Deconfined criticality: SU(2) Dejavu

Nikolay Prokof’ev (UMASS, Amherst)

Anatoly Kuklov (CSI, CUNY)

SF VBSBoris Svistunov (UMASS, Amherst)

Matthias Troyer (ETH)models

m

SF-VBS

+ simulations

PRL 101, 050405 (2008)

models

Munehisa Matsumoto (UC Davis)

, ( )Annals of Physics 321, 1601 (2006)Prog. Theor. Phys. Jap. 160 (2005)PRL 93, 230402 (2004)

= I-order transition

“Quantum critical phenomena” Totonto, September 25-27, 2008

Why Dejavu?

(1) (1)U U× (2)SU

Contributing authors Many of the same authorsContributing authors

Simulations of specific models

Many of the same authors

Same type of models

claims of new criticality claims of new criticality

Problems with scaling for systems

claims of new criticality

Method of solution: flowgrams generic I-order

Dejavu

Solid = insulator with broken translational symmetry (integer filling excluded)

Checkerboard solid (anti-ferromagnet)

Valence-bond solids (VBS) ( g )

Columnar VBS Plaquette VBS

Density-order Current (or bond) orderDensity order

i jn n

Current (or bond) order

2 2i jJ J

Superfluid and Solid orders and transitions:

Ψ M ,x yB B ,x yD D …

order parameters:

”Naïve” Ginzburg-Landau(expansion from the generic quantum disordered phase)

order parameter

?

g gg SF SFirst-order

SF SSFS

SupersolidSF S

Quantum First-order Supersolid disordered

1 2 3 4 5( , , , , ...)S S S S S S= - multi-component order parameter;the symmetry of S is ALWAYSbroken in the ground state

Ψ M ,x yB Bbroken in the ground state

0S ≠

O(3)-case: Heisenberg point = exact O(3)-symmetry

EM

S -sphere

E

θπ

M

θ2

Generic case; I-order SF-S transition

Ψ E

θ2πM−

SF and VBS orders are deeply related … May be even self-dual at the critical point!

O.I. Motrunich and A. Vishwanath, Phys. Rev. 70, 075104 (2004).T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M.P.A. Fisher, Science, 303, 1490 (2004):

Deconfined criticality, does not fit the standard Landau-Ginsburg-Wilson paradigm.

SF VBS

gDeconfined spinons

= vortices in the VBS order x yB iB+

DCP action:

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

1 21,2

aa

S d rd iA z s z u z v z z Aµ µ µνβ ν βτ κ ε=

= ∂ − + + + + ∂∑∫

Ordered SF state for matter fields = (duality mapping) “quantum disordered” for vortex fields

Do LG expansion for vortex fields!

L. Balents, L. Batosch, A. Burkov, S.Sachdev, K. Sengupta, PRB 71, 144508 and 144509 (2005).

Filling factor 1/2 dual magnetic flux/plaquette 1/2 two species of vortexes+ gauge field coupling

at the SF-VBS critical point

DCP action

( )22 2

∫ ( ) ( ) ( )22 22 2 2 22

1 21,2

aa

S d rd iA s u v Aψ µ µ µνβ ν βτ ψ ψ ψ ψ ψ κ ε=

= ∂ − + + + + ∂∑∫2 2 2| | | | | |ψ ψ ψ= +2 1 2| | | | | |ψ ψ ψ= +2[ 1/ ] ( ) g e g L gLκ = = → = Run-away flow to strong coupling

DCP action:

( )22 22 2 2 2

0v =SU(2)

( ) ( ) ( )22 22 2 2 2

1 21,2

ai a

S iA s u v Aψ µ µ µνβ ν βψ ψ ψ ψ ψ κ ε=

= ∂ − + + + + ∂∑ ∑

( ) ( ) ( )21 2

,cos cosXY

rS J A A Aµ µ µ µ

µ

ϕ ϕ κ ⎡ ⎤= − ∆ − + ∆ − + ∇×⎣ ⎦∑ ∑

( ) ( )1

2 2 2 21 2

1,2 with | | | | 1aCP

i aS iA Aµ µ µνβ ν βψ κ ε ψ ψ

=

= ∂ − + ∂ + =∑ ∑

( ) ( )[ ( )] ( ') ( ) ( ) ( ') ( ')S U F j r g Q r r j r j r j r j r = + − + ⋅ +∑ ∑ ( ) ( )1 2 1 2',

