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-rw-r--r--data/cluster-size/cluster-size_2vector3d.dat216
-rw-r--r--data/cluster-size/cluster-size_3potts2d.dat310
-rw-r--r--data/cluster-size/cluster-size_3vector3d.dat184
-rw-r--r--data/cluster-size/cluster-size_4potts2d.dat301
-rw-r--r--data/cluster-size/cluster-size_ising2d.dat310
-rw-r--r--data/cluster-size/cluster-size_ising3d.dat237
-rw-r--r--fig_clusters_ising2d.eps10076
-rw-r--r--fig_correlation-temp.tex4
-rw-r--r--fig_correlation.tex4
-rw-r--r--fig_correlation_collapse-hL.tex4
-rw-r--r--fig_metastable.tex4
-rw-r--r--monte-carlo.bib563
-rw-r--r--monte-carlo.pdfbin185499 -> 211006 bytes
-rw-r--r--monte-carlo.tex49
14 files changed, 9900 insertions, 2362 deletions
diff --git a/data/cluster-size/cluster-size_2vector3d.dat b/data/cluster-size/cluster-size_2vector3d.dat
index 16e71a8..b1d07b7 100644
--- a/data/cluster-size/cluster-size_2vector3d.dat
+++ b/data/cluster-size/cluster-size_2vector3d.dat
@@ -1,54 +1,162 @@
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-8 2.20167 0.01 67.9385416666665 2.299960087698432
-8 2.20167 0.1 105.92954363490816 3.86057588781312
-8 2.20167 1. 213.1218354430377 7.762196801047552
-8 2.20167 10. 368.2691292875991 10.105035534898688
-8 2.20167 100. 470.9830508474578 7.230626947020288
-
-16 2.20167 1.e-7 286.2860304287703 8.98124531728384
-16 2.20167 1.e-6 274.4344131704709 8.678468826714113
-16 2.20167 0.00001 262.02001520141926 8.166182721142784
-16 2.20167 0.0001 233.82716329775104 7.156634142302208
-16 2.20167 0.001 234.45846572854887 7.287514589700096
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-16 2.20167 1. 1872.1783216783197 65.73364751310439
-16 2.20167 10. 3098.8235294117644 78.5758652691415
-16 2.20167 100. 3722.8065573770486 60.9046991010857
-
-32 2.20167 1.e-7 1048.580106979115 29.22072313312051
-32 2.20167 1.e-6 971.4170222854472 26.492627007799296
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-32 2.20167 0.0001 1083.3379648008356 31.228299422072833
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-32 2.20167 10. 22135.92249999999 669.8054971229799
-32 2.20167 100. 30417.365771812078 417.6834673476567
-
-64 2.20167 1.e-7 4289.382018782331 113.41379059607142
-64 2.20167 1.e-6 4021.5975113958357 101.65684041993421
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-64 2.20167 10. 201245.50704225348 4999.55582256939
-64 2.20167 100. 239130.41503267954 3687.9212448362005
-
-128 2.20167 1.e-7 17409.28502540219 666.9019482318438
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+256 2.20167 0.000630957 433035.5322706618 4330.322589835591 \ No newline at end of file
diff --git a/data/cluster-size/cluster-size_3potts2d.dat b/data/cluster-size/cluster-size_3potts2d.dat
index 3b8319d..c205507 100644
--- a/data/cluster-size/cluster-size_3potts2d.dat
+++ b/data/cluster-size/cluster-size_3potts2d.dat
@@ -1,65 +1,255 @@
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diff --git a/data/cluster-size/cluster-size_3vector3d.dat b/data/cluster-size/cluster-size_3vector3d.dat
index 2467b0e..bcbdc89 100644
--- a/data/cluster-size/cluster-size_3vector3d.dat
+++ b/data/cluster-size/cluster-size_3vector3d.dat
@@ -1,23 +1,161 @@
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diff --git a/data/cluster-size/cluster-size_4potts2d.dat b/data/cluster-size/cluster-size_4potts2d.dat
index 4c36fd5..566bc7a 100644
--- a/data/cluster-size/cluster-size_4potts2d.dat
+++ b/data/cluster-size/cluster-size_4potts2d.dat
@@ -1,54 +1,255 @@
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diff --git a/data/cluster-size/cluster-size_ising2d.dat b/data/cluster-size/cluster-size_ising2d.dat
index 8d790cc..5c6dcf5 100644
--- a/data/cluster-size/cluster-size_ising2d.dat
+++ b/data/cluster-size/cluster-size_ising2d.dat
@@ -1,65 +1,255 @@
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diff --git a/data/cluster-size/cluster-size_ising3d.dat b/data/cluster-size/cluster-size_ising3d.dat
index bc3e920..881ba9a 100644
--- a/data/cluster-size/cluster-size_ising3d.dat
+++ b/data/cluster-size/cluster-size_ising3d.dat
@@ -1,54 +1,191 @@
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diff --git a/fig_clusters_ising2d.eps b/fig_clusters_ising2d.eps
index 376cc4c..829e12c 100644
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%%Title: fig_clusters_ising2d.tex
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@@ -441,7 +441,7 @@ SDict begin [
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- /CreationDate (Thu Apr 19 16:11:44 2018)
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diff --git a/fig_correlation-temp.tex b/fig_correlation-temp.tex
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diff --git a/monte-carlo.bib b/monte-carlo.bib
index 789b06d..a624861 100644
--- a/monte-carlo.bib
+++ b/monte-carlo.bib
@@ -1,219 +1,315 @@
-@article{alexandrowicz1989swendsen,
- title={Swendsen-Wang simulation of Ising spins and a precise definition of critical clusters},
- author={Alexandrowicz, Z},
- journal={Physica A: Statistical Mechanics and its Applications},
- volume={160},
- number={3},
- pages={310--320},
- year={1989},
- publisher={Elsevier}
-}
-
-@article{baillie1991comparison,
- title={Comparison of cluster algorithms for two-dimensional Potts models},
- author={Baillie, Clive F and Coddington, Paul D},
- journal={Physical Review B},
- volume={43},
- number={13},
- pages={10617},
- year={1991},
- publisher={APS}
+@article{jose_renormalization_1977,
+ title = {Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model},
+ volume = {16},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.16.1217},
+ doi = {10.1103/PhysRevB.16.1217},
+ abstract = {The classical planar Heisenberg model is studied at low temperatures by means of renormalization theory and a series of exact transformations. A numerical study of the Migdal recursion relation suggests that models with short-range isotropic interactions rapidly become equivalent to a simplified model system proposed by Villain. A series of exact transformations then allows us to treat the Villain model analytically at low temperatures. To lowest order in a parameter which becomes exponentially small with decreasing temperature, we reproduce results obtained previously by Kosterlitz. We also examine the effect of symmetry-breaking crystalline fields on the isotropic planar model. A numerical study of the Migdal recursion scheme suggests that these fields (which must occur in real quasi-two-dimensional crystals) are strongly relevant variables, leading to critical behavior distinct from that found for the planar model. However, a more exact low-temperature treatment of the Villain model shows that hexagonal crystalline fields eventually become irrelevant at temperatures below the Tc of the isotropic model. Isotropic planar critical behavior should be experimentally accessible in this case. Nonuniversal behavior may result if cubic crystalline fields dominate the symmetry breaking. Interesting duality transformations, which aid in the analysis of symmetry-breaking fields are also discussed.},
+ number = {3},
+ urldate = {2018-04-04},
+ journal = {Physical Review B},
+ author = {José, Jorge V. and Kadanoff, Leo P. and Kirkpatrick, Scott and Nelson, David R.},
+ month = aug,
+ year = {1977},
+ pages = {1217--1241},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/RMHXAR3A/PhysRevB.16.html:text/html;José et al. - 1977 - Renormalization, vortices, and symmetry-breaking p.pdf:/home/pants/.zotero/data/storage/M2V99JHC/José et al. - 1977 - Renormalization, vortices, and symmetry-breaking p.pdf:application/pdf}
