Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc)...

60
Bond order in two-dimensional metals with antiferromagnetic exchange interactions HARVARD Gordon Research Conference Les Diablerets May 14, 2013 Subir Sachdev Tuesday, May 14, 13

Transcript of Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc)...

Page 1: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Bond order in two-dimensional metals with antiferromagnetic exchange interactions

HARVARD

Gordon Research ConferenceLes DiableretsMay 14, 2013

Subir Sachdev

Tuesday, May 14, 13

Page 2: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity⇠ ⇢0 +AT↵

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)Tuesday, May 14, 13

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Fermi surface+antiferromagnetism

The electron spin polarization obeys�

⌃S(r, �)⇥

= ⌃⇥(r, �)eiK·r

where K is the ordering wavevector.

+

Metal with “large” Fermi surface

Tuesday, May 14, 13

Page 4: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Metal with “large” Fermi surface

Fermi surface+antiferromagnetism

Tuesday, May 14, 13

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“Hot” spots

Fermi surface+antiferromagnetism

K

Tuesday, May 14, 13

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Fermi surface+antiferromagnetism

Electron and hole pockets in

antiferromagnetic phase

with antiferromagnetic order parameter h~'i 6= 0

Tuesday, May 14, 13

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r

Metal with “large” Fermi surface

h~'i = 0

Metal with electron and hole pockets

Increasing SDW order

h~'i 6= 0

Fermi surface+antiferromagnetism

Tuesday, May 14, 13

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d-wave superconductor: particle-particle pairing at and near hot spots, with

sign-changing pairing amplitude

Dc†k↵c

†�k�

E= "

↵�

S

(cos kx

� cos ky

)

�S

��S

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983)D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)

Tuesday, May 14, 13

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Sign-problem-free Quantum Monte Carlo for antiferromagnetism in metals

−2 −1 0 1 2 3−2

0

2

4

6

8

10 x 10−4

r

P ±(xmax

)L = 10

L = 14

L = 12

rc

P+

_

_P_

|

s/d pairing amplitudes P+/P�as a function of the tuning parameter r

E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

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FluctuatingFermi

pocketsLargeFermi

surface

StrangeMetal

Spin density wave (SDW)

Underlying SDW ordering quantum critical pointin metal at x = xm

Increasing SDW order

T*QuantumCritical

Fermi surface+antiferromagnetism

Tuesday, May 14, 13

Page 11: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

LargeFermi

surface

StrangeMetal

Spin density wave (SDW)

d-wavesuperconductor

Small Fermipockets with

pairing fluctuationsFluctuating, paired Fermi

pockets

QCP for the onset of SDW order is actually within a superconductor

Fermi surface+antiferromagnetism

QuantumCritical

Tuesday, May 14, 13

Page 12: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

TSDW Tc

T0

2.0

0

α"

1.0 SDW

Superconductivity

BaFe2(As1-xPx)2

AF

Resistivity⇠ ⇢0 +AT↵

S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda,

Physical Review B 81, 184519 (2010)Tuesday, May 14, 13

Page 13: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

LETTERdoi:10.1038/nature10345

Magnetic-field-induced charge-stripe order in thehigh-temperature superconductor YBa2Cu3OyTaoWu1, Hadrien Mayaffre1, Steffen Kramer1, Mladen Horvatic1, Claude Berthier1, W. N. Hardy2,3, Ruixing Liang2,3, D. A. Bonn2,3

& Marc-Henri Julien1

Electronic charges introduced in copper-oxide (CuO2) planesgenerate high-transition-temperature (Tc) superconductivity but,under special circumstances, they can also order into filamentscalled stripes1. Whether an underlying tendency towards chargeorder is present in all copper oxides and whether this has anyrelationship with superconductivity are, however, two highly con-troversial issues2,3. To uncover underlying electronic order, mag-netic fields strong enough to destabilize superconductivity can beused. Such experiments, including quantum oscillations4–6 inYBa2Cu3Oy (an extremely clean copper oxide in which chargeorder has not until now been observed) have suggested that super-conductivity competes with spin, rather than charge, order7–9. Herewe report nuclear magnetic resonance measurements showing thathigh magnetic fields actually induce charge order, without spinorder, in the CuO2 planes of YBa2Cu3Oy. The observed static, uni-directional, modulation of the charge density breaks translationalsymmetry, thus explaining quantum oscillation results, and weargue that it is most probably the same 4a-periodic modulationas in stripe-ordered copper oxides1. That it develops only whensuperconductivity fades away and near the same 1/8 hole dopingas in La22xBaxCuO4 (ref. 1) suggests that charge order, althoughvisibly pinned by CuO chains in YBa2Cu3Oy, is an intrinsic pro-pensity of the superconducting planes of high-Tc copper oxides.The ortho II structure of YBa2Cu3O6.54 (p5 0.108, where p is the

hole concentration per planar Cu) leads to two distinct planar CuNMR sites: Cu2F are those Cu atoms located below oxygen-filledchains, and Cu2E are those below oxygen-empty chains10. The maindiscovery of ourwork is that, on cooling in a fieldH0 of 28.5 T along thec axis (that is, in the conditions for which quantum oscillations areresolved; see Supplementary Materials), the Cu2F lines undergo aprofound change, whereas theCu2E lines do not (Fig. 1). To first order,this change can be described as a splitting of Cu2F into two sites havingboth different hyperfine shiftsK5 Æhzæ/H0 (where Æhzæ is the hyperfinefield due to electronic spins) and quadrupole frequencies nQ (related tothe electric field gradient). Additional effects might be present (Fig. 1),but they areminor in comparisonwith the observed splitting. Changesin field-dependent and temperature-dependent orbital occupancy (forexample dx2{y2 versus dz2{r2 ) without on-site change in electronicdensity are implausible, and any change in out-of-plane charge densityor lattice would affect Cu2E sites as well. Thus, the change in nQ canonly arise from a differentiation in the charge density between Cu2Fsites (or at the oxygen sites bridging them). A change in the asymmetryparameter and/or in the direction of the principal axis of the electricfield gradient could also be associated with this charge differentiation,but these are relatively small effects.The charge differentiation occurs below Tcharge5 506 10K for

p5 0.108 (Fig. 1 and Supplementary Figs 9 and 10) and 676 5K forp5 0.12 (Supplementary Figs 7 and 8). Within error bars, for each ofthe samples Tcharge coincides with T0, the temperature at which theHall constant RH becomes negative, an indication of the Fermi surface

reconstruction11–13. Thus, whatever the precise profile of the staticcharge modulation is, the reconstruction must be related to the trans-lational symmetry breaking by the charge ordered state.The absence of any splitting or broadening of Cu2E lines implies a

one-dimensional character of the modulation within the planes andimposes strong constraints on the charge pattern. Actually, only twotypes of modulation are compatible with a Cu2F splitting (Fig. 2). Thefirst is a commensurate short-range (2a or 4a period) modulationrunning along the (chain) b axis. However, this hypothesis is highlyunlikely: to the best of our knowledge, no such modulation has everbeen observed in the CuO2 planes of any copper oxide; it would there-fore have to be triggered by a charge modulation pre-existing in thefilled chains. A charge-density wave is unlikely because the finite-sizechains are at best poorly conducting in the temperature and dopingrange discussed here11,14. Any inhomogeneous charge distributionsuch as Friedel oscillations around chain defects would broaden ratherthan split the lines. Furthermore, we can conclude that charge orderoccurs only for high fields perpendicular to the planes because theNMR lines neither split at 15T nor split in a field of 28.5 T parallelto the CuO2 planes (along either a or b), two situations in whichsuperconductivity remains robust (Fig. 1). This clear competitionbetween charge order and superconductivity is also a strong indicationthat the charge ordering instability arises from the planes.Theonlyother patterncompatiblewithNMRdata is an alternationof

more and less charged Cu2F rows defining a modulation with a periodof four lattice spacings along the a axis (Fig. 2). Strikingly, this corre-sponds to the (site-centred) charge stripes found in La22xBaxCuO4 atdoping levels near p5 x5 0.125 (ref. 1). Being a proven electronicinstability of the planes, which is detrimental to superconductivity2,stripe ordernot onlyprovides a simple explanationof theNMRsplittingbut also rationalizes the striking effect of the field. Stripe order is alsofully consistent with the remarkable similarity of transport data inYBa2Cu3Oy and in stripe-ordered copper oxides (particularly thedome-shaped dependence ofT0 around p5 0.12)11–13. However, stripesmust be parallel from plane to plane in YBa2Cu3Oy, whereas they areperpendicular in, for example, La22xBaxCuO4. We speculate that thisexplains why the charge transport along the c axis in YBa2Cu3Oy

becomes coherent in high fields below T0 (ref. 15). If so, stripe fluctua-tions must be involved in the incoherence along c above T0.Once we know the doping dependence of nQ (ref. 16), the difference

DnQ5 3206 50 kHz for p5 0.108 implies a charge density variationas small as Dp5 0.036 0.01 hole between Cu2Fa and Cu2Fb. Acanonical stripe description (Dp5 0.5 hole) is therefore inadequateat the NMR timescale of ,1025 s, at which most (below T0) or all(above T0) of the charge differentiation is averaged out by fluctuationsfaster than 105 s21. This should not be a surprise: themetallic nature ofthe compound at all fields is incompatible with full charge order, evenif this order is restricted to the direction perpendicular to the stripes17.Actually, there is compelling evidence of stripe fluctuations down tovery low temperatures in stripe-ordered copper oxides18, and indirect

1Laboratoire National des Champs Magnetiques Intenses, UPR 3228, CNRS-UJF-UPS-INSA, 38042 Grenoble, France. 2Department of Physics and Astronomy, University of British Columbia, Vancouver,British Columbia V6T1Z1, Canada. 3Canadian Institute for Advanced Research, Toronto, Ontario M5G1Z8, Canada.

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LETTERdoi:10.1038/nature10345

Magnetic-field-induced charge-stripe order in thehigh-temperature superconductor YBa2Cu3OyTaoWu1, Hadrien Mayaffre1, Steffen Kramer1, Mladen Horvatic1, Claude Berthier1, W. N. Hardy2,3, Ruixing Liang2,3, D. A. Bonn2,3

& Marc-Henri Julien1

Electronic charges introduced in copper-oxide (CuO2) planesgenerate high-transition-temperature (Tc) superconductivity but,under special circumstances, they can also order into filamentscalled stripes1. Whether an underlying tendency towards chargeorder is present in all copper oxides and whether this has anyrelationship with superconductivity are, however, two highly con-troversial issues2,3. To uncover underlying electronic order, mag-netic fields strong enough to destabilize superconductivity can beused. Such experiments, including quantum oscillations4–6 inYBa2Cu3Oy (an extremely clean copper oxide in which chargeorder has not until now been observed) have suggested that super-conductivity competes with spin, rather than charge, order7–9. Herewe report nuclear magnetic resonance measurements showing thathigh magnetic fields actually induce charge order, without spinorder, in the CuO2 planes of YBa2Cu3Oy. The observed static, uni-directional, modulation of the charge density breaks translationalsymmetry, thus explaining quantum oscillation results, and weargue that it is most probably the same 4a-periodic modulationas in stripe-ordered copper oxides1. That it develops only whensuperconductivity fades away and near the same 1/8 hole dopingas in La22xBaxCuO4 (ref. 1) suggests that charge order, althoughvisibly pinned by CuO chains in YBa2Cu3Oy, is an intrinsic pro-pensity of the superconducting planes of high-Tc copper oxides.The ortho II structure of YBa2Cu3O6.54 (p5 0.108, where p is the

hole concentration per planar Cu) leads to two distinct planar CuNMR sites: Cu2F are those Cu atoms located below oxygen-filledchains, and Cu2E are those below oxygen-empty chains10. The maindiscovery of ourwork is that, on cooling in a fieldH0 of 28.5 T along thec axis (that is, in the conditions for which quantum oscillations areresolved; see Supplementary Materials), the Cu2F lines undergo aprofound change, whereas theCu2E lines do not (Fig. 1). To first order,this change can be described as a splitting of Cu2F into two sites havingboth different hyperfine shiftsK5 Æhzæ/H0 (where Æhzæ is the hyperfinefield due to electronic spins) and quadrupole frequencies nQ (related tothe electric field gradient). Additional effects might be present (Fig. 1),but they areminor in comparisonwith the observed splitting. Changesin field-dependent and temperature-dependent orbital occupancy (forexample dx2{y2 versus dz2{r2 ) without on-site change in electronicdensity are implausible, and any change in out-of-plane charge densityor lattice would affect Cu2E sites as well. Thus, the change in nQ canonly arise from a differentiation in the charge density between Cu2Fsites (or at the oxygen sites bridging them). A change in the asymmetryparameter and/or in the direction of the principal axis of the electricfield gradient could also be associated with this charge differentiation,but these are relatively small effects.The charge differentiation occurs below Tcharge5 506 10K for

p5 0.108 (Fig. 1 and Supplementary Figs 9 and 10) and 676 5K forp5 0.12 (Supplementary Figs 7 and 8). Within error bars, for each ofthe samples Tcharge coincides with T0, the temperature at which theHall constant RH becomes negative, an indication of the Fermi surface

reconstruction11–13. Thus, whatever the precise profile of the staticcharge modulation is, the reconstruction must be related to the trans-lational symmetry breaking by the charge ordered state.The absence of any splitting or broadening of Cu2E lines implies a

one-dimensional character of the modulation within the planes andimposes strong constraints on the charge pattern. Actually, only twotypes of modulation are compatible with a Cu2F splitting (Fig. 2). Thefirst is a commensurate short-range (2a or 4a period) modulationrunning along the (chain) b axis. However, this hypothesis is highlyunlikely: to the best of our knowledge, no such modulation has everbeen observed in the CuO2 planes of any copper oxide; it would there-fore have to be triggered by a charge modulation pre-existing in thefilled chains. A charge-density wave is unlikely because the finite-sizechains are at best poorly conducting in the temperature and dopingrange discussed here11,14. Any inhomogeneous charge distributionsuch as Friedel oscillations around chain defects would broaden ratherthan split the lines. Furthermore, we can conclude that charge orderoccurs only for high fields perpendicular to the planes because theNMR lines neither split at 15T nor split in a field of 28.5 T parallelto the CuO2 planes (along either a or b), two situations in whichsuperconductivity remains robust (Fig. 1). This clear competitionbetween charge order and superconductivity is also a strong indicationthat the charge ordering instability arises from the planes.Theonlyother patterncompatiblewithNMRdata is an alternationof

more and less charged Cu2F rows defining a modulation with a periodof four lattice spacings along the a axis (Fig. 2). Strikingly, this corre-sponds to the (site-centred) charge stripes found in La22xBaxCuO4 atdoping levels near p5 x5 0.125 (ref. 1). Being a proven electronicinstability of the planes, which is detrimental to superconductivity2,stripe ordernot onlyprovides a simple explanationof theNMRsplittingbut also rationalizes the striking effect of the field. Stripe order is alsofully consistent with the remarkable similarity of transport data inYBa2Cu3Oy and in stripe-ordered copper oxides (particularly thedome-shaped dependence ofT0 around p5 0.12)11–13. However, stripesmust be parallel from plane to plane in YBa2Cu3Oy, whereas they areperpendicular in, for example, La22xBaxCuO4. We speculate that thisexplains why the charge transport along the c axis in YBa2Cu3Oy

becomes coherent in high fields below T0 (ref. 15). If so, stripe fluctua-tions must be involved in the incoherence along c above T0.Once we know the doping dependence of nQ (ref. 16), the difference

