Magnetic phases and critical points of insulators and superconductors · 2019. 12. 18. · III....

Magnetic phases and critical points of insulators and superconductors

• Colloquium article in Reviews of Modern Physics, July 2003, cond-mat/0211005.

• cond-mat/0109419

Talks online:Sachdev

Quantum Phase TransitionsCambridge University Press

What is a quantum phase transition ?

Non-analyticity in ground state properties as a function of some control parameter g

E

g

True level crossing:

Usually a first-order transition

E

g

Avoided level crossing which becomes sharp in the infinite

volume limit:

second-order transition

T Quantum-critical

Why study quantum phase transitions ?

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

~ zcg gν∆ −

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

characteristic energy scale at other values of g:

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

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

OutlineI. Quantum Ising Chain

II. Coupled Dimer AntiferromagnetA. Coherent state path integral B. Quantum field theory near critical point

III. Coupled dimer antiferromagnet in a magnetic fieldBose condensation of “triplons”

IV. Magnetic transitions in superconductorsQuantum phase transition in a background Abrikosov flux lattice

V. Antiferromagnets with an odd number of S=1/2 spins per unit cell. Class A: Compact U(1) gauge theory: collinear spins,

bond order and confined spinons in d=2Class B: Z2 gauge theory: non-collinear spins, RVB,

visons, topological order, and deconfined spinons

VI. Conclusions

Single order parameter.

Multiple order

parameters.

I. Quantum I. Quantum IsingIsing chainchain

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: xj

jH Jg σ= − ∑ 2Jg

j→

j←

1 1

Coupling between qubits: z zj j

jH J σ σ += − ∑

1 1

Prefers neighboring qubits

are

(not entangle

d)j j j j

either or+ +

↓ ↓↑ ↑

( )( )1 1j j j j+ ++ +← ←→ ←→ →← →

( )0 1 1x z zj j jj

J gH H H σ σ σ += + = − +∑Full Hamiltonian

leads to entangled states at g of order unity

Weakly-coupled qubitsGround 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

ε −

−

= ∆ + +

∆ = − + p

∆

aπ

aπ

−

( )pε

( )1g

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

and momentum

( , )

p

S p

ω

ω→ ←

ω

( ),S p ω( )( )Z pδ ω ε−

Three quasiparticlecontinuum

Quasiparticle pole

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

ϕ ϕ

ϕ

τ τ

τ π−∆

>

=

phase coherence time

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

~3∆

Weakly-coupled qubits ( )1g

Ground states:

2

G

g

=

−

↑ ↑↑↑↑↑↑↑↑↑↑

↑↑↑↑ −↑↑↓↑↑↑

Lowest excited states: domain walls

jjd ↓↓↑ ↓↑↑ ↓ +↑↑= ↓Coupling between qubits creates new “domain-

wall” quasiparticle states at momentum pjipx

jj

p e d= ∑( ) ( )

( )

2 2

2

Excitation energy 4 sin2

Excitation gap 2 2

pap Jg O g

J gJ O g

ε = ∆ + +

∆ = − +

p

∆

aπ

aπ

−

( )pε

Second state obtained by

and mix only at order NG

G G g

↓ ↓

↑↓

⇔↑ 0

Ferromagnetic moment0zN G Gσ= ≠

Strongly-coupled qubits ( )1g


and momentum

( , )

p

S p

ω

ω→ ←

ω

( ),S p ω( ) ( ) ( )220 2N pπ δ ω δ

Two domain-wall continuum

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

ϕ

ϕϕ

τ

ττ π

−∆

>

=

phase coherence time

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

~2∆

Strongly-coupled qubits ( )1g

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

( )1/80 ~ cN g g−P. Pfeuty Annals of Physics, 57, 79 (1970)

ggc

Excitation energy gap ∆ ~ cg g∆ −


and momentum

( , )

p

S p

ω

ω→ ←

ω

( ),S p ω

Critical coupling ( )cg g=

c p

( ) 7 /82 2 2~ c pω −−

No quasiparticles --- dissipative critical continuum

Quasiclassicaldynamics

Quasiclassicaldynamics

1/ 41~z zj k

j kσ σ

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

( )∑ ++−=i

zi

zi

xiI gJH 1σσσ

( ) ( )

( )

0

7 / 4

( ) , 0

1 / ...