}
-@article{coniglio1980clusters,
- title={Clusters and Ising critical droplets: a renormalisation group approach},
- author={Coniglio, A and Klein, W},
- journal={Journal of Physics A: Mathematical and General},
- volume={13},
- number={8},
- pages={2775},
- year={1980},
- publisher={IOP Publishing}
+@article{coniglio_exact_1989,
+ title = {Exact relations between droplets and thermal fluctuations in external field},
+ volume = {22},
+ issn = {0305-4470},
+ url = {http://stacks.iop.org/0305-4470/22/i=17/a=006},
+ doi = {10.1088/0305-4470/22/17/006},
+ abstract = {The authors extend the definition of droplets in Ising and Potts models to the case of an external field different from zero. They also find exact relations between thermal properties and connectivity properties which show why, in mean field, the mean cluster size does not diverge as the susceptibility when the critical temperature is approached from below.},
+ language = {en},
+ number = {17},
+ urldate = {2018-04-04},
+ journal = {Journal of Physics A: Mathematical and General},
+ author = {Coniglio, A. and Liberto, F. de and Monroy, G. and Peruggi, F.},
+ year = {1989},
+ keywords = {cluster-algorithm},
+ pages = {L837},
+ file = {IOP Full Text PDF:/home/pants/.zotero/data/storage/2MABVQ35/Coniglio et al. - 1989 - Exact relations between droplets and thermal fluct.pdf:application/pdf}
}
-@article{coniglio1989exact,
- title={Exact relations between droplets and thermal fluctuations in external field},
- author={Coniglio, A and de Liberto, F and Monroy, G and Peruggi, F},
- journal={Journal of Physics A: Mathematical and General},
- volume={22},
- number={17},
- pages={L837},
- year={1989},
- publisher={IOP Publishing}
+@article{wolff_comparison_1989,
+ title = {Comparison between cluster {Monte} {Carlo} algorithms in the {Ising} model},
+ volume = {228},
+ issn = {0370-2693},
+ url = {http://www.sciencedirect.com/science/article/pii/0370269389915633},
+ doi = {10.1016/0370-2693(89)91563-3},
+ abstract = {We report autocorrelation times for the Swendsen-Wang algorithm and for a recently proposed single cluster variant in the 2D and 3D Ising models at criticality. The new algorithm decorrelates faster in all cases and gains about an order of magnitude on a 643 lattice. Critical slowing down is practically negligible and possibly completely absent in three dimensions. Results on static properties of the 3D model are consistent with published data.},
+ number = {3},
+ urldate = {2018-04-04},
+ journal = {Physics Letters B},
+ author = {Wolff, Ulli},
+ month = sep,
+ year = {1989},
+ pages = {379--382},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/F8LMC2TH/Wolff - 1989 - Comparison between cluster Monte Carlo algorithms .pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/I8GWCEUL/0370269389915633.html:text/html}
}
-@article{destri1992swendsen,
- title={Swendsen-Wang Monte Carlo study of the Ising model with external field},
- author={Destri, C and Di Renzo, F and Onofri, E and Rossi, P and Tecchiolli, GP},
- journal={Physics Letters B},
- volume={278},
- number={3},
- pages={311--316},
- year={1992},
- publisher={Elsevier}
+@article{alexandrowicz_swendsen-wang_1989,
+ title = {Swendsen-{Wang} simulation of {Ising} spins and a precise definition of critical clusters},
+ volume = {160},
+ issn = {0378-4371},
+ url = {http://www.sciencedirect.com/science/article/pii/0378437189904457},
+ doi = {10.1016/0378-4371(89)90445-7},
+ abstract = {A recent ultrafast simulation of Ising spins (σi = ± 1) utilizes a random + to − flip-over of duly defined decoupled blocks of spins. We show that the random dynamics alone suffices to prove the correspondence of the blocks with “critical clusters” describing thermal (magnetic) fluctuation. (The precise requirement is σiσi = 1, for a pair of spins inside a block, and 〈σiσj〉 = 0, for a σi inside and σj, outside.) The present approach helps to extend the study of critical clusters, and also ultrafast simulation, to the case of nonzero magnetization. Finite critical clusters constitute always a ± symmetric set and are very different (much smaller) than continuous domains of similarly oriented spins.},
+ number = {3},
+ urldate = {2018-04-04},
+ journal = {Physica A: Statistical Mechanics and its Applications},
+ author = {Alexandrowicz, Z.},
+ month = oct,
+ year = {1989},
+ pages = {310--320},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/TCTPEINC/Alexandrowicz - 1989 - Swendsen-Wang simulation of Ising spins and a prec.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/ZYI82U3R/0378437189904457.html:text/html}
}
-@article{dotsenko1991cluster,
- title={Cluster Monte Carlo algorithms for random Ising models},
- author={Dotsenko, Vl S and Selke, W and Talapov, AL},
- journal={Physica A: Statistical Mechanics and its Applications},
- volume={170},
- number={2},
- pages={278--281},
- year={1991},
- publisher={Elsevier}
+@article{wolff_collective_1989,
+ title = {Collective {Monte} {Carlo} {Updating} for {Spin} {Systems}},
+ volume = {62},
+ url = {https://link.aps.org/doi/10.1103/PhysRevLett.62.361},
+ doi = {10.1103/PhysRevLett.62.361},
+ abstract = {A Monte Carlo algorithm is presented that updates large clusters of spins simultaneously in systems at and near criticality. We demonstrate its efficiency in the two-dimensional O(n) σ models for n=1 (Ising) and n=2 (x−y) at their critical temperatures, and for n=3 (Heisenberg) with correlation lengths around 10 and 20. On lattices up to 1282 no sign of critical slowing down is visible with autocorrelation times of 1-2 steps per spin for estimators of long-range quantities.},
+ number = {4},
+ urldate = {2018-04-04},
+ journal = {Physical Review Letters},
+ author = {Wolff, Ulli},
+ month = jan,
+ year = {1989},
+ keywords = {monte-carlo, n-component, cluster-algorithm},
+ pages = {361--364},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/P4IULUXD/PhysRevLett.62.html:text/html;Wolff - 1989 - Collective Monte Carlo Updating for Spin Systems.pdf:/home/pants/.zotero/data/storage/ANYNLLMY/Wolff - 1989 - Collective Monte Carlo Updating for Spin Systems.pdf:application/pdf}
}
-@article{du2006dynamic,
- title={Dynamic critical exponents for Swendsen--Wang and Wolff algorithms obtained by a nonequilibrium relaxation method},
- author={Du, Jianqing and Zheng, Bo and Wang, Jian-Sheng},
- journal={Journal of Statistical Mechanics: Theory and Experiment},
- volume={2006},
- number={05},
- pages={P05004},
- year={2006},
- publisher={IOP Publishing}
+@article{destri_swendsen-wang_1992,
+ title = {Swendsen-{Wang} {Monte} {Carlo} study of the {Ising} model with external field},
+ volume = {278},
+ issn = {0370-2693},
+ url = {http://www.sciencedirect.com/science/article/pii/037026939290199E},
+ doi = {10.1016/0370-2693(92)90199-E},
+ abstract = {We present a Monte Carlo study of the scaling limit of the two-dimensional Ising model with external field. While no evidence is found for the E8 mass spectrum, we observe a very good agreement of our numerical data with the theoretical predictions for the magnetization and the correlation length.},
+ number = {3},
+ urldate = {2018-04-04},
+ journal = {Physics Letters B},
+ author = {Destri, C. and Di Renzo, F. and Onofri, E. and Rossi, P. and Tecchiolli, G. P.},
+ month = mar,
+ year = {1992},
