Markov chain sampling of the O(n) loop models on the infinite plane
MMarkov chain sampling of the O(n) loop models on theinfinite plane
Victor HerdeiroDepartment of Mathematics, King’s College, London, United Kingdom.
Abstract
It was recently proposed in [Herdeiro & Doyon Phys. Rev. E (2016)] a numerical method showing a precisesampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extendsthe method to a larger class of models, namely the O ( n ) loop gas models for n ∈ (1 , O ( n ) models canbe numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the onlyrequirement for the present numerical method to work. Within statistical mechanics, critical models form avery rich and exciting subject. Many models are knownto exhibit correlations over infinite distances and di-vergences of the free energy or its derivatives [1]. Therenormalization group (RG) and conformal field theory(CFT) have given us an efficient theoretical frameworkto study and classify such systems [2]. One of the mainsuccess is in explaining how they fall into universal-ity classes where they share universal exponents andidentical emerging collective behaviour. In the case ofcritical lattice spin models, such universal collective be-haviour can be the nucleation of arbitrary large ordereddomains with random loop geometries as boundaries.These random variables have been described by con-formal loop ensembles (CLE) [3, 4] and linked to theCFT algebra [5].Numerical methods, in particular Markov chainMonteCarlo (MCMC), have been fruitful in investi-gating critical models by providing numerical checksand new results beyond analytical reach [6]. They relyon very few assumptions - essentially ergodicity - thushave the advantage of being generalizable to higher di-mensions and many models. It is of great interest to be able to reproduce, through a Markov chain, probabilitydistributions such as the ones discussed in CLE.Common choices of boundary conditions, such asperiodic or generic fixed boundaries, are improper atsampling the infinite volume bulk observables. Indeed,near the critical point the divergence of the correlationlength implies that finite size effects and boundary ef-fects will be carried over infinite range.The work in [7] presented a numerical algorithm ac-curately approximating the finite domain marginal ofan infinite plane distribution in the case of the criti-cal Ising model. These marginals were the restrictionof the degrees of freedom to a finite sublattice A ob-tained after integrating out the fluctuating degrees offreedom from infinitely far away to its boundary ∂A .Such boundary may be seen as “holographic” in thesense that it has the property of encoding all the in-formation in C \ A . The preliminary work done in [7]showed that a chain of discrete lattice dilations, ef-fectively mapping a state inside A to a state on ∂A composed with a rethermalization by lattice updates,reproduced such a holographic boundary, up to effectsmeasured to be negligible. This was a direct conse-quence of the scale invariance of the Gibbs measure.This algorithm was dubbed UV sampler as it is equiv-1 a r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y lent to an inverted RG flow approaching the UV fixedpoint.It offers the possibility to approximate averages ofrandom variables of the infinite plane with finite sup-port. This includes CFT correlation functions with in-sertions of operators - at least for the ones writeable asscaling limit of lattice observables - as well as loop vari-ables such as densities, indicator functions, etc. Ref.[7] showed success in fitting the CFT data of the crit-ical Ising model: central charge, scaling weights andstructure constants.In this paper, we show how these techniques can begeneralized to the O ( n ) loop gas for n ∈ (1 , C \ A , allows us to produce aMCMC that reproduces a holographic boundary. Forthis purpose, we evaluate various scaling dimensions,three-point coupling and four-point functions using ourMCMC, and compare with predicted conformal fieldtheory results. This confirms that a UV sampler onlyrequires scale invariance, and not locality.The plan of the article is as follows: Section 2 ex-plains general aspects of the algorithm, concentratingon the main differences between the Ising and the O ( n )model. Section 3 presents the lattice observables andtheir connection to the CFT operator algebra, and nu-merical checks of the two point functions. In section3, numerical checks for dynamical quantities - namelythe spin four point function and the structure constant C εσσ - are presented. A conclusion is presented in Sec-tion 5. O ( n )critical line. In two dimensions the Ising model is probably the sim-plest lattice model of statistical physics. It consists ofa binary variable σ i taking values in {− , } at eachlattice sites with only first neighbour interactions (forsimplicity we consider the situation without externalmagnetic field). Its Gibbs measure is given by p ( { σ } , β ) = 1 Z ( β ) e β (cid:80) (cid:104) i , j (cid:105) σ i σ j (1)with β the unique coupling, the sum running over everypair of neighbour sites and Z ( β ) the partition function Bold indices will stand for lattice sites variables normalizing the probabilities.This model has been extensively studied in the pastfor being as insightful as it is simple. Its most inter-esting feature is its second order phase transition, fora coupling value β c = 0 .
