Worms Exploring Geometrical Features of Phase Transitions · [N. Prokof’ev & B. Svistunov, Phys....

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Worms Exploring Geometrical Features ofPhase Transitions

Wolfhard Janke

Computational Quantum Field TheoryInstitut für Theoretische PhysikUniversität Leipzig, Germany

Collaboration with Thomas Neuhaus (Jülich) and Adriaan Schakel (Recife)

MECO36Lviv, Ukraine, 05 – 07 April 2011

Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions

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MotivationTheoryResults

Motivation

Develop geometrical picture of phase transitions andcritical phenomena in terms of clusters or clusterboundaries (= loops in 2d) or high-temperature graphs

Relate thermodynamic critical exponents to the fractaldimension of the geometrical objects

Monte Carlo “worm” algorithm: Combined statistics ofloops (= energetic sector) and chains (= magnetic sector)[N. Prokof’ev & B. Svistunov, Phys. Rev. Lett. 87 (2001) 160 601]


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High-temperature series expansion of the Ising model

Partition function (Vs spins, Vb bonds):

Z =∑

{si}

eβP

b si sj =∑

{si}

∑

{nb}

∏

b

βnb

nb!(sisj)

nb

= 2Vs∑

{nb}′

∏

b

βnb

nb!

= 2Vs coshVb β∑

{nb=0,1}′

K Nb

{nb}: on each bond nb = 0, 1, 2, . . .{nb}′:

∑

nb = even at each site (no open chains){nb = 0, 1}′: only nb = 0 (non-active) or nb = 1 (active)Nb =

∑Vbb=1 nb, K = tanh β (partial summation of graphs)


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Leads automatically to a loop gas with fugacity parameterK = tanh β (in arbitrary d )

Zloop =∑

G

K NbNNl

Nb: number of occupied bonds, Nl : number of loops,N: number of loop colors for O(N) model (N = 1 for Ising)

Contains in general intersection points (“knots”)

Pure loop gas on 2d honeycomb lattice (as used here) withcoordination number z = 3 and critical parameter:

Kc =[

2 + (2− N)1/2]−1/2

= 1/√

3 (N = 1)


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Compact honeycomb lattice with periodicity in three directions


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Exploiting duality in 2d

honeycomb lattice←− duality −→ triangular (hexagonal) lattice

K = tanh β = e−2β̃

β on honeycomb lattice (Kc = 1/√

3)β̃ on triangular lattice (β̃c = 1

4 ln 3)

high temperature T ←− duality −→ low temperature T̃high-T graphs ←− duality −→ cluster boundaries (Peierls)


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Spin-spin correlation function 〈si0sj0〉

〈si0sj0〉 =∑

{si}

si0sj0eβP

b si sj /Z ≡ Z (i0, j0)/Z

High-temperature series expansion of Z (i0, j0) containsadditional open chain running from i0 to j0 immersed into thebackground of loops

Pure loop gas is a special case for j0 = i0


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β = 0.30 β = 0.44

β = 0.45 β = 0.70


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Monte Carlo simulations: Worm update

Qualitative description:

Move end points of (open) chain through background ofloopsConnects loop and chain sector by reopening existingloops or creating new ones by self-closing or a “back bite”,leaving a loop behind

Quantitative update prescription (nb = 0, 1):

choose randomly either end point of the chainchoose randomly any of the links attachedupdate nb −→ n′

b = 1− nb with acceptance probability

Paccept = min(

1, K 1−2nb

)

(in effect: Paccept = K < 1 for bond creation, and alwaysaccept bond deletion)


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If start and end point of the chain are nearest neighbors, mixedloop/chain configuration can be turned into a pure loop gas –possible switch between magnetic and energetic sector

We always attempt to close the chain and, if accepted, proceedby randomly choosing a new link among all links of the latticefor the new location of the worm. The bond variable on that linkis accepted/rejected as usual.

Slightly different from Prokof’ev/Svistunov’s original proposal[PRL 87 (2001) 160 601], but close to Wolff’s version [NPB 810(2009) 491].


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Basic scaling properties of loop gases

Distribution of loop length n, similar to percolation:

ℓn ∼ n−d/D−1e−θn , θ ∝ (K − Kc)1/σ .

Given: Tuning parameter K and dimension d (here d = 2)

Measured: Fractal dimension D and exponent σ, governingapproach to criticality at Kc

At criticality: Radius of gyration Rg, end-to-end distance Re (foropen chains):

〈R2g〉 ∝ 〈R2

e〉 ∼ n2/D


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Fractal dimension D for 2d models:

D = 1 + 1/2κ

(SLE classification, Coulomb gas, conformal field theory)

2d Ising: κ = 4/3 −→ D = 11/8 = 1.375

Parameter σ:

σ =1

νD

2d Ising: ν = 1 −→ σ = 8/11 = 0.727272 . . .