+ pages = {311--316},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/UWFALIWT/Destri et al. - 1992 - Swendsen-Wang Monte Carlo study of the Ising model.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/3GWTKW64/037026939290199E.html:text/html}
}
-@article{geyer1992practical,
- title={Practical markov chain monte carlo},
- author={Geyer, Charles J},
- journal={Statistical science},
- pages={473--483},
- year={1992},
- publisher={JSTOR}
+@article{lauwers_critical_1989,
+ title = {The critical 2D {Ising} model in a magnetic field. {A} {Monte} {Carlo} study using a {Swendesen}-{Wang} algorithm},
+ volume = {233},
+ issn = {0370-2693},
+ url = {http://www.sciencedirect.com/science/article/pii/0370269389906412},
+ doi = {10.1016/0370-2693(89)90641-2},
+ abstract = {We determine numerically the spin-spin correlation function in the scaling limit. These data are useful in order to check regularization procedures à la Dotsenko, based on conformal theory, of perturbation series expansions.},
+ number = {1},
+ urldate = {2018-04-04},
+ journal = {Physics Letters B},
+ author = {Lauwers, P. G. and Rittenberg, V.},
+ month = dec,
+ year = {1989},
+ pages = {197--200},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/WBG7GBZA/Lauwers and Rittenberg - 1989 - The critical 2D Ising model in a magnetic field. A.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/I73J45VE/0370269389906412.html:text/html}
}
-@article{hastings1970monte,
- title={Monte Carlo sampling methods using Markov chains and their applications},
- author={Hastings, W Keith},
- journal={Biometrika},
- volume={57},
- number={1},
- pages={97--109},
- year={1970},
- publisher={Biometrika Trust}
+@article{ray_metastability_1990,
+ title = {Metastability and nucleation in {Ising} models with {Swendsen}-{Wang} dynamics},
+ volume = {167},
+ issn = {0378-4371},
+ url = {http://www.sciencedirect.com/science/article/pii/037843719090276X},
+ doi = {10.1016/0378-4371(90)90276-X},
+ abstract = {The cluster numbers of stable phase droplets in the metastable state and nucleation rates for the three-dimensional Ising model with Swendsen-Wang dynamics are measured for T = 0.59 Tc and compared with previous Metropolis results. Both dynamics appear to give the same metastable properties. When the external field is small the results agree with classical nucleation theory. No evidence is found for the existence of a spinodal line.},
+ number = {3},
+ urldate = {2018-04-04},
+ journal = {Physica A: Statistical Mechanics and its Applications},
+ author = {Ray, T. S. and Wang, Jian-Sheng},
+ month = sep,
+ year = {1990},
+ keywords = {monte-carlo, swendsen-wang, metastable, cluster-algorithm},
+ pages = {580--588},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/HD9XTNQ3/Ray and Wang - 1990 - Metastability and nucleation in Ising models with .pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/L7RFGT7S/037843719090276X.html:text/html}
}
-@article{janke1998nonlocal,
- title={Nonlocal Monte Carlo algorithms for statistical physics applications},
- author={Janke, Wolfhard},
- journal={Mathematics and computers in simulation},
- volume={47},
- number={2},
- pages={329--346},
- year={1998},
- publisher={Elsevier}
+@article{wolff_critical_1990,
+ title = {Critical slowing down},
+ volume = {17},
+ issn = {0920-5632},
+ url = {http://www.sciencedirect.com/science/article/pii/092056329090224I},
+ doi = {10.1016/0920-5632(90)90224-I},
+ abstract = {The problem of critical slowing down in Monte Carlo simulations and some methods to alleviate or overcome it are reviewed: overrelaxation, multigrid and cluster algorithms.},
+ urldate = {2018-04-04},
+ journal = {Nuclear Physics B - Proceedings Supplements},
+ author = {Wolff, Ulli},
+ month = sep,
+ year = {1990},
+ pages = {93--102},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/DMY36YUQ/Wolff - 1990 - Critical slowing down.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/Y92EI3HC/092056329090224I.html:text/html}
}
-@article{jose1977renormalization,
- title={Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model},
- author={Jos{\'e}, Jorge V and Kadanoff, Leo P and Kirkpatrick, Scott and Nelson, David R},
- journal={Physical Review B},
- volume={16},
- number={3},
- pages={1217},
- year={1977},
- publisher={APS}
+@article{geyer_practical_1992,
+ title = {Practical {Markov} {Chain} {Monte} {Carlo}},
+ volume = {7},
+ issn = {0883-4237},
+ url = {http://www.jstor.org/stable/2246094},
+ abstract = {Markov chain Monte Carlo using the Metropolis-Hastings algorithm is a general method for the simulation of stochastic processes having probability densities known up to a constant of proportionality. Despite recent advances in its theory, the practice has remained controversial. This article makes the case for basing all inference on one long run of the Markov chain and estimating the Monte Carlo error by standard nonparametric methods well-known in the time-series and operations research literature. In passing it touches on the Kipnis-Varadhan central limit theorem for reversible Markov chains, on some new variance estimators, on judging the relative efficiency of competing Monte Carlo schemes, on methods for constructing more rapidly mixing Markov chains and on diagnostics for Markov chain Monte Carlo.},
+ number = {4},
+ urldate = {2018-04-04},
+ journal = {Statistical Science},
+ author = {Geyer, Charles J.},
+ year = {1992},
+ pages = {473--483},
+ file = {Geyer - 1992 - Practical Markov Chain Monte Carlo.pdf:/home/pants/.zotero/data/storage/UAU5QJNP/Geyer - 1992 - Practical Markov Chain Monte Carlo.pdf:application/pdf}
}
-@article{lauwers1989critical,
- title={The critical 2D Ising model in a magnetic field. A Monte Carlo study using a Swendesen-Wang algorithm},
- author={Lauwers, Paul G and Rittenberg, Vladimir},
- journal={Physics Letters B},
- volume={233},
- number={1-2},
- pages={197--200},
- year={1989},
- publisher={Elsevier}
+@article{janke_nonlocal_1998,
+ title = {Nonlocal {Monte} {Carlo} algorithms for statistical physics applications},
+ volume = {47},
+ issn = {0378-4754},
+ url = {http://www.sciencedirect.com/science/article/pii/S0378475498001098},
+ doi = {10.1016/S0378-4754(98)00109-8},
+ abstract = {After a brief general overview of Monte Carlo computer simulations in statistical physics, special emphasis is placed on applications to phase transitions and critical phenomena. Here, standard simulations employing local update algorithms are severely hampered by the problem of critical slowing down, that is by strong correlations between successively generated data. It is shown that this problem can be greatly reduced by using nonlocal update techniques such as cluster and multigrid algorithms. The general ideas are illustrated for simple lattice spin models and Euclidean path integrals.},
+ number = {2},
+ urldate = {2018-04-05},
+ journal = {Mathematics and Computers in Simulation},
+ author = {Janke, Wolfhard},
+ month = aug,
+ year = {1998},
+ keywords = {Critical phenomena, Cluster algorithms, Importance sampling, Monte Carlo simulations, Multigrid techniques, Phase transitions},
+ pages = {329--346},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/N94BCGSZ/Janke - 1998 - Nonlocal Monte Carlo algorithms for statistical ph.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/U22VXSJZ/S0378475498001098.html:text/html}
}
-@article{liu2014dynamic,
- title={Dynamic scaling at classical phase transitions approached through nonequilibrium quenching},
- author={Liu, Cheng-Wei and Polkovnikov, Anatoli and Sandvik, Anders W},
