Flat-histogram methods in quantum Monte Carlo simulations: Application to the t-j model

Transcription

1 Joural of Physics: Coferece Series PAPER OPEN ACCESS Flat-histogram methods i quatum Mote Carlo simulatios: Applicatio to the t-j model To cite this article: Nikolaos G. Diamatis ad Efstratios Maousakis 216 J. Phys.: Cof. Ser View the article olie for updates ad ehacemets. Related cotet - Optimized broad-histogram esembles for the simulatio of quatum systems Stefa Wessel, Norbert Stoop, Emauel Gull et al. - Dyamics of the Wag Ladau algorithm ad complexity of rare evets for the 3D bimodalisig spi glass Simo Alder, Simo Trebst, Alexader K Hartma et al. - Mote Carlo calculatios of the freeeergy ladscape of vesicle formatio ad growth S.-J. Zhao ad J. T. Kidt This cotet was dowloaded from IP address o 1/7/218 at :42

2 Flat-histogram methods i quatum Mote Carlo simulatios: Applicatio to the t-j model Nikolaos G. Diamatis (1) ad Efstratios Maousakis (1,2) (1) Departmet of Physics, Uiversity of Athes, Paepistimioupolis, Zografos, Athes, Greece (2) Departmet of Physics ad Natioal High Magetic Field Laboratory, Florida State Uiversity, Tallahassee, FL , USA maousakis@maget.fsu.edu Abstract. We discuss that flat-histogram techiques ca be appropriately applied i the samplig of quatum Mote Carlo simulatio i order to improve the statistical quality of the results at log imagiary time or low excitatio eergy. Typical imagiary-time correlatio fuctios calculated i quatum Mote Carlo are subject to expoetially growig errors as the rage of imagiary time grows ad this smears the iformatio o the low eergy excitatios. We show that we ca extract the low eergy physics by modifyig the Mote Carlo samplig techique to oe i which cofiguratios which cotribute to makig the histogram of certai quatities flat are promoted. We apply the diagrammatic Mote Carlo (diag-mc) method to the motio of a sigle hole i the t-j model ad we show that the implemetatio of flat-histogram techiques allows us to calculate the Gree s fuctio i a wide rage of imagiary-time. I additio, we show that applyig the flat-histogram techique alleviates the sig -problem associated with the simulatio of the sigle-hole Gree s fuctio at log imagiary time. 1. Itroductio The quatum Mote Carlo simulatio techique has bee very successful whe dealig with equilibrium properties of system of particles which do ot obey Fermi statistics[1]. This techique caot be applied directly to real-time dyamics ad oe resorts i the calculatio of correlatio fuctios i imagiary-time. From the behavior of these correlatio fuctios i imagiary-time, groud-state properties ca be accurately calculated. I priciple, a accurately kow correlatio fuctio at log imagiary-time scales ca yield useful iformatio about the low-lyig eergy excitatios. However, typical imagiary-time correlatio fuctios calculated i quatum Mote Carlo are subject to expoetially growig errors as the rage of imagiary-time grows. Here, we show that this problem ca be sigificatly alleviated by applyig flat-histogram ideas[2, 3, 4] i the Mote Carlo samplig of cofiguratios which cotribute to makig the histogram of such correlatio fuctios i imagiary-time flat. Flat-histogram ideas have bee successfully applied to classical simulatios of systems udergoig first order phase trasitios, systems with rough eergy ladscapes, etc. The Wag-Ladau algorithm[4] (WLA), a particularly useful flat-histogram algorithm, has bee applied to the simulatio of equilibrium statistical mechaical properties of quatum systems[5]. Followig this work, Gull et al.[6] have applied the idea to the cotiuous-time quatum Mote Cotet from this work may be used uder the terms of the Creative Commos Attributio 3. licece. Ay further distributio of this work must maitai attributio to the author(s) ad the title of the work, joural citatio ad DOI. Published uder licece by Ltd 1

