diff options
| -rw-r--r-- | data/cluster-size/cluster-size_2vector3d.dat | 216 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_3potts2d.dat | 310 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_3vector3d.dat | 184 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_4potts2d.dat | 301 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_ising2d.dat | 310 | ||||
| -rw-r--r-- | data/cluster-size/cluster-size_ising3d.dat | 237 | ||||
| -rw-r--r-- | fig_clusters_ising2d.eps | 9996 | ||||
| -rw-r--r-- | fig_correlation-temp.tex | 4 | ||||
| -rw-r--r-- | fig_correlation.tex | 4 | ||||
| -rw-r--r-- | fig_correlation_collapse-hL.tex | 4 | ||||
| -rw-r--r-- | fig_metastable.tex | 4 | ||||
| -rw-r--r-- | monte-carlo.bib | 563 | ||||
| -rw-r--r-- | monte-carlo.pdf | bin | 185499 -> 211006 bytes | |||
| -rw-r--r-- | monte-carlo.tex | 49 |
14 files changed, 9860 insertions, 2322 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 @@ -8 2.20167 1.e-7 63.59562043795609 2.164973893102592 -8 2.20167 1.e-6 64.87819548872191 2.212648436568576 -8 2.20167 0.00001 67.46940874035968 2.301616822709248 -8 2.20167 0.0001 67.44140423335065 2.30008894357248 -8 2.20167 0.001 71.32843137254912 2.398277793810432 -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 -16 2.20167 0.01 391.3233532934144 13.354160729767935 -16 2.20167 0.1 822.6586126266573 29.32967370950656 -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 -32 2.20167 0.00001 987.9398532419093 27.91466354242355 -32 2.20167 0.0001 1083.3379648008356 31.228299422072833 -32 2.20167 0.001 1151.1507002801193 33.530449835229184 -32 2.20167 0.01 2611.275275590533 85.86586555583693 -32 2.20167 0.1 6339.86940298507 227.132753381163 -32 2.20167 1. 14823.10940170938 515.0581135403254 -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 -64 2.20167 0.00001 4077.0909943129705 103.63447013697126 -64 2.20167 0.0001 4898.0003000525785 125.73227228305818 -64 2.20167 0.001 8836.591433942041 254.10948594506138 -64 2.20167 0.01 21022.64215218081 682.6924817920819 -64 2.20167 0.1 50597.72201630827 1842.307708405678 -64 2.20167 1. 114540.33772652397 4068.2195261694933 -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 -128 2.20167 1.e-6 23689.046527859227 1093.5300645443338 -128 2.20167 0.00001 20089.293792012664 786.2899335338394 -128 2.20167 0.0001 38106.45479910696 1854.133959464583 -128 2.20167 0.001 72399.92470837763 3547.6496046148814 -128 2.20167 0.01 190290.0237010039 10030.696361468363 -128 2.20167 0.1 448240.2068230266 24526.873461304327 -128 2.20167 1. 849276.2519685049 50303.14845552824 -128 2.20167 10. 1.583249774647888e6 67273.13361722566 -128 2.20167 100. 1.7961472880000002e6 61619.78659707237 +8 2.20167 0.00001 64.34711510093926 0.642512063804416 +8 2.20167 0.0000199526 62.958934751413246 0.628436621865984 +8 2.20167 0.0000398107 63.91846160305562 0.638095744595456 +8 2.20167 0.0000794328 66.49236156708199 0.664849522863616 +8 2.20167 0.000158489 64.23076267212288 0.64215616731904 +8 2.20167 0.000316228 63.43043822048307 0.633213766898176 +8 2.20167 0.000630957 64.30047409558733 0.643003238060032 +8 2.20167 0.00125893 61.77718788884173 0.617351998568448 +8 2.20167 0.00251189 66.34295401199564 0.66263997231872 +8 2.20167 0.00501187 65.83620614226945 0.657353339836416 +8 2.20167 0.01 66.61609568036454 0.665747030715904 +8 2.20167 0.0199526 72.61335846281114 0.724902631066112 +8 2.20167 0.0398107 83.04874787884954 0.829709436667392 +8 2.20167 0.0794328 100.23332737350144 1.001081497657856 +8 2.20167 0.158489 125.9836161927849 1.25831117869568 +8 2.20167 0.316228 153.93530927835033 1.537340435948032 +8 2.20167 0.630957 193.17611447440947 1.930651398201856 +8 2.20167 1.25893 234.71404728789915 2.343463704685568 +8 2.20167 2.51189 284.6007060572247 2.841495348130304 +8 2.20167 5.01187 327.4607407407391 3.268970765721088 +8 2.20167 10. 