Testing a Fourier-Accelerated Hybrid Monte Carlo Algorithm

Physics Letters B 528 (2002) 301–305 www.elsevier.com/locate/npe

Testing a Fourier-accelerated hybrid Monte Carlo algorithm

Simon Catterall ∗, Sergey Karamov

Physics Department, Syracuse University, Syracuse, NY 13244, USA Received 18 December 2001; received in revised form 17 January 2002; accepted 20 January 2002 Editor: P.V. Landshoff

Abstract We describe a Fourier-accelerated hybrid Monte Carlo algorithm suitable for dynamical fermion simulations of non-gauge models. We test the algorithm in supersymmetric quantum mechanics viewed as a one-dimensional Euclidean lattice ﬁeld theory. We ﬁnd dramatic reductions in the autocorrelation time of the algorithm in comparison to standard HMC.  2002 Elsevier Science B.V. All rights reserved.

1. Introduction to render the magnitude of the resulting update of a given Fourier component insensitive to its wavevector Dynamical fermion algorithms play a crucial role index. We test this algorithm in supersymmetric quan- in the simulation of lattice field theories. The favorite tum mechanics treated as a Euclidean lattice theory. algorithm for an even number of fermion species is This model is formulated on the lattice in such way the Hybrid Monte Carlo HMC algorithm [1]. In this as to leave intact an exact subgroup of the continuum Letter we introduce and test an improved version of supersymmetry. We are able to demonstrate that the this algorithm in the case of supersymmetric quantum improved algorithm reduces the dynamical critical ex- mechanics. We show that the improved algorithm is ponent to values close to zero on lattices as large as far superior to the usual HMC procedure in combating L = 256 and correlation lengths ξ ∼ 16. the effects of critical slowing down. We first discuss the usual HMC algorithm and show how it may be generalized to allow for Fourier ac- 2. Hybrid Monte Carlo algorithm celeration. The idea is closely related to the usual Fourier acceleration used for Langevin simulations— We will be concerned with simulations of models both the fields and their conjugate momenta are involving scalar and fermion fields. The typical action evolved in momentum space throughout an individ- we will be discussing takes the general form: ual HMC trajectory—the crucial improvement being ¯ = + ¯ to choose a wavelength dependent timestep. The mo- S(x,ψ,ψ) SB (x) ψM(x)ψ, (1) mentum dependence of this timestep is chosen so as containing a bosonic field x and a Dirac field ψ defined on a lattice in Euclidean space. The fermion * Corresponding author. matrix M will contain lattice derivative terms together E-mail address: [email protected] (S. Catterall). with couplings to the bosonic field x. The partition

