Breakdown of the Landau-Ginzburg-Wilson …qpt.physics.harvard.edu/talks/harvard2005.pdfBreakdown of...

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

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. Statement of the problemA. AntiferromagnetsB. Boson lattice models

II. Theory of defects: vortices near the superfluid-insulator transitionBerry phases imply that vortices carry “flavor”

III. The cuprate superconductorsDetection of vortex flavors ?

IV. Defects in the antiferromagnetHedgehog Berry phases and VBS order

I.A Quantum phase transitions of S=1/2 antiferromagnets

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

; 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 =

LGW theory for such a quantum transitionwrite 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 22

1 12 4!x

ud xd rc

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

Three particle continuum

~3∆

( )2 2

; 2

c pp r cε = ∆ + ∆ =∆

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

rS p

ϕ ϕω

> =

ω

( ),S p ω

( )( )Z pδ ω ε−

Pole of a S=1quasiparticle

Structure holds to all orders in u

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

Rev. B 49, 11919 (1994)

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

( ) ( )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


( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


vbsΨ

( ) ( )arbs

cv

tan j iii j


vbs vbs



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

( ) ( )arbs

cv

tan j iii j


vbs vbs



vbsΨ


vbsΨ

( ) ( )arbs

cv

tan j iii j


vbs vbs



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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I.B Quantum phase transitions of boson lattice models

Bose condensationVelocity distribution function of ultracold 87Rb atoms

M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wiemanand E. A. Cornell, Science 269, 198 (1995)

Apply a periodic potential (standing laser beams) to trapped ultracold bosons (87Rb)

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Momentum distribution function of bosons

Bragg reflections of condensate at reciprocal lattice vectors


Superfluid-insulator quantum phase transition at T=0

V0=0Er V0=7Er V0=10Er

V0=13Er V0=14Er V0=16Er V0=20Er

V0=3Er

Bosons at filling fraction f = 1Weak interactions:

superfluidity

Strong interactions: Mott insulator which preserves all lattice

symmetries


Bosons at filling fraction f = 1

sf 0Ψ ≠

Weak interactions: superfluidity


sf 0Ψ =

Strong interactions: insulator

Bosons at filling fraction f = 1/2

sf 0Ψ ≠

Weak interactions: superfluidity


sf 0Ψ =



sf 0Ψ =


Insulator has “density wave” order

12( + )=

Insulating phases of bosons at filling fraction f = 1/2

Valence bond Valence bond solid (VBS) ordersolid (VBS) order


Charge density Charge density wave (CDW) orderwave (CDW) order

( ) .Can define a common CDW/VBS order using a generalized "density" ieρ ρ= ∑ Q rQ

Q

r

sfAll insulators have 0 and 0 for certain ρΨ = ≠Q Q

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)


12( + )=





Q

r



12( + )=






Q

r



12( + )=



Q

r






12( + )=






Q

r



Predictions of LGW theory

(Supersolid)

Coexistence

1 2r r−

First order transitionsfΨ

1 2r r−Superfluid

ρQ

Charge-ordered insulator

Superfluid Charge-ordered insulator

ρQsfΨ

1 2r r−Superfluid

ρQ

sf

( topologically ordered)

" "

0, 0

Disordered

ρ

≠

Ψ = =Q

Charge-ordered insulator

sfΨ

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II. Theory of defects: vortices near the superfluid-insulator transition

Berry phases imply that vortices carry “flavor”

Excitations of the superfluid: Vortices

Observation of quantized vortices in rotating ultracold Na

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, Observation of Vortex Lattices in Bose-Einstein Condensates, Science 292, 476 (2001).

Excitations of the superfluid: Vortices

Central question:In two dimensions, we can view the vortices as

point particle excitations of the superfluid. What is the quantum mechanics of these “particles” ?

In ordinary fluids, vortices experience the Magnus Force

FM

( ) ( ) ( )mass density of air velocity of ball circulationMF = i i

Dual picture:The vortex is a quantum particle with dual “electric”

charge n, moving in a dual “magnetic” field of strength = h×(number density of Bose particles)

A3

A1+A2+A3+A4= 2π fwhere f is the boson filling fraction.