- journal={Physical Review B},
- volume={89},
- number={5},
- pages={054307},
- year={2014},
- publisher={APS}
+@article{coniglio_clusters_1980,
+ title = {Clusters and {Ising} critical droplets: a renormalisation group approach},
+ volume = {13},
+ issn = {0305-4470},
+ shorttitle = {Clusters and {Ising} critical droplets},
+ url = {http://stacks.iop.org/0305-4470/13/i=8/a=025},
+ doi = {10.1088/0305-4470/13/8/025},
+ abstract = {The Migdal-Kadanoff renormalisation group for two-dimensions is employed to obtain the global phase diagram for the site-bond correlated percolation problem. It is found that the Ising critical point (K=K c, H=O) is a percolation point for a range of bond probability rho B such that 1{\textgreater}or= rho B {\textgreater}or=1-e -2 Kc . In particular, as rho B approaches 1-e -2 Kc , the percolation clusters become less compact and coincide with the Ising critical droplets.},
+ language = {en},
+ number = {8},
+ urldate = {2018-04-05},
+ journal = {Journal of Physics A: Mathematical and General},
+ author = {Coniglio, A. and Klein, W.},
+ year = {1980},
+ pages = {2775},
+ file = {IOP Full Text PDF:/home/pants/.zotero/data/storage/XH5C8THH/Coniglio and Klein - 1980 - Clusters and Ising critical droplets a renormalis.pdf:application/pdf}
}
-@article{loos1969symmetric,
- title={Symmetric spaces},
- author={Loos, Ottmar},
- year={1969},
- publisher={Benjamin}
+@article{du_dynamic_2006,
+ title = {Dynamic critical exponents for {Swendsen}–{Wang} and {Wolff} algorithms obtained by a nonequilibrium relaxation method},
+ volume = {2006},
+ issn = {1742-5468},
+ url = {http://stacks.iop.org/1742-5468/2006/i=05/a=P05004},
+ doi = {10.1088/1742-5468/2006/05/P05004},
+ abstract = {Using a nonequilibrium relaxation method, we calculate the dynamic critical exponent z of the two-dimensional Ising model for the Swendsen–Wang and Wolff algorithms. We examine dynamic relaxation processes following a quench from a disordered or an ordered initial state to the critical temperature T c , and measure the exponential relaxation time of the system energy. For the Swendsen–Wang algorithm with an ordered or a disordered initial state, and for the Wolff algorithm with an ordered initial state, the exponential relaxation time fits well to a logarithmic size dependence up to a lattice size L = 8192. For the Wolff algorithm with a disordered initial state, we obtain an effective dynamic exponent z exp = 1.19(2) up to L = 2048. For comparison, we also compute the effective dynamic exponents through the integrated correlation times. In addition, an exact result of the Swendsen–Wang dynamic spectrum of a one-dimensional Ising chain is derived.},
+ language = {en},
+ number = {05},
+ urldate = {2018-04-05},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ author = {Du, Jianqing and Zheng, Bo and Wang, Jian-Sheng},
+ year = {2006},
+ pages = {P05004},
+ file = {IOP Full Text PDF:/home/pants/.zotero/data/storage/NXZR9FP6/Du et al. - 2006 - Dynamic critical exponents for Swendsen–Wang and W.pdf:application/pdf}
}
-@article{metropolis1953equation,
- title={Equation of state calculations by fast computing machines},
- author={Metropolis, Nicholas and Rosenbluth, Arianna W and Rosenbluth, Marshall N and Teller, Augusta H and Teller, Edward},
- journal={The journal of chemical physics},
- volume={21},
- number={6},
- pages={1087--1092},
- year={1953},
- publisher={AIP}
+@article{liu_dynamic_2014,
+ title = {Dynamic scaling at classical phase transitions approached through nonequilibrium quenching},
+ volume = {89},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.89.054307},
+ doi = {10.1103/PhysRevB.89.054307},
+ abstract = {We use Monte Carlo simulations to demonstrate generic scaling aspects of classical phase transitions approached through a quench (or annealing) protocol where the temperature changes as a function of time with velocity v. Using a generalized Kibble-Zurek ansatz, we demonstrate dynamic scaling for different types of stochastic dynamics (Metropolis, Swendsen-Wang, and Wolff) on Ising models in two and higher dimensions. We show that there are dual scaling functions governing the dynamic scaling, which together describe the scaling behavior in the entire velocity range v∈[0,∞). These functions have asymptotics corresponding to the adiabatic and diabatic limits, and close to these limits they are perturbative in v and 1/v, respectively. Away from their perturbative domains, both functions cross over into the same universal power-law scaling form governed by the static and dynamic critical exponents (as well as an exponent characterizing the quench protocol). As a by-product of the scaling studies, we obtain high-precision estimates of the dynamic exponent z for the two-dimensional Ising model subject to the three variants of Monte Carlo dynamics: for single-spin Metropolis updates zM=2.1767(5), for Swendsen-Wang multicluster updates zSW=0.297(3), and for Wolff single-cluster updates zW=0.30(2). For Wolff dynamics, we find an interesting behavior with a nonanalytic breakdown of the quasiadiabatic and diabatic scalings, instead of the generic smooth crossover described by a power law. We interpret this disconnect between the two scaling regimes as a dynamic phase transition of the Wolff algorithm, caused by an effective sudden loss of ergodicity at high velocity.},
+ number = {5},
+ urldate = {2018-04-05},
+ journal = {Physical Review B},
+ author = {Liu, Cheng-Wei and Polkovnikov, Anatoli and Sandvik, Anders W.},
+ month = feb,
+ year = {2014},
+ pages = {054307},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/HUBB74C4/PhysRevB.89.html:text/html;Liu et al. - 2014 - Dynamic scaling at classical phase transitions app.pdf:/home/pants/.zotero/data/storage/C8PPDNT5/Liu et al. - 2014 - Dynamic scaling at classical phase transitions app.pdf:application/pdf}
}
-@article{ray1990metastability,
- title={Metastability and nucleation in Ising models with Swendsen-Wang dynamics},
- author={Ray, TS and Wang, Jian-Sheng},
- journal={Physica A: Statistical Mechanics and its Applications},
- volume={167},
- number={3},
- pages={580--588},
- year={1990},
- publisher={Elsevier}
+@article{wang_cluster_1990,
+ title = {Cluster {Monte} {Carlo} algorithms},
+ volume = {167},
+ issn = {0378-4371},
+ url = {http://www.sciencedirect.com/science/article/pii/037843719090275W},
+ doi = {10.1016/0378-4371(90)90275-W},
+ abstract = {The Swendsen-Wang and Wolff Monte Carlo algorithms are described in some detail, using the Potts model as an example. Various generalizations are then reviewed and some applications are discussed. Two complete Fortran programs for the algorithms are provided.},
+ number = {3},
+ urldate = {2018-04-05},
+ journal = {Physica A: Statistical Mechanics and its Applications},
+ author = {Wang, Jian-Sheng and Swendsen, Robert H.},
+ month = sep,
+ year = {1990},
+ pages = {565--579},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/K6Q27XAZ/Wang and Swendsen - 1990 - Cluster Monte Carlo algorithms.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/QTTQLTT3/037843719090275W.html:text/html}
}
-@article{swendsen1987nonuniversal,
- title={Nonuniversal critical dynamics in Monte Carlo simulations},
- author={Swendsen, Robert H and Wang, Jian-Sheng},
- journal={Physical review letters},
- volume={58},
- number={2},
- pages={86},
- year={1987},
- publisher={APS}
+@article{swendsen_nonuniversal_1987,
