An efficient algorithm for numerical computations of continuous densities of states
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Abstract
In Wang–Landau type algorithms, Monte-Carlo updates are performed with respect to the density of states, which is iteratively refined during simulations. The partition function and thermodynamic observables are then obtained by standard integration. In this work, our recently introduced method in this class (the LLR approach) is analysed and further developed. Our approach is a histogram free method particularly suited for systems with continuous degrees of freedom giving rise to a continuum density of states, as it is commonly found in lattice gauge theories and in some statistical mechanics systems. We show that the method possesses an exponential error suppression that allows us to estimate the density of states over several orders of magnitude with nearly constant relative precision. We explain how ergodicity issues can be avoided and how expectation values of arbitrary observables can be obtained within this framework. We then demonstrate the method using compact U(1) lattice gauge theory as a show case. A thorough study of the algorithm parameter dependence of the results is performed and compared with the analytically expected behaviour. We obtain high precision values for the critical coupling for the phase transition and for the peak value of the specific heat for lattice sizes ranging from \(8^4\) to \(20^4\). Our results perfectly agree with the reference values reported in the literature, which covers lattice sizes up to \(18^4\). Robust results for the \(20^4\) volume are obtained for the first time. This latter investigation, which, due to strong metastabilities developed at the pseudo-critical coupling of the system, so far has been out of reach even on supercomputers with importance sampling approaches, has been performed to high accuracy with modest computational resources. This shows the potential of the method for studies of first order phase transitions. Other situations where the method is expected to be superior to importance sampling techniques are pointed out.
1 Introduction and motivations
Monte-Carlo methods are widely used in theoretical physics, statistical mechanics and condensed matter (for an overview, see e.g. [1]). Since the inception of the field [2], most of the applications have relied on importance sampling, which allows us to evaluate stochastically with a controllable error multi-dimensional integrals of localised functions. These methods have immediate applications when one needs to compute thermodynamic properties, since statistical averages of (most) observables can be computed efficiently with importance sampling techniques. Similarly, in lattice gauge theories, most quantities of interest can be expressed in the path integral formalism as ensemble averages over a positive-definite (and sharply peaked) measure, which, once again, provide an ideal scenario for applying importance sampling methods.
However, there are noticeable cases in which Monte-Carlo importance sampling methods are either very inefficient or produce inherently wrong results for well understood reasons. Among those cases, some of the most relevant situations include systems with a sign problem (see [3] for a recent review), direct computations of free energies (comprising the study of properties of interfaces), systems with strong metastabilities (for instance, a system with a first order phase transition in the region in which the phases coexist) and systems with a rough free energy landscape. Alternatives to importance sampling techniques do exist, but generally they are less efficient in standard cases and hence their use is limited to ad hoc situations in which more standard methods are inapplicable. Noticeable exceptions are micro-canonical methods, which have experienced a surge in interest in the past 15 years. Most of the growing popularity of those methods is due to the work of Wang and Landau [4], which provided an efficient algorithm to access the density of states in a statistical system with a discrete spectrum. Once the density of states is known, the partition function (and from it all thermodynamic properties of the system) can be reconstructed by performing one-dimensional numerical integrals. Histogram-based straightforward generalisations of the Wang–Landau algorithm to models with a continuum spectrum have been shown to break down even on systems of moderate size [5, 6], hence more sophisticate techniques have to be employed, as done for instance in [7], where the Wang–Landau method is used to compute the weights for a multi-canonical recursion (see also [8]).
A very promising method, here referred to as the logarithmic linear relaxation (LLR) algorithm, was introduced in [9]. The potentialities of the method were demonstrated in subsequent studies of systems afflicted by a sign problem [10, 11], in the computation of the Polyakov loop probability distribution function in two-colour QCD with heavy quarks at finite density [12] and—rather unexpectedly—even in the determination of thermodynamic properties of systems with a discrete energy spectrum [13].
The main purpose of this work is to discuss in detail some improvements of the original LLR algorithm and to formally prove that expectation values of observables computed with this method converge to the correct result, which fills a gap in the current literature. In addition, we apply the algorithm to the study of compact U(1) lattice gauge theory, a system with severe metastabilities at its first order phase transition point that make the determination of observables near the transition very difficult from a numerical point of view. We find that in the LLR approach correlation times near criticality grow at most quadratically with the volume, as opposed to the exponential growth that one expects with importance sampling methods. This investigation shows the efficiency of the LLR method when dealing with systems having a first order phase transition. These results suggest that the LLR method can be efficient at overcoming numerical metastabilities in other classes of systems with a multi-peaked probability distribution, such as those with rough free energy landscapes (as commonly found, for instance, in models of protein folding or spin glasses).
