Breakdown of the Landau-Ginzburg-Wilson paradigm at quantum phase transitions

Science 303, 1490 (2004); cond-mat/0312617cond-mat/0401041

Leon Balents (UCSB) Matthew Fisher (UCSB)

S. Sachdev (Yale) T. Senthil (MIT)

Ashvin Vishwanath (MIT)

OutlineOutlineA. Magnetic quantum phase transitions in “dimerized”

Mott insulatorsLandau-Ginzburg-Wilson (LGW) theory

B. Mott insulators with spin S=1/2 per unit cellBerry phases, bond order, and the breakdown of the LGW paradigm

A. 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

close to 0λ Weakly coupled dimers

0iS =Paramagnetic ground state

( )↓↑−↑↓=2

1

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

Excitation: S=1 triplon

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

Excitation: S=1 triplon

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

Excitation: S=1 triplon

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

Excitation: S=1 triplon

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=2

1

Excitation: S=1 triplon

close to 0λ Weakly coupled dimers

( )↓↑−↑↓=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∆ →

TlCuCl3

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

“triplon”

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 i

SS

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 c

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

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 c

ud 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)

Key reason for validity of LGW theory

There is a simple high temperature disordered state with =0 and e

Clas

xpon

sical statistical

entially decaying

mechani

correla

:

t

s

ns

c

ioϕ

There is a " " non-degenerate ground state with =0Quantum mechani

and an energy gap to all excitations

: s

c quantum disorderedϕ

B. Mott insulators with spin S=1/2 per unit cell:

Berry phases, bond order, and the breakdown of the LGW paradigm

Mott insulator with two S=1/2 spins per unit cell

Mott insulator with one S=1/2 spin per unit cell

Mott insulator with one S=1/2 spin per unit cell

Ground state has Neel order with 0ϕ ≠

Mott insulator with one S=1/2 spin per unit cell

Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.

Small ground state has Neel order with 0

Large paramagnetic ground state with 0

g

g

ϕ

ϕ

⇒ ≠

⇒ =

Mott insulator with one S=1/2 spin per unit cell

Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.The strength of this perturbation is measured by a coupling g.

Small ground state has Neel order with 0

Large paramagnetic ground state with 0

g

g

ϕ

ϕ

⇒ ≠

⇒ =

Mott insulator with one S=1/2 spin per unit cell

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bond bond

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

nπΨ ≠ Ψ bond order parameter

Mott insulator with one S=1/2 spin per unit cell

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bond bond

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

nπΨ ≠ Ψ bond order parameter

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bond bond

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

nπΨ ≠ Ψ bond order parameter

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bond bond

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

nπΨ ≠ Ψ bond order parameter

Mott insulator with one S=1/2 spin per unit cell

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

bond bond

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

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

Mott insulator with one S=1/2 spin per unit cell

bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

Mott insulator with one S=1/2 spin per unit cell

bond bond

Another state breaking the symmetry of rotations by / 2 about lattice sites,which also has 0, where is the

nπΨ ≠ Ψ bond order parameter

( ) ( )arond

cb

tan j iii j

iji S S e −Ψ = ∑ i r r

bondΨ

Possible large paramagnetic ground state with 0g ϕ =

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 order

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

AiSAe

Quantum theory for destruction of Neel order

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

sites of a cubic lattice of points a

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

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=

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

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

a a+ 0

oriented area of spherical triangle

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

µϕ ϕ ϕ

→

2 aA µ

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

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

a a+ 0

oriented area of spherical triangle

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

µϕ ϕ ϕ

→

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

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

a a+ 0

oriented area of spherical triangle

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

µϕ ϕ ϕ

→

aγ

a µγ +

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

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

a a+ 0

oriented area of spherical triangle

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

µϕ ϕ ϕ

→

aγ

a µγ +

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

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

a a+ 0

oriented area of spherical triangle

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

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 +

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

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

a a+ 0

oriented area of spherical triangle

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

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→ +

Quantum theory for destruction of Neel order

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.

Quantum theory for destruction of Neel order

,

1a a

ag µµ

ϕ ϕ +

⎛ ⎞⋅⎜ ⎟

⎝ ⎠∑( )2

,

11 expa a a a a aa aa

Z d i Ag µ τ

µ

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

⇒ ≠

⇒ =

Quantum theory for destruction of Neel order

( )2

,

11 expa a a a a aa aa

Z d i Ag µ τ

µ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫

Partition function on cubic lattice

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

⇒ ≠

⇒ =

Simplest large g effective action for the Aaµ

( )2

2 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⎛ ⎞= ∆ − ∆ +⎜ ⎟⎝ ⎠

±

∑ ∑∏∫

N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).

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

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

For 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

⇒⇒

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

D.S. Fisher and J.D. Weeks, Phys. Rev. Lett. 50, 1077 (1983)There is a definite average height of the interfaceGround state has bond order.

.

( )2

,

11 expa a a a a aa aa

Z d i Ag µ τ

µ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫

g0

Neel order 0ϕ ≠

?bond

Bond order0

Not present in LGW theory of orderϕ

Ψ ≠

or