DnQ5 3206 50 kHz for p5 0.108 implies a charge density variationas small as Dp5 0.036 0.01 hole between Cu2Fa and Cu2Fb. Acanonical stripe description (Dp5 0.5 hole) is therefore inadequateat the NMR timescale of ,1025 s, at which most (below T0) or all(above T0) of the charge differentiation is averaged out by fluctuationsfaster than 105 s21. This should not be a surprise: themetallic nature ofthe compound at all fields is incompatible with full charge order, evenif this order is restricted to the direction perpendicular to the stripes17.Actually, there is compelling evidence of stripe fluctuations down tovery low temperatures in stripe-ordered copper oxides18, and indirect

1Laboratoire National des Champs Magnetiques Intenses, UPR 3228, CNRS-UJF-UPS-INSA, 38042 Grenoble, France. 2Department of Physics and Astronomy, University of British Columbia, Vancouver,British Columbia V6T1Z1, Canada. 3Canadian Institute for Advanced Research, Toronto, Ontario M5G1Z8, Canada.

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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

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Tuesday, May 14, 13

Page 14: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry

2. Approximate pseudospin symmetry on the lattice t-J model Pseudogap and bond order in the underdoped cuprates

Tuesday, May 14, 13

Page 15: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry

2. Approximate pseudospin symmetry on the lattice t-J model Pseudogap and bond order in the underdoped cuprates

Tuesday, May 14, 13

Page 16: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

HJ =

X

i<j

Jij ~Si · ~Sj

with

~Si =

12c

†i↵~�↵�ci� is the antiferromagnetic exchange interaction.

Introduce the Nambu spinor

i" =

✓ci"c†i#

◆, i# =

✓ci#�c†i"

Then we can write

HJ =

1

8

X

i<j

Jij

†i↵~�↵� i�

⌘·⇣

†j�~��� i�

which is invariant under independent SU(2) pseudospin transforma-

tions on each site

i↵ ! Ui i↵

This pseudospin (gauge) symmetry is important in classifying spin

liquid ground states of HJ . It is fully broken by the electron hopping

tij but does have remnant consequences in doped spin liquid states.

Pseudospin symmetry of the exchange interaction

I. A✏eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988)E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988)P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

Tuesday, May 14, 13

Page 17: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

HJ =

X

i<j

Jij ~Si · ~Sj

with

~Si =

12c

†i↵~�↵�ci� is the antiferromagnetic exchange interaction.

Introduce the Nambu spinor

i" =

✓ci"c†i#

◆, i# =

✓ci#�c†i"

Then we can write

HJ =

1

8

X

i<j

Jij

†i↵~�↵� i�

⌘·⇣

†j�~��� i�

which is invariant under independent SU(2) pseudospin transforma-

tions on each site

i↵ ! Ui i↵

This pseudospin (gauge) symmetry is important in classifying spin

liquid ground states of HJ . It is fully broken by the electron hopping

tij but does have remnant consequences in doped spin liquid states.

Pseudospin symmetry of the exchange interaction

I. A✏eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988)E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988)P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

HtJ = �X

i,j

tijc†i↵cj↵ +

X

i<j

Jij ~Si · ~Sj

Tuesday, May 14, 13

Page 18: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

HJ =

X

i<j

Jij ~Si · ~Sj

with

~Si =

12c

†i↵~�↵�ci� is the antiferromagnetic exchange interaction.

Introduce the Nambu spinor

i" =

✓ci"c†i#

◆, i# =

✓ci#�c†i"

Then we can write

HJ =

1

8

X

i<j

Jij

†i↵~�↵� i�

⌘·⇣

†j�~��� i�

which is invariant under independent SU(2) pseudospin transforma-

tions on each site

i↵ ! Ui i↵

This pseudospin (gauge) symmetry is important in classifying spin

liquid ground states of HJ . It is fully broken by the electron hopping

tij but does have remnant consequences in doped spin liquid states.

Pseudospin symmetry of the exchange interaction

HtJ = �X

i,j

tijc†i↵cj↵ +

X

i<j

Jij ~Si · ~Sj

We will start with the Neel state, and find important

consequences of the pseudospin symmetry in metals

with antiferromagnetic correlations.

M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, 075127 (2010)Tuesday, May 14, 13

Page 19: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

r

Metal with “large” Fermi surface

h~'i = 0

Metal with electron and hole pockets

Increasing SDW order

h~'i 6= 0

Fermi surface+antiferromagnetism

Tuesday, May 14, 13

Page 20: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

r

Metal with “large” Fermi surface

h~'i = 0

Metal with electron and hole pockets

Increasing SDW order

h~'i 6= 0

Fermi surface+antiferromagnetism

Rest of the talk

K

Tuesday, May 14, 13

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“Hot” spots

Fermi surface+antiferromagnetism

K

Tuesday, May 14, 13

Page 22: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Low energy theory for critical point near hot spots

Fermi surface+antiferromagnetism

K

Tuesday, May 14, 13

Page 23: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Low energy theory for critical point near hot spots

Fermi surface+antiferromagnetism

K

Tuesday, May 14, 13

Page 24: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

v1 v2

�2 fermionsoccupied

�1 fermionsoccupied

kx

ky

Ar. Abanov and A. V. Chubukov,

Phys. Rev. Lett. 84, 5608 (2000).

S =

Zd2rd⌧

†1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4 � �~' ·

⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘�

Tuesday, May 14, 13

Page 25: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

v1 v2

�2 fermionsoccupied

�1 fermionsoccupied

kx

ky

S =

Zd2rd⌧

†1↵ (@⌧ � iv1 ·rr) 1↵ + †

2↵ (@⌧ � iv2 ·rr) 2↵

+1

2(rr ~')

2 +s

2~'2 +

u

4~'4 � �~' ·

⇣ †1↵~�↵� 2� + †

2↵~�↵� 1�

⌘�

M. A. Metlitski and S. Sachdev,

Phys. Rev. B 85, 075127 (2010)

This low-energy theory is invariant under

independent SU(2) pseudospin rotations on each

pair of hot-spots: there is a global

SU(2)⇥SU(2)⇥SU(2)⇥SU(2) pseudospin symmetry.

Tuesday, May 14, 13

Page 26: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

d-wave superconductor: particle-particle pairing at and near hot spots, with

sign-changing pairing amplitude

Dc†k↵c

†�k�

E= "

↵�

S

(cos kx

� cos ky

)

�S

��S

V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983)D.J. Scalapino, E. Loh, and J.E. Hirsch, Phys. Rev. B 34, 8190 (1986)K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, 224505 (2010)

K

Tuesday, May 14, 13

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Incommensurate d-wave bond order: particle-hole pairing at and near hot spots, with

sign-changing pairing amplitude

��Q

�Q

After pseudospin rotation on

half the hot-spots

M. A. Metlitski and S. Sachdev,

Phys. Rev. B 85, 075127 (2010)

Dc†k�Q/2,↵ck+Q/2,↵

E= �Q(cos k

x

� cos ky

)

Q is ‘2kF ’wavevector

K

Tuesday, May 14, 13

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��Q

�Q

Dc†k�Q/2,↵ck+Q/2,↵

E= �Q(cos k

x

� cos ky

)

Incommensurate d-wave bond order

M. A. Metlitski and S. Sachdev,

Phys. Rev. B 85, 075127 (2010)

Tuesday, May 14, 13

Page 29: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Q

Q

�Q

��Q

Dc†k�Q/2,↵ck+Q/2,↵

E= �Q(cos k

x

� cos ky

)

M. A. Metlitski and S. Sachdev,

Phys. Rev. B 85, 075127 (2010)

Incommensurate d-wave bond order

Tuesday, May 14, 13

Page 30: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Consider modulation in an o↵-site “density” like variable at sites ri

and rj

Dc†i↵

cj↵

E⇠

"X

k

�Q(k)eik·(ri�rj)

#eiQ·(ri+rj)/2

The wavevector Q is associated with a modulation in the average co-

ordinate (ri

+ rj

)/2: this determines the wavevector of the neutron/X-ray

scattering peak.

The interesting part is the dependence on the relative co-ordinate ri

� rj

.

Assuming time-reversal, the order parameter �Q(k) can always be ex-

panded as

�Q(k) = cs

+ cs

0(cos k

x

+ cos ky

) + cd

(cos kx

� cos ky

) + . . .

The usual charge-density-wave has only cs

6= 0, and so the density wave is

non-zero only if ri

= rj

.

The bond-ordered state has only cd

non-zero: in this case the density wave

is non-zero only if ri

and rj

are nearest neighbors.

Incommensurate d-wave bond order

relative co-ord. average co-ord.

Tuesday, May 14, 13

Page 31: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Consider modulation in an o↵-site “density” like variable at sites ri

and rj

Dc†i↵

cj↵

E⇠

"X

k

�Q(k)eik·(ri�rj)

#eiQ·(ri+rj)/2

The wavevector Q is associated with a modulation in the average co-

ordinate (ri

+ rj

)/2: this determines the wavevector of the neutron/X-ray

scattering peak.

The interesting part is the dependence on the relative co-ordinate ri

� rj

.

Assuming time-reversal, the order parameter �Q(k) can always be ex-

panded as

�Q(k) = cs

+ cs

0(cos k

x

+ cos ky

) + cd

(cos kx

� cos ky

) + . . .

The usual charge-density-wave has only cs

6= 0, and so the density wave is

non-zero only if ri

= rj

.

The bond-ordered state has only cd

non-zero: in this case the density wave

is non-zero only if ri

and rj

are nearest neighbors.

Incommensurate d-wave bond order

relative co-ord. average co-ord.

Tuesday, May 14, 13

Page 32: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Consider modulation in an o↵-site “density” like variable at sites ri

and rj

Dc†i↵

cj↵

E⇠

"X

k

�Q(k)eik·(ri�rj)

#eiQ·(ri+rj)/2

The wavevector Q is associated with a modulation in the average co-

ordinate (ri

+ rj

)/2: this determines the wavevector of the neutron/X-ray

scattering peak.

The interesting part is the dependence on the relative co-ordinate ri

� rj

.

Assuming time-reversal, the order parameter �Q(k) can always be ex-

panded as

�Q(k) = cs

+ cs

0(cos k

x

+ cos ky

) + cd

(cos kx

� cos ky

) + . . .

The usual charge-density-wave has only cs

6= 0, and so the density wave is

non-zero only if ri

= rj

.

The bond-ordered state has only cd

non-zero: in this case the density wave

is non-zero only if ri

and rj

are nearest neighbors.

Incommensurate d-wave bond order

relative co-ord. average co-ord.

Tuesday, May 14, 13

Page 33: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Incommensurate d-wave bond order

relative co-ord. average co-ord.

Consider modulation in an o↵-site “density” like variable at sites ri

and rj

Dc†i↵

cj↵

E⇠

"X

k

�Q(k)eik·(ri�rj)

#eiQ·(ri+rj)/2

The wavevector Q is associated with a modulation in the average co-

ordinate (ri

+ rj

)/2: this determines the wavevector of the neutron/X-ray

scattering peak.

The interesting part is the dependence on the relative co-ordinate ri

� rj

.

Assuming time-reversal, the order parameter �Q(k) can always be ex-

panded as

�Q(k) = cs

+ cs

0(cos k

x

+ cos ky

) + cd

(cos kx

� cos ky

) + . . .

The usual charge-density-wave has only cs

6= 0, and so the density wave is

non-zero only if ri

= rj

.

The bond-ordered state has only cd

non-zero: in this case the density wave

is non-zero only if ri

and rj

are nearest neighbors.

Tuesday, May 14, 13

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⌦c†r↵cs↵

↵=

X

Q

X

k

eiQ·(r+s)/2e�ik·(r�s)Dc†k�Q/2,↵ck+Q/2,↵

E

where Q extends over Q = (±Q0,±Q0) with Q0 = 2⇡/(7.3) and

Dc†k�Q/2,↵ck+Q/2,↵

E= �Q(cos k

x

� cos ky

)

Note

⌦c†r↵cs↵

↵is non-zero only when r, s are nearest neighbors.

“Bond density” measures amplitude for electrons to be

in spin-singlet valence bond.