274 653 . . . on the triangularlattice. When sitting on this critical point, the sys-tem exhibits scale invariance. The scaling limit canbe taken and all the microscopic details of the latticegeometry become completely irrelevant.We have shown in [7] how to successfully samplea finite subsection of the infinite plane of the criti-cal Ising model approximating to any desired level theholographic boundary.One simple generalization of this model is increasingthe dimension of the local variable to a unit vector ofdimension n , the Ising model being the special case n =1. The product σ i σ j gets promoted to (cid:126)σ i · (cid:126)σ j to conserverotational invariance, and because of this symmetry itis believed that [8] the critical point of this model is inthe same universality class as the model with partitionfunction Z O ( n ) = (cid:88) { σ } (cid:89) (cid:104) i , j (cid:105) (cid:0) x (cid:126)σ i · (cid:126)σ j (cid:1) . (2)Here the first sum represents an integral over the n − x (cid:126)σ i · (cid:126)σ j )( x (cid:126)σ k · (cid:126)σ l ) . . . where sites i , j , k , l , . . . are successively, two by two,neighbours. Such terms only survive the angular inte-gration if i , j , k , l , . . . appear an even number of times.One non-vanishing contribution would be x ( (cid:126)σ i · (cid:126)σ j ) ( (cid:126)σ j · (cid:126)σ k ) ( (cid:126)σ k · (cid:126)σ l ) ( (cid:126)σ l · (cid:126)σ m ) ( (cid:126)σ m · (cid:126)σ n ) ( (cid:126)σ n · (cid:126)σ i ) , which integration gives x n . It can be pictured as aclosed loop which edges join the six lattice sites. Sim-ilarly, any contribution to (2) can be represented as aconfiguration of closed loops. These loops are definedon the hexagonal lattice dual to the triangular latticeholding the spin variables. The partition function takesthe following expression: Z O ( n ) = (cid:88) C ∈ G x || C || n | C | , (3)2ith x ( β ) = e − β , G being the set of all possible con-figurations of non intersecting closed loops on the duallattice, || · || the sum of the lengths of each loop inthe configuration and | · | the total number of loops.In other words, n is a loop-weight and x is an edge-weight. Contrary to (2), in (3) the parameter n doesnot have to be interpreted as the dimension of a vec-tor and is not required to be integer any more. Fromhere on it will be assumed to take any values in [1 , x c ( n ) = 1 (cid:112) √ − n . (4)From (2) to (3), the vector variables were traded forloop variables. In some sense - from a RG point of view- the vector variables were integrated out, and replaced,for larger scale, by emergent fluctuating degrees of free-dom. For MCMC purposes, it is useful to reinterpret(3) as a lattice model of spin variables. It can be lookedat as an Ising model with nearest neighbour coupling β = − ln x c ( n ), and with a non-local contribution tothe Hamiltonian: each (closed) boundary between op-positely oriented spin domains will be a ‘loop’ and willadd a weight equal to n . This has the conveniencethat some numerical methods from the Ising model canbe tuned to accommodate this non-local contribution,Appendix B illustrates how Swendsen-Wang (SW) lat-tice flips can be enhanced to satisfy the equibalanceequation of (3). Further on, Appendix C details how O ( n >
1) samples can be generated from Ising samplesused in [7]. This process is comparable to a physicalquench and aims at reducing computation times.Recall that the UV sampler in [7] manages to con-struct a chain of holographic boundaries under thetwo assumptions of scale invariance and locality. Thismethod takes advantage of the scale invariance to ap-ply lattice dilations on A to effectively map states -in a radial ordered CFT sense - from ∂ ( λ − A ) to ∂A ,where λ > ∂A . Wesuggest the reader to look at [7, II. A.] for details andformalism on this argument.The condition of scale invariance is fulfilled here onthe critical line (4). Compared to the Ising model,the main difference is in the non-locality of the Gibbsmeasure through its dependence on the total numberof loops. When restricted to a finite subsection of the a b ab Figure 1:
We are here looking at a corner of the square do-main A , which boundaries ∂A are given by the thick straightblack lines. The olive and khaki areas are two ordered do-mains with opposite sign thus the white lines between arearcs of loops entering the partition function. This graph il-lustrates a case study for a (Wolff) flip of the C virtual clus-ter from olive to khaki. a and b represent two different,incompatible, connection information beyond the boundary.In the text, we discuss how each information changes theacceptance probability of flipping C . infinite plane it means that the relative weight of twoconfigurations will depend on the loop connections be-yond the boundary. In Fig. 1, an example is given. Thecontribution to