367.4025178632187 3.662625983795712 +8 2.20167 19.9526 397.64438502673664 3.975462423776256 +8 2.20167 39.8107 428.96414852752844 4.261911444578816 +8 2.20167 79.4328 460.12814895947366 4.557434589174272 +8 2.20167 158.489 472.4084919472916 4.69570405902848 +8 2.20167 316.22800000000001 487.43518518518476 4.867775705334272 +8 2.20167 630.95699999999999 487.56 4.854823269999104 +8 2.20167 1258.93000000000006 499.04743083003956 4.863239498466304 +8 2.20167 2511.88999999999987 501.841463414634 4.319170260792832 +8 2.20167 5011.86999999999989 507.2258064516132 2.933977464671744 +8 2.20167 10000. 511.98755186721996 0.012448132782592 +16 2.20167 0.00001 243.1411744110428 2.431011634765824 +16 2.20167 0.0000199526 252.5322712418263 2.520912407564288 +16 2.20167 0.0000398107 253.76404465212622 2.537209270288384 +16 2.20167 0.0000794328 246.4048496315474 2.459453314756608 +16 2.20167 0.000158489 253.212619922432 2.52905513125888 +16 2.20167 0.000316228 248.07934774326068 2.478197899776 +16 2.20167 0.000630957 253.2120674584781 2.531070887374848 +16 2.20167 0.00125893 251.62104915197952 2.513115835588608 +16 2.20167 0.00251189 263.989485057708 2.63932209608704 +16 2.20167 0.00501187 310.58866835501874 3.101933556703232 +16 2.20167 0.01 362.10632829021387 3.619400289759232 +16 2.20167 0.0199526 451.94670070100375 4.513445487394816 +16 2.20167 0.0398107 577.8585142857114 5.775957672071168 +16 2.20167 0.0794328 740.6424753402716 7.401719896977408 +16 2.20167 0.158489 963.2802300602859 9.629850757787649 +16 2.20167 0.316228 1222.2192517936414 12.217774770966528 +16 2.20167 0.630957 1540.389254385959 15.39728969549824 +16 2.20167 1.25893 1882.800861589766 18.80422653820109 +16 2.20167 2.51189 2265.206041394733 22.651750875090944 +16 2.20167 5.01187 2596.588505191989 25.958085128495103 +16 2.20167 10. 2965.618596491227 29.65563780688691 +16 2.20167 19.9526 3192.4773960216985 31.880491410059264 +16 2.20167 39.8107 3448.0937081659968 34.39746678426829 +16 2.20167 79.4328 3571.6885665528994 35.710087698755586 +16 2.20167 158.489 3687.7395048439153 36.69506816500941 +16 2.20167 316.22800000000001 3846.9328493647913 38.018421819891714 +16 2.20167 630.95699999999999 3975.3837638376367 38.96180958455398 +16 2.20167 1258.93000000000006 3968.996183206105 39.22021592267571 +16 2.20167 2511.88999999999987 3997.2430278884435 39.557926898192385 +16 2.20167 5011.86999999999989 4058.5870445344112 22.549504065769472 +16 2.20167 10000. 4080.846774193549 13.085977988919296 +32 2.20167 0.00001 986.1265014644081 9.859696529113087 +32 2.20167 0.0000199526 984.3604448964771 9.837040297869311 +32 2.20167 0.0000398107 1006.0389259506484 10.0341567143936 +32 2.20167 0.0000794328 985.4115314612634 9.83291789000704 +32 2.20167 0.000158489 988.9584489809183 9.860558563180543 +32 2.20167 0.000316228 1053.2916794806108 10.522337285144577 +32 2.20167 0.000630957 1140.1444322337752 11.383942757711871 +32 2.20167 0.00125893 1374.3218044187443 13.724761323536384 +32 2.20167 0.00251189 1623.307196901458 16.213774402650113 +32 