su2hmc: Two Colour Hybrid Monte Carlo with Wilson Fermions

Introduction

Hybrid Monte Carlo algorithm for Two Color QCD with Wilson-Gor'kov fermions based on the algorithm of Duane et al. Phys. Lett. B195 (1987) 216.

There is "up/down partitioning": each update requires one operation of congradq on complex vectors to determine $$ \left(M^\dagger M\right)^{-1}\Phi $$ where $\Phi$ has dimension 4 * kvol * nc * Nf - The matrix M is the Wilson matrix for a single flavor there is no extra species doubling as a result

matrix multiplies done using routines hdslash and hdslashd

Hence, the number of lattice flavors Nf is related to the number of continuum flavors N_f by $$N_f = 2 \text{Nf}$$

Fermion expectation values are measured using a noisy estimator. on the Wilson-Gor'kov matrix, which has dimension 8 * kvol * nc * Nf inversions done using congradp, and matrix multiplies with dslash, dslashd

trajectory length is random with mean dt * stepl The code runs for a fixed number ntraj of trajectories.

Phi	pseudofermion field
bmass	bare fermion mass
fmu	chemical potential
actiona	running average of total action

Fermion expectation values are measured using a noisy estimator. The code produces the following outputs:

File Name	Data type
config.bβββkκκκmuμμμμjJJJsNXtNT.XXXXXX	Lattice configuration for given parameters. Last digits are the configuration number
Output.bβββkκκκmuμμμμjJJJsNXtNT	Number of conjugate gradient steps for each trajectory. Also contains general simulation details upon completion
bose.bβββkκκκmuμμμμjJJJsNXtNT	spatial plaquette, temporal plaquette, Polyakov line
fermi.bβββkκκκmuμμμμjJJJsNXtNT	psibarpsi, energy density, baryon density
diq.bβββkκκκmuμμμμjJJJsNXtNT	real

SJH March 2005

Hybrid code, P.Giudice, May 2013

Converted from Fortran to C by D. Lawlor March 2021

Conversion notes

This two colour implementation was originally written in FORTRAN for: S. Hands, S. Kim and J.-I. Skullerud, Deconfinement in dense 2-color QCD, Eur. Phys. J. C48, 193 (2006), hep- lat/0604004

It has since been rewritten in C and is in the process of being adapted for CUDA. We have sucessfully run on 7000+ Zen 2 cores, as well as A100 GPUs

Some adaptions from the original are:

• Mixed precision conjugate gradient
• Implementation of BLAS routines for vector operations
• Removal of excess halo exchanges
• #pragma omp simd instructions
• Makefiles for Intel, GCC and AMD compilers with flags set for latest machines
• GSL ranlux support
• CUDA implementation.

Other works in progress include:

• Improved action
• SYCL implementation.
• Multi-GPU support
• CMake build system
• yaml input file
• Set lattice volume and CPU grid at runtime
• Higher order integrators. 11 stage 4th order non-gradient integrator implimented but no speedup yet

Getting started

This code is written for MPI on Linux, thus has a few caveats to get up and running

1. In sizes.h, set the lattice size. By default we assume the spatial components to be equal

2. Also in sizes.h set the processor grid size by setting the values of

   npx npy npz npt

   These MUST be divisors of

   nx ny nz nt

   set in step one.

3. Compile the code using the desired Makefile. Please note that the paths given in the Makefiles for BLAS libraries etc. are based on my own system. You may need to adjust these manually.

4. Run the code. This may differ from system to system, especially if a task scheduler like SLURM is being used. On my desktop it can be run locally using the following command

   mpirun -n<nproc> ./su2hmc <input_file>

• nproc is the number of processors, given by the product of npx npy npz npt
• If no input file is given, the programme defaults to midout. The default name is a historical one which goes back generations to the early days of Lattice QCD.

Input parameters

A sample input file looks like

0.00200	1.7	0.1780	0.00	0.000	0.0	0.0	500	20	1	1	100
dt	beta	akappa	jqq	thetaq	fmu	aNf	stepl	ntraj	istart	icheck	iread

where

• dt is the step size for the update
• beta is β, given up to three significant figures
• akappa is hopping parameter, given up to four significant figures
• jqq is the diquark source, given up to three significant figures
• thetaq is the diquark mixing angle
• fmu is the chemical potential
• aNf is ignored. Originating in the Cornell group when Ken Wilson was still there, that molecular dynamics time-discretisation artifacts can be absorbed into renormalisation of the bare parameters of the lattice action
• stepl is the average number of steps per trajectory. For a single trajectory it times dt should equal 1
• ntraj is the number of trajectories
• istart signals a hot start (>=1) or cold start (<=0)
• icheck is how often to print out a configuration. We typically use 5 and tune for 80% acceptance rate
• iread is the starting configuration for continuation runs. If zero, start without reading

The bottom line of the input is ignored by the programme and is just there to make your life easier. Blank space does not matter, so long as there is some gap between the input parameters in the file and they are all on a single line.