competing orders and quantum criticalityindico.ictp.it/event/a01152/session/1/contribution/1/... ·...

Competing orders and quantum criticality

Kwon ParkSubir SachdevMatthias VojtaPeter YoungYing Zhang

Quantum Phase TransitionsCambridge University PressLecture based on the review

article cond-mat/0109419 Science 286, 2479 (1999).

Transparencies online at http://pantheon.yale.edu/~subir

Quantum phase transition: ground states on either side of gc have distinct “order”

T Quantum-critical

• Critical excitations control dynamics in the wide quantum-critical region at non-zero temperatures.

Important property of ground state at g=gc : temporal and spatial scale invariance; characteristic energy scale at other values of g:

• Theory for a quantum system with strong correlations: describe phases on either side of gc by expanding in deviation from the quantum critical point.

• Critical point is a novel state of matter without quasiparticle excitations

ggc

~ zcg g �

� �

OutlineI. Quantum Ising Chain

II. Coupled Ladder Antiferromagnet A. Coherent state path integral B. Quantum field theory for critical point

III. Antiferromagnets with an odd number of S=1/2 spins per unit cell. A. Collinear spins, Berry phases, and bond-order.B. Non-collinear spins and deconfined spinons.

IV. Quantum transition in a BCS superconductor

V. Conclusions

Single order parameter.

Multiple order parameters.

2 2 2 2Transition between and pairingxyx y x yd d id

� �

�

I. Quantum Ising Chain

� � � �

Degrees of freedom: 1 qubits, "large"

,

1 1 or , 2 2

j j

j jj j j j

j N N�

��

�

�

�

� ��

�

0

Hamiltonian of decoupled qubits: x

jj

H Jg �� 2Jg

j�

j�

1 1

Coupling between qubits: z z

j jj

H J � ��

� � �

� �� 1 1j j j j� �

� ��

1 1

Prefers neighboring qubits

are

(not entangle

d)j j j j

either or� �

� ��

� �0 1 1x z zj j j

jJ gH H H � � �

��

Full Hamiltonian

leads to entangled states at g of order unity

Weakly-coupled qubits Ground state:

1 2

G

g

��

��

�

��

� �

� � �

Lowest excited states:

jj ��

Coupling between qubits creates “flipped-spin” quasiparticle states at momentum p

Entire spectrum can be constructed out of multi-quasiparticle states

jipxj

jp e��

�

�

� � � �

� �

2 1

1

Excitation energy 4 sin2

Excitation gap 2 2

pap J O g

gJ J O g

��

�

� ��

�

� � �

�

p

�

a��

a�

�

�

� �p�

� �1g �

Dynamic Structure Factor : Cross-section to flip a to a (or vice versa) while transferring energy

and momentum

( , )

p

S p

�

�

� �

�

�

� �,S p �

� �� Z p� � ��

Three quasiparticlecontinuum

Quasiparticle pole

~3�

Weakly-coupled qubits � �1g �

Structure holds to all orders in 1/g

At 0, collisions between quasiparticles broaden pole to a Lorentzian of width 1 where the

21is given by

Bk TB

T

k Te

� �

�

� �

� �

��

�

�

�

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Strongly-coupled qubits � �1g �

Ground states:

2

G

g

�

�

� ��

��

� �

� � �

Second state obtained by

and mix only at order N

G

G G g

� �

��

�� 0

Ferromagnetic moment0zN G G��

Lowest excited states: domain walls

jjd ��

Coupling between qubits creates new “domain-wall” quasiparticle states at momentum p

jipxj

jp e d��

�

� � � �

� �

2 2

2

Excitation energy 4 sin2

Excitation gap 2 2

pap Jg O g

J gJ O g

��

� � � ��

� � �

�

p

�

a��

a�

�

�

� �p�


and momentum

( , )

p

S p

�

�

� �

�

Strongly-coupled qubits � �1g �

�

� �,S p �

� � � � � �22

0 2N p� � � �

Two domain-wall continuum

~2�Structure holds to all orders in g

At 0, motion of domain walls leads to a finite ,

21and broadens coherent peak to a width 1 where Bk TB

T

k Te

�

�

�

�

�

� �

��

�

�

�

phase coherence time

S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997)