2 tan16

z z i tj k

k

R

BR

i dt t e

AT i

k T

ωχ ω σ σ

ω

π

∞

=

=− Γ +

Γ =

∑∫

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








VI. Conclusions


Multiple order

parameters.

II. Coupled II. Coupled DimerDimer AntiferromagnetAntiferromagnet

S=1/2 spins on coupled dimers

jiij

ij SSJH ⋅= ∑><

10 ≤≤ λ

JλJ

II. Coupled Dimer AntiferromagnetM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).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).

close to 1λSquare lattice antiferromagnetExperimental realization: 42CuOLa

Ground state has long-rangemagnetic (Neel) order

( ) 01 0 ≠−= + NS yx iiiExcitations: 2 spin waves (magnons) 2 2 2 2p x x y yc p c pε = +

close to 0λ Weakly coupled dimers

Paramagnetic ground state 0iS =

( )↓↑−↑↓=2

1


( )↓↑−↑↓=2

1

Excitation: S=1 triplon (exciton, spin collective mode)

Energy dispersion away fromantiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c pε

+= ∆ +

∆spin gap∆ →


( )↓↑−↑↓=2

1

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

λ 1

λc

Quantum paramagnet

0=S

Neelstate

0S N=

Neel order N0 Spin gap ∆

T=0

δ in cuprates ?

Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases

iSAe

A

Path integral for quantum spin fluctuations

II.A Coherent state path integral

iSAe

A

Path integral for quantum spin fluctuations


II.A Coherent state path integral

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

[ ]( )( ) ( ) ( )( )

/

2

Tr

1 exp ( )

H TZ e

iS A d d H Sττ δ τ τ τ τ

−=

= − − − ∫ ∫ ∫

S

N N ND

Path integral for a single spin

( )( ) ( ) 0

Oriented area of triangle on surface of unit sphere bounded

by , , and a fixed reference

A d

dτ τ τ

τ τ τ

=

+N N NAction for lattice antiferromagnet

( ) ( ) ( ), ,j j j jx xτ η τ τ= +N n L1 identifies sublatticesjη = ±

n and L vary slowly in space and time

( ) ( )

( ) ( )( )

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

τ

τ

τ δ η τ τ

τ

η

= − −

− ∂ + ∇

= ±

∑∫ ∫

∫

Dn n

n n

Integrate out L and take the continuum limit

c

c

c

c

g g

g gλ λ

λ λ

< ⇔>

> ⇔

<

Discretize spacetime into a cubic lattice cubic lattice sites, ,

;;

ax yµ τ

→→

( ),

0

oriented area of spherical triangle

formed by and an arbitrary reference poi

11 exp

t

2

, , n

2a a a a a a

a aa

a

a a

iZ d Ag

A

µ τµ

µ

µ

δ η+

+

= − ⋅ −

→

∑ ∑∏∫ n n n n

n n n

( ) ( )

( ) ( )( )

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

τ

τ

τ δ η τ τ

τ

η

= − −

− ∂ + ∇

= ±

∑∫ ∫

∫

Dn n

n n

Integrate out L and take the continuum limit

Discretize spacetime into a cubic lattice cubic lattice sites, ,

;;

ax yµ τ

→→

c

c

c

c

g g

g gλ λ

λ λ

< ⇔>

> ⇔

<

( ),

11 exp2a a a aaa

Z dg µµ

δ +

= − ⋅

∑∏∫ n n n n

Berry phases can be neglected for coupled dimer antiferromagent(justified later)