2.20167 0.00501187 2078.4767117464044 20.77531687665664 +32 2.20167 0.01 2665.160684721832 26.625608278441984 +32 2.20167 0.0199526 3382.701817001935 33.77019860720026 +32 2.20167 0.0398107 4508.303416714723 45.0304281240535 +32 2.20167 0.0794328 5811.758275638198 58.0361292899287 +32 2.20167 0.158489 7497.161990890488 74.85085514933863 +32 2.20167 0.316228 9626.83700367029 96.22968430893465 +32 2.20167 0.630957 12394.018575174861 123.77619739122073 +32 2.20167 1.25893 14952.520494028195 149.37678996088422 +32 2.20167 2.51189 18095.726160029157 180.90646710455502 +32 2.20167 5.01187 21214.461834016325 212.1106554100777 +32 2.20167 10. 23343.19253632752 233.14323590630605 +32 2.20167 19.9526 25547.907207207125 254.32534097618534 +32 2.20167 39.8107 26903.144736842154 268.5509894176113 +32 2.20167 79.4328 29022.12678741659 289.66482235313356 +32 2.20167 158.489 30089.771470160133 300.51542694189465 +32 2.20167 316.22800000000001 31006.274298056112 307.47057616807524 +32 2.20167 630.95699999999999 31952.087121212113 269.31031455645694 +32 2.20167 1258.93000000000006 32463.62698412697 145.24222038481307 +32 2.20167 2511.88999999999987 32104.675889328064 287.3561132376064 +32 2.20167 5011.86999999999989 32237.24899598392 234.80085736269413 +32 2.20167 10000. 32511.387096774182 150.84190567831962 +64 2.20167 0.00001 4064.1647201144015 40.60314739749683 +64 2.20167 0.0000199526 3831.507650574287 38.19021513785344 +64 2.20167 0.0000398107 4188.66782299095 41.814975126175746 +64 2.20167 0.0000794328 4385.068293367005 43.708769321877504 +64 2.20167 0.000158489 4853.758547166494 48.45095767559373 +64 2.20167 0.000316228 5695.764315307246 56.91348827452211 +64 2.20167 0.000630957 7144.635402364125 71.4056318124032 +64 2.20167 0.00125893 9289.727391777424 92.81228705733018 +64 2.20167 0.00251189 11912.868601447843 119.05548221834854 +64 2.20167 0.00501187 15788.644364845843 157.58883634334924 +64 2.20167 0.01 20856.08859245609 208.37557382152193 +64 2.20167 0.0199526 27440.490605042203 274.0263452816179 +64 2.20167 0.0398107 36016.82990342052 359.8593073760174 +64 2.20167 0.0794328 46818.270544934734 467.93899494473726 +64 2.20167 0.158489 60223.83839464863 602.0643708553134 +64 2.20167 0.316228 78153.57714285697 780.832040756183 +64 2.20167 0.630957 97889.77568585273 978.8299562107535 +64 2.20167 1.25893 122566.38974729179 1223.8866702359594 +64 2.20167 2.51189 144807.62783828724 1447.623980018303 +64 2.20167 5.01187 166689.4777162127 1664.0196485088543 +64 2.20167 10. 190408.43825838738 1903.4590165202371 +64 2.20167 19.9526 203667.23594024623 2030.6874611295846 +64 2.20167 39.8107 219313.16305732488 2181.9112891018117 +64 2.20167 79.4328 230769.68900804294 2288.632938228613 +64 2.20167 158.489 238446.67980295594 2356.119678719099 +64 2.20167 316.22800000000001 245372.58348294438 2452.5924606337353 +64 2.20167 630.95699999999999 250755.03309692678 2497.6294915229614 +64 2.20167 1258.93000000000006 256618.89062499988 2139.4119616928156 +64 2.20167 2511.88999999999987 257573.15748031496 2009.5231733897626 +64 2.20167 5011.86999999999989 256347.9367588931 2264.411607160324 +64 2.20167 10000. 