-1

+1

M. A. Metlitski and S. Sachdev,

Phys. Rev. B 85, 075127 (2010)

Incommensurate d-wave bond order

Tuesday, May 14, 13

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1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry

2. Approximate pseudospin symmetry on the lattice t-J model Pseudogap and bond order in the underdoped cuprates

Tuesday, May 14, 13

Page 36: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry

2. Approximate pseudospin symmetry on the lattice t-J model Pseudogap and bond order in the underdoped cuprates

Tuesday, May 14, 13

Page 37: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

HtJ = �X

i,j

tijc†i↵cj↵ +

X

i<j

Jij ~Si · ~Sj

Optimize the free energy w.r.t. a mean field Hamiltonianwhich allows for spin-singlet charge order (�Q(k)):

HMF = �X

i,j

tijc†i↵cj↵ +

X

k,Q

�Q(k)c†k+Q/2,↵ck�Q/2,↵

Expanding the free energy in powers of the order parame-ters we obtain

F = F0 +X

k,Q

�⇤Q(k)MQ(k,k0)�Q(k0)

We compute the eigenvalues, 1 + �Q, and eigenfunctions,�Q(k) of the kernel MQ(k,k0)

Tuesday, May 14, 13

Page 38: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

HtJ = �X

i,j

tijc†i↵cj↵ +

X

i<j

Jij ~Si · ~Sj

Optimize the free energy w.r.t. a mean field Hamiltonianwhich allows for spin-singlet charge order (�Q(k)):

HMF = �X

i,j

tijc†i↵cj↵ +

X

k,Q

�Q(k)c†k+Q/2,↵ck�Q/2,↵

Expanding the free energy in powers of the order parame-ters we obtain

F = F0 +X

k,Q

�⇤Q(k)MQ(k,k0)�Q(k0)

We compute the eigenvalues, 1 + �Q, and eigenfunctions,�Q(k) of the kernel MQ(k,k0)

Tuesday, May 14, 13

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S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Tuesday, May 14, 13

Page 40: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Minimum at Q = (Qm

, Qm

) with

�Q(k) = 0.993(cos kx

� cos ky

)

�0.069(cos(2kx

)� cos(2ky

))

�0.009(cos kx

� cos ky

)

⇥p8 sin k

x

sin ky

Incommensurate

d-wave bond order

Tuesday, May 14, 13

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K(±Q0, Q0)

1.2 1.4 1.6 1.8

0.5

1.

Q0

Qm

Remarkable agreement betweenthe value of Qm from

Hartree-Fock in a metal withshort-range incommensurate

spin correlations,and the value of Q0 fromhot spots of commensurate

antiferromagnetism.

Tuesday, May 14, 13

Page 42: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Minimum at Q = (Qm

, Qm

) with

�Q(k) = 0.993(cos kx

� cos ky

)

�0.069(cos(2kx

)� cos(2ky

))

�0.009(cos kx

� cos ky

)

⇥p8 sin k

x

sin ky

Incommensurate

d-wave bond order

Tuesday, May 14, 13

Page 43: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Tuesday, May 14, 13

Page 44: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Q = (Qm

, 0) with

�Q(k) = 0.963(cos kx

� cos ky

)

�0.231

�0.067(cos(2kx

)� cos(2ky

))

�0.044(cos kx

+ cos ky

)

�0.046(cos(2kx

) + cos(2ky

))

Incommensurate

d +s -wave bond order

Tuesday, May 14, 13

Page 45: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

K

1.2 1.4 1.6 1.8

0.5

1.

Q0

Qm

(0, Q0)

Remarkable agreement betweenthe value of Qm from

Hartree-Fock in a metal withshort-range incommensurate

spin correlations,and the value of Q0 fromhot spots of commensurate

antiferromagnetism.

Tuesday, May 14, 13

Page 46: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Q = (Qm

, 0) with

�Q(k) = 0.963(cos kx

� cos ky

)

�0.231

�0.067(cos(2kx

)� cos(2ky

))

�0.044(cos kx

+ cos ky

)

�0.046(cos(2kx

) + cos(2ky

))

Incommensurate

d +s -wave bond order

Tuesday, May 14, 13

Page 47: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Tuesday, May 14, 13

Page 48: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Q = (⇡,⇡) with

�Q(k) = i(sin kx

� sin ky

)

Orbital currents

Tuesday, May 14, 13

Page 49: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Tuesday, May 14, 13

Page 50: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

S. Sachdev and R. La Placa, arXiv:1303.2114

Charge-ordering eigenvalue �Q/J0.

Q = (0, 0) with

�Q(k) = cos kx

� cos ky

Ising-nematic order

Tuesday, May 14, 13

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Dc†k�Q/2,↵ck+Q/2,↵

E= �Q(cos k

x

� cos ky

)

K. B. Efetov, H. Meier, and

C. Pepin, Nature Physics,

to appear,arXiv:1210.3276

Incommensurate d-wave bond order

High T pseudogap:Fluctuating compositeorder parameter ofnearly degenerated-wave pairing andincommensurate

d-wave bond order.(Approximate) SU(2)symmetry of composite

order preventslong-range order T > 0.

K

Tuesday, May 14, 13

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Incommensurate d-wave bond order

Observed low Tordering.

Our computations showthat the charge order ispredominantly d-wave

also at this Q.

This Q is preferred incomputations of bond

order within thesuperconducting phase.

S. Sachdev and R. La Placa, arXiv:1303.2114 M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999)

M. Vojta and O. Rosch, Physical Review B 77, 094504 (2008)

K

Tuesday, May 14, 13

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0 Doping δ

N (

or J

’/ J

)

p

4

d-wave

A1

A2

q

X broken

broken

FIG. 1. Schematic, proposed, ground state phase diagramof H as a function of the doping ! for physically reasonablevalues of t, J and V . The vertical axis represents a param-eter which measures the strength of quantum spin fluctua-tions—it increases linearly with N but can also be tuned con-tinuously by J !/J . The magnetic M symmetry is brokenin the hatched region, while C symmetry is broken (with ac-companying charge-density modulation) in the shaded region;there are numerous additional phase transitions at which thedetailed nature of the M or C symmetry breaking changes -these are not shown. For ! = 0, M symmetry is broken onlybelow the critical point X, while C symmetry is broken onlyabove X. The superconducting S symmetry is broken for all! > 0 at large N ; for smaller N , the S can be restored atsmall ! by additional C breaking along the vertical axis forthe states in the inset–this is not shown. The superconduc-tivity is pure d-wave only in the large ! region were C andM are not broken. The arrow A1 represents the path alongwhich quantitative results are obtained in this paper, whileA2 is the experimental path. The nature of the C symme-try breaking along path A1 is also sketched: the thick anddashed lines indicate varying values of |Qij | (proportional tothe bond charge density) on the links, while the circles rep-resent b2i (proportional to the site hole density). The chargedensities on the links and sites not shown take values con-sistent with the symmetries of the figures shown. We expectthat the nature of the C symmetry breaking will not changesignificantly as we move from A1 to A2, and across the phaseboundary where M is broken: this suggests the appearanceof collinearly polarized spin-density waves, which break bothC and M, and which undergo an ‘anti-phase’ shift across thehole-rich stripes16.

energy with respect to the site charge density N(1 !b2i ) = "ni# and the complex bond pairing amplitude

NQij = "J !"c†i!c†j"#/(bibj) (where b2i is the hole density

at site i and J denotes the Sp(2N)-invariant antisym-metric tensor), while maintaining certain local and globalconstraints. There have been a number of related earliermean-field studies17, but they have all (with the excep-tion of Ref. 11) restricted attention to the case where bi

and |Qij| are spatially uniform (note that |Qij| has thesame symmetry signature as the bond charge density, andis therefore a measure of its value). However such solu-tions are usually unstable, and at best metastable, at lowdoping; here we have attempted to find the true globalminima of the saddle-point equations, while allowing forarbitrary spatial dependence: such a procedure leads toconsiderable physical insight, and also leads to solutionsin accord with recent experimental observations.

First, at ! = 0 along A1 we find the fully dimerized, in-sulating spin-Peierls (or 2$ 1 bond charge-density wave)solution18 in which |Qij| is non-zero only on the bondsshown in Fig. 1. Moving to small non-zero ! along A1,our numerical search always yielded lowest energy stateswith C broken, consisting of bond-centered charge-densitywaves19 with a p $ 1 unit cell, as shown in Fig. 1. Wealways found p to be an even integer, reflecting the dimer-ization tendency of the ! = 0 solution. Within each p$1unit cell, we find that the holes are concentrated on aq$ 1 region, with a total linear hole density of "#. A keyproperty is that q and "# remain finite, while p % &,as ! % 0. Indeed, the values of q and "# are deter-mined primarily by t, J , and the nearest-neighbor valueof Vij = Vnn, and are insensitive to ! and longer rangeparts of Vij. For ! % 0, we found that q = 2 was optimumfor a wide range of parameter values, while larger valuesof q (q ' 4) appear for smaller values of Vnn; specificallywe had q = 2, "# = 0.42 at t/J = 1.25, Vnn/t = 0.6, andq = 4 , "# = 0.8 at t/J = 1.25, Vnn/t = 0.5. The limitVnn % 0 leads to q % & which reflects the tendency tophase separation in the “bare” t ! J model. The evolu-tion of p with ! is shown in Fig. 2. Note that there is alarge plateau at p = 4 around doping ! = 1/8, and, forsome parameter regimes, this is the last state before Cis restored at large !; indeed p = 4 is the smallest valueof p for which our mean-field theory has solutions withbi not spatially uniform. Experimentally1,2, a pinning ofthe charge order at a wavevector K = 1/4 is observed,and we consider it significant that this value emerges nat-urally from our theory.

Our large-N theory only found states in which theordering wavevector K was quantized at the rationalplateaus in Fig. 2. However, for smaller N we expect thatirrational, incommensurate, values of K will appear, andinterpolate smoothly between the plateau regions.

In our large-N theory, each q-width stripe above isa one-dimensional superconductor, while the intervening(q ! p)-width regions are insulating. However, fluctua-tion corrections will couple with superconducting regions,

2

M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999)S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)

Spinorder

Bondorder

Evidence bond order is along (1,0), (0,1) directionsin low T superconducting phase

Tuesday, May 14, 13

Page 54: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Superconducting d-wave stripes in cuprates: Valence bond order coexistingwith nodal quasiparticles

Matthias Vojta and Oliver RöschInstitut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

!Received 8 January 2008; revised manuscript received 10 January 2008; published 6 March 2008"

We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronicorder in both s- and d-wave channels, with nonlocal Coulomb repulsion suppressing the s-wave component.The resulting bond-charge-density wave, coexisting with superconductivity, is compatible with recent photo-emission and tunneling data and as well as neutron-scattering measurements, once long-range order is de-stroyed by slow fluctuations or glassy disorder. In particular, the real-space structure of d-wave stripes isconsistent with the scanning-tunneling-microscopy measurements on both underdoped Bi2Sr2CaCu2O8+! andCa2!xNaxCuO2Cl2 of Kohsaka et al. #Science 315, 1380 !2007"$.

DOI: 10.1103/PhysRevB.77.094504 PACS number!s": 74.20.Mn, 74.25.Dw, 74.72."h

I. INTRODUCTION

A remarkable aspect of the copper-oxide high-Tc super-conductors is that various ordering phenomena apparentlycompete, including commensurate and incommensuratemagnetisms, superconducting pairing, and charge-density-wave formation. !More exotic states have also been pro-posed, but not verified experimentally beyond doubt." Whilecommensurate magnetism and superconductivity are com-mon phases in essentially all families of cuprates, the role ofother instabilities for the global features of the phase diagramis less clear.

A particularly interesting role is taken by charge-densitywaves. Such states break the discrete lattice translation sym-metry, with examples being stripe, checkerboard, andvalence-bond order. In the compounds, La2!xBaxCuO4 andLa2!xSrxCuO4 !with Nd or Eu codoping" evidence for stripe-like spin and charge modulations with static long-range orderwere detected,1–4 in particular, near one-eighth doping. !Thisis supported, e.g., by strong phonon anomalies seen inneutron-scattering experiments.5" While in other cupratefamilies similar long-range order has not been found, signa-tures of short-range charge order, likely pinned by impurities,have been observed in scanning-tunneling microscopy!STM" on underdoped Bi2Sr2CaCu2O8+! !Refs. 6–9" andCa2!xNaxCuO2Cl2.9,10 The low-energy electronic structure inthe presence of charge order turns out to be remarkable: InLa15/8Ba1/8CuO4, angle-resolved photoemission spectroscopy!ARPES" indicated a quasiparticle !QP" gap with d-wavelikeform, i.e., charge order coexists with gapless !nodal" QP inthe !1,1" direction #while antinodal QP near !0,#" aregapped$.11 STM data on both underdoped Bi2Sr2CaCu2O8+!and Ca2!xNaxCuO2Cl2 and show QP interference arisingfrom coherent low-energy states near the nodes, whereaselectronic states at higher energy and wave vector close tothe antinode are dominated by the real-space modulation ofthe short-range charge order.7,9,12 This dichotomy in momen-tum space has also been found in ARPES experiments inLa2!xSrxCuO4,13 Bi2Sr2CaCu2O8+!,14 and Ca2!xNaxCuO2Cl2!Ref. 15" where well-defined nodal and ill-defined antinodalQP are frequently observed.