Quantum path integral for two-dimensional quantum antiferromagnetPartition function of a classical three-dimensional ferromagnetat a “temperature” g

Quantum transition at λ=λc is related to classical Curie transition at g=gc

⇔

λ close to λc : use “soft spin” field

αφ 3-component antiferromagneticorder parameter

Oscillations of about zero (for λ < λc ) spin-1 collective mode

αφ

T=0 spectrum

ω

Im ( , )pχ ω

( ) ( ) ( )( ) ( )22 22 2 2 212 4!b x cud xd cα τ α α ατ φ φ λ λ φ φ

= ∇ + ∂ + − + ∫S

2 2

2pc pε = ∆ +

∆

II.B Quantum field theory for critical point

c cλ λ∆ = −

ω

( )Im ,pχ ω

Critical coupling ( )cλ λ=

c p

( ) (2 ) / 22 2 2~ c p ηω − −−

No quasiparticles --- dissipative critical continuum

Dynamic spectrum at the critical point








VI. Conclusions


Multiple order

parameters.

III. Coupled III. Coupled DimerDimer AntiferromagnetAntiferromagnet in a magnetic fieldin a magnetic field

Pressure,exchange constant,….

( )( )

cos .

sin .

jj

j

K r

K r

=

+

1

2

S N

N

Collinear spins: 0Non-collinear spins: 0

× =× ≠

1 2

1 2

N NN N

0j =S

T=0

Quantum critical point

Both states are insulators

SDW

H

Evolution of phase diagram in a magnetic field

Effect of a field on paramagnet

Energy of zero

momentum triplon states

H

∆

0

Bose-Einstein condensation of

Sz=1 triplon

H

1/λ

Spin singlet state with a spin gap

SDW

1 Tesla = 0.116 meVRelated theory applies to double layer quantum Hall systems at ν=2

III. Phase diagram in a magnetic field.

gµBH = ∆

H

1/λ


SDW



gµBH = ∆

( )( )

( )( ) ( )( )

2 *

2 2 2 2 2

2

Zeeman term leads to a uniform precession of spins

Take oriented along the direction.

.

, ~ , while for ,

Then

For

c x y c x y

c x c c c

i H i H

H

H H

H zτ α τ α ασρ σ ρ τ α αβγ β γφ φ ε φ φ ε φ

λ λ φ φ λ λ φ φ

λ λ φ λ λ λ λ

∂ ⇒ ∂ − ∂ −

− + ⇒ − − +

> − + < ~ cλ λ= ∆ −

( )( )

( )( ) ( )( )

2 *

2 2 2 2 2

2

Zeeman term leads to a uniform precession of spins

Take oriented along the direction.

.

, ~ , while for ,

Then

For

c x y c x y

c x c c c

i H i H

H

H H

H zτ α τ α ασρ σ ρ τ α αβγ β γφ φ ε φ φ ε φ

λ λ φ φ λ λ φ φ

λ λ φ λ λ λ λ

∂ ⇒ ∂ − ∂ −

− + ⇒ − − +

> − + < ~ cλ λ= ∆ −

H

1/λ


SDW



gµBH = ∆[ ]

[ ]2

Elastic scattering intensity

0

I H

HI aJ

=

+

~c cH λ λ−








VI. Conclusions


Multiple order

parameters.

IV. Magnetic transitions in superconductorsIV. Magnetic transitions in superconductors

Pressure,carrier concentration,….

( )( )

cos .

sin .

jj

j

K r

K r

=

+

1

2

S N

N


× =× ≠

1 2

1 2

N NN N

0j =S

T=0


We have so far considered the case where both states are insulators

SDW

Pressure,carrier concentration,….