261751.60245901643 261.52246707172145 +128 2.20167 0.00001 17218.50733331795 171.69256298001204 +128 2.20167 0.0000199526 18378.8810154216 182.93576751172813 +128 2.20167 0.0000398107 21719.71231105522 216.67560846000129 +128 2.20167 0.0000794328 26683.808660240466 265.9367352134533 +128 2.20167 0.000158489 31502.175400928216 314.121953048789 +128 2.20167 0.000316228 41137.77258249637 410.5978549779825 +128 2.20167 0.000630957 54511.753145356386 543.948611021439 +128 2.20167 0.00125893 72983.17728643809 729.3156350786273 +128 2.20167 0.00251189 95020.88666450547 949.4982691641098 +128 2.20167 0.00501187 122751.97786936312 1226.491100397568 +128 2.20167 0.01 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\ 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 @@ -8 0.994973 1.e-7 28.3823264201984 0.236138387891328 -8 0.994973 1.e-6 27.805823680823615 0.233244623238976 -8 0.994973 0.00001 28.043619527251906 0.23433663419584 -8 0.994973 0.0001 27.385046581398974 0.230684992491712 -8 0.994973 0.001 27.58528428093645 0.231563393720128 -8 0.994973 0.01 28.17325524646157 0.236158926337664 -8 0.994973 0.1 34.48330078124999 0.275121977917568 -8 0.994973 1. 41.996052631579005 0.322332521883392 -8 0.994973 10. 43.341899892357375 0.343088948340544 -8 0.994973 100. 43.08706700548358 0.343120888099008 +8 0.994973 0.00001 28.833012743447934 0.288131927793536 +8 0.994973 0.0000199526 27.48737907261216 0.274842722428992 +8 0.994973 0.0000398107 28.446481024556544 0.284238439891776 +8 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6.870262542690304 +32 0.994973 158.489 684.0973451327417 6.834341808161792 +32 0.994973 316.22800000000001 687.1075949367091 6.870315346255872 +32 0.994973 630.95699999999999 680.431007137194 6.803543693545472 +32 0.994973 1258.93000000000006 675.3311360992625 6.752189528706048 +32 0.994973 2511.88999999999987 683.0683777822352 6.829225651899392 +32 0.994973 5011.86999999999989 685.7667342799197 6.85488348775936 +32 0.994973 10000. 680.0359541683129 6.7933617373952 -64 0.994973 1.e-7 1045.320553473614 8.963296885243905 -64 0.994973 1.e-6 1014.882108700717 8.67390104844288 -64 0.994973 0.00001 1015.2398430933033 8.73498768896 -64 0.994973 0.0001 1007.5999535855084 8.654159380598784 -64 0.994973 0.001 1210.7716902581003 10.254569083936769 -64 0.994973 0.01 1635.5552682133791 13.576588168851456 -64 0.994973 0.1 2161.249827331027 17.837539231420415 -64 0.994973 1. 2670.878363475968 20.079606759276544 -64 0.994973 10. 2726.4530313124455 22.30290342632243 -64 0.994973 100. 2724.6384820239687 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0.994973 0.630957 2613.3575227963515 26.11137835541299 +64 0.994973 1.25893 2676.5019342359633 26.758170033430527 +64 0.994973 2.51189 2733.1189710610925 27.307238220607488 +64 0.994973 5.01187 2714.036709854724 27.131255221878785 +64 0.994973 10. 2701.130712625873 27.0089740645376 +64 0.994973 19.9526 2740.3795472918323 27.398337148817408 +64 0.994973 39.8107 2747.9513436482234 27.45954269478912 +64 0.994973 79.4328 2767.661461586522 27.664404037750785 +64 0.994973 158.489 2712.28305582762 27.111636100911102 +64 0.994973 316.22800000000001 2749.9389920424346 27.48000976755507 +64 0.994973 630.95699999999999 2732.0935309433326 27.316389276172288 +64 0.994973 1258.93000000000006 2750.5 27.479890214432768 +64 0.994973 2511.88999999999987 2735.111445783134 27.336840829607937 +64 0.994973 5011.86999999999989 2763.533678756479 27.62347182630912 +64 0.994973 10000. 