+ title = {Nonuniversal critical dynamics in {Monte} {Carlo} simulations},
+ volume = {58},
+ url = {https://link.aps.org/doi/10.1103/PhysRevLett.58.86},
+ doi = {10.1103/PhysRevLett.58.86},
+ abstract = {A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.},
+ number = {2},
+ urldate = {2018-04-05},
+ journal = {Physical Review Letters},
+ author = {Swendsen, Robert H. and Wang, Jian-Sheng},
+ month = jan,
+ year = {1987},
+ pages = {86--88},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/V6CTHN82/PhysRevLett.58.html:text/html;Swendsen and Wang - 1987 - Nonuniversal critical dynamics in Monte Carlo simu.pdf:/home/pants/.zotero/data/storage/9H7NG55Z/Swendsen and Wang - 1987 - Nonuniversal critical dynamics in Monte Carlo simu.pdf:application/pdf}
}
-@article{wang1989clusters,
- title={Clusters in the three-dimensional Ising model with a magnetic field},
- author={Wang, Jian-Sheng},
- journal={Physica A: Statistical Mechanics and its Applications},
- volume={161},
- number={2},
- pages={249--268},
- year={1989},
- publisher={Elsevier}
+@article{baillie_comparison_1991,
+ title = {Comparison of cluster algorithms for two-dimensional {Potts} models},
+ volume = {43},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.43.10617},
+ doi = {10.1103/PhysRevB.43.10617},
+ abstract = {We have measured the dynamical critical exponent z for the Swendsen-Wang and the Wolff cluster update algorithms, as well as a number of variants of these algorithms, for the q=2 and q=3 Potts models in two dimensions. We find that although the autocorrelation times differ considerably between algorithms, the critical exponents are the same. For q=2, we find that although the data are better fitted by a logarithmic increase in the autocorrelation time with lattice size, they are also consistent with a power law with exponent z≊0.25, especially if there are non-negligible corrections to scaling.},
+ number = {13},
+ urldate = {2018-04-05},
+ journal = {Physical Review B},
+ author = {Baillie, Clive F. and Coddington, Paul D.},
+ month = may,
+ year = {1991},
+ pages = {10617--10621},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/UGDFYRN9/PhysRevB.43.html:text/html;Baillie and Coddington - 1991 - Comparison of cluster algorithms for two-dimension.pdf:/home/pants/.zotero/data/storage/QHJS4V7S/Baillie and Coddington - 1991 - Comparison of cluster algorithms for two-dimension.pdf:application/pdf}
}
-@article{wang1990cluster,
- title={Cluster monte carlo algorithms},
- author={Wang, Jian-Sheng and Swendsen, Robert H},
- journal={Physica A: Statistical Mechanics and its Applications},
- volume={167},
- number={3},
- pages={565--579},
- year={1990},
- publisher={Elsevier}
+@article{wang_clusters_1989,
+ title = {Clusters in the three-dimensional {Ising} model with a magnetic field},
+ volume = {161},
+ issn = {0378-4371},
+ url = {http://www.sciencedirect.com/science/article/pii/0378437189904688},
+ doi = {10.1016/0378-4371(89)90468-8},
+ abstract = {We study the clusters generated in the Swendsen-Wang algorithm in a magnetic field. It is shown that the number of clusters is related to that of Coniglio and Klein by simple factors. With this definition of clusters, infinite size appears whenever the system has a nonzero magnetization. Scaling behavior of the number of clusters near the critical point is confirmed. The number of clusters away from the critical point for large cluster size s is consistent with ln n ≌ {\textbar}h{\textbar}s − Γ s2 3 on the low temperature side of the Coniglio-Klein cluster percolation transition line, and is consistent with ln n≌−({\textbar}h{\textbar} + c)s on the high temperature side. We also argue that this transition line is given by h = ±h̃(T)×(T-Tc)1 near Tc.},
+ number = {2},
+ urldate = {2018-04-05},
+ journal = {Physica A: Statistical Mechanics and its Applications},
+ author = {Wang, Jian-Sheng},
+ month = nov,
+ year = {1989},
+ keywords = {ising},
+ pages = {249--268},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/FQJTTTTH/Wang - 1989 - Clusters in the three-dimensional Ising model with.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/HRKDAKDL/0378437189904688.html:text/html}
}
-@article{wolff1989collective,
- title={Collective Monte Carlo updating for spin systems},
- author={Wolff, Ulli},
- journal={Physical Review Letters},
- volume={62},
- number={4},
- pages={361},
- year={1989},
- publisher={APS}
+@article{loos_symmetric_1969,
+ title = {Symmetric spaces},
+ author = {Loos, Ottmar},
+ year = {1969},
+ file = {Loos - 1969 - Symmetric spaces I.pdf:/home/pants/.zotero/data/storage/GX35WTRQ/Loos - 1969 - Symmetric spaces I.pdf:application/pdf;Loos - 1969 - Symmetric spaces II.pdf:/home/pants/.zotero/data/storage/AJFD39RN/Loos - 1969 - Symmetric spaces II.pdf:application/pdf}
}
-
@article{dimitrovic_finite-size_1991,
title = {Finite-size effects, goldstone bosons and critical exponents in the d = 3 {Heisenberg} model},
volume = {350},
@@ -237,6 +333,7 @@
issn = {0370-1573},
url = {http://www.sciencedirect.com/science/article/pii/0370157379900607},
doi = {10.1016/0370-1573(79)90060-7},
+ abstract = {For beginners: This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied with probability p or empty with probability 1−p. An s-cluster is a group of s occupied sites connected by nearest-neighbor distances; the number of empty nearest neighbors of cluster sites is the perimeter t. For p above pc also one infinite cluster percolates through the lattice. How do the properties of s-clusters depend on s, and how do they feel the influence of the phase transition at p = pc? The answers to these questions are given by various methods (in particular computer simulations) and are interpreted by the so-called scaling theory of phase transitions. The results presented here suggest a qualitative difference of cluster structures above and below pc: Above p c some cluster properties suggest the existence of a cluster surface varying as s23 in three dimensions, but below pc these “surface” contributions are proportional to s. We suggest therefore that very large clusters above pc (but not at and below pc) behave like large clusters of Swiss cheese: Inspite of many internal holes the overall cluster shape is roughly spherical, similar to raindrops. For experts: Scaling theory suggests for large clusters near the percolation threshold pc that the average cluster numbers n s vary as s−τƒ(z), with z ≡ (p − pc)sσ. Analogously the average cluster perimeter is ts = s · (1 − p)/p + sσ · ψ1(z), the average cluster radius Rs varies as sσv · R1(z), and the density profile Ds(r), which depends also on the distance r from the cluster center, varies as s−1δ· D̃1(rs−σv, z). These assumptions relate the seven critical exponents α,β,γ,δ,v,σ,τ in d dimensions through the well-known five scaling laws 2 − α = γ + 2β = βδ + β = dv = β + 1σ = (τ − 1)/σ, leaving only two exponents as independent variables to be fitted by “experiment” and not predicted by scaling theory. For the lattice “animals”, i.e. the number gst of geometrically different cluster configurations, a modified scaling assumption is derived: gstsst1/(s + t)s + 1 ∝ s−τ−12 · ƒ(z), with z ∝ (ac − t/s)sσ and ac = (1 − pc)/pc. All these expressions are variants of the general scaling idea for second-order phase transitions that a function g(x,y) of two critical variables takes the homogeneous form xcG(x/yb) near the critical point, with two free exponents b and c and a scaling function G of a single variable. These assumptions, which may be regarded as generalizations of the Fisher droplet model, are tested “experimentally” by Monte Carlo simulation, series expansion, renormalization group technique, and exact inequalities. In particular, detailed Monte Carlo evidence of Hoshen et al. and Leath and Reich is presented for the scaling of cluster numbers in two and three dimensions. If the cluster size s goes to infinity at fixed concentration p, not necessarily close to pc, three additional exponents ξ, θ, ϱ are defined by: cluster numbers ∝ s−θ exp(−const · sξ) and cluster radii ∝ sϱ. These exponents are different on both sides of the phase transition; for example ξ(p {\textless} pc) = 1 and ξ(p {\textgreater} pc) = 1 − 1/d was found from inequalities, series and Monte Carlo data. The behavior of θ and of ϱ(p {\textless} pc) remains to be explained by scaling theory. This article does not cover experimental applications, correlation functions and “classical” (mean field, Bethe lattice, effective medium) theories. For the reader to whom this abstract is too short and the whole article is too long we recommend sections 1 and 3.},
