Extracting Physics from Topologically Frozen Markov Chains
Urs Gerber, Irais Bautista, Wolfgang Bietenholz, Héctor Mejía-Díaz, Christoph P. Hofmann
aa r X i v : . [ h e p - l a t ] O c t Extracting Physics from Topologically FrozenMarkov Chains
Urs Gerber ∗ , Irais Bautista, Wolfgang Bietenholz, Héctor Mejía-Díaz Instituto de Ciencias NuclearesUniversidad Nacional Autónoma de MéxicoA.P. 70-543, C.P. 04510 Distrito Federal, MexicoE-mail: [email protected],[email protected],[email protected], he − [email protected] Christoph P. Hofmann
Facultad de Ciencias, Universidad de ColimaBernal Díaz del Castillo 340, Colima C.P. 28045, MexicoE-mail: [email protected]
In Monte Carlo simulations with a local update algorithm, the auto-correlation with respect to thetopological charge tends to become very long. In the extreme case one can only perform reliablemeasurements within fixed sectors. We investigate approaches to extract physical informationfrom such topologically frozen simulations. Recent results in a set of s -models and gauge theoriesare encouraging. In a suitable regime, the correct value of some observable can be evaluated to agood accuracy. In addition there are ways to estimate the value of the topological susceptibility. The 32nd International Symposium on Lattice Field Theory,23-28 June, 2014Columbia University New York, NY ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ opologically Frozen Markov Chains
Urs Gerber
1. Introduction
In many relevant models, the configurations are divided into topological sectors (for periodicboundary conditions). This includes the O ( N ) models in d = ( N − ) , all the 2d CP ( N − ) models,2d and 4d Abelian gauge theory, and 4d Yang-Mills theories. The topology persists if we includefermions, hence this class of models also includes the Schwinger model, QED and QCD.In the continuum formulation, a continuous deformation of a configuration (at finite Euclideanaction) cannot change the topological charge Q ∈ Z . On the lattice there are no topological sectorsin this strict sense, but at fine lattice spacing the configurations of the above models occur in distinctsectors with local minima, separated by boundary zones of higher action. Thus it is possible — andoften useful — to introduce topological sectors also in lattice field theory, although the definitionof Q is somewhat ambiguous. For the O ( N ) models that we are going to consider, the geometricdefinition [1] has the virtue that it naturally provides integer values of Q .Most simulations in lattice field theory are performed with local update algorithms, such asthe Metropolis algorithm for spin models, the heat-bath algorithm for pure gauge theories, andthe Hybrid Monte Carlo algorithm for QCD with dynamical quarks. If there are well-separatedtopological sectors, such simulations may face a severe problem: a Markov chain hardly everchanges Q . Thus the simulation tends to get stuck in one topological sector for an extremelylong computation time. Such a tremendous topological auto-correlation time was observed e.g. by the JLQCD Collaboration in their QCD simulations with dynamical overlap quarks [2]. ForQCD simulations with non-chiral quarks ( e.g. given by Wilson fermions) the problem has beenless severe so far, i.e. for lattice spacings a & .
05 fm that have typically been used. However, inthe future even finer lattices will be employed, and then this problem will become manifest.So how can we measure the expectation value of some observable, h W i , or the topologicalsusceptibility c t = V (cid:0) h Q i − h Q i (cid:1) , if only topologically frozen simulations can be performed? Lüscher suggested open boundary conditions, so Q can change continuously [3]. This over-comes the problem, but giving up integer Q has disadvantages, like losing the link to aspects offield theory in the continuum, e.g. regarding the e -regime of QCD.Here we investigate approaches where periodic boundaries, and therefore Q ∈ Z , are pre-served. In the framework of non-linear s -models, we test methods to extract physical results fromMarkov chains, which are permanently trapped in a single topological sector, hence numericalmeasurements are available only at fixed Q . We start with a procedure to determine c t from thecorrelation of the topological charge density, which was introduced by Aoki, Fukaya, Hashimotoand Onogi [4]. Then we probe a way to assemble an expectation value h W i from topologicallyrestricted results h W i | Q | . That approach is based on the Brower-Chandrasekharan-Negele-Wiese(BCNW) formula [5], which also yields an estimate for c t .