2691.7758620689615 26.90687927615488 -128 0.994973 1.e-7 3538.730470981927 29.814243053223937 -128 0.994973 1.e-6 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0.00251189 5436.593093093114 54.33873724574925 +128 0.994973 0.00501187 5942.226028726665 59.4129609878569 +128 0.994973 0.01 6539.906023067058 65.38685399470899 +128 0.994973 0.0199526 7185.313790648009 71.80225094806732 +128 0.994973 0.0398107 7607.125719182262 76.05192419527884 +128 0.994973 0.0794328 8509.232132564795 85.06012612847206 +128 0.994973 0.158489 9084.459018914366 90.79843219664077 +128 0.994973 0.316228 9699.224482988639 96.93695628589465 +128 0.994973 0.630957 10242.548682087121 102.35615435821875 +128 0.994973 1.25893 10690.10528338133 106.77443078466764 +128 0.994973 2.51189 10813.915158591177 108.07136709364941 +128 0.994973 5.01187 11044.638751033934 110.4290307919872 +128 0.994973 10. 10904.528054118613 109.03974017019085 +128 0.994973 19.9526 10876.674689287847 108.71267105333249 +128 0.994973 39.8107 11069.672704559662 110.67788311366861 +128 0.994973 79.4328 10826.279291553137 108.22139588019814 +128 0.994973 158.489 10838.68907563026 108.38421256993178 +128 0.994973 316.22800000000001 10797.59301428024 107.89538156763545 +128 0.994973 630.95699999999999 11011.260842880532 110.0209809042473 +128 0.994973 1258.93000000000006 10700.983060417888 106.99216784757556 +128 0.994973 2511.88999999999987 10879.830039525703 108.79393222693683 +128 0.994973 5011.86999999999989 10937.25276048987 109.36748095443764 +128 0.994973 10000. 10921.908454927025 109.20361795821567 -256 0.994973 1.e-7 10948.04919614151 286.9151418115031 -256 0.994973 1.e-6 10144.75468889555 266.11003730886654 -256 0.994973 0.00001 11373.025692359 302.0328836857856 -256 0.994973 0.0001 14407.5266470835 378.75374401919385 -256 0.994973 0.001 19704.837992013694 512.2242250181837 -256 0.994973 0.01 25847.14064801176 687.3836664888361 -256 0.994973 0.1 36241.574112735005 917.0604850607555 -256 0.994973 1. 41424.834951456294 1019.2338743602709 -256 0.994973 10. 42021.48963730574 1132.0645859317515 -256 0.994973 100. 42350.64099216714 1132.9247419503083 +256 0.994973 0.00001 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41944.16238003834 419.3609795167191 +256 0.994973 1.25893 42418.70357142851 423.96810755538945 +256 0.994973 2.51189 43602.368546940255 435.83846058367385 +256 0.994973 5.01187 43668.05640821205 436.2994766466253 +256 0.994973 10. 43608.58141382043 435.78773340369713 +256 0.994973 19.9526 43561.3560830859 435.2001285881856 +256 0.994973 39.8107 44314.622629714526 442.7160733078651 +256 0.994973 79.4328 43467.43377287938 434.54244928546404 +256 0.994973 158.489 44012.053276178776 439.7769317378949 +256 0.994973 316.22800000000001 43916.89111922154 438.79557867726436 +256 0.994973 630.95699999999999 43827.61387936259 438.1442118457754 +256 0.994973 1258.93000000000006 44871.42643229157 448.62576939925503 +256 0.994973 2511.88999999999987 44387.25498007958 443.7091953455268 +256 0.994973 5011.86999999999989 43899.089785164135 438.79823795401524 +256 0.994973 10000. 43774.37786259522 437.4887027385303 + +512 0.994973 0.00001 42107.256470588094 420.96461915252326 +512 0.994973 0.0000199526 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0.0000398107 190433.27823705642 1903.122096450437 +1024 0.994973 0.0000794328 214057.25213675172 2140.5538196274547 +1024 0.994973 0.000158489 235146.69677257896 2349.882128993354 +1024 0.994973 0.000316228 255624.37879694413 2554.5700820207535 +1024 0.994973 0.000630957 290214.4516816298 2900.3486986072226 +1024 0.994973 0.00125893 311560.98059178976 3113.7958850818213 +1024 0.994973 0.00251189 346328.08076787007 3461.760675130376 +1024 0.994973 