Deconfined quantum criticality - Harvard Universityqpt.physics.harvard.edu/talks/osu_seminar.pdf ·...

Deconfined quantum criticality

Science 303, 1490 (2004); Physical Review B 70, 144407 (2004), 71, 144508 and 71, 144509 (2005), cond-mat/0502002

Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Matthew Fisher (UCSB) Subir Sachdev (Harvard)

Krishnendu Sengupta (HRI, India) T. Senthil (MIT and IISc)

Ashvin Vishwanath (Berkeley)

Talk online at http://sachdev.physics.harvard.edu

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I. Magnetic quantum phase transitions in “dimerized” Mott insulators:Landau-Ginzburg-Wilson (LGW) theory

II. Magnetic quantum phase transitions of Mott insulators on the square latticeA. Breakdown of LGW theoryB. Berry phasesC. Spinor formulation and deconfined criticality

I. Magnetic quantum phase transitions in “dimerized” Mott insulators:

Landau-Ginzburg-Wilson (LGW) theory:Second-order phase transitions described by

fluctuations of an order parameterassociated with a broken symmetry

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


0iS =Paramagnetic ground state

( )↓↑−↑↓=2

1


( )↓↑−↑↓=2

1

Excitation: S=1 quasipartcle


( )↓↑−↑↓=2

1

Excitation: S=1 quasipartcle

Energy dispersion away fromantiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c pε

+= ∆ +

∆spin gap∆ →

TlCuCl3

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

S=1 quasi-

particle

Coupled Dimer Antiferromagnet

close to 1λ Weakly dimerized square lattice

close to 1λ Weakly dimerized square lattice

Excitations: 2 spin waves (magnons)

2 2 2 2p x x y yc p c pε = +

Ground state has long-range spin density wave (Néel) order at wavevector K= (π,π)

0 ϕ ≠

spin density wave order parameter: ; 1 on two sublatticesii iSS

ϕ η η= = ±

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 Pressure in TlCuCl3

Quantum paramagnetNéel state

0ϕ ≠ 0ϕ =

The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,

Phys. Rev. Lett. 89, 077203 (2002))

LGW theory for quantum criticality

write down an effective action for the antiferromagnetic order parameter by expanding in powers

of and its spatial and temporal d

Landau-Ginzburg-Wilson theor

erivatives, while preserving

y:

all s

ϕϕ

ymmetries of the microscopic Hamiltonian

( ) ( ) ( )( ) ( )22 22 2 2 212 4!x cud xd cSϕ ττ ϕ ϕ λ λ ϕ ϕ

⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)






y:

all s

ϕϕ


( ) ( ) ( )( ) ( )22 22 2 2 212 4!x cud xd cSϕ ττ ϕ ϕ λ λ ϕ ϕ

⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

For , oscillations of about 0 constitute the excitation

c

triplonλ λ ϕ ϕ< =

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

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II. Magnetic quantum phase transitions of Mott insulators on the square lattice:

A. Breakdown of LGW theory

Ground state has long-range Néel order

Order parameter 1 on two sublattices

0

i i

i

Sϕ ηη

ϕ

==

≠

±

; spin operator with =1/2ij i j iij

H J S S S S= ⇒∑ iSquare lattice antiferromagnet

; spin operator with =1/2ij i j iij

H J S S S S= ⇒∑ iSquare lattice antiferromagnet

Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.

What is the state with 0 ?ϕ =






y:

all s

ϕϕ


( ) ( )( ) ( )22 22 2 2 212 4!xud xd c rSϕ ττ ϕ ϕ ϕ ϕ

⎡ ⎤= ∇ + ∂ + +⎢ ⎥⎣ ⎦∫

( )