the partition function is looked at withtwo different set of connection information beyond theboundary. If using the information labelled a , theloops are continued by the blue dotted lines. We seethat the initial state has 1 loop while the final one has2. In terms of total number of loops | · | the transitionadds one loop and should be accepted with probabil-ity 1 in our chosen algorithm. If using information binstead, continuation by the pink dotted-dashed lines,the transition removes one loop and should only beaccepted with probability n − .The main point is that provided the information ofthe connections beyond ∂A , ratios of partition func-tions with changes in A are computable.Compared to our successful chain for the planar crit-ical Ising model [7], a MonteCarlo Markov chain sam-pling of this measure will need two extra features: • Through each discrete lattice dilation, it needs totrack the information on the connections - beyondthe border - of the loops touching the border to3alculate faithfully the difference ∆ | · | when at-tempting any flip. Our implementation account-ing for this effect is detailed in Appendix A. • Since at each attempted updates we need to com-pute non-local information, which is the informa-tion on the variation of the total loop number,it can be foreseen that the most global updatesneed to be favoured over the local ones. In thislight, lattice updates of the SW algorithm need tobe chosen instead of single-cluster or single-spinflips. Our modified SW lattice flips, that take intoaccount loop connections, are introduced in Ap-pendix B.In this article, we will present numerical evidencesthat a chain solving the two issues hereabove, in thelattice O ( n ) models, is able to generate the marginalfor the holographic boundaries as it is defined in [7].This will extend the work done in [7] to a larger classof non-local models The details of our implementationof a MonteCarlo Markov chain for a lattice dilation op-eration, a mixing algorithm and the parameters of ourMCMC sampling are in appendices A, B and C respec-tively. The plan of the article is as follows: Section 3will present the lattice observables and their connectionto the CFT operator algebra, and numerical checks ofthe two point functions and the fitted quantities will beintroduced. In section 4, numerical checks for dynami-cal quantities - namely the spin four point function andthe structure constant C εσσ - will be presented. Lattice operators and scaling weights.
The crit-ical line of the O ( n ) model gives a unique CFT minimalmodel for each value of n ∈ [1 , { I, σ, ε } aregiven by [9, 10], see Fig. 2: g ( n ) = 2 − cos − (cid:0) − n (cid:1) π (5)∆ I = 0 (6)∆ σ = 32 g ( n ) − ε = 4 g ( n ) − Z opera-tion σ i → − σ i ∀ i is a symmetry, they can be classified n Theoretical values ∆ σ ∆ ε Figure 2:
The scaling weights of the energy and spin oper-ators have similar profiles, both increase with n ; this meansthat correlations decay faster with increasing n . Regardingthe spin operator, a faster decay with larger n makes sense:the larger n the more energetically favourable it is to createa new loop and thus inducing a flip in the value of σ i σ i + x and thus a decrease of its statistical average. into even { I, ε } or odd { σ } operators. The local latticevariables are the binary lattice variable σ L i ∈ {− , } and the lattice energy density variable ε L i = (cid:88) j ∈N ( i ) σ L i σ L j , with N ( i ) the set of first neighbours of i . The Z sym-metry constrains the CFT operators to be scaling limitsof the lattice variables as follows: σ L i = a ∆ σ N σ σ ( a i ) + . . . , (9) ε L i = (cid:104) ε L (cid:105) I + a ∆ ε N ε ε ( a i ) + . . . , (10)with a the lattice spacing – we will set a = 1 from hereon. Correlators.
We use the sampling method detailedin Appendix C, which is essentially a MCMC withstarting point an element from the Ising samplingmethod in [7] sized 2048x2048, mixing to a critical O ( n )chain by a “quench” done by 8 successive compositionsof lattice dilations followed with the appropriate latticeSW flips.With such chains, lattice two point functions weremeasured for (cid:104) σ i σ j (cid:105) and (cid:104) ε i ε j (cid:105) along horizontal direc-tions. Graphs of the functions are given in Fig. 3 and4 respectively. The exhibited power law behaviour ismanifest. For n = 1 . , . , .
75 and 2, these allowedto fit the non universal quantities:4 | i − j | -2 -1 › σ i σ j fi ( n ) n Figure 3:
Graphs of (cid:104) σ L i σ L j (cid:105) ( n ) for n = 1 . , . , . , | i − j | in [1 , χ minimization ona x → ax b template function. n N σ N ε (cid:104) ε L (cid:105) .
25 0 . .
941 (29) 3 . . . . . .
75 0 . . . . .
205 (85) 1 . n ∆ σ ∆ ε .
25 0 . . . . . . . .
246 (15) . .
75 0 . . . . . . . (12)The bold orange entries are the exact values. Geometrical exponents on the bulk.