0.00501187 378941.4040812399 3789.1978618701087 +1024 0.994973 0.01 410211.89490445645 4100.327491302851 +1024 0.994973 0.0199526 455793.8473193471 4555.927138141733 +1024 0.994973 0.0398107 496729.08733512415 4966.135583728468 +1024 0.994973 0.0794328 540002.7931180311 5395.646331066778 +1024 0.994973 0.158489 587199.0733554172 5867.909570968421 +1024 0.994973 0.316228 624502.688463492 6240.738537572925 +1024 0.994973 0.630957 661338.4471021114 6611.897730461597 +1024 0.994973 1.25893 696978.4716520996 6964.191934196744 +1024 0.994973 2.51189 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b/data/cluster-size/cluster-size_3vector3d.dat @@ -1,23 +1,161 @@ -8 1.44325 0.00001 55.5139971969664 0.554903213121536 -8 1.44325 0.0000199526 56.027379494199806 0.559909155830272 -8 1.44325 0.0000398107 56.192946736956415 0.56169179990272 -8 1.44325 0.0000794328 54.5194957803648 0.544726918441472 -8 1.44325 0.000158489 54.597054353332226 0.545821297943552 -8 1.44325 0.000316228 55.4369717283584 0.5542436232704 -8 1.44325 0.000630957 55.07394505297152 0.55047552796416 -8 1.44325 0.00125893 55.21150950161306 0.551854558490112 -8 1.44325 0.00251189 53.847208515437565 0.53806267020288 -8 1.44325 0.00501187 56.394869546151426 0.563462774585344 -8 1.44325 0.01 58.60907046476697 0.58596544898048 -8 1.44325 0.0199526 60.544246945532414 0.605157448036352 -8 1.44325 0.0398107 69.25171956575693 0.69234327766272 -8 1.44325 0.0794328 82.73071667530547 0.826681271223296 -8 1.44325 0.158489 100.86688560481588 1.007812401429504 -8 1.44325 0.316228 126.55678979886899 1.26487036726528 -8 1.44325 0.630957 161.53304886211532 1.615101040502784 -8 1.44325 1.25893 196.9797927992617 1.969288721453568 -8 1.44325 2.51189 243.51881051175528 2.433779871054848 -8 1.44325 5.01187 284.1764088204406 2.841037843352576 -8 1.44325 10. 331.5890990542551 3.315775558490112 -8 1.44325 19.9526 373.07874564459826 3.727468875063808 -8 1.44325 39.8107 407.2365374937083 4.07083755410944
\ No newline at end of file +8 1.44325 0.00001 55.81130087209267 0.557214466195456 +8 1.44325 0.0000199526 53.62058906227968 0.535732751508992 +8 1.44325 0.0000398107 54.70620958710989 0.54678464494336 +8 1.44325 0.0000794328 55.14115757973811 0.551397629613056 +8 1.44325 0.000158489 54.839998574280195 0.547984963964928 +8 1.44325 0.000316228 55.89314160089344 0.558226670396416 +8 1.44325 0.000630957 55.12730239303782 0.551143283449856 +8 1.44325 0.00125893 54.0407507850409 0.54021822469376 +8 1.44325 0.00251189 56.069402247519236 0.560309421954048 +8 1.44325 0.00501187 57.36534285398784 0.573296458243584 +8 1.44325 0.01 57.27478312448973 0.572603675111424 +8 1.44325 0.0199526 59.583690503044096 0.59514237305344 +8 1.44325 0.0398107 69.7206322444672 0.696694249043968 +8 1.44325 0.0794328 82.50037782185626 0.823941888035328 +8 1.44325 0.158489 102.7730270511954 1.02649059163904 +8 1.44325 0.316228 125.58241258286746 1.255740953616896 +8 1.44325 0.630957 158.91085334198476 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1.44325 0.000316228 853.7443451490141 8.518134632513537 +32 1.44325 0.000630957 934.5505519409234 9.329928573026304 +32 1.44325 0.00125893 1065.9385065176104 10.634770183684097 +32 1.44325 0.00251189 1282.8385029023334 12.810478395686912 +32 1.44325 0.00501187 1636.1877375433114 16.327539234963456 +32 1.44325 0.01 2028.3773902329283 20.277564206219264 +32 1.44325 0.0199526 2725.3914697406217 27.253844727267328 +32 1.44325 0.0398107 3514.783899166507 35.131946704699395 +32 1.44325 0.0794328 4600.096219931246 45.930387735904254 +32 1.44325 0.158489 6182.131688096203 61.75979040473088 +32 1.44325 0.316228 7845.159882079863 78.40590406647809 +32 1.44325 0.630957 10021.836257776336 100.20852809587097 +32 1.44325 1.25893 12289.941632332726 122.86471170962227 +32 1.44325 2.51189 15415.096795907326 154.0501585586094 +32 1.44325 5.01187 18059.611139941237 180.52103880717107 +32 1.44325 10. 