( )( )

cos .

sin .

jj

j

K r

K r

=

+

1

2

S N

N


× =× ≠

1 2

1 2

N NN N

0j =S

T=0


Now both sides have a “background” superconducting (SC) order

SC+SDW SC

If does not exactly connect two nodal points, critical theory is as in an insulator

K

Otherwise, new theory of coupled excitons and nodal quasiparticles

L. Balents, M.P.A. Fisher, C. Nayak, Int. J. Mod. Phys. B 12, 1033 (1998).

Magnetic transition in a d-wave superconductor

ky

•

kx

π/a

π/a0

Insulator

δ~0.12-0.140.055SC

0.020

J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997).

S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999)

S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).

(additional commensurability effects near δ=0.125)

T=0 phases of LSCOInterplay of SDW and SC order in the cuprates

SC+SDWSDWNéel

• •• •

ky

kx

π/a

π/a0

Insulator

δ~0.12-0.140.055SC

0.020





T=0 phases of LSCO

SC+SDWSDWNéel

Interplay of SDW and SC order in the cuprates

•••Superconductor with Tc,min =10 K

•ky

kx

π/a

π/a0

δ~0.12-0.140.055SC

0.020





T=0 phases of LSCO

SC+SDWSDWNéel


Collinear magnetic (spin density wave) order

( ) ( )cos . sin .j jj K r K r= +1 2S N NCollinear spins

( ), 0K π π= =2; N

( )3 4, 0K π π= =2; N

( )

( )3 4,

2 1

K π π=

= −2 1

;

N N

•••Superconductor with Tc,min =10 K

•ky

kx

π/a

π/a0

δ~0.12-0.140.055SC

0.020

T=0 phases of LSCO

SC+SDWSDWNéel

H

Follow intensity of elastic Bragg spots in a magnetic field

Use simplest assumption of a direct second-order quantum phase transition between SC and SC+SDW phases


Dominant effect of magnetic field: Abrikosov flux lattice

2 2

2

Spatially averaged superflow kinetic energy3 ln cs

c

HHvH H

∼ ∼

1sv r∼

r

( )1/ 2 22 2 2 22 2 21 2

0 2 2

T

b rg gd r d c sα τ α α α ατ

= ∇ Φ + ∂ Φ + Φ + Φ + Φ ∫ ∫S

2 22

2cd rd ατ ψ

= Φ ∫Sv

( )4

222

2GL rF d r iA

ψψ ψ

= − + + ∇ −

∫

( ) ( )( )

( )

,

ln 0

GL b cFZ r D r e

Z rr

ψ τ

δ ψδψ

− − −= Φ

=

∫ S S

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

Quantum theory for dynamic and critical spin fluctuations

Static Ginzburg-Landau theory for non-critical superconductivity

1 2N iNα α αΦ = +

D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) proposed static magnetism

(with ∆=0) localized within vortex cores

Triplon wavefunction in bare potential V0(x)

Energy

x0

Spin gap ∆

Vortex cores

( ) ( ) 20

Bare triplon potential

V s ψ= +r rv

( ) ( ) ( ) 20

Wavefunction of lowest energy triplon

after including triplon interactions: V V g

α

α

Φ

= + Φr r r

E. Demler, S. Sachdev, and Y

. Zhang, . , 067202 (2001).A.J. Bray and

repulsive interactions between excitons imply that triplons must be extended as 0.

Phys. Rev. Lett

Strongly relevant∆ →

87 M.A. Moore, . C , L7 65 (1982).

J.A. Hertz, A. Fleishman, and P.W. Anderson, . , 942 (1979).J. Phys

Phys. Rev. Lett15

43

Energy

x0

Spin gap ∆

Vortex cores

( ) ( ) 20

Bare triplon potential

V s ψ= +r rv

2 2

2

Spatially averaged superflow kinetic energy3 ln cs

c

H HvH H

∝

1sv r

∝r

Phase diagram of SC and SDW order in a magnetic field

( ) 2eff2

The suppression of SC order appears to the SDW order as a effective "doping" :3 ln c

c

HHH CH H

δ

δ δ = −

uniform

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


( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

Phase diagram of SC and SDW order in a magnetic field

[ ] [ ]

[ ]

eff

2

2

Elastic scattering intensity, 0,

3 0, ln cc

I H I

HHI aH H

δ δ

δ

≈

≈ +

Structure of long-range SDW order in SC+SDW phase

Magnetic order

parameter

( ) ( ) ( ) ( )3 2Dynamic structure factor

, 2

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

S fω π δ ω δ= − +

→

∑ GG

k k G

G k K

s – sc = -0.3

20 ln(1/ )f H Hδ ∝


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

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).