3 Carlo approach to the quatum impurity solver eeded for all dyamical mea-field theory applicatios. The mai idea used i the preset paper has bee demostrated[7] i the past usig the Fröhligh polaro problem[8] by combiig the so-called multicaoical[2] (MUCA) or the Wag- Ladau algorithm[4] with the diagrammatic Mote Carlo (diag-mc) method[9, 1]. I the preset paper, we illustrate the beefit of applyig flat-histogram techiques to the diag-mc samplig[9, 1] i order to reveal the imagiary-time depedece of the sigle-hole Gree s fuctio movig i a atiferromagetic backgroud as described by the two-dimesioal (2D) t-j model[11, 12, 13, 14, 15]. The t-j model is probably the simplest o-trivial quatum may-body model to capture the iterplay betwee charge-carrier motio ad atiferromagetic fluctuatios ad it is as basic as the Isig model for classical systems. Therefore, we feel that we should ot proceed ay further with quatum simulatio of electroic systems without havig a techique which ca accurately simulate such a model. First, we use the diag-mc method i cojuctio with the flat-histogram techique to the problem of a sigle-hole i a modified soluble versio of t-j model, where, the diag-mc samplig space is restricted to the diagrams which are summed up by the o-crossig approximatio (NCA). This allows us to assess the correct implemetatio ad accuracy of the method ad to have a exact solutio to compare with the results obtaied with ad without applicatio of the flat-histogram method. Next, we sample all the diagrams without ay restrictio which correspods to the fully liearized versio of the t-j model without ay further approximatio. I this case, the samplig techique should sum up amplitudes which are ot positive defiite ad, thus, they caot be simply iterpreted as probability i the Mote Carlo samplig techique. Namely, we ecouter the so-called sig-problem i the applicatio of the Mote Carlo method. We show that eve i this case, where there is the sig-problem, we ca still extract more accurate results for the imagiary-time depedece of the sigle-hole spectral fuctio ad, thus, more accurate results for the low-lyig spectrum of the problem. We will use two differet implemetatios of the diag-mc method. Oe i which the imagiary-time dimesio is ot sampled by the Markov process ad it is treated as a fixed parameter i a particular simulatio. This requires repetitio of the simulatio for each oe of the differet values of imagiary-time eeded. I the secod implemetatio the imagiarytime is oe additioal dimesio of the samplig space. While the latter approach may be more efficiet, we will also use the former approach to demostrate a differet way to apply the flat-histogram idea i which the distributio of the order cotributio to the perturbatio expasio is made flat. 2. The Hamiltoia We use a simplified versio of the 2D t-j model i which the Hamiltoia ad the hole-hoppig terms are liearized withi the spi-wave approximatio to obtai a polaro-like Hamiltoia ([11, 13, 14]), i.e., Ĥ = k, q g( k, q)a k+ q a k b q + H.c. + k hω(k)b k b k, (1) g( k, q) = 4t N (u q γ k q + υ q γ k ), γ k = 2t(cos(k x a)+cos(k y a)), (2) where b q is the Bogoliubov spi-wave creatio operator, ω(k) is the spi-wave dispersio of the square lattice quatum atiferromaget ad a is the hole creatio operator. Also the g( k, q) k is the couplig of the hole to spi waves ad u q ad υ q are the coefficiets of the Bogoliubov trasformatio. For details of the derivatio of this simplified versio of the t-j model, which 2