The rest of the paper is organised as follows. In Sect. 2 we cover the formal general aspects of the algorithm. The investigation of compact U(1) lattice gauge theory is reported in Sect. 3. A critical analysis of our findings, our conclusions and our future plans are presented in Sect. 4. Finally, some technical material is discussed in the appendix. Some preliminary results of this study have already been presented in [14].
2 Numerical determination of the density of states
2.1 The density of states
Owing to formal similarities between the two fields, the approach we are proposing can be applied to both statistical mechanics and lattice field theory systems. In order to keep the discussion as general as possible, we shall introduce notations and conventions that can describe simultaneously both cases. We shall consider a system described by the set of dynamical variables \(\phi \), which could represent a set of spin or field variables and are assumed to be continuous. The action (in the field theory case) or the Hamiltonian (for the statistical system) is indicated by S and the coupling (or inverse temperature) by \(\beta \). Since the product \(\beta S\) is dimensionless, without loss of generality we will take both S and \(\beta \) dimensionless.
2.2 The LLR method
In the following, we will discuss the practical implementation by addressing two questions: (i) How do we solve the non-linear equation? (ii) How do we deal with the statistical uncertainty since the Monte-Carlo method only provides stochastic estimates for the expectation value \(\left\langle \left\langle \Delta E \right\rangle \right\rangle _k (a)\)?
Let us address question (ii) now. We have already pointed out that we have only a stochastic estimate for the expectation value \(\left\langle \left\langle \Delta E \right\rangle \right\rangle _k (a) \) and the convergence of the Newton–Raphson method is necessarily hampered by the inevitable statistical error of the estimator. This problem, however, has been already solved by Robbins and Monroe [15].
2.3 Observables and convergence with \(\delta _E\)
We have already pointed out that expectation values of observables depending on the action only can be obtained by a simple integral over the density of states (see (2.2)). Here we develop a prescription for determining the values of expectations of more general observables by folding with the numerical density of states and analyse the dependence of the estimate on \(\delta _E\).
2.4 The numerical algorithm
So far, we have shown that a piecewise continuous approximation of the density of states that is linear in intervals of sufficiently small amplitude \(\delta _E\) allows us to obtain a controlled estimate of averages of observables and that the angular coefficients \(a_i\) of the linear approximations can be computed in each interval i using the Robbins–Monro recursion (2.30). Imposing the continuity of \(\log \rho (E)\), one can then determine the latter quantity up to an additive constant, which does not play any role in cases in which observables are standard ensemble averages.
\(N_{\mathrm {TH}}\), the number of Monte-Carlo updates in the restricted energy interval before starting to measure expectation values;
\(N_{\mathrm {SW}}\), the number of iterations used for computing expectation values;
\(N_{\mathrm {RM}}\), the number of Robbins–Monro iterations for determining \(a_i\);
\(N_B\), number of final values from the Robbins–Monro iteration subjected to a subsequent bootstrap analysis.
Since the \(a_i\) are determined stochastically, a different reiteration of the algorithm with different starting conditions and different random seeds would produce a different value for the same \(a_i\). The stochastic nature of the process implies that the distribution of the \(a_i\) found in different runs is Gaussian. The generated ensemble of the \(a_i\) can then be used to determine the error of the estimate of observables using analysis techniques such as jackknife and bootstrap.
The parameters \(E_{\mathrm {min}}\) and \(E_{\mathrm {max}}\) depend on the system and on the phenomenon under investigation. In particular, standard thermodynamic considerations on the infinite volume limit imply that if one is interested in a specific range of temperatures and the studied observables can be written as statistical averages with Gaussian fluctuations, it is possible to restrict the range of energies between the energy that is typical of the smallest considered temperature and the energy that is typical of the highest considered temperature. Determining a reasonable value for the amplitude of the energy interval \(\delta _E\) and the other tuneable parameters \(N_{\mathrm {SW}}\), \(N_{\mathrm {TH}}\), \(N_{\mathrm {RM}}\) and \(N_{\mathrm {A}}\) requires a modest amount of experimenting with trial values. In our applications we found that the results were very stable for wide ranges of values of those parameters. Likewise, \(\bar{a}_i\), the initial value for the Robbins–Monro recursion in interval i, does not play a crucial role; when required and possible, an initial value close to the expected result can be inferred inverting \(\langle E (\beta )\rangle \), which can be obtained with a quick study using conventional techniques.