2

2

Solid line - fit ( ) nto : l cc

HHI H aH H

=

See also S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase, Phys. Rev. B 62, R14677 (2000).


Neutron scattering observation of SDW order enhanced by

superflow.

( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

Phase diagram of a superconductor in a magnetic field

Prediction: SDW fluctuations enhanced by superflow and bond order pinned by vortex cores (no

spins in vortices). Should be observable in STM

K. Park and S. Sachdev Physical Review B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Physical Review B 66, 094501 (2002).

( ) ( ) 22

1 triplon energy30 ln c

c

SHHH b

H Hε ε

=

= −

100Å

b7 pA

0 pA

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

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).

Our interpretation: LDOS modulations are

signals of bond order of period 4 revealed in

vortex halo

See also: S. A. Kivelson, E. Fradkin, V. Oganesyan, I. P. Bindloss, J. M. Tranquada, A. Kapitulnik, and C. Howald, cond-mat/0210683.

Fourier Transform of Vortex-Induced LDOS map

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

K-space locations of vortex induced LDOS

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

K-space locations of Bi and Cu atoms

C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik, Phys. Rev. B 67, 014533 (2003).

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

Measured energy dependence of the Fourier component of the density of states which modulates with a period of 4 lattice spacings M. Vojta, Phys. Rev. B 66, 104505 (2002);

D. Podolsky, E. Demler, K. Damle, and B.I. Halperin, Phys. Rev. B in press, 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.








VI. Conclusions


Multiple order

parameters.

V.V. AntiferromagnetsAntiferromagnets with an odd number with an odd number of S=1/2 spins per unit cellof S=1/2 spins per unit cell

Class AClass A

V. Order in Mott insulatorsMagnetic order ( ) ( )cos . sin . j jj K r K r= +1 2S N N

Class A. Collinear spins

( ), 0K π π= =2; N

( )3 4, 0K π π= =2; N

( )

( )3 4,

2 1

K π π=

= −2 1

;

N N

( ) ( )cos . sin . j jj K r K r= +1 2S N NV. Order in Mott insulatorsMagnetic order

Class A. Collinear spins

Order specified by a single vector N.

Quantum fluctuations leading to loss of

magnetic order should produce a paramagnetic state with a vector (S=1) quasiparticle excitation.

Key property

Class A: Collinear spins and compact U(1) gauge theory


iSAe

A

Write down path integral for quantum spin fluctuations


iSAe

A

Write down path integral for quantum spin fluctuations



S=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

µ τµ

µ

δ η

η

η

+

→ ±

= − ⋅ −

→

→

∑ ∑∏∫ n n n n

n

0, e

formed by and an arbitrary reference point ,a a µ+n n n

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 µ

=

Integrate out to obtain effective action forn

n

( ),

11 exp2

2a a a a a a

a aa

iZ d Ag µ τµ

δ η+

= − ⋅ −

∑ ∑∏∫ n n n n

a µ+n

0n

an

aA µ

a µ+n

0n

an

aA µ

aγ a µγ +

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

a a a aA Aµ µ µγ γ+→ − +

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

aA µ

The area of the triangle is uncertain modulo 4π, and the action is invariant under4a aA Aµ µ π→ +

These principles strongly constrain the effective action for Aaµ which provides description of the large g phase

0′n

( )2,

2 2withThis is compact QED in

1 1exp

+1 dimensions with static charges 1 on two sublattic

cos2 22

e

~

.

s

a a a a aaa

d

iZ dA A A

e g

Aeµ µ ν ν µ τµ

η

±

= ∆ − ∆ −

∑ ∑∏∫

Simplest large g effective action for the Aaµ

This theory can be reliably analyzed by a duality mapping.

d=2: The gauge theory is always in a confiningconfining phase and there is bond order in the ground state.

d=3: A deconfined phase with a gapless “photon” is possible.

