Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing...
Transcript of Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing...
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Competing orders in the underdoped cuprates
Talk online: sachdev.physics.harvard.edu
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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
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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.
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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.
![Page 5: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/5.jpg)
The cuprate superconductors
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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.
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The cuprate superconductors
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! !
!
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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
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! !
!
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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
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Nature 450, 533 (2007)
(a) Quantum oscillations
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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
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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"
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
Hc2
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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"
E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).
(b) Phenomenological quantum theory of competing ordersCompetition between superconductivity (SC) and spin-density wave (SDW) order
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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
Hsdw
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.
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!
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
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!
H
dSC (Confinement + VBS order) SDW
+ dSC
MetallicSDW
Exotic metal with “ghost”
Fermi pockets
!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)
Hc2
Hsdw
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!
H
dSC (Confinement + VBS order) SDW
+ dSC
MetallicSDW
Exotic metal with “ghost”
Fermi pockets
!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
Hc2
Hsdw
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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).
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!
H
dSC (Confinement + VBS order) SDW
+ dSC
MetallicSDW
Exotic metal with “ghost”
Fermi pockets
!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
Hc2
Hsdw
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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.
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!
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
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!
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).
Exhibit quantum oscillations without Zeeman splitting
Phase diagramas a function of hole density ! ! t and magnetic field H.
Hc2
Hsdw
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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).
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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).
!
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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).
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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
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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
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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).
!
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Spin density wave theory in hole-doped cuprates
A. J. Millis and M. R. Norman, Physical Review B 76, 220503 (2007). N. Harrison, arXiv:0902.2741.
Incommensurate order in YBa2Cu3O6+x
!!
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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
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Y. Kohsaka et al., Nature 454, 1072, (2008)
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
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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
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
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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
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
![Page 36: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/36.jpg)
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
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
![Page 37: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/37.jpg)
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
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
![Page 38: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/38.jpg)
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
!
(c) Nodal-anti-nodal dichotomy in the underdoped cuprates
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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.
![Page 40: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/40.jpg)
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.
![Page 41: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/41.jpg)
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.
!!!!
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Increasing SDW order
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 !".
!"
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! !
Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates
!"
!
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Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).
Approaching the onset of antiferromagnetism in the spin-fluctuation theory
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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.
![Page 46: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/46.jpg)
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.
![Page 47: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/47.jpg)
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.
![Page 48: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/48.jpg)
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).
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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
!!!!
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Charge carriers in the lightly-doped cuprates with Neel order
Electron pockets Hole
pockets
Increasing SDW order
!
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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
!
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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
!
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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)
!
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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.
!
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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"
![Page 56: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/56.jpg)
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
$g"
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Strong pairing of the g± electron pockets
• Gauge forces lead to a s-wave paired state with a Tc of order
the Fermi energy of the pockets. Inelastic scattering from low
energy gauge modes lead to very singular g± self energy, but
is not pair-breaking.
!g+g!" = !
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Increasing SDW orderElectronoperator
c2!
Electronoperator
c1!
Strong pairing of the g± electron pockets
!
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Increasing SDW orderElectronoperator
c2!
Electronoperator
c1!
Strong pairing of the g± electron pockets
• Transforming back to the physical fermions:!
c1!c1"
"=
!z! !z#"z" z#!
" !g+
g$
";
!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.
!
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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.
!
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Increasing SDW order
g±
!
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Increasing SDW order
g±
f±v !
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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
%
!
![Page 64: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/64.jpg)
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,
![Page 65: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/65.jpg)
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 ++_
_
!
![Page 66: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/66.jpg)
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.
!
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Nature 456, 77 (2008).
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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"
![Page 69: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/69.jpg)
• 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.
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"
![Page 70: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/70.jpg)
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.
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Universal theory of superconductivity
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
• 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"
dSC
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Universal theory of superconductivity
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.
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!
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
![Page 74: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/74.jpg)
!
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.
Exhibit quantum oscillations without Zeeman splitting
Hc2
Hsdw
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★ 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
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★ 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
![Page 77: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/77.jpg)
!
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
![Page 78: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/78.jpg)
★ 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
![Page 79: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/79.jpg)
★ 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
![Page 80: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/80.jpg)
★ 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
![Page 81: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/81.jpg)
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
![Page 82: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/82.jpg)
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)
![Page 83: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/83.jpg)
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
![Page 84: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/84.jpg)
★ 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
![Page 85: Competing orders in the underdoped cupratesqpt.physics.harvard.edu/talks/supergold.pdf · Competing order predictions: • Upper-critical field, H c2, decreases as SDW is enhanced](https://reader033.fdocuments.in/reader033/viewer/2022050610/5fb13f1276691d18ef216abf/html5/thumbnails/85.jpg)
★ 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