theory of the competition between spin density waves and d...
TRANSCRIPT
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