2. Correlation of the topological charge density
Ref. [4] derived an approximate formula for the correlation of the topological charge density q , at topological charge ± Q and large separation | x | (we now use lattice units), h q q x i | Q | , | x |≫ ≈ − c t V + Q V . (2.1) V is the volume, and we will deal with parity symmetric models, where h Q i = opologically Frozen Markov Chains Urs Gerber
The derivation assumes h Q i to be large, and | Q | / h Q i to be small. Therefore we will limit ourconsiderations to the sectors with | Q | ≤
2. We are going to consider the 1d O ( ) model and the 2d O ( ) model, and the explicit condition for h Q i = c t V will be tested.In our simulations we use the Wolff cluster algorithm [7], which performs non-local clusterupdates. Hence we can also measure c t directly, which is useful for testing this method in viewof other models (in particular gauge theories), where no efficient cluster algorithm is available.Preliminary results were anticipated in Ref. [8], and Ref. [9] presented before a related study (withdifferent densities) in 2-flavour QCD.The 1d O ( ) model, or quantum rotor, describes a free quantum mechanical scalar particleon the circle S . We use periodic boundary conditions in Euclidean time over the size L . Thecontinuum formulation deals with an angle j ( x ) , where j ( ) = j ( L ) . The lattice variables are theangles j x , x = , . . . L , with j = j L + . We define the nearest site difference as D j x = ( j x + − j x ) mod 2 p ∈ ( − p , p ] , (2.2) i.e. the modulus function acts such that it minimises the absolute value. This yields the (geometri-cally defined) topological charge density q x and the topological charge Q , q x = p D j x , Q = L (cid:229) x = q x ∈ Z . (2.3)We now give the continuum action and the three lattice actions — standard action, Mantonaction [10] and constraint action [11] — that we studied, S continuum [ j ] = b Z L dx ˙ j ( x ) , S standard [ j ] = b L (cid:229) x = ( − cos D j x ) , S Manton [ j ] = b L (cid:229) x = D j x , S constraint [ j ] = ( D j x < d ∀ x + ¥ otherwise . (2.4)The parameter b corresponds here to the moment of inertia, and d is the constraint angle. Thecontinuum limit is attained at b → ¥ and d →
0, respectively. In the limit L → ¥ , c t and thecorrelation length x are known analytically for all four actions in eqs. (2.4) [6, 11].Figures 1 and 2 show results for the standard action, the Manton action and the constraintaction at different sizes L and parameters b and d . They are all in excellent agreement with theprediction based on eq. (2.1) (horizontal lines), even down to h Q i < O ( ) model, or Heisenberg model, on square lattices of size L × L ,with classical spins ~ e x ∈ S . Here we simulated the standard action and the constraint action, whichare analogous to the formulations (2.4), S standard [ ~ e ] = b (cid:229) x , m ( − ~ e x · ~ e x + ˆ m ) , S constraint [ ~ e ] = ( ~ e x · ~ e x + ˆ m > cos d ∀ x , m = , + ¥ otherwise . (2.5)Also here we use the geometric definition of the topological charge, which is written down explic-itly in Ref. [11]. Figure 3 shows results for the topological charge density correlation. Again thecomparison to the prediction (2.1) works in all cases (in this context we don’t have to worry about Actually this formula also involves a kurtosis term (which vanishes for Gaussian Q -distributions). However, itscontribution is negligible in all examples that we considered, so here we skip that term. opologically Frozen Markov Chains Urs Gerber -0.0002 0 0.0002 0.0004 0.0006 0 20 40 60 80 100 < q q x > x1d O(2), Standard action, b = 2, L = 100Q = 0|Q| = 1|Q| = 2- c t / L- c t / L + 1 / L - c t / L + 4 / L -0.0002 0 0.0002 0.0004 0.0006 0 20 40 60 80 100 < q q x > x1d O(2), Manton action, b = 2, L = 100Q = 0|Q| = 1|Q| = 2- c t / L- c t / L + 1 / L - c t / L + 4 / L Figure 1:
The topological charge density correlation over a separation of x lattice spacings for the standardaction (on the left) and for the Manton action (on the right), both at L =
100 and b =
2. This implies x = . h Q i = .
936 for S standard , and x = . h Q i = .