Entangled states at g of order unity

ggc

“Flipped-spin”Quasiparticle

weight Z

� �1/ 4~ cZ g g�

A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, 11919 (1994)

ggc

Ferromagnetic moment N0

� �1/8

0 ~ cN g g�

P. Pfeuty Annals of Physics, 57, 79 (1970)

ggc

Excitation energy gap � ~ cg g� �


and momentum

( , )

p

S p

�

�

� �

�

Critical coupling � �cg g�

�

� �7 /82 2 2~ c p�

�

�

� �,S p �

c p

No quasiparticles --- dissipative critical continuum

Quasiclassical dynamics

Quasiclassical dynamics

1/ 41~z z

j kj k

� �

�

P. Pfeuty Annals of Physics, 57, 79 (1970)

� ��

i

zi

zi

xiI gJH 1��

� � � �

� �

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

i dt t e

AT i

k T

�

� � � �

�

�

�

� ��

��

� � � �

�

��

�

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 2220 (1997).


II. Coupled Ladder AntiferromagnetA. Coherent state path integral B. Quantum field theory for critical point



V. Conclusions




� �

�

II. Coupled Ladder Antiferromagnet

N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).

S=1/2 spins on coupled 2-leg ladders

jiij

ij SSJH��

��

10 ��

J�J

close to 1�

Square lattice antiferromagnetExperimental realization: 42CuOLa

Ground state has long-rangemagnetic (Neel) order

� � 01 0 �� NS yx ii

i

�

Excitations: 2 spin waves 2 2 2 2p x x y yc p c p� � �

close to 0� Weakly coupled ladders

� ��2

1

Paramagnetic ground state

0iS �

�

Excitation: S=1 exciton(spin collective mode)

S=1/2 spinons are confined by a linear potential.

Energy dispersion away fromantiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c p�

��

�

T=0

�c

Quantum paramagnet

0�S�

Neel state

0S N�

�

Neel order N0 Spin gap �

��in �cuprates �

1�

II.A Coherent state path integralSee Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999).

Path integral for a single spin� ��

/

2

Tr

1 exp ( )

H TZ e

iS A d d H S�

� � � � � �

�

�

� ��

S

N N N�

� �

� � � � 0

Oriented area of triangle on surface of unit sphere bounded

by , ,and a fixed reference

A d

d��

� � �

�

�N N NAction for lattice antiferromagnet

� � � � � �, ,j j j jx x� � � �� N n L

1 identifies sublatticesj� � �

n and L vary slowly in space and time

Integrate out L and take the continuum limit

� � � �

� � � ��

2

2 22 2

, 1 exp ( , )

1 2

1 identifies sublattices

j jj

x

j

Z x iS A x d

d x d cg

�

�

� � � � �

�

�

��

�

��

�

� �

��

�

�n n

n n

c

c

c

c

g g

g g� �

� �

� �

�

� �

�

Berry phases can be neglected for coupled ladder antiferromagent(justified later)Discretize spacetime into a cubic lattice

Quantum path integral for two-dimensional quantum antiferromagnet Partition function of a classical three-dimensional ferromagnet at a “temperature” g

Quantum transition at �=�c is related to classical Curie transition at g=gc

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989).

� �,

1 1 exp 2a a a a

aa

Z dg �

�

��

� ��

� �� n n n n cubic lattice sites; , , ; a x y� ��

�

II.B Quantum field theory for critical point

� close to �c : use “soft spin” field

� � � � � �� 22 22 2 2 21

2 4!b x cud xd c

� � � � ��

� � � � � � �� S

3-component antiferromagnetic order parameter�

�

Oscillations of about zero (for ��c ) spin-1 collective mode�

�

T=0 spectrumIm ( , )p� �

�

2 2

2pc p

� � � ��

c c� ��

Critical coupling � �c� ��

Dynamic spectrum at the critical point

�

� �(2 ) / 22 2 2~ c p

�

�

� �

�

� �Im ,p� �

c p

No quasiparticles --- dissipative critical continuum



III. Antiferromagnets with an odd number of S=1/2 spins per unit cell.A. Collinear spins, Berry phases, and bond-order.B. Non-collinear spins and deconfined spinons.