4 remais a o-trivial quatum may-body problem, as well as for the defiitios of the operators ad the expressio for ω k, u q, v q ad g( k, q), the reader is referred to Refs. [11, 13, 14]. I order to demostrate the importace of applyig flat-histogram techiques, we will cosider two cases. First, o guidace fuctio is used i the diag-dmc simulatio, which we will refer to as bare diag-mc, which is the straightforward way to apply the diag-mc. This is ot the same as the stadard implemetatio of diag-mc[9, 1] where some differet tricks are applied to assist the simulatio at log imagiary time. The secod approach which will discuss is whe a flat-histogram techique is implemeted i the diag-mc simulatio. Whe we apply ay of the flat-histogram techiques i cojuctio with diag-mc, we will refer to it as flat-histogram diag-mc. 3. Fixed-time diagrammatic Mote Carlo The diagrammatic Mote Carlo techique[9, 1] is a Markov process i a space defied by all the terms (or diagrams) which appear i perturbatio theory. For example, it samples the terms of the perturbatio expasio of the imagiary-time sigle-particle Gree s fuctio, which ca be schematically writte as follows: G = D (λ) = O, O = λ D (λ) (3) d x 1 d x 2...d x S (λ) ( x 1, x 2,..., x,), (4) where O represets the sum of all the diagrams of order, O =G is the Gree s fuctio i zeroth order, ad λ is a variable which labels the cotributio D (λ) of a particular diagram of order. As the order of the expasio icreases, the umber of itegratio variables icreases i a similar maer. Notice that whe we refer to the th order i the case of the t-j model we mea that the umber of spi-wave-propagators cotaied i the diagrams is ; therefore, the order i perturbatio expasio is 2. I diag-mc the radom walk makes a series of trasitios betwee states {, λ, R} {,λ, R },wherer =( x 1,..., x ). Through such a Markov process the etire series of terms is sampled. This process geerates a histogram which represets the umber of times N the order appeared i the Markov process. Sice we ca compute a low order diagram aalytically, say for example the zeroth order O, the absolute value of all other orders is computed as follows: O = N N O. (5) 3.1. The problem with bare applicatio of diag-mc First, we restrict our QMC computatio of G to sample the subspace spaed oly by the diagrams icluded i the o-crossig approximatio (NCA). We do that because i this case, the NCA diagrams ca be summed up exactly ([13, 14]) ad we ca use this exact solutio to judge the accuracy of our results. I Fig. 1 O as a fuctio of is show for a fixed value of as calculated for this soluble model. Notice that the distributio of the order is Gaussia-like which peaks at a value of = max. Fig. 1 shows O o a logarithmic scale for = 8, 12 ad 16. Notice that as a fuctio of, max grows almost liearly with, thevalueofo 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 = 3

5 6e+11 MUCA NCA AF NCA AF, muca =5, =6, =7 5e+11 MUCA for =12 1e+6 =5 =6 =7 4e+11 O 3e+11 O 2e+11 1e Figure 1. O as a fuctio of for fixed. O as a fuctio of for three differet values of plotted o a logarithmic scale. 2 NCA AF 5e+6 NCA AF 15 bare diag-mc(1) bare diag-mc(2) bare diag-mc(3) 4e+6 bare diag-mc (1) bare diag-mc (2) bare diag-mc (3) O =5 O 3e+6 =7 2e+6 5 1e Figure 2. O as a fuctio of for = 5 as calculated by repeatig the diag-mc for 3 differet startig cofiguratios. Same as part for =7. is zero or very small, it leads to a fatal situatio i our attempt to calculate the absolute value of O, because this is obtaied usig the formula (7) ad a very small N implies a large ucertaity i the absolute value of all O. This is illustrated i Fig. 2 where the calculated O is show as a fuctio of for =5 ad = 7 as calculated by repeatig a diag-mc simulatio of MC steps for 3 differet startig cofiguratios. Notice that the statistical fluctuatios from oe simulatio to aother affect the values of O uiformly for all by the same fluctuatig scale factor, i.e., 1/N. If we are to calculate the error from these fluctuatios for each value of, the size of the error bars would be much larger tha the size of the fluctuatios of the poits for successive values of. Namely, the poits which represet O form a rather smooth curve. This seems uusual 4

6 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. 7. Note that usig the same umber of MC steps becomes impossible to calculate O beyod this value of =12 because the ratio O max /O becomes much larger tha the umber of MC steps Applicatio of the flat-histogram techique Here, we will solve the problem discussed i the previous subsectio by adoptig flat-histogram techiques which have bee applied i simulatios of classical statistical mechaics[2, 4]. 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 O to the desity of states which correspods to the correspodig cofiguratios. The flat-histogram method reormalizes the desity of states O foreach by kow factors (which ca be easily estimated) ad, the, samples a more-or-less flat-histogram of such populatios. 8 MUCA() MUCA(1) MUCA(3) MUCA(6) =2 MUCA() 2.5e+6 2e+6 =7 NCA MUCA MUCA(1) MUCA(2) MUCA(3) =7 N 6 O 1.5e+6, k=(π/2,π/2) 4 1e+6 2 MUCA(6) MUCA(3) 5e+5 MUCA(1) Figure 3. The evolutio of the re-weighted distributio as a fuctio of the multicaoical steps. Notice that the rage where the distributio has a sigificat value expads as we repeat the multicaoical steps ad after 6 such iteratios (gree curve labeled MUCA(6)) it becomes more or less flat i the etire domai. O as a fuctio of for = 7 as calculated by usig multicaoical diag-mc for 3 differet startig cofiguratios ad by applyig the multicaoical method. Next, we use the idea of the MUCA algorithm[2] as follows: First, for a give fixed value of we carry out a iitial exploratory ru, where we fid that the distributio O of the values of peaks at some value of = max, which depeds o the chose value of. The black solid curve i Fig. 3 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 O 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 show by the red curve i Fig. 3. I the ext step, we divide the probabilities associated with a particular cofiguratio of the th order by the factor f = f max(1,n ). Usig these modified probabilities we carry out a ew set of M diag-mc 5

