quantum entanglement and superconductivityqpt.physics.harvard.edu/talks/perimeterpublic14.pdf ·...

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

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

YBa2Cu3O6+x

High temperature superconductors

Julian Hetel and Nandini Trivedi, Ohio State University

Nd-Fe-B magnets, YBaCuO superconductor

Julian Hetel and Nandini Trivedi, Ohio State University

Nd-Fe-B magnets, YBaCuO superconductor

YBCO cables !

American Superconductor

Corporation

YBa2Cu3O6+x

High temperature superconductors

YBa2Cu3O6+x

High temperature superconductors

CuO2 plane

Cu

O

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

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

?

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

?

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)

Quantum phase transitions

Quantum superposition and

entanglement

Principles of Quantum Mechanics: 1. Quantum Superposition

The double slit experiment

Interference of water waves

The double slit experiment

Send electrons through the slits

Principles of Quantum Mechanics: 1. Quantum Superposition

The double slit experiment

Interference of electrons

Principles of Quantum Mechanics: 1. Quantum Superposition

The double slit experiment

Interference of electrons

Principles of Quantum Mechanics: 1. Quantum Superposition

Unlike water waves,

electrons arrive one-

by-one

The double slit experiment

Interference of electrons

Which slit does an electron

pass through ?

Principles of Quantum Mechanics: 1. Quantum Superposition

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

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

Let |L� represent the statewith the electron in the left slit

|L�

The double slit experiment

Principles of Quantum Mechanics: 1. Quantum Superposition

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

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

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

Hydrogen atom:

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

Hydrogen atom:

=1⌃2

(|⇥⇤⌅ � |⇤⇥⌅)

Hydrogen molecule:

= _

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

_

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

_

Quantum Entanglement: quantum superposition with more than one particle

Principles of Quantum Mechanics: 1I. Quantum Entanglement

_

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

Quantum phase transitions

Quantum superposition and

entanglement

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

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

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

Square lattice of Cu sites

Square lattice of Cu sites

Remove some electrons

Square lattice of Cu sites

Electrons entangle in (“Cooper”) pairs into chemical bonds

= | ⇥⇤⌅ � | ⇤⇥⌅

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 !

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 !

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 !

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 !

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 !

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 !

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 !

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 !

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 !

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 ?

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 ?

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 ?

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 ?

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 ?

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 ?

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 ?

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

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

Scanning Tunneling Microscopy

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

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).

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).

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).

R(r,150mV)  Bi2212  UD45K

x

y

R(r,150mV)  Bi2212  UD45K

x

yCu sites

R(r,150mV)  Bi2212  UD45K

x

yIntricrate pattern of tunneling currents contains

signatures of long-range quantum entanglement !!(?)

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

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 ?

depth ofentanglement

D-dimensionalspace

Tensor network representation of hierarchical entanglement at quantum critical point

G. Vidal, Phys. Rev. Lett. 99, 220405 (2007)

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 . . . . . .

• 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 !

• 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 !

depth ofentanglement

D-dimensionalspace

Tensor network representation of hierarchical entanglement at quantum critical point

String theory near a D-brane

depth ofentanglement

D-dimensionalspace

Emergent spatial direction of string theory Brian Swingle, arXiv:0905.1317

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

States of matter with long-range quantum entanglement

in D dimensions

String theory and Einstein’s General Relativity

in D+1 dimensions

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” ?

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

Objects so massive that light is gravitationally bound to them.

Black Holes

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.

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

_

Quantum Entanglement across a black hole horizon

_

Quantum Entanglement across a black hole horizon

_

Quantum Entanglement across a black hole horizon

Black hole horizon

_

Black hole horizon

Quantum Entanglement across a black hole horizon

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

Black hole horizon

Quantum Entanglement across a black hole horizon

There are special black hole solutions which mimic strange metals and superconductors !

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

Quantum phase transitions

Quantum superposition and

entanglement

Quantum phase transitions

Quantum superposition and

entanglement Quantum critical points

and long-range entanglement of

electrons in crystalsString theory

and black holes

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