Deconfined quantum criticality - Harvard University

Deconfined quantum criticality

Leon Balents (UCSB) Lorenz Bartosch (Frankfurt)

Anton Burkov (Harvard) Matthew Fisher (UCSB) Subir Sachdev (Harvard)

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

Ashvin Vishwanath (Berkeley)

Talks 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.

S=1/2 spins on coupled dimers

jiij

ij SSJH ⋅= ∑><

10 ≤≤ λ

JλJ

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).

Weakly coupled dimersclose to 0λ

close to 0λ Weakly coupled dimers

Paramagnetic ground state 0iS =

( )↓↑−↑↓=2

1

( )↓↑−↑↓=2

1


Excitation: S=1 quasipartcle


( )↓↑−↑↓=2

1



( )↓↑−↑↓=2

1



( )↓↑−↑↓=2

1



( )↓↑−↑↓=2

1



( )↓↑−↑↓=2

1

Energy dispersion away fromantiferromagnetic wavevector

2 2 2 2

2x x y y

p

c p c pε

+= Δ +

Δspin gapΔ →


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 i

SS

ϕ η η= = ±

TlCuCl3

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

λ 1 cλ

Néel state

T=0

Pressure in TlCuCl3

Quantum paramagnet

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

Phys. Rev. B 65, 014407 (2002)

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))

0ϕ ≠ 0ϕ =

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

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

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

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

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

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






y:

all s

ϕϕ


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)

S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

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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= ⇒∑ i

Square 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 =

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 =

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

ud xd c rSϕ ττ ϕ ϕ ϕ ϕ⎡ ⎤= ∇ + ∂ + +⎢ ⎥⎣ ⎦∫






y:

all s

ϕϕ


( )

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


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











Large scale Quantum Monte Carlo studies

A.W. Sandvik, cond-mat/0611343

A.W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett.89, 247201 (2002); A.W. Sandvik and R.G. Melko, cond-mat/0604451.

Easy-plane model

Spin stiffness

Easy-plane model

Valence bond solid (VBS) order in expectation values of plaquette and exchange terms

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

SU(2) invariant model

Strong evidence for a continuous “deconfined”quantum critical point

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


( ) ( )arbs

cv

tan j iii j

iji S S e −Ψ = ∑ i r r

vbs 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

vbsΨ

Characterization of VBS state with 0ϕ =

( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


SU(2) invariant model

Probability distribution of VBS order Ψ at quantum critical point

Emergent circular symmetry is a consequence of a gapless photon excition



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

vbs 0, 0ϕΨ ≠ =


vbs 0, 0ϕΨ ≠ =


vbs 0, 0ϕΨ ≠ =


vbs 0, 0ϕΨ ≠ =


vbs 0, 0ϕΨ ≠ =


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


g

First order transitionϕ

vbsΨ

g

Coexistence

Neel order VBS order

g

"disordered"

Neel order

ϕ

VBS order

vbsΨ

Neel order

ϕvbsΨ

VBS order


g

First order transitionϕ

vbsΨ

g

Coexistence

Neel order VBS order

g

"disordered"

Neel order

ϕ

VBS order

vbsΨ

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

iSAe

A

iSAe

A




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→ +



exp a aa

i A τη⎛ ⎞⎜ ⎟⎝ ⎠

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

,

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

⇒ ≠

⇒ =


( )2

,


Z d i Ag μ τ

μ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫


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 τ

μ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫




*

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 τ

μ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫




*

Rewrite partition function in terms of spino

rs

,

with , and

a

a a a

z

z z

α

α αβ β

α

ϕ σ=↑ ↓

=*

,

Identity fromspherical trigonometry

Arg a aaz z Aα μμ α+⎡ ⎤⎢ ⎥⎣ ⎦

=

( )2

,


Z d i Ag μ τ

μ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫




( )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

( )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⎛ ⎞= Δ − Δ −⎜ ⎟⎝ ⎠

±

∑ ∑∏∫

Large g effective action for the Aaμ after integrating zαμ

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).

g0

Neel order 0ϕ ≠

vbs

VBS order0

Not present in LGW theory of orderϕ

Ψ ≠

or

( )2 *,

,

11 exp aiAa a a a a a a

a aa

Z dz dA z z e z i Ag

μα μ α α μ α τ

μ

δ η+

⎛ ⎞≈ − +⎜ ⎟

⎝ ⎠∑ ∑∏∫

Ordering by quantum fluctuations









g0

Neel order 0ϕ ≠

?vbs

VBS order0

Not present in LGW theory of orderϕ

Ψ ≠

or

( )2 *,

,

11 exp aiAa a a a a a a

a aa

Z dz dA z z e z i Ag

μα μ α α μ α τ

μ

δ η+

⎛ ⎞≈ − +⎜ ⎟

⎝ ⎠∑ ∑∏∫

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.

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

→→


Monopole fugacity

; Arovas-Auerbach state

g

Phase diagram of S=1/2 square lattice antiferromagnet


*

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=

=

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Deconfined quantum criticality - Harvard University