number = {1},
urldate = {2018-04-20},
journal = {Physics Reports},
@@ -247,24 +344,102 @@
file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/KTNA6MWQ/Stauffer - 1979 - Scaling theory of percolation clusters.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/FYC4TRJZ/0370157379900607.html:text/html}
}
-@article{wolff1989comparison,
- title={Comparison between cluster Monte Carlo algorithms in the Ising model},
- author={Wolff, Ulli},
- journal={Physics Letters B},
- volume={228},
- number={3},
- pages={379--382},
- year={1989},
- publisher={Elsevier}
+@article{bruce_coupled_1975,
+ title = {Coupled order parameters, symmetry-breaking irrelevant scaling fields, and tetracritical points},
+ volume = {11},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.11.478},
+ doi = {10.1103/PhysRevB.11.478},
+ abstract = {The phase diagrams of systems described by a Hamiltonian containing an anisotropic quadratic term of the form 12gΣα=1ncα∫→xS2α(→x), and a cubic anisotropic term νΣα=1n∫→xS4α(→x), are studied using mean-field theory, scaling theory, and expansions in ε(=4−d) and 1n. Here, Sα(→x) (a=1, …, n) is a local n-component ordering variable. Systems to which the analysis is applicable include perovskite crystals, stressed along the [100] direction (n=3), anisotropic antiferromagnets in a uniform field, uniaxially anisotropic ferromagnets, ferroelectric ferromagnets and crystalline 4He(n=2). When g=0 and T=Tc these systems undergo a phase transition that may be associated (for small n) with the Heisenberg fixed point (ν∗=0) or (otherwise) with the cubic fixed point (ν∗{\textgreater}0) of the renormalization group. Although ν is an "irrelevant variable" in the former case, it is found to have important effects. For ν{\textless}0, the point g=0, T=Tc represents a bicritical point in the g−T plane, at which a first-order "spin-flop" line (separating two distinct ordered phases) meets two critical lines. For ν{\textgreater}0, the "flop" line splits into two critical lines, associated with transitions between each of the ordered phases and a new intermediate phase; the point T=Tc, g=0 is then tetracritical. The shape of the boundary of the intermediate phase is given by T=T2(g, ν) with [Tc−T2(g, ν)]∼(gν)1ψ2, where ψ2=φg−φν (if the tetracritical point is Heisenberg-like) or ψ2=φCg (if it is cubic). Here, φg, φν, and φCg are appropriate crossover exponents associated with the two symmetry-breaking perturbations. The phase diagram of [111] -stressed perovskites is also discussed and the experimental situation briefly reviewed.},
+ number = {1},
+ urldate = {2018-04-24},
+ journal = {Physical Review B},
+ author = {Bruce, Alastair D. and Aharony, Amnon},
+ month = jan,
+ year = {1975},
+ pages = {478--499},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/9MKKQASE/PhysRevB.11.html:text/html;Bruce and Aharony - 1975 - Coupled order parameters, symmetry-breaking irrele.pdf:/home/pants/.zotero/data/storage/IMSQ5NFW/Bruce and Aharony - 1975 - Coupled order parameters, symmetry-breaking irrele.pdf:application/pdf}
+}
+
+@article{manuel_carmona_$n$-component_2000,
+ title = {\${N}\$-component {Ginzburg}-{Landau} {Hamiltonian} with cubic anisotropy: {A} six-loop study},
+ volume = {61},
+ shorttitle = {\${N}\$-component {Ginzburg}-{Landau} {Hamiltonian} with cubic anisotropy},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.61.15136},
+ doi = {10.1103/PhysRevB.61.15136},
+ abstract = {We consider the Ginzburg-Landau Hamiltonian with a cubic-symmetric quartic interaction and compute the renormalization-group functions to six-loop order in d=3. We analyze the stability of the fixed points using a Borel transformation and a conformal mapping that takes into account the singularities of the Borel transform. We find that the cubic fixed point is stable for N{\textgreater}Nc, Nc=2.89(4). Therefore, the critical properties of cubic ferromagnets are not described by the Heisenberg isotropic Hamiltonian, but instead by the cubic model at the cubic fixed point. For N=3, the critical exponents at the cubic and symmetric fixed points differ very little (less than the precision of our results, which is ≲1\% in the case of γ and ν). Moreover, the irrelevant interaction bringing from the symmetric to the cubic fixed point gives rise to slowly decaying scaling corrections with exponent ω2=0.010(4). For N=2, the isotropic fixed point is stable and the cubic interaction induces scaling corrections with exponent ω2=0.103(8). These conclusions are confirmed by a similar analysis of the five-loop ε expansion. A constrained analysis, which takes into account that Nc=2 in two dimensions, gives Nc=2.87(5).},
+ number = {22},
+ urldate = {2018-04-24},
+ journal = {Physical Review B},
+ author = {Manuel Carmona, José and Pelissetto, Andrea and Vicari, Ettore},
+ month = jun,
+ year = {2000},
+ pages = {15136--15151},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/FMBHR6TG/PhysRevB.61.html:text/html;Full Text PDF:/home/pants/.zotero/data/storage/EZGKPAUL/Manuel Carmona et al. - 2000 - \$N\$-component Ginzburg-Landau Hamiltonian with cub.pdf:application/pdf}
+}
+
+@article{evertz_stochastic_1991,
+ title = {Stochastic cluster algorithms for discrete gaussian ({SOS}) models},
+ volume = {254},
+ issn = {0370-2693},
+ url = {http://www.sciencedirect.com/science/article/pii/037026939190418P},
+ doi = {10.1016/0370-2693(91)90418-P},
+ abstract = {We present new Monte Carlo cluster algorithms which eliminate critical slowing down in the simulation of solid-on-solid models. In this letter we focus on the two-dimensional discrete gaussian model. The algorithms are based on reflecting the integer valued spin variables with respect to appropriately chosen reflection planes. The proper choice of the reflection plane turns out to be crucial in order to obtain a small dynamical exponent z. Actually, the successful versions of our algorithm are a mixture of two different procedures for choosing the reflection plane, one of them ergodic but slow, the other one non-ergodic and also slow when combined with a Metropolis algorithm.},
+ number = {1},
+ urldate = {2018-04-25},
+ journal = {Physics Letters B},
+ author = {Evertz, Hans Gerd and Hasenbusch, Martin and Marcu, Mihail and Pinn, Klaus and Solomon, Sorin},
+ month = jan,
+ year = {1991},
