An Algorithm for Gluinos on the Lattice

TL;DR

The paper develops and tests a lattice algorithm for simulating supersymmetric Yang–Mills theory with a Majorana adjoint fermion, combining Lüscher's local bosonic method with a two-step polynomial approximation and a noisy correction to manage the fermion determinant. It introduces optimized polynomial schemes (single- and two-step, including complex-plane extensions) and applies them to gluino simulations, along with practical use as optimized solvers and an optimized hopping parameter expansion. Numerical experiments in SU(2) demonstrate improved autocorrelation properties and feasible acceptance rates, indicating competitive performance relative to molecular dynamics-based approaches and potential applicability to QCD-like theories. The work suggests broad utility of the optimization framework for efficient, controllable fermionic simulations in lattice field theories, with clear paths for further refinement and larger-scale applications.

Abstract

Lüscher's local bosonic algorithm for Monte Carlo simulations of quantum field theories with fermions is applied to the simulation of a possibly supersymmetric Yang-Mills theory with a Majorana fermion in the adjoint representation. Combined with a correction step in a two-step polynomial approximation scheme, the obtained algorithm seems to be promising and could be competitive with more conventional algorithms based on discretized classical (``molecular dynamics'') equations of motion. The application of the considered polynomial approximation scheme to optimized hopping parameter expansions is also discussed.

Figures (4)

• Figure 1: The relative deviation of $P_{40}(\frac{1}{4};0.03,4.0;x)$ from $x^{-\frac{1}{4}}$. The value of the deviation norm is here: $\delta=2.51.. \cdot 10^{-5}$.
• Figure 2: The positions of the roots of $P_{40}(1;0.1,2.0,1.0;x+iy)$ in the complex plane.
• Figure 3: The square of the residuum vector in a CG iteration on a $4^3 \cdot 8$ configuration at $(\beta=2.0,K=0.150)$. Twelve initial vectors are taken, starting from a randomly chosen point. The two upper bunches of curves belong to the normal CG iteration, whereas in the lower bunch the CG iteration is started after the multiplication of the twelve initial vectors by the optimized solver $P_{32}(1;0.1,2.0,1.0;x+iy)$ having $\delta=0.024..$.
• Figure 4: The square of the residuum vector in the cyclic optimized solver iteration scheme for different polynomial orders $n=40,60,100$. The vector $v$ is always the same and the configuration is at $\beta=2.0, K=0.175$ on $8^3 \cdot 16$ lattice. The interval of polynomial approximations is $[0.01,4.4]$. The dotted curve shows the result of conjugate gradient iteration.