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 bond order.

⇒⇒

V. Order in Mott insulatorsParamagnetic states 0j =S

Class A. Bond order and spin excitons in d=2

( )↓↑−↑↓=2

1

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

S=1/2 spinons are confinedby a linear potential into a

S=1 spin triplon

Spontaneous bond-order leads to vector S=1 spin excitations

A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)

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

( ) ( )2 x x y yi j i j i j k l i j k lij ijkl

H J S S S S K S S S S S S S S+ − + − − + − +⊂

= + − +∑ ∑g=

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








VI. Conclusions


Multiple order

parameters.

V. V. AntiferromagnetsAntiferromagnets with an odd number with an odd number of of SS=1/2 spins per unit cell =1/2 spins per unit cell

Class BClass B

V.B Order in Mott insulatorsMagnetic order

Class B. Noncollinear spins

( )3 4,K π π=(B.I. Shraiman and E.D. Siggia, Phys. Rev. Lett. 61, 467 (1988))

( ) ( )cos . sin . j jj K r K r= +1 2S N N

2 , 0= =22 1 1 2N N N .N

( )4 / 3, 4 3K π π=



( )

( )

2 2

2 2

Solve constraints by expressing in terms of two complex numbers ,

2

Order in ground state specified by a spinor ,

z z

z z

i i z z

z z

z z

↑ ↓

↓ ↑

↓ ↑

↑ ↓

↑ ↓

−

+ = +

1,2

1 2

N

N N

(modulo an overall sign).

This spinor can become a =1/2 spinon in paramagnetic state.S


A. V. Chubukov, S. Sachdev, and T. Senthil Phys. Rev. Lett. 72, 2089 (1994)

2 , 0= =22 1 1 2N N N .N

3 2

2

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

S ZZ z z→ ±




S3

(A) North pole

(B) South pole x

y(A)

(B)

Vortices associated with π1(S3/Z2)=Z2 (visons)

Such vortices (visons) can also be defined in the phase in which spins are “quantum disordered”. A Z2 spin liquid with deconfined spinons must

have visons supressedN. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)

Model effective action and phase diagram

2

h.c.

gau ge fie d l

ij i j ijij

ij

J z z K

Z

α α σ

σ

σ ∗= − +

→

−∑ ∑ ∏S (Derivation using Schwinger bosons on a quantum antiferromagnet: S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991)).

P. E. Lammert, D. S. Rokhsar, and J. Toner, Phys. Rev. Lett. 70, 1650 (1993) ; Phys. Rev. E 52, 1778 (1995). (For nematic liquid crystals)

Magnetically ordered

Confined spinonsFree spinons andtopological order

First order transition

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

A topologically ordered state in which vortices associated with π1(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).

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

V.B Order in Mott insulatorsParamagnetic states 0j =S

Class B. Topological order and deconfined spinons



Number of valence bonds cutting line is conserved

modulo 2. Changing sign of each such bond does not

modify state. This is equivalent to a Z2 gauge transformation

with on sites to the right of dashed line.

Number of valence bonds cutting line is conserved

modulo 2. Changing sign of each such bond does not

modify state. This is equivalent to a Z2 gauge transformation

with on sites to the right of dashed line.

Direct description of topological order with valence bonds

a az z→ −

D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).



Terminating the line creates a plaquette with Z2 flux at the X

--- a vison.

Terminating the line creates a plaquette with Z2 flux at the X

--- a vison.

Direct description of topological order with valence bonds

X

D. Rokhsar and S.A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989).

Effect of flux-piercing on a topologically ordered quantum paramagnetN. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).