V. Conclusions




� �

�

III. Antiferromagnets with an odd number of S=1/2 spins per unit cell

III.A Collinear spins, Berry phases, and bond-orderS=1/2 square lattice antiferromagnet with non-nearest neighbor exchange

ij i ji j

H J S S�

� ��

Include Berry phases after discretizing coherent state path integral on a cubic lattice in spacetime

� �,

a 1 on two square sublattices ;

Neel order parameter; oriented area of spheri

11

cal trian

exp2

~g

l

2a a a a a a

a aa

a a a

a

iZ d Ag

SA

� �

�

�

� �

�

�

�

� �

� ��

�

�

��

�

n n n n

n

0,

e

formed by and an arbitrary reference point ,a a ��n n n

0n

an a ��n

aA�

a� a ��

�

Change in choice of n0 is like a “gauge transformation”

a a a aA A� � �

� ��

� � �

(�a is the oriented area of the spherical triangle formed by na and the two choices for n0 ).

The area of the triangle is uncertain modulo 4��and the action is invariant under4a aA A

� ��

0�n

aA�

an a ��n

0n

Spin-wave theory about Neel state receives minor modifications from Berry phases.

Berry phases are crucial in determining structure of "qua

nt

g

g

�

�

Small

Largeum-disordered" phase with

0

a

a aA�

�

Integrate out to obtain effective action forn

n

These principles strongly constrain the effective action for Aa�

Simplest large g effective action for the Aa�

2 2

2,

withThis is compact QED in 2+1 dime

nsions with Berry

1 1exp cos2 22

pha

ses. ~

a a a aaa

e

iZ dA A Ae

g

� ��

�

� ��

� � � ��

� ��

This theory can be reliably analyzed by a duality mapping.

The gauge theory is always in a confiningconfining phase:

There is an energy gap and the ground state has a bond order wave.

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).

K. Park and S. Sachdev, Phys. Rev. B 65, 220405 (2002).

For large e2 , low energy height configurations are in exact one-to-one correspondence with dimer coverings of the square lattice

2+1 dimensional height model is the path integral of the Quantum Dimer Model�

There is no roughening transition for three dimensional interfaces, which are smooth for all couplings

There is a definite average height of the interfaceGround state has a bond order wave.

�

�

Bond order wave in a frustrated S=1/2 XY magnet

A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, cond-mat/0205270

First large scale numerical study of the destruction of Neel order in S=1/2antiferromagnet with full square lattice symmetry

g=

� � � �2 x x y yi j i j i j k l i j k l

ij ijklH J S S S S K S S S S S S S S� � � � � � � �

�

� � � ��





V. Conclusions




� �

�

III.B Non-collinear spins and deconfined spinons.

� � � � � �1 2

2 21 2 1 2

cos sin

4 4, ; 1 ; 03 3

S N N

N N N N� �

� �

� ��

��

� �

��

�

r Q r Q r

Q

Solve constraints by writing:

1 2

1,2

2 21 2

where are two complexnumbers with

1

abac c bN iN z zz

z z

� ��

� �

��

Magnetically ordered state:

3 2

2

Order parameter space: Physical observables are invariant under the gauge transformation a a

S Z

Z z z� �

Non-magnetic state

P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).S. Sachdev, Phys. Rev. B 45, 12377 (1992). G. Misguich and C. Lhuillier, Eur. Phys. J. B 26, 167 (2002).R. Moessner and S.L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).

N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991). X. G. Wen, Phys. Rev. B 44, 2664 (1991). A.V. Chubukov, T. Senthil and S. Sachdev, Phys. Rev. Lett.

72, 2089 (1994). T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000).

Fluctuations can lead to a “quantum disordered” state in which za are globally well defined. This requires a topologically ordered state in which vortices associated with �1(S3/Z2)=Z2 [“visons”] are gapped

out. This is an RVB state with deconfined S=1/2 spinons za

Recent experimental realization: Cs2CuCl4R. Coldea, D.A. Tennant, A.M. Tsvelik, and Z. Tylczynski, Phys. Rev. Lett. 86, 1335 (2001).





