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).

Talks online:Sachdev

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.

OutlineOutline

A. Coupled dimer antiferromagnetEffect of a magnetic field

B. Magnetic transitions in a superconductorEffect of a magnetic field

C. Spin gap state on the square latticeSpontaneous bond order

(A) InsulatorsCoupled dimer antiferromagnet

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 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 = +

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

λ 1

λc

Quantum paramagnet

0=S

Neelstate

0S N=

Neel order N0 Spin gap ∆

T=0

δ in cuprates ?

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

Phys. Rev. B 65, 014407 (2002)

TlCuCl3M. Matsumoto, B. Normand, T.M. Rice, and

M. Sigrist, cond-mat/0309440.

J. Phys. Soc. Jpn72, 1026 (2003)

Field theory for quantum criticality

λ close to λc : use “soft spin” field

αφ 3-component antiferromagnetic order parameter

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

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

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=

(A) InsulatorsCoupled dimer antiferromagnet:

effect of a magnetic field.

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.

1/λ

Spin singlet state with a spin gap

SDW

1 Tesla = 0.116 meV

HgµBH = ∆

( )( )

( )( ) ( )( )

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= ∆ −

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

Phase diagram in a magnetic field.

( )( )

( )( ) ( )( )

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

Spin singlet state with a spin gap

SDW

1 Tesla = 0.116 meV

gµBH = ∆[ ]

[ ]2

Elastic scattering intensity

0

I H

HI aJ

=

⎛ ⎞+ ⎜ ⎟⎝ ⎠

~c cH λ λ−

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.

(B) SuperconductorsMagnetic transitions in a superconductor:

effect of a magnetic field.

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

( ) ( )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


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 seco een nd-order quantum phase transition betwSC and SC+SDW phases

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 fluctuations1 2N iNα α αΦ = −

( )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) proposed static magnetism

(with ∆=0) localized within vortex cores

( ) ( ) ( ) 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

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

= ⇒

−−

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

=

⎛ ⎞= − ⎜ ⎟⎝ ⎠

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

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).

(C) 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 ?

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

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

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

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⎛ ⎞⎛ ⎞= − ∆ − ∆ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

±

∑ ∑∏∫

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).

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

ConclusionsI. Introduction to magnetic quantum criticality in coupled

dimer antiferromagnet.

II. 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.

III. 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. 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.

III. 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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