QUANTUM MONTE CARLO AND

QUANTUM MONTE CARLO AND NON-GINZBURG-LANDAU TYPE PHASE TRANSITION

NTU Colloquium

Naoki Kawashima

ISSP

December 27, 2011 NTU, Taiwan

Collaborators

Kyoto U. Kenji Harada ... SUN, JQ , BIQ, PWU, K ISSP Synge Todo ... K, JQ, PWU, SUN

Haruhiko Matsuo ... K, JQ, PWU, SUN Jie Lou ... SUN, JQ, K

Akiko Masaki … PWU, OPT, K Takahiro Ohgoe ... OPT, PWU

Hyogo U. Takafumi Suzuki … JQ, K, OPT, PWU, SUN Boston U. Anders Sandvik ... JQ

NEC Kota Sakakura … K

K … Parallelization on “K”

SUN... SU(N) Heisenberg model

BIQ ... Biquadratic Heisenberg model PWU ... Paralllelization of worm update JQ ... SU(N) J-Q model

OPT ... Optical lattice

Ising Model

Critical Slowing Down in Monte Carlo …

The typical size of magnetic domains: ξ ～ a x (T/Tc - 1)^ν The size of the updating unit: (a single site) ～ a

The effect of spin flip at a point propagates

by some diffusion-like process which makes z～2.

So, it takes

τ～ (T-Tc)^νz

Monte Carlo steps for a magnetic domain undertake a substantial change (annihilation, creation, relocation, etc).

Swendsen-Wang Algorithm

S G S’

not flipped flipped

Swendsen-Wang 1987 ... Binding spins together to form a cluster.

Path-Integral Monte Carlo Method

 

   

:

:

S

p p

Z W S

W S w S

S







plaquette



The whole pattern of world-lines

Interaction Vertex（“shaded plaquettes”）

World Line Suzuki 1976

Method Used before 1993

Many Problems --- No change in topological numbers, critical slowing down, high-precision slowing down, etc.

The patterns are updated only locally.

Generalizing SW algorithm to QMC

A "bond" in the Swendsen-Wang algorithm

Loop elements

in the loop algorithm

for QMC

Loop Algorithm for QMC

Cluster Algorithm on Path-Integral Representation:

A graph element (for S=1/2 anti- ferromagnetic Heisenberg model)

Evertz-Lana-Marcu 1993

Magnetic/Non-Magnetic Transition

"Bond-alternation" enforces the transition to the VBS state.

 

   

 

, , , , , :

1 1

x

x

e e

x y x y x y x

H J S S

J S S







  

 



   



x x odd

Conventional Transition

At the transition point, the Skyrmion number is not conserved.

Monopole ("hedgehog")

= The skyrmion-number-changing event . Example: 2+1 D Ｏ（３） Wilson-Fisher f.p.

If the skyrmion number changes at some point of time...

... there must be a singular point in space-time.

Skyrmion number:

 

 ^{   }

 d x n

n

n

Q

4 1



Deconfined Critical Phenomena

If the skyrmion number changes at some point of time...

... there must be a singular point in space-time.

Skyrmion number:

 

 ^{   }

 d x n

n

n

Q

4 1



At the deconfined critical point, the skyrmion number is asymptotically

conserved, and monopoles are prohibited.

Example: non-compact CP(1) model ?

T. Senthil, et al, Science 303, 1490 (2004)

Symmetries Around DCP

VBS

Spin rotation symmetry

Neel

broken not broken

Lattice symmetry

broken not broken

We cannot say one phase has higher symmetry than the other.

T. Senthil, et al., Science 303, 1490 (2004)

SU(2) Symmetric NCCP ¹ Model

Kuklov, Matsumoto, et al, PRL 101, 050405 (2008) g=1.65



 ^{ }

( ) 1,2 1,2

c.c. 1 1

8

iAij

i j j

ij

S t z z e A z

 

g

 

 

       

 

   

1 0, 2 0

i i

z  z  ^{1 2}

* 1 2

0 0,

i i

i i

z z

z z

 



*

1 2 0, 1 2 0

i i i i

z  z  z z 

SU(N) Heisenberg Model

 

 

tion representa some

by d represente rotation

SU(N) of

generators



 r S

R r

S





   

