FLUE_heatbath

SU(3) heatbath update using Cabibbo–Marinari SU(2) subgroups.

This module implements a lattice gauge heatbath update using three embedded SU(2) subgroup updates.

Algorithm:

• Compute staple
• Form W = U * staple
• Apply SU(2) updates in (1,2), (1,3), (2,3)
• Reunitarize

Update strategy:

• Loop over direction mu
• Loop over colour classes
• Parallelise with do concurrent

Supported actions:

• Wilson (2 colours)
• Symanzik (4 colours)
• Iwasaki (4 colours)

Source: src/gauge/heatbath.f90

Dependencies

Contents

• apply_su2_subgroup_update
• build_colour_sites
• updateLinks
• stapleWilson
• stapleRectangle
• stapleSymanzik
• stapleIwasaki
• su3_updated_link

Subroutines

apply_su2_subgroup_update

Apply one embedded SU(2) heatbath update inside SU(3).

Inputs: ULink : current SU(3) link being updated W : current product U * staple, updated in tandem with ULink i1, i2 : the two SU(3) row/column indices defining the subgroup subgroup_id : 1, 2, or 3, used for the Philox substream beta, key : heatbath parameters and Philox key counter0 : counter with site id in counter(1)

Method:

1. Extract the 2x2 block from W.
2. Project it to the SU(2) quaternion representation.
3. Build the SU(2) heatbath matrix.
4. Embed the result back into SU(3).
5. Left-multiply both ULink and W by the embedded matrix.

Attributes: pure

subroutine apply_su2_subgroup_update(ULink, W, i1, i2, subgroup_id, beta, key, counter0)

Arguments

Name	Type	Intent	Attributes	Description
ULink	complex(kind=WC)	inout	dimension(3, 3)	Input link, output link
W	complex(kind=WC)	inout	dimension(3, 3)	Input link, output link
i1	integer	in		Select which 2x2 block
i2	integer	in		Select which 2x2 block
subgroup_id	integer	in		Philox variable
beta	real(kind=WP)	in		beta variables
key	integer(kind=C64)	in	dimension(2)	Philox key
counter0	integer(kind=C64)	in	dimension(4)	Philox keys

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build_colour_sites

Construct checkerboard / multi-colour site lists The output layout is structure-of-arrays (SoA) for optimal memory access in do concurrent loops. Separates sites out into checkerboard Separate arrays for each of t, x, y, z

subroutine build_colour_sites(nt, nx, ny, nz, use_symanzik, sites_it, sites_ix, sites_iy, sites_iz, counts)

Arguments

Name	Type	Intent	Attributes	Description
nt	integer	in		Lattice dimensions
nx	integer	in		Lattice dimensions
ny	integer	in		Lattice dimensions
nz	integer	in		Lattice dimensions
use_symanzik	logical	in		Selects checkerboard scheme
sites_it	integer	out	allocatable, dimension(:, :, 😃	T: Output coordinate arrays (SoA layout)
sites_ix	integer	out	allocatable, dimension(:, :, 😃	X: Output coordinate arrays (SoA layout)
sites_iy	integer	out	allocatable, dimension(:, :, 😃	Y: Output coordinate arrays (SoA layout)
sites_iz	integer	out	allocatable, dimension(:, :, 😃	Z: Output coordinate arrays (SoA layout)
counts	integer	out	allocatable, dimension(:, 😃	Number of sites per colour/checkerboard and direction

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updateLinks

Perform one full heatbath sweep of the lattice

subroutine updateLinks(U, beta, UUpdated, master_key, sweep_id, sites_it, sites_ix, sites_iy, sites_iz, counts, actionTag, xi)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
beta	real(kind=WP)	in		The beta value
UUpdated	complex(kind=WC)	inout	dimension(:,:,:,:,:,:😅	Updated gaugefield [3,3,4,nt,nx,ny,nz]
master_key	integer(kind=C64)	in	dimension(2)	Master key for philox
sweep_id	integer	in		which sweep/trajectory it is. Used for Philox
sites_it	integer	in	dimension(:,:😅	T: Output coordinates array in Structure of Arrays for better memomry access
sites_ix	integer	in	dimension(:,:😅	X: Output coordinates array in Structure of Arrays for better memomry access
sites_iy	integer	in	dimension(:,:😅	Y: Output coordinates array in Structure of Arrays for better memomry access
sites_iz	integer	in	dimension(:,:😅	Z: Output coordinates array in Structure of Arrays for better memomry access
counts	integer	in	dimension(:😅	Number of sites per colour and direction
actionTag	character(len=*)	in	optional	Specifies the action
xi	real(kind=WP)	in	optional	gauge anisotropy