266 for S Manton . For comparison, we showthe prediction based on eq. (2.1), where we insert the measured values of c t . -0.001 0 0.001 0.002 0.003 0.004 0 10 20 30 40 50 < q q x > x1d O(2), Constraint action, d = 2 p / 3, L = 50Q = 0|Q| = 1|Q| = 2- c t / L- c t / L + 1 / L - c t / L + 4 / L -0.0002 0 0.0002 0.0004 0.0006 0.0008 0.001 0 20 40 60 80 100 < q q x > x1d O(2), Constraint action, d = 1, L = 100Q = 0|Q| = 1|Q| = 2- c t / L- c t / L + 1 / L - c t / L + 4 / L Figure 2:
The topological charge density correlation over a separation of x lattice spacings for the constraintaction at d = p / L =
50 (on the left, with x = . h Q i = . d = L =
100 (on the right,with x = . h Q i = . c t . the fact that c t · x diverges logarithmically in the continuum limit). However, we also observethat for increasing volume it becomes soon difficult to resolve a clear signal from the statisticalnoise (as required for the determination of c t ), even with the huge statistics provided by the clusteralgorithm. The quantitative results will be given in Ref. [13].Thus we confirm that formula (2.1) is a valid approximation over a broad set of parameters.Nevertheless, in view of 4d quantum field theory its application is not promising, since for largevolume it becomes very statistics demanding. That limitation is in agreement with the conclusion ofan earlier study in the 2-flavour Schwinger model with dynamical overlap hypercube fermions (andwith the plaquette gauge action) [12]: at b = V = ×
16 and fermion masses m = . . . . .
06a statistics of O ( ) configurations in one topological sector was insufficient to determine c t inthis way; to achieve this to about 2 digits would take at least O ( ) configurations.
3. Applications of the BCNW formula
We now turn to the more ambitious goal of evaluating an observable h W i , when only somevalues h W i | Q | — at various | Q | and volumes — are available. To this end, we use an approximate4 opologically Frozen Markov Chains Urs Gerber < q q x > Q = 0|Q| = 1|Q| = 2- c t / V- c t / V + 1 / V - c t / V + 4 / V
2d O(3), Standard action, L=12, b=1 < q q x > Q = 0|Q| = 1|Q| = 2- c t / V- c t / V + 1 / V - c t / V + 4 / V
2d O(3), Standard action, L=16, b=1 |x| / 2 < q q x > Q = 0|Q| = 1|Q| = 2- c t / V- c t / V + 1 / V - c t / V + 4 / V
2d O(3), Constraint action, L=16, d=0.55p |x| / 2 -1e-0501e-052e-053e-054e-05 < q q x > Q = 0|Q| = 1|Q| = 2- c t / V- c t / V + 1 / V - c t / V + 4 / V
2d O(3), Constraint action, L=32, d=0.55p
Figure 3:
The topological charge density correlation in the 2d O ( ) model on L × L lattices for the standardaction at b = x ≃ . h Q i = .
46 at L =
12 and h Q i = .
39 at L =
16) and for the constraintaction at d = . p (below, x ≃ . h Q i = .
63 at L =
16 and h Q i = .
86 at L = c t . It works well, even at h Q i = .