In [7] amethod was introduced to estimate numerically thefractal dimensions of clusters in bulk subsections, themethod was dubbed bulk finite size scaling (BFSS).These observables have known numerical values [11, 12]on the O ( n ) critical line where the domains becomefractal in the scaling limit. On our samples, the fol-lowing scaling exponent values of the boundary lengthand domain area were estimated: | i − j | -5 -4 -3 -2 -1 ( › ε L i ε L j fi − › ε L fi )( n ) n Figure 4:
Graphs of (cid:0) (cid:104) ε L i ε L j (cid:105) − (cid:104) ε L (cid:105) (cid:1) ( n ) for n =1 . , . , . , | i − j | in [5 , n = 2 curve shows some noise in the tail. Fits wereperformed to extract exponent, amplitude and disconnectedpart (12), (11). These fits were done by χ minimization ona function x → ax b + c . n length mass1 .
25 1 .
381 (3) . .
921 (9) . . .
411 (4) . .
938 (14) . .
75 1 .
427 (9) . .
897 (3) . .
502 (8) . .
876 (11) . The agreement is here satisfying. This is an addi-tional proof that the samples are sitting on the criticalline. O (1 25 0 . . [6 , 11] [65 , . . . [6 , 15] [72 , . 75 0 . . [6 , 11] [30 , . . [6 , 11] [35 , | k − i | and | i − j | respectively, takeninto account in the fit. The selection was made so as tominimize the χ of the fit, in the sense that the rangeis picked where the expansion is valid while the signalstill dominating over the measurement uncertainty. Foreach value of n , the agreement is satisfying. Lattice Stress Energy tensor of the O ( n ) loopmodels. In [7] a lattice representation of the lattice6tress energy tensor for the Ising lattice was introduced.This was done by extracting a Fourier mode of spin 2from the σσ correlator: σ ( x ) σ (0) = 1 | x | h σ (cid:16) I + x h σ c T (0) + O ( x ) (cid:17) + energychannel.termsSince any primary field will have the identity and itsdescendants (this includes T = L − I ) in its self productOPE, and knowing that σ is still a primary operator inthe O ( n > 1) loop model; this construction should stillbe valid. This maintains our definition of the latticestress energy tensor T to be: T i = (cid:88) j ∈(cid:104) i , ·(cid:105) e − iθ ij σ L i σ L j . Lattice Ward identities. The correlator (cid:104) T (0) σ ( x ) σ ( y ) (cid:105) is constrained by the CFT alge-bra to (cid:104) T (0) σ ( x ) σ ( y ) (cid:105) = h σ | x − y | σ ( x − y ) x y . Such functions entirely defined under the constraint ofmeromorphicity and the localization of their poles arecalled Ward identities. In CFT, it includes correlatorsinvolving the stress energy tensor, where knowledge ofthe CFT data is enough to deduce the analytical struc-ture as function of the insertion position of the stressenergy tensor.On the lattice, the Ward identity becomes (cid:104)T k σ L i σ L j (cid:105) = N T N σ h σ | i − j | σ ( i − j ) ( k − i ) ( k − j ) (15)and if using a prescription inserting the spin operatorsdiametrically opposed to the insertion of the stress ten-sor, e.g. i − k = − ( j − k ), (15) simplifies to a powerlaw (cid:104) T σ L n σ L − n (cid:105) = N T N σ h σ − σ | n | − σ − . Similarly to the definitions N σ and N ε , N T is the scal-ing factor between the lattice stress energy tensor andits CFT equivalent. A fit of the power law’s offset andknowledge of N σ from (11) offers a numerical estima-tion of N T . Numerical central charge. The central charge isa key parameter of a CFT. Its numerical estimationhas been in reach of MCMC methods for instance see [15]. Following our measurements of lattice correlationfunctions, it is natural to measure it by looking at theautocorrelations of the lattice stress energy tensor (cid:104)T T n (cid:105) = c N T n . Fitting this power law’s offset and using the previousestimation of N T gives access to a numerical estima-tion of c . Table (16) presents the results of the fits ofthe Ward identity introduced here-above as well as thederived estimations of the central charge for n = 1 . N T - insquare brackets, and a measurement error - due to fit-ting uncertainties - from the power law fit of (cid:104)T T (cid:105) inround brackets.The estimations presented in (16) requiring a sig-nificantly longer computational effort than the resultsintroduced previously, we had to resolve to use an im-provement of the MCMC presented in Appendix C. Todo so, the colouring SW algorithm was replaced by thealgorithm used in [16]; where the authors introduceda Swendsen-Wang algorithm with virtual FK clusterswhich are not confined inside the loops of the O ( n ) lat-tice. In the language introduced in Appendix C, the di-rect consequence is a smaller rejection probability whenflipping each FK cluster, hence a lower autocorrelationand a more efficient MCMC. Beyond this algorithm up-grade, for the runs giving (16), the parameters of theMCMC are identical to the ones given in Appendix Cexcept for the sample size increased to 1 000 000.For the same computational effort, the n = 2 chainis dominated by noise and does not allow to run theequivalent fits with decent precision. This data pointis postponed.7 (cid:104)T T (cid:105) offset N T from (cid:104)T σσ (cid:105) offset c . 