Available olie at www.sciecedirect.com ScieceDirect Physics Procedia 57 (214 ) 48 52 27th Aual CSP Workshops o Recet Developmets i Computer Simulatio Studies i Codesed Matter Physics, CSP 214 Applicatio of flat histogram methods to extract the log imagiary-time behavior i diagrammatic Mote Carlo Nikolaos G. Diamatis a, Efstratios Maousakis a,b a Departmet of Physics, Uiversity of Athes, Paepistimioupolis, Zografos, 157 84 Athes, Greece b Departmet of Physics ad Natioal High Magetic Field Laboratory, Florida State Uiversity, Tallahassee, FL 3236-435, USA Abstract We demostrate that usig flat histogram methods we ca extract the imagiary-time behavior of the Gree s fuctio G from diagrammatic Mote Carlo simulatios very accurately eve whe G chages by may orders of magitude. c 214 The Authors. Published by by Elsevier B.V. B.V. This is a ope access article uder the CC BY-NC-ND licese (http://creativecommos.org/liceses/by-c-d/3./). Peer-review uder resposibility of the Orgaizig Committee of CSP 214 coferece. Peer-review uder resposibility of The Orgaizig Committee of CSP 214 coferece Keywords: Computatioal techiques, computer modellig ad simulatio, Mote Carlo method 1. Itroductio Flat histogram techiques (Berg ad Neuhaus (1991); Oliviera et al. (1996); Wad ad Ladau (21)) have bee show to be very importat methods i order to allow efficiet samplig of the etire phase space ad trasitios betwee cofiguratios i classical systems udergoig a first order phase trasitio. Troyer et al. (Troyer, Wessel ad Alet (23)) have applied flat histogram methods to simulatio of equilibrium quatum statistical mechaics ad showed that it also makes the quatum Mote Carlo algorithm efficiet i hadlig the tuelig problem i first order phase trasitios. Furthermore, followig this idea, Gull et al. (Gull et al. (211)) have applied the flat histogram method to the cotiuous-time quatum Mote Carlo approach to the quatum impurity solver eeded for all dyamical mea-field theory applicatios. The goal of the preset paper is to show that oe ca use flat histogram techiques i order to accurately extract the log imagiary-time behavior of the Gree s fuctio i quatum may-body systems. More precisely, here we apply the flat histogram method to the diagrammatic-mote Carlo (diag-mc) method i order to extract for large. To illustrate the idea we choose the Fröhlich polaro problem (Fröhlich, Pelzer ad Zieau (195)) where the diag-mc has bee previously fruitfully applied (Prokof ev ad Svistiov (1998); Michecko et al. (2)). Some results of the idea preseted i this paper have bee published (Diamatis ad Maousakis (213)) ad the reader is referred to that work because it cotais complemetary iformatio. 2. The problem The diagrammatic Mote Carlo techique (Prokof ev ad Svistiov (1998); Michecko et al. (2)) is a Markov process i a space defied by all the terms (or diagrams) which appear i perturbatio theory. Take for example the 1875-3892 214 The Authors. Published by Elsevier B.V. This is a ope access article uder the CC BY-NC-ND licese (http://creativecommos.org/liceses/by-c-d/3./). Peer-review uder resposibility of The Orgaizig Committee of CSP 214 coferece doi:1.116/j.phpro.214.8.13
Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) 48 52 49 1.4e+8 1.2e+8 =16 1e+8 1e+7 =8 = 12 = 16 I () 1e+8 8e+7 6e+7 I ( ) 1e+6 1e+5 4e+7 1 2e+7 1 1 2 3 1 2 3 4 Fig. 1. I () as a fuctio of for fixed. I () as a fuctio of for three differet values ofplotted o a logarithmic scale. perturbatio expasio of the imagiary-time sigle-particle Gree s fuctio, which ca be schematically writte as follows: = I (), I ()= I (λ) (), I ()=G (), (1) I (λ) ()= λ d x 1 d x 2...d x F (λ) ( x 1, x 2,..., x,), (2) where ca be the order of the diagram adλis a variable which labels the diagrams withi the same order. As the order of the expasio icreases, the umber of itegratio variables icreases i a similar maer. I diag-mc the radom walk makes a series of trasitios betwee states{,λ, R} {,λ, R }, where R=( x 1,..., x ). Through such a Markov process the etire series of terms is sampled. This process geerates a histogram of the umber of times N the order appears i the Markov process. Sice we ca compute a low order diagram aalytically, say for example the zeroth order I (), the absolute value of all other orders is computed as follows: I ()= N N I (). For illustratio of the method we will use a simplified versio (Diamatis ad Maousakis (213)) of the Fröhlich polaro problem (Fröhlich, Pelzer ad Zieau (195)) which has a exact solutio ad, this allows us to compare the Mote-Carlo results with exact results. Whe we refer