V. Conclusions




� �

�

IV. Quantum transitions between BCS superconductors

Evolution of ground state

� � � �2 2arg arg2xy x y

�

�

� � � �

BCS theory fails near quantum critical points

Microscopic study of square lattice model

G. Sangiovanni, M. Capone, S. Caprara, C. Castellani, C. Di Castro, M. Grilli, cond-mat/0111107

2 2 2 2Field theory for transition from to superconductivityxyx y x yd d id

� �

�

12

3 4

xy

Gapless Fermi Points in a d-wave superconductor atwavevectors

K=0.391�

),( KK ��

1†

31

1†

3

ffff

�

�

�

�

� ��

2†

42

2†

4

ffff

�

�

�

�

� ��

� ��

� ��

2†1 12

2†2 22

2

+2

n

n

z xn F x y

z xn F y x

d kS T i v k v k

d k T i v k v k

�

�

� � �

�

� � �

�

� �

�

� � � � � �

� � � � �

��

��

� �

� �

2 † † † † † † † †1 3 1 3 2 4 2 4

2 † †1 1 2 2

Ising order parameter for transition ~

Coupling to low energy fermions

Action for lo

h.c.

=

w ener

xy

y y

i d xd f f f f f f f f

d xd

i

� �

�

�

� ��

� � � � � � � ��

� ��

�

�

�

� � � �

� �

� �

22 22 2 4

†1 1

†2 2

12 2 2 24

gy fluctuation

+

+

s near critical poin

t

z xF x y

z xF y x

c s uS d xd

iv iv

iv iv

�

�

�

� � � � �

� �

� �

�

�

� � � � ��

�

� � � �

� � � �

�

� �† †1 1 2 2 + y y��

��

� �1 2 1 21 2 1 2

{For terms with fermions

=

Chiral symmetry breaking in the Higgs-Yukawa model}

Fv v

� � � ��

�

� � � ��

�

31

2 4 22 3

4

4

1

(3 )2

Momentum shell renormalization group equations

where are functions of / ; similar flow equ

ation f

)

o

(3

rF

d d Cddu d u C u C C ud

C v v

��

� �

� �

��

� � � � �

�

�

* *

/ . Critical point is controlled by fixed point , , = , which obeys hyperscaling, and is

Lorentz invariant.

F

F

v vu u v v� �

�

� �

��

Crossovers near transition in d-wave superconductor

ssc

Superconducting Tc

Quantumcritical

2 2

superconductivityxyx y

d id�

�2 2

superconductivityx y

d�

T

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000).

2 2

2 2

2

3

In a Fermi liquid, quasiparticle relaxation rate ~ In a BCS superconductor, quasiparticle relaxation rate ~

In a gapped BCS superconductor, (as in the

gapped Ising chain) quasipa

x y

xyx y

Td T

d id�

��

/rticle relaxation rate ~ Te�

In quantum critical region:

� ��

�

��

��

�

�

�

TkTkck

TkkG

BBBF F

F�

��

��,, 1

Nodal quasiparticle Green’s functionk wavevector separation from node

Damping of Nodal Quasiparticles

Photoemission on BSSCO (Valla et al Science 285, 2110 (1999))

Width of quasiparticle energy distribution curve (EDC) ~ Bk T

Observations of splitting of the ZBCPYoram Dagan and Guy Deutscher, Phys. Rev. Lett. 87, 177004 (2001).

Spontaneous splitting (zero field) Magnetic field splitting

Covington, M. et al. Observation of Surface-Induced Broken Time-Reversal Symmetry in YBa2Cu3O7-� Tunnel Junctions, Phys. Rev. Lett. 79, 277-281 (1997)

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

g [mV

]

1 0

1.5

2.0

2.5

3.0�

/kT c

�/kTc

Zero Field splitting and �-1 versus [�max-�]1/2 All YBCO samples

Conclusions: Phase transitions of BCS superconductors

Examined general theory of all possible candidates for zero momentum, spin-singlet order parameters which can induce a

second-order quantum phase transitions in a d-wave superconductor

Only cases

have renormalization group fixed points with a non-zero interaction strength between the bosonic order parameter mode and the nodal fermions, and so are candidates for

producing damping ~ kBT of nodal fermions.