25 0 . . 243 (84) 0 . 626 (19) [85] . . . . . 733 (85) [27] . . 75 0 . . . 834 (132) [67] . (16) The results presented here assert of the successful gen-eralization of the recipe introduced for the Ising modelin [7] to a class of models highly non-local in the spinvariables. This numerical study is not exhaustive andmany numerical checks are still missing such as the fourpoint correlators (cid:104) σσεε (cid:105) , (cid:104) εεεε (cid:105) , the structure constant C εεε , or the precise profile of the Ward identities.Other generalizations are being studied by the au-thor. Extending the recipe to higher dimensions - forinstance the 3d Ising model - could offer a new numer-ical approach to 3d CFT [17].Another generalization would include perturbedCFTs on a lattice with finite correlation length ξ com-parable or larger than the linear lattice size L . Suchnumerical simulations usually suffer from strong finitesize and boundary effects. Even though the perturba-tion will break the scale invariance - which is the mainingredient to this numerical recipe taking advantage oflattice dilations - preliminary results show that a tun-ing of the perturbation coupling after each dilation isenough to construct a Markov chain sampling the bulkmarginal of this massive QFT. Acknowledgements The author would like to thank B. Doyon for sug-gesting this problem as well as his continuous guid-ance throughout. The author is funded by a Gradu-ate Teaching Assistantship from KCL Department ofMathematics. A Discrete lattice dilations andtracking loop connections In ref. [7] to take advantage of the scaling invarianceand to approach the UV fixed point, we introduced an“inverse Kadanoff block-spin transformation”. Essen-tially it is a discrete lattice dilation with parameter λ > 1, mapping λ − A → A . More precisely it mapped σ i ← σ (cid:98) λ − i (cid:101) (17)where (cid:98) x (cid:101) means the closest lattice site to x . On adiscrete system it is obvious that A holds more infor-mation than λ − A , implying that (17) cannot be donedirectly in a one-to-one fashion. A prescription detailedin [7, II. C. & Fig. 4] forces (17) to be performed one-to-one and fills in the missing information, e.g assigninga spin value on the sites with no preimage, using a heat-bath weighted assignation. It appeared to be the mosteffective choice for the Ising model. This step takesplace between the dilation and the rethermalization.In the O ( n ) case, the main concern of the discretedilation procedure needs to be to track the informa-tion of the loop connections before/after dilating. Thebest effort should entirely conserve this information.Heat-bath assignations as we used in [7] cannot fulfilthis requirement as it implies a non-zero probability ofseeing a boundary spin given a value opposite to allits neighbours. Such occurrence would create a loop asan artefact of the dilation. Such loop would touch theboundary but would have no antecedent through thedilation and thus no connection information.Another prescription relaxing the one-to-one require-ment on (17) and allowing a one-to-many mapping, e.g.duplication of a spin value by being mapped to all thespins sharing the same dilation preimage, resolves theissue. No artefact loop can be created this way.One necessary improvement to the dilation algo-rithm is to track the ( C \ A ) connections existing be-tween edges touching the border. On the technicallevel, the data container we chose for storing such in-formation is a symmetrical map object: a binary treewith a hash table mapping an edge object to another8ne. This gives a map:edge X → edge Y , edge Y → edge X . We call this a connection map object. It makes it easywhen tracking a loop by jumping along its edges tojump from an edge touching the border (edge X) tothe next one on the same loop and inside A (edge Y).The next step has to be filling this map with theinformation we get after dilating. It means trackingthe information just before cropping the central do-main. This is explained by Fig. 6. When updating theboundary connection information post-dilation we seetwo scenarios here: • case labelled 1 on the graph: the loop is entirelyinside λA and when reducing to ∂A we add to theconnection map that the two edges touching ∂A are connected. Connection information is ‘created’here. • case 2 : this loop touches ∂ ( λA ) thus it is neededto use the information from the pre-dilation con-nection map