deconfined criticality: su(2) dejavu · deconfined criticality: su(2) dejavu nikolay prokof’ev...
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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