A2A4

A1


• At f=1, the “magnetic” flux per unit cell is 2π, and the vortex does not pick up any phase from the boson density.

• The effective dual “magnetic” field acting on the vortex is zero, and the corresponding component of the Magnus force vanishes.

Bosons at rational filling fraction f=p/q

Quantum mechanics of the vortex “particle” in a periodic potential with f flux quanta per unit cell

Space group symmetries of Hofstadter Hamiltonian:

, : Translations by a lattice spacing in the , directions

: Rotation by 90 degrees.x yT T x y

R

2

1 1 1 4

Magnetic space group: ;

; ; 1

ifx y y x

y x x y

T T e T T

R T R T R T R T R

π

− − −

=

= = =

The low energy vortex states must form a representation of this algebra

At filling = / , there are species of vortices, (with =1 ), associated with degenerate minima inthe vortex spectrum. These vortices realizethe smallest, -dimensional, representation of the

f p q qq

q

q

ϕ …

magnetic algebra.

Hofstadter spectrum of the quantum vortex “particle” with field operator ϕ

Vortices in a superfluid near a Mott insulator at filling f=p/q

21

2

1

: ; :

1 :

i fx y

qi mf

mm

T T e

R eq

π

π

ϕ ϕ ϕ ϕ

ϕ ϕ

+

=

→ →

→ ∑

Boson-vortex duality The vortices characterize superconducting and density wave orders

q bothϕ

Superconductor insulator : 0 0 ϕ ϕ= ≠

Boson-vortex duality The vortices characterize superconducting and density wave orders

q bothϕ

( )

ˆ

* 2

1

Density wave order:

Status of space group symmetry determined by 2density operators at wavevectors ,

: ;

:

qi mnf i mf

mn

i xx

n

y

mn

p m n

e

T

e

q

e T

π π

πρ

ρ ρ

ρ ϕ ϕ +=

=

→

= ∑i

Q

QQ Q

Q

( ) ( )

ˆ

:

i ye

R R

ρ ρ

ρ ρ

→

→

iQQ Q

Q Q

Field theory with projective symmetry

Degrees of freedom: complex vortex fields 1 non-compact U(1) gauge field

qAµ

ϕ

Fluctuation-induced, weak, first order transitionsfΨ

1 2r r−

ρQ Superfluid

0, 0mnϕ ρ= =Charge-ordered insulator

0, 0mnϕ ρ≠ ≠

Field theory with projective symmetry

1 2r r−

ρQCharge-ordered insulator

0, 0mnϕ ρ≠ ≠

1 2r r−

ρQ

Charge-ordered insulator0, 0mnϕ ρ≠ ≠

Supersolid0, 0mnϕ ρ= ≠

Field theory with projective symmetryFluctuation-induced,

weak, first order transitionsfΨ Superfluid

0, 0mnϕ ρ= =

sfΨ Superfluid

0, 0mnϕ ρ= =

1 2r r−

ρQCharge-ordered insulator

0, 0mnϕ ρ≠ ≠

1 2r r−

ρQ


Supersolid0, 0mnϕ ρ= ≠

Field theory with projective symmetryFluctuation-induced,

weak, first order transitionsfΨ Superfluid

0, 0mnϕ ρ= =

sfΨ Superfluid

0, 0mnϕ ρ= =

Second order transition

1 2r r−

ρQ


sfΨ Superfluid

0, 0mnϕ ρ= =


The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.

The orientation of the vortex in flavor space imp

qi

i lies a particular configuration of density-wave order in its vicinity.

12( + )=

Mott insulators obtained by condensing vortices at f = 1/2





Q

r

All insulators have 0 and 0 for certain ρΨ = ≠Q Q


Mott insulators obtained by condensing vortices at f = 1/4, 3/4

unit cells;

, , ,

all integers

a bq q ab

a b q

×


The excitations of the superfluid are described by the quantum mechanics of flavors of low energy vortices moving in zero dual "magnetic" field.

The orientation of the vortex in flavor space imp

qi

i lies a particular configuration of VBS order in its vicinity.

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III. The cuprate superconductors

Detection of dual vortex wavefunctionin STM experiments ?

