breakdown of the landau-ginzburg-wilson paradigm at...

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)

T. Senthil (MIT) Ashvin Vishwanath (MIT)

Parent compound of the high temperature superconductors: 2 4La CuO

A Mott insulator

spin operator withangular momentum =1/2

ij i jij

H J S S

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

ϕ η η= = ±

A Mott insulator

ij i jij

H J S S

ϕ η η= = ±

A Mott insulator

ij i jij

H J S S

ϕ η η= = ±

Doped state is a paramagnet with 0 and also a high temperature superconductor with the BCS pairing order parameter 0

With increasing , there must be one or more qua

Ψ ≠

um phase transitions involving ( ) onset of a non-zero ( ) restoration of spin rotation invariance by a transition from 0 to 0

Introduce mobile carriers of density δby substitutional doping of out-of-plane

ions e.g. 2 4La Sr CuOδ δ−

Superconductivity in a doped Mott insulator

First study magnetic transition in Mott insulators………….

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

C. Technical detailsDuality and dangerously irrelevant operators

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

ij SSJH ⋅= ∑><

10 ≤≤ λ

close to 0λ Weakly coupled dimers

0, 0iS ϕ= =Paramagnetic ground state

( )↓↑−↑↓=2

Excitation: S=1 triplon

( )↓↑−↑↓=2

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

Energy dispersion away fromantiferromagnetic wavevector

2 2 2 2

2x x y y

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”

For quasi-one-dimensional systems, the triplon linewidth takesthe exact universal value 1.20 at low TBk T

Bk Te−∆=K. Damle and S. Sachdev, Phys. Rev. B 57, 8307 (1998)

This result is in good agreement with observations in CsNiCl3 (M. Kenzelmann, R. A. Cowley, W. J. L. Buyers, R. Coldea, M. Enderle, and D. F. McMorrow Phys. Rev. B 66, 174412 (2002)) and Y2NiBaO5 (G. Xu, C. Broholm, G. Aeppli, J. F. DiTusa, T.Ito, K. Oka, and H. Takagi, preprint).

Coupled Dimer Antiferromagnet

close to 1λ Weakly dimerized square lattice

Excitations: 2 spin waves (magnons)

2 2 2 2p x x y yc p c pε = +

0 ϕ ≠

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

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))

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

Phys. Rev. B 65, 014407 (2002)T=0

λ 1 cλ δ in cuprates ?

Quantum paramagnet

0ϕ ≠ 0ϕ =

Néel state

Magnetic order as in La2CuO4 Electrons in charge-localized Cooper pairs

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 criticalitywrite 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

ymmetries of the microscopic Hamiltonian

( ) ( ) ( ) ( )22 22 2 22

1 12 4!x c

ud xdc

Sϕ ττ ϕ ϕ λ λ ϕ ϕ⎡ ⎛ ⎞ ⎤= ∇ + ∂ + − +⎜ ⎟⎢ ⎥⎝ ⎠ ⎦⎣∫

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

( )2 2

c pp cε λ λ= ∆ + ∆ = −∆

( )For oscillations of about 0 lead to the following structure in the dynamic structure factor ,

S pλ λ ϕ ϕ

( ),S p ω

( )( )Z pδ ω ε−

Three triplon continuumTriplon pole

Structure holds to all orders in u

A.V. Chubukov, S. Sachdev, and J.Ye, Phys.

Rev. B 49, 11919 (1994)

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

Ground state has Neel order with 0ϕ ≠

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

⇒ ≠

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.

Large paramagnetic ground state with 0

⇒ ≠

Possible large paramagnetic gr Class ound state ( ) with 0A g ϕ =

bondΨ

( ) ( )arond

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

( ) ( )arond

tan j iii j

bond bond

bondΨ

ossible large paramagnetic gr Class ound state ( ) with 0A g ϕ =P

( ) ( )arond

tan j iii j

bond bond

bondΨ

( ) ( )arond

tan j iii j

bond bond

bondΨ

bond bond

( ) ( )arond

tan j iii j

bond bond

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

( ) ( )arond

tan j iii j

bondΨ

bond bond

( ) ( )arond

tan j iii j

bondΨ

bond bond

( ) ( )arond

tan j iii j

bondΨ

bond bond

( ) ( )arond

tan j iii j

bondΨ

bond bond

( ) ( )arond

tan j iii j

bondΨ

Resonating valence bonds

Different valence bond pairings resonate with each other, leading to Cl

a resoass B

nating valenparamagnet)

ce bond , ( with 0

liquidΨ =

P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974); P.W. Anderson 1987

Such states are associated with non-collinear spin correlations, Z2 gauge theory, and topological order.N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991).