63 the result is reasonable, although slight deviations from the prediction show up. But the lastplot illustrates that for increasing volume the signal get lost in the statistical noise. formula, which was derived in Ref. [5], h W i | Q | ≈ h W i + cV c t (cid:16) − Q V c t (cid:17) . (3.1)Our input are measured values for the left-hand-side in various | Q | and V , and a fit determines h W i , c t and c , where the former two are of interest. We refer to a regime of moderate V , where thesethree quantities practically take their infinite-volume values, but the h W i | Q | are still well distinct.This is the beginning of an expansion in 1 / h Q i , hence h Q i should be large, but what thatmeans has to explored numerically. Moreover the assumption of a small value of | Q | / h Q i isinvolved again (see also the re-derivation in Ref. [12]), hence we only use sectors with | Q | ≤ h W i and c t was addressed in Refs. [14–16], and will be discussed further in Ref. [13].As our observables we consider the action density s = h S i / V and the magnetic susceptibility5 opologically Frozen Markov Chains Urs Gerber s = < S > / V LStandard action, b = 1all sectorsQ = 0|Q| = 1|Q| = 2 35 36 37 38 39 40 60 80 100 120 140 c m LConstraint action, d = 0.55 p all sectorsQ = 0|Q| = 1|Q| = 2 Figure 4:
The action density s = h S i / V for the 2d O ( ) model on L × L lattice (standard action, b = x ≃ . d = . p ( x ≃ . | Q | = , c m = h ~ M i / V (where ~ M = (cid:229) x ~ e x is the magnetisation, and h ~ M i = ~ O ( ) model in V = L × L are shown in the plots of Figure 4, which reveal the aforementioned regimesof “moderate V ”. The fitting results involving the sectors | Q | = , ,
2, and various ranges in L inthose regimes, are given in Table 1. In particular we see an impressive precision of the values for c m , and also the fitting results for s and c t are quite good. Standard action directly measuredfitting range for L −
24 16 −
28 16 −
32 in all sectors at L = s c t Constraint action directly measuredfitting range for L −
64 48 −
96 48 −
128 in all sectors at L = c m c t Table 1:
Above: the action density s = h S i / V extracted from fits to the BCNW formula (3.1), at | Q | ≤ L . For L ≥
16 the directly measured s stabilises. It is close to the fitting results.Below: the susceptibilities c m and c t , extracted from fits in various ranges of the L . For L ≥
48 the directlymeasured c m stabilises, and the results in distinct sectors converge quite well around L = . . . c m . For both observables, also the fitting resultsfor c t are correct within less than 2 s .
4. Conclusions
In simulations with local update algorithms and fine lattices, the Monte Carlo history tends tobe confined to a single topological sector for an extremely long (simulation) time. This rises ques-tions about the ergodicity (even within one sector). Here we do not address this conceptual issue;we trust the topologically restricted measurements of h W i | Q | , and try to interpret them physically.In very large volumes V , the restricted expectation values all coincide with the physical result, h W i | Q | ≡ h W i , cf. eq. (3.1), but in practical simulations such large volumes are often inaccessible.For smaller V , where h W i is well converged to its large- V limit, but the h W i | Q | are still significantlydistinct, the BCNW formula (3.1) often allows us to determine h W i to a good accuracy, and it also6 opologically Frozen Markov Chains Urs Gerber provides useful results for c t . It is favourable to employ only the sectors with | Q | ≤
2, and (roughlyspeaking) the method is successful if h Q i & . h Q i & /
3, we can still measure c t from the topological chargedensity correlation h q q x i | Q | . The (theoretical) condition of a large separation | x | turns out to beharmless in practice, but for increasing V the wanted signal decreases very rapidly. Therefore thatmethod is hardly promising for 4d models, where — in reasonable volumes — the signal wouldmost likely be overshadowed by statistical noise.On the other hand, the BCNW formula is promising for applications in QCD, where typicalsimulations take place at h Q i = O ( ) . This observation is supported by studies in the Schwingermodel [12, 15, 17], the quantum rotor with a potential [14] and in 4d SU ( ) gauge theory [16],which will be reported in detail in Ref. [13]. Acknowledgements:
We are indebted to Christopher Czaban, Arthur Dromard, Lilian Prado andMarc Wagner for valuable communication and collaboration. This work was supported by the Mex-ican
Consejo Nacional de Ciencia y Tecnología (CONACyT) through project 155905/10 “Físicade Partículas por medio de Simulaciones Numéricas”, as well as DGAPA-UNAM. The simulationswere performed on the cluster of the Instituto de Ciencias Nucleares, UNAM.
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