Cu

O

LaLa2CuO4

La2CuO4

Mott insulator: square lattice antiferromagnet

jiij

ij SSJH ⋅= ∑><

La2-δSrδCuO4

Superfluid: condensate of paired holes

0S =

Many experiments on the cupratesuperconductors show:

• Tendency to produce modulations in spin singlet observables at wavevectors (2π/a)(1/4,0) and (2π/a)(0,1/4).

• Proximity to a Mott insulator at hole density δ =1/8 with long-range charge modulations at wavevectors(2π/a)(1/4,0) and (2π/a)(0,1/4).

The cuprate superconductor Ca2-xNaxCuO2Cl2

T. Hanaguri, C. Lupien, Y. Kohsaka, D.-H. Lee, M. Azuma, M. Takano, H. Takagi, and J. C. Davis, Nature 430, 1001 (2004).

Possible structure of VBS order

This strucuture also explains spin-excitation spectra in neutron scattering experiments

Tranquada et al., Nature 429, 534 (2004)

xy

Bond operator (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) theory of coupled-ladder model, M.

Vojta and T. Ulbricht, Phys. Rev. Lett. 93, 127002 (2004)




Do vortices in the superfluid “know” about these “density” modulations ?

Consequences of our theory:

Information on VBS order is contained in the vortex flavor space

( )

* 2

1

2Density operators at wavevectors ,

q

i mnf i mfmn

mn

n

p m nq

e eπ π

πρ

ρ ϕ ϕ +=

=

=

∑

Q Q

Each pinned vortex in the superfluid has a halo of density wave order over a length scale ≈ the zero-point quantum motion of the

vortex. This scale diverges upon approaching the insulator

Vortex-induced LDOS of Bi2Sr2CaCu2O8+δ integrated from 1meV to 12meV at 4K

100Å

b7 pA

0 pA

Vortices have halos with LDOS modulations at a period ≈ 4 lattice spacings

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).

Prediction of VBS order near vortices: K. Park and S. Sachdev, Phys. Rev. B 64, 184510 (2001).

Distinct experimental charcteristics of underdoped cuprates at T > Tc

Measurements of Nernst effect are well explained by a model of a liquid of vortices and anti-vortices

N. P. Ong, Y. Wang, S. Ono, Y. Ando, and S. Uchida, Annalender Physik 13, 9 (2004).

Y. Wang, S. Ono, Y. Onose, G. Gu, Y. Ando, Y. Tokura, S. Uchida, and N. P. Ong, Science299, 86 (2003).

STM measurements observe “density” modulations with a period of ≈ 4 lattice spacings

LDOS of Bi2Sr2CaCu2O8+δ at 100 K.M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004).


Our theory: modulations arise from pinned vortex-anti-vortex pairs –these thermally excited vortices are also responsible for the Nernst effect

LDOS of Bi2Sr2CaCu2O8+δ at 100 K.M. Vershinin, S. Misra, S. Ono, Y. Abe, Y. Ando, and A. Yazdani, Science, 303, 1995 (2004).


Superfluids near Mott insulators

• Dual description using vortices with flux h/(2e) which come in multiple (usually q) “flavors”

• The lattice space group acts in a projective representation on the vortex flavor space.

• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”

• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.

Superfluids near Mott insulators

• Dual description using vortices with flux h/(2e) which come in multiple (usually q) “flavors”

• The lattice space group acts in a projective representation on the vortex flavor space.

• These flavor quantum numbers provide a distinction between superfluids: they constitute a “quantum order”

• Any pinned vortex must chose an orientation in flavor space. This necessarily leads to modulations in the local density of states over the spatial region where the vortex executes its quantum zero point motion.

The Mott insulator has average Cooper pair density, f = p/qper site, while the density of the superfluid is close (but need

not be identical) to this value

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IV. Defects in the antiferromagnet

Hedgehog Berry phases and VBS order

Defects in the Neel state

rϕ ∝Monopoles are points in spacetime, representing tunnelling events

Breakdown of the Landau-Ginzburg-Wilson …qpt.physics.harvard.edu/talks/harvard2005.pdfBreakdown of...

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