An Exact Local Hybrid Monte Carlo Algorithm. for Gauge Theories. A. D. Kennedy and K. M. Bitar. Supercomputer Computations Research Institute

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1 An Exact Local Hybrid Monte Carlo Algorithm for Gauge Theories A. D. Kennedy and K. M. Bitar Supercomputer Computations Research Institute Florida State University Tallahassee, FL 06{405, USA Internet: FSU{SCRI{9{17 November 16, 199 Abstract We introduce a new Monte Carlo method for pure gauge theories. It is not intended for use with dynamical fermions. It belongs to the class of Local Hybrid Monte Carlo (LHMC) algorithms, which make use of the locality of the action by updating individual sites or links by following a classical mechanics trajectory in ctitious time. We choose to update a one-parameter subgroup of the gauge eld on each link of the lattice, and the classical trajectory can be found in closed form in terms of elliptic functions for this case. We show that this gives an overrelaxation algorithm with a tunable parameter which, unlike some previous methods, does not reuire the numerical integration of the euations of motion. To appear in the Proceedings of Lattice '9, Dallas, TX; October

2 1 LHMC FOR FREE FIELD THEORY Let us rst uickly review the Local Hybrid Monte Carlo (LHMC) and Adler's overrelaxation (AOR) algorithms [1] for free eld theory. See [] for further references. Consider the Gaussian model dened by the free eld action S() = 1 X X (x) m (x) : A single-site AOR update with parameter replaces (x) by 0 (x) = (1 )(x) F ( ) ; where D m, F P jx yj=1 (y), and is a Gaussian random number. The lattice may be updated using a checkerboard scheme. For = 1 this is the heatbath (HB) algorithm, because 0 (x) does not depend upon (x). The exponential autocorrelation time is which corresponds to z =. HB = D m p (p 0); If we adjust so as to minimize the autocorrelation time we nd that = m p D O(m ); AOR p D m ; which gives z = 1. Consider the LHMC algorithm applied to the Gaussian model: We introduce the Hamiltonian H = 1 p S() on \ctitious" phase space. The corresponding euation of motion are x = x F; whose solution in terms of the Gaussian distributed random initial momentum p x and the initial eld value x is x (t) = x cos t 1 cos t F p x sin t; which is exactly the AOR update considered before if we identify 1 cos t and p x. If we use this exact solution of the euations of motion to generate candidate congurations for LHMC then the acceptance rate will be unity.

3 LHMC FOR PURE GAUGE THEORY We now turn to the case of pure gauge theory. The Wilson action is S(U`) = Re Tr V Tr 1 where the matrix V P P@ U ` P. Our update will be U` gu` where g is a group element. There is no reason why all the degrees of freedom on a link need to be updated simultaneously; we are free to update just a single degree of freedom at a time as long as we can guarantee ergodicity. Therefore we shall restrict g to lie in a one-dimensional subgroup. For U in the fundamental representation of SU(N) or SO(N) any group element can be written as a conjugate of 6 4 e i e i or 6 4 cos sin sin cos respectively, so updating with just this subgroup will be ergodic if random gauge transformations are carried out from time to time. In other words, selecting an element from these subgroups is euivalent to choosing a completely general group element up to gauge transformations. If we introduce the parameters e i 8 >< >:... V 11 V for SU(N) (V 11 V ) i(v 1 V 1 ) then we may write the action in the simple form for SO(N) S() = f1 cos( )g: N As usual, we introduce a ctitious momentum p and a Hamiltonian H(p; ) = 1 p S() = 1 p V 0 sin where V 0 =N is the maximum potential energy of the \pendulum". Our LHMC algorithm consists of repeatedly applying the following steps in any order: Refresh the momentum p from a Gaussian heatbath; 7 5

4 Integrate the euations of motion for a time (E) and then reverse the sign of the nal momentum; Perform a random gauge transformation. The only subtlety is that we can allow the trajectory length depend upon the constants of motion (specically the energy E) because for an exact solution of the euations of motion this still satises detailed balance. We now have to solve the euations of motion, which are = V 0 sin cos subject to the initial conditions (0) = 0 and p(0) = p 0. It is convenient to introduce a new variable y sin and a rescaled time u t E=, whence dy du = (1 y )(1 k y ) where the parameter k V 0 =E. The solution of this euation is the Jacobian elliptic function y = sn(u sgn p 0 u 0 ; k) where sn u 0 = sin. Remember that we can choose any value ~u(k) for the trajectory length. One particularly interesting choice is to take the trajectory length to be a xed fraction of half the period of an orbit, u, which corresponds to overrelaxation; ~u(k) = 1 1 where is (proportional to) the longest relevant correlation length of the gauge theory. A pendulum exhibits two dierent types of motion depending on the value of the parameter k (for k = 1 the solution is a Sine{Gordon instanton) u = 8 < : where K is the complete elliptic integral K(k) u; K(k) if k < 1, E > V 0 K 1 k Z 1 0 k if k > 1, E < V0, dy (1 y )(1 k y ) : With this choice of trajectory length the value of selected may be written as where the amplitude am u sin 1 sn u. ~ = am u 0 u sgn p 0 ; k 4 ;

5 H(,p) p Figure 1: The Hamiltonian as a function on phase space (with ) with the corresponding classical trajectories <Plauette> m = 0. (hot) m = 0.5 (hot, gf=1) m = 0.5 (hot, gf=5) m = 0.7 (hot) heatbath Sweeps Figure : Preliminary results for two-dimensional SU() on an 18 lattice at = 0:0. Convergence for various values of m = 1= and for hot and cold starting congurations is demonstrated. Studies of the dynamical critical behaviour are in progress. 5

6 CONCLUSIONS We may hope that the LHMC algorithm, with optimally chosen trajectory lengths, will exhibit a dynamical critical exponent z 1, if the results from free eld theory prove to be applicable to pure gauge theories. In this regard it is similar to the Hybrid Overrelaxation algorithm analysed by Wol. The advantages of the LHMC method as compared to previous overrelaxation methods for pure gauge theories is that it allows a true tunable overrelaxation parameter without the need for explicit numerical integration. ACKNOWLEDGEMENTS This research was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the U.S. Department of Energy through contracts #DE{FC05{85ER50000 and #DE{FG05{9ER4074. References [1] C. Whitmer. Overrelaxation methods for Monte Carlo simulations of uadratic and multiuadratic actions. Phys. Rev., D9:06{11, [] A. D. Kennedy. Progress in lattice eld theory algorithms. In Jan Smit and Pierre van Baal, editors, Lattice '9, volume B0 of Nuclear Physics (Proceedings Supplements), pages 96{107, March 199. Proceedings of the International Symposium on Lattice Field Theory, Amsterdam, the Netherlands, 15{19 September