The average \(\left\langle \left\langle \dots \right\rangle \right\rangle \) imposes an update that restricts configurations to those with energies in a specific range. In most of our studies, we have imposed the constraint analytically at the level of the generation of the newly proposed variables, which results in a performance that is comparable with that of the unconstrained system. Using a simple-minded more direct approach, in which one imposes the constraint after the generation of the proposed new variable, we found that in most cases the efficiency of Monte-Carlo algorithms did not drop drastically as a consequence of the restriction, and even for systems like SU(3) (see Ref. [9]) we were able to keep an efficiency of at least 30 % and in most cases no less than 50 % with respect to the unconstrained system.
2.5 Ergodicity
Our implementation of the energy restricted average \(\left\langle \left\langle \cdots \right\rangle \right\rangle \) assumes that the update algorithm is able to generate all configurations with energy in the relevant interval starting from configurations that have energy in the same interval. This assumption might be too strong when the update is local^{2} in the energy (i.e. each elementary update step changes the energy by a quantity of order one for a system with total energy of order V) and there are topological excitations that can create regions with the same energy that are separated by high energy barriers. In these cases, which are rather common in gauge theories and statistical mechanics^{3}, generally in order to go from one acceptable region to the other one has to travel through a region of energies that is forbidden by an energy-restricted update method such as the LLR. Hence, by construction, in such a scenario our algorithm will get trapped in one of the allowed regions. Therefore, the update will not be ergodic.
As already noticed in [18], the replica exchange step is amenable to parallelisation and hence can be conveniently deployed in calculations on massively parallel computers. Note that the replica exchange step adds another tuneable parameter to the algorithm, which is the number \(N_{\mathrm {SWAP}}\) of configurations swaps during the Monte-Carlo simulation at a given Monte-Carlo step. A modification of the LLR algorithm that incorporates this step can easily be implemented.
2.6 Reweighting with the numerical density of states
This procedure of using a modified ensemble followed by re-weighting is inspired by the multi-canonical method [20], the only substantial difference being the recursion relation for determining the weights. Indeed for U(1) lattice gauge theory a multi-canonical update for which the weights are determined starting from a Wang–Landau recursion is discussed in [7]. We also note that the procedure used here to restrict ergodically the energy interval between \(E_{\mathrm {min}}\) and \(E_{\mathrm {max}}\) can easily be implemented also in the replica exchange method analysed in the previous subsection.
3 Application to compact U(1) lattice gauge theory
3.1 The model
The existence of topological sectors and the presence of a transition with exponentially suppressed tunnelling times can provide robust tests for the efficiency and the ergodicity of our algorithm. This motivates our choice of compact U(1) for the numerical investigation presented in this paper.
3.2 Simulation details
The study of the critical properties of U(1) lattice gauge theory is presented in this section. In order to test our algorithm, we investigated the behaviour of specific heat as a function of the volume. This quantity has been carefully investigated in previous studies, and as such provides a stringent test of our procedure. In order to compare data across different sizes, our results will be often provided normalised to the number of plaquette \(6 L^4 = 6V\).
Values of the tuneable parameters of the LLR algorithm used in our numerical investigation, in the last column we report the total number of global MC steps needed to perform the entire investigation
L | \(E_{\mathrm {min}}/(6V)\) | \(E_{\mathrm {max}}/(6V)\) | \(N_{SW}\) | \(N_{RM}\) | \((E_{\mathrm {max}}-E_{\mathrm {min}})/\delta _E\) | MC Steps |
---|---|---|---|---|---|---|
8 | 0.5722222 | 0.67 | 250 | 600 | 512 | \(7.7 \,10^7\) |
10, 12, 14, | 0.59 | 0.687777 | 200 | 400 | 512 | \(4.1\,10^7\) |
16, 18, 20 |
One of our first analyses was a screening for potential ergodicity violations with the LLR approach. As detailed in Sect. 2.5, these can emerge for LLR simulations using contiguous intervals as is the case for the U(1) study reported in this paper. To this aim, we calculated the action expectation value \(\langle E \rangle \) for a \(12^4\) lattice for several values using the LLR method and using the re-weighting with respect to the estimate \(\tilde{\rho }\). Since the latter approach is conceptually free of ergodicity issues, any violations by the LLR method would be flagged by discrepancy. Our findings are summarised in Fig. 3 and the corresponding table. We find good agreement for the results from both methods. This suggests that topological objects do not generate energy barriers that trap our algorithm in a restricted section of configuration space. Said in other words, for this system the LLR method using contiguous intervals seems to be ergodic.
