![Page 1: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/1.jpg)
January 2013, ScientificAmerican.com 45Illustration by Artist NameIllustration by Artist Name Photograph by Zachary Zavislak
MAGNET is being levitated by an unseen superconductor in which
countless trillions of electrons form a vast inter connected quan-
tum state. Astoundingly, the quantum state of many modern
materials is subtly related to the mathematics of black holes.
sad0113Sach3p.indd 45 11/16/12 6:20 PM
Quantum Entanglement and Superconductivity
Subir Sachdev, Perimeter Institute and Harvard University
![Page 2: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/2.jpg)
January 2013, ScientificAmerican.com 45Illustration by Artist NameIllustration by Artist Name Photograph by Zachary Zavislak
MAGNET is being levitated by an unseen superconductor in which
countless trillions of electrons form a vast inter connected quan-
tum state. Astoundingly, the quantum state of many modern
materials is subtly related to the mathematics of black holes.
sad0113Sach3p.indd 45 11/16/12 6:20 PM
Superconductor, levitated by an unseen magnet, in which countless trillions of electrons form a vast interconnected quantum state. Scientific American, January 2013
Quantum Entanglement and Superconductivity
Subir Sachdev, Perimeter Institute and Harvard University
![Page 3: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/3.jpg)
YBa2Cu3O6+x
High temperature superconductors
![Page 4: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/4.jpg)
Julian Hetel and Nandini Trivedi, Ohio State University
Nd-Fe-B magnets, YBaCuO superconductor
![Page 5: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/5.jpg)
Julian Hetel and Nandini Trivedi, Ohio State University
Nd-Fe-B magnets, YBaCuO superconductor
![Page 6: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/6.jpg)
YBCO cables !
American Superconductor
Corporation
![Page 7: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/7.jpg)
YBa2Cu3O6+x
High temperature superconductors
![Page 8: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/8.jpg)
YBa2Cu3O6+x
High temperature superconductors
CuO2 plane
Cu
O
![Page 9: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/9.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
?T.Wu, H. Maya↵re, S. Kramer, M. Horvatic, C. Berthier, W.N. Hardy, R. Liang,D.A. Bonn, and M.-H. Julien, Nature 477, 191 (2011).
Superconductor
![Page 10: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/10.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
T.Wu, H. Maya↵re, S. Kramer, M. Horvatic, C. Berthier, W.N. Hardy, R. Liang,D.A. Bonn, and M.-H. Julien, Nature 477, 191 (2011).
Superconductor
Antiferromagnet
?
![Page 11: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/11.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
T.Wu, H. Maya↵re, S. Kramer, M. Horvatic, C. Berthier, W.N. Hardy, R. Liang,D.A. Bonn, and M.-H. Julien, Nature 477, 191 (2011).
Superconductor
Strange metalAntiferromagnet
?
![Page 12: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/12.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
T.Wu, H. Maya↵re, S. Kramer, M. Horvatic, C. Berthier, W.N. Hardy, R. Liang,D.A. Bonn, and M.-H. Julien, Nature 477, 191 (2011).
Superconductor
Strange metalAntiferromagnet
?
“Quantum critical point” with
“long-range quantum entanglement”. !
Can this help us understand high temperature
superconductivity ? (and the rest of the phase diagram)
![Page 13: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/13.jpg)
Quantum phase transitions
Quantum superposition and
entanglement
![Page 14: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/14.jpg)
Principles of Quantum Mechanics: 1. Quantum Superposition
The double slit experiment
Interference of water waves
![Page 15: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/15.jpg)
The double slit experiment
Send electrons through the slits
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 16: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/16.jpg)
The double slit experiment
Interference of electrons
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 17: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/17.jpg)
The double slit experiment
Interference of electrons
Principles of Quantum Mechanics: 1. Quantum Superposition
Unlike water waves,
electrons arrive one-
by-one
![Page 18: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/18.jpg)
The double slit experiment
Interference of electrons
Which slit does an electron
pass through ?
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 19: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/19.jpg)
The double slit experiment
Interference of electrons
Which slit does an electron
pass through ?
No interference when you watch the electrons
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 20: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/20.jpg)
The double slit experiment
Interference of electrons
Which slit does an electron
pass through ?
Each electron passes
through both slits !