These results suggest that momentum-space differentia-tion and tendencies toward charge ordering are common to

underdoped cuprates.16–18 The concept of fluctuating stripes,i.e., almost charge-ordered states, has been discussed earlyon.1,3,16,18,19 This concept, appropriate for compounds with-out static long-range order, assumes the existence of a nearbystripe-ordered state, with physical observables being influ-enced by the low-lying collective modes associated withcharge-ordering instability. Following this idea, we have re-cently calculated20 the spin excitation spectrum of slowlyfluctuating !or disordered" stripes. We were able to show thatfluctuating stripes give rise to an “hour-glass” magneticspectrum, very similar to that observed in neutron-scatteringexperiments both on La2!xBaxCuO4 !Ref. 21" andYBa2Cu3O6+!.22,23

The focus of this paper is on the electronic structure ofstripe states. We introduce the concept of “d-wave stripes”:Here, the modulation of charge densities has primarily ad-wave form factor, i.e., lives more on the bonds than on thesites of the square lattice, leaving nodal QP unaffected. Weillustrate that a picture of such bond-centered charge order,coexisting with superconductivity !this state may be dubbed“valence-bond supersolid”", is consistent with various fea-tures seen in both ARPES and STM measurements. In par-ticular, the real-space pattern of d-wave stripes !Fig. 1" isstrikingly similar to the STM results of Ref. 9, obtained onunderdoped Bi2Sr2CaCu2O8+! and Ca2!xNaxCuO2Cl2.

FIG. 1. Schematic real-space structure of a stripe state withprimarily d-wave character and a 4$1 unit cell, i.e., Q= !%# /2,0". Cu lattice sites are shown as circles, with their sizerepresenting the on-site hole densities. The line strengths indicatethe amplitude of bond variables such as kinetic and magnetic ener-gies. The modulation in the site charge densities is small, whereasthe one in the bond densities is large and of d-wave type !Ref. 24".Note the similarity of the bond modulation with the STM data ofRef. 9.

PHYSICAL REVIEW B 77, 094504 !2008"

1098-0121/2008/77!9"/094504!5" ©2008 The American Physical Society094504-1

Evidence bond order is along (1,0), (0,1) directionsin low T superconducting phase

Superconducting d-wave stripes in cuprates: Valence bond order coexistingwith nodal quasiparticles

Matthias Vojta and Oliver RöschInstitut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

!Received 8 January 2008; revised manuscript received 10 January 2008; published 6 March 2008"

We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronicorder in both s- and d-wave channels, with nonlocal Coulomb repulsion suppressing the s-wave component.The resulting bond-charge-density wave, coexisting with superconductivity, is compatible with recent photo-emission and tunneling data and as well as neutron-scattering measurements, once long-range order is de-stroyed by slow fluctuations or glassy disorder. In particular, the real-space structure of d-wave stripes isconsistent with the scanning-tunneling-microscopy measurements on both underdoped Bi2Sr2CaCu2O8+! andCa2!xNaxCuO2Cl2 of Kohsaka et al. #Science 315, 1380 !2007"$.

DOI: 10.1103/PhysRevB.77.094504 PACS number!s": 74.20.Mn, 74.25.Dw, 74.72."h

I. INTRODUCTION

A remarkable aspect of the copper-oxide high-Tc super-conductors is that various ordering phenomena apparentlycompete, including commensurate and incommensuratemagnetisms, superconducting pairing, and charge-density-wave formation. !More exotic states have also been pro-posed, but not verified experimentally beyond doubt." Whilecommensurate magnetism and superconductivity are com-mon phases in essentially all families of cuprates, the role ofother instabilities for the global features of the phase diagramis less clear.

A particularly interesting role is taken by charge-densitywaves. Such states break the discrete lattice translation sym-metry, with examples being stripe, checkerboard, andvalence-bond order. In the compounds, La2!xBaxCuO4 andLa2!xSrxCuO4 !with Nd or Eu codoping" evidence for stripe-like spin and charge modulations with static long-range orderwere detected,1–4 in particular, near one-eighth doping. !Thisis supported, e.g., by strong phonon anomalies seen inneutron-scattering experiments.5" While in other cupratefamilies similar long-range order has not been found, signa-tures of short-range charge order, likely pinned by impurities,have been observed in scanning-tunneling microscopy!STM" on underdoped Bi2Sr2CaCu2O8+! !Refs. 6–9" andCa2!xNaxCuO2Cl2.9,10 The low-energy electronic structure inthe presence of charge order turns out to be remarkable: InLa15/8Ba1/8CuO4, angle-resolved photoemission spectroscopy!ARPES" indicated a quasiparticle !QP" gap with d-wavelikeform, i.e., charge order coexists with gapless !nodal" QP inthe !1,1" direction #while antinodal QP near !0,#" aregapped$.11 STM data on both underdoped Bi2Sr2CaCu2O8+!and Ca2!xNaxCuO2Cl2 and show QP interference arisingfrom coherent low-energy states near the nodes, whereaselectronic states at higher energy and wave vector close tothe antinode are dominated by the real-space modulation ofthe short-range charge order.7,9,12 This dichotomy in momen-tum space has also been found in ARPES experiments inLa2!xSrxCuO4,13 Bi2Sr2CaCu2O8+!,14 and Ca2!xNaxCuO2Cl2!Ref. 15" where well-defined nodal and ill-defined antinodalQP are frequently observed.

These results suggest that momentum-space differentia-tion and tendencies toward charge ordering are common to

underdoped cuprates.16–18 The concept of fluctuating stripes,i.e., almost charge-ordered states, has been discussed earlyon.1,3,16,18,19 This concept, appropriate for compounds with-out static long-range order, assumes the existence of a nearbystripe-ordered state, with physical observables being influ-enced by the low-lying collective modes associated withcharge-ordering instability. Following this idea, we have re-cently calculated20 the spin excitation spectrum of slowlyfluctuating !or disordered" stripes. We were able to show thatfluctuating stripes give rise to an “hour-glass” magneticspectrum, very similar to that observed in neutron-scatteringexperiments both on La2!xBaxCuO4 !Ref. 21" andYBa2Cu3O6+!.22,23

The focus of this paper is on the electronic structure ofstripe states. We introduce the concept of “d-wave stripes”:Here, the modulation of charge densities has primarily ad-wave form factor, i.e., lives more on the bonds than on thesites of the square lattice, leaving nodal QP unaffected. Weillustrate that a picture of such bond-centered charge order,coexisting with superconductivity !this state may be dubbed“valence-bond supersolid”", is consistent with various fea-tures seen in both ARPES and STM measurements. In par-ticular, the real-space pattern of d-wave stripes !Fig. 1" isstrikingly similar to the STM results of Ref. 9, obtained onunderdoped Bi2Sr2CaCu2O8+! and Ca2!xNaxCuO2Cl2.

FIG. 1. Schematic real-space structure of a stripe state withprimarily d-wave character and a 4$1 unit cell, i.e., Q= !%# /2,0". Cu lattice sites are shown as circles, with their sizerepresenting the on-site hole densities. The line strengths indicatethe amplitude of bond variables such as kinetic and magnetic ener-gies. The modulation in the site charge densities is small, whereasthe one in the bond densities is large and of d-wave type !Ref. 24".Note the similarity of the bond modulation with the STM data ofRef. 9.

PHYSICAL REVIEW B 77, 094504 !2008"

1098-0121/2008/77!9"/094504!5" ©2008 The American Physical Society094504-1

Superconducting d-wave stripes in cuprates: Valence bond order coexistingwith nodal quasiparticles

Matthias Vojta and Oliver RöschInstitut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937 Köln, Germany

!Received 8 January 2008; revised manuscript received 10 January 2008; published 6 March 2008"

We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronicorder in both s- and d-wave channels, with nonlocal Coulomb repulsion suppressing the s-wave component.The resulting bond-charge-density wave, coexisting with superconductivity, is compatible with recent photo-emission and tunneling data and as well as neutron-scattering measurements, once long-range order is de-stroyed by slow fluctuations or glassy disorder. In particular, the real-space structure of d-wave stripes isconsistent with the scanning-tunneling-microscopy measurements on both underdoped Bi2Sr2CaCu2O8+! andCa2!xNaxCuO2Cl2 of Kohsaka et al. #Science 315, 1380 !2007"$.

DOI: 10.1103/PhysRevB.77.094504 PACS number!s": 74.20.Mn, 74.25.Dw, 74.72."h

I. INTRODUCTION

A remarkable aspect of the copper-oxide high-Tc super-conductors is that various ordering phenomena apparentlycompete, including commensurate and incommensuratemagnetisms, superconducting pairing, and charge-density-wave formation. !More exotic states have also been pro-posed, but not verified experimentally beyond doubt." Whilecommensurate magnetism and superconductivity are com-mon phases in essentially all families of cuprates, the role ofother instabilities for the global features of the phase diagramis less clear.

A particularly interesting role is taken by charge-densitywaves. Such states break the discrete lattice translation sym-metry, with examples being stripe, checkerboard, andvalence-bond order. In the compounds, La2!xBaxCuO4 andLa2!xSrxCuO4 !with Nd or Eu codoping" evidence for stripe-like spin and charge modulations with static long-range orderwere detected,1–4 in particular, near one-eighth doping. !Thisis supported, e.g., by strong phonon anomalies seen inneutron-scattering experiments.5" While in other cupratefamilies similar long-range order has not been found, signa-tures of short-range charge order, likely pinned by impurities,have been observed in scanning-tunneling microscopy!STM" on underdoped Bi2Sr2CaCu2O8+! !Refs. 6–9" andCa2!xNaxCuO2Cl2.9,10 The low-energy electronic structure inthe presence of charge order turns out to be remarkable: InLa15/8Ba1/8CuO4, angle-resolved photoemission spectroscopy!ARPES" indicated a quasiparticle !QP" gap with d-wavelikeform, i.e., charge order coexists with gapless !nodal" QP inthe !1,1" direction #while antinodal QP near !0,#" aregapped$.11 STM data on both underdoped Bi2Sr2CaCu2O8+!and Ca2!xNaxCuO2Cl2 and show QP interference arisingfrom coherent low-energy states near the nodes, whereaselectronic states at higher energy and wave vector close tothe antinode are dominated by the real-space modulation ofthe short-range charge order.7,9,12 This dichotomy in momen-tum space has also been found in ARPES experiments inLa2!xSrxCuO4,13 Bi2Sr2CaCu2O8+!,14 and Ca2!xNaxCuO2Cl2!Ref. 15" where well-defined nodal and ill-defined antinodalQP are frequently observed.

These results suggest that momentum-space differentia-tion and tendencies toward charge ordering are common to

underdoped cuprates.16–18 The concept of fluctuating stripes,i.e., almost charge-ordered states, has been discussed earlyon.1,3,16,18,19 This concept, appropriate for compounds with-out static long-range order, assumes the existence of a nearbystripe-ordered state, with physical observables being influ-enced by the low-lying collective modes associated withcharge-ordering instability. Following this idea, we have re-cently calculated20 the spin excitation spectrum of slowlyfluctuating !or disordered" stripes. We were able to show thatfluctuating stripes give rise to an “hour-glass” magneticspectrum, very similar to that observed in neutron-scatteringexperiments both on La2!xBaxCuO4 !Ref. 21" andYBa2Cu3O6+!.22,23

The focus of this paper is on the electronic structure ofstripe states. We introduce the concept of “d-wave stripes”:Here, the modulation of charge densities has primarily ad-wave form factor, i.e., lives more on the bonds than on thesites of the square lattice, leaving nodal QP unaffected. Weillustrate that a picture of such bond-centered charge order,coexisting with superconductivity !this state may be dubbed“valence-bond supersolid”", is consistent with various fea-tures seen in both ARPES and STM measurements. In par-ticular, the real-space pattern of d-wave stripes !Fig. 1" isstrikingly similar to the STM results of Ref. 9, obtained onunderdoped Bi2Sr2CaCu2O8+! and Ca2!xNaxCuO2Cl2.

FIG. 1. Schematic real-space structure of a stripe state withprimarily d-wave character and a 4$1 unit cell, i.e., Q= !%# /2,0". Cu lattice sites are shown as circles, with their sizerepresenting the on-site hole densities. The line strengths indicatethe amplitude of bond variables such as kinetic and magnetic ener-gies. The modulation in the site charge densities is small, whereasthe one in the bond densities is large and of d-wave type !Ref. 24".Note the similarity of the bond modulation with the STM data ofRef. 9.

PHYSICAL REVIEW B 77, 094504 !2008"

1098-0121/2008/77!9"/094504!5" ©2008 The American Physical Society094504-1

Tuesday, May 14, 13

Page 55: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Incommensurate d-wave bond order

Observed low Tordering.

Our computations showthat the charge order ispredominantly d-wave

also at this Q.

This Q is preferred incomputations of bond

order within thesuperconducting phase.

S. Sachdev and R. La Placa, arXiv:1303.2114 M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999)

M. Vojta and O. Rosch, Physical Review B 77, 094504 (2008)

K

Tuesday, May 14, 13

Page 56: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

An Intrinsic Bond-Centered ElectronicGlass with Unidirectional Domainsin Underdoped CupratesY. Kohsaka,1 C. Taylor,1 K. Fujita,1,2 A. Schmidt,1 C. Lupien,3 T. Hanaguri,4 M. Azuma,5M. Takano,5 H. Eisaki,6 H. Takagi,2,4 S. Uchida,2,7 J. C. Davis1,8*

Removing electrons from the CuO2 plane of cuprates alters the electronic correlations sufficientlyto produce high-temperature superconductivity. Associated with these changes are spectral-weighttransfers from the high-energy states of the insulator to low energies. In theory, these should bedetectable as an imbalance between the tunneling rate for electron injection and extraction—atunneling asymmetry. We introduce atomic-resolution tunneling-asymmetry imaging, findingvirtually identical phenomena in two lightly hole-doped cuprates: Ca1.88Na0.12CuO2Cl2 andBi2Sr2Dy0.2Ca0.8Cu2O8+d. Intense spatial variations in tunneling asymmetry occur primarily at theplanar oxygen sites; their spatial arrangement forms a Cu-O-Cu bond-centered electronic patternwithout long-range order but with 4a0-wide unidirectional electronic domains dispersedthroughout (a0: the Cu-O-Cu distance). The emerging picture is then of a partial hole localizationwithin an intrinsic electronic glass evolving, at higher hole densities, into complete delocalizationand highest-temperature superconductivity.