Resonance in benzene leads to a symmetric configuration of valence

bonds (F. Kekulé, L. Pauling)

Excitations of the paramagnet with non-zero spin

bond 0; Class AΨ ≠

S=1/2 spinons, , are confined into a S=1

triplon, ϕ*~ z zα αβ βϕ σ

bond 0; Class AΨ ≠ bond 0; Class BΨ =

zα S=1/2 spinons can propagate

independently across the lattice

Quantum theory for destruction of Neel order

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

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

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

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

( ), ,x yµ τ=

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

oriented area of spherical triangle

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

µϕ ϕ ϕ

2 aA µ

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

µϕ ϕ ϕ

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

µϕ ϕ ϕ

a µγ +

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

µϕ ϕ ϕ

a µγ +

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

µϕ ϕ ϕ

Change in choice of is like a “gauge transformation”

2 2a a a aA Aµ µ µγ γ+→ − +

a µγ +

square subl

2attices ;

a aSϕ η ϕη

→→ ±=

a+µϕ

a a+ 0

µϕ ϕ ϕ

Change in choice of is like a “gauge transformation”

2 2a a a aA Aµ µ µγ γ+→ − +

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

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

exp a aa

i A τη⎛ ⎞⎜ ⎟⎝ ⎠

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

with "current" of

charges 1 on sublattices

static

µ µµ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

11 expa a a a a aa aa

Z d i Ag µ τ

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

Large paramagnetic ground state with 0 Berry phases lead to large cancellations between different time histories need an effective action for a

⇒ ≠

→ at large gµ

Simplest large g effective action for the Aaµ

withThis is compact QED in 3 spacetime dimensions with

static charges 1 on

1exp c

two sublat

tices.

sa a a a aaa

Z dA A A i Ae

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

There is a definite average height of the interfaceGround state has bond order.

⇒⇒

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

Z d i Ag µ τ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫

Neel order 0ϕ ≠

Bond order0

Not present in LGW theory of orderϕ

Ψ ≠

Naïve approach: add bond order parameter to LGW theory “by hand”

First order

transitionbond

Neel order0, 0ϕ ≠ Ψ = bond

Bond order0, 0ϕ = Ψ ≠

Coexistence0,

ϕ ≠

Ψ ≠bond

Neel order0, 0ϕ ≠ Ψ = bond

"disordered"0,

Ψ =bond

Neel order0, 0ϕ ≠ Ψ =

Bond order in a frustrated S=1/2 XY magnet

A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)

First large scale (> 8000 spins) numerical study of the destruction of Neel order in a S=1/2 antiferromagnet with full square lattice symmetry

( ) ( )2 x x y yi j i j i j k l i j k l

ij ijklH J S S S S K S S S S S S S S+ − + − − + − +

= + − +∑ ∑g=

Z d i Ag µ τ

ϕ δ ϕ ϕ ϕ η+

⎛ ⎞= − ⋅ +⎜ ⎟

⎝ ⎠∑ ∑∏∫

Neel order 0ϕ ≠

Bond order0

Not present in LGW theory of orderϕ

Ψ ≠

g0S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990). S. Sachdev and K. Park, Annals of Physics 298, 58 (2002).

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, cond-mat/0311222.

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

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

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

S spinons z z z

α α αβ β

• →

pagating degrees of freedom.

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

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

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

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

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

Neel order

~ 0z zα αβ βϕ σ ≠

(associated with condensation of monopoles in ),Aµ

bondBond order 0Ψ ≠

1/ 2 spinons confined, 1 triplon excitations

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004). S. Sachdev cond-mat/0401041.

(Quadrupled) monopole fugacity

Bond order

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

Berry phases, bond order, and the breakdown of the LGW paradigmOrder parameters/broken symmetry

+Emergent gauge excitations, fractionalization.

C. Technical details

Duality and dangerously irrelevant operators

Nature of quantum critical point

Use a sequence of simpler models which can be analyzed byduality mappings

A. Non-compact QED with scalar matterB. Compact QED with scalar matterC. N=1: Compact QED with scalar matter and Berry phasesD. theoryE. Easy plane case for N=2

N → ∞

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

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

N → ∞

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

N → ∞

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

N → ∞

Identical critical theories !

N → ∞

Identical critical theories !

D. , compact U(1), Berry phasesN → ∞

E. Easy plane case for N=2

Phase diagram of S=1/2 square lattice antiferromagnet

Neel order

~ 0z zα αβ βϕ σ ≠

(associated with condensation of monopoles in ),Aµ

bondBond order 0Ψ ≠

1/ 2 spinons confined, 1 triplon excitations

T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004). S. Sachdev cond-mat/0401041.

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