3.3 Volume dependence of \(\log \tilde{\rho }\) and computational cost of the algorithm
As mentioned before, the issue facing importance sampling studies at first order phase transitions are connected with tunnelling times that grow exponentially with the volume. With the LLR method, the algorithmic cost is expected to grow with the size of the system as \(V^2\), where one factor of V comes from the increase of the size and the other factor of V comes from the fact that one needs to keep the width of the energy interval per unit of volume \(\delta _E/V\) fixed, as in the large-volume limit only intensive quantities are expected to determine the physics. One might wonder whether this apparently simplistic argument fails at the first order phase transition point. This might happen if the dynamics is such that a slowing down takes place at criticality. In the case of compact U(1), for the range of lattice sizes studied here, we have found that the computational cost of the algorithm is compatible with a quadratic increase with the volume.
3.4 Numerical investigation of the phase transition
\(\beta _c(L)\) evaluated with the LLR algorithm and reference data from [32]
L | \( \beta _{c}(L)\) present method | \(\beta _{c}(L)\) reference values |
---|---|---|
8 | 1.00744(2) | 1.00741(1) |
10 | 1.00939(2) | 1.00938(2) |
12 | 1.010245(1) | 1.01023(1) |
14 | 1.010635(5) | 1.01063(1) |
16 | 1.010833(4) | 1.01084(1) |
18 | 1.010948(2) | 1.010943(8) |
20 | 1.011006(2) |
Estimates of \(\beta _c\) for various choices of the fit parameters. In bold the best fits
\( L_{\mathrm {min}} \) | \( k_{\mathrm {max}} \) | \( \beta _{c} \) | \(\chi ^{2}_{\mathrm {red}}\) |
---|---|---|---|
14 | 1 | 1.011125(3) | 0.91 |
12 | 1 | 1.011121(3) | 2.42 |
12 | 2 | 1.011129(4) | 0.67 |
10 | 1 | 1.011116(5) | 7.44 |
10 | 2 | 1.011127(3) | 0.60 |
8 | 1 | 1.011093(5) | 90.26 |
8 | 2 | 1.011126(2) | 0.62 |
\(C_V(\beta _c(L))\) evaluated with the LLR algorithm and reference data from [32]. Results for a \(20^4\) lattice have never been reported before in the literature
L | \( C_V/(6V) \) peak present work | \( C_V/(6V) \) peak from [32] |
---|---|---|
8 | 0.000551(2) | 0.000554(1) |
10 | 0.000384(2) | 0.000385(1) |
12 | 0.0002971(11) | 0.000298(1) |
14 | 0.0002537(8) | 0.000254(1) |
16 | 0.0002272(7) | 0.000226(2) |
18 | 0.0002097(5) | 0.000211(2) |
20 | 0.0002007(4) |
Estimates of G for various choices of the fit parameters. In bold the best fits
\( L_{\mathrm {min}} \) | \( k_{\mathrm {max}} \) | G | \(\chi ^{2}_{\mathrm {red}}\) |
---|---|---|---|
14 | 1 | 0.02712(9) | 4.6 |
12 | 1 | 0.0273(2) | 31 |
12 | 2 | 0.02688(7) | 1.4 |
10 | 1 | 0.0276(2) | 74 |
10 | 2 | 0.02710(12) | 9.7 |
10 | 3 | 0.02681(9) | 1.4 |
8 | 1 | 0.0281(4) | 335 |
8 | 2 | 0.02731(15) | 26 |
8 | 3 | 0.02703(11) | 6.7 |
Location of the peak of the probability density in the two meta-stable phases
L | \( E_1/(6V) \) | \(E_2/(6V)\) |
---|---|---|
12 | 0.6263(5) | 0.65580(14) |
14 | 0.6264(2) | 0.65532(5) |
16 | 0.6272(2) | 0.65512(4) |
18 | 0.6274(4) | 0.65495(6) |
20 | 0.6275(2) | 0.65491(7) |
3.5 Discretisation effects
Values of \(\delta _E\) used to perform the study of the discretisation effects. The other simulation parameters are kept identical to the one reported in Table 1
L | \((E_{\mathrm {max}}-E_{\mathrm {min}})/\delta _E\) |
---|---|
10 | 8, 16, 32, 64, 128, 512 |
12 | 8, 16, 20, 32, 64, 128, 512 |
14 | 16, 32, 64, 512 |
16 | 16, 32, 64, 128, 512 |
The coefficient \(b_{dis}\) for different lattice sizes
L | \( b_{dis} \) |
---|---|
8 | −3.1(2) \(10^{-10}\) |
10 | −5.9(4) \(10^{-11}\) |
12 | −1.8(1) \(10^{-11}\) |
14 | −4(1) \(10^{-12}\) |
16 | −9(3) \(10^{-13}\) |
4 Discussion, conclusions and future plans
The density of states \(\rho (E)\) is a measure of the number of configurations on the hyper-surface of a given action E. Knowing the density of states relays the calculation of the partition function to performing an ordinary integral. Wang–Landau type algorithms perform Markov chain Monte-Carlo updates with respect to \(\rho \) while improving the estimate for \(\rho \) during simulations. The LLR approach, first introduced in [9], uses a non-linear stochastic equation (see (2.17)) for this task and is particularly suited for systems with continuous degrees of freedom. To date, the LLR method has been applied to gauge theories in several publications, e.g. [10, 11, 12, 14], and it has turned out in practice to be a reliable and robust method. In the present paper, we have thoroughly investigated the foundations of the method and have presented high-precision results for the U(1) gauge theory to illustrate the excellent performance of the approach.
