum criticality in the cuprates

Underlying SDW ordering quantum critical pointin metal at x = xm

R. Daou et al., Nature Physics 5, 31 - 34 (2009)

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Theory of quantum criticality in the cuprates

Onset of d-wave superconductivityhides the critical point x = xm

Fluctuating, paired Fermi

pockets

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Theory of quantum criticality in the cuprates

Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

Fluctuating, paired Fermi

pockets

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuations

Theory of quantum criticality in the cuprates

Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

Fluctuating, paired Fermi

pockets

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin density wave (SDW)

Spin gap

Thermallyfluctuating

SDW

d-wavesuperconductor

Theory of quantum criticality in the cuprates

Competition between SDW order and superconductivitymoves the actual quantum critical point to x = xs < xm.

Fluctuating, paired Fermi

pockets

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

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

Write down a Landau-Ginzburg action for the quantum fluctua-tions of the SDW order (!") and superconductivity (#):

S =!

d2rd$

"12(%! !")2 +

c2

2(!x!")2 +

r

2!"2 +

u

4#!"2

$2

+ & !"2 |#|2%

+!

d2r

"|(!x " i(2e/!c)A)#|2 " |#|2 +

|#|4

2

%

where & > 0 is the repulsion between the two order parameters,and !#A = H is the applied magnetic field.

E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).See also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004);S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, 144516 (2002).

Write down a Landau-Ginzburg action for the quantum fluctua-tions of the SDW order (!") and superconductivity (#):

S =!

d2rd$

"12(%! !")2 +

c2

2(!x!")2 +

r

2!"2 +

u

4#!"2

$2

+ & !"2 |#|2%

+!

d2r

"|(!x " i(2e/!c)A)#|2 " |#|2 +

|#|4

2

%

where & > 0 is the repulsion between the two order parameters,and !#A = H is the applied magnetic field.

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

E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).See also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004);S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, 144516 (2002).

Write down a Landau-Ginzburg action for the quantum fluctua-tions of the SDW order (!") and superconductivity (#):

S =!

d2rd$

"12(%! !")2 +

c2

2(!x!")2 +

r

2!"2 +

u

4#!"2

$2

+ & !"2 |#|2%

+!

d2r

"|(!x " i(2e/!c)A)#|2 " |#|2 +

|#|4

2

%

where & > 0 is the repulsion between the two order parameters,and !#A = H is the applied magnetic field.

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

E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).See also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004);S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, 144516 (2002).

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

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

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

• SDW order is more stable in the metalthan in the superconductor: xm > xs.

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

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

• SDW order is more stable in the metalthan in the superconductor: xm > xs.

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

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

• For doping with xs < x < xm, SDW order appearsat a quantum phase transition at H = Hsdw > 0.

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

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

Neutron scattering on La1.855Sr0.145CuO4

J. Chang et al., Phys. Rev. Lett. 102, 177006 (2009).

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

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, Phys. Rev. Lett. 102, 177006 (2009).

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

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

Neutron scattering on YBa2Cu3O6.45

D. Haug et al., Phys. Rev. Lett. 103, 017001 (2009).

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

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

0.3 0.4 0.5 0.6 0.7

20

40

60

80

100

(b)

(b)(a)

Inte

nsity (

a.u

.)

H = 0T

H = 13.5T

Inte

nsity (

counts

/ ~

12 m

in)

QH (r.l.u.)

H = 0T

H = 14.9T

0 1 2 3 4

100

1000

T = 2K

T = 2K

Energy E (meV)

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, Phys. Rev. Lett. 103, 017001 (2009)

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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). S. E. Sebastian, N. Harrison, C. H. Mielke, Ruixing Liang, D. A. Bonn, W.~N.~Hardy, and G. G. Lonzarich, arXiv:0907.2958

Quantum oscillations without Zeeman splitting

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

Nature 450, 533 (2007)

Quantum oscillations

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

Change in frequency of quantum oscillations in electron-doped materials identifies xm = 0.165

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

Nd2!xCexCuO4

T. Helm, M. V. Kartsovni, M. Bartkowiak, N. Bittner,

M. Lambacher, A. Erb, J. Wosnitza, R. Gross, arXiv:0906.1431

Increasing SDW orderIncreasing SDW order

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

Neutron scattering at H=0 in same material identifies xs = 0.14 < xm

0 100 200 300 4005

10

20

50

100

200

500

Temperature (K)

Spin

cor

rela

tion

leng

th !