 

' ,

 ^



r r

r S

r N S

H J

 ^{S , S}  ^{  S  }

^S

^{ ,  ,  ,   1 , 2 ,  , N}

 

 

 

 



A general extension of the SU(2) anti-ferromagnetic Heisenberg model

， ， ， ，etc

Representation:

n=1

(fundamental representation.)

n=2 n=3 n=4

2D Analogue of "Haldane" States

Nc ～ ５．３ n

Read & Sachdev (1989) Arovas & Auerbach (1988)

2D Isotropic Case (Tanabe, N.K.) Prediction from 1/N expansion

New challenge --- 「京」

京 = 10

0.64 M cores

Parallelization of loop algorithm

S. Todo & H. Matsuo

Binary-tree algorithm for cluster identification

... Relative overhead is negligible at very low temperatures

Parallelization of loop algorithm

S. Todo & H. Matsuo

Asynchronous lock-free union-find algorithm

(1) find root of each cluster/tree (2) unify two clusters

(3) compress path to the new root Locking whole clusters is no good.

(reduces parallelization efficiency) Finding root and path compression are “thread-safe”

Lock-free unification can be achieved by using CAS (compare-and-swap) atomic operation

S. Todo & H. Matsuo

Fundamental Representation (n=1)

   

 

   

 ^{/ / 4 2}    ^{0 0}

4 /

2 /

M L

M

M L

M

L C

L L C

R

M M M





4<Nc<5

  ^{R S}

  ^{R S}

  ^R

M  

Neel order dissapears at Nc

Tanabe & N.K.: PRL 98 057202 (2007)

2D Isotropic Case (Tanabe, N.K.)

SU(N) Model (n=2)

Nc～９

     

     























B

y x

n n n

V

V V

A

1 2

1

;

   

 R R R e

R R

R



  

A

Very small but finite LRO is present.

Lattice rotation symmetry is broken.

R

N

L

2D Isotropic Case (Tanabe, N.K.)

Ground-State Manifold is U(1) Symmetric

Tanabe & N.K.: PRL 98 057202 (2007)

n=1, N=6

   

 

      ^



 



   













 N

S S

P

P V P

D

1

,

, 1 ,













R R

R R

e R R e

R R

R

^  ^{D ,}

^D



The system is asymptotically U(1) symmetric though the original microscopic model

does not possess this symmetry.

pure columnar

pure plaquette

Ｎ＝１０，ｎ＝１，Ｌ＝３２，β＝２０

SU(N) Heisenberg Model (n=1)

D

D

2D Isotropic Case (Tanabe, N.K.) ... Reflection of the U(1)

symmetry at DCP

Multi-spin Interactions

2D JQ-Model (Lou, Sandvik, N.K.)

J. Lou, A. Sandvik, N.K.: PRB 80, 180414R (2009)

A. W. Sandvik, Phys. Rev. Lett. 98, 227202 (2007)

SU(2) J-Q Model

A. W. Sandvik, PRL98, 227202 (2007)

U(1) Nature is confirmed near the critical point.

(Slightly on the dimer

order side.)

VBS - VBS Crossover

SU(2) J-Q3 SU(3) J-Q2

Large Q Small Q

2D JQ-Model (Lou, Sandvik, N.K.) J. Lou, A. Sandvik, N.K (2009)

Scaling Properties of Anisotropy

SU(2) J-Q3 Model

SU(3) J-Q2 Model

) 5 ( 20 . 1 ),

2 ( 69 . 0 ),

2 ( 20 .

0 ₄

d    a 



) 2 ( 6 . 1 ),

3 ( 65 . 0 ),

3 ( 42 .

0 ₄

d    a 



SU(2) J-Q3 SU(3) J-Q2

 

  ^{cos   4 }

,

2 2

2 4

y x

y x

y x

D D

D D

P dD dD

D





 

CF: J. Lou, A. W. Sandvik, and L. Balents, PRL (2007).

2D JQ-Model (Lou, Sandvik, N.K.)

J. Lou, A. Sandvik, N.K. (2009)

Recovery of Discreteness

The exponent nu on the VBS side may be affected by the additional fixed point and can be differ from the Neel side.