7 steps which yields a ew histogram with populatios N, etc. The blue ad gree curves i Fig. 3 are obtaied i the third ad sixth iteratio of this process. Notice that already at the 6 th step the histogram is reasoably flat. 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) factor f (k) we determie O adg., ad, by re-weightig the observables by the biasig Fig. 3 shows the results of applyig the MUCA 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 curves i Fig. 2. Notice the sigificat reductio of the statistical fluctuatios betwee the three differet simulatios. Furthermore, the flat-histogram approach allows us to calculate O for almost ay, somethig which is ot possible usig bare diag-mc. 4. Samplig the imagiary time i the diag-mc Here we discuss the usual implemetatio of the diag-mc where the imagiary time variable is also a dimesio of the samplig space. I this case the radom walk makes a series of trasitios betwee states {, λ, R, i} {,λ, R,i },where R =( x 1,..., x ), is the perturbatio order, λ is a particular diagram, ad i (or i ) is the label of a particular imagiary time iterval ( i δ/2, i + δ/2) (where δ = i+1 i ). Namely, i this case we sample the histogram of G by icludig the imagiary time as a extra dimesio of the samplig space. I this case, the value G i of the histogram G( i )ithei th -iterval is foud by usig the kow value of G i the first iterval, amely, G i = N i N G, (6) where N i is the umber of occurreces of the MC radom walk i the i th iterval. I Fig. 4 we illustrate what the problem is with the bare diag-mc. We first otice that the error i G for a give umber of MC steps grow expoetially with the maximum value of used i this simulatio. The reaso is that because of the expoetial depedece of G o itself (See Fig. 4), give a fixed umber of MC steps, whe is sampled, durig the Markov process, the umber of occurreces N i the = iterval is expoetial small with the value of used. Sice the value of N eters i the Eq. 6 for G i the fluctuatios i G i icrease expoetially with icreasig i this case. This is illustrated i Fig. 4 where the relative errors i G i are give for various values of. For compariso, the relative errors obtaied by usig the Wag-Ladau techique to make the histogram of G i flat ad for approximately the same amout of CPU time to that used i the bare diag-mc simulatio, is also show i Fig. 4. Notice that by icreasig the error i the bare diag-mc icreases more or less uiformly for all values of ad it is much larger tha the error obtaied after the applicatio of the WLA. 5. Results for the full t-j model Here, we preset the results for the full t-j model, where we sample all the diagrams usig diag-mc. First, we will keep the imagiary time fixed ad i the secod part we will preset the case whe is also sampled Fixed-time diagrammatic Mote Carlo First, we ote that some terms i a give order O have a positive cotributio, while some other terms have a egative cotributio. We separate these cotributios to O + ad O such that O = O + O where both O ± are positive. 6

