magnetic phases and critical points of insulators and...

Magnetic phases and critical points of insulators and superconductors

Colloquium article:Reviews of Modern Physics, 75, 913 (2003).

Reviews:http://onsager.physics.yale.edu/qafm.pdf

cond-mat/0203363

Talks online:Sachdev

http://onsager.physics.yale.edu/qafm.pdf

What is a quantum phase transition ?Non-analyticity in ground state properties as a function of some control parameter g

Why study quantum phase transitions ?

T Quantum-critical

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. • 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.

OutlineOutlineA. “Dimerized” Mott insulators with a spin gap

Tuning quantum transitions by applied pressure

B. Spin gap state on the square latticeSpontaneous bond order

C. Tuning quantum transitions by a magnetic field1. Mott insulators2. Cuprate superconductors

(A) “Dimerized” Mott Insulators with a spin gapTuning quantum transitions by applied pressure

TlCuCl3

M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.

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).

S=1/2 spins on coupled dimers

JλJ

jiij

ij SSJH ⋅= ∑><

10 ≤≤ λ

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

0iS =Paramagnetic ground state


( )↓↑−↑↓=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

TlCuCl3

N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001).

“triplon” or spin exciton

close to 1λSquare lattice antiferromagnetExperimental realization: 42CuOLa

Ground state has long-rangemagnetic (Neel or spin density wave) order

( ) 01 0 ≠−= + NS yx iii

Excitations: 2 spin waves (magnons) 2 2 2 2p x x y yc p c pε = +

TlCuCl3

J. Phys. Soc. Jpn 72, 1026 (2003)

λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,

Phys. Rev. B 65, 014407 (2002)T=0

λ 1 cλ

Neel state

0S N=

δ in cuprates ?Pressure in TlCuCl3

Quantum paramagnet0=S

PHCC – a two-dimensional antiferromagnet

b

c

PHCC = C4H12N2Cu2Cl6

a

cCu

ClC

N

M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).


ω(m

eV)

Dispersion to “chains”Dispersion to “chains”

Not chains but planesNot chains but planes

⊥ M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).


ω(m

eV)

Dispersion to “chains”Dispersion to “chains”

Not chains but planesNot chains but planes

⊥

ω(m

eV)

0

1 0

1

h

M. B. Stone, I. A. Zaliznyak, D. H. Reich, and C. Broholm, Phys. Rev. B 64, 144405 (2001).

Triplon dispersion

S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990).

Quantitative theory of experiments and simulations: method of bond operators

Operators algebra for all states on a single dimer

( )† 102

s s≡ ≡ ↑↓ − ↓↑

( )

( )

( )

†

†

†

102

02

102

x x

y y

z z

t t

it t

t t

≡ ≡ ↑↑ − ↓↓

≡ ≡ ↑↑ + ↓↓

≡ ≡ ↑↓ + ↓↑

† †

†

†

1

, 1

,

s s t t

s s

t t

α α

α β αβδ

+ =

⎡ ⎤ =⎣ ⎦⎡ ⎤ =⎣ ⎦

Canonical Bose operators with a hard

core constraint

( )

( )

† † †1

† † †1

1212

S s t t s i t t

S s t t s i t t

α α α αβγ β γ

α α α αβγ β γ

ε

ε

= + −

= − − −

Spin operators on both sites can be expressed in terms of bond operators


S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990). A. V. Chubukov and Th. Jolicoeur, Phys. Rev. B 44, 12050 (1991).

†

Hamiltonian for coupled dimers

Solve c 1 ,

o

nstraint by s t tα α= −

( ) ( ) ( )

( ) ( ) ( )

† † †

2 2

Triplon

2

d

ispe

rs

io

n

:

t k k k k k kk

B kH A k t t t t t t

k A k B k

α α α α α α

ε

− −

⎛ ⎞= + + +⎜ ⎟

⎝ ⎠

= −

∑

( ) ( )222 2 2

Transition to magnetically ordered state occurs when 0 and the bosons condense, lea

ding to 0

x x x y y yc k

t t

K c k K

α α

= ∆ + −

∆ →

−

≠

+


S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990).