/a

x=0.154x=0.150x=0.145x=0.134x=0.129x=0.106x=0.075x=0.038

E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K. Mang, and M. Greven,Nature 445, 186 (2007).

Nd2!xCexCuO4

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

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

Experiments on Nd2!xCexCuO4 show that at lowfields xs = 0.14, while at high fields xm = 0.165.

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Fluctuating, paired Fermi

pockets

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Fluctuating, paired Fermi

pockets

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Fluctuating, paired Fermi

pockets

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Theory of theonset of d-wave

superconductivityfrom a

large Fermi surface

Fluctuating, paired Fermi

pockets

Increasing SDW order

!!! !

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

!"

D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

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

Increasing SDW order

++_

_

!

d-wave pairing of the large Fermi surface

!ck!c"k#" # !k = !0(cos(kx)$ cos(ky))

!"

K

D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

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.

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.

“Nodal/anti-nodal dichotomy” in STM measurements

(0,π) (π,0)

E

kxky

J.C. Davis and collaborators

A. Pushp, C. V. Parker, A. N. Pasupathy, K. K. Gomes, S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani,

Science 324, 1689 (2009)

STM in BSCCO

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

1. Phenomenological quantum theory of competition between superconductivity and SDW order Survey of recent experiments

2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange

3. Superconductivity in the underdoped regimeU(1) gauge theory of fluctuating SDW order

Outline

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Theory of theonset of d-wave

superconductivityfrom small

Fermi pockets

Fluctuating, paired Fermi

pockets

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

Physics ofcompetition:

d-wave SC andSDW “eat up’same pieces ofthe large Fermi

surface.

Begin with SDW ordered state, and rotate to a framepolarized along the local orientation of the SDW order !̂"

!c!c"

"= R

!#+

##

"; R† !̂" · !$R = $z ; R†R = 1

Theory of underdoped cuprates

Increasing SDW order

!!! !

With R =!

z! !z"#z# z"!

"

the theory is invariant under theU(1) gauge transformation

z! " ei"z! ; !+ " e$i"!+ ; !$ " ei"!$

and the SDW order is given by

"̂# = z"!"$!#z#

Theory of underdoped cuprates

Theory of underdoped cuprates

Starting from the “SDW-fermion” modelwith Lagrangian

L =!

k

c†k!

"!

!"! #k

#ck!

!Esdw

!

i

c†i! $̂%i · $&!"ci"eiK·ri

+12t

$!µ $̂%

%2

Theory of underdoped cuprates

L =!

k,p=±

"!†kp

#"

"#! ipA! + $k!pA

$!kp

! Esdw!†kpp!k+K,p

%

we obtain a U(1) gauge theory of

• fermions !p with U(1) charge p = ±1 and pocket Fermisurfaces,

• relativistic complex scalars z! with charge 1, represent-ing the orientational fluctuations of the SDW order

• Shraiman-Siggia term

Theory of underdoped cuprates

+1t

!|(!! ! iA! )z"|2 + v2|"! iA)z"|2 + i"(|z"|2 ! 1)

"

we obtain a U(1) gauge theory of

• fermions !p with U(1) charge p = ±1 and pocket Fermisurfaces,

• relativistic complex scalars z! with charge 1, represent-ing the orientational fluctuations of the SDW order

• Shraiman-Siggia term

Theory of underdoped cuprateswe obtain a U(1) gauge theory of

• fermions !p with U(1) charge p = ±1 and pocket Fermisurfaces,

• relativistic complex scalars z! with charge 1, represent-ing the orientational fluctuations of the SDW order

• Shraiman-Siggia term

!

k,p,q

zq!p/2,"zq+p/2,#

"p · !"(k)

!k

##†

k+q,!#k!q,+ + c.c.