to the th order i this case we mea that the umber of phoo-propagators cotaied i the diagrams is ; therefore, the order i perturbatio expasio is 2, because there are 2 vertices whe there are phoo propagators. I Fig. 1 the calculated I () as a fuctio of is show for a fixed value ofas calculated for this model. Notice that the fuctio I is a Gaussia-like distributio which peaks at a value of = max (). Fig. 1 shows I () o a logarithmic scale for=8, 12 ad 16. Notice that as a fuctio of, max () grows almost liearly with, the value of I () at the maximum grows dramatically with icreasig. Namely, as icreases higher ad higher order diagrams give the most sigificat cotributio. As a result, for large eough, for ay give limited umber of Mote Carlo steps, the umber of walks ladig i small values of becomes very small or o-existet. However, whe the umber of MC steps which lad i the state = is zero or very small, it leads to a fatal situatio i our attempt to calculate the absolute value of I (), because this is obtaied usig the formula (3) ad a very small N implies a large ucertaity i the absolute value of all I (). This is illustrated i Fig. 2 where the red poits with error bars are obtaied with the diag-mc process. Notice that the (3)
5 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) 48 52 I () 1.6e+6 1.4e+6 1.2e+6 1e+6 8e+5 DMC + Multicaoical DMC I ( ) 1e+5 1 1 6 5 4 3 2 1 6e+5 4e+5 1 2e+5 1 2 3 1 1 2 3 4 5 6 7 Fig. 2. Demostratio of the applicatio of the multicaoical algorithm. See text for details. size of the error bars is much larger tha the size of the fluctuatios of the poits for successive values of. Namely, the poits which represet I () form a rather smooth curve. This seems uusual give the size of the error bars. This ca be explaied by the fact that the error is due to the poor estimatio of N which propagates through the formula give by Eq. 3. Note that usig the same umber of MC steps becomes impossible to calculate I () beyod this value of=12 because the ratio I max ()/I becomes much larger tha the umber of MC steps. 3. How to solve the problem Here, we will solve the problem discussed i the previous sectio by adoptig the flat histogram techiques which have bee applied to simulatio of classical statistical mechaics (Berg ad Neuhaus (1991); Wad ad Ladau (21)). We map the particular value of to the eergy level i stadard flat histogram methods for classical statistical mechaics ad the sum of the terms givig I () to the desity of states which correspods to the correspodig cofiguratios. The flat histogram method reormalizes the desity of states I () for each by kow factors (which ca be easily estimated) ad, the, samples a more-or-less flat histogram of such populatios. Next, we use the idea of the multicaoical algorithm (Berg ad Neuhaus (1991)) as follows: First, for a give fixed value ofwe carry out a iitial exploratory ru, where we fid that the distributio I of the values of peaks at some value of = max, which depeds o the chose value of. The curve labeled i Fig. 2 shows the result obtaied for the histogram usig M = 1 6 diag-mc steps. This distributio falls off rapidly for > max, ad, thus, we ca determie the maximum value c of visited by the Markov process. We choose a value of m safely greater tha c, such that the value of I m is practically zero. The, we modify the probabilities associated with a particular cofiguratio of the th order by dividig the origial probabilities by a factor f () = max(1, N ). Usig these modified probabilities we carry out aother set of M diag-mc steps which yields a ew histogram with populatios N (1) show by curve labeled by 1 i Fig. 2. I the ext step, we divide the probabilities associated with a particular cofiguratio of the th order by the factor f (1) = f () max(1, N (1) ). Usig these modified probabilities we carry out a ew set of M diag-mc steps which yields a ew histogram with populatios N (2) show by curve labeled by 2 i Fig. 2. We cotiue this process several times where the probabilities at the i th step are divided by a factor f (i) = f (i 1) max(1, N (i) where N (i) is the populatio of the th order at the i th step. The curves labeled 3,4,5 ad 6 i Fig. 2 are obtaied by repeatig this process four more times. Notice that already at the 6 th step the histogram is reasoably flat icludig for small. Whe, the histogram becomes more-or-less flat at some k th step, we begi a Markov process for a relatively large umber of MC steps, by dividig the origial probabilities by the factor f (k), ad, by re-weightig the observables by the biasig factor f (k) we determie I () ad.
Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) 48 52 51 1 28 1 26 Multicaoical 1 28 1 26 Wag-Ladau 1 24 1 24 1 22 1 2 1 22 1 2 1 18 1 18 1 16 1 14 25 3 35 4 45 5 1 16 1 14 25 3 35 4 45 5 Fig. 3. The results obtaied usig the fixed- diag-mc with applicatio of the multicaoical algorithm to make the histogram of I () asa fuctio of flat. with applicatio of the WL algorithm to make the histogram of I () as a fuctio of flat. The results i both cases are compared with the exact results. Fig. 2 shows the results of applyig the multicaoical algorithm as discussed i the previous paragraph for the same umber of MC steps ad approximately the same amout of CPU time as i the calculatio with the straightforward applicatio of the diag-mc to obtai the red curve i Fig. 2. Notice the sigificat reductio of the error bars. Furthermore, the flat histogram approach allows us to calculate I () for almost ay, somethig which is ot possible usig simple diag-mc. I Fig. 3 we compare with the exact results the results for obtaied for various values ofusig the multicaoical method to make the histogram flat (Fig. 3) or the W-L algorithm (Fig. 3). Notice that the agreemet holds over a rage where chages by 14 orders of magitude. 4. Samplig the histogram of Istead of fixig, we ca divide the iterval ofi small itervals of fixed duratioδ ad we defie g l as the time average of iside the particular iterval l. I this case, what we regard as the state i the Markov process acquires a additioal label l ad becomes{l,,λ, R} (where is the order,λaparticular diagram of order, ad R=( x 1,.., x ), the itegratio variables. Thus, we ca also have trasitios betwee differet l values. I this case i order to calculate g l we create a listn l of the umber of times the Markov radom walker lads i the iterval l ad we ca calculate the absolute value of g l as g l = N l N g, where g = 1. Notice that g l /g ca be a very large umber, ad ifis large eough it ca be much larger tha the total umber of MC steps. I this case becausen is expected to be either or a very small umber (ad, thus, with large error), it makes sese i order to apply the flat histogram method, to map the eergy level i the classical MC ad the desity of states to the particular iterval l ad g l respectively. This way, the histogram of g l becomes flat. Fig. 3 compares the results obtaied without applicatio of the flat histogram method (red circles with error bars) with those obtaied usig the W-L method (blue squares, the size of the error i this case is smaller tha the symbol-size ad it omitted for clarity) to make the histogram of g l flat. We have limited the maximum value ofi order to make it possible to obtai results with the simple diag-mc method, i.e., to obtai a o-zero value ofn. Agai the reaso for the errors i the stadard diag-mc without applyig the flat histogram method, is the fact that (4)
52 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) 48 52 1 4 1 4 1 3 1 3 1 2 1 2 1 Wag-Ladau DMC without guidace 1 1 1 2 4 6 8 1 2 4 6 8 Fig. 4. Compariso of the stadard diag-mc with diag-mc+wl ad with the exact results. See text for details. N is very small ad, by usig the expressio (4) its error propagates to all other values of l. This is show i Fig. 3 where the results of differet rus, each with the same umber of MC steps are compared. Notice that the various lies are almost parallel to each other startig with differet value at l=. 5. Coclusios We have combied the idea of flat histogram methods with diagrammatic MC quatum simulatio techique i order to extract the sigle Gree fuctio at log imagiary time. We have show that a) Simple applicatio of the stadard diag-mc, without some a priori kowledge about the behavior of at log, leads to either very large errors or it becomes impossible to estimate if is large eough. We cosiderto be large whe the rage of values of I () for various values of or the rage of values of i the iterval (,) ivolves may orders of magitude. b) To cure this problem, we used the idea of flat histogram methods: I the case where we eed to apply the diag- MC by keepigfixed, we made the histogram of I () flat for various values of. I the case where we allowedto vary, which eables us to samplealso, we made flat the histogram of as fuctio of. c) We fid that this is a crucial improvemet over the stadard diag-mc i order to extract the imagiary time behavior of the Gree s fuctio. This allows us to extract the low-eergy physics of may-body systems. Refereces B. A. Berg, ad T. Neuhaus, Phys. Lett. B, 267, 249 (1991). P. M. C. de Oliviera et al.,jphys.26, 677 (1996). F. Wag ad D. P. Ladau, Phys. Rev. Lett. 86, 25 (21). M. Troyer, S. Wessel, ad F. Alet, Phys. Rev. Lett. 9, 1221 (23). E. Gull, A. J. Millis, A. I. Lichtestei, A. N. Rubstov, M. Troyer, P. Werer, Rev. Mod. Phys. 83, 349 (211). G. Li, W. Werer, ad A. N. Rubstov, S. Base, M. Potthoff, Phys. Rev. B 8, 195118 (29). E. Gull, Ph.D. Thesis. H. Fröhlich, H. Pelzer, ad S. Zieau, Phil. Mag. 41, 221 (195). N. V. Prokof ev ad B. V. Svistuov, Phys. Rev. Lett. 81, 2514 (1998). A. S. Mishcheko, N. V. Prokof ev, A. Sakamoto, B. V. Svistuov, Phys. Rev. B, 62, 6317 (2). N. G. Diamatis ad E. Maousakis, Phys. Rev. E, 88, 4332 (213).