Field theory for quantum criticality

αϕ 3-component antiferromagnetic order parameter

( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c

ud xd cSϕ α τ α α ατ ϕ ϕ λ λ ϕ ϕ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫

( )For oscillations of about 0 lead to the following structure in the dynamic structure factor ,

c

S pα αλ λ ϕ ϕ

ω< =

( )2 2

; 2 c

c pp cε λ λ= ∆ + ∆ = −∆

ω

( ),S p ω

( )( )Z pδ ω ε−

Three triplon continuum

Triplon pole

~3∆Structure holds to all orders in u


αϕ 3-component antiferromagnetic order parameter

( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c

ud xd cSϕ α τ α α ατ ϕ ϕ λ λ ϕ ϕ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫

( )0For oscillations of about 0 lead to the

following dynamic structure factor ,c z

zz

Nlongitudinal S p

αλ λ ϕ ϕω

> = ≠

ω

( ),zzS p ω

Two spin-wave continuum

Structure holds to all orders in u~3∆

( ) ( ) ( )220 2N pπ δ ω δ

Entangled states at λ of order λc

1/λλc

( )~ cZ ηνλ λ−Triplonquasiparticle

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

Antiferromagneticmoment N0

( )0 ~ cN βλ λ−

1/λλc

Triplon energy gap ∆ ( )~ c

νλ λ∆ −

1/λλc

Field theory for quantum criticalityDynamic spectrum at the critical point

ω

Critical coupling ( )cλ λ=

c p

( ) (2 ) / 22 2 2~ c pη

ω− −

−( ),S p ω

No quasiparticles --- dissipative critical continuum


Quantum criticality described by strongly-coupled critical theory with universal dynamic response functions dependent on

Triplon scattering amplitude is determined by kBT alone, and not by the value of microscopic coupling u

S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

Bk Tω

( ) ( ), BT T g k Tηχ ω ω=

(B) Spin gap state on the square lattice:Spontaneous bond order

Paramagnetic ground state of coupled ladder model

Can such a state with bond order be the ground state of a system with full square lattice symmetry ?

Can such a state with bond order be the ground state of a system with full square lattice symmetry ?

Need additional exchange interactions with full square lattice symmetry to move out of Neel state into

paramagnet e.g. a second neighbor exchange J2. This defines a dimensionless coupling g = J2 / J

Collinear spins and compact U(1) gauge theory

Write down path integral for quantum spin fluctuations

Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases

AiSAe

0n

a µ+nan

aA µ

a

Neel order parameter; 1 on two square sublattices ;

oriented area of spherical triangle

formed by and an ar

~

, ,

a a a

a

a a

S

A µ

µ

ηη

+

→→ ±

→

n

n n 0bitrary reference poi tn n

Discretize imaginary time: path integral is over fields on the sites of a

cubic lattice of points a

Collinear spins and compact U(1) gauge theory

( ),

11 exp2

2a a a a a a

a aa

iZ d Ag µ τ

µ

δ η+

⎛ ⎞= − ⋅ −⎜ ⎟

⎝ ⎠∑ ∑∏∫ n n n n

Partition function on square lattice

Modulus of weights in partition function: those of a classical ferromagnet at “temperature” g

0Small ground state has Neel order with 0

Large paramagnetic ground state with 0 Berry phases lead to large cancellations between different time histories need an effective action for

a

a

g N

g

⇒ = ≠

⇒ =

→

n

n

at large aA gµ

0n

a µ+nan

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 ).

0′n

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

Simplest large g effective action for the Aaµ

( ),

2 2

2

withThis is compact QED in

1 1e

+1 dimensions with static char

xp co

ges 1 on two sublattice

s2

~

s.

22

a a a a aaa

d

iZ dA A A Ae

e g

µ µ ν ν µ τµ

η⎛ ⎞⎛ ⎞= − ∆ − ∆ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

±

∑ ∑∏∫

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).

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

For S=1/2 and large e2 , low energy height configurations are in exact one-to-one correspondence with nearest-neighbor valence bond pairings of the sites square lattice

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.

