magnetic phases and critical points of insulators and superconductors colloquium article: reviews of...

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

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. • 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. “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 insulators

2. 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.

S=1/2 spins on coupled dimers

jiij

ij SSJH

10

JJ

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 0 Weakly coupled dimers

Paramagnetic ground state 0iS

2

1


2

1

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

Energy dispersion away from antiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c p

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 1Square 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 2

p x x y yc p c p

TlCuCl3

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

1c

Neel state

0S N

T=0

in cupratesPressure in TlCuCl3

Quantum paramagnet

0S

c = 0.52337(3) M.

Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)

b

c

PHCC – a two-dimensional antiferromagnet

PHCC = C4H12N2Cu2Cl6

a

cCu

ClC

N

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

(

meV

)

Dispersion to “chains”Dispersion to “chains”

Not chains but planesNot chains but planes



(

meV

)

Dispersion to “chains”Dispersion to “chains”

Not chains but planesNot chains but planes

(m

eV)

0

1 0

1

h

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

† 10

2s s

†

†

†

10

2

02

10

2

x x

y y

z z

t t

it t

t t

† †

†

†

1

, 1

,

s s t t

s s

t t

Canonical Bose operators with a hard

core constraint

† † †1

† † †1

1

21

2

S s t t s i t t

S s t t s i t t

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 t

† † †

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

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


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

3-component antiferromagnetic order parameter

22 22 2 2 21

2 4!x c

ud xd cS

2 2

; 2 c

c pp c

Field theory for quantum criticality

For oscillations of about 0 lead to the

following structure in the dynamic structure factor ,c

S p

,S p Z p

Three triplon continuum

Triplon pole

Structure holds to all orders in u~3

3-component antiferromagnetic order parameter

22 22 2 2 21

2 4!x c

ud xd cS


0For oscillations of about 0 lead to the

following dynamic structure factor ,c z

zz

N

longitudinal S p

,zzS p

Two spin-wave continuum

Structure holds to all orders in u~3

220 2N p

Entangled states at of order c

1/c

Triplon quasiparticle

weight Z

~ cZ

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

Antiferromagnetic moment N0

0 ~ cN

1/c

Triplon energy gap ~ c

1/c

,S p

Critical coupling c

c p

(2 ) / 22 2 2~ c p

No quasiparticles --- dissipative critical continuum

Dynamic spectrum at the critical point



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 T

, BT T g k T

(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

Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases

iSAe

A

Write down path integral for quantum spin fluctuations


iSAe

A

Write down path integral for quantum spin fluctuations

Key ingredient: Spin Berry PhasesKey ingredient: Spin Berry Phases

a n

0n

an

aA

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

n

n n 0bitrary reference poi t n n

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

cubic lattice of points a


,

11 exp

22

a a a a a aa aa

iZ d A

g

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 g

a n

0n

an

aA

a n

0n

an

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 ).

0n

aA

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

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

,

2 2

2

with

This 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 A

e

e g

Simplest large g effective action for the Aa

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 interface Ground state has bond order.


Smooth interface with average height 3/8

0

1/2

1/4

3/4

0

1/2

1/4

3/4

0 1/4 0 1/4

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


1

1/2

1/4

3/4

1

1/2

1/4

3/4

1 1/4 1 1/4



1

1/2

5/4

3/4

1

1/2

5/4

3/4

1 5/4 1 5/4



0

1/2

1/4

-1/4

0

1/2

1/4

-1/4

0 1/4 0 1/4


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

1/2

1/4

3/4 1/2

1/4

3/4

1/4 1/4


0 1 0 1

0 1 0 1


1/23/4 1/23/4


1/ 4

5 / 4

1 1

1 1

1/ 4

5 / 4

1/ 4

5 / 4

1/ 4

5 / 4

1/4

-1/4 -1/4


1/ 2

-1/ 2

0 0

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

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

Distinct lines represent different values of on linksi jS S


g0

,

a 1 on two square sublattices ;


oriented area of spheri

11

cal trian

exp2

~

g

l

2a a a a a a

a aa

a a a

a

iZ d A

g

S

A

n n n n

n

0,

e

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

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 l

ij 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=1/2 antiferromagnet with full square lattice symmetry

g0

,

a 1 on two square sublattices ;


oriented area of spheri

11

cal trian

exp2

~

g

l

2a a a a a a

a aa

a a a

a

iZ d A

g

S

A

n n n n

n

0,

e

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

g0

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

g

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

Phase diagram of S=1/2 square lattice antiferromagnet

or

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

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

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

A. Non-compact QED with scalar matterB. Compact QED with scalar matter

C. N=1: Compact QED with scalar matter and Berry phases D. theory

E. Easy plane case for N=2N

Nature of quantum critical point

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

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

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


A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phases

D. theoryE. Easy plane case for N=2

N



A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry

phases D. theoryE. Easy plane case for N=2

N


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






Identical critical theories!