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 21: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/21.jpg)
Let |L� represent the statewith the electron in the left slit
|L�
The double slit experiment
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 22: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/22.jpg)
And |R� represents the statewith the electron in the right slit
Let |L� represent the statewith the electron in the left slit
|L� |R�
The double slit experiment
Principles of Quantum Mechanics: 1. Quantum Superposition
![Page 23: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/23.jpg)
And |R� represents the statewith the electron in the right slit
Let |L� represent the statewith the electron in the left slit
|L� |R�
The double slit experiment
Principles of Quantum Mechanics: 1. Quantum Superposition
Actual state of each electron is
|Li + |Ri
![Page 24: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/24.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
![Page 25: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/25.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
Hydrogen atom:
![Page 26: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/26.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
Hydrogen atom:
=1⌃2
(|⇥⇤⌅ � |⇤⇥⌅)
Hydrogen molecule:
= _
![Page 27: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/27.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
_
![Page 28: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/28.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
_
![Page 29: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/29.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
_
![Page 30: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/30.jpg)
Quantum Entanglement: quantum superposition with more than one particle
Principles of Quantum Mechanics: 1I. Quantum Entanglement
_
Einstein-Podolsky-Rosen “paradox”: Measurement of one particle instantaneously
determines the state of the other particle arbitrarily far away
![Page 31: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/31.jpg)
Quantum phase transitions
Quantum superposition and
entanglement
![Page 32: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/32.jpg)
Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
![Page 33: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/33.jpg)
Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
![Page 34: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/34.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
Superconductor
Strange metalAntiferromagnet
?Quantum critical point
![Page 35: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/35.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
T.Wu, H. Maya↵re, S. Kramer, M. Horvatic, C. Berthier, W.N. Hardy, R. Liang,D.A. Bonn, and M.-H. Julien, Nature 477, 191 (2011).
Superconductor
Strange metalAntiferromagnet
?Spins of electrons on Cu sites
![Page 36: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/36.jpg)
Square lattice of Cu sites
![Page 37: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/37.jpg)
Square lattice of Cu sites
Remove some electrons
![Page 38: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/38.jpg)
Square lattice of Cu sites
Electrons entangle in (“Cooper”) pairs into chemical bonds
= | ⇥⇤⌅ � | ⇤⇥⌅
![Page 39: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/39.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 40: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/40.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 41: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/41.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 42: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/42.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 43: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/43.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 44: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/44.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 45: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/45.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 46: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/46.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 47: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/47.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
Superconductivity !
![Page 48: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/48.jpg)
Square lattice of Cu sites
Cooper pairs form quantum superpositions at different locations: “Bose-Einstein condensation” in which all pairs are “everywhere at the same time”
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 49: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/49.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 50: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/50.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 51: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/51.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 52: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/52.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 53: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/53.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 54: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/54.jpg)
Square lattice of Cu sites
Electrons entangle by exchanging partners, and there is long-range quantum entanglement near the quantum critical point.
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
![Page 55: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/55.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
Superconductor
Strange metalAntiferromagnet
?Quantum critical point
![Page 56: Quantum Entanglement and Superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf · 0.02 0.04 0.06 15 T 0 50 100 0 50 100 1.0 1.5 2.0 33.5 T 28.5 T T (K) T (K) a c e](https://reader034.vdocument.in/reader034/viewer/2022043002/5f7dbeb922eda44b175629d9/html5/thumbnails/56.jpg)