2 2 2 2

2 2 2 2

pairing an(A)

(B

d

pairing)x y x y

xyx y x y

d d is

d d id� �

� �

� �

� �

Independent evidence for (B) from tunneling experiments.

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000).





V. Conclusions




� �

�

Competing orders in the cuprate superconductors

Eugene Demler (Harvard)Kwon Park

Anatoli PolkovnikovSubir Sachdev

Matthias Vojta (Augsburg)Ying Zhang

Talk online at http://pantheon.yale.edu/~subir

Lecture based on the article cond-mat/0112343 and the reviews

cond-mat/0108238 and cond-mat/0203363

� �† †0 cos cos

0

Hole-doped cuprates are BCS superconductors w

-wa

ith

ve pairing

spin-single t

x y dc c k k

S

� ��

�

��

kk -k

yk

xk

0 0

2 2

Low enerSuperflow

1/ 2 fermion

gy excitat

ic quasiparticles

:

:

ons:

ii

S

e

E

�

�

� �

��

�

� �k k k

0

0

BCS theory also predicts that the Fermi surface, with gapless quasiparticles, will reveal itself when 0, eitherlocally or globally at low temperatures. can be suppressedby a strong magnetic fiel

� �

�

d, and near vortices, impurities and interfaces.

Superconductivity in a doped Mott insulator

Hypothesis: cuprate superconductors have low energy excitations associated with additional order parameters

Theory and experiments indicate that the most likely candidates are spin density waves and associated

“charge” order

Superconductivity can be suppressed globally by a strong magnetic field or large current flow.

Competing orders are also revealed when superconductivity is suppressed locally, near impurities or around vortices.

S. Sachdev, Phys. Rev. B 45, 389 (1992); N. Nagaosa and P.A. Lee, Phys. Rev. B 45, 966 (1992);

D.P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang Phys. Rev. Lett. 79, 2871 (1997); K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001).

OutlineI. Experimental introduction

II. Spin density waves (SDW) in LSCO Tuning order and transitions by a magnetic field.

III. Connection with “charge” order – phenomenological theorySTM experiments on Bi2Sr2CaCu2O8+�

IV. Connection with “charge” order – microscopic theoryTheories of magnetic transitions predict bond-centered modulation of exchange and pairing energies with even periods---a bond order wave

V. Conclusions

I. Experimental introduction

The doped cuprates

2-D CuO2 planeNéel ordered ground state at zero doping

2-D CuO2 planewith finite hole doping

Phase diagram of the doped cuprates

T

d-waveSC

3DAFM

0�

T = 0 phases of LSCO

ky

•

kx

�/a

�/a0

Insulator• •• ••••

Superconductor with Tc,min =10 K•

0.055SC+SDWNéel SDW

0.020SC

~0.12-0.14 �

J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996) S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999).

S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).

Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999).

SDW order parameter for general ordering wavevector

� � � � c.c.iS e� �

�

� � �K rr r

� � is a field and =(3 / 4, )Spin density wave is (and not spiral):

�� complex

longitudi nalr K

ie n�

� ��

Bond-centered

Site-centered


II. Spin density waves (SDW) in LSCOTuning order and transitions by a magnetic field.



V. Conclusions

II. Effect of a magnetic field on SDW order with co-existing superconductivity

H

0.055SC+SDW SCNéel SDW

0.020

ky

•

kx

�/a

�/a0

InsulatorSuperconductor with Tc,min =10 K

••• •

~0.12 �

Effect of the Zeeman term: precession of SDWorder about the magnetic field

H

�

Spin singlet state

SDW

�c

� �~ zcH �

� ��

2

Elastic scattering intensity

( ) (0) HI H I aJ

� ��

� �

Characteristic field g�BH = �, the spin gap1 Tesla = 0.116 meV

Effect is negligible over experimental field scales

Dominant effect: uniformuniform softening of spin excitations by superflow kinetic energy

2 2

2

Spatially averaged superflow kinetic energy3 ln c

sc

HHvH H

� �

1sv

r�

r

Competing order is enhanced in a

“halo” around each vortex

� � 2

2

The presence of the field replaces by3 ln c

effc

HHH CH H

�

� ��

� � � ��

E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Effect of magnetic field on SDW+SC to SC transition (extreme Type II superconductivity)

Infinite diamagnetic susceptibility of non-criticalsuperconductivity leads to a strong effect.