Ｊ－Ｑ Model

Phys. Rev. B 80, 180414 (2009)

 

 



 



    































1 ,

2 4 for

1

, ,

, 3 3

) ( 2 2

) ( 1

3 2

1

n N

C

C C C Q

H

C C Q

H

C J

H

H H

H H

j i

j i ij

ijklmn

mn kl ij ijkl

kl ij ij

ij

S S











Continuous Transition is suggested.

SU(2) J-Q Model

2D JQ-Model (Lou, Sandvik, N.K.)

SU(3) and SU(4) J-Q2 Models

) 3 ( 65 . 0 ),

3 ( 38 .

s

 0  



) 2 ( 70 . 0 ),

5 ( 42 .

s

 0  



SU(3) J-Q2

SU(4) J-Q2

Jie Lou, A. Sandvik and N.K.:

Phys. Rev. B 80, 180414 (2009)

Universality?

T. Senthil, et al, Science 303, 1490 (2004)

M. Levin and T. Senthil, Phys. Rev. B 70, 220403R (2004).

. 1 ,

1

For N  



For N  1, 

 N .

M. A. Metlitski, et al, PRB 78, 214418 (2008);

G. Murthy and S. Sachdev, Nucl. Phys. B 344, 557 (1990).

J. Lou, A. Sandvik, N.K.: PRB 80, 180414R (2009)

2D JQ-Model (Lou, Sandvik, N.K.)

Scaling Dimension (CP ^N-1 Model)

Murthy and Sachdev, Nucl. Phys. B 344 557 (1990) Metlitski, et al, PRB78 214418 (2008)

 

 

           

 

1

2 2

VBS VBS 2

1 1

1

1

: 1 gauge field

monopole scaling dimension

0 0 0 1

1 2

lim 0.2492 0.062296 Murthy & Sachdev

2

q

D N

L D z i z

g

D iA

A U

D R D R v R v

R

N N N



  





 







 

    

 

  

 

    

          

 

is non-compact

conservation of the gauge current

absense of monopoles

G

A

J A









 



2D JQ-Model (Lou, Sandvik, N.K.)

1

2

2 1 1 1

0.2492 0.32

D

O

N N N N

          

 

Monopole Scaling Dimension up to O(N ^-1 )

D

N



2D JQ-Model (Lou, Sandvik, N.K.) Jie Lou, A. Sandvik

and N.K.: Phys. Rev.

B 80, 180414 (2009)

Recent Refinement by Kaul & Sandvik

R. Kaul and A. Sandvik, arXiv:1110.4130v1

Quantum Spin System

Yip

  ^{ on - site Coulomb repulsion }

)

( , 1,0,1







  











 ij

i j

j

i

b b b

b t

H



    

Effective Hamiltonian to the 2nd order in t/U.

   

 

] is spin total

when the repulsion

site -

on the [

3 4 3

2 , 2

const

0 2

2 2

2 2 )

(

2

S U

U t U

J t U

J t

J J

H

S

Q L

ij

j i Q j

i L





















  ^{S S S S}

23

Na: 0< U

< U

( J

< J

< 0)

Bilinear-Biquadratic Model in 2D with

strong spatial anisotropy (Phase Diagram)

   

 



















sin cos ,

Q L

2 Q

L

J J

J J

y x

S S J S S J H























Diversing Correlation Lengths

θ＝－π／２

2 correlation length diverges at the same point.

1 1



, 

VBS Neel

y x

J J

θ＝ー０．５π （ＳＵ（３） symmetric）

Jx

 Jy /



R

R

x y

J

 J



  ⁵

125 .

c

 0



A single transition is likely...

Cannot obtain a reliable finite-size scaling plot.

Binder Ratio

Quasi-1D SU(N) (Harada, Troyer, N.K.)

Conclusion

(1) Isotropic SU(N) Heisenberg Model ✔ VBS Ground State

✔ Proximity to DCP critical phenomena (2) Multi-Spin Interactions (J-Q Models) ✔ Consistent with DCP

✔ η

proportional to N (The correction term is estimated)

(3) Quasi-1D SU(3) and SU(4) Models ✔ Direct transition is likely

✔ Still not clear if the transition is of the 2nd order

(We need bigger machines, and a better strategy.)