Metallicity of the cuprate CuO2 planesderives (1) from both oxygen 2p andcopper 3d orbitals (Fig. 1A). Coulomb

interactions lift the degeneracy of the relevantd-orbital, producing lower and upper d-states sepa-rated by the Mott-Hubbard energy U (Fig. 1B).The lower d-states and oxygen p-state becomehybridized, yielding a correlated insulator withcharge-transfer gapD (Fig. 1B). The “hole-doping”process, which generates highest-temperature su-perconductivity, then removes electrons from theCuO2 plane, creating new hole-like electronicstates with predominantly oxygen 2p character(2). This is a radically different process than hole-doping a conventional semiconductor because,when an electron is removed from a correlatedinsulator, the states with which it was correlatedare also altered fundamentally. Numericalmodeling of this process (3) indicates that whenn holes per unit cell are introduced, the correlationchanges generate spectral-weight transfers fromboth filled and empty high-energy bands—resulting in the creation of ~2n new empty states

just above the chemical potential m (Fig. 1B). Butprecisely how these spectral-weight transfersresult in cuprate high-temperature supercon-ductivity remains controversial.

Recently, it has been proposed that thesedoping-induced correlation changes might beobservable directly as an asymmetry of electrontunneling currents with bias voltage (4, 5)—electron extraction at negative sample bias beingstrongly favored over electron injection at posi-tive sample bias. Such effects should be de-tectable with a scanning tunneling microscope(STM). The STM tip-sample tunneling currentis given by

I!r!, z, V " # f !r!, z"Z

eV

0N!r!, E"dE !1"

where z is the tip’s surface-normal coordinate, Vis the relative sample-tip bias, and N!r!,E" isthe sample’s local-density-of-states (LDOS) atlateral locations r! and energy E. Unmeasurableeffects due to the tunneling matrix elements, thetunnel-barrier height, and z variations from elec-tronic heterogeneity are contained in f !r!, z" (seesupporting online text 1). For a simple metallicsystem where f !r!, z" is a featureless constant,Eq. 1 shows that spatial mapping of the dif-ferential tunneling conductance dI=dV !r!,V "yields N!r!,E # eV ". However, for the stronglycorrelated electronic states in a lightly hole-doped cuprate, the situation is much morecomplex. In theory (4), the correlations causethe ratio Z(V) of the average density-of-statesfor empty states N!E # $eV " to that of filledstates N!E # !eV " to become asymmetric byan amount

Z!V " " N!E # $eV "N!E # !eV "

#2n

1$ n!2"

Spectral-weight sum rules (5) also indicate thatthe ratio R!r!" of the energy-integrated N!r!,E"for empty states E > 0 to that of filled states E < 0is related to n by

R!r!" "

ZWc

0N!r!,E" dE

Z0

!$N!r!,E"dE

# 2n!r!"1 ! n!r!"

$ OntU

! "!3"

Here t is in-plane hopping rate and Wc satisfies“all low-energy scales” < Wc < U.

As a test of such ideas, we show in Fig. 1Cthe predicted evolution of the tunnelingasymmetry (TA) with n from (4), and in Fig.1D we show the measured evolution of spatiallyaveraged TA in a sequence of lightly hole-dopedCa2-xNaxCuO2Cl2 samples with different x. Wesee that the average TA is indeed large at low x

RESEARCHARTICLE

1Laboratory of Atomic and Solid State Physics, Departmentof Physics, Cornell University, Ithaca, NY 14853, USA.2Department of Advanced Materials Science, University ofTokyo, Kashiwa, Chiba 277-8651, Japan. 3Département dePhysique, Université de Sherbrooke, Sherbrooke, QC J1K2R1, Canada. 4Magnetic Materials Laboratory, RIKEN,Wako, Saitama 351-0198, Japan. 5Institute for ChemicalResearch, Kyoto University, Uji, Kyoto 601-0011, Japan.6National Institute of Advanced Industrial Science andTechnology, Tsukuba, Ibaraki 305-8568, Japan. 7Depart-ment of Physics, University of Tokyo, Bunkyo-ku, Tokyo113-0033, Japan. 8Condensed Matter Physics and Materi-als Science Department, Brookhaven National Laboratory,Upton, NY 11973, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

Fig. 1. (A) Relevant electronicorbitals of the CuO2 plane: Cu 3dorbitals are shown in orange andoxygen 2p orbitals are shownin blue. A single plaquette offour Cu atoms is shown withinthe dashed square box, and asingle Cu-O-Cu unit is withinthe dashed oval. (B) Schematicenergy levels in the CuO2 planeand the effects of hole dopingupon it. (C) The expected tun-neling asymmetry between elec-tron extraction (negative bias)and injection (positive bias) from(4) where low values of Z occur atlow hole densities n. (D) Mea-sured doping dependence ofaverage tunneling asymmetryin Ca2-xNaxCuO2Cl2. a.u., arbi-trary units.

9 MARCH 2007 VOL 315 SCIENCE www.sciencemag.org1380

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An Intrinsic Bond-Centered ElectronicGlass with Unidirectional Domainsin Underdoped CupratesY. Kohsaka,1 C. Taylor,1 K. Fujita,1,2 A. Schmidt,1 C. Lupien,3 T. Hanaguri,4 M. Azuma,5M. Takano,5 H. Eisaki,6 H. Takagi,2,4 S. Uchida,2,7 J. C. Davis1,8*

Removing electrons from the CuO2 plane of cuprates alters the electronic correlations sufficientlyto produce high-temperature superconductivity. Associated with these changes are spectral-weighttransfers from the high-energy states of the insulator to low energies. In theory, these should bedetectable as an imbalance between the tunneling rate for electron injection and extraction—atunneling asymmetry. We introduce atomic-resolution tunneling-asymmetry imaging, findingvirtually identical phenomena in two lightly hole-doped cuprates: Ca1.88Na0.12CuO2Cl2 andBi2Sr2Dy0.2Ca0.8Cu2O8+d. Intense spatial variations in tunneling asymmetry occur primarily at theplanar oxygen sites; their spatial arrangement forms a Cu-O-Cu bond-centered electronic patternwithout long-range order but with 4a0-wide unidirectional electronic domains dispersedthroughout (a0: the Cu-O-Cu distance). The emerging picture is then of a partial hole localizationwithin an intrinsic electronic glass evolving, at higher hole densities, into complete delocalizationand highest-temperature superconductivity.

Metallicity of the cuprate CuO2 planesderives (1) from both oxygen 2p andcopper 3d orbitals (Fig. 1A). Coulomb

interactions lift the degeneracy of the relevantd-orbital, producing lower and upper d-states sepa-rated by the Mott-Hubbard energy U (Fig. 1B).The lower d-states and oxygen p-state becomehybridized, yielding a correlated insulator withcharge-transfer gapD (Fig. 1B). The “hole-doping”process, which generates highest-temperature su-perconductivity, then removes electrons from theCuO2 plane, creating new hole-like electronicstates with predominantly oxygen 2p character(2). This is a radically different process than hole-doping a conventional semiconductor because,when an electron is removed from a correlatedinsulator, the states with which it was correlatedare also altered fundamentally. Numericalmodeling of this process (3) indicates that whenn holes per unit cell are introduced, the correlationchanges generate spectral-weight transfers fromboth filled and empty high-energy bands—resulting in the creation of ~2n new empty states

just above the chemical potential m (Fig. 1B). Butprecisely how these spectral-weight transfersresult in cuprate high-temperature supercon-ductivity remains controversial.

Recently, it has been proposed that thesedoping-induced correlation changes might beobservable directly as an asymmetry of electrontunneling currents with bias voltage (4, 5)—electron extraction at negative sample bias beingstrongly favored over electron injection at posi-tive sample bias. Such effects should be de-tectable with a scanning tunneling microscope(STM). The STM tip-sample tunneling currentis given by

I!r!, z, V " # f !r!, z"Z

eV

0N!r!, E"dE !1"

where z is the tip’s surface-normal coordinate, Vis the relative sample-tip bias, and N!r!,E" isthe sample’s local-density-of-states (LDOS) atlateral locations r! and energy E. Unmeasurableeffects due to the tunneling matrix elements, thetunnel-barrier height, and z variations from elec-tronic heterogeneity are contained in f !r!, z" (seesupporting online text 1). For a simple metallicsystem where f !r!, z" is a featureless constant,Eq. 1 shows that spatial mapping of the dif-ferential tunneling conductance dI=dV !r!,V "yields N!r!,E # eV ". However, for the stronglycorrelated electronic states in a lightly hole-doped cuprate, the situation is much morecomplex. In theory (4), the correlations causethe ratio Z(V) of the average density-of-statesfor empty states N!E # $eV " to that of filledstates N!E # !eV " to become asymmetric byan amount

Z!V " " N!E # $eV "N!E # !eV "

#2n

1$ n!2"

Spectral-weight sum rules (5) also indicate thatthe ratio R!r!" of the energy-integrated N!r!,E"for empty states E > 0 to that of filled states E < 0is related to n by

R!r!" "

ZWc

0N!r!,E" dE

Z0

!$N!r!,E"dE

# 2n!r!"1 ! n!r!"

$ OntU

! "!3"

Here t is in-plane hopping rate and Wc satisfies“all low-energy scales” < Wc < U.

As a test of such ideas, we show in Fig. 1Cthe predicted evolution of the tunnelingasymmetry (TA) with n from (4), and in Fig.1D we show the measured evolution of spatiallyaveraged TA in a sequence of lightly hole-dopedCa2-xNaxCuO2Cl2 samples with different x. Wesee that the average TA is indeed large at low x

RESEARCHARTICLE

1Laboratory of Atomic and Solid State Physics, Departmentof Physics, Cornell University, Ithaca, NY 14853, USA.2Department of Advanced Materials Science, University ofTokyo, Kashiwa, Chiba 277-8651, Japan. 3Département dePhysique, Université de Sherbrooke, Sherbrooke, QC J1K2R1, Canada. 4Magnetic Materials Laboratory, RIKEN,Wako, Saitama 351-0198, Japan. 5Institute for ChemicalResearch, Kyoto University, Uji, Kyoto 601-0011, Japan.6National Institute of Advanced Industrial Science andTechnology, Tsukuba, Ibaraki 305-8568, Japan. 7Depart-ment of Physics, University of Tokyo, Bunkyo-ku, Tokyo113-0033, Japan. 8Condensed Matter Physics and Materi-als Science Department, Brookhaven National Laboratory,Upton, NY 11973, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

Fig. 1. (A) Relevant electronicorbitals of the CuO2 plane: Cu 3dorbitals are shown in orange andoxygen 2p orbitals are shownin blue. A single plaquette offour Cu atoms is shown withinthe dashed square box, and asingle Cu-O-Cu unit is withinthe dashed oval. (B) Schematicenergy levels in the CuO2 planeand the effects of hole dopingupon it. (C) The expected tun-neling asymmetry between elec-tron extraction (negative bias)and injection (positive bias) from(4) where low values of Z occur atlow hole densities n. (D) Mea-sured doping dependence ofaverage tunneling asymmetryin Ca2-xNaxCuO2Cl2. a.u., arbi-trary units.

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direct test of such ideas has not been possiblebecause neither the real-space electronic struc-ture of the ECG state, nor that of an individual“cluster,” could be determined directly as nosuitable imaging techniques existed.

Design of TA studies in Ca1.88Na0.12CuO2Cl2and Bi2Sr2Dy0.2Ca0.8Cu2O8+d. STM-based im-aging might appear an appropriate tool to ad-dress such issues. But dI/dV imaging is fraught

with problems in lightly doped cuprates. Forexample, a standard dI/dV image, although welldefined, is not a direct image of the LDOS (seesupporting online text 1). Moreover, there aretheoretical concerns that, in Ca2-xNaxCuO2Cl2,the topmost CuO2 plane may be in an “extraor-dinary” state (34) or that interference betweentwo tunneling trajectories through the 3pz-Clorbitals adjacent to a dopant Na+ ion may cause

rotational symmetry breaking in the tunnelingpatterns (35).

The new proposals (4, 5) for tunnelingasymmetry measurements provide a notablesolution to problems with standard dI/dVimaging because Eqs. 2 and 3 have a crucialpractical advantage. If we define the ratiosZ!r!, V " and R!r!, V " in terms of the tunnelingcurrent

Z!r!,V " !dIdV !r

!, z, #V "dIdV !r

!, z,"V "!4a"

R!r!, V " ! I!r!, z, #V "I!r!, z, "V " !4b"

we see immediately from Eq. 1 that the un-known effects in f !r!, z" are all canceled outby the division process. Thus, Z!r!, V " andR!r!, V " not only contain important physicalinformation (4, 5) but, unlike N!r!, E", are alsoexpressible in terms of measurable quantitiesonly. We have confirmed that the unknownfactors f !r!, z" are indeed canceled out in Eq. 4(see supporting online text and figures 2).

To address the material-specific theoret-ical concerns (34, 35), we have designed asequence of identical TA-imaging exper-iments in two radically different cuprates:strongly underdoped Ca1.88Na0.12CuO2Cl2(Na-CCOC; critical temperature Tc ~ 21 K)and Bi2Sr2Dy0.2Ca0.8Cu2O8+d (Dy-Bi2212; Tc ~45 K). As indicated schematically in Fig. 2, Band C, they have completely different crystal-lographic structure, chemical constituents, anddopant species and sites in the terminationlayers lying between the CuO2 plane and theSTM tip. Na-CCOC has a single CuO2 layer

Fig. 4. (A and D) R maps of Na-CCOC and Dy-Bi2212, respectively (taken at 150 mV from areas inthe blue boxes of Fig. 3, C and D). The fields of view are (A) 5.0 nm by 5.3 nm and (B) 5.0 nm by5.0 nm. The blue boxes in (A) and (D) indicate areas of Fig. 4, B and C, and Fig. 4, E and F,respectively. (B and E) Higher-resolution R map within equivalent domains from Na-CCOC and Dy-Bi2212, respectively (blue boxes of Fig. 4, A and D). The locations of the Cu atoms are shown asblack crosses. (C and F) Constant-current topographic images simultaneously taken with Fig. 4, Band E, respectively. Imaging conditions are (C) 50 pA at 600 mV and (F) 50 pA at 150 mV. Themarkers show atomic locations, used also in Fig. 4, B and E. The fields of view of these images areshown in Fig. 3, A and B, as orange boxes.