[ ( )] ( ) ( ) ( ) ( ) ( )J ar rr

S U F j r g Q r r j r j r j r j rµ µ µ µ µµ

= + + +∑ ∑

1 2( ) i ( / ) ( ) /∑1 2( ) sin ( /2) ( ) 1/Q q q Q r rµµ

− = → ∑ ∼

Mappings: “High-T” expansion (expansion in kinetic energy)

( )'24 2 *

''

. . [ ]rriAr r r r

r rrS U t e c c Aψ µ ψ ψ ψ κ< >

< >

= − − + + ∇×∑ ∑ ∑

( ) ( )b b

S

n m

Z DAD e

t t

ψ

∞ ∞

= ∫∫

( )* *' '

0 0'

( ) ( )( ) ( )! !

b bb b b br

b

Ab

b

n mn m iA n mS S

r r r r rn mr b rr b b

t tDA d e e en m

ψ ψ ψ ψ ψ∞ ∞

−

= ==< >

− −= ∑ ∑∏ ∏∫∫ ∫

' , with ' , with{ } { }

e (xp{ ( ' { }) } ) J

b b b bb b b b

Sr r

n m rr n mj n m loops j n m l p

b

oo

g Q eW j= − ∈ = − ∈

− − = =∑ ∑ r r j js

∑

loop configurationsof oriented currents (DCP)1 2r r r = +j j j

INSULATOR (NORMAL) SUPERFLUID

Winding numbers = MC “blessing”2 2 2 /dTL W W L−Λ

(Pollock, Ceperley ’87)

Superfluid stiffness: 2 2 2 /ds TL W W LΛ = →

j-current through any cross-section

Probe system properties at the largest scales; ideal for studies of critical phenomena

Continuous critical point:the entire disctibution

is universalI-order SF-S transition:( )P W

( )P Wis universal

( )P W

2 1W ∼≈

2

SFW L∼

W1 2W1 20

Since 0( )

(0)W

P W

P≠ =

∑ in 0 I

SFi

∞ 0( )

#(0)

WP W

P≠ =

∑Transition point Def:

(0)P 0 Iin (0)P

Quantity to study at I order∞Quantity to study at the transition point:

2 ( )W f L = →c

I-order continuous onst

∞(scale invariance)

Phase diagram of the U(1)xU(1) DCP action

( ') ( ')gQ r r V r rδ− → −short-range version

g ~ 0.5 - weak I-order

small g maps to g > 0.5

RG flow: ( )effg L gL∝

Flowgrams of . What to expect across the tricitical point.2W

L

2 2W I-orderdivergence

TP criticality

criticalitybelow TP

Lshort-range CP1

Flowgrams of for the run-away flow to strong coupling and I-order.2W

L

2 ( )P S2W

S

( )P SSFI

I-orderdivergence

Slarge g

criticalitycriticalityat g=e =02

L

Flowgram collapse! 2W

2 2

criticality

2W 2W

l ( )L

criticalityat g=e =02

( )P Sln( )L

S

( )P SSFI

Different g are identical on

S

glength scales ( ) 1L gξ≥ >>criticality

at g=e =02criticalityat g=e =02

ln( / ( ))L gξln( / ( ))L gξ

(1) (1) caseU U×

2.1

2.2

2.3

1.7

1.8

1.9

2.0

0.58

1.3

1.4

1.5

1.6

<W2 >

0.5 I-order proved

0 9

1.0

1.1

1.2

1.3

g=0.125 0.25

0.375

-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.9

ln [(g+g2)L]

Proof of first-order transition at g=0.58

Probability distributions at the critical point for g=0.58

Scaling of peak positions with system size L

‘07

(2) caseSU

07

‘07

‘07

1(2) case, - modelSU CP

“Smoking gun” of the first-order transition

(2) caseSU

O.I. Motrunich and A. Vishwanath ‘08

Conclusions:

1. So far, all 2+1 dim. models of the SF-S transitions observe either I-order, or problems with scaling

in agreement with both (!!)

GWL using 1 2 3 4 5( , , , , ...) 0S S S S S S= ≠

Ψ CBS ,x yB B g SFS

DCP action = a continuous theory of weak I-order scenario

U

SF S

U

2. Strong renormalized pairing of vortices on large scales may be the mechanism ti d fi d iti lit f h ipreventing deconfined criticality from happening