8 bare diag-mc (max=3.8).1 =6. Occurreces/bi 1 Relative Error.1 =5. =3.8 bare diag-mc, =3.8 bare diag-mc, =5. bare diag-mc, =6. diag-mc+wl, = diag-mc+wl bis (time slices) 5 1 Figure 4. G calculated usig bare diag-mc. The relative errors as a fuctio of for various values of whe we use the bare diag-mc method ad whe usig the Wag-Ladau method to make the histogram of G flat. The diag-mc process for fixed discussed i Sec. 3 geerates a histogram which represets the umber of times N +, ad N agiveorder appeared i the Markov process with positive or egative sig. The, we ca compute O ± usig the ratio of these occurreces, i.e., O ± = N ± N + O+. (7) Notice that we have used O + as a referece i both cases because O =. I Fig. 5 we illustrate both cotributios O ± obtaied by applyig the diag-mc+muca samplig for MC steps for = 4 ad = 5. Notice that the two cotributios O + ad O are close ad their relative differece decreases expoetially with, thus, the statistical error icreases expoetially with. This is a maifestatio of the so-called sig-problem i quatum Mote Carlo simulatio. I this case the problem with the bare applicatio of the diag-mc discussed i Sec. 3.1, is sigificatly ehaced due to the icreased fluctuatios i each of the N ± distributios themselves with icreasig. We ca apply the flat-histogram techique as discussed i Sec.3.2 to each of the distributios O ± simultaeously as illustrated i Fig. 6. This approach by reducig the statistical fluctuatios i O ±, for large, allows us to obtai a more accurate G as illustrated i Fig. 6. I Fig. 6 we preset G for the full t-j model usig J/t =.3 ad for k =(π/2,π/2) calculated for 2 differet diag-mc rus usig diag-mc with (red crosses) ad without (ope blue circles) the applicatio of the multicaoical approach ad for the same umber of MC steps i each oe of these MC simulatios. Notice that as icreases the statistical fluctuatios of G obtaied with bare diag-mc are larger tha those obtaied with the applicatio of the flat-histogram techique Samplig the histogram of G. We ca also apply the flat-histogram approach whe we iclude as a dimesio of the samplig space for the full t-j model. I this case, as i Sec. 4, we make the histogram of G flat. Oce agai we wish to demostrate the eed for the applicatio of flat-histogram techiques. I Fig. 7 we preset the positive ad egative cotributios G ± tog calculated with 7

9 6 =4 j=.3, k=(p/2,p/2), t=4 4e+5 =5 j=.3, k=(π/2,π/2), =5 5 positive egative total 3e+5 positive egative total 4 J/t=.3, k=(π/2,π/2) J/t=.3, k=(π/2,π/2) O 3 O 2e Figure 5. The calculated O ± for the full t-j model usig J/t =.3 ad for k =(π/2,π/2) for =4ad =5. =5 j=.3, k=(π/2,π/2), =5 AF FULL, DMC(), MUCA j=.3, k=(π/2,π/2) p-muca() -MUCA() p-muca(1) -MUCA(1) p-muca(2) -MUCA(2) MUCA() J/t=.3, k=(π/2,π/2) N MUCA(2) G 1 1 MUCA(1) bare diag-mc MUCA Figure 6. Demostratio of the applicatio of the flat-histogram approach to make both histograms N ± flat as a fuctio of. This is illustrated for = 5 after oe applicatio of multicaoical step (MUCA(1)) ad after a secod multicaoical step (MUCA(2)). G calculated for 2 diag-mc rus usig diag-mc without (ope blue circles) ad with (red crosses) the applicatio of flat-histogram techique. the bare diag-mc ad i Fig. 7 we preset the results of the calculatio of G adthe G ± cotributios usig diag-mc with the applicatio of the WLA. We ote that while both calculatios were carried out for approximately the same amout of CPU time (2 repetitios of 1 8 MC steps each), for the bare diag-mc case we used =3.8, while for diag-mc+wla we used =4.8. If we were to exted the value of for the former calculatio to the value used i the latter calculatio, most like there would be o occurreces of the Markov 8