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stapleWilson

Compute the Wilson staple for one site

V = sum_{nu != mu} [ path(nu,-mu,-nu) + path(-nu,-mu,nu) ] with the starting coordinate shifted to x + mu.

Attributes: pure

subroutine stapleWilson(U, V, coord, mu, xi)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
V	complex(kind=WC)	out	dimension(3, 3)	Output Wilson staple at this coordinate
coord	integer	in	dimension(4)	Coordinate of the staple [it,ix,iy,iz]
mu	integer	in		Staple is along mu direction
xi	real(kind=WP)	in		The gauge anisotropy

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stapleRectangle

Computes the rectangle staple (1x2) as in the Symanzik improved action for one site

Attributes: pure

subroutine stapleRectangle(U, V, coord, mu, xi)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
V	complex(kind=WC)	out	dimension(3, 3)	Output rectangle staple at this coordinate
coord	integer	in	dimension(4)	Coordinate of the staple [it,ix,iy,iz]
mu	integer	in		Staple is along mu direction
xi	real(kind=WP)	in		The gauge anisotropy

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stapleSymanzik

Compute the tree-level Symanzik staple for one SU(3) link.

This consists of:

• the plaquette staple contribution (Wilson staple),

Attributes: pure

subroutine stapleSymanzik(U, V, coord, mu, xi)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
V	complex(kind=WC)	out	dimension(3, 3)	Output tree-level Symanzik staple at this coordinate
coord	integer	in	dimension(4)	Coordinate of the staple [it,ix,iy,iz]
mu	integer	in		Staple is along mu direction
xi	real(kind=WP)	in		The gauge anisotropy

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stapleIwasaki

Compute the Iwasaki staple for one SU(3) link.

This consists of:

• the plaquette staple contribution (Wilson staple),
• minus the weighted rectangle contribution.

With c0= 3.648, c1 = -0.331

Attributes: pure

subroutine stapleIwasaki(U, V, coord, mu, xi)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
V	complex(kind=WC)	out	dimension(3, 3)	Output Iwasaki staple at this coordinate
coord	integer	in	dimension(4)	Coordinate of the staple [it,ix,iy,iz]
mu	integer	in		Staple is along mu direction
xi	real(kind=WP)	in		The gauge anisotropy

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Functions

su3_updated_link

Compute updated SU(3) link at a single lattice site

Compute the updated SU(3) link at one site and one direction.

Steps:

1. Extract the current link U_mu(coord).
2. Compute either the Wilson or Symanzik staple.
3. Form W = U * staple.
4. Apply the three Cabibbo-Marinari SU(2) subgroup updates: (1,2), (1,3), (2,3)
5. Reunitarize the final matrix.

Attributes: pure

Returns: complex(kind=WC)

function su3_updated_link(U, beta, coord, mu, key, dims4, stapleKernel, xi) result(Uout)

Arguments

Name	Type	Intent	Attributes	Description
U	complex(kind=WC)	in	dimension(:, :, :, :, :, :, 😃	Input gaugefield [3,3,4,nt,nx,ny,nz]
beta	real(kind=WP)	in		beta value
coord	integer	in	dimension(4)	Coordinate of link to be updated [it,ix,iy,iz]
mu	integer	in		Direction of link
key	integer(kind=C64)	in	dimension(2)	Philox variable
dims4	integer	in	dimension(4)	Lattice dimensions [nt,nx,ny,nz]
stapleKernel	procedure(stapleInterface)	in	pointer	Pointer used to switch which action is used
xi	real(kind=WP)	in		The gauge anisotropy

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