- (i)
It solves an overlap problem in the sense that the method can specifically target the action range that is of particular importance for an observable. This range might easily be outside the regime for which standard MC methods would be able to produce statistics.
- (ii)
It features exponential error suppression: although the density of states \(\rho \) spans many orders of magnitude, \(\tilde{\rho }\), the density of states defined from the linear approximation of its log, has a nearly constant relative error (see Sect. 2.2) and the numerical determination of \(\tilde{\rho }\) preserves this level of accuracy.
Key ingredient for the LLR approach is the double-bracket expectation value [9] (see (2.13)). It appears as a standard Monte-Carlo expectation value over a finite action interval of size \(\delta _E\) and with the density of states as a re-weighting factor. The derivative of the density of states a(E) emerges from an iteration involving these Monte-Carlo expectation values. This implies that their statistical errors interfere with the convergence of the iteration. This might introduce a bias preventing the iteration to converge to the true derivative a(E). We resolved this issue by using the Robbins–Monro formalism [15]: we showed that a particular type of under-relaxation produces a normal distribution of the determined values a(E) with the mean of this distribution coinciding with the correct answer (see Sect. 2.2).
- (1)
The LLR simulations restrict the Monte-Carlo updates to a finite action interval and might therefore be prone to ergodicity violations.
- (2)
The LLR approach seems to be limited to the calculation of action dependent observables only.
To address issue (2), we first point out that the latter re-weighting approach produces a sequence of configurations that can be used to calculate any observable by averaging with the correct weight. Second, we have developed in Sect. 2.2 the formalism to calculate any observable by a suitable sum over a combination of the density of states and double-bracket expectation values involving the observable of interest. We were able to show that the order of convergence (with the size \(\delta _E\) of the action interval) for these observables is the same as for \(\rho \) itself (i.e., \(\mathcal{O}(\delta _E^2)\)).
In view of the features of the density of states approach, our future plans naturally involve investigations that either are enhanced by the direct access to the partition function (such as the calculation of thermodynamical quantities) or that are otherwise hampered by an overlap problem. These, most notably, include complex action systems such as cold and dense quantum matter. The LLR method is very well equipped for this task since it is based upon Monte-Carlo updates with respect to the positive (and real) estimate of the density of states and features an exponential error suppression that might beat the resulting overlap problem. Indeed, a strong sign problem was solved by LLR techniques using the original degrees of freedom of the \(Z_3\) spin model [10, 11]. We are currently extending these investigations to other finite density gauge theories. QCD at finite densities for heavy quarks (HDQCD) is work in progress. We have plans to extend the studies to finite density QCD with moderate quark masses.
Footnotes
- 1.
The most general case in which \(O(\phi )\) cannot be written as a function of E is discussed in Sect. 2.3.
- 2.
This is for instance the case for the popular heat-bath and Metropolis update schemes.
- 3.
For instance, in a d-dimensional Ising system of size \(L^d\), to go from one ground state to the other one needs to create a kink, which has energy growing as \(L^{d-1}\).
Notes
Acknowledgments
We thank Ph. de Forcrand for discussions on the algorithm that led to the material reported in Sect. 2.6. The numerical computations have been carried out using resources from HPC Wales (supported by the ERDF through the WEFO, which is part of the Welsh Government) and resources from the HPCC Plymouth. KL and AR are supported by the Leverhulme Trust (Grant RPG-2014-118) and STFC (Grant ST/L000350/1). BL is supported by STFC (Grant ST/L000369/1). RP is supported by STFC (Grant ST/L000458/1).
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