D. , compact U(1), Berry phasesN

E. Easy plane case for N=2

g

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

Phase diagram of S=1/2 square lattice antiferromagnet

or

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

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

OutlineOutline

A. “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 insulators


(C) Tuning quantum transitions by a magnetic field

1. Mott insulators

1c

Neel state

0S N

T=0

in cupratesPressure in TlCuCl3

Quantum paramagnet

0S

c = 0.52337(3) M.

Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002)

Effect of a field on paramagnet

Energy of zero

momentum triplon states

H

0

Bose-Einstein condensation of

Sz=1 triplon

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/

Spin singlet state with a spin gap

Canted magnetic order

gBH =

H

1/

Spin singlet state with a spin gap


1 Tesla = 0.116 meV

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

gBH =

2

Elastic scattering

intensity

0

I H

HI a

J

~c cH

Phase diagram in a magnetic field

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


At very large H, magnetization

saturates

1



H

Spin gap


M

1

1/2

Respulsive interactions between triplons can lead to

magnetization plateau at any rational fraction

ij zi zji j

J S S


H


Spin gap


M

1

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


ky

•

kx

/a

/a0

Insulator

~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 =0.125)

T=0 phases of LSCO

Interplay of SDW and SC order in the cuprates

SC+SDWSDWNéel

• •• •

ky

kx

/a

/a0

Insulator

~0.12-0.140.055SC

0.020





T=0 phases of LSCO

SC+SDWSDWNéel


••

•Superconductor with Tc,min =10 K•

ky

kx

/a

/a0

~0.12-0.140.055SC

0.020





T=0 phases of LSCO

SC+SDWSDWNéel


Collinear magnetic (spin density wave) order

.Re ; order parameter is complex vector ji K rj e

��

S

, 0K ��

;

3 4, 0K ��

;

3 4, / 8K ��

;

Collinear spins ie n

••


ky

kx

/a

/a0

~0.12-0.140.055SC

0.020

T=0 phases of LSCO

SC+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

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

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

22 † is allowed

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

d rd

2 22 2rd rd c V S

Similar terms present in action for SDW ordering in the insulator

Magnetic transition in a d-wave superconductor

••


ky

kx

/a

/a0

~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


Recall, in an insulator intensity would increase ~ H2

Dominant effect of magnetic field: Abrikosov flux lattice

2 2

2

Spatially averaged superflow kinetic energy

3 ln c

sc

HHv

H H

1sv

r

r

1/ 2 22 2 2 22 2 21 2

0 2 2

T

b r

g gd r d c s S

2 22

2c d rd Sv

4

222

2GL rF d r iA

,

ln 0

GL b cFZ r D r e

Z r

r

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

Triplon wavefunction in bare potential V0(x)

Energy

x0

Spin gap

Vortex cores

2

0

Bare triplon potential

V s 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)

2

0

Wavefunction of lowest energy triplon

after including triplon interactions: V V g

r 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. Lett

15

43

Energy

x0

Spin gap

Vortex cores

2

0

Bare triplon potential

V s r rv

TlCuCl3

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

and M. Sigrist, cond-mat/0309440.


Spin gap paramagnet

2 2

2

Spatially averaged superflow kinetic energy

3 ln c

sc

H Hv

H H

1sv

r

r

Phase diagram of SC and SDW order in a magnetic field

2eff

2

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

3 ln c

c

HHH C

H H

uniform

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


eff

( )~

ln 1/

c

c

c

H

H

Phase diagram of SC and SDW order in a magnetic field

eff

2

2

Elastic scattering intensity

, 0,

3 0, ln c

c

I H I

HHI a

H H

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 a

H H

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

2

2

2

Solid line --- fit to :

is the only fitting parameter

Best fit value - = 2.4 with

3.01 l

= 6

n

0 T

0

c

c

c

I H HH

H

a

aI H

a H

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

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


Neutron scattering observation of SDW order enhanced by

superflow.

eff

( )~

ln 1/

c

c

c

H

H

Phase diagram of a superconductor in a magnetic field

2

2

1 triplon energy

30 ln c

c

S

HHH b

H 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, ) 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

3 4, 0K ��

;

3 4, / 8K ��

;

Collinear spins , and there is modulation

in the parameter at

wavevector 2

x

i

j j j a

e

Qbond order r

K

��

n

S S


Neutron scattering observation of SDW order enhanced by

superflow.

eff

( )~

ln 1/

c

c

c

H

H

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).

2

2

1 triplon energy

30 ln c

c

S

HHH b

H H

STM around vortices induced by a magnetic field in the superconducting state

J. 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

rential C

onducta

nce (

nS

)

Sample Bias (mV)

Local density of states

1Å spatial resolution image of integrated

LDOS of Bi2Sr2CaCu2O8+

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

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

100Å

b7 pA

0 pA

Vortex-induced LDOS of Bi2Sr2CaCu2O8+ 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.

Conclusions

I. 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.

Conclusions

I. 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 superconductors colloquium article: reviews of...

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