Fig. 5. (A) Locations relative tothe O and Cu orbitals in the CuO2plane where each dI/dV spectrumat the surfaces of Fig. 4, C and F,and shown in Fig. 5B, is mea-sured. Spectra are measuredalong equivalent lines labeled1, 2, 3, and 4 in both domainsof Fig. 4, B and E, and Fig. 5A.(B) Differential tunneling con-ductance spectra taken alongparallel lines through equiv-alent domains in Na-CCOC andDy-Bi2212. All spectra weretaken under identical junction conditions (200 pA, 200 mV). Numbers (1 to 4)correspond to trajectories where these sequences of spectra were taken.Locations of the trajectories, relative to the domains, are shown betweenFig. 4B (C) and 4E (F) by arrows.

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Page 57: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa2Cu3O6!!

Superconductors Identified by Resonant Elastic X-ray Scattering

A. J. Achkar,1 R. Sutarto,2,3 X. Mao,1 F. He,3 A. Frano,4,5 S. Blanco-Canosa,4 M. Le Tacon,4 G. Ghiringhelli,6

L. Braicovich,6 M. Minola,6 M. Moretti Sala,7 C. Mazzoli,6 Ruixing Liang,2 D. A. Bonn,2 W.N. Hardy,2

B. Keimer,4 G.A. Sawatzky,2 and D.G. Hawthorn1,*1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z43Canadian Light Source, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0X44Max-Planck-Institut fur Festkorperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany

5Helmholtz-Zentrum Berlin fur Materialien und Energie, Albert-Einstein-Straße 15, D-12489 Berlin, Germany6CNR-SPIN, CNISM and Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

7European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France(Received 16 July 2012; published 17 October 2012)

Recently, charge density wave (CDW) order in the CuO2 planes of underdoped YBa2Cu3O6!! was

detected using resonant soft x-ray scattering. An important question remains: is the chain layer responsible

for this charge ordering? Here, we explore the energy and polarization dependence of the resonant

scattering intensity in a detwinned sample of YBa2Cu3O6:75 with ortho-III oxygen ordering in the chain

layer. We show that the ortho-III CDW order in the chains is distinct from the CDW order in the planes.

The ortho-III structure gives rise to a commensurate superlattice reflection at Q " #0:33 0L$ whose

energy and polarization dependence agrees with expectations for oxygen ordering and a spatial modu-

lation of the Cu valence in the chains. Incommensurate peaks at [0.30 0 L] and [0 0.30 L] from the CDW

order in the planes are shown to be distinct in Q as well as their temperature, energy, and polarization

dependence, and are thus unrelated to the structure of the chain layer. Moreover, the energy dependence of

the CDW order in the planes is shown to result from a spatial modulation of energies of the Cu 2p to

3dx2%y2 transition, similar to stripe-ordered 214 cuprates.

DOI: 10.1103/PhysRevLett.109.167001 PACS numbers: 74.72.Gh, 61.05.cp, 71.45.Lr, 78.70.Dm

Direct evidence for charge density wave (CDW) orderin YBa2Cu3O6!! (YBCO) was recently observed inhigh magnetic field using nuclear magnetic resonance [1]and in zero-field diffraction, first with resonant softx-ray scattering (RSXS) [2] and subsequently with hardx-ray scattering [3]. Prior to these measurements,density wave order [4,5] had been observed in 214cuprates [La2%x%y&Ba; Sr'x&Eu;Nd'yCuO4] [6] as wellas Ca2%xNaxCuO2Cl2 [7] and Bi2Sr2CaCu2O8!! [8].However, density wave order in YBCO—a material longconsidered a benchmark cuprate due to its low disorder andhigh Tc;max ’ 94:2 K—had only been inferred indirectly,being offered as an explanation for Hall effect measure-ments [9] and the electron pockets observed in quantumoscillation experiments [10–12]. The observation of den-sity wave order in YBCO thus marks an important mile-stone in efforts to determine whether density wave order isgeneric to the cuprates while providing new opportunitiesto identify common features of CDWorder in the cuprates.

RSXS is well suited to give direct insight into the natureof CDW order in YBCO. RSXS involves diffraction withthe photon energy tuned through an x-ray absorption edge.This gives significant energy dependence to the atomicscattering form factor, f&!', enhancing the scatteringfrom weak ordering and providing sensitivity to the charge,

spin, and orbital occupation of specific elements. At the CuL absorption edge, the scattering is sensitive to modula-tions in the unoccupied Cu 3d states that are central to thelow energy physics of the cuprates [13–17]. The recentRSXS measurements of Ghiringhelli et al. at the Cu Labsorption edge identified superlattice peaks at Q "#0:31 0L$ and [0 0.31 L] indicative of CDW order [2].They also showed that the intensity of the superlatticereflections peak at (Tc and decrease in intensity forT < Tc, providing a clear link between the density waveorder and superconductivity [2]. Importantly, based on theenergy dependence of the scattering intensity and thepresence of peaks at H " 0:31 and K " 0:31 in a det-winned sample, Ghiringhelli et al. also demonstrate thatthe CDW superlattice peaks originate from modulations inthe CuO2 planes.However, the possible role of the charge reservoir layer

in stabilizing the CDWorder is not yet clear. In YBCO, thecharge reservoir for the CuO2 planes is composed of CuOchains. The Cu sites in the chains (Cu1) and planes (Cu2)have different orbital symmetries and contribute differ-ently to x-ray absorption spectroscopy (XAS) and RSXSmeasurements [18,19]. In addition to making the structureorthorhombic (a ! b), the chain layer can be oxygenordered into a variety of ‘‘ortho’’ ordered phases [20,21].

PRL 109, 167001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

19 OCTOBER 2012

0031-9007=12=109(16)=167001(5) 167001-1 ! 2012 American Physical Society

Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa2Cu3O6!!

Superconductors Identified by Resonant Elastic X-ray Scattering

A. J. Achkar,1 R. Sutarto,2,3 X. Mao,1 F. He,3 A. Frano,4,5 S. Blanco-Canosa,4 M. Le Tacon,4 G. Ghiringhelli,6

L. Braicovich,6 M. Minola,6 M. Moretti Sala,7 C. Mazzoli,6 Ruixing Liang,2 D. A. Bonn,2 W.N. Hardy,2

B. Keimer,4 G.A. Sawatzky,2 and D.G. Hawthorn1,*1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z43Canadian Light Source, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0X44Max-Planck-Institut fur Festkorperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany

5Helmholtz-Zentrum Berlin fur Materialien und Energie, Albert-Einstein-Straße 15, D-12489 Berlin, Germany6CNR-SPIN, CNISM and Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

7European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France(Received 16 July 2012; published 17 October 2012)

Recently, charge density wave (CDW) order in the CuO2 planes of underdoped YBa2Cu3O6!! was

detected using resonant soft x-ray scattering. An important question remains: is the chain layer responsible

for this charge ordering? Here, we explore the energy and polarization dependence of the resonant

scattering intensity in a detwinned sample of YBa2Cu3O6:75 with ortho-III oxygen ordering in the chain

layer. We show that the ortho-III CDW order in the chains is distinct from the CDW order in the planes.

The ortho-III structure gives rise to a commensurate superlattice reflection at Q " #0:33 0L$ whose

energy and polarization dependence agrees with expectations for oxygen ordering and a spatial modu-

lation of the Cu valence in the chains. Incommensurate peaks at [0.30 0 L] and [0 0.30 L] from the CDW

order in the planes are shown to be distinct in Q as well as their temperature, energy, and polarization

dependence, and are thus unrelated to the structure of the chain layer. Moreover, the energy dependence of

the CDW order in the planes is shown to result from a spatial modulation of energies of the Cu 2p to

3dx2%y2 transition, similar to stripe-ordered 214 cuprates.

DOI: 10.1103/PhysRevLett.109.167001 PACS numbers: 74.72.Gh, 61.05.cp, 71.45.Lr, 78.70.Dm

Direct evidence for charge density wave (CDW) orderin YBa2Cu3O6!! (YBCO) was recently observed inhigh magnetic field using nuclear magnetic resonance [1]and in zero-field diffraction, first with resonant softx-ray scattering (RSXS) [2] and subsequently with hardx-ray scattering [3]. Prior to these measurements,density wave order [4,5] had been observed in 214cuprates [La2%x%y&Ba; Sr'x&Eu;Nd'yCuO4] [6] as wellas Ca2%xNaxCuO2Cl2 [7] and Bi2Sr2CaCu2O8!! [8].However, density wave order in YBCO—a material longconsidered a benchmark cuprate due to its low disorder andhigh Tc;max ’ 94:2 K—had only been inferred indirectly,being offered as an explanation for Hall effect measure-ments [9] and the electron pockets observed in quantumoscillation experiments [10–12]. The observation of den-sity wave order in YBCO thus marks an important mile-stone in efforts to determine whether density wave order isgeneric to the cuprates while providing new opportunitiesto identify common features of CDWorder in the cuprates.

RSXS is well suited to give direct insight into the natureof CDW order in YBCO. RSXS involves diffraction withthe photon energy tuned through an x-ray absorption edge.This gives significant energy dependence to the atomicscattering form factor, f&!', enhancing the scatteringfrom weak ordering and providing sensitivity to the charge,

spin, and orbital occupation of specific elements. At the CuL absorption edge, the scattering is sensitive to modula-tions in the unoccupied Cu 3d states that are central to thelow energy physics of the cuprates [13–17]. The recentRSXS measurements of Ghiringhelli et al. at the Cu Labsorption edge identified superlattice peaks at Q "#0:31 0L$ and [0 0.31 L] indicative of CDW order [2].They also showed that the intensity of the superlatticereflections peak at (Tc and decrease in intensity forT < Tc, providing a clear link between the density waveorder and superconductivity [2]. Importantly, based on theenergy dependence of the scattering intensity and thepresence of peaks at H " 0:31 and K " 0:31 in a det-winned sample, Ghiringhelli et al. also demonstrate thatthe CDW superlattice peaks originate from modulations inthe CuO2 planes.However, the possible role of the charge reservoir layer

in stabilizing the CDWorder is not yet clear. In YBCO, thecharge reservoir for the CuO2 planes is composed of CuOchains. The Cu sites in the chains (Cu1) and planes (Cu2)have different orbital symmetries and contribute differ-ently to x-ray absorption spectroscopy (XAS) and RSXSmeasurements [18,19]. In addition to making the structureorthorhombic (a ! b), the chain layer can be oxygenordered into a variety of ‘‘ortho’’ ordered phases [20,21].

PRL 109, 167001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

19 OCTOBER 2012

0031-9007=12=109(16)=167001(5) 167001-1 ! 2012 American Physical Society

Analysis of the energy and polarization dependence ofthe integrated scattering intensities (Fig. 3) demonstratesthat the H ! 0:30 and K ! 0:30 peaks are due to modu-lations in the CuO2 planes, whereas the H ! 0:33 peaksare due to ortho-III ordering in the chain layer. To modelthe scattering intensity of the H ! 0:33 peak, we followedthe procedure in Ref. [19] which illustrated that the

scattering intensity and polarization dependence of theoxygen order superstructure in ortho-II ordered YBCO(full-empty-full-empty chains) could be calculated by ac-counting for the impact of the oxygen dopants on the Cu1 dstates in the full and empty chains. This was done byexperimentally determining the energy dependence ofthe atomic scattering tensor, Fi, for Cu in full, FCu1f"!#,and empty, FCu1e"!#, chains using polarization dependentx-ray absorption measurements in YBCO prepared witheither an entirely full (YBa2Cu3O7) or an entirely empty(YBa2Cu3O6) chain layer. Here we use the same analysisfor the H ! 0:33 peak with FCu1f"!# and FCu1e"!# fromRef. [19] and Isc;o$III"H ! 0:33; ~!# ! jfCu1f"!; ~!# % fO $fCu1e"!; ~!#j2. As shown in Fig. 3(a), this analysis repro-duces the energy and polarization dependence of the H !0:33 peak, providing confirmation that this peak is domi-nated by the oxygen order in the chain layer.In contrast, both the polarization and energy dependence

of the H ! 0:30 and K ! 0:30 peaks are consistent with aspatial modulation of the Cu 3dx2$y2 states in the CuO2

planes. First, one must note that the incident " and #polarizations couple to different components of the scat-tering tensor. For # polarization, the photon polarization isentirely along the b"a# axis for the H"K# ! 0:30 peakand is therefore sensitive to the bb"aa# components ofthe scattering tensor. However, for " polarized light,the polarization has components along both the a andc axes that depend on the scattering geometry. For modu-lations of Cu 3dx2$y2 states, faa;Cu2 ’ fbb;Cu2 & fcc;Cu2and Isc"""0#=Isc"##0# ! 'sin"$# sin"%#!faa(2, where $and % are the angles of the incident and scattered lightrelative to the sample surface [see Fig. 1(b)] [30]. For thevalues of $ and % in our measurement, one would expectthe ratio of Isc"""0#=Isc"##0# ! 0:46 for a modulation ofCu 3dx2$y2 states. As shown in Fig. 3(b), the K ! 0:30peak is in good agreement with this ratio.A final intriguing aspect of the energy dependence of the

scattering intensity is that the line shape can be describedby a simple phenomenological model for the scatteringintensity based on a spatial modulation of the energy of theCu 2p to 3dx2$y2 transition. The energy of this transition is

determined by the energy of the 3dx2$y2 states, as well as

the core hole energy and the interaction energy of the corehole with the d electrons, all of which may be spatiallymodulated. This energy shift model was recently shown toaccount for the energy dependence of the scattering inten-sity of the [1=4 0 L] charge stripe ordering peak inLa1:475Nd0:4Sr0:125CuO4, unlike models based on latticedisplacements or charge density modulations [17].Although in YBCO we do not know the structure factorthat accounts for the [0.30 0 L] and [0 0.30L] peaks, we cannaively invoke the same energy shift model and assume thatIsc'0:30 0L("!# / Isc'0 0:30L("!# / jfCu2a"@!% !E# $fCu2b"@!$ !E#j2, where Cu2a and Cu2b represent twosites in the CuO2 planes with f"!# that is identical apart

FIG. 3 (color online). (a) The measured energy dependence ofthe [0.33 0 1.4] oxygen ordering peak with # and " polarizedincident light along with the calculated spectra for ortho-IIIoxygen ordering of the chain layer. (b) The energy dependenceof the [0 0.30 1.44] peak measured with # and " polarized light.(c) The energy dependence of the [0.30 0 1.44] peak with "polarized light compared to the energy shift model calculation.The energy shift calculation captures the correct peak positionand energy width of the scattering intensity. (d) The energy shiftmodel calculation compared to the [0 0.30 1.44] peak with #polarized light.