⇒⇒


0

1/2

1/4

3/4

0

1/2

1/4

3/4

0 1/4 0 1/4

Smooth interface with average height 3/8

W. Zheng and S. Sachdev, Phys. Rev. B 40, 2704 (1989)

1/41 1/41

3/4 1/2 3/4 1/2

1/41/4 11



5/41 5/41

3/4 1/2 3/4 1/2

5/45/4 11



1/40 1/40

1/2-1/4 -1/4 1/2

1/41/4 00



1/4

“Disordered-flat” interface with average height 1/2

1/23/4 1/2

1/40 1↔0 1↔

3/4

1/41/4 0 1↔0 1↔



1/ 4 5 / 4

↔ 1/ 4 5 / 4

↔1 1

3/4 1/2 3/4 1/2

1 1 1/ 4 5 / 4

↔1/ 4 5 / 4

↔


0 01/4

-1/4 -1/4


1/ 2-1/ 2

↔

0 0

“Disordered-flat” interface with average height 0

1/4

1/4

1/ 2-1/ 2

↔

1/4

1/4


3 / 4 -1/ 4

↔

0 0

0 0


1/4

1/4

3/ 4-1/ 4

↔

1/4

1/2 1/2

Two possible bond-ordered paramagnets for S=1/2

Distinct lines represent different values of on linksi jS Si

There is a broken lattice symmetry, and the ground state is at least four-fold degenerate.


( ),

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

g0

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

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

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

( ) ( )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+ − + − − + − +

⊂

= + − +∑ ∑g=

( ),

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

g0

g0S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990). K. Park and S. Sachdev, Phys. Rev. B 65, 220405 (2002).

Phase diagram of S=1/2 square lattice antiferromagnet

g

or

Spontaneous bond order, confined spinons, and “triplon” excitations

Neel order

Critical theory is not expressed in terms of order parameter of either phase, but instead contains spinons interacting the a non-compact U(1) gauge force

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, submitted to Science

Nature of quantum critical point

Use a sequence of simpler models which can be analyzed byduality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2

N → ∞

A. N=1, non-compact U(1), no Berry phases

C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981).




N → ∞

B. N=1, compact U(1), no Berry phases


Use a sequence of simpler models which can be analyzed by duality mappings


N → ∞

C. N=1, compact U(1), Berry phases




N → ∞

Identical critical theories!

D. , compact U(1), Berry phasesN → ∞

E. Easy plane case for N=2

Phase diagram of S=1/2 square lattice antiferromagnet

g

or

Spontaneous bond order, confined spinons, and “triplon” excitations

Neel order

Critical theory is not expressed in terms of order parameter of either phase, but instead contains spinons interacting the a non-compact U(1) gauge force

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, submitted to Science

OutlineOutlineA. “Dimerized” Mott insulators with a spin gap

Tuning quantum transitions by applied pressure

B. Spin gap state on the square latticeSpontaneous bond order

C. Tuning quantum transitions by a magnetic field1. Mott insulators2. Cuprate superconductors

(C) Tuning quantum transitions by a magnetic field

1. Mott insulators

λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,

Phys. Rev. B 65, 014407 (2002)T=0

λ 1 cλ

Neel state

0S N=

δ in cuprates ?Pressure in TlCuCl3

Quantum paramagnet0=S

Effect of a field on paramagnet

Energy of zero

momentum triplon states

∆

0

Bose-Einstein condensation of

Sz=1 triplon

H

TlCuCl3

Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht, and P. Vorderwisch, Nature 423, 62 (2003).

TlCuCl3

Ch. Rüegg, N. Cavadini, A. Furrer, H.-U. Güdel, K. Krämer, H. Mutka, A. Wildes, K. Habicht, and P. Vorderwisch, Nature 423, 62 (2003).

“Spin wave (phonon) above critical field

Phase diagram in a magnetic field

H

1/λ

Spin singlet state with a spin gap

Canted magnetic order

gµBH = ∆

Phase diagram in a magnetic field

H

1/λ

Spin singlet state with a spin gap

Canted magnetic order gµBH = ∆[ ]

[ ]2

Elastic scattering intensity

0

I H

HI aJ

=

⎛ ⎞+ ⎜ ⎟⎝ ⎠

~c cH λ λ−

1 Tesla = 0.116 meV

Related theory applies to double layer quantum Hall systems at ν=2

TlCuCl3

M. Matsumoto, B. Normand, T.M. Rice,

and M. Sigrist, cond-mat/0309440.