evidence (explaining the rotational symmetry breaking) over a broadtemperature range in YBa2Cu3Oy (refs 14, 19–22). Therefore, insteadof being a defining property of the ordered state, the small amplitude ofthe charge differentiation is more likely to be a consequence of stripeorder (the smectic phase of an electronic liquid crystal17) remainingpartly fluctuating (that is, nematic).In stripe copper oxides, charge order at T5Tcharge is always accom-
panied by spin order at Tspin,Tcharge. Slowing down of the spin
fluctuations strongly enhances the spin–lattice (1/T1) and spin–spin(1/T2) relaxation rates between Tcharge and Tspin for
139La nuclei. Forthemore strongly hyperfine-coupled 63Cu, the relaxation rates becomeso large that the Cu signal is gradually ‘wiped out’ on cooling belowTcharge (refs 18, 23, 24). In contrast, the 63Cu(2) signal here inYBa2Cu3Oy does not experience any intensity loss and 1/T1 does notshow any peak or enhancement as a function of temperature (Fig. 3).Moreover, the anisotropy of the linewidth (SupplementaryInformation) indicates that the spins, although staggered, align mostlyalong the field (that is, c axis) direction, and the typical width of thecentral lines at base temperature sets an uppermagnitude for the staticspin polarization as small as gÆSzæ# 23 1023mB for both samples infields of,30T. These consistent observations rule out the presence ofmagnetic order, in agreement with an earlier suggestion based on thepresence of free-electron-like Zeeman splitting6.In stripe-ordered copper oxides, the strong increase of 1/T2 on
cooling below Tcharge is accompanied by a crossover of the time decayof the spin-echo from the high-temperature Gaussian formexp(2K(t/T2G)2) to an exponential form exp(2t/T2E)18,23. A similarcrossover occurs here, albeit in a less extreme manner because of theabsence ofmagnetic order: 1/T2 sharply increases belowTcharge and thedecay actually becomes a combination of exponential and Gaussiandecays (Fig. 3). In Supplementary Information we provide evidencethat the typical values of the 1/T2E below Tcharge imply that antiferro-magnetic (or ‘spin-density-wave’) fluctuations are slow enough toappear frozen on the timescale of a cyclotron orbit 1/vc< 10212 s.In principle, such slow fluctuations could reconstruct the Fermi sur-face, provided that spins are correlated over large enough distances25,26
(see also ref. 9). It is unclear whether this condition is fulfilled here. The
0.04 0.08 0.12 0.160
40
80
120
Superconducting
Spinorder
T (K
)
p (hole/Cu)
Field-inducedcharge order
Figure 4 | Phase diagram of underdoped YBa2Cu3Oy. The charge orderingtemperature Tcharge (defined as the onset of the Cu2F line splitting; blue opencircles) coincides with T0 (brown plus signs), the temperature at which the Hallconstant RH changes its sign. T0 is considered as the onset of the Fermi surfacereconstruction11–13. The continuous line represents the superconductingtransition temperature Tc. The dashed line indicates the speculative nature ofthe extrapolation of the field-induced charge order. The magnetic transitiontemperatures (Tspin) are frommuon-spin-rotation (mSR) data (green stars)27.T0and Tspin vanish close to the same critical concentration p5 0.08. A scenario offield-induced spin order has been predicted for p. 0.08 (ref. 8) by analogy withLa1.855Sr0.145CuO4, for which the non-magnetic ground state switches toantiferromagnetic order in fields greater than a few teslas (ref. 7 and referencestherein).Ourwork, however, shows that spin order does not occur up to,30T.In contrast, the field-induced charge order reported here raises the question ofwhether a similar field-dependent charge order actually underlies the fielddependence of the spin order in La22xSrxCuO4 and YBa2Cu3O6.45. Error barsrepresent the uncertainty in defining the onset of theNMR line splitting (Fig. 1fand Supplementary Figs 8–10).
0 20 40 60 80 1000
4
8
100
10–1
10–2
1/T 1
(ms–
1 )1/T 2
(�s–
1 )
33.5 T28.5 T
15 T15 T
T (K)
Inte
nsity
(arb
. uni
ts)
15 T
0
0.02
0.04
0.06
15 T
0 50 100 0 50 1001.0
1.5
2.0
33.5 T
28.5 T
T (K)
T (K)
a
c
e
b
d
f
g
�
Figure 3 | Slow spin fluctuations instead of spin order. a, b, Temperaturedependence of the planar 63Cu spin-lattice relaxation rate 1/T1 for p5 0.108(a) and p5 0.12 (b). The absence of any peak/enhancement on cooling rulesout the occurrence of a magnetic transition. c, d, Increase in the 63Cu spin–spinrelaxation rate 1/T2 on cooling below,Tcharge, obtained from a fit of the spin-echo decay to a stretched form s(t) / exp(2(t/T2)
a), for p5 0.108 (c) andp5 0.12 (d). e, f, Stretching exponent a for p5 0.108 (e) and p5 0.12 (f). Thedeviation from a5 2 on cooling arises mostly from an intrinsic combination ofGaussian and exponential decays, combined with some spatial distribution ofT2 values (Supplementary Information). The grey areas define the crossovertemperature Tslow below which slow spin fluctuations cause 1/T2 to increaseand to become field dependent; note that the change of shape of the spin-echodecay occurs at a slightly higher (,115K) temperature than Tslow. Tslow isslightly lower thanTcharge, which is consistentwith the slow fluctuations being aconsequence of charge-stripe order. The increase of a at the lowesttemperatures probably signifies that the condition cÆhz2æ1/2tc= 1, where tc isthe correlation time, is no longer fulfilled, so that the associated decay is nolonger a pure exponential. We note that the upturn of 1/T2 is already present at15T, whereas no line splitting is detected at this field. The field therefore affectsthe spin fluctuations quantitatively but not qualitatively. g, Plot of NMR signalintensity (corrected for a temperature factor 1/T and for the T2 decay) againsttemperature. Open circles, p5 0.108 (28.5T); filled circles, p5 0.12 (33.5T).The absence of any intensity loss at low temperatures also rules out the presenceof magnetic order with any significant moment. Error bars represent the addeduncertainties in signal analysis, experimental conditions andT2measurements.All measurements are with H | | c.