FIG. 2 (color online). The [H 0 L] [(a) and (b)] and [0 K L][(c) and (d)] normalized scattering intensity, Isc=I0, in arbitraryunits. The scattering intensity was measured with # [(a) and (c)]and " [(b) and (d)] incident photon polarization at T ! 60 K.r.l.u., reciprocal lattice units.

PRL 109, 167001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

19 OCTOBER 2012

167001-3

Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa2Cu3O6!!

Superconductors Identified by Resonant Elastic X-ray Scattering

A. J. Achkar,1 R. Sutarto,2,3 X. Mao,1 F. He,3 A. Frano,4,5 S. Blanco-Canosa,4 M. Le Tacon,4 G. Ghiringhelli,6

L. Braicovich,6 M. Minola,6 M. Moretti Sala,7 C. Mazzoli,6 Ruixing Liang,2 D. A. Bonn,2 W.N. Hardy,2

B. Keimer,4 G.A. Sawatzky,2 and D.G. Hawthorn1,*1Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

2Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z43Canadian Light Source, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0X44Max-Planck-Institut fur Festkorperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany

5Helmholtz-Zentrum Berlin fur Materialien und Energie, Albert-Einstein-Straße 15, D-12489 Berlin, Germany6CNR-SPIN, CNISM and Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy

7European Synchrotron Radiation Facility, BP 220, F-38043 Grenoble Cedex, France(Received 16 July 2012; published 17 October 2012)

Recently, charge density wave (CDW) order in the CuO2 planes of underdoped YBa2Cu3O6!! was

detected using resonant soft x-ray scattering. An important question remains: is the chain layer responsible

for this charge ordering? Here, we explore the energy and polarization dependence of the resonant

scattering intensity in a detwinned sample of YBa2Cu3O6:75 with ortho-III oxygen ordering in the chain

layer. We show that the ortho-III CDW order in the chains is distinct from the CDW order in the planes.

The ortho-III structure gives rise to a commensurate superlattice reflection at Q " #0:33 0L$ whose

energy and polarization dependence agrees with expectations for oxygen ordering and a spatial modu-

lation of the Cu valence in the chains. Incommensurate peaks at [0.30 0 L] and [0 0.30 L] from the CDW

order in the planes are shown to be distinct in Q as well as their temperature, energy, and polarization

dependence, and are thus unrelated to the structure of the chain layer. Moreover, the energy dependence of

the CDW order in the planes is shown to result from a spatial modulation of energies of the Cu 2p to

3dx2%y2 transition, similar to stripe-ordered 214 cuprates.

DOI: 10.1103/PhysRevLett.109.167001 PACS numbers: 74.72.Gh, 61.05.cp, 71.45.Lr, 78.70.Dm

Direct evidence for charge density wave (CDW) orderin YBa2Cu3O6!! (YBCO) was recently observed inhigh magnetic field using nuclear magnetic resonance [1]and in zero-field diffraction, first with resonant softx-ray scattering (RSXS) [2] and subsequently with hardx-ray scattering [3]. Prior to these measurements,density wave order [4,5] had been observed in 214cuprates [La2%x%y&Ba; Sr'x&Eu;Nd'yCuO4] [6] as wellas Ca2%xNaxCuO2Cl2 [7] and Bi2Sr2CaCu2O8!! [8].However, density wave order in YBCO—a material longconsidered a benchmark cuprate due to its low disorder andhigh Tc;max ’ 94:2 K—had only been inferred indirectly,being offered as an explanation for Hall effect measure-ments [9] and the electron pockets observed in quantumoscillation experiments [10–12]. The observation of den-sity wave order in YBCO thus marks an important mile-stone in efforts to determine whether density wave order isgeneric to the cuprates while providing new opportunitiesto identify common features of CDWorder in the cuprates.

RSXS is well suited to give direct insight into the natureof CDW order in YBCO. RSXS involves diffraction withthe photon energy tuned through an x-ray absorption edge.This gives significant energy dependence to the atomicscattering form factor, f&!', enhancing the scatteringfrom weak ordering and providing sensitivity to the charge,

spin, and orbital occupation of specific elements. At the CuL absorption edge, the scattering is sensitive to modula-tions in the unoccupied Cu 3d states that are central to thelow energy physics of the cuprates [13–17]. The recentRSXS measurements of Ghiringhelli et al. at the Cu Labsorption edge identified superlattice peaks at Q "#0:31 0L$ and [0 0.31 L] indicative of CDW order [2].They also showed that the intensity of the superlatticereflections peak at (Tc and decrease in intensity forT < Tc, providing a clear link between the density waveorder and superconductivity [2]. Importantly, based on theenergy dependence of the scattering intensity and thepresence of peaks at H " 0:31 and K " 0:31 in a det-winned sample, Ghiringhelli et al. also demonstrate thatthe CDW superlattice peaks originate from modulations inthe CuO2 planes.However, the possible role of the charge reservoir layer

in stabilizing the CDWorder is not yet clear. In YBCO, thecharge reservoir for the CuO2 planes is composed of CuOchains. The Cu sites in the chains (Cu1) and planes (Cu2)have different orbital symmetries and contribute differ-ently to x-ray absorption spectroscopy (XAS) and RSXSmeasurements [18,19]. In addition to making the structureorthorhombic (a ! b), the chain layer can be oxygenordered into a variety of ‘‘ortho’’ ordered phases [20,21].

PRL 109, 167001 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

19 OCTOBER 2012

0031-9007=12=109(16)=167001(5) 167001-1 ! 2012 American Physical Society

Resonant X-Ray Scattering Measurements of a Spatial Modulation of the Cu 3d and O 2pEnergies in Stripe-Ordered Cuprate Superconductors

A. J. Achkar,1 F. He,2 R. Sutarto,3 J. Geck,4 H. Zhang,5 Y.-J. Kim,5 and D.G. Hawthorn1

1Department of Physics and Astronomy, University of Waterloo, Waterloo N2L 3G1, Canada2Canadian Light Source, University of Saskatchewan, Saskatoon, Saskatchewan S7N 0X4, Canada

3Department of Physics and Astronomy, University of British Columbia, Vancouver V6T 1Z4, Canada4Leibniz Institute for Solid State and Materials Research IFW Dresden, Helmholtzstraße 20, 01069 Dresden, Germany

5Department of Physics, University of Toronto, Toronto M5S 1A7, Canada(Received 12 March 2012; published 2 January 2013)

A prevailing description of the stripe phase in underdoped cuprate superconductors is that the charge

carriers (holes) phase segregate on a microscopic scale into hole-rich and hole-poor regions. We report

resonant elastic x-ray scattering measurements of stripe-ordered La1:475Nd0:4Sr0:125CuO4 at the Cu L and

O K absorption edges that identify an additional feature of stripe order. Analysis of the energy dependence

of the scattering intensity reveals that the dominant signature of the stripe order is a spatial modulation in

the energies of Cu 3d and O 2p states rather than the large modulation of the charge density (valence)

envisioned in the common stripe paradigm. These energy shifts are interpreted as a spatial modulation of

the electronic structure and may point to a valence-bond-solid interpretation of the stripe phase.

DOI: 10.1103/PhysRevLett.110.017001 PACS numbers: 74.72.Gh, 61.05.cp, 71.45.Lr, 78.70.Dm

Static stripe order in cuprates was first theoreticallypredicted by mean-field Hubbard model calculations[1–4] and subsequently observed in lanthanum-based cup-rates by neutron and x-ray diffraction [5–11]. Althoughstill a matter of debate, more recent work has indicated thatstripe-like density wave order is generic to the cuprates[12–17] and plays a significant role in competing with orpossibly causing superconductivity [18].

Microscopically, stripes in the cuprates have beenwidely described as rivers of charge—hole-rich antiphasedomain walls that separate undoped antiferromagneticregions. However, alternate models with different under-lying physics, such as the valence bond solid (VBS),have also been proposed to explain stripe order [19–21].VBS models involve singlet formation between neigh-boring spins and, in contrast to other models of stripeorder, may occur with a small modulation of the chargedensity [20].

Distinguishing which of these models is most relevantto stripe order in the cuprates is challenging since themodels share many symmetries and experimental signa-tures. In particular, direct evidence for charge-densitymodulations, which may distinguish various models,has been elusive. Neutron and conventional x-ray scat-tering are only sensitive to lattice displacements. It istherefore only inferred indirectly that these lattice dis-placements are induced by modulations in charge density(valence). Resonant soft x-ray scattering (RSXS) offers ameans to couple more directly to modulations in theelectronic structure, including charge-density modula-tions. By performing an x-ray diffraction measurementon resonance (at an x-ray absorption edge), the atomicscattering form factor f!!" is enhanced and made

sensitive to the valence, orbital orientation, and spin stateof specific elements. A key feature of RSXS is that theenergy dependence of the scattering intensity through anabsorption edge differs for lattice distortions, charge-density modulations, or other forms of electronic order-ing, providing a means to distinguish these differenttypes of order.In the cuprates, RSXS of the [2", 0, L] charge-density

wave (CDW) superlattice peak has been measured in stripe-ordered La2#xBaxCuO4 (LBCO) [10], La2#x#yEuySrxCuO4

(LESCO) [11,22], and La1:475Nd0:4Sr0:125CuO4 (LNSCO)[23] at the O K (1s!2p) and Cu L (2p ! 3d) absorptionedges, which provide sensitivity to the O 2p and Cu 3dorbitals that are central to the physics of the cuprates. Thesemeasurements have been interpreted as direct evidence fora large valence modulation on the O sites [10]. Moreover,it is argued that a modulation of the valence occurs primarilyon the O sites and not on the Cu sites, which are insteadsubject to lattice distortions induced by the valence modu-lation on the O sites [10,11]. However, efforts to model theenergy dependence of the scattering intensity based on thispicture are not truly reconciled with experiment, leavingthis interpretation open to question [11].In this Letter, we present O K and Cu L edge RSXS

measurements of LNSCO. The energy dependence of thescattering intensity is modeled using x-ray absorptionmeasurements to determine f!!" at different sites in thelattice, a procedure that has proven effective in describingthe scattering intensity of valence modulations in the chainlayer of ortho-II YBa2Cu3O6$! (YBCO) [24]. Contrary toprevious analysis of LESCO [11] and LBCO [10], we showthat the resonant scattering intensity is best described bysmall energy shifts in the O 2p and Cu 3d states at different

PRL 110, 017001 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

4 JANUARY 2013

0031-9007=13=110(1)=017001(5) 017001-1 ! 2013 American Physical Society

Tuesday, May 14, 13

Page 58: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Electron spectral function

4

Finally, note that Fig. 2 also has a broad local minimumnear Q = (⇡, ⇡). Here �Q(k) is found to have the odd-paritypx,y form in Table I. Condensation of this mode will breaktime-reversal symmetry, and lead to the state with sponta-neous orbital currents proposed by Chakravarty et al. [25].

We now study the electronic spectral function in the Q =(Q0, 0) charge-ordered state. We will work with the state withbi-directional charge order [13]; in our theory the degeneracybetween the uni-directional and bi-directional charge orderedstate is broken only by terms quartic in the �Q, and we havenot accounted for these here. Choosing the largest 2 compo-nents from Table I, we have the order parameter

�Q(k) =(�s + �d(cos kx � cos ky) , Q = (±Q0, 0)�s � �d(cos kx � cos ky) , Q = (0,±Q0) (9)

with �s/�d = �0.234. We computed the imaginary part ofthe single-electron Green’s function, ImGk,k(! + i⌘), and theresults are shown in Fig. 4. The stability of the Fermi arc in

FIG. 4: Electron spectral density in the phase with bidirectionalcharge order at Q = (Q0, 0) and (0,Q0) with Q0 = 4⇡/11. Theleft panel show ImGk,k(! + i⌘) at ! = 0 and ⌘ = 0.02; the rightpanel shows log

⇥ImGk,k(! + i⌘)

⇤for the same parameters, as a way

of enhancing the low intensities. The dashed line is the underlyingFermi surface of the metal without charge order. The charge order isas in Eq. (9) with �d = 0.3, and other parameters as in Fig. 2.

‘nodal’ region (kx ⇡ ky) is enhanced [30] because of the weakcoupling to the charge order, arising from the predominantd character of Eq. (9). However, there is strong coupling inthe anti-nodal region, and any Fermi surfaces appearing in thelatter region should be easily broadened by impurity-inducedphase-shifts in the charge ordering.

