Post on 21-Jul-2020
CQ T
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]
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
Compact honeycomb lattice with periodicity in three directions
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
β = 0.30 β = 0.44
β = 0.45 β = 0.70
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
β = 0.30 β = 0.44
β = 0.45 β = 0.70
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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].
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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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 . . .
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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)
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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).
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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 !
Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions
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MotivationTheoryResults
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Wolfhard Janke Worms Exploring Geometrical Features of Phase Transitions