+ pages = {185--191},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/I8XRWWAF/Evertz et al. - 1991 - Stochastic cluster algorithms for discrete gaussia.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/2HIT7KGW/037026939190418P.html:text/html}
+}
+
+@article{blankschtein_fluctuation-induced_1982,
+ title = {Fluctuation-induced first-order transitions and symmetry-breaking fields: {The} \$n=3\$-component cubic model},
+ volume = {25},
+ shorttitle = {Fluctuation-induced first-order transitions and symmetry-breaking fields},
+ url = {https://link.aps.org/doi/10.1103/PhysRevB.25.6939},
+ doi = {10.1103/PhysRevB.25.6939},
+ abstract = {The effects of an off-diagonal quadratic symmetry-breaking field, g, on a three-component (n=3) cubic model with no accessible fixed points are studied. It is shown that this perturbation induces a crossover from first-order to continuous transition. Depending upon the initial values of the parameters characterizing the model, two types of (g,T) phase diagrams are possible, both of which are rather complex, exhibiting tricritical, critical, and critical end points. The (g,T) phase diagrams are studied using large-g expansion, mean-field theory, and renormalization-group analysis. A universal amplitude ratio associated with the critical end points is calculated to leading (zeroth) order in ε=4−d. The phase diagrams are predicted to be realizable in certain n=3 cubic crystals undergoing structural phase transitions, such as BaTiO3, RbCaF3, and KMnF3.},
+ number = {11},
+ urldate = {2018-04-25},
+ journal = {Physical Review B},
+ author = {Blankschtein, Daniel and Mukamel, David},
+ month = jun,
+ year = {1982},
+ pages = {6939--6951},
+ file = {APS Snapshot:/home/pants/.zotero/data/storage/KE9V2NH2/PhysRevB.25.html:text/html;Blankschtein and Mukamel - 1982 - Fluctuation-induced first-order transitions and sy.pdf:/home/pants/.zotero/data/storage/UXSNRJXE/Blankschtein and Mukamel - 1982 - Fluctuation-induced first-order transitions and sy.pdf:application/pdf}
}
-@article{wolff1990critical,
- title={Critical slowing down},
- author={Wolff, Ulli},
- journal={Nuclear Physics B-Proceedings Supplements},
- volume={17},
- pages={93--102},
- year={1990},
- publisher={Elsevier}
+@article{caracciolo_wolff-type_1993,
+ title = {Wolff-type embedding algorithms for general nonlinear σ-models},
+ volume = {403},
+ issn = {0550-3213},
+ url = {http://www.sciencedirect.com/science/article/pii/055032139390044P},
+ doi = {10.1016/0550-3213(93)90044-P},
+ abstract = {We study a class of Monte Carlo algorithms for the nonlinear σ-model, based on A Wolff-type embedding of Ising spins into the target manifold M. We argue heuristically that, at least for an asymptotically free model, such an algorithm can have a dynamic critical exponent z « 2 only if the embedding is based on an (involutive) isometry of M whose fixed-point manifold has codimension 1. Such an isometry exist only if the manifold is a discrete quotient of a product of spheres. Numerical simulations of the idealized codimension-2 algorithm for the two-dimensional O(4)-symmetric σ-model yield zint,M2 = 1.5±0.5 (sujective 68\% confidence interval), in agreement with our heuristic argument.},
+ number = {1},
+ urldate = {2018-04-25},
+ journal = {Nuclear Physics B},
+ author = {Caracciolo, Sergio and Edwards, Robert G. and Pelissetto, Andrea and Sokal, Alan D.},
+ month = aug,
+ year = {1993},
+ pages = {475--541},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/WSZ7RUI5/Caracciolo et al. - 1993 - Wolff-type embedding algorithms for general nonlin.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/MEJNDW2B/055032139390044P.html:text/html}
}
+@article{caracciolo_generalized_1991,
+ title = {Generalized {Wolff}-type embedding algorithms for nonlinear σ-models},
+ volume = {20},
+ issn = {0920-5632},
+ url = {http://www.sciencedirect.com/science/article/pii/092056329190883G},
+ doi = {10.1016/0920-5632(91)90883-G},
+ abstract = {We study a class of Monte Carlo algorithms for the nonlinear σ-model, based on a Wolff-type embedding of Ising spins into the target manifold M. We argue heuristically that such an algorithm can have dynamic critical exponent z ⪡ 2 only if the embedding is based on an involutive isometry of M whose fixed-point manifold has codimension 1. Such an isometry exists only if the manifold is a product of spheres and discrete quotients of spheres. Numerical simulations of the codimension-2 algorithm for the two-dimensional O(4)-symmetric σ-model yield z = 1.5 ± 0.3, in agreement with our heuristic argument.},
+ urldate = {2018-04-25},
+ journal = {Nuclear Physics B - Proceedings Supplements},
+ author = {Caracciolo, Sergio and Edwards, Robert G. and Pelissetto, Andrea and Sokal, Alan D.},
+ month = may,
+ year = {1991},
+ pages = {72--75},
+ file = {ScienceDirect Full Text PDF:/home/pants/.zotero/data/storage/CADBLAPP/Caracciolo et al. - 1991 - Generalized Wolff-type embedding algorithms for no.pdf:application/pdf;ScienceDirect Snapshot:/home/pants/.zotero/data/storage/XW9KUK53/092056329190883G.html:text/html}
+} \ No newline at end of file
diff --git a/monte-carlo.pdf b/monte-carlo.pdf
index c8a15de..a8dcab7 100644
--- a/monte-carlo.pdf
+++ b/monte-carlo.pdf
Binary files differ
diff --git a/monte-carlo.tex b/monte-carlo.tex
index 96e5f91..222c5e8 100644
--- a/monte-carlo.tex
+++ b/monte-carlo.tex
@@ -115,23 +115,23 @@ of doing this, approximating thermodynamic quantities by sampling the
distribution of systems states. For a particular system, a Monte Carlo
algorithm is better the faster it arrives at a statistically independent
sample. This is typically a problem at critical points, where critical slowing
-down \cite{wolff1990critical} results in power-law divergences of any dynamics. Celebrated cluster
+down \cite{wolff_critical_1990} results in power-law divergences of any dynamics. Celebrated cluster
algorithms largely addressed this for many spin systems in the absence of
-external fields by using nonlocal updates \cite{janke1998nonlocal} whose clusters undergo a percolation
-transition at the critical point of the system \cite{coniglio1980clusters} and that in relatively small
-dynamic exponents \cite{wolff1989comparison,du2006dynamic,liu2014dynamic,wang1990cluster},
-including the Ising, $n$-component \cite{wolff1989collective}, and Potts
-\cite{swendsen1987nonuniversal,baillie1991comparison} models. These
+external fields by using nonlocal updates \cite{janke_nonlocal_1998} whose clusters undergo a percolation
+transition at the critical point of the system \cite{coniglio_clusters_1980} and that in relatively small
+dynamic exponents \cite{wolff_comparison_1989,du_dynamic_2006,liu_dynamic_2014,wang_cluster_1990},
+including the Ising, $\mathrm O(n)$ \cite{wolff_collective_1989}, and Potts
+\cite{swendsen_nonuniversal_1987,baillie_comparison_1991} models. These
algorithms rely on the natural symmetry of the systems in question under
global rotations, so the general application of external fields is not
trivial. Some
success has been made in extending these algorithms to systems in certain
external fields based on applying the ghost site representation
-\cite{coniglio1989exact} of certain
+\cite{coniglio_exact_1989} of certain
spin systems that returns global rotation invariance to spin Hamiltonians at
the cost of an extra degree of freedom, but these results only allow the application of a narrow
category of fields
-\cite{alexandrowicz1989swendsen,destri1992swendsen,lauwers1989critical,wang1989clusters}.