0370-2693/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII:S0370-2693(02)01217-0 302 S. Catterall, S. Karamov / Physics Letters B 528 (2002) 301–305 function for this system is then just afieldx with associated momentum p can be written − ¯ as [2] Z = Dx Dψψe¯ S(x,ψ,ψ). (2) xr(δt) = xr(0) + δtArr pr (0) In order to simulate this action we first replace 2 (δt) T the fermion field by a bosonic pseudofermion field φ + A A F (0), 2 rr r r r whose action is δt T 1 − = + + T 1 pr(δt) pr(0) Arr Fr (0) Fr (δt) , (3) φi M M φj . 2 2 ij ij where F =−∂H/∂x is an associated force and A is This is an exact representation of the original boson an arbitrary matrix. Notice our notation—the fields effective action in the case where detM>0(which are indexed by a integer vector giving their lattice will be the case for SUSY QM). The resultant nonlocal position. This update, indeed, satisfies the time re- action S(x,φ) can be simulated using the Hybrid versibility criterion: inverting the sign of momen- Monte Carlo (HMC) algorithm [1]. In the HMC tum as pn(δt) =−pn(δt) and updating the resultant scheme momentum fields (p, π) conjugate to (x, φ) configuration leads to φn(2δt) = φn(0), pn(2δt) = are added and a Hamiltonian H constructed from the −pn(0). original action and additional terms depending on the To implement our improved algorithm we choose momenta: amatrixArr whose elements depend only on the 1 difference of the lattice vectors between two sites r = + = 2 + 2 H S S, S p π . = − 2 i i and r . Arr a(r r ). i We then take the lattice Fourier transform of Eq. (3) − The corresponding partition function Z = e H is, up to arrive at an update equation for the Fourier compo- to a constant factor, identical to the original Z.The nents xˆr and pˆr. The Fourier components of the field augmented system (x,p,φ,π)is now associated with xr are given by the discrete transform a classical dynamics depending on an auxiliary time 1 ik·r variable t xˆk = √ e xr, (4) ∂x ∂p ∂H V r = p, =− , ∂t ∂t ∂x with similar expressions for pˆk and aˆk. Because of ∂φ ∂π ∂H = π, =− . the lattice convolution theorem the dependence of this ∂t ∂t ∂φ momentum space update on the Fourier coefficients aˆk Introducing a finite time step t allows us to may then be completely absorbed into the definition simulate this classical evolution and to produce a of a wavelength dependent time step δtk = δtaˆk.The sequence of configurations (x(t), φ(t)). final update equations take the form = If t 0thenH would be conserved along such a 2 (δtk) trajectory. As t is finite H is not exactly conserved. xˆ (t + δt ) =ˆx (t) + δt pˆk(t) + F (t), k k k k 2 k However, a finite length of such an approximate δtk trajectory can be used as a global move on the fields pˆk(t + δtk) =ˆpk(t) + Fk(t) + Fk(t + δtk) . (5) (x, φ) which may then be subject to a metropolis 2 step based on H . Provided the discrete, classical The final step in applying Fourier acceleration to dynamics is reversible and care is taken to ensure this update is to choose the acceleration kernel a¯k ergodicity the resulting move satisfies detailed balance so as eliminate the wavelength dependence of the and hence this dynamics will provide a simulation of update in the free theory [3]. For the update of a Z and hence also of the original partition function Z. bosonic field this implies that we should utilize the Ergodicity is taken care of by drawing new mo- square root of the momentum space lattice propagator. menta from a Gaussian distribution after each trajec- Such a propagator contains a mass parameter macc tory. The reversibility criterion can be satisfied by us- which can be tuned to optimize the update. In the ing a leapfrog integration scheme. Its general form for free theory it should clearly be set to the bare lattice S. Catterall, S. Karamov / Physics Letters B 528 (2002) 301–305 303 mass, but in a general interacting theory it is left as a Notice that this model employs a non-standard boson parameter. In this Letter we provide evidence that the action containing not the usual scalar lattice Laplacian autocorrelation time is optimized by setting macc to the but the square of the symmetric difference operator. approximate position of the massgap. The fermionic This is done in order to treat the fermions and bosons kernel is then chosen to be the inverse of the bosonic in a symmetric manner—indeed, because of this the kernel. action is invariant under a single SUSY-like symmetry Notice the occurrence of the square root of the = ¯ = + = momentum space lattice propagator here—rather than δxi ψi ξ, δψi (Dij xj Pi )ξ, δψi 0. the propagator itself which would be usual in Fourier- Doubles in both bosonic and fermionic sectors are accelerated Langevin algorithms—this reflects the fact eliminated by means of the Wilson mass term K.The that here the HMC update corresponds to a discrete physics results from this study were published in [5]. second order differential equation in time unlike the For bosonic and pseudo-fermionic field updates first order Langevin equation. This choice of the respectively we use the following timesteps which are square root propagator was not made in [2] and may simple inverses of each other explain, in part, why acceleration led to only small t(m + 2) reductions in the autocorrelation time. B = acc δtk , 2 2 2 sin (2πk/L)+ (macc + 2sin (πk/L)) 3. Model t sin2(2πk/L)+ (m + 2sin2(πk/L))2 F = acc δtk . macc + 2 A simple way to demonstrate the effectiveness of this update is to consider supersymmetric quantum mechanics [4] which contains a real scalar field x 4. Autocorrelation time and two independent real fermionic fields ψ and ψ¯ defined on a one-dimensional lattice of L sites with Suppose that Q is some observable and Qt is a periodic boundary conditions imposed on both scalar measurement of Q in configuration corresponding to and fermion fields: Monte Carlo time t. Then the autocorrelation function 1 is defined as: S = (Dij xj + Pi)(Dij xj + Pi) 2 Q Q − Q2 ij c(t) = 0 t . Q2− Q2 1 ¯ + ψi Dij + P ψj . (6) 2 ij Typically the function c(t) is approximately exponen- ij tial c(t) = e−t/τ where τ is the autocorrelation time— For our simulations the quantity Pi and its derivative the time between decorrelated configurations. Clearly, are chosen as: we can define τ also from the relation ∞ P = K x + gx3,P= K + 3gx2δ . i ij j i ij ij i ij = −t/τ = j τ e dt c(t) t 0 The symmetric difference operator Dij and the Wilson mass matrix Kij are defined as: and this yields a robust way to estimate τ even when c(t) is not exactly an exponential—measured 1 Dij = [δj,i+1 − δj,i−1], this way it is sometimes referred to as the integrated 2 autocorrelation time. It also provides a convenient way 1 to find the autocorrelation time from a sequence of N Kij = mδij − (δi,j+1 + δi,j−1 − 2δij ). 2 Monte Carlo measurements Here dimensionless lattice units are m = m a, phys N j