The ground state for 0 has no broken symmetry and a gapped S=1 quasiparticle excitation oscillations of about 0

r

ϕ ϕ

>

=

Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries

Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries

“Liquid” of valence bonds has fractionalized S=1/2 excitations

Another possible state, with 0, is the valence bond solid (VBS)ϕ =


vbsΨ

( ) ( )arbsc

vtan j ii

i jij

i S S e −Ψ = ∑ i r rvbs vbs

Such a state breaks the symmetry of rotations by / 2 about lattice sites,and has 0, where is the

nπΨ ≠ Ψ VBS order parameter


( ) ( )arbsc

vtan j ii

i jij




vbsΨ


vbsΨ

( ) ( )arbsc

vtan j ii

i jij




A )nother possible state, with 0, is the valence bond solid (VBSϕ =

( ) ( )arbsc

vtan j ii

i jij




vbsΨ


vbsΨ

( ) ( )arbsc

vtan j ii

i jij




The VBS state does have a stable S=1 quasiparticle excitation

vbs 0, 0ϕΨ ≠ =

LGW theory of multiple order parameters

[ ] [ ][ ]

[ ]

vbs vbs int

2 4vbs vbs 1 vbs 1 vbs

2 42 2

2 2int vbs

F F F F

F r u

F r u

F v

ϕ

ϕ

ϕ

ϕ ϕ ϕ

ϕ

= Ψ + +

Ψ = Ψ + Ψ +

= + +

= Ψ +

Distinct symmetries of order parameters permit couplings only between their energy densities

LGW theory of multiple order parametersFirst order transitionϕ

vbsΨ

Neel order VBS orderg

g

Coexistence

Neel order

ϕ

VBS order

vbsΨ

g

"disordered"

Neel order

ϕvbsΨ

VBS order

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B. Berry phases

Quantum theory for destruction of Neel order

Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases

AiSAe

Quantum theory for destruction of Neel orderDiscretize imaginary time: path integral is over fields on the

sites of a cubic lattice of points a


sites of a cubic lattice of points aa a

a

Recall = (0,0,1) in classical Neel state; 1 on two

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

( ), ,x yµ τ=



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0

oriented area of spherical triangle

formed by and an arbitrary reference , poi, nta hA alfµ

µϕ ϕ ϕ

→

2 aA µ



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0



µϕ ϕ ϕ

→



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0



µϕ ϕ ϕ

→

aγ

a µγ +



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0



µϕ ϕ ϕ

→

aγ

a µγ +



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0



µϕ ϕ ϕ

→

aγ

Change in choice of is like a “gauge transformation”

2 2a a a aA Aµ µ µγ γ+→ − +

0ϕ

a µγ +



a


square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

0ϕ

aϕ

a a+ 0



µϕ ϕ ϕ

→

aγ

Change in choice of is like a “gauge transformation”

2 2a a a aA Aµ µ µγ γ+→ − +

0ϕ

The area of the triangle is uncertain modulo 4π, and the action has to be invariant under 2a aA Aµ µ π→ +


Ingredient missing from LGW theory: Ingredient missing from LGW theory: Spin Berry PhasesSpin Berry Phases

exp a aa

i A τη⎛ ⎞⎜ ⎟⎝ ⎠

∑Sum of Berry phases of all spins on the square lattice.


,

1a a

agµ

µ

ϕ ϕ +⎛ ⎞

⋅⎜ ⎟⎝ ⎠

∑( )2,

11 expa a a a a aa aa

Z d i Ag µ τµ

ϕ δ ϕ ϕ ϕ η+⎛ ⎞

= − ⋅ +⎜ ⎟⎝ ⎠

∑ ∑∏∫

Partition function on cubic lattice

LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g

Small ground state has Neel order with 0

Large paramagnetic ground state with 0

g

g

ϕ

ϕ

⇒ ≠

⇒ =


( )2,


Z d i Ag µ τµ


= − ⋅ +⎜ ⎟⎝ ⎠

∑ ∑∏∫


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

Berry phases lead to large cancellations between different time histories need an effective action for at large aA gµ→

S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

Small ground state has Neel order with 0

Large paramagnetic ground state with 0

g

g

ϕ

ϕ

⇒ ≠

⇒ =

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C. Spinor formulation and deconfinedcriticality