LETTER RESEARCH
8 S E P T E M B E R 2 0 1 1 | V O L 4 7 7 | N A T U R E | 1 9 3
Macmillan Publishers Limited. All rights reserved©2011
Superconductor
Strange metalAntiferromagnet
?Quantum critical point
Scanning tunneling microscopy
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Scanning Tunneling Microscopy
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Ultra low vibration cryostat.
to Pumps
Load Lock
Lead Filled Legs
High-field SC Magnet
Lead Filled Table
Turbo Pump
Air Springs
Fridge
Cleaver
L4He
J. C. Davis group, Rev. Sci. Inst. 70, 1459 (1999).SI-STM System
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STM Head
•Massive ULV Cryostat •Subkelvin Fridge •STM + High Field Magnet •Sample Exchange from RT •Cryogenic UHV Cleave
to Pumps
Load Lock
Lead Filled Legs
High-field SC Magnet
Turbo Pump
Air Springs
Cleaver
L4He
Sample insert
SI-STM SystemJ. C. Davis group, Rev. Sci. Inst. 70, 1459 (1999).
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xy
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007).
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xy
Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma,M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007).
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R(r,150mV) Bi2212 UD45K
x
y
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R(r,150mV) Bi2212 UD45K
x
yCu sites
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R(r,150mV) Bi2212 UD45K
x
yIntricrate pattern of tunneling currents contains
signatures of long-range quantum entanglement !!(?)
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Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
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Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
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Square lattice of Cu sites
Long-range entanglement has a heierarchical structure: electrons entangle in pairs, pairs entangle with pairs, and so on..…
= | ⇥⇤⌅ � | ⇤⇥⌅
High temperature superconductivity ?
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depth ofentanglement
D-dimensionalspace
Tensor network representation of hierarchical entanglement at quantum critical point
G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)
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String theory
• Allows unification of the standard model of particlephysics with Einstein’s theory of gravitation(general relativity).
• Vibrations of a string (its “musical notes”) corre-spond to quarks, gravitons, the Higgs boson, pho-tons, gluons . . . . . .
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• A D-brane is a D-dimensional surface on which strings
can end.
• If we focused only on the blue points on theD-dimensional
surface, they would appear to us to have long-range
quantum entanglement !
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• A D-brane is a D-dimensional surface on which strings
can end.
• If we focused only on the blue points on theD-dimensional
surface, they would appear to us to have long-range
quantum entanglement !
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depth ofentanglement
D-dimensionalspace
Tensor network representation of hierarchical entanglement at quantum critical point
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String theory near a D-brane
depth ofentanglement
D-dimensionalspace
Emergent spatial direction of string theory Brian Swingle, arXiv:0905.1317
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depth ofentanglement
D-dimensionalspace
Brian Swingle, arXiv:0905.1317
Emergent spatial direction of string theory
Tensor network representation of hierarchical entanglement at quantum critical point
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States of matter with long-range quantum entanglement
in D dimensions
String theory and Einstein’s General Relativity
in D+1 dimensions
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States of matter with long-range quantum entanglement
in D dimensions
String theory and Einstein’s General Relativity
in D+1 dimensions
Are there solutions of Einstein’s General Relativity in D+1 dimensions which correspond to superconductors and “strange metals” ?
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Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
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Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
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Objects so massive that light is gravitationally bound to them.
Black Holes
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Horizon radius R =2GM
c2
Objects so massive that light is gravitationally bound to them.
Black Holes
In Einstein’s theory, the region inside the black hole horizon is disconnected from
the rest of the universe.
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Around 1974, Bekenstein and Hawking showed that the application of the
quantum theory across a black hole horizon led to many astonishing
conclusions
Black Holes + Quantum theory
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Quantum Entanglement across a black hole horizon
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Quantum Entanglement across a black hole horizon
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Quantum Entanglement across a black hole horizon
Black hole horizon
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Black hole horizon
Quantum Entanglement across a black hole horizon
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Black hole horizon
Quantum Entanglement across a black hole horizon
There is long-range quantum entanglement between the inside
and outside of a black hole
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Black hole horizon
Quantum Entanglement across a black hole horizon
There are special black hole solutions which mimic strange metals and superconductors !
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Black hole horizon
Quantum Entanglement across a black hole horizon
There are special black hole solutions which mimic strange metals and superconductors !
These black hole states of General Relativity in D+1 dimensions correspond to (and allow us
to compute the properties of) superconductors and strange metals in
D dimensions
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Quantum phase transitions
Quantum superposition and
entanglement
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Quantum phase transitions
Quantum superposition and
entanglement Quantum critical points
and long-range entanglement of
electrons in crystalsString theory
and black holes
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January 2013, ScientificAmerican.com 45Illustration by Artist NameIllustration by Artist Name Photograph by Zachary Zavislak
MAGNET is being levitated by an unseen superconductor in which
countless trillions of electrons form a vast inter connected quan-
tum state. Astoundingly, the quantum state of many modern
materials is subtly related to the mathematics of black holes.
sad0113Sach3p.indd 45 11/16/12 6:20 PM
Superconductor, levitated by an unseen magnet, in which countless trillions of electrons form a vast interconnected quantum state. Scientific American, January 2013
Quantum Entanglement and Superconductivity
Subir Sachdev, Perimeter Institute and Harvard University