Fig. 1. ln τ vs ln macc for supersymmetric quantum mechanics with Fig. 2. ln τ vs ln L for supersymmetric quantum mechanics with L = 64, m = 10, g = 100. macc ∼∞, m = 10, g = 100. Straight line fit yields z = 1.7. where we measure the autocorrelation function only out to M steps (we must choose M τ ). The efficiency of different simulation algorithms can be described in terms of the dynamic critical exponent z: 1 z L z τ ∼ = ∼ Lz. Mlattice Mphysical An algorithm which resulted in z = 0issaidtosuffer no critical slowing down while most local algorithms such as conventional HMC typically yield z ∼ 2. Fig. 1 represents the autocorrelation time τ as a function of macc for lattice SUSY QM with m = 10, g = 100 for a lattice with L = 64 sites. Notice that the dimensionless parameter characterizing the interaction 2 Fig. 3. ln τ vs ln L for supersymmetric quantum mechanics with strength g/m is unity in this case—we are in a strong macc = 15, m = 10, g = 100. coupling regime. The massgap of the theory for these values of the bare parameters can be calculated via Hamiltonian methods and yields a value mg ∼ 16.0. To derive a dynamic critical exponent we have also For large macc →∞ the update can be shown compared the dependence of the autocorrelation time to reduce to the usual local HMC algorithm and, on lattice size L both for the usual local HMC and the as the plot makes clear, leads to a maximal value Fourier-accelerated algorithm with macc = 15.0. For for τ . Indeed the minimal value of τ is can be the case macc =∞Fig. 2 shows a plot of ln τ against seen to be achieved by setting macc approximately ln L together with a straight line fit yielding z ∼ 1.7. equal to the massgap of the model ln macc ∼ 2.7. The plot makes it clear that this is a lower bound for In the vicinity of this point, one can see that the z so that in all likelihood z achieves a value of z = 2 minimum autocorrelation τmin is more than an order for very large lattices. This is to be contrasted to Fig. 3 of magnitude smaller than the τ achieved by the usual which shows the same plot for macc = 15. A fit in this local algorithm for this lattice size. case yields a value of z consistent with zero. For the S. Catterall, S. Karamov / Physics Letters B 528 (2002) 301–305 305 largest lattice size L = 256 we see that more than two Acknowledgements orders of magnitude decrease in τ are gained by use of the accelerated algorithm. Simon Catterall was supported in part by DOE grant DE-FG02-85ER40237.

5. Discussion References We have shown that the use of Fourier acceleration can yield large payoffs in the simulation of non- [1] S. Duane, A. Kennedy, B. Pendleton, D. Roweth, Phys. gauge ﬁeld theories with dynamical fermions. We Lett. B 195 (1987) 216. [2] D. Espriu, V. Koulovassilopoulos, A. Travesset, Phys. Rev. D 56 have presented results which support this conclusion (1997) 6885. for supersymmetric quantum mechanics in which the [3] G. Katz, G. Batrouni, C. Davies, A. Kronfeld, P. Lepage, HMC algorithms can be pushed to large (L = 256) P. Rossi, B. Svetitsky, K. Wilson, Phys. Rev. D 37 (1988) 1589. lattice size and correlation length (ξ ∼ 16). We have [4] E. Witten, Nucl. Phys. B 185 (1981) 513. obtained similar, though less quantitative, results for [5] S. Catterall, E. Gregory, Phys. Lett. B 487 (2000) 349. [6] S. Catterall, S. Karamov, hep-lat/0108024, submitted to Phys. two-dimensional Wess–Zumino models [6]. Rev. D.