21844.195376994976 218.41339794204262 +32 1.44325 19.9526 23864.580251221196 238.6133592013865 +32 1.44325 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1.44325 630.95699999999999 1.980097142023344e6 19540.927657961063 +128 1.44325 1258.93000000000006 2.02198996039604e6 19857.524576760105 +128 1.44325 2511.88999999999987 2.0259372646048097e6 19879.654187178392 +128 1.44325 5011.86999999999989 2.0418392758620698e6 19846.025820302737 +128 1.44325 10000. 2.0769803904382477e6 9952.510969695437 +256 1.44325 0.00001 60161.45099983107 599.3263746074542 +256 1.44325 0.0000199526 77901.23797637496 777.8785821259203 +256 1.44325 0.0000398107 101219.99686797833 1009.7149891387064 +256 1.44325 0.0000794328 136260.0666018779 1358.5048457738978 +256 1.44325 0.000158489 180213.0812666957 1798.52251791727 +256 1.44325 0.000316228 244606.69981441813 2439.4315712125995
\ No newline at end of file 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 @@ -8 0.910239 1.e-7 31.25128528907744 0.246253494231872 -8 0.910239 1.e-6 30.78990703851277 0.244874662488896 -8 0.910239 0.00001 32.308278361051585 0.25217886251904 -8 0.910239 0.0001 31.30189867722483 0.246826664226944 -8 0.910239 0.001 32.18340571006183 0.251616343170432 -8 0.910239 0.01 32.33067211288518 0.251779013153728 -8 0.910239 0.1 38.88447177685043 0.280855693400128 -8 0.910239 1. 47.08413939313638 0.313063502786432 -8 0.910239 10. 47.468546958062845 0.336828361986496 -8 0.910239 100. 48.136758538046976 0.334695743642496 +8 0.910239 0.00001 31.432848837209217 0.314280438098624 +8 0.910239 0.0000199526 32.11704986563174 0.320821499650752 +8 0.910239 0.0000398107 31.281828703703745 0.312529607784768 +8 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62 V +59 61 V +59 64 V +59 59 V +60 63 V +59 67 V +59 52 V +59 64 V +59 55 V +60 51 V +59 49 V +59 36 V +59 34 V +59 28 V +60 15 V +59 13 V +59 10 V +59 11 V +26 4 V +.4839 g 9705 1129 M +8346 796 L +60 8 V +59 40 V +59 43 V +59 57 V +60 65 V +59 56 V +59 67 V +59 59 V +59 71 V +60 59 V +59 65 V +59 59 V +59 65 V +59 57 V +60 58 V +59 56 V +59 49 V +59 48 V +60 40 V +59 29 V +59 27 V +59 22 V +56 13 V +1 g 9705 1002 M +8494 844 L +59 58 V +59 59 V +60 66 V +59 63 V +59 69 V stroke LTw % End plot #2 diff --git a/fig_correlation-temp.tex b/fig_correlation-temp.tex index 95893df..6661e20 100644 --- a/fig_correlation-temp.tex +++ b/fig_correlation-temp.tex @@ -8,7 +8,7 @@ {\GNUPLOTspecial{" %!PS-Adobe-2.0 EPSF-2.0 %%Creator: gnuplot 5.2 patchlevel 2 -%%CreationDate: Thu Feb 22 13:53:46 2018 +%%CreationDate: Thu Apr 26 20:22:42 2018 %%DocumentFonts: %%BoundingBox: 0 0 246 151 %%EndComments @@ -448,7 +448,7 @@ SDict begin [ /Creator (gnuplot 5.2 patchlevel 2) % /Producer (gnuplot) % 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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 Binary files differindex c8a15de..a8dcab7 100644 --- a/monte-carlo.pdf +++ b/monte-carlo.pdf 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} |