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Results

Distribution ℓn of loop length n at criticality

1

10

100

1000

10000

100000

1e+006

1e+007

1e+008

1e+009

1e+010

10 100 1000 10000

n

ℓ n

Honeycomb lattice (L = 352); straight line: ℓn ∝ n−2/D−1 withD = 11/8 (bump for n > 2000: winding loops)


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Distribution zn of open chains at criticality (L = 352)Inset: Log-log plot of L2/

∑

n zn = L2/χ ∝ L2−γ/ν = Lη vs L

0

50000

100000

150000

200000

250000

300000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

3 3.5 4 4.5 5 5.5 6 6.5

n

z n

Inset: 2-parameter fit: η = 0.2498(26) ≈ 1/4 (χ2/dof= 1.87)


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Distribution Pn(r) of the end-to-end distance r of open chains atcriticality with # steps n = 615, 1230, 1845 fixed (L = 352)

0

1

2

3

4

5

6

0 50 100 150 200 250 300

61512301845

r

105P

n(r

)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2

61512301845

r/n1/Dn2/

DP

n

Perfect scaling collapse with D = 11/8. Scaling function is∝ tϑ exp(−btδ); t = r/n1/D , ϑ = 3/8, δ = (1− 1/D)−1 = 11/3


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Average squared end-to-end distance (of chains) and radius ofgyration of chains and loops ∝n2/D, D =11/8 =1.375 (L= 352)

0

500

1000

1500

2000

2500

3000

200 400 600 800 1000 1200 1400 1600 1800

end-to-endopen

closed

n

1 5〈R

2 e〉,〈R

2 g〉

〈R2e〉: D = 1.3755(31) = 1.0004(23)Dexact

〈R2g〉: D = 1.3789(55) = 1.0028(40)Dexact (for chains)


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Average length of winding loops 〈nw 〉 ∝ LD on a honeycomblattice of linear extent L at criticality

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

3 3.5 4 4.5 5 5.5 6 6.5

ln L

ln〈n

w〉

2-parameter fit: D = 1.37504(32) = 1.00003(24)Dexact

(χ2/dof= 1.13)


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Probability ΠL that one or more loops wind the honeycomblattice of linear size L at criticality

0.506

0.508

0.51

0.512

0.514

0.516

0.518

0.52

0 50 100 150 200 250 300 350 400

L

ΠL

Pw for L = 160:

P0 = 0.48802(74)P1 = 0.49960(73)P2 = 0.01236(14)P3 = 0.0000299(64)P4 = 0

P0 + P2 + P4 + . . .= 0.50038

Average: ΠL = 0.51257(27) > 0.5 (χ2/dof= 1.08)


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Binary variable: 0 = even, 1 = odd winding numbers

Average IL(β̃) (L = 32, 64, 96):

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.25 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295 0.3

326496

β̃

I L

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-1 -0.5 0 0.5 1

326496

(β̃ − β̃c)L1/νI L

crossing point at β̃c : scaling plot with ν = 1IL(β̃c) = 0.50024(21) straight line at 3/4:

# “odd”/# “even” configs


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Derivative of IL(β̃) at criticality: I′L(β̃) ∝ L1/ν

0

50

100

150

200

250

300

350

400

0 50 100 150 200 250 300 350 400

0.494 0.496 0.498 0.500 0.502 0.504 0.506 0.508

0 100 200 300 400

L

I′ L(β̃

c)

2-parameter fit: 1/ν = 1.0001(15) (χ2/dof= 1.01)Inset: IL(β̃c) = 0.50024(21).


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Summary

Worm update perfect tool for loop-gas approachto fluctuating fields on a lattice

Concepts developed for self-avoiding walks(N = 0) generalized to Ising (N = 1) loops

Extend study in future work to other numbers ofcolors N

Collaboration with Thomas Neuhaus (FZ Jülich) andAdriaan Schakel (then Leipzig, now Recife, Brazil)

WJ, T. Neuhaus & A.M.J. Schakel, Nucl. Phys. B 829 [FS] (2010) 573


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Acknowledgements

eee

Deutsche Forschungsgemeinschaft (DFG)

EU RTN Network “ENRAGE”

Graduate School “BuildMoNa”

DFH–UFA Graduate School Nancy–Leipzig

NIC Computer time, JSC, Forschungszentrum Jülich

THANK YOU !


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Worms Exploring Geometrical Features of Phase Transitions · [N. Prokof’ev & B. Svistunov, Phys....

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