object (dotted red line going out of ∂ ( λA ) telling us of a connection between the twopieces of the same loop). This information is stillrelevant to the connectedness of the loop partscontained inside ∂A . Here connection informationis updated. B Swendsen-Wang colouring algo-rithm Evolution algorithm As stated before, the motivation in describing the O ( n )loop models with spin variables is in the re-usabilityof the numerical methods known on the Ising model.Here we present how the SW evolution algorithm canbe generalized for a O ( n ) Markov chain.In the case of a MCMC on the Ising model, the lit-erature offers many options when choosing the evo-lution algorithm. We usually distinguish local up-dates changing the value of a single lattice spin at thetime (Metropolis, Glauber, ...) from non-local updatesupdating one or many clusters at each step (Wolff,Swendsen-Wang, ...). At the critical point, the latterhave proven to be very efficient against critical slow-down [18, 19]. Nonetheless these algorithms are not Figure 6: Picture of A being dilated to λA , λ ≈ . ∂ ( λ − A ), the dashed contour square, gets mappedto ∂A , the continuous contour square. Before integratingout (cid:0) ∂A, ∂ ( λA ) (cid:1) - the light gray area - by cropping it, allinformation in the loops crossing ∂A needs to be stored in anew connection map object. Cases 1 and 2 are describedin the text. O ( n ) loop gas mod-els. The culprit is obviously the non-local contributionof the total number of loops into the Hamiltonian to theflip acceptance probability. This implies directly thatany flip, for instance Metropolis or Wolff, will be ac-cepted depending on the variation of the total numberof loops. In the pessimistic scenario, calculating thisshift in the number of loops requires a computationaleffort scaling with the lattice area for each attemptedflip. This is very ineffective for local updates.The work done in [20] offers an efficient and simpleway of circumventing this obstacle. For n > 1, we knowthat the measure (3) favours configurations countingmore loops. It can be done using a n -dependent freez-ing of a subset of the loops. Freezing here means thatthe spins on both sides of each edges of a given loopbecome non-dynamical, e.g. cannot change value, withrespect to the next spin- or cluster- flip update. Qual-itatively this makes the deletion of a loop less likelyhence rewards the configurations with more loops. Thisaddon enables any algorithm - such as Metropolis,Wolff or Swendsen-Wang - to be “biased” as to sat-isfy the equibalance derivable from (3). This can beproven to happen for a freezing probability of p freezing = 1 − n . The authors of [20] dubbed this prescription “colouringalgorithm”, where colouring a loop means freezing itas defined by the paragraph here above. The namecolouring algorithm will be used from here on. Ourimplementation choice was to couple it to Swendsen-Wang (SW) lattice updates [19].From the lattice spin variables point of view, the O ( n ) loop model can be pictured as an - off critical -Ising models with random fluctuating boundaries.Our lattice flip update procedure follows two steps: • First, we read the loops individually, connectingdifferent segments of a same loop using the con-nection map information for the loops touchingthe border. For each loop an independent randomchoice is made: with a probability p colouring = 1 − n the loop is coloured, meaning that for each of itsedges, the spins on both sides will be frozen withrespect to the next step. • Second, we will run a complete SW lattice flipswhere only the unfrozen and non boundary siteswill be part of virtual FK ( flippable ) clusters. TheFK bonding probability is: p bond = 1 − x c ( n )with x c ( n ) defined in (4). For each virtual cluster,we will calculate the energy difference ∆ E involv-ing all the edges linking the spins inside the vir-tual cluster to fixed edges, namely the ones joiningthem to a border spin or to a frozen spin. Finally,the cluster may be flipped depending on a Glauberacceptance ratio: p flip = 11 + e ∆ E . Numerical Checks In [20] the colouring prescription is coupled toMetropolis updates while our implementation choicewas to add them on top of SW lattice flips. It seemsworth providing a numerical evidence of the correctnessof the implementation.An easily accessible result is the scaling weight of thespin operator. It is predicted by Coulomb gas methodsto give (7). We fitted the scaling exponent by finitesize scaling. When using conformal plus boundaries -equivalent to an external field h = + ∞ applied on theboundary spins - RG arguments say that the averagemagnetization m = L (cid:80) i σ i should scale as m = 1 L ∆ σ + subleading termswith L the linear lattice size. For ≈ 30 different val-ues of