Spin gap paramagnet

Phase diagram in a strong magnetic field.

Magnetization =density of triplons

H∆

Spin gap



1

∆


H

Spin gapAt very large H, magnetization

saturatesCanted magnetic order


M

∆

1

1/2


H

Spin gap ij zi zji j

J S S<∑

Respulsive interactions between triplons can lead to

magnetization plateau at any rational fraction



∆

1

M

1/2

Quantum transitions in and out of plateau are

Bose-Einstein condensations of “extra/missing”

triplons


H

Partial magnetization plateau observed in SrCu2(BO3)2 and NH4CuCl3

Spin gap


(C) Tuning quantum transitions by a magnetic field

2. Cuprate superconductors

Interplay of SDW and SC order in the cuprates

T=0 phases of LSCOky

•

kx

π/a

π/a0

Insulator

δ~0.12-0.140.0550.020SCSC+SDWSDWNéel

(additional commensurability effects near δ=0.125)

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).



• •• •

kxπ/a0

Insulatorπ/a








••• • Superconductor with Tc,min =10 K

kxπ/a0

π/a






Collinear magnetic (spin density wave) order

.Re ; order parameter is complex vector ji K rj e⎡ ⎤= ⎣ ⎦S Φ Φ

( ), 0K π π θ= =;

( )3 4, 0K π π θ= =;

( )3 4, / 8K π π θ π= =;

Collinear spins ie θ⇒ = nΦ



••• • Superconductor with Tc,min =10 K

kxπ/a0

π/a

δ~0.12-0.140.055SC

0.020SC+SDWSDWNéel

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

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

K

Magnetic transition in a d-wave superconductor

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

( )2 22 2rd rd c Vα τ α ατ ⎡ ⎤= ∇ Φ + ∂ Φ + Φ⎣ ⎦∫S

Similar terms present in action for SDW ordering in the insulator

Coupling to the S=1/2 Bogoliubov quasiparticles of the d-wave superconductor

Trilinear “Yukawa” coupling

is prohibited unless ordering wavevector is fine-tuned.

2d rd ατ Φ ΨΨ∫

22 † is allowed

Scaling dimension of (1/ - 2) 0 irrelev t.an

d rd αα

κ τ

κ ν

Φ Ψ Ψ

= < ⇒

∑∫


T=0 phases of LSCO

•••Superconductor with Tc,min =10 K•

ky

kx

π/a

π/a0

δ~0.12-0.140.055SC

0.020SC+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

Recall, in an insulator intensity would increase ~ H2

Dominant effect of magnetic field: Abrikosov flux lattice

1sv

r∼

r

2 2

2

Spatially averaged superflow kinetic energy3 ln c

sc

HHvH H

∼ ∼

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

Quantum theory for dynamic and critical spin fluctuations

( )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

( ) ( )( )

( )

,

ln 0

GL b cFZ r D r e

Z rr

ψ τ

δ ψδψ

− − −= Φ⎡ ⎤⎣ ⎦

⎡ ⎤⎣ ⎦ =

∫ S S

2 22

2c d rd ατ ψ⎡ ⎤= Φ⎢ ⎥⎣ ⎦∫S v

( )4

222

2GL rF d r iAψ

ψ ψ⎡ ⎤

= − + + ∇ −⎢ ⎥⎢ ⎥⎣ ⎦

∫

Static Ginzburg-Landau theory for non-critical superconductivity

Triplon wavefunction in bare potential V0(x)

Energy

x0

Spin gap ∆

Vortex cores

( ) ( ) 20

Bare triplon potential

V s ψ= +r rv

D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) suggested nucleation of static magnetism (with ∆=0) within vortex scores in a first-order transition. However,

given the small size of the vortex cores, the magnetism must become dynamic as in a spin gap state.