10 process i the crucial iterval of =. The reaso is that the expoetial growth of G pushes all the occurreces of the Markov process at high value of. This is evidet for this smaller value of by the large fluctuatios which are more clearly oticeable at low values of. I Fig. 8 we show the relative errors for various values of whe calculated with the bare diag-mc method. Notice that the error i G ± for a give umber of MC steps grows expoetially with the maximum value of used i this simulatio because of the expoetial depedece of G ± o itself. I Fig. 8 we also preset the error obtaied whe the WLA is applied i cojuctio with the diag-mc for approximately the same amout of CPU time. This compariso suggests that the gai of the applicatio of the WLA is sigificat. 1e+6 G + G - G DMC 1e+6 G, Positive Part,Negative Part j=.3,k=(pi/2,pi/2) G + G - G G 1 1 G = = Figure 7. G for the full t-j model (for J/t =.3 ad k =(π/2,π/2)) calculated usig bare diag-mc ad diag-mc+wla. I Fig. 8 the relative errors of G ± adg are show. We ca uderstad the reductio of error whe applyig the flat-histogram techique i a very simple way as follows. For a give value of the error o both G + adg is sigificatly reduced whe combiig the diag- MC techique with the flat-histogram techique. The reaso for this is essetially the same as i the case of the NCA (Sec. 4) where every diagrammatic cotributio is positive defiite. As a result the error i the differece G =G + G is also sigificatly reduced. Thus, the flat-histogram techique is reducig the error i G itself. 6. Discussio We demostrated that the combiatio of the flat-histogram techiques with the diag-mc method yields a sigificat improvemet over the bare diag-mc method. This combiatio has bee applied to extract the imagiary-time sigle-hole Gree s fuctio G ithet-j model. First, we restricted our samplig space to oly those o-crossig diagrams whose cotributio is positive-defiite ad, thus, ca be regarded as a probability distributio. We foud that the combiatio of flat-histogram techiques with the diag-mc method yields much more accurate results for G over a wide rage of the imagiary-time. Secod, whe we sampled the etire diagrammatic space, without the NCA restrictio, there are both positive ad egative cotributios from o NCA diagrams. The positive ad 9

11 .1 DMC Relative Errors j=.3, k=(pi/2,pi/2) =3.8 =4.2.1 G G - G + Relative Error.1.1 diag-mc+wla =3.2 bare diag-mc =3.2 bare diag-mc =3.8 bare diag-mc =4.2 diag-mc+wla Relative Error Figure 8. Relative errors of G obtaied by usig the bare diag-mc for differet values of are compared to the error obtaied by applyig diag-mc+wla. The relative errors to the quatities show i part Fig. 7. egative cotributios approach each other expoetially as we icrease, thus, the error i the differece, i.e., G, grows expoetially with icreasig. This is very similar to the sig problem i other quatum Mote Carlo simulatios of fermios where the statistical error icreases expoetially as a fuctio of the particle umber. The applicatio of the flathistogram techique with the diag-mc allows us to obtai more accurate results o both the positive ad the egative cotributios to G. This eables us to compute G iawider rage of. 7. Ackowledgemets This work was supported i part by the U.S. Natioal High Magetic Field Laboratory, which is fuded by NSF DMR ad the State of Florida. Refereces [1] D. M. Ceperley, Rev. Mod. Phys. 67, 279 (1995). Rev. Mod. Phys. 71, S , (1999). [2] B. A. Berg, ad T. Neuhaus, Phys. Lett. B, 267, 249 (1991). [3] P. M. C. de Oliviera et al., J Phys. 26, 677 (1996). [4] F. Wag ad D. P. Ladau, Phys. Rev. Lett. 86, 25 (21). [5] M. Troyer, S. Wessel, ad F. Alet, Phys. Rev. Lett. 9, 1221 (23). [6] 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, (29). E. Gull, Ph.D. Thesis. [7] N.G.DiamatisadE.Maousakis,Phys.Rev.E,88, 4332 (213). N. G. Diamatis, ad E. Maousakis, Phys. Procedia, 57, 48 (214). [8] H. Fröhlich, H. Pelzer, ad S. Zieau, Philos. Mag. 41, 221 (195). [9] N. V. Prokof ev ad B. V. Svistuov, Phys. Rev. Lett. 81, 2514 (1998). [1] A. S. Mishcheko, N. V. Prokof ev, A. Sakamoto, B. V. Svistuov, Phys. Rev. B, 62, 6317 (2). [11] C. L. Kae, P.A. Lee, ad N. Read, Phys. Rev. B 39, 688 (1989). [12] E. Maousakis, Rev. Mod. Phys. 61, 1 (1991). [13] Z. Liu ad E. Maousakis, Phys. B 44, 2414 (1991). [14] Z. Liu ad E. Maousakis, Phys. B 45, 2425 (1992). [15] E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). 1