Increasing SDW order

! f±v

!± = f±v near(±"/2,±"/2). Theseare spinless fermions,and are invisible to

photoemission.

Normal state above Tc

Increasing SDW order

!

Normal state above Tc

Gauge forces lead tobound states of f±v

and z! which form“banana Fermi

surfaces” of spinfulfermions visible to

photoemission.

R. K. Kaul, Yong-Baek Kim, S. Sachdev, and T. Senthil, Nature Physics 4, 28 (2008).

Increasing SDW order

!

Pairing across Tc

Focus on pairing near (!, 0), (0, !), where "± ! g±,and the electron operators are

!c1!c1"

"= Rz

!g+

g#

";

!c2!c2"

"= Rz

!g+

"g#

"

Rz !!

z! "z$"z" z$!

".

Electron c1!,spinless fermion g±

Electron c2!,spinless fermion g±

Increasing SDW order

!

Pairing across Tc

Focus on pairing near (!, 0), (0, !), where "± ! g±,and the electron operators are

!c1!c1"

"= Rz

!g+

g#

";

!c2!c2"

"= Rz

!g+

"g#

"

Rz !!

z! "z$"z" z$!

".

Electron c1!,spinless fermion g±

Electron c2!,spinless fermion g±

Fluctuating pocket theory forelectrons near (0, !) and (!, 0)

Attractive gauge forces lead to simple s-wave pairing of the g±

!g+g!" = !

For the physical electron operators, this pairing implies

!c1"c1#" = !!|z!|2

"

!c2"c2#" = #!!|z!|2

"

i.e. d-wave pairing !

Increasing SDW order

++_

_

!

Strong pairing of the g± electron pockets

g±

!g+g!" = !

!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 ++_

_

!

Weak pairing of the f± hole pockets

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

Increasing SDW order

++_

_

!

Features of superconductivity

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

Increasing SDW order

++_

_

!

Strong“s-wave”pairing

Weak“p-wave”pairing

Features of superconductivity

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

Features of superconductivity

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and possible valence bondsolid (VBS) order.

Features of superconductivity

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

Features of superconductivity

R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

• d-wave superconductivity.

• Nodal-anti-nodal dichotomy: strong pairing near (!, 0),(0, !), and weak pairing near zone diagonals.

• Tc decreases as spin correlation increases (competingorder e!ect).

• Shift in quantum critical point of SDW ordering:gauge fluctuations are stronger in the superconduc-tor.

• After onset of superconductivity, monopoles condenseand lead to confinement and nematic and/orvalence bond solid (VBS) order.

Features of superconductivity

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

Magneticquantumcriticality

Spin gapThermallyfluctuating

SDW

d-waveSC

T

Fluctuating, paired Fermi

pockets

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

d-waveSC

T

Tsdw

R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, 045110 (2008).

Fluctuating, paired Fermi

pockets

VBSand/ornematic

H

SC

M

SC+SDW

SDW(Small Fermi

pockets)

"Normal"(Large Fermi

surface)

Hc2

Hsdw

SDWinsulator

Include metal-insulator transition

VBS and/ornematic

Theory of competition between spin density wave order and superconductivity:

Conclusions

Spin density waves and d-wave superconductivity compete for the same portion of the Fermi surface: expressed by U(1) gauge theory of “pocket” fermions and CP1 model

Identified quantum criticality in cuprate superconductors with a critical point at optimal

doping associated with onset of spin density wave order in a metal

Conclusions

Elusive optimal doping quantum critical point has been “hiding in plain sight”.

It is shifted to lower doping by the onset of superconductivity

H

SC

M"Normal"

(Large Fermisurface)

SDW(Small Fermi

pockets)

SC+SDW

Small Fermipockets with

pairing fluctuationsLargeFermi

surface

StrangeMetal

d-waveSC

T

Tsdw

Fluctuating, paired Fermi

pockets

VBSand/ornematic