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

( ) ( ) ( ) 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

TlCuCl3

M. Matsumoto, B. Normand, T.M. Rice,

and M. Sigrist, cond-mat/0309440.


Spin gap paramagnet

Phase diagram of SC and SDW order in a magnetic field

2 2

2

Spatially averaged superflow kinetic energy3 ln c

sc

H HvH H

∝

1sv

r∝

r

( ) 2eff

2

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).

Phase diagram of SC and SDW order in a magnetic field


( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

[ ] [ ]

[ ]

eff

2

2

Elastic scattering intensity, 0,

3 0, ln c

c

I H I

HHI aH H

δ δ

δ

≈

⎛ ⎞≈ + ⎜ ⎟⎝ ⎠

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 c

c

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

( )( )

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

⎛ ⎞= + ⎜ ⎟⎝ ⎠

H (Tesla)

2 4

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

Elastic neutron scatt

neau, B , 014528 (2002)

ering off La C O

.

u y

Phys. Rev.

+

66

Phase diagram of a superconductor in a magnetic field


Neutron scattering observation of SDW order enhanced by

superflow.

( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

( ) ( ) 2

2

1 triplon energy30 ln c

c

SHHH b

H Hε ε

=

⎛ ⎞= − ⎜ ⎟⎝ ⎠

Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field

B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason,

and A. Schröder, Science 291, 1759 (2001).

Peaks at (0.5,0.5) (0.125,0)and (0.5,0.5) (0,0.125)

dynamic SDW of period 8

±±

⇒

2- 4Neutron scattering off La Sr CuO ( 0.163, ) δ δ δ = SC phasered dat lo otsw temperatures in =0 ( ) and =7.5T blue d )s( otH H

Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field

B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason,

and A. Schröder, Science 291, 1759 (2001).

2- 4Neutron scattering off La Sr CuO ( 0.163, ) δ δ δ = SC phase

Peaks at (0.5,0.5) (0.125,0)and (0.5,0.5) (0,0.125)

dynamic SDW of period 8

±±

⇒

red dat lo otsw temperatures in =0 ( ) and =7.5T blue d )s( otH H

Collinear magnetic (spin density wave) order

.Re ; order parameter is complex vector ji K rj e⎡ ⎤= ⎣ ⎦S Φ Φ

( )Collinear spins , and there is modulation

in the parameter at

wavevector 2x

i

j j j a

e

Qbond order r

K

θ

+

⇒ =

≡ i

n

S S

Φ

( )3 4, 0K π π θ= =;

( )3 4, / 8K π π θ π= =;

Phase diagram of a superconductor in a magnetic field


Neutron scattering observation of SDW order enhanced by

superflow.

( )

( )( )

eff

( )~ln 1/

c

c

c

H

H

δ δ

δ δδ δ

= ⇒

−−

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).

( ) ( ) 2

2

1 triplon energy30 ln c

c

SHHH b

H Hε ε

=

⎛ ⎞= − ⎜ ⎟⎝ ⎠

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

Diff

eren

tial C

ondu

ctan

ce (n

S)

Sample Bias (mV)

Local density of states

1Å spatial resolution image of integrated

LDOS of Bi2Sr2CaCu2O8+δ

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

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

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

100Å

b7 pA

0 pA

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.

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).

ConclusionsI. Introduction to magnetic quantum criticality in coupled

dimer antiferromagnet.

II. Berry phases and bond order in square lattice antiferromagnets.

III. Theory of quantum phase transitions provides semi-quantitative predictions for neutron scattering measurements of spin-density-wave order in superconductors; theory also proposes a connection to STM experiments.

IV. Spontaneous bond order in spin gap state on the square lattice: possible connection to modulations observed in vortex halo.

ConclusionsI. Introduction to magnetic quantum criticality in coupled

dimer antiferromagnet.

II. Berry phases and bond order in square lattice antiferromagnets.

III. Theory of quantum phase transitions provides semi-quantitative predictions for neutron scattering measurements of spin-density-wave order in superconductors; theory also proposes a connection to STM experiments.

IV. Spontaneous bond order in spin gap state on the square lattice: possible connection to modulations observed in vortex halo.

magnetic phases and critical points of insulators and...

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