Discussion. We have described the features of a simplemodel of the underdoped cuprates. We began with a metalwith antiferromagnetic spin correlations. Exchange of the an-tiferromagnetic fluctuations leads to an attractive force in thespin-singlet d channel of the particle-particle sector, and a cor-responding instability to superconductivity. Ref. [20] notedthat the same antiferromagnetic fluctuations also lead to anenhancement in the spin-singlet d channel of the particle-holesector, and a sub-dominant instability to bond order. Here wehave studied the momentum-space structure of the latter in-stability across the entire Brillouin zone, without any assump-tions of particle-hole symmetry or the continuum limit, and

found that the spin-singlet, d character persists to the exper-imentally observed wavevectors. This leads to our proposalfor charge ordering in the underdoped cuprates, summarizedin Eq. (9).

The charge order here, and its connection to spin order,should be distinguished from that of the “stripe” model [31];in the latter model, the charge order is tied to the squareof incommensurate spin order, and occurs at twice the spin-ordering wavevector. Instead, in our model, bond orderappears in a regime of “quantum-disordered” antiferromag-netism [26]. This is consistent with Ref. [7], which showedthat the spin order and charge order are in distinct dopingregimes in YBa2Cu3Oy, with the charge-ordering regime co-inciding with regime of quantum oscillations [16, 17].

We also presented computations of the spectral function ofthe charge-ordered metal, and showed that it contains “Fermiarc” features across the diagonals of the Brillouin zone. It wasargued by Harrison and Sebastian [15] that the same Fermiarcs can be combined to explain the quantum oscillations.

In our computations here, the strongest instability to chargeordering was at wavevector ±(Q0,±Q0); but notice also theblue “valleys” in Fig. 2 extending from this wavevector to(±Q0, 0), (0,±Q0). Other approaches to charge orderingdue to antiferromagnetic interactions [23, 26], which includestrong-coupling e↵ects, do find ordering along the (1, 0) and(0, 1) directions. Specifically, we expect that extending ourpresent computation to include pairing e↵ects will enhancecharge order correlations along the (1, 0) and (0, 1) directions,as was the case in the computations of Ref. [26]. It would alsobe interesting to include an applied magnetic field in such anextension.

For the phase diagram of the hole-doped cuprates, ourmodel has a quantum-critical point near optimal doping as-sociated with disappearance of bond order [7, 26] describedby Eq. (9). An important challenge is to use such a criticalpoint to describe the evolution of the Fermi surface [16], andthe ‘strange’ metal.

Acknowledgments. We thank for A. Chubukov, D. Chowd-hury, J. C. Davis, E. Demler, K. Efetov, D. Hawthorn,P. Hirschfeld, J. Ho↵man, H. Meier, W. Metzner, C. Pepin,and L. Taillefer useful discussions. This research was sup-ported by the NSF under Grant DMR-1103860, and by theU.S. Army Research O�ce Award W911NF-12-1-0227.

[1] J. E. Ho↵man et al., Science 295, 466 (2002).[2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and

A. Yazdani, Science 303, 1995 (2004).[3] Y. Kohsaka, et al., Science 315, 1380 (2007).[4] W. D. Wise et al. Nature Physics 4, 696 (2008).[5] M. J. Lawler et al., Nature 466, 374 (2010).[6] A. Mesaros et al., Science 333, 426 (2011).[7] T. Wu et al., Nature 477, 191 (2011).[8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012).[9] G. Ghiringhelli et al., Science 337, 821 (2012).

3

� �(k) Q = Q = Q = Q = �S (k)(1.15,1.15) (1.15, 0) (0,0) (⇡, ⇡)

s 1 0 -0.231 0 0 0s0 cos kx + cos ky 0 0.044 0 0 0s00 cos(2kx) + cos(2ky) 0 -0.046 0 0 0d cos kx � cos ky 0.993 0.963 0.997 0 0.997d0 cos(2kx) � cos(2ky) - 0.069 -0.067 -0.057 0 -0.056d00 2 sin kx sin ky 0 0 0 0 0px

p2 sin kx 0 0 0 0.706 0

pyp

2 sin ky 0 0 0 -0.706 0g (cos kx � cos ky) -0.009 0 0 0 0⇥p

8 sin kx sin ky

TABLE I: Values of cQ,� in the expansion in Eq. (9) for various val-ues Q and �. The last column shows the coe�cients in the corre-sponding expansion for �S (k). We used µ = �1.2, ⇠ = 2, T = 0.06,and L = 80.

write

�Q(k) =X

cQ,� �(k) (9)

where � labels various orthonormal basis functions, and cQ,�are numerical coe�cients that we determine. Thus we havethe s basis function s(k) = 1, the extended s function s0 (k) = cos kx + cos ky, the d function d(k) = cos kx � cos ky

and so on, as shown in Table I. Depending upon the symmetryof Q (in particular, the little group of the wavevector Q) andof the eigenvector, some of the cQ,� may be exactly zero. Butfor a generic Q, only time-reversal constrains the values ofcQ,�, and we are allowed to have an admixture of many basisfunctions. Nevertheless, we will see that only a small numberof basis functions have appreciable coe�cients, and so Eq. (9)represents a useful expansion.

Dc†k�Q/2,↵ck+Q/2,↵

E/ �Q(k) = �0(cos kx � cos ky)

with Q = (±Q, 0), (0,±Q).

Acknowledgments. We thank for A. Chubukov, K. Efe-tov, H. Meier, W. Metzner, and C. Pepin useful discus-sions. This research was supported by the NSF under GrantDMR-1103860, and by the U.S. Army Research O�ce AwardW911NF-12-1-0227.

[1] J. E. Ho↵man et al., Science 295, 466 (2002).[2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and

A. Yazdani, Science 303, 1995 (2004).[3] Y. Kohsaka, et al., Science 315, 1380 (2007).[4] W. D. Wise et al. Nature Physics 4, 696 (2008).[5] M. J. Lawler et al., Nature 466, 374 (2010).[6] A. Mesaros et al., Science 333, 426 (2011).[7] T. Wu et al., Nature 477, 191 (2011).[8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012).[9] G. Ghiringhelli et al., Science 337, 821 (2012).

[10] J. Chang et al., Nature Phys. 8, 871 (2012).[11] A. J. Achkar et al., Phys. Rev. Lett. 109, 167001 (2012).[12] D. LeBoeuf, S. Kramer, W. N. Hardy, Ruixing Liang,

D. A. Bonn, and C. Proust, Nature Physics 9, 79 (2013).[13] N. Doiron-Leyraud et al., Nature 447, 565 (2007).[14] N. Harrison and S. E. Sebastian, Phys. Rev. Lett. 106, 226402

(2011).[15] S. E. Sebastian, N. Harrison and G. G. Lonzarich, Rep. Prog.

Phys. 75, 102501 (2012).[16] B. Vignolle, D. Vignolles, M.-H. Julien, and C. Proust,

C. R. Physique 14, 39 (2013).[17] M. A. Metlitski and S. Sachdev, Phys. Rev. B 82, 075128

(2010).[18] Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608

(2000).[19] T. Holder and W. Metzner, Phys. Rev. B 85, 165130 (2012).[20] C. Husemann and W. Metzner, Phys. Rev. B 86, 085113 (2012).[21] K. B. Efetov, H. Meier, and C. Pepin, arXiv:1210.3276.[22] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak,

Phys. Rev. B 63, 094503 (2001).

4

Finally, note that Fig. 2 also has a broad local minimumnear Q = (⇡, ⇡). Here �Q(k) is found to have the odd-paritypx,y form in Table I. Condensation of this mode will breaktime-reversal symmetry, and lead to the state with sponta-neous orbital currents proposed by Chakravarty et al. [25].

We now study the electronic spectral function in the Q =(Q0, 0) charge-ordered state. We will work with the state withbi-directional charge order [13]; in our theory the degeneracybetween the uni-directional and bi-directional charge orderedstate is broken only by terms quartic in the �Q, and we havenot accounted for these here. Choosing the largest 2 compo-nents from Table I, we have the order parameter

�Q(k) =(�s + �d(cos kx � cos ky) , Q = (±Q0, 0)�s � �d(cos kx � cos ky) , Q = (0,±Q0) (9)

with �s/�d = �0.234. We computed the imaginary part ofthe single-electron Green’s function, ImGk,k(! + i⌘), and theresults are shown in Fig. 4. The stability of the Fermi arc in

FIG. 4: Electron spectral density in the phase with bidirectionalcharge order at Q = (Q0, 0) and (0,Q0) with Q0 = 4⇡/11. Theleft panel show ImGk,k(! + i⌘) at ! = 0 and ⌘ = 0.02; the rightpanel shows log

⇥ImGk,k(! + i⌘)

⇤for the same parameters, as a way

of enhancing the low intensities. The dashed line is the underlyingFermi surface of the metal without charge order. The charge order isas in Eq. (9) with �d = 0.3, and other parameters as in Fig. 2.

‘nodal’ region (kx ⇡ ky) is enhanced [30] because of the weakcoupling to the charge order, arising from the predominantd character of Eq. (9). However, there is strong coupling inthe anti-nodal region, and any Fermi surfaces appearing in thelatter region should be easily broadened by impurity-inducedphase-shifts in the charge ordering.

Discussion. We have described the features of a simplemodel of the underdoped cuprates. We began with a metalwith antiferromagnetic spin correlations. Exchange of the an-tiferromagnetic fluctuations leads to an attractive force in thespin-singlet d channel of the particle-particle sector, and a cor-responding instability to superconductivity. Ref. [20] notedthat the same antiferromagnetic fluctuations also lead to anenhancement in the spin-singlet d channel of the particle-holesector, and a sub-dominant instability to bond order. Here wehave studied the momentum-space structure of the latter in-stability across the entire Brillouin zone, without any assump-tions of particle-hole symmetry or the continuum limit, and

found that the spin-singlet, d character persists to the exper-imentally observed wavevectors. This leads to our proposalfor charge ordering in the underdoped cuprates, summarizedin Eq. (9).

The charge order here, and its connection to spin order,should be distinguished from that of the “stripe” model [31];in the latter model, the charge order is tied to the squareof incommensurate spin order, and occurs at twice the spin-ordering wavevector. Instead, in our model, bond orderappears in a regime of “quantum-disordered” antiferromag-netism [26]. This is consistent with Ref. [7], which showedthat the spin order and charge order are in distinct dopingregimes in YBa2Cu3Oy, with the charge-ordering regime co-inciding with regime of quantum oscillations [16, 17].

We also presented computations of the spectral function ofthe charge-ordered metal, and showed that it contains “Fermiarc” features across the diagonals of the Brillouin zone. It wasargued by Harrison and Sebastian [15] that the same Fermiarcs can be combined to explain the quantum oscillations.

In our computations here, the strongest instability to chargeordering was at wavevector ±(Q0,±Q0); but notice also theblue “valleys” in Fig. 2 extending from this wavevector to(±Q0, 0), (0,±Q0). Other approaches to charge orderingdue to antiferromagnetic interactions [23, 26], which includestrong-coupling e↵ects, do find ordering along the (1, 0) and(0, 1) directions. Specifically, we expect that extending ourpresent computation to include pairing e↵ects will enhancecharge order correlations along the (1, 0) and (0, 1) directions,as was the case in the computations of Ref. [26]. It would alsobe interesting to include an applied magnetic field in such anextension.

For the phase diagram of the hole-doped cuprates, ourmodel has a quantum-critical point near optimal doping as-sociated with disappearance of bond order [7, 26] describedby Eq. (9). An important challenge is to use such a criticalpoint to describe the evolution of the Fermi surface [16], andthe ‘strange’ metal.

Acknowledgments. We thank for A. Chubukov, D. Chowd-hury, J. C. Davis, E. Demler, K. Efetov, D. Hawthorn,P. Hirschfeld, J. Ho↵man, H. Meier, W. Metzner, C. Pepin,and L. Taillefer useful discussions. This research was sup-ported by the NSF under Grant DMR-1103860, and by theU.S. Army Research O�ce Award W911NF-12-1-0227.

[1] J. E. Ho↵man et al., Science 295, 466 (2002).[2] M. Vershinin, S. Misra, S. Ono, Y. Abe, Yoichi Ando, and

A. Yazdani, Science 303, 1995 (2004).[3] Y. Kohsaka, et al., Science 315, 1380 (2007).[4] W. D. Wise et al. Nature Physics 4, 696 (2008).[5] M. J. Lawler et al., Nature 466, 374 (2010).[6] A. Mesaros et al., Science 333, 426 (2011).[7] T. Wu et al., Nature 477, 191 (2011).[8] Y. Kohsaka, et al., Nature Physics 8, 534 (2012).[9] G. Ghiringhelli et al., Science 337, 821 (2012).

ImG(k,! + i⌘)log ImG(k,! + i⌘)

S. Sachdev and R. La Placa, arXiv:1303.2114Tuesday, May 14, 13

Page 59: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Summary

Antiferromagnetism in metals and the high temperature superconductors

Antiferromagnetic quantum criticality leads to d-wave superconductivity (supported by sign-problem-free Monte Carlo simulations)

Tuesday, May 14, 13

Page 60: Bond order in two-dimensional metals with ...qpt.physics.Harvard.edu/talks/grc13.pdfc) superconductivity but, under special circumstances, they can also order into filaments called

Metals with antiferromagnetic spin correlations have nearly degenerate instabilities: to d-wave superconductivity, and to a charge density wave with a d-wave form factor. This is a promising explanation of the pseudogap regime.

Summary

Antiferromagnetism in metals and the high temperature superconductors

Antiferromagnetic quantum criticality leads to d-wave superconductivity (supported by sign-problem-free Monte Carlo simulations)

Tuesday, May 14, 13