n in [1 , 2] and sizes L in { , , , } , usingcoloured SW updates as detailed above and taking 100000 measurements each separated by 10 updates gaveus Fig. 7.This graph is a first supporting evidence of thecolouring SW (cSW) evolution algorithm to satisfy theright equibalance equations. On top of that, the chainwas fairly effective with very short mixing time andnegligible autocorrelations.For an additional proof, we check if a compositionof lattice dilations and rethermalization through cSWupdates can create a sample with scale invariance, thissuccession of steps is more extensively detailed in Ap-pendix C. This is meant here in the sense that the σσ .0 1.2 1.4 1.6 1.8 2.0 n ∆ σ Coulomb Gas predictionsNumerical estimations Figure 7: The red line is the graph of the theoretical ex-pression given by (7). The orange points are the fitted scal-ing exponent values. Each point is the exponent result ofa power law fit in the lattice linear size. The uncertaintybars are included but thinner than the data points’ width.Here as well we fitted by χ minimization. The agreementis obvious and extremely accurate. correlations will be power law behaved, in the fashionof the results exhibited in [7].For n = 1 . 5, a sample of ∼ λ = 1 . ≈ 50 coloured SW lattice flips. Appendix Cwill present numerical proof that “quenching” an Isingsublattice is an efficient way of generating O ( n ) sam-ples. The correlators were checked with insertions atleast 500 lattice units from the boundaries and withseparation distances between 1 and 50 to give Fig. 8.These are are first two proofs of the efficiency of ourdilation implementation and of the proper scale invari-ance of our sample. The next appendix presents adeeper analysis of the mixing of the chain as well asa quantitative study of the bulkiness of its samples. C Sample parameters Using the two ingredients above, we present here thedetails of the MCMC used for all the measurements dis-closed in this paper. Ref. [7] presented a chain whichend products were subsections of the planar criticalIsing model on the triangular lattice. Its starting pointwas the critical Ising model on the torus. For the O ( n )we decided to use as starting point these critical Isingsamples, since these had been stored and were readilyavailable. The idea is that a fixed number of cycles of | i − j | -1 › σ i σ j fi FitExpectationSample Data Figure 8: On the x-axis, the separation between the twospin insertions. On the y-axis, the value of the average spincorrelator. Both axes are log scaled. The power law be-haviour seems obvious, the fitted exponent is 2∆ σ (1 . 5) =0 . . . . . The fit χ is ≈ − , a very low value. lattice dilations followed by coloured SW rethermaliza-tion steps, tuned to desired n value, should move thesamples by a “quench” along the O ( n ) critical line. Inthe following, this MCMC will be showed to mix atthis precise value of n .In the case of a MCMC to generate samples of O ( n =2) we propose the following timeline: • It starts with a subsection of the triangular criticalIsing model, sized 2048x2048. • It is followed by N SW = 80 cSW lattice flips, tunedto n = 2 (with fixed boundaries). • It will undergo a succession of 8 discrete latticedilations with parameters λ taking values 1.2, 1.2,1.2, 1.15, 1.1, 1.06, 1.04 and finally 1.02. The mo-tivation is to use large dilations at the beginningto send far away the initial boundaries marginallydistributed according to the Ising Gibbs distribu-tion, but they also leave a large number of dupli-cates on the border ( ∼ λ to dampen thisexcess of correlations on and near the boundary. • Inbetween each of these dilations, N SW cSW up-dates are performed to rethermalize the sample. • To present a proof that the chain does mix at the O ( n = 2) critical point, we monitored the (cid:104) σ i σ i + n (cid:105) correlations. The prescription was to take i and i + n in a central domain, at least 50 lattice sites11igure 9: Monitoring the mixing of a O ( n = 2) MCMC.Along the x-axis, the chronology of the evolution steps. Thevertical dashed lines show the occurrences of a lattice dila-tion. The y-axis is duplicated: the blue curve shows thevalue of the fitted ∆ σ ( t ) with axis markers on the left side;whereas the red curve shows the fits χ with units on theright, the latter is log scaled. The two horizontal lines showthe value of ∆ σ for the Ising and the O (2) model, orangeand chocolate respectively. away from the border, and n along horizontal di-rection with | n | ∈ [1 , 50] for computational effi-ciency. One measurement was taken every 10 cSWflips. This was repeated over ∼ (cid:104) σ i σ i + n (cid:105) ( t ) at the t step of themixing process. At each value of t we get a graphof the correlator, (cid:104) σ i σ i + n (cid:105) as a