( )2,


Z d i Ag µ τµ


= − ⋅ +⎜ ⎟⎝ ⎠

∑ ∑∏∫


*

Rewrite partition function in terms of spino

rs

,

with , and

a

a a a

z

z z

α

α αβ β

α

ϕ σ=↑ ↓

=



( )2,


Z d i Ag µ τµ


= − ⋅ +⎜ ⎟⎝ ⎠

∑ ∑∏∫


*

Rewrite partition function in terms of spino

rs

,

with , and

a

a a a

z

z z

α

α αβ β

α

ϕ σ=↑ ↓

=*

,

Remarkable identity fromspherical trigonometry

Arg a aaz z Aα µµ α+⎡ ⎤⎢ ⎥⎣ ⎦ =



( )2,


Z d i Ag µ τµ


= − ⋅ +⎜ ⎟⎝ ⎠

∑ ∑∏∫



( )2

*,

,

1

1 exp a

a a aa

iAa a a a

a a

Z dz dA z

z e z i Ag

µ

α µ α

α µ α τµ

δ

η+

≈ −

⎛ ⎞× +⎜ ⎟

⎝ ⎠

∏∫

∑ ∑

Partition function expressed as a gauge theory of spinordegrees of freedom

Large g effective action for the Aaµ after integrating zαµ

( )22 2

,

withThis is compact QED in 3 spacetime dimensions with

static charges 1 on

1exp c

two sublat

tices.

o2

~

sa a a a aaa

Z dA A A i Ae

e g

µ µ ν ν µ τµ

η⎛ ⎞= ∆ − ∆ −⎜ ⎟⎝ ⎠

±

∑ ∑∏∫

This theory can be reliably analyzed by a duality mapping.

The gauge theory is in a confining phase, and there is VBS order in the ground state. (Proliferation of monopoles in

the presence of Berry phases).

This theory can be reliably analyzed by a duality mapping.

The gauge theory is in a confiningconfining phase, and there is VBS order in the ground state. (Proliferation of monopoles in

the presence of Berry phases).

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).

( )2 * ,,

11 exp aiAa a a a a a aa aa

Z dz dA z z e z i Ag

µα µ α α µ α τ

µ

δ η+⎛ ⎞

≈ − +⎜ ⎟⎝ ⎠

∑ ∑∏∫

vbs

VBS order0

Not present in LGW theory of orderϕ

Ψ ≠

or

Neel order 0ϕ ≠

g0

Ordering by quantum fluctuations

( )2 * ,,

11 exp aiAa a a a a a aa aa

Z dz dA z z e z i Ag

µα µ α α µ α τ

µ

δ η+⎛ ⎞

≈ − +⎜ ⎟⎝ ⎠

∑ ∑∏∫

g0

Neel order 0ϕ ≠

?vbs

VBS order0

Not present in LGW theory of orderϕ

Ψ ≠

or

Second-order critical point described by emergent fractionalized degrees of freedom (Aµ and zα );

Order parameters (ϕ and Ψvbs ) are “composites” and of secondary importance

Second-order critical point described by emergent fractionalized degrees of freedom (Aµ and zα );

Order parameters (ϕ and Ψvbs ) are “composites” and of secondary importance

→

S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990); G. Murthy and S. Sachdev, Nuclear Physics B 344, 557 (1990); C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, 134510 (2001); S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002); O. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004)

Theory of a second-order quantum phase transition between Neel and VBS phases

*

At the quantum critical point: +2 periodicity can be ignored

(Monopoles interfere destructively and are dangerously irrelevant). =1/2 , with ~ , are globally

pro

A A

S spinons z z z

µ µ

α α αβ β

π

ϕ σ

• →

•

pagating degrees of freedom.

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

Phase diagram of S=1/2 square lattice antiferromagnet

g*

Neel order

~ 0z zα αβ βϕ σ ≠

(associated with condensation of monopoles in ),Aµ

or

vbsVBS order 0Ψ ≠

1/ 2 spinons confined, 1 triplon excitations

S zS

α=

=

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

Conclusions

• New quantum phases induced by Berry phases: VBS order in the antiferromagnet

• Critical resonating-valence-bond states describes the quantum phase transition from the Neel to the VBS

• Emergent gauge fields are essential for a full description of the low energy physics.

Conclusions

• New quantum phases induced by Berry phases: VBS order in the antiferromagnet

• Critical resonating-valence-bond states describes the quantum phase transition from the Neel to the VBS

• Emergent gauge fields are essential for a full description of the low energy physics.

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