DD

a DΨ = ∑D =

1 2 3Lx-1Lx-2 Lx

Ly

Φ

Effect of flux-piercing on a topologically ordered quantum paramagnetN. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002).

DD

a DΨ = ∑D =

1 2 3Lx-1Lx-2 Lx

( )Number of bonds cutting dashed line

After flux insertion

1 ;

D

D

⇒

−

vison

Equivalent to inserting a inside hole of the torus.

This leads to a ground state degeneracy.

vison

Ly

VI. ConclusionsI. Quantum Ising Chain







VI. Cuprates are best understood as doped class A Mott insulators.


Multiple order

parameters.

Competing order parameters in the cuprate superconductors

1. Pairing order of BCS theory (SC)

(Bose-Einstein) condensation of d-wave Cooper pairs

Orders (possibly fluctuating) associated with Orders (possibly fluctuating) associated with proximate Mott insulator in class Aproximate Mott insulator in class A

2. Collinear magnetic order (CM)

3. Bond/charge/stripe order (B)

(couples strongly to half-breathing phonons)

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); M. Vojta, Phys. Rev. B 66, 104505 (2002).

Evidence cuprates are in class A


• Neutron scattering shows collinear magnetic order co-existing with superconductivity

J. M. Tranquada et al., Phys. Rev. B 54, 7489 (1996). Y.S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). S. Wakimoto, R.J. Birgeneau, Y.S. Lee, and G. Shirane, Phys. Rev. B 63, 172501 (2001).



• Proximity of Z2 Mott insulators requires stable hc/e vortices, visongap, and Senthil flux memory effect

S. Sachdev, Physical Review B 45, 389 (1992)N. Nagaosa and P.A. Lee, Physical Review B 45, 966 (1992)T. Senthil and M. P. A. Fisher, Phys. Rev. Lett. 86, 292 (2001).D. A. Bonn, J. C. Wynn, B. W. Gardner, Y.-J. Lin, R. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Nature 414, 887 (2001).J. C. Wynn, D. A. Bonn, B. W. Gardner, Y.-J. Lin, R. Liang, W. N. Hardy, J. R. Kirtley, and K. A. Moler, Phys. Rev. Lett. 87, 197002 (2001).




• Non-magnetic impurities in underdoped cuprates acquire a S=1/2 moment

Effect of static non-magnetic impurities (Zn or Li)

Zn

Zn

Spinon confinement implies that free S=1/2 moments form near each impurity

Zn

Zn

impurity( 1)( 0)3 B

S STk T

χ +→ =

J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (2001).

Inverse local susceptibilty

in YBCO

7Li NMR below Tc

impurityMeasured with 1/ 2 in underdoped sample.

This behavior does not emerge out of BCS the

( 1)(

ory

0)3

.B

S STk T

Sχ =+→ =

A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitel’baum, Physica C 168, 370 (1990).

Spatially resolved NMR of Zn/Li impurities in the superconducting state





• Tests of phase diagram in a magnetic field



superflow.

( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−




superflow.


Possible STM observation of predicted bond order in halo

around vortices

K. Park and S. Sachdev Physical Review B 64, 184510 (2001); Y. Zhang, E. Demler and S. Sachdev, Physical Review B 66, 094501 (2002).

( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

Doping a paramagnetic bond-ordered Mott insulatorsystematic Sp(N) theory of translational symmetry breaking, while

preserving spin rotation invariance.

S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991).

d-wave superconductor

Superconductor with co-existing

bond-order

Mott insulator with bond-order

T=0

VI. Doping Class A

A phase diagram

•Pairing order of BCS theory (SC)

•Collinear magnetic order (CM)

•Bond order (B)

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); M. Vojta, Phys. Rev. B 66, 104505 (2002).

Microscopic theory for the interplay of bond (B) and d-wave

superconducting (SC) order

Vertical axis is any microscopic parameter which suppresses

CM order

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