+\cite{alexandrowicz_swendsen-wang_1989,destri_swendsen-wang_1992,lauwers_critical_1989,wang_clusters_1989}.
We show that the scaling of correlation
time near the critical point of several models suggests that this approach is
a natural one, e.g., that it extends the celebrated scaling of dynamics in
@@ -147,10 +147,10 @@ set of states accessible by a spin, and $R$ is the \emph{symmetry group} of
$X$. The set $X$ must admit a measure $\mu$ that is invariant under the action of $R$, e.g., for any
$A\subseteq X$ and $r\in R$, $\mu(r\cdot A)=\mu(A)$. This trait is shared by
the counting measure on any discrete set, or by any group acting by isometries
-on a Riemannian manifold, such as $O(n)$ on $S^{n-1}$ in the $n$-component
-model. Finally, the subset of elements in $R$ of order two must act
+on a Riemannian manifold, such as $\mathrm O(n)$ on $S^{n-1}$ in the $\mathrm O(n)$
+model \cite{caracciolo_wolff-type_1993}. Finally, the subset of elements in $R$ of order two must act
transitively on $X$. This property, while apparently obscure, is shared by any
-symmetric space \cite{loos1969symmetric} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural
+symmetric space \cite{loos_symmetric_1969} or by any transitive, finitely generated isometry group. In fact, all the examples listed here have spins spaces with natural
metrics whose symmetry group is the set of isometries of the spin spaces.
We put one spin at each site of the lattice described by $G$, so that the
state of the entire spin system is described by elements $\vec s\in X\times\cdots\times
@@ -176,7 +176,7 @@ well for these cases, but we will drop the additional index notation for clarity
\hline\hline
Ising & $\{-1,1\}$ & $\Z/2\Z$ & $0\cdot
s\mapsto s$, $1\cdot s\mapsto -s$ & $st$ & $Hs$ \\
- $n$-component & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\
+ $\mathrm O(n)$ & $S^{n-1}$ & $\mathrm O(n)$ & $M\cdot s\mapsto Ms$ & $s^{\mathrm T}t$ & $H^{\mathrm T}s$\\
Potts & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
s=-m-s$ & $\delta(s,t)$ & $\sum_mH_m\delta(m,s)$\\
Clock & $\mathbb Z/q\mathbb Z$ & $D_n$ & $r_m\cdot s=m+s$, $s_m\cdot
@@ -189,7 +189,7 @@ well for these cases, but we will drop the additional index notation for clarity
their external fields are also given. Other fields are possible, of course:
for instance, some are interested in modulated fields $H\cos(2\pi k\theta(s))$ for
integer $k$ and $\theta(s)$ giving the angle of $s$ to some axis applied
- to $n$-component models \cite{jose1977renormalization}.}
+ to $\mathrm O(n)$ models \cite{jose_renormalization_1977}.}
\label{table:models}
\end{table*}
@@ -222,7 +222,7 @@ $\vec s'$ in the ensemble).
While any several related cluster algorithms can be described for this
system, we will focus on the Wolff algorithm in particular
-\cite{wolff1989collective}. We will first describe a generalized version of the celebrated Wolff algorithm
+\cite{wolff_collective_1989}. We will first describe a generalized version of the celebrated Wolff algorithm
in the standard case where $B(s)=0$. After reflecting on the technical
requirements of that algorithm, we will introduce a transformation to our
system and Hamiltonian that allows the same algorithm to be applied with
@@ -401,11 +401,11 @@ elements, performing the algorithm on the Ising model in a field is very
accurately described by simply adding an extra spin coupled to all others and
running the ordinary algorithm. The ghost spin version of the algorithm has
been applied by several researchers previously
-\cite{wang1989clusters,ray1990metastability,destri1992swendsen,lauwers1989critical}
+\cite{wang_clusters_1989,ray_metastability_1990,destri_swendsen-wang_1992,lauwers_critical_1989}
-\subsection{The $n$-component Model}
+\subsection{The $\mathrm O(n)$ Model}
-In the $n$-component model, spins are described by vectors on the $(n-1)$-sphere,
+In the $\mathrm O(n)$ model, spins are described by vectors on the $(n-1)$-sphere,
so that $X=S^{n-1}$. The symmetry group of this model is $O(n)$, $n\times n$
orthogonal matrices. The symmetry group acts on the spins by matrix
multiplication. The elements of $O(n)$ that are order two are reflections
@@ -460,7 +460,8 @@ changes so big that the whole system is always flipped, it is better to select
random reflections about integers close to the average state of the system.
Continuous roughening models---where the spin states are described by real
numbers and the symmetry group is $\mathrm E(1)$, the Euclidean group for
-one-dimensional space---are equally well described.
+one-dimensional space---are equally well described. A variant of the algorithm has been
+applied without a field before \cite{evertz_stochastic_1991}.
%\begin{figure}
@@ -494,8 +495,8 @@ If a given dynamics for a system at zero field results in scaling like
$t^{-z\nu}$, one should expect its natural extension in the presence of a
field to scale like $h^{-z\nu/\beta\delta}$. We measured the autocorrelation
time for the 2D square-lattice model at a variety of system sizes,
-temperatures, and fields using methods here
-\cite{geyer1992practical}. The resulting scaling behavior, plotted in
+temperatures, and fields $B(s)=hs/\beta$ using methods here
+\cite{geyer_practical_1992}. The resulting scaling behavior, plotted in
Fig.~\ref{fig:correlation_time-collapse}, is indeed consistent with the
zero-field scaling behavior.
@@ -554,7 +555,10 @@ should go as $(hL^{\beta\delta})^{2/\delta}$ for large argument. We further
conjecture that this scaling behavior should hold for other models whose
critical points correspond with the percolation transition of Wolff clusters.
This behavior is supported by our numeric work along the critical isotherm for various Ising, Potts, and
-$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. As can be
+$\mathrm O(n)$ models, shown in Fig.~\ref{fig:cluster_scaling}. Fields for the
+Potts and $\mathrm O(n)$ models take the form
+$B(s)=(h/\beta)\sum_m\cos(2\pi(s-m)/q)$ and $B(s)=(h/\beta)[1,0,\ldots,0]s$
+respectively. As can be
seen, the average cluster size collapses for each model according to the
scaling hypothesis, and the large-field behavior likewise scales as we expect
from the na\"ive Ising conjecture.
@@ -587,7 +591,8 @@ prior methods. Instead of simply applying a spin-like field, this method
allows for the application of \emph{arbitrary functions} of the spins. For
instance, theoretical predictions for the effect of symmetry-breaking
perturbations on spin models can be tested numerically
-\cite{jose1977renormalization}.
+\cite{jose_renormalization_1977}
+\cite{blankschtein_fluctuation-induced_1982,bruce_coupled_1975,manuel_carmona_$n$-component_2000}.
\begin{acknowledgments}
\end{acknowledgments}