function of | n | , onwhich a power law fit is performed to extract aneffective value of the scaling weight and an uncer-tainty on the fit. Here, we use the uncertainty onthe fitted scaling weight, linked to the χ uncer-tainty of the fit. • The graph of these monitored quantities along thetimeline of the chain is given in Fig. 9. The graphshows that the fitted exponents indeed travel fromthe value it takes in the critical Ising model to itsvalue in the O (2) model. The monitoring showsthat the dilations and the rethermalization stepswork hand in hand: each dilation brings a jumptowards the O (2) endpoint. The uncertainty lineshows an increase during the transitional periodwhere the correlations have no motivation to bepower law behaved. After mixing its magnitudeis similar to what it was on the Ising sample, thishints that the end product is of similar quality or bulkiness as the chain generating the Ising sam-ples. • For each value of n in { . , . , . , } , the nu-merical estimations of the present paper were ranover a sample of size ∼ 50 000. Each sample ele-ment was independently generated by a chain justlike the one described above. D C εσσ from (cid:104) σσσσ (cid:105) In the limit where z → z , we have the known OPE: σ ( z ) σ ( z ) = 1 | z | h σ (cid:18) C εσσ ε ( z ) | z | ∆ ε + . . . (cid:19) . Taking the same limit in (13), an OPE on the lefthand side gives: (cid:104) σ ( z ) σ ( z ) σ ( z ) σ ( z ) (cid:105) = (cid:10) | z | h σ (cid:16) C εσσ ε ( z ) | z | h ε + . . . (cid:17) σ ( z ) σ ( z ) (cid:11) = 1 | z z | h σ (cid:18) C εσσ (cid:12)(cid:12)(cid:12)(cid:12) z z z z (cid:12)(cid:12)(cid:12)(cid:12) h ε + . . . (cid:19) . For simplicity we set: z = 0 , z → , z = 1 , z = 2 , giving us: (cid:104) σ ( z ) σ ( z ) σ ( z ) σ ( z ) (cid:105) = 1 | z | h σ (cid:18) C εσσ (cid:12)(cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12)(cid:12) h ε + . . . (cid:19) . Expanding the right hand side of (13) with η = z O ( z )gives: 1 | z | h σ (cid:18) . . . + B ( κ ) (cid:12)(cid:12)(cid:12)(cid:12) z (cid:12)(cid:12)(cid:12)(cid:12) h ε + . . . (cid:19) . The coefficients in front of (cid:12)(cid:12) z (cid:12)(cid:12) h ε read: C εσσ ( n ) = B ( κ ( n )) . eferences [1] G. Mussardo. Statistical field theory: an introduc-tion to exactly solved models in statistical physics .Oxford University Press, 2010.[2] P. Francesco, P. Mathieu, and D. S´en´echal. Con-formal field theory . Springer Science & BusinessMedia, 2012.[3] S. Sheffield. Exploration trees and conformal loopensembles. Duke Math. J. , 147:79, 2009.[4] S. Sheffield and W. Werner. Conformal loop en-sembles: the markovian characterization and theloop-soup construction. Ann. Math. , 176:1827,2012.[5] D. Doyon. Conformal loop ensembles and thestress-energy tensor. Lett. Math. Phys. , 103:233,2013.[6] M. Newman and G. Barkema. Monte CarloMethods in Statistical Physics . Oxford UniversityPress: New York, USA, 1999.[7] V. Herdeiro and B. Doyon. Monte carlo methodfor critical systems in infinite volume: The planarIsing model. Phys. Rev. E , 94:043322, Oct 2016.[8] M. Henkel and D. Karevski. Conformal in-variance: an introduction to loops, interfacesand stochastic Loewner evolution , volume 853.Springer Science & Business Media, 2012.[9] B. Nienhuis. Exact critical point and critical ex-ponents of O(n) models in two dimensions. Phys.Rev. Lett. , 49(15):1062, 1982.[10] B. Nienhuis. Critical behavior of two-dimensionalspin models and charge asymmetry in the coulombgas. J. Stat. Mech. , 34(5-6):731–761, 1984.[11] H. Saleur and B. Duplantier. Exact determina-tion of the percolation hull exponent in two dimen-sions. Phys. Rev. Lett. , 58:2325–2328, Jun 1987.[12] B. Nienhuis. Phase transitions and critical phe-nonema vol 11, ed. Domb and Lebowitz (NewYork: Academic). 1987.[13] A. A. Belavin, A. M. Polyakov, and A. B.Zamolodchikov. Infinite conformal symmetry intwo-dimensional quantum field theory. Nucl.Phys. B , 241(2):333–380, 1984. [14] A. Gamsa and J. Cardy. Correlation functionsof twist operators applied to single self-avoidingloops. J. Phys. A , 39(41):12983, 2006.[15] P. Bastiaansen and H. Knops. Monte carlo methodto calculate the central charge and critical expo-nents. Phys. Rev. E , 57:3784–3796, Apr 1998.[16] Y. Deng, W. Guo, and H. W. J. Blote. Clustersimulation of the O(n) loop model on the honey-comb lattice. arXiv preprint cond-mat/0605165 ,2006.[17] V. Herdeiro. in preparation .[18] U. Wolff. Collective monte carlo updating for spinsystems. Phys. Rev. Lett. , 62(4):361, 1989.[19] R. H. Swendsen and J. Wang. Nonuniversal crit-ical dynamics in monte carlo simulations. Phys.Rev. Lett. , 58(2):86, 1987.[20] C. Ding, Y. Deng, W. Guo, Xi. Qian, and H. W. J.Bl¨ote. Geometric properties of two-dimensionalO(n) loop configurations.