• Theory should account for dynamic quantum spin fluctuations

• All effects are ~ H2 except those associated with H induced superflow.

• Can treat SC order in a static Ginzburg-Landau theory

� �1/ 2 22 2 2 22 2 21 2

0 2 2

T

b rg gd r d c s

� � � � � ��

�� S

2 22

2c d rd�

� ��

� �� S v

� �4

222

2GL rF d r iA�

� �

� ��

� � �

� � � �

� �

� �

,

ln 0

GL b cFZ r D r e

Z rr

� �

� �

��

� � ��

� ��

�S S

Main resultsT=0

� �� ( )~

ln 1/c

c

H � �

� �

�

�

2

2

Elastic scattering intensity3( ) (0) ln c

c

HHI H I aH H

� ��

� �

� � � � 2

2

1 exciton energy30 ln c

c

SHHH b

H H� �

�

� ��

� �

��c

E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Neutron scattering measurements of static spin correlations of the superconductor+spin-density-wave (SC+SDW) in a magnetic field

2 4

B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner,

Elas

and

tic neutron scatterin

R.J. Birgeneau, cond-

g off La C

mat/01

uO

12505.

y�

� �

� �2

2

2

Solid line --- fit to :

is the only fitting parameterBest fit value - = 2.4 with

3.01 l

= 6

n

0 T

0

c

c

c

I H HHH

a

aI H

a H

� ��

� �

2- 4Neutron scattering of La Sr CuO at =0.1x x x

2

2

Solid line - fit ( ) nto : l c

c

HHI H aH H

� ��

� �

B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, T. E. Mason,Nature, 415, 299 (2002).

Structure of long-range SDW order in SC+SDW phaseE. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Magnetic order

parameter

� � � � � � � �3 2

Dynamic structure factor

, 2

reciprocal lattice vectors of vortex lattice. measures deviation from SDW ordering wavevector

S f� � � � ��

�

� �GG

k k G

G k K

s – sc = -0.3

20 ln(1/ )f H H� �

D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) discussed static magnetism within the vortex cores in the SC phase.Their model implies a

~H dependence of the intensity





V. Conclusions

III. Connections with “charge” order – phenomenological theorySpin density wave order parameter for general ordering wavevector

� � � � c.c.iS e� �

�

� � �K rr r

� � is a field and =(3 / 4, )Spin density wave is (and not spiral):

�� complex

longitudi nalr K

ie n�

� ��

Bond-centered

Site-centered

A longitudinal spin density wave necessarily has an accompanyingmodulation in the site charge densities, exchange and pairing

energy per link etc. at half the wavelength of the SDW

� � � � � �2 2 2 c.c.iS e� �

�

��

� � � ��K rr r r

J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). H. Schulz, J. de Physique 50, 2833 (1989). K. Machida, Physica 158C, 192 (1989). O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57, 1422 (1998).

“Charge” order: periodic modulation in local observables invariant under spin rotations and time-reversal.

Order parmeter � �2~�

�

�� r

Prediction: Charge order should be pinned in halo around vortex core

K. Park and S. Sachdev Phys. Rev. B 64, 184510 (2001).E. Demler, S. Sachdev, and Ying Zhang, Phys. Rev. Lett. 87, 067202 (2001).

STM around vortices induced by a magnetic field in the superconducting stateJ. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan,

H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

-120 -80 -40 0 40 80 1200.0

0.5

1.0

1.5

2.0

2.5

3.0

Regular QPSR Vortex

Diffe

renti

al Co

nduc

tance

(nS)

Sample Bias (mV)

Local density of states

S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).

1Å spatial resolution image of integrated

LDOS of Bi2Sr2CaCu2O8+�

( 1meV to 12 meV) at B=5 Tesla.

Vortex-induced LDOS of Bi2Sr2CaCu2O8+� integrated from 1meV to 12meV

100Å

b7 pA

0 pA

J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).

Fourier Transform of Vortex-Induced LDOS map

K-space locations of vortex induced LDOS

K-space locations of Bi and Cu atoms

Distances in k –space have units of 2�/a0a0=3.83 Å is Cu-Cu distance

J. Hoffman et al. Science, 295, 466 (2002).

(extreme Type II superconductivity)Summary of theory and experimentsT=0

STM observation of CDW fluctuations enhanced by superflow

and pinned by vortex cores.

Neutron scattering observation of SDW order enhanced by

superflow.

� �� ( )~

ln 1/c

c

H � �

� �

�

�

��cE. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

Quantitative connection between the two experiments ?

Pinning of CDW order by vortex cores in SC phase

� � � � � �22 *

1 1, , ,vd� � �

� � � � �� r r r

� �2 ,�

�� r

Y. Zhang, E. Demler, and S. Sachdev, cond-mat/0112343.

� ��

1/2

pinAll where 0 0

c.c.v v

Ti

vd e �

�

�

� ��

� �� Sr r

r

low magnetic fieldhigh magnetic field

near the boundary to the SC+SDW phase

�

�

�

�

Vortex-induced LDOS of Bi2Sr2CaCu2O8+� integrated from 1meV to 12meV

100Å

b7 pA

0 pA

J. Hoffman E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science 295, 466 (2002).





V. Conclusions

Long-range charge order without spin

order

g

IV. Microscopic theory of the charge order: Mott insulators and superconductors “Large N” theory in region

with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev.Lett. 83, 3916 (1999).M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000).

Hatched region --- spin orderShaded region ---- charge order

See also J. Zaanen, Physica C 217, 317 (1999),S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998),

S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).C. Castellani, C. Di Castro, and M. Grilli, Phys.Rev. Lett. 75, 4650 (1995).

S. Mazumdar, R.T. Clay, and D.K. Campbell, Phys. Rev. B 62, 13400 (2000).

Charge order is bond-centered and has an even period.

IV. STM image of pinned charge order in Bi2Sr2CaCu2O8+� in zero magnetic field

Charge order period = 4 lattice spacings


C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, cond-mat/0201546

Spectral properties of the STM signal are sensitive to the microstructure of the charge order

M. Vojta, cond-mat/0204284. D. Podolsky, E. Demler, K. Damle, and B.I. Halperin, cond-mat/0204011

Theoretical modeling shows that this spectrum is best obtained by a modulation of bond variables, such as the exchange, kinetic or pairing energies.

Measured energy dependence of the Fourier component of the density of states which modulates with a period of 4 lattice spacings

C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, cond-mat/0201546

H. A. Mook, Pengcheng Dai, and F. DoganPhys. Rev. Lett. 88, 097004 (2002).

IV. Neutron scattering observation of static charge order in YBa2Cu3O6.35(spin correlations are dynamic)



g

IV. Bond order waves in the superconductor. “Large N” theory in region with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev.Lett. 83, 3916 (1999).M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000).

Hatched region --- spin orderShaded region ---- charge order

See also J. Zaanen, Physica C 217, 317 (1999),S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998),

S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).C. Castellani, C. Di Castro, and M. Grilli, Phys.Rev. Lett. 75, 4650 (1995).

S. Mazumdar, R.T. Clay, and D.K. Campbell, Phys. Rev. B 62, 13400 (2000).

ConclusionsI. Cuprate superconductivity is associated with doping

Mott insulators with charge carriers

II. The correct paramagnetic Mott insulator has bond-order and confinement of spinons

III. Mott insulator reveals itself vortices and near impurities. Predicted effects seen recently in STM and NMR experiments.

IV. Semi-quantitative predictions for neutron scattering measurements of spin-density-wave order in superconductors; theory also establishes connection to STM experiments.

V. Future experiments should search for SC+SDW to SC quantum transition driven by a magnetic field.

VI. Major open question: how does understanding of low temperature order parameters help explain anomalous behavior at high temperatures ?

competing orders and quantum criticalityindico.ictp.it/event/a01152/session/1/contribution/1/... ·...

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